Durham E-ThesesRovibrational excitation in collisions involving diatomicmoleculesBaker, D. J.How to cite:Baker, D. J. (1984) Rovibrational excitation in collisions involving diatomic molecules, Durham theses,Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/7499/Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission orcharge, for personal research or study, educational, or not-for-proﬁt purposes provided that:• a full bibliographic reference is made to the original source• a link is made to the metadata record in Durham E-Theses• the full-text is not changed in any wayThe full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full Durham E-Theses policy for further details.Academic Support Oﬃce, Durham University, University Oﬃce, Old Elvet, Durham DH1 3HPe-mail: e-theses.admin@dur.ac.uk Tel: +44 0191 334 6107http://etheses.dur.ac.ukROVIBRATIONAL EXCITATION IN COLLISIONS INVOLVING DIATOMIC MOLECULES by D. J. BAKER, B.Sc. ( B r i s t o l ) The copyright of this thesis rests with the author. No quotation from it should be published without his prior written consent and information derived from it should be acknowledged. A t h e s i s s u b m i t t e d t o the U n i v e r s i t y of Durham f o r the Degree o f Doctor of Philosophy December, 1984. w r i t i n g o f many books i s e n d l e s s , and e x c e s s i v e d e v o t i o n t o books i s wearying t o the body' . E c c l e s i a s t e s 12,12. ABSTRACT The a p p l i c a b i l i t y o f a s y m p t o t i c s e r i e s expansions o f the type i n t r o d u c e d by G a i l i t i s (1976) i n heavy p a r t i c l e s c a t t e r i n g c a l c u l a t i o n s i s examined. The convergence o f the s o l u t i o n s , w i t h r e s p e c t t o the va l u e o f the s c a t t e r i n g c o o r d i n a t e i s found t o be g r e a t l y improved i n a model problem o f the r o v i b r a t i o n a l e x c i t a t i o n o f H 2 by H +. The r e s u l t s o f i n f i n i t e o r d e r sudden (IOS) and b r e a t h i n g sphere (BS) c a l c u l a t i o n s o f cross s e c t i o n s 12 16 f o r the r o v i b r a t i o n a l e x c i t a t i o n of C 0 by para-( j - ^ = 0 o n l y ) are p r e s e n t e d . Comparisons are made w i t h e x i s t i n g t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . A l l o w i n g f o r r e m a i n i n g u n c e r t a i n t i e s i n the i n t e r a c t i o n p o t e n t i a l o f Poulsen (1 9 8 2 ) , our r e s u l t s are i n reasonable accord w i t h the e x p e r i m e n t a l r e s u l t s o f Andrews and Simpson (1976) f o r the v i b r a t i o n a l r e l a x a t i o n of CO (v^ = 1) by ortho-H2. IOS c a l c u l a t i o n s o f the v i b r a t i o n a l d e a c t i v a t i o n 12 16 of C O = 1 ) by H2 ( j | = 0 or 2 1 which s i m u l t a n e o u s l y undergoes the r o t a t i o n a l t r a n s i t i o n s Aj^ = 0,2, or 4 are p r e s e n t e d . Of major i n t e r e s t i s the near-resonance process CO (vx»0 •+ — * CO(vz = ©)•/• ^ (jt = Q^ 87-03 em Comparison i s made w i t h both r e c e n t quantum mechanical and s e m i - c l a s s i c a l c a l c u l a t i o n s o f t h i s p r o c e s s , and w i t h e x p e r i m e n t . The near-resonance process i s i n s u f f i c i e n t t o account f o r the e x p e r i m e n t a l l y d e t e r m i n e d d i f f e r e n c e i i between the r a t e s o f the d e a c t i v a t i o n o f CO = 1) by para- and o r t h o - ^ (Andrews and Simpson ( 1 9 7 6 ) ) . IOS c a l c u l a t i o n s o f the v i b r a t i o n a l d e a c t i v a t i o n of CO = 1) by HD ( j | = 0 o r 1) are p r e s e n t e d and d i s c u s s e d . The agreement w i t h the e x p e r i m e n t a l r e s u l t s (Andrews and Simpson (1976)) i s a c c e p t a b l e i n the tem-p e r a t u r e range 1 4 0 ^ 1 ^ 240 K. i i i ACKNOWLEDGEMENTS I am inde b t e d t o Dr. D.K. Elower f o r h i s e x c e l Lent s u p e r v i s i o n of t h i s r e s e a r c h . I am ex t r e m e l y g r a t e f u l f o r h i s guidance, a s s i s t a n c e , and encouragement throughout the course o f t h i s work. Also I would l i k e t o thank P r o f e s s o r B.H. Bransden f o r a c c e p t i n g me as a gradu a t e s t u d e n t at Durham. I am g r a t e f u l t o a l l those who have g i v e n me h e l p and a d v i c e d u r i n g t h i s r e s e a r c h , i n p a r t i c u l a r Drs. D. Dewangan, D. K i rkp.i t r i ck, C. Newby, and R. S h i n g a l . I wish t o thank rny c o l l e a g u e , (.'. Danby, f o r numerous s t i m u l a t i n g d i s c u s s i o n s . I would l i k e t o thank the SERC f o r the p r o v i s i o n o f a s t u d e n t s h i p , and of the g r a n t s which made p o s s i b l e the use of the Cray-1 computer. F i n a l l y , I am ver y g r a t e f u l t o Ms. M. Chipchase f o r her e f f i c i e n t and p a t i e n t t y p i n g of the m a n u s c r i p t , and wish t o express my g r a t i t u d e t o Dr. C. Newby f o r h i s a s s i s t a n c e i n p r o o f r e a d i n g the typed m a n u s c r i p t . CONTENTS Page CHAPTER I - INTRODUCTION 1 1. M o l e c u l a r Processes i n the I n t e r s t e l l a r Medium 1 2. Cross S e c t i o n s and Rate C o e f f i c i e n t s 5 3. E x p e r i m e n t a l D e t e r m i n a t i o n o f Cross Sec t I o n s 8 4. T h e o r e t i c a l D e t e r m i n a t i o n o f Cross S e c t i o n s 12 4.1 The D e t e r m i n a t i o n o f P o t e n t i a l Energy Surfaces 13 4.2 S c a t t e r i n g t h e o r y 18 4.2.1 Quantum mechanics 18 4.2.2 C l a s s i c a l and s e m i c l a s s i c a l methods 20 CHAPTER I I - QUANTUM THEORY OF MOI.KCIJI.AK COLLISIONS 2 5 1 . In t rodnc L i on 2 5 2. C l o s e - C o u p l i n g Theory 26 3. Approximate Methods 39 3.1 The c e n t r i f u g a l sudden a p p r o x i m a t i o n 40 3.2 The energy sudden a p p r o x i m a t i o n 49 3.3 The i n f i n i t e o r d e r sudden a p p r o x i m a t i o n 56 4. Other Quantum Mechanical D e c o u p l i n g A p p r o x i m a t i o n s 65 4.1 The e f f e c t i v e p o t e n t i a l method 65 4.2 The L-dominant and decoupled L-dominant a p p r o x i m a t i o n s 68 CHAPTER I I I - NUMERICAL SOLUTION OF THE COUPLED " F^QTJATTTJ^TS • ~ ~ " 70 1. I n t r o d u c t i o n 70 2. General Comparison o f the Methods 72 Page 3. S u i t a b i l i t y o f the R-matrix Propagator Method f o r IOS C a l c u l a t i o n s 74 4. The R-matrix P r o p a g a t i o n Method 78 4.1 General t h e o r y 78 4.2 Boundary c o n d i t i o n s 84 4.3 Long range i n t e r a c t i o n p o t e n t i a l s -a s y m p t o t i c s e r i e s expansions 87 4.4 D e r i v a t i o n o f the s e c t o r R-matrix 93 4.5 The step s i z e a l g o r i t h m 100 4.6 P r o p a g a t i n g v a r i a b l e numbers o f c l o s e d channels 103 CHAPTER IV - ROVIBRATIONAL EXCITATION OF H 0 BY H + , n , 1. I n t r o d u c t i o n 106 2. The I n t e r a c t i o n P o t e n t i a l 109 3. The H2 V i b r a t i o n a l Wavefunctions 112 4. Numerical D e t a i l s 117 4.1 The v i b r a t i o n a l q u a d r a t u r e 118 4.2 S o l u t i o n o f the coupled e q u a t i o n s f o r f i x e d o r i e n t a t i o n s 122 4.3 The a n g u l a r q u a d r a t u r e 128 5. R e s u l t s and D i s c u s s i o n 133 Appendix. The use of a s y m p t o t i c s e r i e s expansions i n heavy p a r t i c l e s c a t t e r i n g 139 CHAPTER V - ROVIBRATIONAL EXCITATION OF 1 2 C 1 6 0 = " BY PARA -H 2 °~ 147 1. I n t r o d u c t i o n 147 2. The I n t e r a c t i o n P o t e n t i a l s 151 2.1 The p o t e n t i a l o f Poulsen (1982) 151 2.2 The p o t e n t i a l o f van Hemert (1983) 159 Page 3. D e s c r i p t i o n o f the System 164 3.1 IOS c a l c u l a t i o n s 164 3.2 B r e a t h i n g Sphere c a l c u l a t i o n s 167 4. Numerical Methods 169 4.1 IOS c a l c u l a t i o n s 169 4.1.1 E v a l u a t i o n o f the p o t e n t i a l m a t r i x elements 170 4.1.2 S o l u t i o n o f the coup l e d e q u a t i o n s f o r f i x e d o r i e n t a t i o n s 173 4.1.3 D e s c r i p t i o n of the CO molecule 185 4.2 B r e a t h i n g sphere c a l c u l a t i o n s 189 4.3 Angular q u a d r a t u r e 192 5. R e s u l t s and D i s c u s s i o n 198 6. Summary 216 CHAPTER VI - NEAR RESONANCE VIBRATIONAL " RELAXATION OF 12cl6Q IN COLLISIONS WITH PARA -H 2 219 1. I n t r o d u c t i o n 219 2. D e s c r i p t i o n o f the System 222 3. Numerical Methods 226 3.1 S o l u t i o n o f the coupled e q u a t i o n s at f i x e d o r i e n t a t i o n 228 3.2 D e s c r i p t i o n o f the molecule 233 3.3 D e s c r i p t i o n o f the CO molecule 234 3.4 Quadrature over the CO o r i e n t a t i o n s 238 4. R e s u l t s and D i s c u s s i o n 239 4.1 The dependence o f the v i b r a t i o n a l d e - e x c i t a t i o n o f CO on t h e r o t a t i o n a l s t a t e o f ^ 240 4.2 The near-resonance process 247 4.2.1 D i s c u s s i o n o f the model c a l c u l a t i o n 4.2.2 R e s u l t s 5. Summary CHAPTER V I I - ROVIBRATIONAL EXCITATION OF ~" 1 ZC l f e0 BY HP 1. I n t r o d u c t i o n 2. The I n t e r a c t i o n P o t e n t i a l 3. Numerical D e t a i l s 4. R e s u l t s and d i s c u s s i o n CHAPTER V I I I - CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK REFERENCES APPENDIX 1 - The d e r i v a t i o n and the s o l u t i o n o f the d i f f e r e n t i a l e q u a t i o n s f o r the a s y m p t o t i c s e r i e s expansions o f G a i l i t i s ( 1 9 7 6 ) . APPENDIX 2 - The n u m e r i c a l values o f the expansion c o e t t i c i e n t s, C v 2 i , used i n the c a l c u l a t i o n o f exact Morse o s c i l l a t o r w a v e f u n c t i o n s f o r H 0 APPENDIX 3 - D e t a i l s o f the c a l c u l a t i o n o f CO Morse o s c i l l a t o r w a v e f u n c t i o n s . APPENDIX 4 - The e r r o r i n the paper o f Poulsen ( 1 9 8 2 ) , a d i s c u s s i o n o f i t s e f f e c t on the H~ + CO r e s u l t s o f Chapter V I . APPENDIX 5 - The e r r o r i n the paper o f Poulsen {1982) , a d i s c u s s i o n o f i t s e f f e c t on the HD + CO r e s u l t s o f Chapter V I I . CHAPTER I INTRODUCTION 1. M o l e c u l a r Processes i n the I n t e r s t e l l a r Medium I n r e c e n t years i t has been determined t h a t the i n t e r s t e l l a r gas, once c o n s i d e r e d t o c o n s i s t almost e n t i r e l y o f H atoms and i o n s , c o n t a i n s a s i g n i f i c a n t amount and a c o n s i d e r a b l e v a r i e t y o f m o l e c u l a r s p e c i e s . The m a j o r i t y o f the molecules have been d e t e c t e d i n the r a d i o r e g i o n of the spectrum u s u a l l y i n e m i s s i o n , but o c c a s i o n a l l y i n a b s o r p t i o n , a l t h o u g h some molecules have been d i s c o v e r e d by t h e i r a b s o r p t i o n o f v i s i b l e (CH, CH+ and CN) and u l t r a v i o l e t ( H 2 ana CO) s t a r l i g h t . The molecules are g e n e r a l l y found i n c o m p a r a t i v e l y dense extended r e g i o n s . Also the most dense m o l e c u l a r clouds are r e g i o n s o f a c t i v e s t a r f o r m a t i o n . M o l e c u l a r processes do not o n l y p r o v i d e us w i t h i n f o r m a t i o n about the e v o l u t i o n o f these c l o u d s , but are a l s o a means o f d e t e r m i n i n g t h e i r c o m p o s i t i o n and p h y s i c a l p r o p e r t i e s ( e . g . , d e n s i t y , t e m p e r a t u r e , e t c . ) . The abundance o f a p a r t i c u l a r m o l e c u l a r species r e p r e s e n t s the c o m p e t i t i o n between the chemical processes r e s p o n s i b l e f o r the c r e a t i o n and d e s t r u c t i o n o f the mole c u l e . The p r i n c i p a l methods n o r m a l l y c o n s i d e r e d f o r the f o r m a t i o n o f i n t e r s t e l l a r molecules from i n d i v i d u a l atoms are d i r e c t r a d i a t i v e attachment and c a t a l y t i c f o r m a t i o n on the s u r f a c e o f dust g r a i n s . I n the f i r s t p r o c e s s , two atoms, f o r example, can s t i c k t o g e t h e r t o form a m o l e c u l e , w i t h t h e i r b i n d i n g energy g o i n g o f f as a photon. I n the second p r o c e s s , s u c c e s s i v e atoms h i t a g r a i n , s t i c k t o the s u r f a c e m o m e n t a r i l y , and then combine t o form a m o l e c u l e , g i v i n g o f f t h e i r b i n d i n g energy as heat t r a n s m i t t e d t o the g r a i n ; t he molecule t h e n e i t h e r escapes by e v a p o r a t i o n or by some o t h e r p r o c e s s , perhaps i n v o l v i n g photon e x c i t a t i o n o f th e molecule t o a l e v e l which i s not bound t o the g r a i n . The d e s t r u c t i o n of i n t e r s t e l l a r molecules i s due t o the a b s o r p t i o n of photons. D i r e c t p h o t o d i s s o c i a t i o n can occur when the energy of the photon i s l a r g e r than the b i n d i n g energy o f the molecule. Lower energy photons can cause d i s s o c i a t i o n by e x c i t i n g the molecule t o an i n t e r m e d i a t e s t a t e which subsequently d i s s o c i a t e s ( e . g . , i n p r e d i s s o c i a t i o n , spontaneous r a d i a t i v e d i s s o c i a t i o n , and p h o t o i o n i z a t i o n ) . Both f o r m a t i o n and d i s r u p t i o n can be m o d i f i e d by exchange i n t e r a c t i o n s between molecules and e i t h e r atoms or o t h e r m o l e c u l e s . For d e t a i l e d accounts of f o r m a t i o n and d i s r u p t i o n see the reviews o f Dalgarno (1975) and Watson (1974). The o n l y a v a i l a b l e i n f o r m a t i o n about m o l e c u l a r c l o u d s i s t h e i r s p e c t r a , i . e . the r a d i a t i o n added t o or removed from the r a d i a t i o n - f i e l d a l o n g the l i n e o f the processes by which molecules emit and absorb photons. I f t h e c l o u d i s i n e q u i l i b r i u m w i t h a r a d i a t i o n - f i e l d , the number o f photons e m i t t e d w i l l equal the number absorbed. I n t h i s s i t u a t i o n one w i l l not "see" a s p e c t r a l C r u c i a l t o t h i s task, i s an u n d e r s t a n d i n g o f - 3 -l i n e s i n c e no net photons w i l l t h e n be added t o or removed from the f i e l d a l o n g the l i n e o f s i g h t . The processes which d i s t u r b t h i s e q u i l i b r i u m are m o l e c u l a r c o l l i s i o n s . The s p e c t r a a c t u a l l y observed w i l l depend on the com-p o s i t i o n and p h y s i c a l c o n d i t i o n s i n the c l o u d s . T h e r e f o r e i f we have s u f f i c i e n t knowledge about the process of s p e c t r a l l i n e f o r m a t i o n , t h i s may be used i n c o n j u n c t i o n w i t h the o b s e r v a t i o n s t o i n f e r the c o n d i t i o n s i n the i n t e r s t e l l a r c l o u d s . The t e m p e r a t u r e of i n t e r s t e l l a r c l o u d s i s q u i t e low , 100 K. At these low t e m p e r a t u r e s , v i b r a t i o n a l e x c i t a t i o n due t o c o l l i s i o n processes i s h i g h l y i m p r o b a b l e . For example, the energy o f the f i r s t e x c i t e d v i b r a t i o n a l l e v e l o f CO above the ground s t a t e corresponds t o a t e m p e r a t u r e o f Cii. 3000 K. The l e v e l s can, however, be p o p u l a t e d by the passage of a shock wave when the d e n s i t y and t e m p e r a t u r e o f the c l o u d are h i g h f o r a s h o r t p e r i o d o f time (Aannestad and F i e l d ( 1 9 7 3 ) , H o l l e n b a c h and S h u l l ( 1 9 7 7 ) ) . Other mechanisms are u l t r a v i o l e t f l u o r e s e n c e ( B l a c k and Dalgarno ( 1 9 7 3 ) ) , and the a b s o r p t i o n of h i g h energy photons ( B l a c k and Dalgarno ( 1 9 7 6 ) ) . Another source of v i b r a t i o n a l e x c i t a t i o n i s the m o l e c u l a r f o r m a t i o n process i t s e l f ( S p i t z e r and Cochran ( 1 9 7 3 ) ) . The f o r m a t i o n mechanism may be more e f f i c i e n t i n p o p u l a t i n g v e r y h i g h r o v i b r a t i o n a l s t a t e s , but i t occurs o n l y once d u r i n g the l i f e t i m e of the molecule whereas the o t h e r e x c i t a t i o n processes would be expected t o occur more - 4 -f r e q u e n t l y . The d i f f e r e n t e x c i t a t i o n mechanisms r e s u l t i n d i f f e r e n t energy l e v e l p o p u l a t i o n s , and t h e r e f o r e the i n t e n s i t y of the s p e c t r a l l i n e s g i v e s i n f o r m a t i o n as t o the source o f the e x c i t a t i o n ( G a u t i e r et a l . ( 1 9 7 6 ) ) . I f the e x c i t a t i o n has been caused by a shock wave, the l o c a t i o n o f the v i b r a t i o n a l e m i s s i o n r e g i o n can be used t o d e t e r m i n e the o r i g i n o f the shock. Also t he v e l o c i t y o f t he shock may be determined by the d e t a i l s o f the em i s s i o n s p e c t r a (Simon et a l . ( 1 9 7 9 ) ) . I n t e r s t e l l a r clouds l o s e energy by the c o n v e r s i o n of k i n e t i c energy of e x c i t a t i o n o f the c l o u d c o n s t i t u e n t s . The e x c i t e d atoms and molecules emit photons which e v e n t u a l l y escape from the c l o u d . I n d i f f u s e c l o u d s the major energy l o s s a r i s e s from the e x c i t a t i o n o f the f i n e - s t r u c t u r e l e v e l s o f i o n i z e d carbon (e.g. see Dalgarno and McCray ( 1 9 7 2 ) ) . The r o t a t i o n a l e x c i t a t i o n o f molecules i s an i m p o r t a n t process i n d e t e r m i n i n g the c o o l i n g o f c l o u d s . The sources o f the h e a t i n g o f i n t e r s t e l l a r c l o u d s are m a n i f o l d . I n a d d i t i o n t o the macroscopic such as hydromagnetic waves (Arons and Max (1975)) and the d i s -s i p a t i o n of t u r b u l e n t motion (Dalgarno and McCray ( 1 9 7 2 ) ) , atomic and m o l e c u l a r processes w o r k i n g i n response t o th e a b s o r p t i o n o f i n t e r s t e l l a r u l t r a v i o l e t r a d i a t i o n and g a l a c t i c cosmic rays are p o t e n t i a l l y l a r g e sources. The v a r i o u s c o o l i n g and h e a t i n g mechanisms g i v e r i s e t o t h e r m a l and p r e s s u r e g r a d i e n t s t h a t a f f e c t the dynamical e v o l u t i o n o f the c l o u d and may indeed t r i g g e r i t s c o l l a p s e t o the p o i n t where s t a r f o r m a t i o n b e g i n s . D e t a i l s o f these processes are g i v e n i n the r e v i e w o f Flower (1983). I n c o n c l u s i o n r o v i b r a t i o n a l e x c i t a t i o n c r o s s s e c t i o n s are needed not o n l y t o d e r i v e the energy l o s s from a c l o u d , but a l s o t o i n t e r p r e t the r a d i o data on m o l e c u l a r l i n e e m i s s i o n and a b s o r p t i o n . A l l these a s t r o p h y s i c a l processes are d i s c u s s e d by Dalgarno ( 1 9 7 5 ) . 2. Cross Se c t i o n s and Rate C o e f f i c i e n t s Consider the simple case of atom-diatomic molecule s c a t t e r i n g . The p r o b a b i l i t y t h a t the molecule w i l l change l e v e l s by energy t r a n s f e r d u r i n g a c o l l i s i o n , and t h a t the p r o j e c t i l e w i l l be s c a t t e r e d at a g i v e n a n g l e , i s expressed i n terms o f a d i f f e r e n t i a l cross s e c t i o n , which has the u n i t s o f area and which has the f o l l o w i n g p h y s i c a 1 i n t e r p r e t a t i o n . The cross s e c t i o n w i l l depend on the i n i t i a l ( ]f) and f i n a l ( ^ ) quantum numbers f o r the c o l l i s i o n , and a l s o on the r e l a t i v e c o l l i s i o n v e l o c i t y J/j. I f we assume t h a t the o r i g i n o f c o o r d i n a t e s i s a t the t a r g e t and a l l o w the p r o j e c t i l e t o approach a l o n g the z-ax i s ( F i g u r e 1) the d i f f e r e n t i a l c r o s s s e c t i o n f o r a c o l l i s i o n v e l o c i t y V, i s g i v e n da- / = number of p a r t i c l e s g i v i n g r i s e t o the t r a n s i t i o n ft-^}f d e f l e c t e d i n t o s o l i d angle dJ2.{&, ) . I f one c o n s t r u c t s a c i r c u l a r r e g i o n o f area = = / \ F i g u r e 2 an -2 The cross s e c t i o n CF( £ f 2 f ^ I V" ) thus g i v e s " e f f e c t i v e i n t e r a c t i o n r a d i u s " b m a x = ( G^CX^X1 |' v*")/7t) 2such t h a t a t r a n s i t i o n w i l l occur i f the p r o j e c t i l e approaches c l o s e r t o the t a r g e t than D m a x • The r a t e at which t r a n s i t i o n s ^ occur i s g i v e n by the f l u x o f p a r t i c l e s p a s s i n g t h r o u g h the area 1.2.2 where n i s the number d e n s i t y o f p r o j e c t i l e s . I n g e n e r a l t h e r e w i l l be a d i s t r i b u t i o n o f c o l l i s i o n v e l o c i t i e s , so t h a t /fib - a -£{V) i s u s u a l l y taken t o be the M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n at temp e r a t u r e T , 3A - & f>-** (gffi fyjg) 1.2.4 where k i s Boltzmann's c o n s t a n t andJU- i s the reduced mass f o r the c o l l i s i o n . m a v be then expressed <*9 where we have c o n v e r t e d from v e l o c i t y t o k i n e t i c energy u s i n g 3. E x p e r i m e n t a l D e t e r m i n a t i o n o f Cross S e c t i o n s There are e s s e n t i a l l y two e x p e r i m e n t a l t e c h n i q u e s f o r d e t e r m i n i n g r o v i b r a t i o n a l c r oss s e c t i o n s , m o l e c u l a r o r i o n beam s c a t t e r i n g e x p e r i m e n t s , and b u l k r e l a x a t i o n t e c h n i q u e s . S c a t t e r i n g experiments have the advantage t h a t s t a t e t o s t a t e cross sec I ions may \w di' ( c rm i n n I , h o w v - r ', He ii < • x | )<• r i in< • 111 \ a i <• I i m i i i - < I I n • <- i i .1 I 1 > m> 11 1 - 1 11 I m 1 and atomic species by t e c h n i c a l d i f f i c u l t i e s . In b u l k r e l a x a t i o n experiments none of those d i f f i c u l t i e s are encountered, however these t e c h n i q u e s a l l o w o n l y the d e t e r m i n a t i o n o f r e l a x a t i o n r a t e c o e f f i c i e n t s . I n m o l e c u l a r or i o n beam s c a t t e r i n g experiments two beams o f p a r t i c l e s i n t e r s e c t each o t h e r , u s u a l l y a t r i g h t a n g l e s . The pre s s u r e o f the beams and i n - 9 -the apparatus are o f the o r d e r o f 1 0 ~ ^ t o r r c o r r e s p o n d i n g t o a mean f r e e p a t h of 10 t o 100 metres. Thus the molecules i n the two beams do not undergo c o l l i s i o n s except i n the r e g i o n o f the i n t e r s e c t i o n (known as the s c a t t e r i n g r e g i o n ) . A d e t e c t o r , which can be r o t a t e d about the s c a t t e r i n g r e g i o n , measures the i n t e n s i t y o f the s c a t t e r e d p a r t i c l e s as a f u n c t i o n o f the s c a t t e r i n g a n g l e . E s s e n t i a l l y two f u n d a m e n t a l l y d i f f e r e n t e x p e r i m e n t a l t e c h n i q u e s have been employed t o d e t e c t i n e l a s t i c s c a t t e r i n g . One i s the s t a t e s e l e c t i o n method and the o t h e r t he energy change method. I n the s t a t e s e l e c t i o n experiment the molecules are p r e p a r e d i n a d e f i n i t e s t a t e b e f o r e s c a t t e r i n g and are then analysed f o r t h e i r f i n a l quantum s t a t e a f t e r s c a t t e r i n g by an a p p r o p r i a t e " s t a t e f i l t e r " , w hich p e r m i t s o n l y molecules i n the d e s i r e d s t a t e t o a r r i v e at the d e t e c t o r . S t a t e - s e l e c t i o n experiments use f o c u s i n g p r o p e r t i e s o f e l e c t r i c f i e l d s , or v a r i o u s s p e c t r o s c o p i c t e c h n i q u e s . The r e s o l v i n g power i s u s u a l l y v e r y h i g h and l a r g e l y independent of the beam v e l o c i t i e s . However, the t e c h n i q u e s are o n l y a p p l i c a b l e t o heavy molecules w i t h l a r g e d i p o l e moments (e.g. T j F , C s F , K C i ) . I n the energy change method the i n e l a s t i c events are d e t e c t e d i n d i r e c t l y by making use of energy conser-v a t i o n . I f b o t h beams are monoenergetic and are w e l l - 10 -c o l l i m a t e d , then the c o n v e r s i o n o f t r a n s l a t i o n a l t o i n t e r n a l energy i n an i n e l a s t i c event w i l l l ead t o a change i n the r e l a t i v e v e l o c i t y a f t e r s c a t t e r i n g . This w i l l be observed by a s m a l l change i n the l a b o r a t o r y v e l o c i t y o f bo t h s c a t t e r e d p a r t i c l e s . The i n t e n s i t y o f molecules w i t h an a l t e r e d v e l o c i t y i s a measure o f the i n e l a s t i c c r o s s s e c t i o n . Beam experiments i n t h e i r p r e s e n t s t a t e o f development s u f f e r from s e v e r a l d i s a d v a n t a g e s . S t a t e s e l e c t i o n e x periments are l i m i t e d t o s p e c i a l cases and the lower r e s o l u t i o n o f energy change methods makes the measurement of s t a t e t o s t a t e r o v i b r a t i o n a l c r o ss s e c t i o n s e x t r e m e l y d i f f i c u l t , e s p e c i a l l y f o r n e u t r a l beams. The use of i o n s i n beam experiments has the s i g n i f i c a n t advan-tage t h a t they are e a s i l y a c c e l e r a t e d t o h i g h e n e r g i e s , where v i b r a t i o n a l e x c i t a t i o n o c c u r s . Moreover, they are e a s i l y energy analysed and d e t e c t e d . T h e r e f o r e beam experiments can r e s o l v e r o v i b r a t i o n a l c r o ss s e c t i o n s i n systems such as L i + + ^ because o f the ease of d e t e c t i o n o f the L i + , and the h i g h r o t a t i o n a l c o n s t a n t o f H2. However, f o r most systems (e.g. H2 t CO) v i b r a t i o n a l r e l a x a t i o n r a t e s are de t e r m i n e d by c o l l i s i o n s w i t h p r o b a b i l i t i e s o f the o r d e r o f 10 - ^ , and beam experiments are o n l y s e n s i t i v e t o f a i r l y l a r g e t r a n s i t i o n p r o b a b i l i t i e s o f the o r d e r o f IX or g r e a t e r . Bulk r e l a x a t i o n experiments do not s u f f e r from any o f these problems,such experiments have the common 11 f e a t u r e o f d i s t u r b i n g a system from i t s e q u i l i b r i u m d i s t r i b u t i o n and measuring the r a t e o f r e t u r n t o e q u i l i -b r i u m . T y p i c a l experiments i n c l u d e l a s e r Raman e x c i t a t i o n , sound a b s o r p t i o n , n u c l e a r magnetic resonance s p i n - l a t t i c e r e l a x a t i o n , and double-resonance s p e c t r a l t echniques. I n l a s e r Raman e x p e r i m e n t s , Raman a c t i v e molecules are s t i m u l a t e d by a s h o r t l a s e r p u l s e t o the f i r s t v i b r a t i o n a l s t a t e i n a low temperature gas c e l l . As the molecules r e l a x t o the ground s t a t e t h e r e i s a s m a l l t e m p e r a t u r e i n c r e a s e , which leads t o a d e n s i t y change, which i s d e t e c t e d by a l a s e r S c h l i e r e n t e c h n i q u e (Ducuing et a l . ( 1 9 7 0 ) ) . The method has the advantage t h a t n o n p o l a r molecules such as H2 and N2 are e x c i t e d i n t o a d e f i n e d v i b r a t i o n a l s t a t e i n a low t e m p e r a t u r e heat b a t h . I n sound a b s o r p t i o n e x p e r i m e n t s , one measures the a t t e n u a t i o n o f u l t r a s o n i c waves as a f u n c t i o n of d i s t a n c e t r a v e l l e d i n a gas. Part o f the a t t e n u a t i o n comes from c o n v e r t i n g t r a n s l a t i o n a l i n t o r o v i b r a t i o n a l m o l e c u l a r energy, and t h e r e f o r e , the r a t e c o e f f i c i e n t may be measured. I n a n u c l e a r magnetic resonance s p i n - l a t t i c e r e l a x a t i o n , a non-thermal d i s t r i b u t i o n o f n u c l e a r s p i n s t a t e s i s c r e a t e d by magnetic f i e l d s and pulses of resonant r a d i o f r e q u e n c i e s . The r a t e o f r e t u r n t o e q u i l i b r i u m can then be m o n i t o r e d . I f t h i s experiment 12 i s performed i n the gas phase, the r e l a x a t i o n occurs m a i n l y by the c o u p l i n g of n u c l e a r s p i n and m o l e c u l a r r o t a t i o n . Thus c o l l i s i o n s which change the r o t a t i o n a l s t a t e w i l l a l s o t h e r m a l i z e the n u c l e a r s p i n s t a t e s , and one measures the r o t a t i o n a l r a t e s w e i g h t e d by th e c o u p l i n g c o n s t a n t s , which are known. I n the double-resonance one d i s t u r b s the e q u i l i b r i u m r o t a t i o n a l p o p u l a t i o n s by pumping w i t h s t r o n g r a d i a t i o n at a resonant frequency which d i s t u r b s t he p o p u l a t i o n i n resonant l e v e l s . T his d i s t u r b a n c e propagates t o o t h e r l e v e l s t h r o u g h c o l l i s i o n a l r e d i s t r i b u t i o n and i s sampled by n o t i n g the change i n i n t e n s i t y i n o t h e r t r a n s i t i o n s . These changes i n i n t e n s i t y are r e l a t e d t o the r e l a t i v e r a t e s o f c o l l i s i o n a l t r a n s f e r between a l l the l e v e l s . For a more d e t a i l e d account o f these and o t h e r e x p e r i m e n t a l t e c h n i q u e s see the r e v i e w of Toennies (1 9 7 6 ) . 4. T h e o r e t i c a l D e t e r m i n a t i o n of Cross S e c t i o n s One of the most u s e f u l s i m p l i f y i n g a p p r o x i m a t i o n s f o r m o l e c u l a r c o l l i s i o n s i s the Born-Oppenheimer assumption o f separable e l e c t r o n i c and n u c l e a r motions. Because the e l e c t r o n s are much l i g h t e r than the n u c l e i , t hey move much more r a p i d l y so t h a t we may expect them t o a d i a b a t i c a l l y a d j u s t t o the i n s t a n t a n e o u s p o s i t i o n o f the n u c l e i . T h e r e f o r e , we have s i m p l i f i e d t h e problem by d i v i d i n g i t i n t o two separate problems. A d e t e r m i n a t i o n o f the p o t e n t i a l energy s u r f a c e due - 13 -t o the mo t i o n o f the e l e c t r o n s and then we c o n s i d e r the m o t i o n o f the n u c l e i on t h i s p o t e n t i a l s u r f a c e . 4.1 The d e t e r m i n a t i o n o f P o t e n t i a l Energy Surfaces The i n t e r a c t i o n p o t e n t i a l between a d i a t o m i c molecule AB, approximated as a r i g i d r o t o r , and a d i a t o m i c molecule CD, approximated as a v i b r a t i n g r o t o r i s g i v e n by V(^c) - + CZ> {f ) ~ ^c^(c)-EAB where R i s the v e c t o r j o i n i n g the c e n t r e s o f mass of the molecules AB and CD, and r l i e s a l o n g the i n t e r -n u c l e a r a x i s o f CD . Eftg+zi) i s t n e t o t a ^ e l e c t r o n i c energy o f the t o t a l system f o r p o s i t i o n v e c t o r s R and r , and / t ) a n d £flg are the t o t a l e n e r g i e s o f the i s o l a t e d molecules ( i . e . f o r R = CO ). I n t e r a c t i o n p o t e n t i a l s m a n i f e s t themselves i n a v a r i e t y o f s t a t i c and dynamic phenomena. For example, the e q u i l i b r i u m s t r u c t u r e o f s o l i d s , v i r i a l c o e f f i c i e n t s of gases, v i s c o s i t y , e t c . Measurements o f such phenomena are o f t e n used t o o b t a i n i n f o r m a t i o n about t he i n t e r -a c t i o n p o t e n t i a l , b u t , d e s p i t e the v a r i e t y o f a v a i l a b l e methods, r e l a t i v e l y l i t t l e i n f o r m a t i o n has been o b t a i n e d t h i s way. The p o t e n t i a l i s o b t a i n e d o n l y by comparing o b s e r v a t i o n s w i t h p r e d i c t i o n s based on model p o t e n t i a l s . Because the models are r a t h e r i n f l e x i b l e , and because d i f f e r e n t experiments tend t o sample d i f f e r e n t p a r t s or d i f f e r e n t averages o f the i n t e r a c t i o n , i t i s o f t e n found t h a t a p o t e n t i a l which models one type o f data w i l l be inadequate f o r another (e.g. see Shafer and Gordon ( 1973 ) ) . 14 The problem o f d e t e r m i n i n g the i n t e r a c t i o n p o t e n t i a l between two systems i s e s s e n t i a l l y the quantum chemical problem o f d e t e r m i n i n g the t o t a l energy o f the e l e c t r o n s and n u c l e i o f the systems. Since the e l e c t r o n i c m o t i o n i s v e r y much f a s t e r than the n u c l e a r m o t i o n , the problem reduces t o d e t e r m i n i n g the e l e c t r o n i c energy as a f u n c t i o n o f f i x e d n u c l e a r geometry (Born-Oppenheimer) a p p r o x i m a t i o n . The major c o n t r i b u t i o n s t o the energy are the k i n e t i c energy o f the e l e c t r o n s and the Coulomb i n t e r a c t i o n among the e l e c t r o n s and n u c l e i . The i n t e r -a c t i o n energy i s the d i f f e r e n c e between the t o t a l energy o f the combined system and t h a t o f the se p a r a t e d system. This can l e a d t o l a r g e c a n c e l l a t i o n w i t h subsequent l o s s o f accuracy. I n g e n e r a l , d i f f e r e n t t e c h n i q u e s are u s e f u l at d i f f e r e n t s e p a r a t i o n s , and i t i s c o n v e n i e n t t o d i s t i n g u i s h between l o n g range, s h o r t range, and i n t e r m e d i a t e d i s t a n c e s . At l a r g e d i s t a n c e s AB and CD can be d e s c r i b e d as n o n - o v e r l a p p i n g change d i s t r i b u t i o n s , and the i n t e r -a c t i o n reduces t o the e l e c t r o s t a t i c problem o f i n t e r -a c t i n g permanent and induced mult i p o l e moment s . The p o t e n t i a l c o n s i s t s o f t h r e e terms. The e l e c t r o s t a t i c energy due t o the i n t e r a c t i o n o f permanent m u l t i p o l e moments, the i n d u c t i o n energy, due t o the i n t e r a c t i o n o f permanent moments w i t h those induced i n the o t h e r c o l l i s i o n p a r t n e r , and the d i s p e r s i o n energy. Because the i n t e r a c t i o n i s s m a l l , i t i s a c c u r a t e l y r e p r e s e n t e d - 15 -as a p e r t u r b a t i o n o f the se p a r a t e d systems: t he i n t e r -a c t i o n energy can be c a l c u l a t e d d i r e c t l y by p e r t u r b a t i o n t e c h n i q u e s a v o i d i n g the problem o f c a n c e l l a t i o n . At s m a l l d i s t a n c e s , the AB and CD change d i s t r i -b u t i o n s o v e r l a p s t r o n g l y and the i n t e r a c t i o n becomes r e p u l s i v e . I n t h i s r e g i o n the system i s best d e s c r i b e d as a s i n g l e m o l e c u l e , and m o l e c u l a r o r b i t a l t e c h n i q u e s such as the Hartree-Fock method are a p p l i c a b l e . I n the Hartree-Fock or s e l f c o n s i s t e n t f i e l d method, each e l e c t r o n i s c o n s i d e r e d t o move i n the e l e c t r o s t a t i c f i e l d c r e a t e d by the o t h e r e l e c t r o n s . To d e s c r i b e the m o tion o f one e l e c t r o n one must know the s o l u t i o n s f o r a l l the o t h e r e l e c t r o n s which d e t e r m i n e the e l e c t r o -s t a t i c f i e l d . I n p r a c t i c e , one achieves t h i s by an i t e r a t i v e t e c h n i q u e . A reasonable i n i t i a l guess i s made t o the s o l u t i o n s and these are used i n the H a r t r e e -Fock e q u a t i o n s t o produce new s o l u t i o n s which become the next guess. This c y c l e i s re p e a t e d u n t i l the s o l u t i o n s are the same as the i n p u t . Hence the name s e l f c o n s i s t e n t f i e l d . The w a v e f u n c t i o n s o f the e l e c t r o n s ( o r o r b i t a l s ) must be expanded i n some s o r t o f b a s i s f u n c t i o n s . For m o l e c u l a r systems one o f t e n uses atomic o r b i t a l s - or at l e a s t o r b i t a l s c e n t r e d on the v a r i o u s atoms - f o r the b a s i s . This leads t o the commonly seen SCF-LCAO f o r s e l f c o n s i s t e n t f i e l d - l i n e a r c o m b i n a t i o n of atomic o r b i t a l s . One o f t e n sees "MO" appended - 16 -t o t h i s t o i n d i c a t e t h a t the LCAO are formed i n t o m o l e c u l a r o r b i t a l s . The Hartree-Fock method has c e r t a i n i n h e r e n t l i m i t a t i o n s , f o r example i t does not c o r r e c t l y d e s c r i b e the i n s t a n t a n e o u s c o r r e l a t i o n o f e l e c t r o n m o t i o n s . The r e s u l t i n g e r r o r i n the energy i s c a l l e d c o r r e l a t i o n energy. However a t s h o r t range t h i s c o r r e l a t i o n energy i s much s m a l l e r than t he e l e c t r o s t a t i c energy, and t h e r e f o r e t h i s i s not a s e r i o u s problem. At i n t e r m e d i a t e d i s t a n c e s the l o n g range a t t r a c t i v e f o r c e s and the s h o r t range r e p u l s i v e f o r c e s compete t o form a p o t e n t i a l w e l l , and t h i s i s the most d i f f i c u l t r e g i o n f o r which t o o b t a i n an a c c u r a t e i n t e r a c t i o n . The l o n g range p e r t u r b a t i o n e xpansion f a i l s as the change d i s t r i b u t i o n s b e g i n t o o v e r l a p . M o l e c u l a r o r b i t a l methods g e n e r a l l y become inadequate i n t h i s r e g i o n because the c o r r e l a t i o n energy i s comparable t o the i n t e r a c t i o n energy and moves r a p i d l y w i t h d i s t a n c e as the o r b i t a l s change f r o m a t o m i c t o m o l e c u l a r i n c h a r a c t e r . To o b t a i n a c c u r a t e r e s u l t s at i n t e r m e d i a t e d i s t a n c e s , c o n f i g u r a t i o n i n t e r a c t i o n t e c h n i q u e s which e x p l i c i t l y i n c l u d e c o r r e l a t i o n e f f e c t s must be used. I n Hartree-Fock methods o n l y one s e t , or c o n f i g u r a t i o n , o f m o l e c u l a r o r b i t a l s i s employed, whereas i n c o n f i g u r a t i o n i n t e r a c t i o n c a l c u l a t i o n s , t h e w a v e f u n c t i o n s employed are l i n e a r c o m b i n a t i o n s o f a l l p o s s i b l e c o n f i g u r a t i o n s . Hence c o n f i g u r a t i o n i n t e r a c t i o n c a l c u l a t i o n s a l l o w 17 a b e t t e r d e s c r i p t i o n o f the w a v e f u n c t i o n s as they change from m o l e c u l a r t o atomic. However, such c a l c u l a t i o n s r e q u i r e r o u g h l y an o r d e r of magnitude more computer time than Hartree-Fock c a l c u l a t i o n s , and t h i s r e p r e s e n t s a major p r o j e c t f o r a l l but the s m a l l e s t systems. These techniq u e s o u t l i n e d f o r the c a l c u l a t i o n o f p o t e n t i a l energy s u r f a c e s are d e s c r i b e d i n more d e t a i l by Schaefer (1972 ) . The ab i n i t i o methods d e s c r i b e d above r e l y on approximate s o l u t i o n o f the a p p r o p r i a t e Schrb'dinger e q u a t i o n . A v a r i e t y of l e s s r i g o r o u s t e c h n i q u e s f o r o b t a i n i n g i n t e r a c t i o n p o t e n t i a l s have a l s o been developed, such as the e l e c t r o n gas model o f Gordon and Kim ( 1 9 7 2 ) . The energy of a u n i f o r m e l e c t r o n gas i s a f u n c t i o n o n l y of i t s d e n s i t y . Gordon and Kim (1972) use t h i s f a c t p l u s two o t h e r assumptions. I n each volume element, the e l e c t r o n d e n s i t y i s a p p r o x i m a t e l y c o n s t a n t , and the energy c o n t r i b u t i o n i s r e l a t e d t o t h i s d e n s i t y by the u n i f o r m e l e c t r o n gas f o r m u l a e ; the t o t a l energy o f the system i s o b t a i n e d by i n t e g r a t i n g over a l l space. Secondly the e l e c t r o n d e n s i t y of the combined system i s taken as the sum of the Hartree-Fock d e n s i t i e s o f the s e p a r a t e d systems; t h i s assumption e f f e c t i v e l y l i m i t s the method t o i n t e r a c t i o n s between c l o s e d s h e l l systems where the charge d i s t r i b u t i o n s do not r e a r r a n g e t o form a chemical bond. The method f a i l s at l o n g range. The i n d u c t i o n energy i s not o b t a i n e d s i n c e - 1 H -the charge d i s t r i b u t i o n s are not a l l o w e d t o p o l a r i s e , and the d i s p e r s i o n energy i s not o b t a i n e d as the i n t e r a c t i o n v a nishes f o r n o n o v e r l a p p i n g charge d i s t r i b u t i o n s . There i s e v i d e n c e , however, t h a t the model i s r e l i a b l e f o r t he s h o r t range a n g u l a r dependence o f the p o t e n t i a l ( e . g . see Green ( 1 9 7 4 ) ) . 4.2 S c a t t e r i n g Theory Once the i n t e r a c t i o n p o t e n t i a l has been d e t e r m i n e d , the e q u a t i o n s d e s c r i b i n g the dynamics o f the c o l l i s i o n p a r t n e r s i n t h a t p o t e n t i a l must be s o l v e d . T h i s i s known as s c a t t e r i n g t h e o r y , and a number of t e c h n i q u e s f o r s o l v i n g the e q u a t i o n s are a v a i l a b l e . The approach one uses can be f u l l y quantum m e c h a n i c a l , i t can make use o f b o t h quantum and c l a s s i c a l mechanics, or can be a p u r e l y c l a s s i c a l method. 4.2.1 Quantum Mechanics I n t h i s approach the e q u a t i o n t o be s o l v e d i s the time independent Schrb'dinger e q u a t i o n c o n t a i n i n g the f u l l H a m i l t o n i a n of each c o l l i s i o n p a r t n e r . This i s reduced t o a set of coupled second o r d e r d i f f e r e n t i a l e q u a t i o n s . I n the c o n v e n t i o n a l c l o s e - c o u p l i n g t e c h n i q u e , the e q u a t i o n s are s o l v e d i n the s p a c e - f i x e d c o o r d i n a t e system. The t o t a l w a v e f u n c t i o n of the system i s expanded i n terms o f the b a s i s s t a t e s which are e i g e n f u n c t i o n s o f the t o t a l a n g u l a r momentum j and the v i b r a t i o n a l H a m i l t o n i a n . Since J i s compounded from the r o t a t i o n a l a n g u l a r momenta and J_2 > a n c^ t n e o r b i t a l a n g u l a r - 19 -momentum X, each b a s i s s e t i s indexed by the r o t a t i o n a l a n g u l a r momenta and the a n g u l a r momentum X 5 and the v i b r a t i o n a l quantum number v 2 ? ( j ^ j 2^-v 7 ^ • The p o t e n t i a l couples t o g e t h e r a l l the b a s i s s t a t e s ( I . e . J •- 1 1 -L) • J_ i s conserved t h e r e f o r e the c o u p l i n g m a t r i x i s d i a g o n a l i n the t o t a l a n g u l a r momentum. Because the o r i e n t a t i o n o f the whole system i n space i s i r r e l e v a n t the c o u p l i n g m a t r i x i s independent of t he z - p r o j e c t i o n of J. However f o r l a r g e o r ^2, t h e number o f e q u a t i o n s t o be s o l v e d can become e x t r e m e l y l a r g e , because of the 2 j ^ + l ( i = 1,2) d e g e n e r a c i e s . Depending on the a l g o r i t h m used t o s o l v e the coupled e q u a t i o n s , the computing time i n c r e a s e s 2 3 as N or N where N i s the number of channels. T h e r e f o r e except f o r the s i m p l e s t systems, a p p r o x i m a t i o n s must be made t o the e q u a t i o n s i n o r d e r t o reduce the number of channels r e t a i n e d . I n r e c e n t years a number o f d e c o u p l i n g a p p r o x i m a t i o n s have been developed which i n v o l v e a s i m p l i f i e d t r e a t m e n t of one or more of the terms i n the f u l l c l o s e - c o u p l i n g e q u a t i o n s , and hence t h e i r v a l i d i t y depends on the importance o f those terms i n the c o l l i s i o n . I n Chapter I I o f t h i s t h e s i s , a f u l l d e r i v a t i o n of the c l o s e - c o u p l i n g e q u a t i o n s i s g i v e n , and some of the d e c o u p l i n g approximations i n c u r r e n t use are d i s c u s s e d . 20 -4.2.2 C l a s s i c a l and S e m i - C l a s s i c a l methods There are e s s e n t i a l l y two extremes; a t one extreme, quantum mechanics i s s u i t e d t o c o l l i s i o n s i n v o l v i n g l i g h t p a r t i c l e s a t low energy ( i . e . few quantum s t a t e s are e x c i t e d ) . At the o t h e r extreme c l a s s i c a l mechanics i s best s u i t e d t o h i g h energy c o l l i s i o n s ( i . e . where we c o n s i d e r almost c o n t i n u o u s energy l e v e l s ) . G e n e r a l l y speaking most systems s t u d i e d f a l l between these two extremes, and f o r these a s e m i - c l a s s i c a l approach i s the most a p t . I n t h i s s e c t i o n we o u t l i n e a number of the most commonly used s e m i - c l a s s i c a l and c l a s s i c a l approaches. I t i s w o r t h m e n t i o n i n g t h a t o n l y the s t r o n g - c o u p l i n g correspondence p r i n c i p l e o f P e r c i v a l and Richards ( 1 9 7 0 ) , and the s e m i - c l a s s i c a l method o f B i l l i n g (1974) have been extended from atom-diatom s c a t t e r i n g t o d i a t o m - d i a t o m s c a t t e r i n g . I n the time-dependent c l o s e - c o u p l i n g method the r e l a t i v e m o t i o n i s t r e a t e d c l a s s i c a l l y and the i n t e r n a l m o t i o n i s t r e a t e d u s i n g quantum mechanics. For c o n s i s t e n c y th e r e l a t i v e m otion should be determined i n a p o t e n t i a l independent of the t a r g e t o r i e n t a t i o n . This i s the p r i n c i p a l source o f e r r o r i n the a p p r o x i m a t i o n s i n c e the back c o u p l i n g from the t a r g e t t o the t r a j e c t o r y i s n e g l e c t e d . T h e r e f o r e , the use o f c l a s s i c a l t r a j e c t o r i e s i s o n l y v a l i d i f the i n e l a s t i c t r a n s i t i o n s which occur do not s i g n i f i c a n t l y a f f e c t t he r e l a t i v e m o t i o n . - 2L -One assumes t h a t the p r o j e c t i l e f o l l o w s a c l a s s i c a l p a t h d e t e r m i n e d by a s p h e r i c a l l y s y m m e t r i c - p o t e n t i a l , or a n u l l p o t e n t i a l ( s t r a i g h t l i n e p a t h s ) . Once the t r a j e c t o r y has been determined i t can be used t o c o n s t r u c t a time dependent i n t e r a c t i o n p o t e n t i a l . The problem t h e n c o n s i s t s o f c a l c u l a t i n g the p r o b a b i l i t y o f r o -v i b r a t i o n a l t r a n s i t i o n s due t o t h i s time dependent i n t e r a c t i o n e x e r t e d by the p a s s i n g p r o j e c t i l e . As i n quantum mechanical c a l c u l a t i o n s , v a r i o u s s i m p l i f i c a t i o n s of the dynamics can be made r e s u l t i n g i n , f o r i n s t a n c e , the time-dependent sudden and t i m e -dependent coupled s t a t e s a p p r o x i m a t i o n s . These dynamical approaches are reviewed by B a l i n t - K u r t i (1975) and D i c k i n s o n (1979). The major f a u l t of p u r e l y c l a s s i c a l d e s c r i p t i o n s o f the dynamics i s the n e g l e c t o f quantum mechanical e f f e c t s . These are accounted f o r i n the c l a s s i c a l S -matrix method o f M i l l e r (1974) and Marcus ( 1 9 7 2 ) , t h i s i s a g e n e r a l i z a t i o n of the s e m i c l a s s i c a l t r e a t m e n t o f e l a s t i c s c a t t e r i n g of Ford and Wheeler (19 5 9 ) . One determines a l l those c l a s s i c a l t r a j e c t o r i e s which l e a d from the i n i t i a l t o the f i n a l s t a t e of i n t e r e s t . The quantum a m p l i t u d e f o r the process i s o b t a i n e d by summing c o n t r i b u t i o n s from each t r a j e c t o r y w i t h a w e i g h t i n g r e l a t e d t o the d e n s i t y o f c l a s s i c a l paths i n t h a t neighbourhood and w i t h r e l a t i v e phase between the c o n t r i b u t i o n s a l s o d e t e r m i n e d c l a s s i c a l l y from 22 the a c t i o n a l o n g t h a t p a t h . However, f o r a system w i t h s e v e r a l degrees of freedom, the n u m e r i c a l e f f o r t i n v o l v e d i n the search f o r a l l t r a j e c t o r i e s s a t i s f y i n g a g i v e n set o f double-ended boundary c o n d i t i o n s becomes p r o h i b i t i v e l y l a r g e . T his problem i s reduced, however, i f one o n l y r e q u i r e s c r oss s e c t i o n s averaged over some quantum number ( M i l l e r ( 1 9 7 1 ) ) . The s e m i - c l a s s i c a l s t r o n g - c o u p l i n g correspondence p r i n c i p l e o f P e r c i v a l and Richards (1970) approximates the s o l u t i o n o f the time-dependent c l o s e - c o u p l i n g e q u a t i o n s u s i n g a c l a s s i c a l d e s c r i p t i o n of the m o t i o n o f the t a r g e t . I t i n c o r p o r a t e s the use o f c l a s s i c a l p e r t u r b a t i o n t h e o r y t o determine the change i n a c t i o n o f the t a r g e t . I t i s expected t o be most s u c c e s s f u l f o r l a r g e quantum numbers but comparisons w i t h quantum c a l c u l a t i o n s have shown s a t i s f a c t o r y r e s u l t s f o r c r o s s s e c t i o n s f o r t r a n s i t i o n s between l o w - l y i n g r o v i b r a t i o n a l s t a t e s ( C l a r k ( 1 9 7 7 ) ) . However the computing time f o r c a l c u l a t i o n s employing t h i s method i s l a r g e l y independent o f the quantum numbers i n v o l v e d and a r b i t r a r y l a r g e quantum numbers may be handled r o u t i n e l y . T his method i s d i s c u s s e d i n d e t a i l by C l a r k et a l . ( 1 9 7 7 ) . For v i b r a t i o n a l e x c i t a t i o n c a l c u l a t i o n s t h e r e are two f r e q u e n t l y used s e m i - c l a s s i c a l methods. An a p p r o x i m a t i o n developed by B i l l i n g (1974) t r e a t s one i n t e r n a l degree o f freedom quantum m e c h a n i c a l l y , w h i l e a l l the r e m a i n i n g degrees of freedom are t r e a t e d 23 c l a s s i c a l l y . For example, i n an a t o m - v i b r o t o r c a l c u l a t i o n the v i b r a t i o n a l m otion o f the molecule c o u l d be t r e a t e d quantum m e c h a n i c a l l y w h i l e the r o t a t i o n , and the t r a n s -l a t i o n a l motion are t r e a t e d c l a s s i c a l l y . The o t h e r method i s based on the correspondence between the c l a s s i c a l and quantum f o r c e d harmonic o s c i l l a t o r . Exact c l a s s i c a l t r a j e c t o r i e s are employed t o o b t a i n t h e c l a s s i c a l energy t r a n s f e r as a f u n c t i o n o f a n g l e . Using t he Poisson d i s t r i b u t i o n p r e d i c t e d by the f o r c e d o s c i l l a t o r model, v i b r a t i o n a l e x c i t a t i o n p r o b a b i l i t i e s can be c a l c u l a t e d (Giese and Gentry I f quantum mechanical e f f e c t s are u n i m p o r t a n t (e . g . i n t e r f e r e n c e and t u n n e l 1 i n g ) > a p u r e l y c l a s s i c a l method may be used. Completely c l a s s i c a l c a l c u l a t i o n s are a t t r a c t i v e s i n c e a l l the c o u p l i n g s are t r e a t e d e s s e n t i a l l y e x a c t l y w i t h o u t the problems o f e n s u r i n g convergence endemic i n a quantum c l o s e - c o u p l i n g expansion. A major problem i n o b t a i n i n g r e s u l t s from a p u r e l y c l a s s i c a l c a l c u l a t i o n t o be compared t o quantum c a l -c u l a t i o n s o r w e l l r e s o l v e d measurements i s the procedure t o be used t o " q u a n t i s e " the c o n t i n u o u s c l a s s i c a l v a r i a b l e s such as a n g u l a r momentum. I n r o t a t i o n a l e x c i t a t i o n , a w i d e l y used t e c h n i q u e i s t o d e f i n e a f i n a l c l a s s i c a l a n g u l a r momentum j t h r o u g h the energy ( 1 9 7 4 ) ) . C/<0= 3j&Ue + l^ 24 where B i s the r o t a t i o n a l c o n s t a n t of the molecule and t h e n a s s o c i a t e a f i n a l r o t a t i o n a l quantum number j<5 w i t h J c b y where = 1 f o r h e t e r o n u c l e a r m o l e c u l e s , and - 2 f o r homonuclear molecules t o a l l o w f o r the A j = 2 s e l e c t i o n r u l e , 'techniques f o r p e r f o r m i n g c l a s s i c a l c a l c u l a t i o n s have been reviewed by Bunker (1971) and P o l a n y i and S c h r e i b e r ( 1 9 7 4 ) . - 25 -CHAPTER I I QUANTUM THEORY OF MOLECULAR COLLISIONS 1 . I n t r o d u c t i on I n t h i s c h a p t e r we present the quantum mechanical d e s c r i p t i o n o f c o l l i s i o n s o f atoms and d i a t o m i c molecules w i t h d i a t o m i c m o l e c u l e s . Once the i n t e r a c t i o n p o t e n t i a l has been o b t a i n e d , the e q u a t i o n s d e s c r i b i n g the dynamics o f the c o l l i s i o n p a r t n e r s i n t h a t p o t e n t i a l must be s o l v e d . The e q u a t i o n i n q u e s t i o n i s the t i m e -independent S c h r o d i n g e r e q u a t i o n c o n t a i n i n g the f u l l H a m i l t o n i a n o f each c o l l i s i o n p a r t n e r . This may be reduced t o a set of coupled second o r d e r d i f f e r e n t i a l e q u a t i o n s . The s o l u t i o n o f the L a t t e r i s r e f e r r e d t o as the Close-C o u p l i n g (CC) method. This i s pr e s e n t e d i n S e c t i o n 2. However, f o r a l l but the s i m p l e s t systems, the n u m e r i c a l e f f o r t i n v o l v e d i n s o l v i n g the CC e q u a t i o n s i s p r o h i b i t i v e l y l a r g e even u s i n g f a s t modern computers. The l a r g e number o f eq u a t i o n s t h a t must be s o l v e d i s e s s e n t i a l l y due t o the c o u p l i n g between r o t a t i o n a l and o r b i t a l a n g u l a r momenta. However, i n r e c e n t years t h e r e has been a c o n s i d e r a b l e amount o f work i n the development o f a p p r o x i m a t i o n s which lead t o the t o t a l or p a r t i a l d e c o u p l i n g o f the a n g u l a r momenta. I n S e c t i o n 3 we present some a n g u l a r momentum d e c o u p l i n g a p p r o x i m a t i o n s . We beg i n by d i s -c u s s i n g the C e n t r i f u g a l sudden a p p r o x i m a t i o n i n S e c t i o n - 2 b -3.1 , wc d i s c u s s the Knergy sudden a pprox i nt.i t i on i n S e c t i o n 3.2 , and i n S e c t i o n 3.3 we d i s c u s s t he I n f i n i t e o r d e r sudden a p p r o x i m a t i o n (a h y b r i d o f the c e n t r i f u g a l and energy sudden a p p r o x i m a t i o n s ) , and f i n a l l y i n S e c t i o n 4 we touch on some o t h e r a l t e r n a t i v e s which may be used when none o f t h e above a p p r o x i m a t i o n s are a p p r o p r i a t e (namely t he L-Dominant, the decoupled L-Dominant, and the E f f e c t i v e p o t e n t i a l a p p r o x i m a t i o n s ) . For the sake of g e n e r a l i t y the t h e o r y i s pr e s e n t e d f o r c o l l i s i o n s between two d i a t o m i c m o l e c u l e s , except f o r the energy sudden a p p r o x i m a t i o n which has never been employed i n d i a t o m - d i a t o m c a l c u l a t i o n s , and we t h e r e f o r e , c o n s i d e r the atom-diatom t h e o r y t o be adequate. 2. C l o s e - C o u p l i n g Theory We w i l l c o n s i d e r t he d e r i v a t i o n o f the problem f o r the c o l l i s i o n between s t a t e d i a t o m i c m o l e c u l e s . We w i l l approximate the p r o j e c t i l e as a r i g i d r o t o r (denoted by 1 ) , and the t a r g e t as a v i b r a t i n g r o t o r (denoted by 2 ) . This d e r i v a t i o n i s i n the space-fixed(SF) frame and f o l l o w s the d e r i v a t i o n p r e s e n t e d by Takayanagi ( 1 9 6 5 ) . We take ^ = ( ^ , ^ , ^ 1 , and T 2 = ( r 2 ,# 2 ,2 ) l y i n g a l o n g the m o l e c u l a r axes o f 1 and 2 r e s p e c t i v e l y , and R = (R,©",<^ ) t o be the p o s i t i o n v e c t o r o f the i n t e r -m o l e c u l a r a x i s ( F i g u r e 1 ) . - 27 F i g u r e 1 I n t h i s c o o r d i n a t e system the Schrb'dinger e q u a t i o n may be w r i t t e n I I . 2 .1 where j£ denotes the t o t a l w a v e f u n c t i o n and E i s the t o t a l energy o f the system. The H a m i l t o n i a n , H, may be p a r t i t i o n e d as where the reduced mass of the system i s g i v e n by = {Mat (fnc^Mp^) I I .2.2 I I .2. 3 V i s the i n t e r a c t i o n p o t e n t i a l as determined by the Born-Oppenheimer a p p r o x i m a t i o n , 11. 2 .4 and h^ and h2 are the H a m i l t o n i a n s o f the i s o l a t e d - 28 -molecules 1 and 2 r e s p e c t i v e l y where 9 ^V-**- , I I . 2.5a and h -i » — ^ V£f|,~> . 11. 2 . 5b v ( r 2 > a n d ^ i ^ a r e t n e i n t e r n u c l e a r p o t e n t i a l and reduced mass o f molecule 2 r e s p e c t i v e l y . B i s the r o t a t i o n a l c o n s t a n t o f molecule 1 i n i t s ground v i b r a t i o n a l s t a t e . The k i n e t i c energy o p e r a t o r s may be se p a r a t e d i n t o r a d i a l and a n g u l a r p a r t s » ^t&, S^LgJ31- I I . 2 . 6 a and _M 7 | - - _ Z — ^ fL + Ji , 3 * . % u ^ < ^ 2 ^ r - i i . 2 . 6 b where jL i s the o r b i t a l a n g u l a r momentum o p e r a t o r , and and ^ a r e t n e r o t a t i o n a l a n g u l a r momentum o p e r a t o r s of molecules 1 and 2 r e s p e c t i v e l y . Use i s made of the c o n s e r v a t i o n o f a n g u l a r momentum of the system i n the c o l l i s i o n , H^^J -tj*. + A ^ J j + — ' I I . 2 . 7 where t h e primes denote values a f t e r the c o l l i s i o n . The boundary c o n d i t i o n s o f I I . 2.1 are (choosing the i n i t i a l r e l a t i v e m o t i o n t o be i n the z d i r e c t i o n ) >-^*> , 11.2.8a **P (ik*9&*>f ( V m ^ ^ ^ a W i A l . 2 . 8 b ***** * Xty fc> Yj>„<(^ . - 29 -The f i r s t Lerm i s c o n s i s t e n t w i t h the i n c i d e n t d i r e c t i o n of r e l a t i v e m otion b e i n g the z - a x i s , ^ f ^ j . ( ' ' i ) ^ / M j ^ . ) are t he r o v i b r a t i o n a l e i g e n s t a t e s o f molecule 2 and s a t i s f y u ^ )Xy^ (f€) \j3f*^ (30 =• o , " I I . 2 . 9 a where Sv^j^ i s the r o v i b r a t i o n a l eigenenergy o f molecule 2 i n v i b r a t i o n a l s t a t e ^2 and r o t a t i o n a l s t a t e Yjt/Y\( n ) i s the r o t a t i o n a l e i g e n s t a t e o f molecule 1 ( i n i t s ground v i b r a t i o n a l s t a t e ) V w I I . 2 . 9 b The magnitude of t h e i n c i d e n t wavevector kyyjj^ i s d e f i n e d by The second term i n I I . 2 . 8 b d e s c r i b e s the s c a t t e r e d wave, the s c a t t e r i n g a m p l i t u d e y i e l d s t h e s t a t e t o s t a t e d i f f e r e n t i a l c r o s s s e c t i o n V V I I . 2 . 1 1 Since t he t o t a l a n g u l a r momentum J and i t s p r o j e c t i o n M on the SF z- a x i s are conserved i t i s con v e n i e n t t o expand the t o t a l w a v e f u n c t i o n i n terms o f e i g e n s t a t e s o f J and M. The r o t a t i o n a l a n g u l a r momentum o f the 30 -two molecules j ^ and a r e coupled t o form j _ ^ 2 w i t h p r o j e c t i o n m^2 o n t n e SF z - a x i s , i . e . , rfx-^Y^., U 2 l 2 a and M n 5 mf ~¥ ^ 2 = , I I . 2 . 1 2 b which i s s u b s e q u e n t l y coupled t o the o r b i t a l a n g u l a r momentum v e c t o r JL t o form the t o t a l a n g u l a r momentum J (e.g. Alexander and D e p r i s t o ( 1 9 7 7 ) ) . x &«, e2) YJmjL d~), where C(j) m^rtlz. j ^ / i a ^ x " ) i s a C 1 e b s h " G o r d a n c o e f f i c i e n t S u b s t i t u t i o n o f I I . 2 . 1 4 i n t o I I . 2 . 1 and u s i n g I I . 2 . 9 a , b and the o r t h o n o r m a l p r o p e r t i e s o f # (j*jxjn..JLTM I A A ft~) and dLv^ (r£) , v / I / t e ^ ^ J-?**)oil / I I . 2 . 1 5 b we f i n d t h a t the r a d i a l f u n c t i o n s i n I I . 2 . 1 4 s a t i s f y the f o l l o w i n g s et o f d i f f e r e n t i a l e q u a t i o n s : 31 fnmtoi ,^7^ ft & = c; CC * V * ^ & } ^ / g ) , I I . 2 . 1 8 b where ^ + + / ^ 1 2 i s a n e v e n i n t e g e r -Using s t a n d a r d a n g u l a r momentum a l g e b r a i c t e c h n i q u e s the p o t e n t i a l m a t r i x V"1 may be w r i t t e n (Messiah (1962)) 32 , (^i"i')(yJ'(*0 , VJ=*> 0 and the s o l u t i o n s o f I I . 2 . 1 6 w i t h V J = 0 a r e , f o r k. vjj.j j-L ^ ® (Abramowitz and Stegun (1965 ) ) , I I . 2 . 2 0 a w h e r e ^ ^ / and HjJ are S p h e r i c a l Bessel f u n c t i o n s o f the f i r s t and second k i n d . We may, i n s t e a d , use S p h e r i c a l 33 Hankel f u n c t i o n s o f the f i r s t and second k i n d (these are S p h e r i c a l Bessel f u n c t i o n s of the t h i r d k i n d ) . to The boundary c o n d i t i o n (as R —$> 0O ) f o r iH can be expressed as a l i n e a r c o m b i n a t i o n o f kJ^-'irul I I . 2 . 2 0 b I I . 2 . 2 1 a a l s o ti%*frf«--+. o • E q u a t i o n I I . 2 . 2 1 a d e f i n e s the S-matrix which i s d i a g o n a l i n J, s i n c e J i s conserved. Channels w i t h itv^y/j^ ^ O are known as c l o s e d c h annels, w h i l e kv^^'j^ ^ O d e f i n e s open channels. I I . 2.21a may be r e w r i t t e n 34 employing the a s y m p t o t i c forms of the Spherical. Hankel f u n c t i o n s Thus lH i s expressed i n terms o f an incoming i n c i d e n t wave, and o u t g o i n g s c a t t e r e d waves. We r e q u i r e the c o r r e c t l i n e a r c o m b i n a t i o n of Y jsJ^J ^ which s a t i s f i e s the boundary c o n d i t i o n I I . 2 . 8 a , b. We employ the expansion of a plane wave i n t o S p h e r i c a l Bessel f u n c t i o n s (Abramowitz and Stegun) and t he e x p r e s s i o n f o r the product of two s p h e r i c a l harmonics ( t w i c e ) , and o b t a i n £'*vy\^ ^ ( V ) X)smv ($) foMx (^) I I . 2 . 2 3 - 35 -I n o r d e r f o r the f i r s t term i n the t o t a l w a v e f u n c t i o n t o go over t o t h i s a s y m p t o t i c a l l y , t he expansion used i s S u b s t i t u t i o n o f 11.2.21b, I I . 2 . 1 3 and I I . 2 . 1 4 i n t o I I . 2 . 2 4 g i v e s t he a s y m p t o t i c form v Ytiy Y.mt (6~) ^ (£) Jvy x (rC). n . 2 . 2 5 Comparison o f I I . 2 . 2 5 w i t h I I . 2 . 8 b g i v e s the f o l l o w i n g e x p r e s s i o n f o r the s c a t t e r i n g a m p l i t u d e x lx'x'tZU^CQ^m^m^j^,^ C Qt2.mn.JL0 /VAi) x C (jlntynl /jW^C^Ukt'/™') i i . 2.26 - 36 -where i s Che t r a n s i t i o n m a t r i x , r e l a t e d t o the S-matrix We may impose o t h e r boundary c o n d i t i o n s f o r open c h a n n e l s , namely V * / * ' I I . 2. 27a Using I I . 2 . 2 7 a we o b t a i n the T - m a t r i x d i r e c t l y , however I I . 2 . 2 7 b has the advantage t h a t the f u n c t i o n s are r e a l (except the - 2 i f a c t o r ) , and i t i s t h e r e f o r e , u s u a l p r a c t i c e t o o b t a i n the K - m a t r i x (known as the r e a c t a n c e m a t r i x ) , and c a l c u l a t e and "H from i t u s i n g ^ I I . 2 . 2 8 T* J J J where 1 i s a u n i t m a t r i x . The m a t r i c e s K , S , and T are symmetric, t h i s f o l l o w s from the symmetry of the m a t r i x , and r e f l e c t s the i n v a r i a n c e of the dynamics under time r e v e r s a l . The commonly used t o t a l cross s e c t i o n i s o b t a i n e d from I I . 2.11 by a v e r a g i n g over i n i t i a l itk , summing over - 37 -' A f i n a l , and i n t e g r a t i n g over angle dR 3 Because of the symmetry of T J , ^(V%^jJ^~¥ Vi.Ji'ji ~) s a t i s f i e s t h e d e t a i l e d balance c o n d i t i o n 30 As a l r e a d y s t a t e d t h i s d e r i v a t i o n a p p l i e s t o the s c a t t e r i n g o f a r o t o r (molecule 1) by a v i b r o t o r (molecule 2 ) , however r e s u l t s f o r a t o m - v i b r o t o r c o l l i s i o n s may be e a s i l y o b t a i n e d by s e t t i n g = 0 (hence = ^12 = ^2^' ^ n e P°tential expansion takes a f a r s i m p l e r form ( w i t h T^/^jU/^O) - < fa O O 9 I I . 2 . 3 1 and t h e p o t e n t i a l m a t r i x elements take the form where ^CWjL i VJfi , fO " f ^ l e ^ fa O ^jl'tidoOi - 38 -and the ^^(jjj^^fjfjj'i ET) a r e C h e p e r c i v a l Seaton c o e f f i c i e n t s ( P e r c i v a l and Seaton (1957)) known a n a l y t i c a l l y i n terms of 3- j and 6-j symbols. S o l u t i o n o f I I . 2 . 1 6 t o o b t a i n i s known as the c l o s e - c o u p l i n g (CC) method. The number o f coupled e q u a t i o n s i n I I . 2 . 1 6 f o r each J v a l u e i s i n t h e o r y unbounded, however n it ii ii i n p r a c t i c e the summation over J 2 V 2 ^ m u s ^ D e f i n i t e , ii ii ii and one i n c r e a s e s ^ a n d V2 (and t h e r e f o r e ") u n t i l t h e t r a n s i t i o n s of i n t e r e s t i n a g i v e n problem have converged The problem w i t h CC e q u a t i o n s i s the r a p i d i n c r e a s e i n the number o f channels as ^ ( and i n c r e a s e , due e s s e n t i a l l y t o the ( 2 j . + 1 ) , i = 1,2, degenerate o r i e n t a t i o n s i n l e v e l j . ( i = 1,2). A p a r t i a l s o l u t i o n i s p r o v i d e d by the p r o p e r t i e s o f p a r i t y , (-1 ^J^^ ? s o l u t i o n s o f even and odd p a r i t y are uncoupled, and may be s o l v e d s e p a r a t e l y . F u r t h e r s i m p l i c a t i o n can be made when s t u d y i n g c o l l i s i o n s o f i d e n t i c a l m o l e c u l e s , e.g. HF-HF, or i d e n t i c a l homonuclear m o l e c u l e s , e.g. H2-H2. For c o l l i s i o n s o f two homonuclear d i a t o m i c s , the p o t e n t i a l must be i n v a r i a n t under t h e o p e r a t i o n s r,~* —r*s •) f"2.~9",Z • F r o m I I . 2.19 t h i s f o r c e s b o t h 9^ and ^ 2 t o be even. A consequence as i n d i c a t e d by the second, and t h i r d 3 - j symbols i n I I . 2 . 1 9 , i s t h a t b o t h j - ^ and ^ c a n o n l y change by even i n c r e m e n t s . A d e t a i l e d e x a m i n a t i o n of t h i s has been made by Alexander and De P r i s t o (1977 ) . Because o f these problems few CC c a l c u l a t i o n s have been c a r r i e d out ( i n f a c t about 50 channels i s the maximum p o s s i b l e on the f a s t e s t modern c o m p u t e r s ) . This means - 39 -t h a t a p a r t from hydrogen and the h y d r i d e s , CC c a l c u l a t i o n s o f r o v i b r a t i o n a l c r oss s e c t i o n s are not p o s s i b l e . At e n e r g i e s h i g h enough t o study v i b r a t i o n a l e x c i t a t i o n a l a r g e number of r o t a t i o n a l l e v e l s are a c c e s s i b l e . Consider the c o l l i s i o n s o f N2 and H2 w i t h an atom at e n e r g i e s h i g h enough t o e x c i t e the f i r s t e x c i t e d v i b r a t i o n a l l e v e l . I n the case o f N2 t h e r e are around 30 r o t a t i o n a l l e v e l s below the f i r s t e x c i t e d v i b r a t i o n a l s t a t e . Since N2 i s homonuclear t h i s means 256 channels f o r one p a r i t y , and 240 channels f o r the o t h e r . We must, i n a d d i t i o n , c o n s i d e r the r o t a t i o n a l l e v e l s below the second v i b r a t i o n a l s t a t e . Such a c a l c u l a t i o n i s c l e a r l y not p o s s i b l e . Since H2 i s a l i g h t e r molecule the r o t a t i o n a l l e v e l s are more w i d e l y spaced, t h e r e are about 8 l e v e l s below the f i r s t e x c i t e d v i b r a t i o n a l s t a t e g i v i n g 25 and 20 c h a n n els. C o n s i d e r i n g the r o t a t i o n a l l e v e l s below the second e x c i t e d v i b r a t i o n a l s t a t e i n c r e a s e s t h e number o f channels t o 50 and 40; t h i s i s a f e a s i b l e c a l c u l a t i o n . 3 . Approximate methods Depending on the a l g o r i t h m used t o s o l v e the coupled 2 e q u a t i o n s I I . 2 . 1 6 the computing time i n c r e a s e s as N 3 or N , where N i s the number o f c h a n n e l s . I t i s obvious from the d i s c u s s i o n i n the p r e v i o u s s e c t i o n t h a t except f o r the s i m p l e s t systems (e.g. r o v i b r a t i o n a l c a l c u l a t i o n s f o r atom-H2 systems) CC c a l c u l a t i o n s are not p o s s i b l e , and a p p r o x i m a t i o n s must be made t o the e q u a t i o n s i n o r d e r t o reduce the number o f channels r e t a i n e d . - 40 -The d e c o u p l i n g a p p r o x i m a t i o n s d i s c u s s e d i n t h i s s e c t i o n i n v o l v e a s i m p l i f i e d t r e a t m e n t of one or more of the terms i n the f u l l CC e q u a t i o n s , and t h e i r v a l i d i t y I s t h e r e f o r e , dependent upon the importance o f these terms i n the c o l l i s i o n . For example i n the energy sudden a p p r o x i m a t i o n one assumes t h a t the r o t a t i o n a l l e v e l s o f a molecule are degenerate, t h e r e f o r e i t w i l l be a c c u r a t e i n c o l l i s i o n s where the r o t a t i o n a l l e v e l s are not i m p o r t a n t . I n g e n e r a l , however, the accuracy o f a g i v e n a p p r o x i m a t i o n i s d e t e r m i n e d by comparison w i t h f u l l CC c a l c u l a t i o n s . 3.1 The C e n t r i f u g a l Sudden a p p r o x i m a t i o n T h i s a p p r o x i m a t i o n i s w i d e l y used i n heavy p a r t i c l e s c a t t e r i n g c a l c u l a t i o n s , and i s c o n s i d e r e d t o be the most a c c u r a t e of a h i e r a r c h y o f p o s s i b l e a p p r o x i m a t i o n s f o r t r e a t i n g a g e n e r a l c l a s s of c o l l i s i o n systems. We w i l l d i s c u s s the s u f f i c i e n c y c o n d i t i o n s f o r the method l a t e r . The c e n t r i f u g a l sudden (CS) a p p r o x i m a t i o n (sometimes known as the c o u p l e d s t a t e s o r ^ c o n s e r v i n g a p p r o x i m a t i o n ) was s i m u l t a n e o u s l y i n t r o d u c e d by McGuire and K o u r i ( 1 9 7 4 ) , and by Pack ( 1 9 7 4 ) . This approach f o r the atom-molecule system has been w i d e l y t e s t e d ( e . g . , McGuire and K o u r i ( 1 9 7 4 ) , and K o u r i e t a l . (1976)) and has been g e n e r a l i s e d t o m o l e c ule-molecule systems (e.g. H e i l and K o u r i (1976) , H e i l et a l . (1978), and G o l d f l a m and K o u r i ( 1 9 7 9 ) ) . This d e r i v a t i o n o f the CS a p p r o x i m a t i o n i s i n a r o t a t i n g r e f e r e n c e frame i n which the z - a x i s always l i e s a l o n g R ( i . e . a b o d y - f i x e d c o o r d i n a t e (BF) system as shown i n F i g u r e 2 ) . I n such a frame the p o t e n t i a l i s a x i a l l y symmetric, and r A/ and r 4 1 A "2= K A F i g u r e 2. The S c h r b d i n g e r e q u a t i o n f o r t h e system may be w r i t t e n I I . 3 . 1 where H i s d e f i n e d by ( c f . I I . 2 . 2 ) I I . 3 . 2 a and U? may be expanded i n a r o t a t i o n a l b a s i s I I . 3 . 2 b where i s the p r o j e c t i o n o f ] ^ o n the b o d y - f i x e d z - a x i s . The r o t a t i o n a l b a s i s f u n c t i o n i s g i v e n by 4 2 where N j ^ ^ i s a n o r m a l i s e d symmetric top e i g e n f u n c t i o n (Rose ( 1 9 5 7 ) ) , and D M j j _ i s a r o t a t i o n m a t r i x element. S u b s t i t u t i n g I I . 3 . 2 b i n t o I I . 3 . 1 we f i n d indites. * FUdf^2.^-L(f?\ 11. 3. 3 I n the BF frame t he c e n t r i f u g a l b a r r i e r i s no l o n g e r d i a g o n a l , but g i v e n by (Launay ( 1 9 7 6 ) , Alexander and De P r i s t o (1977 ) ) 43 and the i n t e r m o l e c u l a r p o t e n t i a l i s expanded I I . 3 . 5 a where I I . 3 . 5 b where O ^ ^T" ^ MlA* ( £ 0 <|,a. ) . T his i s i n v a r i a n t under r o t a t i o n s about the z - a x i s . Using t h i s e x p r e s s i o n we are a b l e t o g i v e an e x p r e s s i o n f o r the p o t e n t i a l m a t r i x elements ( e . g . see Launay (19 7 7 ) , Danby (198 3 ) ) $ & o J \ o 0 o o I I . 3 . 6 44 -The d e l t a f u n c t i o n V V ' e n s u r e s t n e d i a g o n a l i t y o f t h e V - m a t r i x elements i n the _J2. i n d e x ; t h i s i s a consequence of the i n v a r i a n c e o f the p o t e n t i a l under r o t a t i o n about R. The summation over q i 0 i s such t h a t 1 l q 2 q 1 2 i s even The key t o the CS a p p r o x i m a t i o n i s the assumption t h a t t h e nond i a g o n a l c e n t r i f u g a l p o t e n t i a l terms i n I I . 3 . 4 may be i g n o r e d . Such o f f d i a g o n a l elements a r i s e from the c o r i o l i s f o r c e s due t o the non i n e r t i a l frame. T h i s leads t o a s i g n i f i c a n t r e d u c t i o n i n the number o f coupled e q u a t i o n s s i n c e , as mentioned b e f o r e , t he p o t e n t i a l m a t r i x I I . 3 . 6 i s s t r i c t l y d i a g o n a l i n the m u l t i p l i c i t y ~SlJ I n a d d i t i o n we s e t the d i a g o n a l elements o f t h e c e n t r i f u g a l p o t e n t i a l equal t o L ( L + 1 )~K*"**where L i s now a parameter. P h y s i c a l l y one i s assuming t h a t the exact v a l u e o f the o r b i t a l a n g u l a r momentum i s u n i m p o r t a n t , s i n c e L can be thought o f as an e f f e c t i v e o r b i t a l a n g u l a r momentum. The name j c o n s e r v i n g a p p r o x i m a t i o n i s now e v i d e n t , s i n c e t he p r o j e c t i o n quantum number, JlJ , i s conserved d u r i n g t he c o l l i s i o n . I n s t e a d o f e q u a t i o n s coupled i n i\i J2' v 2 ' a n c^ w e h a v e n o w e q u a t i o n s coupled i n j 1 ? j 2 and v 2 . The e q u a t i o n s I I . 3 . 3 may be s o l v e d s u b j e c t t o the boundary c o n d i t i o n s o ' I ' < ' „ I o k I\ • •> I I . 3 . 7 T h i s y i e l d s an S-matrix which i s b l o c k d i a g o n a l i n , because of the symmetry o f the p o t e n t i a l o n l y need be c o n s i d e r e d , and the coupled e q u a t i o n s are s o l v e d f o r —T2_ ^ min ( j ^ 2 ' j -j 2 ^ * ^ e m a ^ o b t a i n an approximate s p a c e - f i x e d S - m a t r i x , S , by employing the u n i t a r y t r a n s -f o r m a t i o n (Launay ( 1977 ) ) p I I . 3.8 The phase f a c t o r i ^ g uarantees t h a t the approximate r a d i a l f u n c t i o n , p. %JlMrn vi( £\ w i n f u i f n t he exact a s y m p t o t i c boundary c o n d i t i o n s I I . 2 . 2 1 b . I t i s i r r e l e v a n t f o r i n t e g r a l cross s e c t i o n s , but must be i n c l u d e d t o o b t a i n c o r r e c t d i f f e r e n t i a l c r o ss s e c t i o n s (e.g. see Secrest ( 1 9 7 5 ) , and Schinke and McGuire ( 1 9 7 8 a ) ) . The c e n t r i f u g a l parameter L i s s t i l l a r b i t r a r y and i t must be i d e n t i f i e d w i t h a p h y s i c a l q u a n t i t y b e f o r e v a r i o u s c ross s e c t i o n f ormulae can be d e r i v e d . The two n a t u r a l c h o i c e s f o r L are jL and j£, , the i n i t i a l and f i n a l o r b i t a l a n g u l a r momentum r e s p e c t i v e l y . Only - 46 -f o r the choices I, j£ , J^> can s i g n i f i c a n t s i m p l i f i c a t i o n s be made i n the s c a t t e r i n g a m p l i t u d e . The c h o i c e o f L has been f u l l y d i s c u s s e d by Pack ( 1 9 7 7 ) , G o l d f l a m et a l . ( 1 9 7 7 ) , and o t h e r s . I d e n t i f y i n g L w i t h , when Is i s s u b s t i t u t e d i n t o I I . 2 . 2 6 one o b t a i n s ( H e i l e t a l . (1978)) KfiM^M^ ^ vjfimfyml lit t Jit-Jin, / k; /«* 1.-. \ v. uiu ) ' I I . 3 . 9 a w hich i s a g e n e r a l i s a t i o n o f the atom-diatom d i f f e r e n t i a l s c a t t e r i n g a m p l i t u d e (McGuire and K o u r i ( 1 9 7 4 ) ) . I f we i d e n t i f y L w i t h j£ , the a m p l i t u d e i s g i v e n by ( G o l d f l a m and K o u r i (1979 ) ) 4x C vtf M> fan* =p vdj}t*ifi!Ml I } c G * H 'AW Uii^L I I . 3 . 9 b where - 47 -wher re p l i t u d e s ' and The main d i f f e r e n c e between the am i s t h a t f o r L = jt> , A n w h i l e w i t h m /^m^2 / 0« T h i s i s s i g n i f i c a n t as i t i s known from CC c a l c u l a t i o n s ( e .g. see T a r r e t a l . ( 1 9 7 7 ) , and Alexander et a l . (1977)) t h a t the exact s c a t t e r i n g a m p l i t u d e i s not d i a g o n a l i n m1^. I t makes no d i f f e r e n c e whether and e a s i l y shown from the u n i t a r i t y o f the D m a t r i c e s , b o t h s c a t t e r i n g a m p l i t u d e s g i v e i d e n t i c a l r e s u l t s f o r a l l degeneracy averaged cross s e c t i o n s . I t i s w o r t h n o t i n g t h a t n e i t h e r c h o i c e o f L g i v e s a f u l l y symmetric S J (Hunter ( 1 9 7 5 ) ) . Formal t h e o r e t i c a l c r i t e r i a f o r the a p p l i c a b i l i t y o f the CS a p p r o x i m a t i o n have not y e t been f o r m u l a t e d , t h e accuracy of the a p p r o x i m a t i o n f o r a g i v e n system i s u s u a l l y assessed by comparison w i t h CC c a l c u l a t i o n s . The a p p r o x i m a t i o n has been s u c c e s s f u l l y employed i n atom-d i a t o m , and d i a t o m - d i a t o m c a l c u l a t i o n s . G e n e r a l l y speaking we use because, as noted by Khare ( 1.977), - 4 8 the energy o f the c o l l i s i o n s hould be s u f f i c i e n t l y h i g h so t h a t t h e p r e c i s e v a l u e o f the c e n t r i f u g a l term i s u n i m p o r t a n t . For p u r e l y r e p u l s i v e e l e c t r o s t a t i c i n t e r a c t i o n p o t e n t i a l s t h e p o s i t i o n o f the c l a s s i c a l t u r n i n g p o i n t w i l l n o t v a r y p a r t i c u l a r l y r a p i d l y , however i f the p o t e n t i a l has an a t t r a c t i v e w e l l t h e r e may be t h r e e t u r n i n g p o i n t s a l l v a r y i n g r a p i d l y w i t h , and the assumption o f ;m e f f e c t i v e a n g u l a r momentum w i l l not be j u s t i f i e d . (The importance o f the c e n t r i f u g a l term i s determined by t he v a r i a t i o n o f the t u r n i n g p o i n t s w i t h r e s p e c t t o X ' • K o u r i e t a l ( 1 9 7 6 ) found t h a t the impact parameter s h o u l d be s m a l l e r than the c l a s s i c a l t u r n i n g p o i n t , and the energy should be w e l l above t h r e s h o l d f o r the t r a n s i t i o n b e i n g s t u d i e d , and sho u l d be a t l e a s t comparable w i t h t h e w e l l d e p t h . The l a t t e r f i n d i n g s l i m i t s t h e CS a p p r o x i m a t i o n t o s h o r t range p o t e n t i a l s , t h i s i s demonstrated i n the r e s u l t s o f Kouri and McCuiro ( 1 9 7 4 ) f o r the l . i 1 i system, and i n the r e s u l t s o f H e i l et a l . ( 1 9 7 8 ) f o r the H2 + HCj^system. I n b o t h systems the l o n g range i n t e r a c t i o n s are c a u s i n g t he r e s u l t s t o be i n e r r o r . I t i s w o r t h n o t i n g t h a t by r e t a i n i n g some o f the c o r i o l i s terms K i n n e r s l e y ( 1 9 7 9 ) has o b t a i n e d s a t i s f a c t o r y r e s u l t s f o r L i + + H2. The r e s u l t s o f H e i l and K o u r i ( 1 9 7 6 ) f o r H2 + i n d i c a t e s t h a t t h e CS a p p r o x i m a t i o n works as w e l l f o r d i a t o m - d i a t o m c o l l i s i o n s as f o r atom-diatom systems. Since v i b r a t i o n a l e x c i t a t i o n occurs t h r o u g h hard s h o r t ranged c o l l i s i o n s , the CS a p p r o x i m a t i o n would be - 4 9 -expected t o be u s e f u l . In f a c t the CS c a l c u l a t i o n s o f Alexander and McGuire (1976) f o r the He + H2 system are i n e x c e l l e n t agreement w i t h the CC r e s u l t s of Flower and K i r k p a t r i c k ( 1 9 8 2 ) . I n c o n c l u s i o n one would expect the CS a p p r o x i m a t i o n t o be v a l i d i n s t u d i e s where the c o l l i s i o n energy i s r e a s o n a b l y h i g h ( t o p r e v e n t the s i t u a t i o n of t h r e e t u r n i n g p o i n t s ) , and f o r systems f o r which the p o t e n t i a l s u r f a c e c o n t a i n s no l o n g range a n i s o t r o p i e s . 3-2 The Energy Sudden A p p r o x i m a t i o n The energy sudden (ES ) a p p r o x i m a t i o n i s v a l i d i n c o l l i s i o n s where the r o t a t i o n a l p e r i o d o f the t a r g e t i s much l o n g e r t h a n the c o l l i s i o n t i m e ( i . e . the t a r g e t o n l y r o t a t e s s l i g h t l y d u r i n g the c o l l i s i o n ) . T his a p p r o x i m a t i o n i s v a l i d f o r c o l l i s i o n s i n v o l v i n g heavy t a r g e t m o l e c u l e s , and h i g h c o l l i s i o n e n e r g i e s , i t was i n t r o d u c e d by Chase (1956) and has been w i d e l y used i n e l e c t r o n - m o l e c u l e s c a t t e r i n g t h e o r y where i t i s known as the a d i a b a t i c -n u c l e i a p p r o x i m a t i o n . Note t h a t the s m a l l mass of the e l e c t r o n makes i t w e l l s u i t e d t o such c o l l i s i o n s (e.g. see C o l l i n s and Norcross ( 1 9 7 8 ) ) . Few c a l c u l a t i o n s have been performed employing j u s t the ES a p p r o x i m a t i o n . Two a t o m - r o t o r c a l c u l a t i o n s (Chu and Dalgarno (1975a), and Khare ( 1978)), and one r o t o r -r o t o r c a l c u l a t i o n (H2 + N2 s t u d i e d by Bergeron et a l . ( 1 9 7 8 ) , however the H2 was c o n s t r a i n e d i n i t s ground r o t a t i o n a l s t a t e and i t was t h e r e f o r e an atom-molecule c a l c u l a t i o n ) . 50 N e v e r t h e l e s s the ES a p p r o x i m a t i o n i s w o r t h d i s c u s s i n g as i t i s a component of o t h e r d e c o u p l i n g a p p r o x i m a t i o n s ( n o t a b l y the i n f i n i t e o r d e r sudden a p p r o x i m a t i o n d i s c u s s e d i n Sect i o n 2.3). The molecule i s a t r e s t d u r i n g the c o l l i s i o n , and t h e r e f o r e one s o l v e s the s c a t t e r i n g problem f o r a l l s t a t i o n a r y r o t o r o r i e n t a t i o n s , and uses t h i s i n f o r m a t i o n t o d e t e r m i n e the s t a t e t o s t a t e s c a t t e r i n g a m p l i t u d e s . The l a t t e r problem i s the s i m p l e r , i f y^^, (jOz.) & ^ ) i s t h e s c a t t e r i n g a m p l i t u d e f o r f i x e d r o t o r o r i e n t a t i o n , I I . 3 . 1 0 T h i s f o r m u l a was d e r i v e d by Chase ( 1 9 5 6 ) , and one must c a l c u l a t e -^p f { ^ - ^ ) a t s u f f i c i e n t o r i e n t a t i o n s t o o b t a i n a s a t i s f a c t o r y r e s u l t . Note however, t h a t t h e s c a t t e r i n g c a l c u l a t i o n i s independent o f the o r i e n t a t i o n o f t he system i n space, and i n s t e a d t h i s can be c o n s i d e r e d as a l l o w i n g t h e atom t o approach the molecule from a l l d i r e c t i o n s . We are f r e e t o choose as our p o l a r a x i s which means t h a t the i n t e r a c t i o n p o t e n t i a l V w i l l be a x i a l l y symmetric, and t h e r e f o r e jL^ w i l l be conserved, a l t h o u g h Js i s not ( F i g u r e 3 ) . 51 8 A JL8, F i g u r e 3 p l a c e d b y £ ' ~ & I n t h i s new c o o r d i n a t e system R i s r e Since the molecule i s a t r e s t d u r i n g the c o l l i s i o n , the r o t o r s t a t e s may be c o n s i d e r e d degenerate, and we may denote them by ^ = 0- T h e r e f o r e ^V^a, a n d fi-V^^ are now independent o f J2 and the s u b s c r i p t may be o m i t t e d ( & a ^ kv^o ^ kv^ ). Expanding the t o t a l w a v e f u n c t i o n as 11.3.11 the r a d i a l f u n c t i o n s s a t i s f y (fO. f t ) Co 11.3.12 - 52 2- ° I I . 3.13a Using the s i n g l e c e n t r e expansion, I I . 2 . 3 1 , t h i s reduces t o I I . 3 . 1 3 b . Since i s conserved, the e q u a t i o n s are d i a g o n a l i n . T h e r e f o r e , i n s t e a d o f e q u a t i o n s coupled i n j 2 , v 2 , and we now have s e t s of e q u a t i o n s indexed by v 2 and jls o n l y . I f the t h e o r y had been p r e s e n t e d f o r r o t o r - v i b r o t o r c o l l i s i o n s , t he same c o n c l u s i o n would have been reached ( i . e . e q u a t i o n s indexed by j ^ , j 2 , v 2 , and Jls reduce t o e q u a t i o n s indexed by j ^ , v 2 and Ji, ) . However, s i n c e the r o t o r ( molecule 1) would not have been s t a t i o n a r y , Jbg would not have been conserved. Note t h a t the above c o n c l u s i o n , namely t h a t i s conserved, was drawn w i t h o u t any a p p r o x i m a t i o n s t o the p o t e n t i a l s u r f a c e or c e n t r i f u g a l p o t e n t i a l o p e r a t o r . - 53 -T h e r e f o r e i n the r o t o r - v i b r o . t o r case t h i s a p p r o x i m a t i o n leads t o t h e c o n s e r v a t i o n o f ( JL'Z, f ) (cf.CS a p p r o x i -m a t i o n which leads t o t h e c o n s e r v a t i o n o f ^\ -g: -=h ^ S. "Z, The e q u a t i o n s I I . 3 . 1 2 may be s o l v e d s u b j e c t t o the boundary c o n d i t i o n s . A*. &A>vJ%' t?o ° Using t he sudden S-matrix , $ ^ ^jLv^ jhj) an approximate space - f i x e d S-matrix can be e v a l u a t e d Kv>> I I . 3 . 1 4 u s i n g T h i s r e s u l t comes from I I . 3.10, the i n t e g r a l may be per-formed a n a l y t i c a l l y by r o t a t i o n t o a new c o o r d i n a t e system and t he p r o p e r t i e s o f t h e r o t a t i o n m a t r i c e s employed (Khare ( 1 9 7 8 ) , Secrest ( 1 9 7 5 ) ) . The space f i x e d S-matrix may be used i n the f u l l CC e q u a t i o n I I . 2 . 2 6 w i t h j ^ = 0 t o o b t a i n t h e cr o s s s e c t i o n s . The coupled e q u a t i o n s must be s o l v e d f o r a l l the values o f and f o r l a r g e j£. t h i s i s h i g h l y p r o b l e m a t i c (e . g . see Chu and Dalgarno (1975a)). However u s i n g p h y s i c a l arguments Khare (1978) has shown t h a t i t i s o n l y necessary t o c o n s i d e r ( J 2 m a x + 1)» s i n c e o n l y the f i r s t few terms c o n t r i b u t e s i g n i f i c a n t l y (j2max i s the l a r g e s t j a a c c e s s i b l e from the ^ 2 ~ ^ s t a t e ) . Note because of the symmetry of the p o t e n t i a l m a t r i x o n l y ^ ^ O need be c o n s i d e r e d . Using t he Clebsh-Gordan s e r i e s f o r s p h e r i c a l harmonics i n I I . 3 . 1 0 an i m p o r t a n t s i m p l i f i c a t i o n can be made, K < / x 6 -does not incLude the i n f l u e n c e o f c l o s e d ( o r almost c l o s e d ) channels. I f the w e l l d e p t h i s l a r g e i n comparison w i t h t he r e l e v a n t k i n e t i c energy, and the a n i s o t r o p y of the i n t e r a c t i o n p o t e n t i a l i s s t r o n g , then c o u p l i n g t o s t a t e s which are c l o s e d or almost c l o s e d can be i m p o r t a n t However, as the c o l l i s i o n energy i n c r e a s e s the k i n e t i c energy o f s t r o n g l y coupled s t a t e s becomes g r e a t e r t h a n o r equal t o the w e l l depth o f the p o t e n t i a l , and the c l o s e d channels become l e s s i m p o r t a n t even i n s t r o n g l y coupled systems, and the KS a p p r o x i m a t i o n becomes reasonable 3.3 The I n f i n i t e Order Sudden A p p r o x i m a t i o n T h i s a p p r o x i m a t i o n i s a c o m b i n a t i o n o f the energy sudden, and the c e n t r i f u g a l sudden a p p r o x i m a t i o n s . I t was i n t r o d u c e d by T s i e n and Pack ( 1 9 7 0 ) , and was l a t e r g e n e r a l i s e d i n d e p e n d e n t l y by Secrest (1975) and by Hunter (19 7 5 ) . There are many d e r i v a t i o n s i n t h e l i t e r a t u r e f o r the atom-molecule approach i n b o t h t he s p a c e - f i x e d frame ( e .g. T s i e n and Pack (197 0 ) , Secrest ( 1 9 7 5 ) , and Parker and Pack ( 1 9 7 8 ) ) , and the b o d y - f i x e d frame (e.g. Pack ( 1 9 7 4 ) , and Bowman and Leasure ( 1 9 7 7 ) ) . The a p p r o x i -mation has been g e n e r a l i s e d t o d i a t o m - d i a t o m systems, r o t o r - r o t o r c o l l i s i o n s i n which one r o t o r was t r e a t e d w i t h i n t h e i n f i n i t e o r d e r sudden (IOS) a p p r o x i m a t i o n and where the o t h e r r o t o r was t r e a t e d w i t h i n the CS a p p r o x i m a t i o n ( G o l d f l a m and K o u r i ( 1 9 7 9 ) , and Andres et a l . ( 1 9 8 2 ) ) , and i n a c a l c u l a t i o n i n which b o t h r o t o r s where t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n ( G o l d f l a m - 57 -and K o u r i ( 1 9 7 9 ) ) . The present d e r i v a t i o n i s i n the b o d y - f i x e d frame as i n S e c t i o n 3.1 , o n l y the v i b r o t o r ( m olecule 2) i s t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n , the r o t o r ( molecule 1) i s t r e a t e d w i t h i n the CS a p p r o x i m a t i o n The IOS a p p r o x i m a t i o n can be c o n s i d e r e d as an energy sudden a p p r o x i m a t i o n t o the c e n t r i f u g a l sudden a p p r o x i m a t i o n . I n the CS, the d i a g o n a l elements of the c e n t r i f u g a l p o t e n t i a l 2 are set equal t o L ( L + 1 )-h , and i n the ES the r o t o r s t a t e s o f the molecule are assumed t o be degenerate ( i . e . we r e p l a c e kv^J^, by Kvy, ' • The p o t e n t i a l may be expanded as i n 11.3.5a, however we w i l l employ a s i m p l i f i e d v e r s i o n of t h i s i n which the terms w i t h w / 0 are n e g l e c t e d ( t h i s w i l l be j u s t i f i e d l a t e r ) , the p o t e n t i a l expansion may be w r i t t e n , I I . 3 . 1 9 Expanding the t o t a l w a v e f u n c t i o n IP as r% I I . 3.20 we o b t a i n 58 -fnmeol I I . 3 . 2 1 The p o t e n t i a l m a t r i x elements are g i v e n by X Because the o p e r a t o r s which depend on the a n g l e , 0^ between R and ^ have been removed, we see t h a t I I . 3.21 depends on o n l y p a r a m e t r i c a l l y t h r o u g h J&/> V ^ j ^ ( a n d c a n be s o l v e d f o r each t^g d e s i r e d h o l d i n g / f i x e d . C o m p u t a t i o n a l l y I I . 3 . 2 1 i s f a r s i m p l e r t o s o l v e than e i t h e r I I . 3 . 3 or I I . 3 . 1 2 s i n c e now the cou p l e d e q u a t i o n s are o n l y indexed by ^2 and j ^ . The e q u a t i o n s I I . 3 . 2 1 may be s o l v e d s u b j e c t t o the boundary c o n d i t i o n s 59 -y i e l d i n g a s c a t t e r i n g m a t r i x ^ J2y *> V i ^ y ^ J 2 . / / ^ which i s b l o c k d i a g o n a l i n ^ s q u a d r a t u r e over @ £ y i e l d s J0 I I . 3.24 We note at t h i s p o i n t t h a t the c o n s e r v a t i o n o f —^z, , i m p l i e d by I I . 3 . 2 4 t o g e t h e r w i t h the c o n s e r v a t i o n o f jTL 9 i m p l i e d by the i n v a r i a n c e o f the i n t e r a c t i o n p o t e n t i a l I I . 3 . 1 9 under r o t a t i o n s about the i n t e r m o l e c u l a r a x i s , r e s u l t i n the c o n s e r v a t i o n o f J~Lj . I t f o l l o w s t h a t o n l y those terms w i t h w = 0 need t o be r e t a i n e d i n the p o t e n t i a l expansion I I . 3.5a. The S-matrix d e f i n e d by I I . 3 . 2 4 can not be used i n I I . 3 . 8 t o o b t a i n an approximate s p a c e - f i x e d S-matrix, s i n c e the t r a n s f o r m a t i o n , I I . 3 . 8 . assumes t h a t the e i g e n -f u n c t i o n s j^/J^/Vf ') a n d fyLj*%(&-0 a r e c o u P l e d t o g e t h e r ( n o t e however t h a t the t r a n s f o r m a t i o n can be used i n atom-diatom c a l c u l a t i o n s , s i n c e = 0 ) . To - 60 -o b t a i n the s c a t t e r i n g a m p l i t u d e we f i r s t f i n d the body-f i x e d s c a t t e r i n g a m p l i t u d e f o r f i x e d v i b r o t o r o r i e n t a t i o n • The a s y m p t o t i c form o f the wave f u n c t i o n I I . 3 . 2 0 may be w r i t t e n F JC^ *#p( i kty R) f( i y t JL, -* vJjlsiJ, &x'/i") I I . 3.25 we f i n d t h e c o r r e c t l i n e a r t o f i n d c o m b i n a t i o n o f j£ which goes over t o I I . 3 . 2 5 asymp-t o t i c a l l y , and f o l l o w the procedure o u t l i n e d i n S e c t i o n 2. I f we t a k e the o r b i t a l a n g u l a r momentum parameter equal t o the i n i t i a l v a l u e , J^, , one f i n d s t h a t the f i x e d v i b r o t o r s c a t t e r i n g a m p l i t u d e i s g i v e n by * it ( l & O - n.3.26 The SF s c a t t e r i n g a m p l i t u d e i s th e n g i v e n by 61 T h i s i s a g e n e r a l i s a t i o n o f the r e s u l t g i v e n by Parker and Pack (1978) f o r atom-diatom s c a t t e r i n g . The d i f f e r e n t i a l cross s e c t i o n s are g i v e n by ^ I J* I I . 3 . 2 8 where JZ^ &/*U,^/J$0 and Jl^^ f>M*L Q^jJ^ 0 Since the ES a p p r o x i m a t i o n has been employed i t i s o n l y necessary t o c a l c u l a t e r e s u l t s f o r t r a n s i t i o n s f o r which t h e i n i t i a l v a l u e o f ^ = 0, and t h e r e f o r e t h e summation over c o l l a p s e s . The i n t e g r a l c r oss s e c t i o n s are found u s i n g 'Jut ^ ^fofy-** 0o>*'fa/L'<>)/ I I . 3 . 2 9 and r o t a t i o n a l l y summed cross s e c t i o n s are e v a l u a t e d u s i n g (e.g. see G o l d f l a m e t a l . (1977) and Parker and Pack (1978) ) f J/ I I . 3 . 3 0 - 62 -To e v a l u a t e $L(^2$^{J^zffi-Q-t w e m u s t s o l v e the c o u p l e d e q u a t i o n s a t a s u f f i c i e n t number o f o r i e n t a t i o n s t o o b t a i n s a t i s f a c t o r y r e s u l t s . I t i s usual t o employ Gaussian q u a d r a t u r e , however i f we have t o c a l c u l a t e v e r y many cross s e c t i o n s , we can decrease the number of i n t e g r a l s by expanding the T - m a t r i x i n Legendre p o l y -nomials ( f o r each p a r t i a l wave). Tt fait H^ /La/ /%) A I I . 3 . 3 1 T h i s a l l o w s the i n t e g r a l i n I I . 3 . 2 4 t o be e v a l u a t e d a n a l -y t i c a l l y , however, i t does not a l l o w a r e d u c t i o n i n the number o f o r i e n t a t i o n s , O t L , t h a t must be c o n s i d e r e d . The number o f terms i n the s e r i e s i s d e t e r m i n e d by the number of v a l u e s c o n s i d e r e d . I f t h e S-matrix i s s t r o n g l y dependent on Si c a l c u l a t i o n s a t many o r i e n t a t i o n s may be r e q u i r e d . A n u m e r i c a l t e c h n i q u e t o reduce the number of v a l u e s r e q u i r e d has been d i s c u s s e d by Secrest (1979). The approach i s t o i n t e r p o l a t e the a m p l i t u d e and phase of the S-matrix which v a r y more s l o w l y w i t h than the r e a l and i m a g i n a r y p a r t s. I n atom-molecule c a l c u l a t i o n s a u s e f u l f e a t u r e o f the IOS a p p r o x i m a t i o n i s t h a t a s i n g l e c e n t r e expansion o f the p o t e n t i a l has no p a r t i c u l a r advantage. I n CC - 63 -and i n the o t h e r approximate methods ( a l s o i n the IOS a p p r o x i m a t i o n when / 0) the p o t e n t i a l m a t r i c e s i n v o l v e i n t e g r a l s over t r i p l e p r o d u c t s o f s p h e r i c a l harmonics which can be e v a l u a t e d a n a l y t i c a l l y . I n IOS atom-molecule c a l c u l a t i o n s t h e r e are no such i n t e g r a l s . Green (1978) has t e s t e d t h e IOS a p p r o x i m a t i o n a g a i n s t CC or CS r e s u l t s f o r HCjL + Ar, HC + He, CO + He, and HCN -+• He. He found reasonable agreement f o r a l l t he systems a t E^ = 100 c m - \ except HC.£^ + Ar. T h i s i s not s u r p r i s i n g s i n c e at a g i v e n energy we would expect the sudden a p p r o x i m a t i o n t o be l e s s v a l i d f o r the HCX t Ar system (Ar I s h e a v i e r than Ho, and would t h e r e f o r e approach the t a r g e t at a lower speed). G e n e r a l l y speaking one would expect the IOS a p p r o x i -m a t i o n t o be w e l l s u i t e d t o problems o f r o v i b r a t i o n a l e x c i t a t i o n . V i b r a t i o n a l e x c i t a t i o n occurs t h r o u g h h a r d s h o r t range c o l l i s i o n s , and t h e r e f o r e we would expect the CS a p p r o x i m a t i o n t o be v a l i d , a l s o v i b r a t i o n a l e x c i t a t i o n r e q u i r e s h i g h c o l l i s i o n e n e r g i e s , we would expect t he ES a p p r o x i m a t i o n t o be v a l i d . I n a d d i t i o n n o t e t h a t CC c a l c u l a t i o n s are c o m p u t a t i o n a l l y expensive f o r heavy molecules w i t h c l o s e l y packed r o t a t i o n a l l e v e l s , the ES a p p r o x i m a t i o n would be i d e a l l y s u i t e d t o such molecules (and hence the IOS). The l a r g e r o t a t i o n a l energy s p a c i n g o f the H2 molecule means t h a t c o l l i s i o n s i n v o l v i n g t h i s molecule p r o v i d e a s t r i n g e n t t e s t o f the IOS method. Schinke and McGuire - 64 -(1978a) have compared IOS and CC r e s u l t s f o r 1 ^ + H + a t E = 3.7eV and the g e n e r a l l e v e l o f agreement i s s a t i s -f a c t o r y , however the IOS r e s u l t s f o r the -^2T t r a n s i t i o n a re i n e r r o r . A l t h o u g h the CS a p p r o x i m a t i o n i s not s u i t e d t o long range p o t e n t i a l s ( t h e r e are long range i s o t r o p i c and a n i s o t r o p i c terms due t o the charge on the p r o t o n ) , the agreement between the CS r e s u l t s and the CC r e s u l t s o f McGuire (1976) i s good f o r the ^ = 0—^2 t r a n s i t i o n . The c o n c l u s i o n t h a t may be drawn from t h i s i s t h a t the sudden a p p r o x i m a t i o n i s f a i l i n g f o r 2_ t r a n s i t i o n s ; t h e l o n g range i n t e r a c t i o n means t h a t t h e H + i s spending a r e l a t i v e l y l o n g time i n the i n t e r a c t i o n r e g i o n . The i n c r e a s i n g f a i l u r e o f the IOS as the a n g u l a r momentum in c r e a s e s i s c o n s i s t e n t w i t h t h i s . As expected the agreement i s improved at h i g h e r c o l l i s i o n e n e r g i e s . The 1 ^ + H + system i s used t o t e s t our IOS programmes (Chapter I V ) . I n the IOS method, we are making b o t h the CS and ES a p p r o x i m a t i o n s . T h e r e f o r e the two p r e c e d i n g d i s c u s s i o n s a p p l y ( S e c t i o n s 3*1 and 3 . 2 ) ) . However i t i s w o r t h m e n t i o n i n g t h a t t h e f a c t o r i z a t i o n r e l a t i o n s p r e s e n t e d i n S e c t i o n 3.2 a p p l y p r o v i d e d o n l y the ES c o n d i t i o n i s met. Also note t h a t the method s h o u l d be c o n s i d e r e d even i n cases where the v a l i d i t y c o n d i t i o n s are not w e l l s a t i s f i e d ( p r o v i d i n g one r e q u i r e s o n l y q u a l i t a t i v e l y a c c u r a t e r e s u l t s ) . The a p p r o x i m a t i o n i s v e r y s i m p l e , and can y i e l d r e s u l t s w i t h v e r y l i t t l e c o m p u t a t i o n a l e f f o r t , e s p e c i a l l y s i n c e o n l y the cross s e c t i o n s out o f the j£ = 0 l e v e l are r e q u i r e d . The f a c t o r i z a t i o n r e l a t i o n I I . 3 . 1 7 - 65 -g i v e n i n S e c t i o n 3.2 may be g e n e r a l i z e d t o d i a t o m - d i a t o m s c a t t e r i n g w i t h o u t any a d d i t i o n a l f a c t o r s ( e . g. see Andres e t a l . (1982)) I n t h i s s e c t i o n we d i s c u s s some l e s s i m p o r t a n t d e c o u p l i n g a p p r o x i m a t i o n s t h a t may be employed when the a p p r o x i m a t i o n s d i s c u s s e d i n S e c t i o n 3 are not v a l i d . 4.1 The E f f e c t i v e P o t e n t i a l Methods The b a s i c d i f f i c u l t y i n h a n d l i n g quantum mechanical d i a t o m ( o r atom) - d i a t o m c a l c u l a t i o n s i s the l a r g e number of i n t e r n a l m o l e c u l a r s t a t e s t h a t can be i n v o l v e d . The l a r g e s t c o n t r i b u t i o n t o the number of m o l e c u l a r s t a t e s g e n e r a l l y comes from the 2 j + 1 degenerate r o t a t i o n a l p r o j e c t i o n m s t a t e s f o r each r o t a t i o n a l l e v e l l a b e l l e d by j . Note t h a t most experiments do not d i r e c t l y measure a l l o f the p o t e n t i a l l y l a r g e amounts o f data generated by c o n v e n t i o n a l c o l l i s i o n c a l c u l a t i o n s . T y p i c a l e x p e r i -ments measure c r o s s s e c t i o n s or r a t e s t h a t are a p p r o p r i a t e averages over the m s t a t e s . T h e r e f o r e , the numerous m s t a t e s are the r o o t o f the c o m p u t a t i o n a l d i f f i c u l t i e s , but t h e i r i n f o r m a t i o n c o n t e n t i s f r e q u e n t l y averaged over t o compare w i t h e x p e riment. The presence of v i b r a t i o n a l s t a t e s does not a l t e r these arguments. I I . 3 . 3 2 4. Other quantum mechanical d e c o u p l i n g a p p r o x i m a t i o n s U h The e f f e c t i v e p o t e n t i a l (EP) approach was developed by R a b i t z (1972) t o handle the a f o r e m e n t i o n e d d i f f i c u l t i e s . The EP approach reduces the d i m e n s i o n a l i t y o f the problem by p r o j e c t i n g out or p r e a v e r a g i n g over t he m s t a t e s b e f o r e c a r r y i n g out the dynamics. This was the f i r s t o f the d e c o u p l i n g schemes f o r r o t a t i o n a l e x c i t a t i o n . T his method i s r e viewed by R a b i t z (1976). For a p o t e n t i a l o f the form I I . 2 . 1 8 a , the e f f e c t i v e p o t e n t i a l m a t r i x elements may be w r i t t e n ( Z a r u r and R a b i t z (1974)) ^zTin II-4-1-where where V***- = fc) < / / . ' / / llM*> and t he reduced m a t r i x element <^ 4J

0 - _ 2 L - <, (21+1) ITx {Vk-> V u I I . 4 . 3 The BS a p p r o x i m a t i o n i s capable o f g i v i n g p h y s i c a l l y u s e f u l r e s u l t s , but r o t a t i o n a l degrees o f freedom are i m p o r t a n t i n v i b r a t i o n a l energy t r a n s f e r (see Chapter V) . 4.2 The L-Dominant and Decoupled L-Dominant a p p r o x i m a t i o n s The L-dominant (LD) and decoupled L-dominant a p p r o x i -mations were developed by D e p r i s t o and Alexander (1975 and 1976 r e s p e c t i v e l y ) and l a t e r g e n e r a l i s e d t o d i a t o m -d i a t o m c o l l i s i o n s ( D e p r i s t o and Alexander ( 1 9 7 7 ) ) . They were designed t o take advantage of s i t u a t i o n s where the i n t e r a c t i o n i s dominated by the c e n t r i f u g a l terms, where l o n g range e l e c t r o s t a t i c i n t e r a c t i o n s s t i l l a ct at l a r g e v a l u e s o f the o r b i t a l a n g u l a r momentum ( s i t u a t i o n s where the a p p r o x i m a t i o n s d e s c r i b e d i n S e c t i o n 3 are not v a l i d ) . 69 The LD a p p r o x i m a t i o n i s based upon the o b s e r v a t i o n t h a t i n pure r o t a t i o n a l e x c i t a t i o n problems at l a r g e J, t he l a r g e s t elements o f the s t a n d a r d CC S - m a t r i x , ^7fylJ*^?J'J^J^/) ' th°Se W h L C h ^>^/ t r e a t e d w i t h i n the IPS a p p r o x i m a t i o n , and a r o t o r t r e a t e d w i t h i n the CS a p p r o x i m a t i o n , the c o u p l i n g m a t r i x (R) may be w r i t t e n , f o r a f i x e d v a l u e o f Jft»f 9 I I . 3 . 1 I n d e t e r m i n i n g the s u i t a b i l i t y of the R-matrix p r o p a g a t o r method t o s o l v e t h i s problem we must c o n s i d e r two f a c t o r s , F i r s t we n o t i c e t h a t the coupled e q u a t i o n s w i l l be c oupled i n b o t h \?2 a n ^ j ^ - A t each v a l u e o f J7.f t h a t - 1 •> -we c o n s i d e r we w i l l , have a c o u p l i n g m a t r i x o f dimension (N x N) where N i s the product (v2max x j^max). V2max i s the h i g h e s t v i b r a t i o n a l s t a t e c o n s i d e r e d , and j^max i s t h e maximum v a l u e o f j - ^ p e r m i s s i b l e at a g i v e n v a l u e of JZj • Secondly n o t i c e t h a t t h e a n g u l a r momentum parameter, L, w i l l have the same v a l u e i n each c h a n n e l , and o n l y e n t e r s i n t o t h e d i a g o n a l elements o f the c o u p l i n g mat r i x . I f we are o n l y i n t e r e s t e d i n a t o m - v l b r o t o r c o l l i s i o n s , we may set j | - - 0, and the e q u a t i o n s are o n l y coupled i n V2. G e n e r a l l y speaking the v i b r a t i o n a l energy l e v e l s o f molecules are r e l a t i v e l y w i d e l y spaced, and s i n c e we are o n l y i n t e r e s t e d i n re a s o n a b l y low c o l l i s i o n e n e r g i e s we need r e t a i n o n l y a few v i b r a t i o n a l s t a t e s . For such a s m a l l number o f channels t he n u m e r i c a l e f f o r t o f an AP method w i l l not be l a r g e l y i n excess of t h a t r e q u i r e d by an AS method. I n a d d i t i o n an AP method has the advantage of r e q u i r i n g l a r g e r s t e p s i z e s than an AS method. However, i f we now r e i n t r o d u c e the r o t a t i o n a l l e v e l s o f the r o t o r ( ) the choi c e of method appears l e s s c l e a r c u t , t he a d d i t i o n o f o n l y two r o t a t i o n a l channels i n c r e a s e s the dimensions o f the c o u p l i n g m a t r i x by a f a c t o r o f two. We must t h e r e f o r e c o n s i d e r o t h e r f a c t o r s b e f o r e a reasonable c h o i c e can be p r o p e r l y made. Since t he a n g u l a r momentum parameter, L, comes i n t o t h e coup l e d e q u a t i o n s i n a s i m i l a r manner t o the 76 t o t a l c o l l i s i o n energy, E, the p r o p e r t i e s of AP methods which a l l o w the g e n e r a t i o n o f r e s u l t s at many e n e r g i e s , can a l s o be used t o generate r e s u l t s at many L v a l u e s . To o b t a i n the t o t a l c r o ss s e c t i o n i t i s necessary t o s o l v e the c o u p l e d e q u a t i o n s a t many va l u e s o f L. We may use i n f o r m a t i o n c a l c u l a t e d a t the i n i t i a l v a l u e o f L t o e v a l u a t e r e s u l t s at subsequent v a l u e s o f L. However, t h e r e i s one e s s e n t i a l d i f f e r e n c e between the ways L and E come i n t o the c a l c u l a t i o n . E i s independent ULtO/02-o f the i n t e g r a t i o n c o o r d i n a t e , R, b u t . i s n o t . For the r o t o r - v i b r o t o r systems s t u d i e d + CO, HD + CO) i t t u r n s out t h a t the s t e p s i z e I s not v e r y s t r o n g l y dependent upon I , , t h e r e f o r e the g e n e r a t i o n o f s o l u t i o n s f o r l a r g e numbers of L values w i t h l i t t l e n u m e r i c a l e f f o r t i s p o s s i b l e . I n the case of HD + CO, i n a d d i t i o n t o the t h r e e v i b r a t i o n a l channels t o d e s c r i b e the CO m o l e c u l e , i t was necessary t o r e t a i n s i x r o t a t i o n a l channels t o d e s c r i b e the HD molecule c o r r e c t l y . Consequently the c o u p l i n g m a t r i x f o r = 0 was of dimension (18 x 1 8 ) , at subsequent p a r t i a l waves we found t h a t the computer time r e q u i r e d was s\y 50% of t h a t r e q u i r e d at the i n i t i a l v a l u e o f L. However, we r e a l i s e d t h a t f o r such l a r g e m a t r i c e s , m a t r i x m u l t i p l i c a t i o n was e x t r e m e l y c o s t l y , and when the a p p r o p r i a t e r o u t i n e s were o p t i m i s e d f o r t h e Cray-1 computer system a f a r g r e a t e r s a v i n g was o b t a i n e d . The HD + CO c a l c u l a t i o n s are d i s c u s s e d i n Chapter V I I . - 77 -I n some systems the R dependence o f the L term i s more i m p o r t a n t i n the choi c e o f s t e p s i z e . I n such cases, t he approximate p o t e n t i a l used can be m o d i f i e d t o d e a l w i t h t h i s dependence a n a l y t i c a l l y . The step s i z e can be made v i r t u a l l y independent o f L, and the g e n e r a t i o n o f l a r g e numbers o f L v a l u e s i s p o s s i b l e . This procedure i s discus s e d i n S e c t i o n 4 .4. These methods of e v a l u a t i n g r e s u l t s at l a r g e numbers of p a r t i a l waves c o u l d aLso be employed i n e f f e c t i v e p o t e n t i a l c a l c u l a t i o n s . This d e c o u p l i n g a p p r o x i m a t i o n a l s o makes use of an e f f e c t i v e a n g u l a r momentum parameter. Since we are d e a l i n g w i t h the IOS, and CS a p p r o x i -mations (which are j u s t t h a t - a p p r o x i m a t i o n s ) t he accuracy o f the r e s u l t s i s r e q u i r e d t o no more tha n t h r e e t o f o u r s i g n i f i c a n t f i g u r e s . The AP methods are capable o f a c h i e v i n g t h i s l e v e l o f accuracy w i t h o u t employing e x c e s s i v e l y smaLl step s i z e s . I n view o f the mathematical form of the I OS e q u a t i o n s and t he l e v e l o f accuracy r e q u i r e d we c o n s i d e r t h a t an AP method r a t h e r than an AS approach i s more s u i t e d t o our c a l c u l a t i o n s . I n a d d i t i o n we p r e f e r t o employ an i n v a r i a n t imbedding t e c h n i q u e r a t h e r than a s o l u t i o n f o l l o w i n g t e c h n i q u e , due t o t h e i r i n h e r e n t s t a b i l i t y . There i s o n l y one a l g o r i t h m i n t h i s c a t e g o r y , t h e R-m a t r i x p r o p a g a t o r t e c h n i q u e o f S t e c h e l , Walker, and L i g h t ( 1 9 7 8 ) , and t h e r e f o r e t h i s a l g o r i t h m was employed i n our c a l c u l a t i o n s . 78 -4. The R-matrix P r o p a g a t i o n Method 4.1 General Theory The R-matrix p r o p a g a t o r method was o r i g i n a l l y developed f o r r e a c t i v e s c a t t e r i n g c a l c u l a t i o n s by L i g h t and Walker ( 1 9 7 6 ) , and was l a t e r extended t o t r e a t i n e l a s t i c s c a t t e r i n g problems by S t e c h e l , Walker and L i g h t (1978) ( h e r e a f t e r r e f e r r e d t o as SWL). This method i s based upon the p r o p a g a t i o n o f the R-matrix. T h i s i s the m a t r i x r e l a t i n g f u n c t i o n s t o t h e i r d e r i v a t i v e s a t a g i v e n v a l u e o f the i n t e g r a t i o n c o o r d i n a t e . Since i t i s an i n v a r i a n t - i m b e d d i n g t e c h n i q u e i t i s not s e n s i t i v e t o the n u m e r i c a l problems a s s o c i a t e d w i t h the p r o p a g a t i o n o f c l o s e d channel s o l u t i o n s , and i n a d d i t i o n SWL r e p o r t t h a t i t i s f a s t and a c c u r a t e . I t i s f a i r l y easy t o u n d e r s t a n d , and r e l a t i v e l y s i m p l e t o programme. I t i n v o l v e s v a r i o u s m a t r i x o p e r a t i o n s ( e . g . , m u l t i p l i c a t i o n , d i a g o n a l i z a t i o n , i n v e r s i o n , e t c . ) which can be achieved by means o f s t a n d a r d r o u t i n e s . E s s e n t i a l l y one s t a r t s by s o l v i n g the problem f o r a p a r t o f the p o t e n t i a l , t a k i n g the p o t e n t i a l t o be zero beyond c e r t a i n l i m i t s . Then one s o l v e s the problem f o r an a d j a c e n t p a r t o f the p o t e n t i a l . One t h e n combines the two s o l u t i o n s t o o b t a i n the s o l u t i o n f o r the p o t e n t i a l c o n s i s t i n g o f both p a r t s . Proceeding i n t h i s way, c o n c a t e n a t i n g p a r t s o f the p o t e n t i a l , e v e n t u a l l y the e n t i r e problem i s s o l v e d . We may w r i t e the g e n e r a l form o f the coupled e q u a t i o n s I I I . 1 . 1 i n m a t r i x f o r m , I I I . 4 . 1 - 79 -The b a s i s o f the R-matrix p r o p a g a t i o n method i s the Magnus e x p o n e n t i a l method. I n t h i s method the e q u a t i o n s g i v e n i n I I I . 4 . 1 a re w r i t t e n as t w i c e as many co u p l e d f i r s t o r d e r e q u a t i o n s s I I I . 4 . 2 where A i s a (2N x 2N) m a t r i x d e f i n e d by where 0 and I are the zero and u n i t m a t r i x r e s p e c t i v e l y , A p r o p a g a t o r £ i s d e f i n e d t o r e l a t e s o l u t i o n s across an i n t e r v a l A R * I I I .4.3 where the (2N x 2N) m a t r i x U i s r e l a t e d t o the f i r s t o r d e r c o u p l i n g m a t r i x A by the Magnus e x p o n e n t i a t i o n % * em ftesQA-CA*-)3 fk>' o 1i — / At L " J L I I I . 4 . 4 where faf* i s the d e r i v a t i v e o f Ia) . The method of SWL i s t o d i a g o n a l i z e £J and t o r e t a i n o n l y the f i r s t t erm o f the e x p o n e n t i a l s e r i e s , t h i s corresponds t o t a k i n g a c o n s t a n t r e f e r e n c e p o t e n t i a l across each s e c t o r . However SWL do not recommend the e x p o n e n t i a l method 80 ( i n (.his method of s o l u t i o n I he s t e p prop,ig,i t ors .'ire m u l t i p l i e d t o g e t h e r t o form the p r o p a g a t o r over the e n t i r e i n t e g r a t i o n r a n g e ) . I f t h e r e are l o c a l l y c l o s e d channels i n t h e c a l c u l a t i o n , the e x p o n e n t i a l g rowth o f the s o l u t i o n s a s s o c i a t e d w i t h them may l e a d t o a l o s s o f l i n e a r independence i n the f u l l set o f s o l u t i o n s . I n f a c t any p r o p a g a t o r may be used, and i n s t e a d o f e q u a t i o n I I I . 4 . 3 , SWL employ t o propagate the s c a t t e r i n g i n f o r m a t i o n . The R - matrix r e l a t e s the t r a n s l a t i o n a l f u n c t i o n s t o t h e i r d e r i v a t i v e s at the boundaries o f the r e g i o n over which the S c h r o d i n g e r e q u a t i o n has been s o l v e d . A,B,C... denote a set o f s u r f a c e s i n c o n f i g u r a t i o n space on which i n f o r m a t i o n r e l a t i n g w a v e f u n c t i o n s and t h e i r ( o u t w a r d l y normal) d e r i v a t i v e s i s r e q u i r e d . For example f o r the t h r e e d i m e n s i o n a l r e a c t i v e s c a t t e r i n g o f an atom and d i a t o m i c m o l e c u l e , t h e r e are t h r e e asymptotic rearrangement channels and t h e r e f o r e the R-matrix i s b l o c k e d 3 x 3 , however i n e l a s t i c s c a t t e r i n g s u f f i c i e n t i n f o r m a t i o n i s a v a i l a b l e on a s i n g l e s u r f a c e . We are now i n a p o s i t i o n t o d e s c r i b e C R R R AC C ^AA A i >CA G J R K BA A^C B B *0 R R R /CCA /~CB CC I I I . 4 . 5 - 81 how the R-matrix s o l u t i o n t o the coupled e q u a t i o n s may be c o n s t r u c t e d . The approach i s t o d i v i d e the i n t e g r a t i o n range i n t o s m a l l e r r e g i o n s i n which the l o c a l R-matrix may be d e t e r m i n e d . T h i s i s shown s c h e m a t i c a l l y i n F i g u r e 1. The g l o b a l R - m a t r i x , which i s o f dimension (N x N) i s c o n s t r u c t e d by assembling the l o c a l R-matrices d e t e r -mined i n each s e c t o r . The g l o b a l R - m a t r i x i s an (N x N) m a t r i x because i t o n l y r e t a i n s i n f o r m a t i o n about a s i n g l e s u r f a c e , but the s e c t o r R-matrices are (2N x 2N) m a t r i c e s s i n c e they c a r r y i n f o r m a t i o n from one s u r f a c e t o an o t h e r . I n accordance w i t h n e g l e c t i n g the h i g h e r o r d e r terms i n I I I . 4 . 4 we assume t h a t the c o u p l i n g m a t r i x U) i s c o n s t a n t over the w i d t h o f a s e c t o r ( i . e . (AT^~ O )• The c o u p l i n g m a t r i x i s d i a g o n a l i s e d at the c e n t r e o f each s e c t o r , I I I . 4 . 6 where j j ^ ' i s the transpose and i n v e r s e o f t ^ ' s i n c e jX) i s assumed t o be symmetric ( i . e . ' i s o r t h o g o n a l ) We thus t r a n s f o r m from the o r i g i n a l b a s i s (j> t o a new basi s \js which are l i n e a r c o m b i n a t i o n s o f the o r i g i n a l j, 0 (0 s W i t h i n a g i v e n s e c t o r ( i ) the v a l u e s o f the t r a n s l a t i o n a l f u n c t i o n s a t the r i g h t and l e f t hand boundaries o f the s e c t o r are g i v e n by (assembling the f u n c t i o n s R ^ ^(R) i n a column v e c t o r ) Since t he c o u p l i n g has been removed from I I I . 4 . 1 t h e s e c t o r R -matrix r ^ ^ i n the s e c t o r ( i ) may be w r i t t e n ( i n the l o c a l l y d i a g o n a l r e p r e s e n t a t i o n ) X ( 0 ? Fr*'° -p-<0 I I I .4.10 The s i m i l a r i t y w i t h the Magus e x p o n e n t i a l method I I I . 4 . 3 i s now c l e a r . The va l u e s o f the elements n^'^, ti}p ... depend upon the v a l u e o f / ^ ^ j T his i s d i s c u s s e d i n S e c t i o n 4.4 . Hi One must ensure the c o n t i n u i t y o f t h e t o t a l wave-f u n c t i o n and i t s normal d e r i v a t i v e a t each s e c t o r boundary, we t h e r e f o r e t r a n s f o r m from the£(i - 1) r e p r e s e n t a t i o n t o the r e p r e s e n t a t i o n and back t o the £( i ) r e p r e s e n t a t i o n , I I I . 4 . 1 1 f I I .4.12 where $ ( ^ - t •> tl ^ T T ^ . Note t h a t s i n c e t he t r a n s f o r m a t i o n m a t r i c e s ~ ^ and T^^^ are o r t h o g o n a l , the m a t r i x ^ i s a l s o o r t h o g o n a l . The method f u n c t i o n s by assembling the s e c t o r R-matrices g i v e n by I I I . 4 . 1 0 r e c u r s i v e l y b e g i n n i n g near t he o r i g i n , and p r o g r e s s i n g towards the a s y m p t o t i c r e g i o n . At each step a new g l o b a l R-matrix i s e v a l u a t e d by assembling a new s e c t o r R-matrix w i t h the o l d g l o b a l K -matrix. The g l o b a l R-matrix i s always d e f i n e d so t h a t i t r e l a t e s the r a d i a l f u n c t i o n s t o i t s d e r i v a t i v e s at the o u t e r boundary o f the l a s t s e c t o r , i . e . , ^ ,- , - - . . . . . . . . I I I . 4 . 1 3 The R-matrix i s the i n v e r s e o f the l o g - d e r i v a t i v e m a t r i x d e f i n e d by Johnson (1973). To f i n d the new g l o b a l R -matrix we employ the r e l a t i o n - 84 -I I . 4.14a where Z-^ i Q 1 °+ Q ( I > ^ (C'0(H (t-l, Oj' I I I . 4.14b Note t h a t the r e c u r s i o n r e l a t i o n s I I I . 4 . 1 4 a , b produce a symmetric R-matrix ( p r o v i d e d and rL are s y m m e t r i c ) . 4 • 2 Boundary C o n d i t i o n s The R-matrix r e c u r s i o n i s i n i t i a l i z e d near the o r i g i n ( i n t he c l a s s i c a l l y f o r b i d d e n r e g i o n ) , w i t h the i n i t i a 1 m a t r i x d e f i n e d by I I I . 4 . 1 5 The r e l a t i o n I I I . 4 . 1 5 assumes the p o t e n t i a l i s l a r g e and r e p u l s i v e near the o r i g i n (which i s u s u a l l y the c a s e ) , WO f o r o t h e r types o f p o t e n t i a l s , (K, must be chosen t o r e f l e c t the behaviour of the r e g u l a r s o l u t i o n near the o r i g i n . When one reaches the a s y m p t o t i c r e g i o n , one t r a n s f o r m s back t o the o r i g i n a l b a s i s set , u s i n g t h e t r a n s f o r m a t i o n " J?*>ZR(M~) H I . 4 . 1 6 a 85 -where M i s the number of s e c t o r s . Combining I I I . 4 . 1 6 a , b w i t h t he e q u a t i o n f o r the l a s t R-matrix d e t e r m i n e d by the r e c u r s i o n t o the r i g h t hand s i d e o f the l a s t s e c t o r M as i n I I I . 4 . 1 3 g i v e s t he e q u a t i o n f o r the f i n a l g l o b a l R-matrix 9 K#~im jl*)^(m) j(*> III 417a \*(M) - $£(»0 • 111 •"•17b The a s y m p t o t i c t r a n s l a t i o n a l f u n c t i o n s w i l l have the form Gr ~ A — B S° I I I . 4 . 1 8 a Or's f\' =. B'S° i n . 4 . 1 8 b <"v PC * where A and B are d i a g o n a l m a t r i c e s o f incoming and o u t g o i n g a s y m p t o t i c f u n c t i o n s r e s p e c t i v e l y , and S° *—\s i s e s s e n t i a l l y t he s c a t t e r i n g m a t r i x . Using e q u a t i o n s I I I . 4 . 1 8 a , b we o b t a i n A - 3 = ^ ' ^ / A ^ g ^ O I I I . 4 . 1 9 fromwhich we can o b t a i n the " S - m a t r i x " One w i l l r e c a l l t h a t i n Chapter I I , t h e boundary c o n d i t i o n s were w r i t t e n e x p l i c i t l y i n terms o f S p h e r i c a l Bessel f u n c t i o n s , and S p h e r i c a l Hankel f u n c t i o n s . To d e t e r m i n e the S-matrix i n the IOS a p p r o x i m a t i o n we have ( c f * i i . 3 . 2 3 ) Q- / - i k„L/RA, « 0 ) . _.././ ^ > O I I I . 4 . 2 1 86 -Comparing I I I . 4 . 2 1 and I I I . 4 . 1 8 a we f i n d A l t e r n a t i v e l y we c o u l d work i n terms o f the rea c t a n c e m a t r i x ( K - m a t r i x ) . T h i s i s c o m p u t a t i o n a l l y e a s i e r as a l l t h e f u n c t i o n s a re r e a l (cf„ I I . 2 . 2 7 b ). We f i n d I f the channel i s c l o s e d ( i . e . fcy'f( 0 ) the a s y m p t o t i c boundary c o n d i t i o n may be expressed as a l i n e a r c o m b i n a t i o n o f e x p o n e n t i a l l y g r o w i n g s o l u t i o n s , g,^ ,^ where M~/^YJJ,^/ > a n c^ e x p o n e n t i a l l y decaying s o l u t i o n s €.= K . However p h y s i c a l l y the e x p o n e n t i a l l y g rowing s o l u t i o n i s not a c c e p t a b l e s i n c e a l l the f l u x e n t e r s v i a open c h a n n e l s , and t h e r e f o r e we are j u s t i f i e d i n e x p r e s s i n g the boundary c o n d i t i o n s j u s t i n terms o f i . e . , £VIA ^ o **V%U ^ 111. 4. 22 87 In p r a c t i c e we f L t the boundary c o n d i t i o n i n terms o f m o d i f i e d Bessel f u n c t i o n s o f the t h i r d k i n d (Abramowitz and Stegun (1965)) which are e x p o n e n t i a l l y d e c a y i n g f u n c t i o n s , and so I I I . 4 . 2 2 may be r e w r i t t e n —^ AM vji,' z. / j L KL+j. (^) , V y ^ 2H I I I . 4 . 2 3 where ^ ^ V ^ / / ^ . 4»3 Long Range I n t e r a c t i o n P o t e n t i a l s -Asym p t o t i c S e r i e s Expansions For v e r y l o n g range p o t e n t i a l s such as those g o i n g as l/R^, i t i s necessary t o i n t e g r a t e q u i t e f a r b e f o r e the p o t e n t i a l has decayed s u f f i c i e n t l y so t h a t the wave f u n c t i o n can be f i t t e d to the S p h e r i c a l Bessel f u n c t i o n s or S p h e r i c a l Hankel f u n c t i o n s . I f a l l t h e l o n g range terms o f the p o t e n t i a l are o f the form I/R", where n i s an i n t e g e r (as i s u s u a l l y t he c a s e ) , i t i s p o s s i b l e t o f i n d a complete set of a s y m p t o t i c s o l u t i o n s f o r the l o n g range p o t e n t i a l and match these t o the s o l u t i o n t o d e t e r m i n e t he s c a t t e r i n g i n f o r m a t i o n . I n t h i s s e c t i o n we f i r s t take a c l o s e r l o o k at the c o n v e n t i o n a l method of f i t t i n g t h e a s y m p t o t i c boundary c o n d i t i o n and su b s e q u e n t l y extend our d i s c u s s i o n t o the use o f a s y m p t o t i c s e r i e s expansions i n d e a l i n g w i t h l o n g range p o t e n t i a l s . For s i m p l i c i t y we w i l l o n l y c o n s i d e r a t o m - v i b r o t o r s c a t t e r i n g ( i . e . j - ^ = 0 ) . - 88 -I n the IOS a p p r o x i m a t i o n the coupled d i f f e r e n t i n I. e q u a t i o n s take the form ( c f , [1.3.21 ) I I I . 4 . 2 4 I f R i s s u f f i c i e n t l y l a r g e so t h a t a l l the s h o r t range ( e x p o n e n t i a l ) terms of the i n t e r a c t i o n p o t e n t i a l have become n e g l i g i b l e , we may w r i t e 7\ fcn I I I . 4 . 2 5 and I I I . 4 . 2 4 has the a s y m p t o t i c form I I I . 4 . 2 6 o, O + where I f we i n t e g r a t e the coupled e q u a t i o n s s u f f i c i e n t l y f a r i n t o the a s y m p t o t i c r e g i o n so t h a t the term ^ ( ^ + 1)/R i s n e g l i g i b l e , the boundary c o n d i t i o n may be w r i t t e n where Ku - K U faj, , Jl, / 9*. ) [ I T .4.27 - 89 -I n p r a c t i c e , however, Che convergence of the s o l u t i o n s o b t a i n e d by I I I . 4 . 2 7 i s e x t r e m e l y slow, and i t i s more e f f i c i e n t t o employ Bessel f u n c t i o n s o f the f i r s t and second k i n d (denoted by and r e s p e c t i v e l y ) , Abramowitz and Stegun (1965) 9 Lv I I I . 4 . 2 8 I f we t a k e "V£ = 0, and y\ = L, t h e n we o b t a i n a boundary c o n d i t i o n t h a t may be used when the i n t e r a c t i o n p o t e n t i a l i s n e g l i g i b l e , and not the c e n t r i f u g a l term ( c f . I I . 2 . 2 7 b ) , I I I .4.29 I f , however, the l o n g range p o t e n t i a l I I I . 4 . 2 5 c o n t a i n s terms w i t h n ^ 3, i t i s necessary, as mentioned p r e v i o u s l y , t o i n t e g r a t e the coupled e q u a t i o n s out t o e x t r e m e l y l a r g e R v a l u e s b e f o r e the boundary con-d i t i o n can be f i t t e d u s i n g I I I . 4 . 2 9 . T h i s problem may be overcome by the use o f a s y m p t o t i c s e r i e s expansions. When a l l the l o n g range terms are o f the form R n , i t i s p o s s i b l e t o i n t e g r a t e s u f f i c i e n t l y out i n t o t h e - 90 -a s y m p t o t i c r e g i o n t o where a l l the s h o r t range terms i n the p o t e n t i a l have become n e g l i g i b l e . One then f i n d s a complete set of a s y m p t o t i c s o l u t i o n s f o r the l o n g range p o t e n t i a l and matches these t o the s o l u t i o n t o o b t a i n t h e K-m a t r i x . I n t h i s case the a s y m p t o t i c form I I I . 4 . 2 5 may be w r i t t e n I I I . 4 . 3 0 The a s y m p t o t i c s o l u t i o n f o r the l o n g range p o t e n t i a l i n s c a t t e r i n g c a l c u l a t i o n s was f i r s t employed i n e l e c t r o n s c a t t e r i n g by Burke and Schey ( 1 9 6 2 ) , and was l a t e r adapted t o m o l e c u l a r c o l l i s i o n s by Brandt and T r u h l a r ( 1 9 7 3 ) . Secrest (1979) has suggested a s y m p t o t i c s e r i e s expansions o f the form ceo t e . fr(*0J ] Q*x~ j?m(-$-QC '• • I I I .4. 31 O " ' J J I J , 4 , , v 7 where (t h e s u b s c r i p t V^l has been dropped) A s y m p t o t i c s e r i e s expansions of t h i s type are based on the assumption t h a t convergence w i l l be improved by f i t t i n g n ot t o s i n x and cos x, as i n I I I . 4 . 2 7 , but t o these f u n c t i o n s m o d i f i e d by t h e i r d e r i v a t i v e s - 91 -While such expansions have proved t o be o f g r e a t v a l u e i n heavy p a r t i c l e s c a t t e r i n g c a l c u l a t i o n s (e.g. P f e f f e r and Secrest ( 1 9 8 3 ) , and G e r r a t t a n d Roberts (1983) -b o t h w i l l be d i s c u s s e d i n Chapter IV) the t r i g o n o m e t r i c f u n c t i o n s do not i n c l u d e the e f f e c t s o f the c e n t r i f u g a l t e r m , and we c o n s i d e r t h a t a s y m p t o t i c s e r i e s expansions i n c o r p o r a t i n g S p h e r i c a l Bessel f u n c t i o n s wouLd be more s a t i s f a c t o r y . Such expansions are suggested by G a i l i t i s ( 1 9 7 6 ) , »herefi.(kx)=kflfL(kiC) and rrL(kR~)^ kRn,_(kR). To o b t a i n the s e r i e s expansions f o r A(R) and B(R) we f i r s t o b t a i n t he d i f f e r e n t i a l e q u a t i o n s f o r A(R) and B(R) by s u b s t i t u t i n g I I I . 4 . 3 2 i n t o I I I . 4 . 3 0 . AY*Q- 28//OL(L+r> _ a.B'(ft) />*- £ * % n I I I . 4 . 3 3 a / v ' TF^ &n I I I . 4 . 3 3 b As R =^^0 we r e q u i r e t h a t A(R) - > 1 and B(R) =^ 0, and t h e r e f o r e w r i t e the s e r i e s expansions f o r A(R) and B(R) 92 B (R) - ^2 Q>mH~m~^ i > / , I H . 4 . 3 4 b where p, and q are c o n s t a n t s . I d e n t i f y i n g c o e f f i c i e n t o f the same power o f R i n I I I . 4. 33a, b, we o b t a i n : -p = n, q = n - 1 and the r e c u r s i o n r e l a t i o n s f o r the h i g h e r o r d e r c o e f f i c i e n t s i n I I I . 4 . 3 4 a , b I l l . 4 . 3 5 b I f a d d i t i o n a l terms are r e t a i n e d i n the l o n g range p o t e n t i a l , > 3 I I . 4 . 3 6 - 93 -i n g e n e r a l O and the r e c u r s i o n r e l a t i o n s are appro-p r i a t e l y m o d i f i e d . The d e r i v a t i o n o f I I I . 4 . 3 3 a , b , and t h e i r s o l u t i o n i s shown i n Appendix 1 at the back o f t h i s t h e s i s . I t i s c l e a r t h a t the use of a s y m p t o t i c s e r i e s expansions i n systems w i t h l o n g range p o t e n t i a l s v a r y i n g as R - N (where n ^ 3 ) has one v e r y i m p o r t a n t advantage. Since one need o n l y i n t e g r a t e the coupled e q u a t i o n s t o where the s h o r t range terms of the i n t e r a c t i o n p o t e n t i a l have become n e g l i g i b l e we have a means o f r e d u c i n g the computer time r e q u i r e d t o s o l v e the f i x e d angle coupled e q u a t i o n s . I n Chapter IV we s h a l l i l l u s t r a t e the advantage o f the f u n c t i o n s I I I . 4 . 3 2 over the more c o n v e n t i o n a l S p h e r i c a l Bessel f u n c t i o n s w i t h r e f e r e n c e t o + H + c o l l i s i o n s . The ^ + H + p o t e n t i a l s u r f a c e c o n t a i n s l o n g range i s o t r o p i c and a n i s o t r o p i c terms due t o the charge on the p r o t o n . 4.4 D e r i v a t i o n o f the Sector R-matrix I n t h i s s e c t i o n we p r e s e n t the d e r i v a t i o n o f the s e c t o r R-matrix. W i t h i n a s e c t o r the e q u a t i o n s are c o m p l e t e l y uncoupled, the e q u a t i o n s i n q u e s t i o n are w r i t t e n I I I .4.37 - 94 -We b e g i n by assuming t h a t t h e e i g e n v a l u e s ^ N ( R ) of the c o u p l i n g m a t r i x are c o n s t a n t across a g i v e n s e c t o r , and l a t e r g e n e r a l i z e t o more complex s i t u a t i o n s . The ei g e n v a l u e s can have any known form, however, the t r a n s -f o r m a t i o n m a t r i x T must remain c o n s t a n t across the s e c t o r t o m a i n t a i n the decoupling o f the e q u a t i o n s . The elements of r . , r 0 , r 0 and r, form a 2 x 2 ~ 1 ' -\,2 ' ^_3 b l o c k m a t r i x , 'Fn&i= r^»» lF~Fn,'(*V , Fn (ln>J i (rOhn (ni^H JL rhf^)J I I I . 4 . 3 8 where a and b are the s e c t o r boundaries ( ) . The elements o f the s e c t o r R-matrix ( T| , r 2, £'3* XM ^ m u s t ^e d''''K0'1-'11 L o prevent m i x i n g of the s o l u t i o n s . l i q u a t i o n I I I . 4. 38 i s a sim p l e second o r d e r d i f f e r e n t i a l e q u a t i o n , and has two l i n e a r l y independent s o l u t i o n s A^NR) and B ^ ^ ( R ) , and thus the g e n e r a l s o l u t i o n may be w r i t t e n I I I . 4 . 3 9 a dK dK I I I . 4 . 3 9 b where ©C and ^ are a r b i t r a r y c o n s t a n t s . The Wronskian o f two l i n e a r l y independent s o l u t i o n s A ^ ^ ( R ) and i s denoted byjW*and i s a c o n s t a n t . - 9 5 -I I I .4.40 The R - matrix r e l a t i o n I I I . 4 . 3 8 h o l d s f o r any s o l u t i o n of I I I . 4 . 3 7 , and t h e r e f o r e i s independent o f o( and ^3. We choose two s o l u t i o n s s a t i s f y i n g the f o l l o w i n g s i m p l e boundary c o n d i t i o n s p ; (a) - / f^i'(a) - C 111.4.4ia P*. (6> I rlfO-yO , I I I . 4 . 4 1 b where 1, and 2 r e f e r t o two s e p a r a t e boundary c o n d i t i o n s of the F's, and the s u b s c r i p t n has been dropped. The R-matrix i s now e a s i l y o b t a i n e d i n terms o f and F2 from n = - ft. (*) 1*1 to 1 = i/p, fa I I I . 4 . 4 2 a I I I . 4 . 4 2b The s o l u t i o n s s a t i s f y i n g the boundary c o n d i t i o n s I I I . 4 . 3 9 a , b are F,(R) = Uf'1 QS> f(oC) A (R) -f> '( A ftp-A '(tr> 8 f ^V*0 > 8 fa) ' S^ a) can be generated u s i n g t he c o r r e s p o n d i n g v a l u e s used i n t h e p r e v i o u s p a r t i a l wave. The major draw-back o f t h i s t e c h n i q u e _ 100 -i s t h e amount of s t o r a g e r e q u i r e d , we would need t o know the v a l u e s o f f\(aj, f\f(e\j , g (a) and 5t<*) a t each s e c t o r boundary f o r each q , and i n a d d i t i o n we r e q u i r e the v a l u e of R at each s e c t o r boundary and i ho values o f q n at the c e n t r e o f each s e c t o r . Therefore f o r an N-channel problem we would need t o s t o r e (5N + 1) numbers. I n r o t o r - v i b r o t o r c a l c u l a t i o n s t h i s amount o f s t o r a g e might be p r o h i b i t i v e l y l a r g e however, i n atom-v i b r o t o r c a l c u l a t i o n s when we o n l y r e t a i n a few v i b r a t i o n a l c h a n n e l s , t h i s s t o r a g e should p r e s e n t no problems. 4 .5 The Step s i z e A l g o r i t h m I n the R-matrix p r o p a g a t o r method one assumes t h a t the c o u p l i n g m a t r i x feT i s c o n s t a n t over the w i d t h of the s e c t o r , t h i s i n t u r n loads t o the assumption t h a t the d e r i v a t i v e {/J* i s zero over the w i d t h o f the s e c t o r . T h e r e f o r e a scheme i s r e q u i r e d i n which one reduces the m a t r i x o f d e r i v a t i v e s U/f i n t o a s i n g l e parameter s u i t a b l e f o r d e t e r m i n i n g the s e c t o r w i d t h . We w i l l assume t h a t t he f u n c t i o n i s a s a t i s f a c t o r y measure of t h e r a t e o f change o f the c o u p l i n g p o t e n t i a l , and the new s e c t o r w i d t h i s g i v e n by where BETA i s a c o n s t a n t . The r a t e o f change o f the average e i g e n v a l u e i s a s a t i s f a c t o r y measure of the r a t e o f change o f the p o t e n t i a l V f&). - 101 I I I . 4 . 5 5 U n f o r t u n a t e l y , at l o c a l minima, and maxima i n the p o t e n t i a l (where y 0) , I I I . 4 . 5 5 o v e r e s t i m a t e s the next s e c t o r w i d t h . T h i s can be overcome by l i m i t i n g t h e new s e c t o r w i d t h r e l a t i v e t o the l a s t , employing kui GST*?) = hi X FACT I I I . 4 . 5 6 Another measure o f the change of the c o u p l i n g m a t r i x i s the o v e r l a p m a t r i x ^ . For s m a l l s e c t o r s i s a l m o s t a u n i t m a t r i x , i f T^ "J^t) t h e n ~^ -^w The r a t e o f change o f the t r a n s f o r m a t i o n m a t r i x ^ ^ from s e c t o r t o s e c t o r i s l i m i t e d v i a the r e l a t i o n hi+i ( s CS>TEP*) = COP MAX x f N-l Xa" ( a *CUPLR I I I . 4 . 5 7 where CUPLK - T (j = Cg") I n I I I . 4 . 5 7 CUPMAX i s an i n p u t t o l e r a n c e s p e c i f y i n g the maximum magnitude o f a t y p i c a l o f f d i a g o n a l element o f A f i n a l f i n e c o n t r o l upon the s t e p s i z e i s imposed by s p e c i f y i n g the parameters STPMIN and STPMAX. The new s e c t o r i s r e q u i r e d t o f a l l w i t h i n the p e r m i s s i b l e - 102 -range o f s e c t o r w i d t h s (STPMIN, STPMAX). Thus the a l g o r i t h m i s dependent upon f i v e parameters, namely:-STPMIN, STPMAX, BETA, FACT, CUPMAX where BETA c o n t r o l s t h e s e c t o r w i d t h s , and i s based on the r a t e o f change o f the e i g e n v a l u e s o f hO . FACT l i m i t s the maximum r a t e o f change i n the s e c t o r w i d t h and CUPMAX c o n t r o l s the new s e c t o r w i d t h , and i s based on the r a t e o f change o f the m a t r i x Jg. A l l these parameters must be s p e c i f i e d t o o b t a i n the best r e s u l t s f o r the problem b e i n g s t u d i e d . Of these parameters, o n l y VSTEP w i l l be a f f e c t e d by the c h o i c e o f the e i g e n v a l u e ^h(*^) • ^ n S e c t i o n 4.4 we d i s c u s s e d two p o s s i b l e forms f o r 7^(\(^f • We f i r s t assumed t h a t (it) was co n s t a n t over the s e c t o r w i d t h , where, A ^ n < R > = 4 L < L 1 1 > R 2 Secondly, we c o n s i d e r e d a scheme i n which the c e n t r i f u g a l 2 term was t r e a t e d a n a l y t i c a l l y , and was take n as co n s t a n t across the w i d t h o f the s e c t o r , B > n = *n • r-A places the g r e a t e s t c o n s t r a i n t upon VSTEP i n t h a t i t assumes t h a t b o t h the d i a g o n a l i z a t i o n and the c e n t r i f u g a l term are co n s t a n t w i t h i n the s e c t o r , whereas B o n l y assumes the former t o be co n s t a n t across t h e w i d t h o f the s e c t o r . - 103 -As d i s c u s s e d i n S e c t i o n 4.4 scheme B has t h e advantage t h a t the s t e p s i z e i s v i r t u a l l y independent o f L, and c o n s e q u e n t l y l a r g e numbers o f p a r t i a l waves may be generated w i t h l i t t l e n u m e r i c a l e f f o r t . T h i s can be achieved by employing scheme A, however we have the a d d i t i o n a l c o n s t r a i n t t h a t the s t e p s i z e must be s u f f i c i e n t l y s m a l l t o ensure n u m e r i c a l accuracy a t t h e h i g h e s t p a r t i a l wave, and t h i s can t h e n be used at a l l the p a r t i a l waves c o n s i d e r e d . I n s i t u a t i o n s where the s t e p s i z e i s h i g h l y dependent upon L, i t may be necessary t o d i v i d e the range o f p a r t i a l waves c o n s i d e r e d i n the c a l c u l a t i o n i n t o s m a l l e r subgroups, and d e t e r m i n e an a p p r o p r i a t e s t e p s i z e f o r each group. The s t e p s i z e a l g o r i t h m makes no mention o f L o r the t o t a l c o l l i s i o n energy, E. However i t i s w o r t h m e n t i o n i n g t h a t t h e s t e p s i z e w i l l be weakly dependent upon L (and E) even when scheme B i s employed. As s t a t e d i n S e c t i o n 4.4, as t h e d i a g o n a l elements o f the c o u p l i n g m a t r i x fej i n c r e a s e the o f f d i a g o n a l elements w i l l become l e s s i m p o r t a n t . 4.6 P r o p a g a t i n g V a r i a b l e Numbers of Closed Channels I n s o l v i n g a s c a t t e r i n g problem, one i s u s u a l l y o n l y i n t e r e s t e d i n t r a n s i t i o n s among a c e r t a i n set o f s t a t e s . The t r a n s i t i o n s t o s t a t e s f a r i n energy from t h e i n i t i a l s t a t e s of i n t e r e s t are o n l y n e g l i g i b l y e x c i t e d . N e v e r t h e l e s s , i t i s f r e q u e n t l y found t h a t t hey must be i n c l u d e d i n the c a l c u l a t i o n t o o b t a i n 104 a c c u r a t e v a l u e s f o r the t r a n s i t i o n s t o s t a t e s which are o f i n t e r e s t . I n a d d i t i o n i t i s o f t e n necessary t o c a r r y c l o s e d channels t o o b t a i n f u l l y converged t r a n s i t i o n s . The c l o s e d c h annels, and the channels b e l o n g i n g t o s t a t e s which are weakly e x c i t e d are necessary t o d e s c r i b e the system p r o p e r l y i n the p e r t u r b e d r e g i o n where the i n t e r a c t i o n i s s t r o n g . I n p a r t i c u l a r the c l o s e d channels are r e q u i r e d t o a c c u r a t e l y d e s c r i b e the d e f o r m a t i o n o f the c o l l i s i o n p a r t n e r s . The t r a n s l a t i o n a l f u n c t i o n s o f such s t a t e s c a r r y no u s e f u l s c a t t e r i n g i n f o r m a t i o n as they are so d e e p l y b u r i e d under the p o t e n t i a l b a r r i e r t h a t they never g a i n any a p p r e c i a b l e a m p l i t u d e . I n the R-matrix p r o p a g a t o r a l g o r i t h m i t i s p o s s i b l e t o c a r r y the optimum number o f channels i n a g i v e n s e c t o r . Dropping channels from the p r o p a g a t i o n i s e x t r e m e l y s i m p l e , and i s achieved by t r u n c a t i n g the R -matrix ( t h e r e l e v a n t row and column are removed). The t r u n c a t e d R-matrix w i l l s t i l l be symmetric, and c o n s e q u e n t l y t h e f i n a l S-matrix w i l l be u n i t a r y . " P i c k i n g up" new channels i s e a s i l y a c h i e v e d by p r o p a g a t i n g the o r i g i n a l g l o b a l R-matrix on the o u t e r s u r f a c e and b r i n g i n g i t up t o the l a r g e r dimension by adding elements u s i n g the a p p r o p r i a t e i n i t i a l c o n d i t i o n s . SWL s p e c i f y two methods f o r d e t e r m i n i n g the number of c l o s e d channels needed t o be propagated across a s e c t o r . The f i r s t i s a r a t h e r s i m p l e energy-based 105 -c r i t e r i o n . I l I s p o s s i b l e l o s p e c i f y .1 I o I e r . i r i c e l ) . i s e r l upon t he e x t e n t t o which a channel i s c l o s e d , any channel more de e p l y b u r i e d than s p e c i f i e d by the parameter may be dropped. I n a d d i t i o n we may dete r m i n e the e x t e n t t o which a de e p l y b u r i e d channel i s coupled t o l e s s d e e p l y b u r i e d c h a n n e l s , t h i s may be achieved by con-s i d e r i n g the o f f d i a g o n a l elements o f the m a t r i x jg,. The q u a n t i t y ^ j i s an e s t i m a t e o f the e x t e n t t o which channel ( j ) i s coupled t o channels ( 1 , 2 ... j - 1 ) [ 2 + - 106 -CHAPTER IV ROVIBRATIONAL EXCITATION OF H 2 BY H + 1. I n t r o d u c t i o n Due t o i t s l a r g e r o t a t i o n a l energy l e v e l s p a c i n g t h e H2 molecule p r o v i d e s a r i g o r o u s t e s t o f approximate t r e a t m e n t s o f the r o t a t i o n a l m o t i o n of m o l e c u l e s , p a r t i c u l a r l y the energy sudden and the i n f i n i t e o r d e r sudden a p p r o x i m a t i o n s , (see Chapter I I , S e c t i o n s 3.2 and 3.3). The i n t e r a c t i o n p o t e n t i a l f o r H 2 + H + contains l o n g range i s o t r o p i c , and a n i s o t r o p i c components due t o t h e charge on t h e p r o t o n . The presence o f such terms means t h a t t h e accuracy o f the coupled s t a t e s component o f t h e IOS a p p r o x i m a t i o n w i l l be reduced. The accuracy o f t h e sudden component i s a l s o reduced, s i n c e t h i s a p p r o x i m a t i o n r e q u i r e s t h a t the r o t a t i o n a l , p e r i o d of the molecule be l a r g e compared t o the c o l l i s i o n t i m e , which i s p r o p o r t i o n a l t o t h e range o f t h e i n t e r a c t i o n p o t e n t i a l . The l o n g range i n t e r a c t i o n s have the r e s u l t o f i n c r e a s i n g t h e c o l l i s i o n t i m e , and t h e p r o t o n spends a l o n g t i m e i n the i n t e r a c t i o n r e g i o n . The presence o f l o n g range i n t e r a c t i o n terms a l s o makes t h e c a l c u l a t i o n o f c r o s s s e c t i o n s f o r the H2 + H + system d i f f i c u l t s i n c e l a r g e i n t e g r a t i o n ranges are necessary. T h i s means a l a r g e number o f steps i n the a l g o r i t h m , which may be s m a l l steps t o min i m i z e e r r o r i n each s e c t o r , and hence reduce t he accumulated round-o f f e r r o r i n the f i n a l r e s u l t . Three ab i n i t i o p o t e n t i a l s u r f a c e s f o r the H2 + H + system (Csizmadia et a l . ( 1 9 7 0 ) , B a u s c h l i c h e r et a l . ( 1 9 7 3 ) , 10 7 and Carney and P o r t e r ( 1 9 7 4 ) ) , have been compared and d i s c u s s e d by Giese and Gentry ( 1 9 7 4 ) . Giese and Gentry concluded t h a t an a n a l y t i c f i t t o a r e s t r i c t e d set o f p o i n t s from t h e SCF-MO-configuration i n t e r a c t i o n c a l c u l a t i o n o f Csizmadia et a l . , s u i t a b l y a d j u s t e d , would p r o v i d e the best r e p r e s e n t a t i o n o f the l ^ + H + p o t e n t i a l s u r f a c e . The c a l c u l a t i o n s o f Csizmadia et a l . are by f a r the most e x t e n s i v e and cover a comprehensive range of n u c l e a r g e o m e t r i e s . Ciese and Gentry (1974) have performed s e m i - c l a s s i c a l c a l c u l a t i o n s o f v i b r a t i o n a l e x c i t a t i o n o f H2 by H + u s i n g t h e i r DECENT model ( D i s t r i b u t i o n (among quantum s t a t e s ) o f Exact C l a s s i c a l Energy T r a n s f e r ) . The c l a s s i c a l energy t r a n s f e r (as a f u n c t i o n o f a n g l e ) i s o b t a i n e d by means o f exact c l a s s i c a l 3-d t r a j e c t o r i e s . By employing t he correspondence p r i n c i p l e between a c l a s s i c a l , and quantum f o r c e d harmonic o s c i l l a t o r they o b t a i n e d d i f f e r e n t i a l and i n t e g r a l c r o ss s e c t i o n s . A l t h o u g h the r e s u l t s o f Giese and Gentry agree w e l l w i t h the e x p e r i m e n t a l data o f Udseth et a l . (1973) i t was p o s t u l a t e d by Schinke (1977) t h a t t h e r e are some aspects ( e . g . rainbow s t r u c t u r e s ) t h a t might be improved by the a p p l i c a t i o n o f more quantum l i k e s c a t t e r i n g t h e o r i e s . Consequently Schinke (1977) performed t i m e dependent c l o s e - c o u p l i n g c a l c u l a t i o n s which employed an energy sudden t r e a t m e n t o f the r o t a t i o n . The o v e r a l l agreement w i t h t h e r e s u l t s o f Giese and Gentry i s s a t i s f a c t o r y and, i n a d d i t i o n , the r e s u l t s o f Schinke c o n t a i n a d d i t i o n a l rainbow s t r u c t u r e s . Schinke concluded t h a t t h e use o f the s t r a i g h t l i n e impact parameters r e s t r i c t e d 108 the time df.'|)(M)donl c l o s e c o u p l i n g met h o d i n c o l l i s i o n e n e r g i e s ^ 1 '> eV. McGuire (1976) has performed CC and CS c a l c u l a t i o n s t r e a t i n g the H2 molecule as a r i g i d r o t o r . The p o t e n t i a l s u r f a c e o f Giese and Gentry was employed w i t h t h e H2 i n t e r -n u c l e a r s e p a r a t i o n set a t i t s e q u i l i b r i u m v a l u e . The o b j e c t of the c a l c u l a t i o n s was t o a s c e r t a i n t he f e a s i b i l i t y o f a c l o s e - c o u p l i n g s tudy i n t h e approximated coupled s t a t e s form. C a l c u l a t i o n s c a r r i e d out by Schinke and McGuire (1978a) employing the IOS a p p r o x i m a t i o n were compared w i t h the CS r e s u l t s and i t was concluded t h a t the IOS a p p r o x i m a t i o n was v a l i d f o r H2 + H + f o r c o l l i s i o n e n e r g i e s ^ 3.7 eV. These IOS c a l c u l a t i o n s were extended by Schinke and McGuire (1978b), by the i n c l u s i o n o f the v i b r a t i o n a l degree o f freedom which was t r e a t e d by t h e c l o s e - c o u p l i n g t e c h n i q u e . Programmes which p e r f o r m IOS c a l c u l a t i o n s o f r o -v i b r a t i o n a l e x c i t a t i o n c r o ss s e c t i o n s were w r i t t e n i n FORTRAN IV, and developed on the IBM 370/168 computer at Newcastle U n i v e r s i t y . The aim o f t h i s study was t o t e s t these programmes by r e p e a t i n g the c a l c u l a t i o n o f Schinke and McGuire (1978b) a t E = 4.67 eV. I t i s w o r t h m e n t i o n i n g t h a t the r e s u l t s o f t h i s c a l c u l a t i o n are o f l i t t l e v a l u e , a p a r t from a s c e r t a i n i n g t h a t the computer programmes, which are used t o o b t a i n t he r e s u l t s p r e s e n t e d i n the f o l l o w i n g c h a p t e r s , are f u n c t i o n i n g c o r r e c t l y ; the p o t e n t i a l s u r f a c e o f Giese and Gentry (1974) (GG) employed by Schinke and McGuire has been shown t o be i n e r r o r by - 109 -Schinke et a l . ( 1 9 8 0 ) . I n an appendix at the end of t h i s c h a p t e r we w i l l present r e s u l t s i l l u s t r a t i n g the advantage of a s y m p t o t i c s e r i e s expansions of the type i n t r o d u c e d by G a i l i t i s ( 1 9 7 6 ) , over the c o n v e n t i o n a l method o f f i t t i n g the a s y m p t o t i c boundary c o n d i t i o n . 2. The I n t e r a c t i o n P o t e n t i a l The c a l c u l a t i o n o f Schinke and McGuire (1978b) employs the a n a l y t i c p o t e n t i a l s u r f a c e o f Giese and Gentry (1974) (GG). GG r e p o r t t h e t o t a l p o t e n t i a l o f the H 2 + H + system ( t h e y i n c l u d e the p o t e n t i a l o f the i s o l a t e d H 2 molecule i n t h e i r c a l c u l a t i o n ) . T h e r e f o r e , one advantage o f the GG p o t e n t i a l i s t h a t i t may be used t o d e t e r m i n e the exact v i b r a t i o n a l w a v e f u n c t i o n s of the H 2 m o l e c u l e . Such a c a l c u l a t i o n has been u n d e r t a k e n by Schinke ( 1 9 7 7 ) , and these w a v e f u n c t i o n s are employed by Schinke and McGuire (1978b). U n f o r t u n a t e l y , the method used by Schinke t o d e t e r m i n e these w a v e f u n c t i o n s d e s t r o y s t h e i r o r t h o n o r m a l i t y , and leads t o u n p h y s i c a l b e h a v iour f o r the h i g h e r v i b r a t i o n a l s t a t e s . Consequently we p r e f e r t o c a l c u l a t e our own exact w a v e f u n c t i o n s , the c a l c u l a t i o n o f which i s d e s c r i b e d i n S e c t i o n 3, t o g e t h e r w i t h a d i s c u s s i o n o f the problems a s s o c i a t e d w i t h S c h i n k e 1 s w a v e f u n c t i o n s . The p o t e n t i a l o f GG i s a f i t t o 138 ab i n i t i o con-f i g u r a t i o n i n t e r a c t i o n e n e r g i e s of Csizmadia et a l . (1970) t o the f o l l o w i n g t e n parameter a n a l y t i c f u n c t i o n . A l l t e n parameters ( u n d e r l i n e d ) were o p t i m i z e d by an i t e r a t i v e l e a s t - s q u a r e s f i t o f t h i s f u n c t i o n t o the 138 c a l c u l a t e d p o i n t s . The c o - o r d i n a t e system used i s shown i n F i g u r e 1. - 110 -H F i g u r e 1 9 The p o t e n t i a l i s expressed i n terms o f the t h r e e i n t e r a t o m i c d i s t a n c e s ^ (k = 1,2,3), where i s the H2 i n t e r n u c l e a r s e p a r a t i o n ( t h u s = f ) , and i s g i v e n i n atomic u n i t s by -3 k s l IV.2.1 H(R^) i s a minor v a r i a n t o f a Hu1 h u r t - H i r s c h f c 1 d e r p o t e n t i a l f u n c t i o n ( H u l b u r t and H i r s c h f e l d e r (1941) ) and may be w r i t t e n as = A [-ze + e o / / V 5 Z 3 r Y/ - z ) 7 , i v. 2.2 where E = e - z Z = B ( R k / R e - l ) B = 1.4426 - 0.126871F, R = 1.40083 + 0.27923F, H = 0.17449 - (0.014665 + 0.022721 ) F 4 and P i s the charge-induced d i p o l e c o n t r i b u t i o n -fro ¥ - A z ^ ^ ^ ) > RT* IV. 2. 3 A Q and A2 are determined from cubic f i t s t o the s p h e r i c a l and angle-dependent p o l a r i z a b i l i t i e s versus R2 (Kolos and Wolniewicz ( 1 9 6 5 ) , T r u h l a r ( 1 9 7 2 ) ) . I l l The n u m e r i c a l v a l u e s are A Q = 2.6091 + £2.246 + (0.3181 - 0 . 1 1 9 4 ^ ) ^ 7 ( 0 , A 2 = 0.60735 + £i.3588 + (0.5573 - 0 . 3 1 7 0 ^ > ) f > 3 p , where p = R 2 - 1.40083. IV.2.4 The charge-quadrupole c o n t r i b u t i o n Q i s g i v e n by = Q 2P 2R~ » where 6( 2 i - s d e t e r m i n e d from a c u b i c f i t t o t he quadrupole moment versus R 2 * Q 2 - 0.45883 i- [h. 53223 t (0.03234 - 0 . 091 4 74jO) ^ . F i n a l l y , , F 2, F^ and F^ are r o l l - o f f and r o l l -on f u n c t i o n s g i v e n by F, ~ R5I (/33- ^ 7 ^ 1 - f £ 5 ) IV.2.5a ^ K ^ / f x r j o g * - * * * ) iv.i.5b f 3 * I j f l + w b HQ5 MZQJ] i v . 2 . 5 c Fij. = / ///•/ o- ooolbH-2.8') ft*). IV.2.5d The use o f t h e summation over the d i a t o m i c p o t e n t i a l f u n c t i o n s H(R^) a l l o w s the w i d t h , depth and p o s i t i o n o f the p o t e n t i a l minimum t o v a r y s l o w l y as the p r o t o n approaches The above f i t i s a good r e p r e s e n t a t i o n of the H 2 i H4 p o t e n t i a l , however, o n l y the q u a n t i t i e s A , A 2, and Q 2 are independent o f R and can be c a l c u l a t e d a t a l l t h e r e q u i r e d q u a d r a t u r e p o i n t s once and used f o r a l l v a l u e s o f R. The c o m p u t a t i o n a l l y expensive t a s k s such as expon-e n t i a t i o n are c o n t a i n e d i n H(R^). At each s t e p i n the i n t e g r a t i o n range these m a t r i x elements must be e v a l u a t e d n u m e r i c a l l y . However, these c a l c u l a t i o n s are o n l y necessary a t the i n i t i a l p a r t i a l wave, subsequent p a r t i a l waves r e q u i r e no e x p l i c i t r e f e r e n c e t o the p o t e n t i a l . I n CC and CS c a l c u l a t i o n s the form o f the p o t e n t i a l 112 i s a more severe problem. To und e r t a k e such c a l c u l a t i o n s the p o t e n t i a l s u r f a c e Ls more c o n v e n i e n t l y expressed i n terms o f a s i n g l e c e n t r e expansion s V(%>r,e*)^ C)- V(S > r ) - V W l (r), i v . 2 . 7 where V ^ ( r ) i s t n e p o t e n t i a l o f the i s o l a t e d H 2 m o l e c u l e , and i s g i v e n by IV.2.1 w i t h R = CO where A' = 0.17449, E' = e" Z', Z = 1 . 4 4 2 6 ( r / l . 4 0 0 8 3 - 1 ) . 3. The H 2 v i b r a t i o n a l w a v e f u n c t i o n s The p o t e n t i a l o f GG i n c l u d e s t he p o t e n t i a l o f the i s o l a t e d H 2 molecule ( I V . 2 . 8 ) , t h e r e f o r e the c a l c u l a t i o n o f exact v i b r a t i o n a l w a v e f u n c t i o n s i s p o s s i b l e . T o d e t e r m i n e exact H~ bound s t a t e s w and e n e r g i e s g we must s o l v e ( y * r # + vH, (r> - eg] 9 2= and are g i v e n by ^ fV) s VKp^-^^Ni Hi IV.3.4. where <^ = (JUlKJ^ (T~ Pe~> , i s a n o r m a l i s a t i o n c o e f f i c i e n t s and i s a Hermite p o l y n o m i a l . S u b s t i t u t i n g IV.3.2 i n t o IV.3.1 and making use o f IV.3.3 and the o r t honorma 1 p r o p e r t i e s o f ^ ^ ' ( ) we o b t a i n I ho homo-geneous set o f l i n e a r e q u a t i o n s f o r the c o e f f i c i e n t s c^ * /s* /^f ' IV. 3. 5. where the m a t r i c e s V and H are d e f i n e d by AO1 = c e f - £ e x>ty. ex The d e s i r e d e n e r g i e s ^ v are the n t h e s o l u t i o n s o f th e s e c u l a r e q u a t i o n / X + & I "°- 3.6. As Schinke (1977) p o i n t s out the harmonic w a v e f u n c t i o n s are o f a sim p l e f o r m , and t h i s i n c r e a s e s the use of h i s exact w a v e f u n c t i o n s i n f u r t h e r a p p l i c a t i o n s . However, harmonic w a v e f u n c t i o n s have two s e r i o u s d i s a d v a n t a g e s f o r t h i s t y p e o f c a l c u l a t i o n . The exact ^ w a v e f u n c t i o n s 114 -are s u b s t a n t i a l l y d i f f e r e n t from the HO f u n c t i o n s f o r r e a s o n a b l y h i g h v i b r a t i o n a l s t a t e s , and a l a r g e number of b a s i s f u n c t i o n s are r e q u i r e d i n the expansion. The major problem i s t h a t f o r h i g h e r v i b r a t i o n a l s t a t e s ( i ^ 1 0 ) the HO f u n c t i o n s are non-zero f o r /** < 0. Th i s i s c l e a r l y u n p h y s i c a l as i t corresponds t o the two n u c l e i o f t h e H 2 molecule p a s s i n g t h r o u g h each o t h e r . T h i s c o u l d be e a s i l y overcome by e x t e n d i n g t h e i n t e g r a t i o n range i n t h e V ^ j elements, and indeed t h i s i s e s s e n t i a l i f the o r t h o n o r m a l p r o p e r t i e s o f the wave f u n c t i o n s are t o be p r e s e r v e d . W h i l s t the V^ p o t e n t i a l i s not i n f i n i t e HO f o r r < 0 i t i s l a r g e r t h a n the harmonic p o t e n t i a l V T h e r e f o r e e x t e n d i n g t h e i n t e g r a t i o n range would r e s u l t i n a s h i f t o f emphasis from t h e p o t e n t i a l w e l l i n which we are p r i m a r i l y i n t e r e s t e d , t o t h e r e g i o n f o r which r < 0. Schinke does not extend the i n t e g r a t i o n range ( K i r k p a t r i c k (1983)1, and hence d e s t r o y s t h e o r t h o n o r m a l i t y o f the HO w a v e f u n c t i o n s which are employed t o d e r i v e the s e c u l a r e q u a t i o n ( I V . 3.6). Schinke uses 20 HO wavef unct ions i n h i s expansion. The expansion c o e f f i c i e n t s used i n h i s c a l c u l a t i o n are shown i n Table 1. For the h i g h e r o r d e r w a v e f u n c t i o n s , V 2 ^ 4, the b a s i s used i s i n s u f f i c i e n t , t h i s i s most e v i d e n t f o r V 2 = 6; compare C^/tj w i t h Cfc^b-I n the p r e s e n t IOS c a l c u l a t i o n t h e exact w a v e f u n c t i o n s were r e c a l c u l a t e d u s i n g Morse o s c i l l a t o r w a v e f u n c t i o n s . The Morse p o t e n t i a l and c o r r e s p o n d i n g e i g e n f u n c t i o n s and TAW,I-; 1 Expansion c o e f f i c i e n t s , C v o i , o f the v i b r a t i o n a l b a s i s s t a t e s employed Dy Schinke (1977). i v 2 = 0 v 2 = l v 2 = 2 v 2 = 3 v 2 = 4 v 2 = 5 v 2 = 6 0 0.98924 -0.13199 0.02928 -0.03512 0.03022 -0.02126 0.01426 1 0.14160 0.91099 -0.34437 0.11093 -0.07474 0.06783 -0.05556 2 -0.00421 0.37450 0.67933 -0.51369 0.24862 -0.14282 0.11137 3 0.03460 0.06605 0.57768 0.26929 -0.50248 0.37348 -0.23332 4 0.01112 0.07537 0.21227 0.60622 -0.20469 -0.23465 0.34202 5 -0.00047 0.04585 0.14929 0.36869 0.36577 -0.49440 0.17604 6 0.00266 0.01193 0.11484 0.25262 0.39956 -0.05132 -0.38364 7 0.00130 0.00918 0.05492 0.21649 0.32015 0.21168 -0.34572 8 -0.0006 0.00666 0.03190 0.14259 0.30800 0.24808 -0.09666 9 0.00027 0.00217 0.02320 0.08986 0.25168 0.29905 0.03075 10 0.00019 0.00137 0.01262 0.06508 0.18779 0.31351 0.11111 11 -0.00001 0.00111 0.00713 0.04374 0.14519 0.28292 0.24079 12 0.00003 0.00042 0.00500 0.02754 0.10864 0.23235 0.22695 13 0.00003 0.00024 0.00297 0.01920 0.08017 0.21120 0.31950 14 0.0 0.00020 0.00165 0.01253 0.05454 0.14742 0.18046 15 0.0 0.00009 0.00118 0.00858 0.04404 0.14298 0.32688 16 0.00001 0.00004 0.00068 0.00504 0.02542 0.07681 0.09873 17 0.0 0.00004 0.00043 0.00403 0.02301 0.09175 0.28077 18 0.0 0.00002 0.00022 0.00173 0.00935 0.02699 0.00861 19 0.0 0.00001 0.00018 0.00168 0.01132 0.05487 0.23006 115 e i g e n v a l u e s are g i v e n by (Mies 1964) < f > T ( ^ * *c i / ' ^ A ( y)"-3.7b where - *C € and £/ (^) ^ S a ^ h i t t a k e r f u n c t i o n (Abramowitz and Stegun ( 1 9 6 5 ) ) . The parameters D, and f~e are chosen t o approximate V M 0 as V H2. Values ( i n a.uj are D = 0.17449 w e l l d e p t h , f e - 1.40083 e q u i l i b r i u m s e p a r a t i o n , j> = 0.9900134 . The parameter 100 SM r e p o r t t he v i b r a t i o n a l c o u p l i n g i s n e g l i g i b l e , and they C M w I—1 CO < f—1 VJ D cO 3 CT z cO 3 u r-l, -—' CO C o x> z •r-l > .-— X> '— C ••-I O N x: o II C O z o CO X O C M X) O w O CM s: > 0) i—i CO > • x> c CD e CD CD •r-l 1—1 oo 0) 0) •"-I c •r^ CD > c 0) CD 00 x: •I-l QJ CD 0) 4-1 X CO 1—1 X CO . U > •r-l CD 3 o 4-' O |. O o (. o r-~ CNI C O T - l 00 o- C O 00 CSl Nl" oo 00 O O o o T H T - l T - l • • • • o O o o T—1 1 T H 1 1 T H 1 1 o 1 ON 1 N t 1 \0 T- l ON ON o o n o o CSJ T H ON C O N t C O < f O N ON ON ON • o « a o o o o T - l 1 T - l 1 1 T H 1 1 ON 1 1 UO 1 CM r~- m o C O CM r-t <}• UN m VJO L O O N ON ON ON r--• • • • o O O O T - l 1 T H I T - l 1 T - l 1 1 r-t 1 00 1 1 « • • • O O o O T - l 1 T H • T H 1 T H 1 1 m 1 C O 1 m 1 C O m T H UO r~-CM o o 00 00 00 -X) vD • • • • o o O o T H T - l T - l 1 T H 1 C O 1 T - l 1 CM 1 CM T - l T - l T - l T H <(- >£> ON ON o o 00 o o oo CM CM CNJ CM • • • o o o o CNI 1 CM i CM i CNI 1 I CNI 1 vX5 I 1 o ON T - l X X [J fcrj O C M O C M z > x: > 118 _ d e t e r m i n e the v i b r a t i o n a l l y e l a s t i c S - m atrix elements u s i n g WKB phase s h i f t s c a l c u l a t e d on the \/cc p o t e n t i a l m a t r i x element- The v i b r a t i o n a l l y averaged s u r f a c e , Voo )&z)> i s g i v e n by We, however, r e t a i n two v i b r a t i o n a l channels i n the f i x e d a n g le c o u p l e d e q u a t i o n s t o d e t e r m i n e such S-matrix elements. T h i s d i f f e r e n c e i n approach i s u n l i k e l y t o obscure the comparison as t h e WKB method has been shown t o be e x t r e m e l y a c c u r a t e f o r l a r g e L v a l u e s (e.g. see P f e f f e r and Secrest ( 1 9 7 7 ) ) , and i s a p p l i c a b l e under c o n d i t i o n s f o r which th e IOS method i s n o r m a l l y expected t o be v a l i d . The o t h e r n u m e r i c a l d e t a i l s employed i n t h i s c a l c u l a t i o n are d i s c u s s e d i n d e t a i l , i n the r e s t o f t h i s s e c t i o n , and where p o s s i b l e have been compared w i t h the methods used by SM. The reduced mass o f the ^ + H + system was taken t o be y«6= 1224.101013a.u. as employed by SM ( K i r k p a t r i c k ( 1 9 8 3 ) ) . 4.1 The v i b r a t i o n a l q u a d r a t u r e The e v a l u a t i o n o f the p o t e n t i a l m a t r i x elements i n an IOS c a l c u l a t i o n r e q u i r e the s o l u t i o n o f the i n t e g r a l o f the t y pe ( c f . I I . 3 . 2 2 ) V( %?r7e^otvldr . i v . 4 , 2 SM r e p o r t t h a t i n t h e i r c a l c u l a t i o n which employs t h e HOEX w a v e f u n c t i o n s , the p o t e n t i a l m a t r i x elements were c a l c u l a t e d u s i n g a 2 8 - p o i n t Gauss-Legendre q u a d r a t u r e i n the range 0.2 4l f~ g 3.0a.u. - 119 -I n the present c a l c u l a t i o n we employ the MOEX wave-f u n c t i o n s , and u s i n g these the i n t e g r a l may be r e w r i t t e n where A^,- a C l f c i A; f - / ) V Laguerre p o l y n o m i a l / . • ( ^ / has been w r i t t e n i n terms o f a g e n e r a l i s e d I . e (Abramowitz and Stegun ( 1 9 6 5 ) ) . Since the magnitude of the w a v e f u n c t i o n s i s zero f o r f ~ < f 0 , the range o f i n t e g r a t i o n can be extended t o — o° t o CO , and u s i n g the f a c t t h a t ^ = ^Cr~^) the i n t e g r a l may be r e -w r i t t e n as T h i s makes the i n t e g r a l f o r Vy yf(K)&z) i d e a l l y s u i t e d t o e v a l u a t i o n by Gauss-Leguerre q u a d r a t u r e which employs the weight f u n c t i o n fi. $ . This scheme has the advantage o f e l i m i n a t i n g the use o f i n p u t l i m i t s s i n c e the weight f u n c t i o n f o r c e s t he q u a d r a t u r e p o i n t s over t h e c o r r e c t range. The weight f u n c t i o n 6T& need n o t be c a l c u l a t e d . The q u a n t i t i e s ^ - * ' ' ~ \ / ^ ~ 2 L , ~ 7 ^ ) , and lf,~ 2' are independent o f R and can be c a l c u l a t e d at the r e q u i r e d q u a d r a t u r e p o i n t s on i n i t i a l i z a t i o n o f the p o t e n t i a l r o u t i n e and used r e p e a t e d l y f o r a l l v a l u e s o f R. Table 3 c o n t a i n s a comparison o f p o t e n t i a l m a t r i x elements V0]/f (^}9L) f o r &^ = 71-/2, = 0,6 and R= 3 ,10 , 20a . u . C O - J PQ < U J O X T J QJ CO CO o CO C o T 3 CO o u C U-l QJ > CO 3 X w o 2: 00 c •i-l LO 3 C M CM oi 00 z 0 0 CM L W O 0) 1—1 CO > 0 0 C M 3 2 II Z 3 _ 0) VJ CO T ) CO cr >-< ^ 3 00 CO - J i n (/) CO O T3 U 3 4_) CO u CO cr C OJ 00 0» ,—1 10 CO O a 1 z > ^ X - a CO I—I 3 C J t—I CO O to c o U c 3 M-l QJ > CO 3 3 CO a-. r o c o si NO r i c i 10 m r - l r - i O O C O C O I I O N N£> O rH 0 0 0 0 o o i - l i-H C O C O o o CM I o C O C O o CM C O o CM c i O N o O N r H 0 0 d I I 0 0 T H L O C O N£) * £ > O I C - i 0 0 \f C M 0 0 o > C O < f C M N O r H 0 0 O 0 0 0 0 C M r-- r-- O N o > C O C M v£) L O 0 O N O N C M - C M <± • J - J 0 0 C O C O C M 0 O O O L O L O L O C O 1-1 1~l r H m 10 L O C O O N O N O N 1-1 0 0 0 0 0 0 O O 0 0 0 C O L O • L O 1 1 1 1 1 1 1 < J -1 i-H 1 1 1 r~-1 1 1 1 CM r-- r-~ CM CM CM r - l L O r^ - L O L O CM i O L O L O < t C O C O C O 1-1 CM CM CM 1^-< h ^: x> L O O I C O I 1-1 v£> 0 0 O N O C O d 1 L O 1 O N I S C M C O 0 O d I I o C M O N O N c o o 1-1 - 3 -d I o C M L O I C O C O L O CM L O CM C O 0 0 d I o C O < t O N C O o o I - 120 -c a l c u l a t e d by 28-, 48-, and 64- p o i n t Gauss Leguerre quad-r a t u r e . The d i s c r e p a n c i e s between v a l u e s o b t a i n e d u s i n g 28 and 64 p o i n t s are e x t r e m e l y s m a l l even at l a r g e values of R, where the p o t e n t i a l i s e x t r e m e l y s m a l l . Also shown a 28- p o i n t Gauss Legendre q u a d r a t u r e i n the range 0.2 f -2? 3.0a.u. (emplo y i n g the HOEX w a v e f u n c t i o n s c a l -c u l a t e d by K i r k p a t r i c k ( 1 9 8 3 ) ) . For the m a t r i x elements of major i n t e r e s t (V^-S 3 ) , t h e r e are no s i g n i f i c a n t d i f -f e r e n c e s between t h e r e s u l t s o b t a i n e d u s i n g t h e Gauss Leguerre q u a d r a t u r e s , and the Gauss Legendre q u a d r a t u r e . SM r e p o r t t h a t seven v i b r a t i o n a l s t a t e s must be i n c l u d e d t o o b t a i n r e l i a b l e r e s u l t s f o r &V2m^ 3 t r a n s i t i o n s , and t h e r e f o r e the h i g h e r m a t r i x elements must a l s o be c o n s i d e r e d . As can be seen I n Table 3 the 2 8 - p o i n t Gauss Legendre q u a d r a t u r e can f a i l b a d l y f o r s m a l l m a t r i x elements w i t h a h i g h l y o s c i l l a t o r y i n t e g r a n d , such as V o^(R = 20,7172). I t was d e c i d e d t o employ a 6 4 - p o i n t Gauss Leguerre q u a d r a t u r e i n t h i s work. The e v a l u a t i o n o f the p o t e n t i a l m a t r i x elements accounts f o r a l a r g e percentage of the t o t a l computer time f o r t h i s c a l c u l a t i o n (^507 o f o r a 2 8 - p o i n t q u a d r a t u r e ) . However, the use of an approximate p o t e n t i a l a l g o r i t h m a l l o w s the e f f i c i e n t g e n e r a t i o n of many p a r t i a l waves, the e v a l u a t i o n o f the m a t r i x elements i s o n l y r e q u i r e d a t the i n i t i a l p a r t i a l wave. Subsequent values o f L r e q u i r e no e x p l i c i t r e f e r e n c e t o the pot ent i a l mat rix e lements. The m a t r i x elements can be c a l c u l a t e d u s i n g are the values o f 0,6 c a l c u l a t e d u s i n g 121 a l a r g e number ol q u a d r a t u r e p o i n t s w i t h o u t s i g n i f i c a n t l y I n c r e a s i n g the t o t a l time r e q u i r e d . SM employ de Vogelaere's a l g o r i t h m t o s o l v e the coupled e q u a t i o n s , and t h e r e f o r e have t o e v a l u a t e t h e m a t r i x elements a t each p a r t i a l wave. The d i f f e r e n c e s between t h e r e s u l t s o b t a i n e d u s i n g a 2 8 - p o i n t Gauss Legendre, and a 6 4 - p o i n t Gauss Laguerre q u a d r a t u r e w i l l a f f e c t the c r o s s s e c t i o n s t o a degree, however, the most severe d i f f e r e n c e s occur at l a r g e R values ( R ^ 2 0 a . u . ) and i t i s o n l y necessary t o i n t e g r a t e the f i x e d a n gle coupled e q u a t i o n s out t o R v 20a.u. t o o b t a i n s a t i s f a c t o r y r e s u l t s f o r the v i b r a t i o n a l l y i n e l a s t i c t r a n s i -s i g n i f i c a n t f i g u r e s ( i . e . v i b r a t i o n a l e x c i t a t i o n i s not p a r t i c u l a r l y i m p o r t a n t a t l a r g e R v a l u e s ) . Cross s e c t i o n s f o r t h e v i b r a t i o n a l l y e l a s t i c t r a n s i t i o n i n the v i b r a t i o n a l ground s t a t e r e q u i r e a much l a r g e r i n t e g r a t i o n range t o a c h i e v e the same l e v e l o f convergence ( t h e s o l u t i o n of the coupled e q u a t i o n s f o r f i x e d o r i e n t a t i o n s i s d i s c u s s e d i n S e c t i o n 4 . 2 ) . However, such c r o s s s e c t i o n s are e s s e n t i a l l y d e t e r m i n e d by the h i g h e r p a r t i a l waves. As L i n c r e a s e s v i b r a t i o n a l c o u p l i n g becomes l e s s i m p o r t a n t and i t i s p o s s i b l e t o propagate fewer channels t o o b t a i n s a t i s f a c t o r y r e s u l t s . SM r e p o r t t h a t a t h i g h e r p a r t i a l waves ( t h e y do not s t a t e what they mean by h i g h e r ) i t was o n l y necessary t o r e t a i n f i v e v i b r a t i o n a l s t a t e s and f o r L ^ 100 i t i s w o r t h r e p e a t i n g t h a t the v i b r a t i o n a l l y e l a s t i c S m a t r i x elements were d e t e r m i n e d u s i n g t h e WKB a p p r o x i m a t i o n . t i o n s , m a i n t a i n i n g accuracy i n t o t h r e e 122 -4.2 S o l u t i o n o f the coupled e q u a t i o n s f o r f i x e d o r i e n t a t i o n The c a l c u l a t i o n o f cross s e c t i o n s employing the IOS a p p r o x i m a t i o n i n v o l v e s two o p e r a t i o n s . The coupled second o r d e r d i f f e r e n t i a l , e q u a t i o n s are s o l v e d at g i v e n r o t o r o r i e n t a t i o n s t o produce the S-m a t r i c e s , which are the n used t o c a l c u l a t e t h e cross s e c t i o n s . The c a l c u l a t i o n o f t h e S-matrices i s by f a r the most p r o b l e m a t i c o p e r a t i o n and takes about 957c of t h e t o t a l computer t i m e , and t h e r e f o r e , i s d i s c u s s e d i n d e t a i l i n t h i s s e c t i o n . A computed v e r s i o n o f the R - m a t r i x p r o p a g a t o r method t o s o l v e c o u p l e d d i f f e r e n t i a l e q u a t i o n s was w r i t t e n i n FORTRAN IV and developed on the IBM 370/168 computer at NUMAC by K I r k p a t r i c k ( 1 9 8 3 ) . T h i s i n t e g r a t i o n r o u t i n e I n c o r p o r a t e s t h e s t e p l e n g t h a l g o r i t h m as d i s c u s s e d i n Chapter I I I . The form o f the s e c t o r R-matrix was chosen t o be t h a t a p p r o p r i a t e t o a c o n s t a n t p o t e n t i a l across the s e c t o r ( i . e . I I I . 4 . 4 6 ) . The programme i s able t o e f f i c i e n t l y g e n e r a t e many e n e r g i e s and p a r t i a l waves. T h i s r o u t i n e has been checked by K i r k p a t r i c k (1983) by r e p r o d u c i n g t h e r e s u l t s o f SWL o b t a i n e d f o r the model atom - f o r c e d harmonic o s c i l l a t o r system d e s c r i b e d by S e c r e s t and Johnson ( 1 9 6 6 ) , and a l s o MOLSCAT was used t o v e r i f y t h e a l g o r i t h m f o r r o v i b r a t i o n a l IOS c a l c u l a t i o n s ( t h e system s t u d i e d was 1 ^ + He). I t was de c i d e d i n t h i s c a l c u l a t i o n t o use the a l -g o r i t h m w i t h a f i x e d s t e p l e n g t h . I t i s d i f f i c u l t t o assess t he p o s s i b l e advantage o f employing t h e v a r i a b l e - 12 3 -s t e p s i z e a l g o r i t h m i n t h i s c a l c u l a t i o n . K i r k p a t r i c k (1983) notes t h a t i t i s e x c e s s i v e l y dependent on the r a t e o f change of the p o t e n t i a l . I n c a l c u l a t i o n s employing the H2 + He p o t e n t i a l o f Gordon and Secrest ( 1 9 7 9 ) , K i r k p a t r i c k (1983) found t h a t a v a r i a b l e s t e p s i z e gave o n l y a 207o r e d u c t i o n i n computer t i m e over a f i x e d s t e p s i z e ( m a i n t a i n i n g accuracy t o 4 f i g u r e s ) . The GG p o t e n t i a l i s f a r more complex th a n t h a t of Gordon and Secrest (1979) (wh i c h i s a s m o o t h l y decaying e x p o n e n t i a l ) , and t h e r e f o r e the use o f the a l g o r i t h m might have r e s u l t e d i n a sub-s t a n t i a l r e d u c t i o n of computer t i m e . However, K i r k p a t r i c k (1983) a l s o r e p o r t s t h a t the a l g o r i t h m i s h i g h l y prone t o f a i l u r e . At one extreme the a l g o r i t h m a l l o w s l a r g e steps i n the a s y m p t o t i c r e g i o n , but a l s o f o r c e s the s t e p s i z e i n the c l a s s i c a l l y f o r b i d d e n r e g i o n t o be f a r s m a l l e r t h a n r e q u i r e d , w h i l e a t the o t h e r extreme i f the t o l e r a n c e s are r e l a x e d a l l o w i n g l a r g e r steps near R = 0, the l a r g e r s t e p s i z e s f u r t h e r out become too l a r g e t o m a i n t a i n accuracy. I t was d e c i d e d , t h e r e f o r e , t o employ a c o n s t a n t s t e p s i z e i n t h i s and i n a l l subsequent work (H2 + CO, HD t CO). I n a l l our c a l c u l a t i o n s , the s e c t o r R-matrices used were a p p r o p r i a t e t o a p p r o x i m a t i n g t h e elements o f the l o c a l l y d i a g o n a l m a t r i x as c o n s t a n t ( I I I . 4 . 4 6 ) . As d i s c u s s e d i n Chapter I I I . 4 . 4 t h e r e are advantages i n employing more complex s e c t o r R-matrices which are a p p r o p r i a t e t o a p p r o x i m a t i n g the elements of the l o c a l l y d i a g o n a l m a t r i x as f c o n s t a n t + L{L+1)/R^ J. Indeed the H 2 + H + s y s t em i s i d e a l l y s u i t e d as a t e s t case w i t h which 1 2 4 -t o I n v e s t i g a t e the e f f i c i e n c y o f Bessel f u n c t i o n p r o p a g a t o r s , i t s p o t e n t i a l s u r f a c e has a deep w e l l and i s long ranged. I n a d d i t i o n at t h e e n e r g i e s o f i n t e r e s t l a r g e numbers of p a r t i a l waves are r e q u i r e d ( ^ 2 0 0 a t E = lOeV, and g^, 1 0 0 a t E = 4.67eV). However, t h e aim of t h e present c a l c u l a t i o n i s t o check the computer programmes f o r use w i t h o t h e r systems, e.g. H2 + CO, and HD + CO, whose p o t e n t i a l s u r f a c e s do not warre n t t he use of such pro p -a g a t o r s so t h e r e f o r e any m o d i f i c a t i o n o f the e x i s t i n g r o u t i n e s was not c o n s i d e r e d w o r t h w h i l e . Both these schemes r e q u i r e t h a t the s t e p s i z e be s u f f i c i e n t l y s m a l l so t h a t t he d i a g o n a l i z a t i o n , performed a t the c e n t r e o f the s e c t o r , be a c c u r a t e over t h e e n t i r e w i d t h o f t h e s e c t o r . I n the form e r scheme i t must be s u f f i c i e n t l y s m a l l so t h a t t he c o u p l i n g m a t r i x ( i n c l u d i n g t he L ( L + 1 ) / R term) can be a c c u r a t e l y a p p r o x i m a t e d as c o n s t a n t w i t h i n t h e s e c t o r . The r a t e o f change of t h e term L ( L + 1 ) / R i n c r e a s e s w i t h L and t h e r e f o r e a s t e p s i z e which i s s u f f i c i e n t l y s m a l l t o a c c u r a t e l y c a l c u l a t e r e s u l t s f o r the h i g h e s t L v a l u e r e q u i r e d w i l l a l s o be s u f f i c i e n t f o r t h e lower v a l u e s . I t i s t h e r e f o r e , w o r t h w h i l e u s i n g a s m a l l e r s t e p s i z e t h a n r e q u i r e d at the f i r s t p a r t i a l waves. P r e l i m i n a r y c a l c u l a t i o n s were c a r r i e d out by K i r k p a t r i c k ( 1 9 8 3 ) f o r the H2 + H + system. He s t u d i e d t he system a t E = lOeV, and d e t e r m i n e d t h a t i n o r d e r t o m a i n t a i n I St, (0)^2. / ^ t ) / 2 ' a c c u r a t e t o t n r e e s i g n i f i c a n t f i g u r e s f o r Vji = 0 , 1 , 2 and 3 i t was necessary t o employ about 3 0 0 0 - 125 -steps o f f i x e d l e n g t h i n the range 0.15^R^"30a.U. Since we employ an approximate p o t e n t i a l a l g o r i t h m t h e s t e p s i z e should be e s s e n t i a l l y independent o f the c o l l i s i o n energy (Chapter 111. 2 ) , t h e r e f o r e t h a t d e t e r m i n e d at E = lOeV should be s a t i s f a c t o r y at E = 4.67eV. A f t e r c o n s i d e r a b l e e x p e r i m e n t a t i o n i t was found t h a t t o m a i n t a i n the same l e v e l o f accuracy t h a t t h e above i n t e g r a t i o n range and s t e p s i z e were s a t i s f a c t o r y . However, an e x c e p t i o n was t h e v i b r a t i o n a l l y e l a s t i c t r a n s i t i o n i n the v i b r a t i o n a l ground s t a t e ( f o r t h e l i n e a r o r i e n t a t i o n , cos &2 = 1 ) ; here t he l e a d i n g l o n g range a n i s o t r o p i c term i n the p o t e n t i a l s u r f a c e has maximum e f f e c t ( I V . 2 . 4 ) . Table 4 shows the convergence o f the p a r t i a l c r o ss s e c t i o n s , CTL (Vz'(yZ-0 VJ!^ 0 I » f o r t h e l i n e a r o r i e n -t a t i o n , $2 = 0J w i t h r e s p e c t t o the i n t e g r a t i o n c o o r d i n a t e , R. To o b t a i n these one o r i e n t a t i o n p a r t i a l c r o s s - s e c t i o n s the programme e v a l u a t e s t he p r o d u c t I //_ t h i s i s e q u i v a l e n t t o e v a l u a t i n g t h e i n t e g r a l ( c f . I I . 3 . 2 4 ) employing a one o r i e n t a t i o n q u a d r a t u r e w i t h t h e weight f u n c t i o n set equal t o one. The r e s u l t i n g one o r i e n t a t i o n c r o ss s e c t i o n s are p r o p o r t i o n a l t o I t i s c l e a r from Table 4 t h a t t h e convergence of (0&"^OojS^o) i s e x t r e m e l y slow, however, i t was not c o n s i d e r e d w o r t h w h i l e e x t e n d i n g t he i n t e g r a t i o n range i n t h e p r e s e n t c a l c u l a t i o n s as p h y s i c a l l y near- c o l l i n e a r c o l l i s i o n s have a s m a l l p r o b a b i l i t y o f a c t u a l l y o c c u r r i n g , TAHI.E 4 The convergence o f v i b r a t i o n a l l y e l a s t i c p a r t i a l c r o s s s e c t i o n s , (T^ (OO-^O 0 I &2~ i a t t n e ^ i - n e a r o r i e n t a t i o n , 0^ = 0. I n t e g r a t i o n range 0.15 ^ R X a.u. Cross s e c t i o n s i n 8^ . X(a.u.) L = 0 50 75 125 20 0. 21849600- 3 0. 13904351+0 0. 64909899+0 30 0. 21809306- 3 0. 13286757+0 0. 63955445+0 0 .92426370+0 40 0. 21855717- 3 0. 13054751+0 0. 63581842+0 0 .93338700+0 50 0. 12943995+0 0. 63397099+0 0 .93743308+0 TABLE 5 The approximate computer times r e q u i r e d f o r the s o l u t i o n o f the coupled e q u a t i o n s at one o r i e n t a t i o n u s i n g the Cray-1 computer, (a) I n i t i a l p a r t i a l wave, (b) Subsequent p a r t i a l waves. 7 v i b r a t i o n a l channels 2 v i b r a t i o n a l channels (a) 17.5 4.8 Time(s ) (b) 7.5 0.6 (3000 steps of f i x e d l e n g t h i n the range 0.154 R^ 30 a.u. ) 126 -and indeed e x a c t l y c o l l l n e a r c o l l i s i o n s have a zero p r o b a b i l i t y o f o c c u r r i n g due t o the s i n ^ w e i g h t i n g i n the i n t e g r a l f o r t h e S-matrix ( I V . 4 . 5 ) . I n Table 5 are the approximate computer times r e q u i r e d f o r t h e s o l u t i o n o f the coupled e q u a t i o n s at one o r i e n t a t i o n , u s i n g t h e Cray-1 computer, r e t a i n i n g 7 and 2 v i b r a t i o n a l c h a n n e l s . The times f o r the i n i t i a l and subsequent p a r t i a l waves are g i v e n . The coupled e q u a t i o n s are t o be s o l v e d at 14 o r i e n t a t i o n s i n these c a l c u l a t i o n s , and seven channels are t o be r e t a i n e d f o r 0 ^ L ^ > 100, so t h a t t h e i n c r e a s e i n computer time due t o an extended i n t e g r a t i o n range would have been u n a c c e p t a b l e . Once the e q u a t i o n s have been i n t e g r a t e d out t o the upper l i m i t o f the i n t e g r a t i o n range, t he code c o n t i n u e s outwards u n t i l t h e l a r g e s t r e l a t i v e change i n the S-m a t r i x elements i s l e s s than a g i v e n t o l e r a n c e . This ensures t h a t t h e i n t e g r a t i o n range employed i s s u f f i c i e n t . SM r e p o r t p l o t s o f jSL (o,v{ versus L at E = 4.67eV f o r © 2 = 0°, 51.43°, and 90°, and = 1 and 2. Our p r e l i m i n a r y c a l c u l a t i o n s i n v o l v e d r e p r o d u c i n g these r e s u l t s t o check t h a t our d e s c r i p t i o n o f the system was adequate and t o f i n d programming e r r o r s b e f o r e u n d e r t a k i n g t h e f u l l s c a t t e r i n g c a l c u l a t i o n s . The r e s u l t s o f SM are shown i n F i g u r e 3, t o g e t h e r w i t h t h e r e s u l t o b t a i n e d u s i n g t he MOEX w a v e f u n c t i o n s . For a l l t h r e e o r i e n t a t i o n s and f o r = 1, and 2, the o v e r a l l agreement i s s a t i s f a c t o r y . The d i s c r e p a n c i e s are seen t o i n c r e a s e w i t h y{_ and w i t h d e c r e a s i n g L. Here the d i f f e r e n c e s i n the p o t e n t i a l m a t r i x (I>) (a) 0-5 0-4 8- 90 0-3 0 2 0 1 0-4 51.43 03 02 0-1 0 ' 0-4 0-3 0-1 20 80 20 40 40 60 60 80 0 L F i g u r e 3 T r a n s i t i o n p r o b a b i l i t i e s f o r v i b r a t i o n a l e x c i t a t i o n J &L ' v e r s u s p a r t i a l wave I. at K 4 . 6 7 eV f o r t h r e e o r i e n t at i o n s $ 2 • F u l l curves : Present c a l c u l a t i o n . Broken curves : SM. Column (a) v 2 = 1, column (b) v 2 = 2 . \SL(0)V^\Q^)\L in a.u. - 127 -elements due t o the use o f a 6 4 - p o i n t Gauss Leguerre q u a d r a t u r e r a t h e r than a 2 8 - p o i n t Gauss Legendre q u a d r a t u r e a r e h a v i n g g r e a t e s t e f f e c t , s i n c e the p o t e n t i a l i s more i m p o r t a n t at these low L v a l u e s . The e x c e p t i o n t o t h i s r u l e i s the p l o t o f / SL (o, I / © i ) / 2 f o r B2 = 0°, where l a r g e d i s c r e p a n c i e s are seen at l a r g e L v a l u e s . T h i s poor agreement would seem t o i n d i c a t e e i t h e r the presence of a programming e r r o r , which i s becoming more apparent as L i n c r e a s e s , or the d e t e r i o r a t i o n o f our n u m e r i c a l accuracy w i t h L. The s a t i s f a c t o r y agreement at l a r g e L f o r the o t h e r two o r i e n t a t i o n s would appear t o r u l e out b o t h the above p o s s i b i l i t i e s ( a l s o n o t i c e t he s a t i s f a c t o r y agreement at B2 = 0° f o r L ^ 8 0 ) . Schinke (1980) has shown t h a t p a r t i a l c r o s s s e c t i o n s w i t h L ^ 75 are e x t r e m e l y s e n s i t i v e t o the form o f the i n t e r a c t i o n p o t e n t i a l . Schinke p r e s e n t s a p l o t o f v i b r a -t i o n a l l y e l a s t i c p a r t i a l c r oss s e c t i o n s , t h e ?2 (cos 62^ p o l y n o m i a l has i t s maximum v a l u e , and hence a t t h i s o r i e n t a t i o n t h e ?2 a n i s o t r o p y w i l l have maximum e f f e c t . I n view o f the reasonable agreemnt o b t a i n e d at the va l u e s o f © 2 s t u d i e d we c o n s i d e r t h a t our d e s c r i p t i o n o f the system and our n u m e r i c a l methods are a c c u r a t e . 4.3 The Angular Quadrature Once the o r i e n t a t i o n S-mat r i c e s have been o b t a i n e d i t i s necessary t o i n t e g r a t e them over o r i e n t a t i o n t o o b t a i n t he body f i x e d S-matrix ( I I . 3 . 2 6 ) . The o r i e n t a t i o n S-matrices are c a l c u l a t e d at the p o i n t s r e l e v a n t t o t h e q u a d r a t u r e scheme used. I n t h i s c a l c u l a t i o n we o n l y need c o n s i d e r i n t e g r a l s a p p r o p r i a t e t o the e v a l u a t i o n o f (T (v^O — 7 > Vij^) • A l l o t h e r c r o s s s e c t i o n s can be d e r i v e d employing ( I I . 3 . 1 7 ) . ' &vJt,)J'-L [000/ IV.4.6 To c a l c u l a t e a l l cross s e c t i o n s i n v o l v i n g s t a t e s up t o j / r m x w e r e c l u i r e a 1 1 ^ 6 4 0 -f^Z/l'^x) because of the p r o p e r t i e s o f the t h r e e j symbols, f o r a complete summation. The e v a l u a t i o n o f i,}> IV.4.11 I-1 where 2 , W = 0,2 N* , N* £ 2(N - 1 ) . ( ^ 2 * 1 = ^° a n <^ ^ ^2^N = because o n l y Legendre p o l y -nomials o f even o r d e r are i n c l u d e d due t o the symmetry about 9 0 3 . Two advantages of t h i s q u a d r a t u r e scheme are t h a t t h e expansion o r d e r may be e n l a r g e d by adding more o r i e n t a t i o n s ( ^ 2 ' i w i - t n o u t : n e g l e c t i n g the p r e v i o u s l y c a l c u l a t e d v a l u e s , a l s o the values o f fy^ chosen can be c o n c e n t r a t e d i n the r e g i o n where t h e i n t e g r a n d i s v a r y i n g most r a p i d l y . However SM r e p o r t t h a t t h i s scheme does not a l l o w a r e d u c t i o n i n the number of o r i e n t a t i o n angles - 131 -© 2 t h a t must be c o n s i d e r e d i n the i n t e g r a n d IV.4.6. The computer programmes w r i t t e n t o p e r f o r m these c a l c u l a t i o n s employ the more usual s c a t t e r i n g f o r m a l i s m , and c o n s e q u e n t l y we p r e f e r t o use Gauss I.egendre q u a d r a t u r e . I n t h e i r paper SM compare t h e i r 1 2 - p o i n t q u a d r a t u r e w i t h a 1 4 - p o i n t Gauss Legendre q u a d r a t u r e ( t h e y do not s t a t e which type t h i s i s ) . The comparison i s made by examining r i g i d r o t o r d i f f e r e n t i a l c r o ss s e c t i o n s f o r t h e r o t a t i o n a l t r a n s i t i o n s j = 1=¥ 5 and j = 1=^7 (SM are not i n t e r e s t e d i n t r a n s i t i o n s f o r w h i c h A j > 6 , as t h e r e i s no e x p e r i m e n t a l d a t a f o r such t r a n s i t i o n s ) . The l e v e l o f agreement between the two q u a d r a t u r e s f o r the j = 1-W t r a n s i t i o n i n d i c a t e s t h a t t he use o f a 1 4 - p o i n t Gauss Legendre q u a d r a t u r e r a t h e r t han the 1 2 - p o i n t q u a d r a t u r e w i l l not obscure the comparison between the v i b r a t i o n a l l y e l a s t i c cross s e c t i o n s , however SM do not compare these q u a d r a t u r e s f o r the v i b r a t i o n a l l y i n e l a s t i c c r o s s s e c t i o n s . I n F i g u r e 4 we show the v a r i a t i o n o f the r e a l and i m a g i n a r y p a r t s o f SL(o)V^/ 6 ^ ) f o r L = 25 and = 0,1, 2, and 3. Values o f SL(o}V-l IQ%) have been c a l c u l a t e d at 20 e q u i d i s t a n t v a l u e s o f I t I S c l e a r t h a t at t h i s low v a l u e o f L, where v i b r a t i o n a l e x c i t a t i o n i s i m p o r t a n t , t h a t f o r the H2 + H + system the v i b r a t i o n a 1 l y i n e l a s t i c S -matrix elements become s t r o n g e r f u n c t i o n s o f ^ a s v/ i n c r e a s e s . T h e r e f o r e i t i s d i f f i c u l t t o know, a p r i o r i , j u s t how the use o f the Gaussian scheme r a t h e r than t h a t used by SM w i l l a f f e c t the l e v e l o f agreement between r 0 8 v v 0-2 02 0-4 0-6 0 8 1 0 0 0 0 2 0-4 0-6 0-8 1 0 1 2 K 0 0 02 04 0 6 08 10 12 1 e F i g u r e 4 The v a r i a t i o n of (^\^0 > /O^,^ a s a f u n c t i o n o f &• 2 f o r L = 25 and v/ = 0,1,2,3 at E = 4.67 eV. F u l l curves : Real p a r t o f S-matrix; broken curves i m a g i n a r y p a r t of S-matrix. S-matrices i n a.u., 0"^ i n r a d i a n s . rz t h e v i b r a t i o n a l L y i n e l a s t i c c r oss s e c t i o n s . I t i s hoped t h a t a d i f f e r e n c e o f two q u a d r a t u r e p o i n t s w i l l not have a s i g n i f i c a n t e f f e c t , even f o r K i r k p a t r i c k (1983) has compared the e f f i c i e n c y o f the two types o f Gaussian q u a d r a t u r e s d e s c r i b e d e a r l i e r , w i t h r e g a r d t o the ^ + He system, and r e p o r t s t h a t i t i s more e f f i c i e n t t o employ h a l f the p o i n t s o f a 2N q u a d r a t u r e For H2 + He the S-matrix elements are smoothly v a r y i n g f u n c t i o n s o f ©2 (even at L = 0 ) , however i n sharp c o n t r a s t , f o r H2 + H+ the S-matrix elements are h i g h l y o s c i l l a t o r y f u n c t i o n s o f the o r i e n t a t i o n angle ^ > f ° r L ^ 2 5 (as i l l u s t r a t e d i n F i g u r e 4 ) . So at f i r s t s i g h t the work of K i r k p a t r i c k may not appear r e l e v a n t . However one may show t h a t u s i n g h a l f t he p o i n t s o f a 2N q u a d r a t u r e i s more e f f i c i e n t by c o n s i d e r i n g the e r r o r f u n c t i o n s o f the two types o f Gauss Legendre q u a d r a t u r e s . I t can be shown t h a t i f i s c o n t i n u o u s f o r -1.«-3c. ) ' J 3 / / ) - - 2N IV.4.12a which decreases r a p i d l y as 2N i n c r e a s e s . For the Gaussian r u l e ( I V . 3 . 9 b ) , i t may be shown t h a t f o r an N p o i n t q u a d r a t u r e t h e e r r o r i s g i v e n by (Abramowitz and Stegun) A=f\r IV.4.12b - 133 -where b = 1, and a = 0. For a g i v e n v a l u e o f N the e r r o r i n u s i n g h a l f the p o i n t s o f a 2N q u a d r a t u r e i s f a r l e s s than t he e r r o r i n v o l v e d i n u s i n g the a d j u s t e d p o i n t s and w e i g h t s o f an N-point q u a d r a t u r e . T h e r e f o r e we choose t o employ the former scheme, and employ 1 4 - p o i n t from a 2 8 - p o i n t q u a d r a t u r e . 5. R e s u l t s and d i s c u s s i o n I n Table 6 we present t he i n t e g r a l c r oss s e c t i o n s f o r the v i b r a t i o n a l l y e l a s t i c p r ocess, f o r b o t h para ~^2 = ^ a n <^ o r t h o - l ^ J i , = 1 ) - The agreement between the r e s u l t s o f the present c a l c u l a t i o n and the r e s u l t s of SM i s s a t i s f a c t o r y f o r a l l r o t a t i o n a l t r a n s i t i o n s except J x = 0->2 and J v = 1—^3. The range o f p a r t i a l waves (L-^175) employed i n the c a l c u l a t i o n was i n s u f f i c i e n t t o converge the i n t e g r a l cross s e c t i o n f o r t h e = 0->2 t r a n s i t i o n (and c o n s e q u e n t l y the i n t e g r a l c r oss s e c t i o n f o r the = l - > 3 t r a n s i t i o n which i s e s s e n t i a l l y d e t e r m i n e d by ^(0O=>oi)) • As p o i n t e d out by Schinke and McGuire (1978a) the sudden a p p r o x i m a t i o n o v e r e s t i m a t e s the l o n g range p a r t o f the A j 2 = 2 t r a n s i t i o n s caused by the charge-quadrupole i n t e r a c t i o n . As d i s c u s s e d p r e v i o u s l y (Chapter I I . 3.2) the sudden a p p r o x i m a t i o n r e q u i r e s t h a t the r o t a t i o n a l p e r i o d o f the t a r g e t be l a r g e compared t o t he c o l l i s i o n t i m e (which i s p r o p o r t i o n a l t o the range of t he i n t e r a c t i o n p o t e n t i a l ) . The charge-quadrupole i n t e r a c t i o n has the e f f e c t o f i n c r e a s i n g the e f f e c t i v e c o l l i s i o n t i m e and making the IOS a p p r o x i m a t i o n i n a p p l i c a b l e TABLE 6 V i b r a t i o n a l l y e l a s t i c i n t e g r a l cross s e c t i o n s 0- (OJi Ojl) i n S2 at E = 4.6 7eV. (a) Schinke and McGuire ( 1978b). (b) Present c a l c u l a t i o n . ( c ) R i g i d r o t o r coupled s t a t e s r e s u l t s , McGuire (1976) JI p a r a - H 2 ( j x = 0) : 0 (a) 67.7 (b) 65.25 (c ) 61.9 2 14.0 8. 72 + 2.34* 8.10 4 3.04 2.88 1.50 6 1.27 1.20 0.614 8 0. 567 0.481 0.443 10 0.293 0. 26 3 0. 3 70 ortho-H2 (^ 2, = 1) : 1 73.3 67 .69 64.2 3 9.77 7.35 4.97 5 2.28 2.16 1.25 7 0.938 0.874 0.610 " E s t i m a t e d c o n t r i b u t i o n f o r L > 175 134 f o r £ty% 2. I r.ins I I. I on;! . I'or SM show ( l i . i l o n l y the ?2 a n i s o t r o p y o f the GG p o t e n t i a l s u r f a c e s u r v i v e s , t h e r e f o r e h i g h e r r o t a t i o n a l t r a n s i t i o n s ( e . g . = 0-^4, 0-^6 ...) must be the r e s u l t o f m u l t i p l e ^J-j. = 2 t r a n s i t i o n s i n t h i s r e g i o n . The o v e r e s t i m a t i o n o f the r o t a t i o n a l t r a n s -i t i o n s i n v o l v i n g hj% = 2 i s c l e a r l y seen when one compares th e r e s u l t s o f SM ( o r the p r e s e n t c a l c u l a t i o n ) w i t h t h e c o u p l e d - s t a t e s r i g i d r o t o r c r oss s e c t i o n s o f McGuire (1976) which are a l s o shown i n Table 6. As Schinke and McGuire (1978a) p o i n t out the above e f f e c t i s more severe f o r J j . = 0-^2, and c o n s e q u e n t l y t h i s i s the o n l y unconverged t r a n s i t i o n f o r para - l ^ . The c o n t r i b u t i o n t o the i n t e g r a l c r oss s e c t i o n f o r th e J^. = 0~^2 t r a n s i t i o n , from p a r t i a l cross s e c t i o n s w i t h L *> 175 was e s t i m a t e d i n the f o l l o w i n g manner. The p a r t i a l c r oss s e c t i o n s G~L(oO-}oi) f o r 150.$ 175, were p l o t t e d versus L. I n t h i s r e g i o n t he v a r i a t i o n w i t h r e s p e c t t o L i s e s s e n t i a l l y l i n e a r and may be r e p r e s e n t e d by \ 7 5 . I t i s w o r t h m e n t i o n i n g t h a t Schinke (1980) has shown t h a t the rainbow maxima i n the = 1—>6 and J x = 1-^7 d i f f e r e n t i a l c r o s s s e c t i o n s are caused by the P2 term o f t he i n t e r a c t i o n p o t e n t i a l ( i . e . he has shown t h a t the rainbow maxima r e s u l t from p a r t i a l waves L^,55). These rainbow maxima are s p u r i o u s , and are reduced when the more a c c u r a t e p o t e n t i a l s u r f a c e o f Schinke e t a l . (1980) i s employed. The p r e s e n t c a l c u l a t i o n s c o n f i r m t h a t t h e maxima are due t o i n a c c u r a c i e s i n the GG p o t e n t i a l and t he form the w a v e f u n c t i o n s have no e f f e c t whatsoever on t he maxima. This i s i n c o n t r a s t t o K i r k p a t r i c k (1983) who p o s t u l a t e d t h a t the maxima might be due t o the " i n a c c u r a t e " HOEX w a v e f u n c t i o n s . 10 1=*1 i 1 10 d - f l 10 1 10" 10 r i I 1~*>5 1-W J2 J2 1 10 10 1 10 10 1 1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 (degrees) F i g u r e 5 V i b r a t i o n a l l y e l a s t i c d i f f e r e n t i a l c r o ss s e c t i o n s ^ ^ %"^) w i t h the r e s u l t s o f SM ( ) . Cross s e c t i o n s i n 136 R o v i b r a t i o n a l c ross s e c t i o n s f o r t h e v i b r a t i o n a l l y i n e l a s t i c process = 0 — > 1 , 2 , 3 are p r e s e n t e d i n Table 7, The agreement between the r e s u l t s o f the present c a l c u l a t i o n s and those o f SM i s s a t i s f a c t o r y . The o n l y e x c e p t i o n i s the c r o s s s e c t i o n f o r the process = 0 J x = 0 = 3 j£ = 2 . T h i s i s most l i k e l y due t o a t y p o g r a p h i c a l e r r o r i n the paper o f SM, s i n c e f o r a l l o t h e r g i v e n r o t a t i o n a l t r a n s i t i o n s the cross s e c t i o n s decrease m o n o t o n i c a l l y w i t h i n c r e a s i n g V^. I t appears t h a t n e i t h e r the use o f MOEX w a v e f u n c t i o n s r a t h e r than HOEX w a v e f u n c t i o n s , or t h e use of a Gaussian q u a d r a t u r e r a t h e r t h a n t h e 1 2 - p o i n t e q u i d i s t a n t q u a d r a t u r e o f SM have a s i g n i f i c a n t e f f e c t on the n u m e r i c a l v a lues o f the v i b r a t i o n a l y i n e l a s t i c c r oss s e c t i o n . However, g e n e r a l l y speaking i n t e g r a l cross s e c t i o n s are not a p a r t i c u l a r l y s e n s i t i v e measure of the phase o f the S-m a t r i x elements, and hence the q u a d r a t u r e employed i n a c a l c u l a t i o n , a f a r more r i g o r o u s t e s t of d i f f e r e n t q u a d r a t u r e schemes are d i f f e r e n t i a l c r o ss s e c t i o n s (e.g. see P f e f f e r and Secrest (1977)). I t i s p r o b a b l e t h a t i f the d i f f e r e n t i a l c r o ss s e c t i o n s f o r the t r a n s i t i o n s g i v e n i n t a b l e 7 were compared t h a t the agreement would not be as s a t i s f a c t o r y ( e s p e c i a l l y i n the h i g h e r angle r e g i o n $ ^ 4 0 ° , where the cross s e c t i o n s are d e t e r m i n e d e s s e n t i a l l y by the lower p a r t i a l waves). D e s p i t e the d i f f e r e n t n u m e r i c a l methods employed i n t he p r e s e n t c a l c u l a t i o n s and by SM t h e o v e r a l l agreement between a l l the r e s u l t s p r e s e n t e d i s s a t i s f a c t o r y . I t TABLE 7 V i b r a t i o n a l l y i n e l a s t i c i n t e g r a l c r o s s s e c t i o n s G*(°J*-'**v£jz)in ^ 2 a t E = 4-6 7eV. (a) Schinke and McGuire (1978b). (b) Present c a l c u l a t i o n . v 2 = 1 2 3 para-H2 (J^ = 0) : 0 (a) 0 .107 0.027 0.025 (b) 0 .110 0.031 0.019 2 0 .188 0.048 0.057 0 .183 0.048 0.020 4 0 .232 0.058 0.025 0 .224 0.056 0.028 0 .215 0.0 54 0.026 0 . 204 0.06 2 0.02 7 8 0 .136 0.043 0.026 0 .133 0.060 0.026 10 0 .072 0.034 0.017 0 .081 0.042 0.020 o r t h o - H 2 ( = 1 ) : 1 0 .183 0.046 0.036 0 .184 0.051 0.028 3 0 . 216 0.054 0.027 0 . 186 0.054 0.025 5 0 . 228 0.057 0.026 0 .219 0.060 0.028 7 0 .180 0.049 0.027 0 .173 0.062 0.027 13 7 -i s e v i d e n t t h a t the use of the MOEX w a v e f u n c t i o n s r a t h e r t h a n the HOEX w a v e f u n c t i o n s does not s i g n i f i c a n t l y a f f e c t the r e s u l t s o f t h e s c a t t e r i n g c a l c u l a t i o n (as p o s t u l a t e d i n S e c t i o n 3 ) . T h i s i s not p a r t i c u l a r l y s u r p r i s i n g as the w a v e f u n c t i o n s are q u a l i t a t i v e l y v e r y s i m i l a r , t h i s however, i s i n sharp c o n t r a s t w i t h t h e c o n c l u s i o n s o f K i r k p a t r i c k ( 1 9 8 3 ) , who p o s t u l a t e s t h a t a l l the cross s e c t i o n s ( b o t h v i b r a t i o n a I Iy e l a s t i c and i n e l a s t i c ) would be s i g n i f i c a n t l y a f f e c t e d by the use of "improved" wave-f u n c t i o n s . He s t u d i e s f i x e d angle o r i e n t a t i o n S-matrices at L = 25, 50, 75 and 100 a t E =DeV, c a l c u l a t e d u s i n g b o t h the Morse and HOEX o s c i l l a t o r w a v e f u n c t i o n s . He admits t h a t h i s c a l c u l a t i o n s are by no means c o n c l u s i v e , f o r example, i t i s not obvious what e f f e c t d i f f e r e n c e s i n f i x e d angle S-matrices w i l l have on cross s e c t i o n s , and i t i s w o r t h n o t i n g t h a t h i s s tudy i s at E - 1OeV not 4.6 7eV. The c o n c l u s i o n s he draws are not d i r e c t l y a p p l i c a b l e t o t h i s c a l c u l a t i o n , f o r example v i b r a t i o n a l e x c i t a t i o n i s more i m p o r t a n t at E = lOeV. F o r t u n a t e l y the d i f f e r e n t q u a d r a t u r e schemes used i n the two c a l c u l a t i o n s have not obscured the comparison. I f , as mentioned above, d i f f e r e n t i a l cross s e c t i o n s had been compared^the d i f f e r e n c e s i n t h e two qu a d r a t u r e s would p r o b a b l y have been more app a r e n t . There i s no guarantee t h a t t h e 12 p o i n t q u a d r a t u r e employed by SM i s s u f f i c i e n t t o converge any of the cross s e c t i o n s . S M i n d i c a t e t h a t t h i s q u a d r a t u r e was not c o n s i d e r e d - 1 38 -c o m p l e t e l y s a t i s f a c t o r y but was the maximum p r a c t i c a b l e due t o the l a r g e amount of computer time r e q u i r e d f o r the c a l c u l a t i o n s . I t i s p r o b a b l e t h a t t w e l v e o r i e n t a t i o n s w i l l be i n s u f f i c i e n t t o g i v e an a c c u r a t e i n t e g r a t i o n o f the S-matrix elements over f o r L ^ 25, where the f i x e d angle S-matrices are h i g h l y o s c i l l a t o r y f u n c t i o n s o f the o r i e n t a t i o n a n g l e , . I n c o n c l u s i o n , i n view of the l e v e l of agreement between our own c a l c u l a t i o n s and those of SM we b e l i e v e t h a t we can be r e a sonably sure t h a t our IOS programmes are w o r k i n g c o r r e c t l y . 139 APPENDIX The Use of A s y m p t o t i c S e r i e s Expansions i n Heavy p a r t i c l e Scat t e r i n t The t h e o r y o f a s y m p t o t i c s e r i e s expansions was d i s -cussed i n Chapter I I I . 4 . 3 . I n t h i s appendix we s h a l l present the r e s u l t o f c a l c u l a t i o n s i l l u s t r a t i n g the ad-vantages of a s y m p t o t i c s e r i e s expansions of the type i n t r o d u c e d by Gai. l i t i s (1 976) over the c o n v e n t i o n a l method of f i t t i n g t h e boundary c o n d i t i o n ( c f . I I . 2 . 2 7 b ) . + l £ p J IV.A.1 E q u a t i o n IV.A.1 may be employed once t h e i n t e r a c t i o n p o t e n t i a l has become n e g l i g i b l e , however i n the case of \\. i H1 i t i s necessary t o i n t e g r a t e the coupled e q u a t i o n s out t o e x t r e m e l y l a r g e H values b e f o r e the boundary con-d i t i o n s can be f i t t e d . The i n t e r a c t i o n p o t e n t i a l c o n t a i n s l o n g range i s o t r o p i c , and a n i s o t r o p i c components due t o t h e charge on the p r o t o n . The charge-induced d i p o l e c o n t r i b u t i o n ( I V . 2 . 3 ) and the charge-quadrupole c o n t r i b u t i o n (IV.2.4 IV.A.2 3 - ^ ^ O ^ - 3 IV.A.3 - 140 -As d i s c u s s e d i n Chapter I I I .4. 3 i I a I I the long range terms i n the i n t e r a c t i o n p o t e n t i a l are of the form 1/Rn, where n i s an i n t e g e r , i t i s p o s s i b l e t o i n t e g r a t e s u f f i c i e n t l y out i n t o the a s y m p t o t i c r e g i o n t o where a l l t h e s h o r t range terms i n the p o t e n t i a l have become n e g l i g i b l e . One then f i n d s a complete set of a s y m p t o t i c s o l u t i o n s f o r the l o n g range p o t e n t i a l and matches these t o the s o l u t i o n t o o b t a i n the K - m a t r i x . I n t h i s case the c oupled d i f f e r e n t i a l e q u a t i o n s have the a s y m p t o t i c form ( c f . I J I .4 . 30 ) . n > 3 . iv.A.4 The a s y m p t o t i c s o l u t i o n f o r t h e l o n g range p o t e n t i a l has a l r e a d y been s u c c e s s f u l l y used i n heavy p a r t i c l e s c a t t e r i n g ( P f e f f e r and Secrest ( 1 9 8 3 ) , G e r r a t t and Roberts ( 1 9 8 3 ) ) . Both these c a l c u l a t i o n s employed a s y m p t o t i c s e r i e s expansions (ASE) o f the type i n t r o -duced by Secrest (1979) ( c f . E I A . 3 1 ) . 9 CO IV.A.5 The s e r i e s expansions f o r a(R) and b(R) are found by s u b s t i t u t i n g f ( R ) or g(R) i n t o IV.A.4. P f e f f e r and Secrest (1983) have s t u d i e d the ^ + L i + system whose p o t e n t i a l s u r f a c e c o n t a i n s l o n g range terms which decay as 1/R 3, 1/R^, and VR5. Using the above - 141 -t e c h n i q u e P f e f f e r and Secrest (1983) found t h a t I t was p o s s i b l e t o s t o p the i n t e g r a t i o n a f t e r i n t e g r a t i n g o n l y one t e n t h of the range r e q u i r e d t o a r r i v e at the r e g i o n i n which they c o u l d t e r m i n a t e u s i n g R i c c a t i Bessel f u n c t i o n s . G e r r a t t ( p r i v a t e communication) r e p o r t s t h a t these f u n c t i o n s have been s u c c e s s f u l l y employed i n a study o f v i b r a t i o n a l e x c i t a t i o n i n atom-diatom c o l l i s i o n s ( G e r r a t t and Roberts ( 1 9 8 3 ) ) . The R-matrix method (B o c c h e t t a (1983)) was employed i n a r e g i o n 0-^R^R^ (R^ was d e t e r m i n e d by the requirement t h a t o n l y the van der Waals (^/R^) c o n t r i b u t i o n t o the p o t e n t i a l should be s i g n i f i c a n t f o r R >R^). The use o f these f u n c t i o n s e l i m i n a t e d the need t o i n t e g r a t e the c o u p l e d d i f f e r e n t i a l e q u a t i o n s from R = R^ t o a p o i n t where the van der Waals p o t e n t i a l had become n e g l i g i b l y s m a l l . We p r e f e r t o employ ASE o f the type i n t r o d u c e d by G a i l i t i s (1976) ( i l l . 4 . 3 2 ) . ]^(kfC) J Wi (kfC) . IV.A.6 As mentioned i n Chapter I I I . 4 these f u n c t i o n s are p r e f e r -a b l e t o IV.A.5 i n t h a t the e f f e c t of the l o n g range term L(L + 1)/R i s a l r e a d y i n c o r p o r a t e d i n the S p h e r i c a l Bessel f u n c t i o n s . I n t h i s study we r e t a i n the l e a d i n g l o n g range term of the + H + p o t e n t i a l ( I V . A . 3 ) , and t h e r e f o r e the S c h r o d i n g e r e q u a t i o n takes the a s y m p t o t i c form I dR WR3 / * i IV.A.7 p(fO 142 who rr The c a l c u l a t i o n s are e s s e n t i a l l y a c o n t i n u a t i o n o f the study p r e s e n t e d i n the main body o f t h i s c h a p t e r , and t h e r e f o r e the n u m e r i c a l d e t a i l s developed f o r t h a t c a l c u l a t i o n are employed i n the p r e s e n t work. I n t h i s s t u d y we are concerned w i t h the e v a l u a t i o n o f f i x e d o r i e n t a t i o n p a r t i a l c r o ss sect ions , C*~ C^zji^ ^\f^ / ^ 2. ) ' i t w i l l be r e c a l l e d t h a t the convergence o f the v i b r a -t i o n a l l y e l a s t i c c r o s s s e c t i o n s i n the v i b r a t i o n a l ground stair.' was cx( roino I y slow, e s p e c i a l l y n l I ho col l i n e a r o r i e n t a t i o n (Table 4, S e c t i o n 4.2). T h e r e f o r e we examine i n g r e a t e r d e t a i l , and use the r e s u l t s i n o r d e r t o i l l u s t r a t e t h e advantages o f the ASE over the c o n v e n t i o n a l Bessel f u n c t i o n s . To c a l c u l a t e the s e r i e s A(R) and B(R) and t h e i r d e r i v a t i v e s , a r o u t i n e was w r i t t e n i n FORTRAN IV and developed on the IBM 370/168 computer at NUMAC. Thi s r o u t i n e i n c o r p o r a t e s a c a p a b i l i t y t o m o n i t o r the behaviour of t h e terms of b o t h s e r i e s . The s e r i e s we are d e a l i n g w i t h are a s y m p t o t i c , t h a t i s t o say, the s e r i e s g i v e n by A(R) and B(R) do not converge. As more and more terms are added t o the s e r i e s , the added terms become s m a l l e r and the s e r i e s approach the c o r r e c t s o l u t i o n up t o a p o i n t . Beyond t h a t p o i n t the added terms begin t o get l a r g e r , and the s e r i e s d i v e r g e . Thus i t i s i m p o r t a n t t o s t o p computing terms i n the s e r i e s at the p r o p e r p l a c e . the convergence of such cross s e c t i o n s , a1- f^O ~r oo 143 -The programme stops adding terms t o each s e r i e s i f the terms o f e i t h e r A(R) andB(R) s t a r t t o d i v e r g e . This i s s u f f i c i e n t t o ensure t h a t the f u n c t i o n s IV.A.6 and t h e i r d e r i v a t i v e s are convergent (e.g. see K r e y s z i g ( 1 9 7 2 ) ) . I t i s d i f f i c u l t t o know j u s t how a c c u r a t e the s o l u t i o n i s at t h i s p o i n t , when e q u a t i o n IV.A.7 i s s o l v e d one o b t a i n s an R-matrix o f q u e s t i o n a b l e accuracy. I f however, t h i s p o i n t i s not reached i t i s not necessary t o c o n t i n u e adding terms which do not form a s i g n i f i c a n t c o n t r i b u t i o n t o the s e r i e s , and t h e r e f o r e t h e s e r i e s must be t r u n c a t e d at s u i t a b l e p o i n t s . The r e l a t i v e i m p ortance o f each term i s det e r m i n e d by (e.g. f o r B(R)) TesT> / J ^ — / , / SOW I ' IV.A.8 where b i s the c u r r e n t ( n t h ) term o f the s e r i e s , and n ' SUM ^ (&m • "TEST" i s an i n p u t t o l e r a n c e f h » o whi ch must be s p e c i f i e d . I f the quotient on t h e r i g h t h a n d s i d e o f IV.A.8 i s l e s s than TEST, then t h e s e r i e s i s t e r m i n a t e d . I n the p r e s e n t c a l c u l a t i o n s the for m e r c o n d i t i o n was never reached, and the l a t t e r t e s t was used t o t r u n c a t e the s e r i e s . Table 8 shows v a l u e s o f e v a l u a t ed at E = 4.67 eV, and at L = 0 and 5 w i t h ©2 = 0, u s i n g d i f f e r e n t v a l u e s o f TEST. As can be seen t he agreement -9 -10 between r e s u l t s e v a l u a t e d u s i n g TEST = 1 0 , 1 0 , and -11 10 i s e x c e l l e n t . I n c a l c u l a t i o n s employing the IOS TABLE 8 A comparison o f at 8 2 = O'eval uated u s i n g d i f f e r e n t v a l u e s o f the parameter TEST at L = 0 and 5. !a) 10 " 9 , (b) 1 0 ~ 1 0 , ( c ) 1 0 " 1 1 L = 0 5 (a) 0.37304935 0.31158844 (b) c) ^3000 steps i n the range 0.15 ^ R 4 3 0 a.u. 144 -a p p r o x i m a t i o n i t i s s u f f i c i e n t t o e v a l u a t e r e s u l t s c o r r e c t t o no more than t h r e e or f o u r s i g n i f i c a n t f i g u r e s . E v i d e n t l y TEST c o u l d have been r e l a x e d t o m a i n t a i n t h i s l e v e l o f accuracy, however, i t was found t h a t v a r y i n g t h e v a l u e o f TEST had an i n s i g n i f i c a n t e f f e c t on the t o t a l computer t i m e , t h e vast m a j o r i t y o f which i s used _q i n t h e R-matrix r o u t i n e . A v a l u e o f TEST = 10 was employed i n a l l these c a l c u l a t i o n s . I n Table 9 we compare the convergence, w i t h r e s p e c t t o the i n t e g r a t i o n c o o r d i n a t e R, o f (T f o r L = 0, 5 and 10 at E = 4.67eV, e v a l u a t e d u s i n g f i t s t o the f u n c t i o n s IV.A.6 or t o S p h e r i c a l Bessel f u n c t i o n s i n the a s y m p t o t i c r e g i o n . I t i s apparent from Table 9 t h a t the ASE r e s u l t s are 'more or l e s s ' converged a t R£: 40a.u. , whereas the S p h e r i c a l Bessel f u n c t i o n r e s u l t s are not - even at R = 140a.u. G e r r a t t ( p r i v a t e com-m u n i c a t i o n ) r e p o r t s t h a t he o b t a i n e d s i m i l a r r e s u l t s i n h i s study o f v i b r a t i o n a l e x c i t a t i o n i n atom-diatom c o l l i s i o n s . One w i l l r e c a l l however, t h a t such c o l l i n e a r c o l l i s i o n s have a zero p r o b a b i l i t y o f o c c u r r i n g ( t h e r e s u l t s p r e s e n t e d i n Table 9 are p u r e l y f o r the purpose o f i l l u s t r a t i o n ) . N e v e r t h e l e s s t he r e s u l t s o b t a i n e d f o r near c o l l i n e a r c o l l i s i o n s (which are not p a r t i c u l a r l y i m p o r t a n t , but do c o n t r i b u t e t o the f i n a l r e s u l t s ) w i l l be very s i m i l a r i n bot h t h e i r magnitude and convergence p r o p e r t i e s . I n the K2 f H + c a l c u l a t i o n s p r e s e n t e d i n t h i s c h a p t e r ON UJ PQ < O II M o o * o o - t to c O •1-1 J_J U 0) 0) to tn O o C o o >-. CO X3 C D O PQ O ( /) C o •1-1 4-J u c • 4-1 U4 CO 1-1 < OJ — (/) 10 c/) C O • i - i to C CO C L X 1—1 \JD CO • •r-l • i - l U 4-J M-l O O J - J a w e E > , r-l t/) 0 ) CO 4-J (4-J c o •1-1 ' 0 0 ) If) o a e 1—1 1 m 1-1 1 < t r- . 1 oo T-l 1 r -UJ o o o o C > •r-l 0 0 C/) QJ C 1-1 1-1 i — i 1—1 1-1 1—1 1-H T H r-i r- l O 1 1 1 i 1 1 1 1 1 CO CD -,-4 vD x> C O 1-1 ON 01 4-J O T- l i — i ON 00 O O 00 C/) to U i - l 00 «X> V40 vD VJD C OJ c m 1 1 1 1 o ON 00 UJ ON o ON ON CO oo CNJ 1-H 1—1 < CNJ CNJ CNJ CNJ O o o o o D C oo oo C O c o C O C O C O oo C O r - l O 1 1 1 1 1 1 1 1 1 0) 'r-l O m T- l o CNJ c o N T m 00 4-J oo 00 00 ON ON ON ON ON 05 u m i - i r-l i - l 1— i 1-1 1-1 T- l a) e CNI CNJ CNJ CNJ CNJ CNJ CNJ CNi CNI PQ 3 tM o o O O o o o o O II r J • o o O o o o O o o CO 1-1 CNJ C O < t 00 o CNJ -—- 1—1 i — i 1—1 pel 145 i t w i l l be r e c a l l e d t h a t the coupled d i f f e r e n t i a l e q u a t i o n s were i n t e g r a t e d out t o R ^ 30a.u., and the boundary c o n d i t i o n s were f i t t e d employing S p h e r i c a l Bessel f u n c t i o n s . T his upper l i m i t on the i n t e g r a t i o n range was imposed by t h e computer time a v a i l a b l e f o r those c a l c u l a t i o n s . I t i s d i f f i c u l t t o know how the f i n a l cross s e c t i o n s would have been changed by e x t e n d i n g the i n t e g r a t i o n range, however i t i s e v i d e n t t h a t r e s u l t s o b t a i n e d f o r the t r a n s i t i o n = 0 J x = 0 t o V£ = 0 j l = 0 were not f u l l y converged. Employing the ASE, the same i n t e g r a t i o n range would have ensured 'more or l e s s ' converged r e s u l t s . I n F i g u r e 6 we p l o t t h e increment i n the p a r t i a l c r o s s s e c t i o n s , j\o~L - \&l(n>) ~ < H - ^ - L > / 9 as a f u n c t i o n o f the l o c a l s c a t t e r i n g c o o r d i n a t e . and > f~i . The b e t t e r convergence p r o p e r t i e s o f the ASE are e v i d e n t from t h i s f i g u r e . P f e f f e r and Secrest (1983) r e p o r t t h a t i n t h e i r s tudy o f the r o v i b r a t i o n a l e x c i t a t i o n o f ^ by L i + at E = 4.23eV, the use of the f u n c t i o n s IV.A.5 reduced t he i n t e g r a t i o n range by a f a c t o r t e n . I t i s wo r t h n o t i n g t h a t as i n our c a l c u l a t i o n s t h e v i b r a t i o n a l motion o f the t a r g e t molecule was t r e a t e d w i t h i n the c l o s e c o u p l i n g a p p r o x i m a t i o n , and the r o t a t i o n a l motion was t r e a t e d w i t h i n the 10 S a p p r o x i m a t i o n . Also the l e a d i n g l o n g range term o f the i n t e r a c t i o n p o t e n t i a l i s the charge-quadrupole i n t e r a c t i o n v a r y i n g as R . However, one x 10 x 1 0 -6 35 30 25 20 15 10 5 0 4 i — n F i g u r e 6 P l o t s o f A C T a g a i n s t ( a . u . ) , f o r t h r e e v a l u e s o f L. F u l l curves : u s i n g S p h e r i c a l Bessel f u n c t i o n s ; broken curves : u s i n g a s y m p t o t i c s e r i e s expans i o n s . 20 18 16 14 °€ 12 b 10 O 8 6 4 2 0 x IC 20 18 16 14 12 10 8 6 4 7 L = 0 1 1 1 1 I I 20 40 60 80 100 120 140 _ l 1 1 I 1 1 1 L = 5 _ — \ \ \ 20 40 60 80 20 40 60 80 100 120 140 r 2 (a.u.) 146 would expect the ASE t o be advantageous when a p p l i e d t o N2 + L i * f o r the f o l l o w i n g reason. The p o t e n t i a l term i n IV.A.7 i s more i m p o r t a n t due t o the h i g h e r reduced mass of the L i 4 + N 2 system, and the h i g h e r quadrupole moment of the N2 molecule. I n Table 10 we g i v e the r e l e v a n t n u m e r i c a l v a l u e s . I t i s i n t e r e s t i n g t h a t P f e f f e r and Secrest (1983) a l s o r e p o r t t h a t the ASE method worked best at s m a l l L ( t h e y do not s p e c i f y what they mean by s m a l l ) . As mentioned p r e v i o u s l y , we would expect the ASE of the type i n t r o d u c e d by G a i l i t i s (1976) t o be advantageous over a l a r g e r range o f L than the ASE suggested by Secrest ( 1 9 7 9 ) , s i n c e i n the former f u n c t i o n s IV.A.6 the e f f e c t o f the c e n t r i f u g a l term, L, i s i n c l u d e d i n the S p h e r i c a l Bessel f u n c t i o n s . TABLE 10 H 2 + H + Reduced mass o f system = 1224.101013 a.u. Quadrupole moment of H 2 = 0.459 a.u. N 2 + L i + ( P f e f f e r and Secrest (1983)) Reduced mass of system = 12347.587078 a.u. Quadrupole moment of N 2 = 0.943 a.u. The quadrupole moments are g i v e n f o r the H 2 and N molecules set a t t h e i r e q u i l i b r i u m g e o m e t r i e s . - 147 -CHAPTER V ROVIBRATIONAL EXCITATION OF 1 2 C 1 6 0 by PARA-H2 1. I n t r o d u c t i o n The study o f the r o v i b r a t i o n a l e x c i t a t i o n o f CO i s i m p o r t a n t f o r two reasons. The CO molecule i s the most abundant heavy molecule i n the i n t e r s t e l l a r medium, and i t can be c o l l i s i o n / i l l y e x c i t e d by the o t h e r most abundant species i n the i n t e r s t e l l a r c l o u d s , l i k e H 2, H, H +, and He. Secondly, a knowledge o f r e l a x a t i o n r a t e s and d e a c t i v a t i o n cross s e c t i o n s i s an e s s e n t i a l p r e r e q u i s i t e f o r the dynamical study o f CO l a s e r s . One o f the systems t h a t has been most t h o r o u g h l y s t u d i e d e x p e r i m e n t a l l y i n r e c e n t years i s H 2 + CO. The f i r s t s tudy o f the v i b r a t i o n a l a c t i v a t i o n of CO ( v 2 =•• 0) i n c o l l i s i o n s w i t h was c a r r i e d out by Hooker and M I M I k a n (19 0 3 ) . The II., i CO system was used i n the f i r s t measurement o f v i b r a t i o n a l r e l a x a t i o n times i n the low tem p e r a t u r e range 100 - 300K by M i l l e r and M i l l i k a n ( 1 9 7 0 ) . Subsequent t o these s t u d i e s , c o n s i d e r a b l e e f f o r t has been devoted t o d e t e r m i n i n g the r a t e o f v i b r a t i o n a l d e a c t i v a t i o n o f CO by H o (e.g. S t a r r et a l . ( 1 9 7 4 ) , Stephenson and Mosburg (197 4 ) , Andrews and Simpson ( 1 9 7 5 ) , ( 1 9 7 6 ) , and A l l e n e t a l . ( 1 9 7 9 ) ) . Andrews and Simpson (1976) have o b t a i n e d r e s u l t s f o r normal -H 2, para-H 2 and o r t h o -H 2 i n the temperature range 77-340K u s i n g l a s e r f l u o r e s c e n c e e x p e r i m e n t s . The CO gas i s v i b r a t i o n a l l y - 148 -e x c i t e d u s i n g a p u l s e d chemical CO l a s e r , and the r a t e of f l u o r e s c e n c e decay i s measured. S t a r r et a l . (1974) have o b t a i n e d r e s u l t s f o r normal -H^ i n the temperature range 100 - 6r)0 K u s i n g a frequency doubled C()^ l a s e r . I n comparison very l i t t l e t h e o r e t i c a l work has been devoted t o the r o v i b r a t i o n a l e x c i t a t i o n o f CO by H2. The major reason f o r t h i s has been the absence o f a s u i t a b l e p o t e n t i a l energy s u r f a c e . I n o r d e r t o study the r o v i b r a t i o n a l e x c i t a t i o n o f CO one r e q u i r e s a p o t e n t i a l energy s u r f a c e i n c o r p o r a t i n g the a n g u l a r , i n t e r m o l e c u l a r , and i n t e r n u c l e a r v a r i a t i o n o f the i n t e r a c t i o n . Such p o t e n t i a l energy s u r f a c e s have been r e c e n t l y p u b l i s h e d (Poulsen (1 9 8 2 ) , van llemert ( 1 9 8 3 ) , and Baci c , Schinke and D i e r c k s e n ( 1 9 8 4 a ) ) . Poulsen (1982) has o b t a i n e d an a n a l y t i c a l r e p r e s e n t a t i o n o f the a b i n i t i o SCF p o i n t s o f P r i s s e t t e et a l . (1978) and Flower e t a l . ( 1 9 7 9 ) . The dependence o f the i n t e r a c t i o n on the CO and H2 i n t e r -n u c l e a r d i s t a n c e s was o b t a i n e d by r e p r e s e n t i n g the i n t e r -a c t i o n as a s u p e r p o s i t i o n o f atom-atom i n t e r a c t i o n s . Poulsen (1982) o n l y employs the s h o r t range p a r t of the SCF c a l c u l a t i o n s . The m u l t i p o l e p a r t o f the p o t e n t i a l was g i v e n as the u s u a l i n t e r a c t i o n between permanent m u l t i -p o l e s . For a l l the m u l t i p o l e s , f u n c t i o n a l forms t h a t modelled t h e i r dependence on the i n t e r n u c l e a r c o o r d i n a t e s were suggested. The l o n g range d i s p e r s i o n was chosen from the c o e f f i c i e n t s f o r CO and as g i v e n i n the l i t e r a t u r e ( Parker and Pack ( 1 9 7 6 ) , Meyer ( 1 9 7 6 ) ) . - 149 -Van Hemert (1983) has employed the e l e c t r o n gas model of Gordon and Kim (1972) t o c a l c u l a t e the + CO i n t e r a c t i o n at 4158 p o i n t s . Van Hemert (1983) a l s o r e p o r t s an a n a l y t i c a l e x p r e s s i o n o f the H2 + CO i n t e r a c t i o n d e r i v e d from h i s e l e c t r o n gas c a l c u l a t i o n s and the second o r d e r l o n g range ( d i s p e r s i o n and i n d u c t i o n ) i n t e r a c t i o n s o f the system c a l c u l a t e d by p e r t u r b a t i o n t h e o r y . The e l e c t r o n gas model c a l c u l a t i o n s e x p l i c i t l y g i v e the v a r i a t i o n of the i n t e r a c t i o n w i t h the i n t e r n u c l e a r s e p a r a t i o n s . Very r e c e n t l y , a f t e r these c a l c u l a t i o n s had been completed, a H2 + CO p o t e n t i a l s u r f a c e was c o n s t r u c t e d by B a c i c , Schinke, and D i e r c k s e n (1984a). E x t e n s i v e SCF c a l c u l a t i o n s ( i n c l u d i n g the e l e c t r o s t a t i c , i n d u c t i o n , and charge t r a n s f e r e f f e c t s ) performed f o r a number of H2 and CO bonds l e n g t h s by D i e r c k s e n and Kraemer (1984) were combined w i t h damped l o n g range d i s p e r s i o n e n e r g i e s t o d e t e r m i n e an a n a l y t i c a l p o t e n t i a l s u r f a c e i n c l u d i n g the v a r i a t i o n o f the CO i n t e r n u c l e a r d i s t a n c e . Poulsen and B i l l i n g (1982) have employed the p o t e n t i a l of Poulsen (1982) i n s e m i - c l a s s i c a l c a l c u l a t i o n s i n which the v i b r a t i o n of the CO and the r o t a t i o n of H2 a r e t r e a t e d quantum m e c h a n i c a l l y , and the r o t a t i o n o f CO and the r e l a t i v e t r a n s l a t i o n a l m otion are t r e a t e d c l a s s i c a l l y . The r a t e o f v i b r a t i o n a l d e a c t i v a t i o n o f CO (V2 = 1) i n c o l l i s i o n s w i t h p a r a - and o r t h o which s i m u l t a n e o u s l y undergoes the r o t a t i o n a l t r a n s i t i o n s Aj.: = 0 , 2 or 4 i s s t u d i e d . Good agreement i s found - 150 -w i t h the e x p e r i m e n t a l r e l a x a t i o n r a l e s i n the t e m p e r a t u r e range 100 ^ T ^ 7 000K. Drol sh.igen and G i a n t u r c o (198 3, 1984) have performed quantum mechanical c a l c u l a t i o n s of cross s e c t i o n s f o r the v i b r a t i o n a l e x c i t a t i o n of CO by and o f by CO. Both the CO and molecules are t r e a t e d w i t h i n the b r e a t h i n g sphere (BS) a p p r o x i m a t i o n . Cross s e c t i o n s f o r t h e v i b r a t i o n a l e x c i t a t i o n o f H2 by CO are found t o be much s m a l l e r than the c o r r e s p o n d i n g cross s e c t i o n s f o r the e x c i t a t i o n o f CO by i ^ . C a l c u l a t i o n s are c a r r i e d out i n the energy range K£ 4 eV. The agreement w i t h the e x p e r i m e n t a l r e l a x a t i o n r a t e of CO ( v ^ I ) i s un-sat i s f a c t o r y . Bacic, Schinke and D i e r c k s e n (1984a) have c a r r i e d out quantum mechanical c a l c u l a t i o n s employing the p o t e n t i a l s u r f a c e c o n s t r u c t e d from the SCF c a l c u l a t i o n s of D i e r c k s e n and Kraemer ( 1 9 8 4 ) . The r o t a t i o n o f the CO i s t r e a t e d w i t h i n the IOS, BS and CS a p p r o x i m a t i o n s . The H2 molecule i s c o n s t r a i n e d t o i t s ground r o t a t i o n a l s t a t e , and the d e - e x c i t a t i o n of CO (V2 = 1) i s s t u d i e d i n the energy range £ 0.5 eV. The IOS and CS r e l a x a t i o n r a t e s agree w e l l w i t h the e x p e r i m e n t a l measurements i n the temperature range 100 £ T ^ 300K, however the r e s u l t s computed u s i n g the BS a p p r o x i m a t i o n are found t o be u n s a t i s f a c t o r y . C o m p u t a t i o n a l l y the CO molecule i s not s u i t e d t o r i g o r o u s CC c a l c u l a t i o n s . Since CO has a s m a l l r o t a t i o n a l c o n s t a n t , t h e r e are many c l o s e l y packed r o t a t i o n a l 151 l e v e l s ; the 2 j 2 + 1 r o t a t i o n a l degeneracy means t h a t the number o f coupled e q u a t i o n s becomes e x c e s s i v e l y l a r g e . However, because the r o t a t i o n a l l e v e l s are c l o s e l y packed CO i s i d e a l l y s u i t e d t o approximate t r e a t m e n t s o f the r o t a t i o n a l m o t i o n , p a r t i c u l a r l y t he energy sudden and the i n f i n i t e o r d e r sudden a p p r o x i m a t i o n s (Chapter I T , S e c t i o n s 3.2 and 3.3 r e s p e c t i v e l y ) . I n the present c a l c u l a t i o n s we w i l l t r e a t the r o t a t i o n a l degree o f freedom o f the CO w i t h i n the IOS, and BS a p p r o x i m a t i o n s . Since the p o t e n t i a l energy s u r f a c e s (van Hemert ( 1 9 8 3 ) , and Poulsen ( 1 9 8 2 ) ) , t he dynamical a p p r o x i m a t i o n s and the d e s c r i p t i o n o f the v i b r a t i o n a l m o t i o n o f the CO are the major o b j e c t s o f t h i s s t udy we w i l l c o n f i n e o u r s e l v e s t o H2 c o n s t r a i n e d i n i t s ground r o t a t i o n a l s t a t e ( j 1 = 0 ) . 2. The I n t e r a c t i o n P o t e n t i a l s As s t a t e d i n S e c t i o n 1 t h e r e are t h r e e i n t e r a c t i o n p o t e n t i a l s u r f a c e s a v a i l a b l e i n the l i t e r a t u r e s u i t a b l e t o s t u dy the r o v i b r a t i o n a l e x c i t a t i o n o f CO by H2 (Poulsen (1982) , van Hemert ( 1 9 8 3 ) , and Baci6, Schinke and D i e r c k s e n ( 1 9 8 4 a ) ) . However, when these c a l c u l a t i o n s were begun the l a t t e r p o t e n t i a l was not a v a i l a b l e , and c o n s e q u e n t l y o n l y t he p o t e n t i a l s o f Poulsen (1982) and van Hemert (1983) are employed i n our s t u d y . 2.1 The P o t e n t i a l o f Poulsen The p o t e n t i a l o f Poulsen (1982) ( h e r e a f t e r r e f e r r e d t o as the P p o t e n t i a l ) i s an a n a l y t i c a l f i t t o the ab i n i t i o SCF c a l c u l a t i o n s o f P r i s s e t t e et a l . (1978) and 152 Flower et a l . (19 7 9 ) . This c a l c u l a t i o n c o n s i d e r s 19 c o n f i g u r a t i o n s at 10 i n t e r m o l e c u l a r d i s t a n c e s from 4 t o 10 a.u. keeping the two molecules at t h e i r e q u i l i b r i u m g e o m e t r i e s . The t o t a l i n t e r a c t i o n p o t e n t i a l has t h r e e SR p a r t s , a s h o r t range p a r t V , the l o n g range m u l t i p o l e i n t e r a c t i o n , and the d i s p e r s i o n energy V< '^'"SP, i . e . , the SCF c a l c u l a t i o n s g i v e the f i r s t two p a r t s V.2.2 The s h o r t range p o t e n t i a l was t h e r e f o r e , o b t a i n e d by SCF MP f i t t i n g an a n a l y t i c a l e x p r e s s i o n t o V - V The s h o r t range p a r t i s e x t r a c t e d from the SCF p o i n t s u s i n g X/SCF VMl>_ l / S C F ^ S C P SCF V .v -v - v v ^ 9 t t v 2 3 where t h e s u p e r s c r i p t SCF i n d i c a t e s t h a t the d i p o l e (pL) and quadrupole (Q) moments o b t a i n e d w i t h the SCF b a s i s s et were used. An a n a l y t i c a l f u n c t i o n i s f i t t e d t o the s h o r t range p o t e n t i a l d e f i n e d by V.2.3. When f i t t i n g , t h e e q u i l i b r i u m bond l e n g t h s o f H 2 and CO used i n the SCF c a l c u l a t i o n were employed. As mentioned above the ab i n i t i o c a l c u l a t i o n s c o n s i d e r the e q u i l i b r i u m bond l e n g t h s . A model was t h e r e f o r e used from which the dependence of the i n t e r a c t i o n on the i n t r a n u c l e a r d i s t a n c e s was d e r i v e d . The c o o r d i n a t e system used i s 153 -shown i n F i g u r e 1 X 1 c I / fa' . . K — 1 ?• F i g u r e 1. The a n a l y t i c a l r e p r e s e n t a t i o n was expanded i n R and i n the s p h e r i c a l harmonics ^ ^ ( 0 ^ , ^ ) f ° r ^ w i t h L = 0 and 2. One o f the uses o f t h i s p o t e n t i a l s u r f a c e w i l l be t o examine the near-resonance process V.2.4 T h i s t r a n s i t i o n may be induced t o f i r s t o r d e r by a P^(cos6)^) term and t o second o r d e r by P 2 ( c o s & ^ ) . The c o e f f i c i e n t c o u l d not be o b t a i n e d by f i t t i n g an expanded form t o the SCF p o i n t s , because the number o f c o n f i g u r a t i o n s i s t oo l i m i t e d t o determine b o t h the P2 and P^ c o e f f i c i e n t s . An e s t i m a t e was o b t a i n e d by a f u r t h e r expansion o f the a n a l y t i c a l form i n c l u d i n g now w i t h L = 4 and u s i n g the same parameters as o b t a i n e d i n the f i t w i t h o n l y L = 0 and 2. 1 54 The s h o r t range p o t e n t i a l , i g n o r i n g terms w i t h A , i s g i v e n by ( i n 1^1 V.2.5 where ~.J-Vic = (Wl£)*V*i 2 ^ / 3 , i«/> ^ 34> f 4 > a n c ' ^5 a r e g i v e n by - ^ c w ( 9 z - 0 - 5 e V f i ^ * * . * " ' == ^ o t h e r w i s e . The f u n c t i o n f ^ was i n c l u d e d t o improve the a n i s o t r o p y around t he C end o f the CO mo l e c u l e , v.2.6 where m and m are the masses o f the C and 0 atoms c o r e s p e c t i v e l y , t he f o l l o w i n g v a l u e s were assumed ( P r i s s e t t e et a l . ( 1 9 7 8 ) ) . The parameters A^ t o are g i v e n i n Table 1. TABLE 1 Parameters o f the s h o r t range H2 + CO p o t e n t i a l A x = 248.55 eV A 2 = 3.0168 8 _ 1 A 3 = 1.0493 eV A 4 = 1.549 8" 1 A 5 = 338.81 eV A 4 = 3.5776A 5 - 1 A ? = 0.35584 eV A g = 1 . 9 6 0 l 8 _ 1 Ag = 2.0749 A 1 Q = 0.00 A 1 ; L = 0.24 - 156 -The m u l t i p o l e p a r t o f t h e p o t e n t i a l i s taken as the u s u a l i n t e r a c t i o n between the permanent m u l t i p o l e s o f the two molecules ( e . g . see Flower et a l . ( 1 9 7 9 ) , L e a v i t t ( 1 9 8 0 ) ) , where V.2.7 V - V/p. -tVtv+ V<,¥A-t !/<»„<* , /- y w Ct> , -/A. VMM yOO \ * (3Ylo Yio •* 3 'i*' * 1*1/ V* ' « > * • Y Z k ~ * IB"- Y & R,%.) here ' yHH ^Oo HH HH where Q and are the quadrupole and hexadecapole moments of the H2 molecule r e s p e c t i v e l y , and wherejUp4^ CO and Q are the d i p o l e and quadrupole moments o f t h e CO molecule r e s p e c t i v e l y . I n the paper by Poulsen (1982) the f a c t o r (>i~i%f^o>) i s m i s s i n g ( B a c i c , Schinke, and D i e r c k s e n ( 1 9 8 4 b ) ) . This i s u n i m p o r t a n t f o r t h e pr e s e n t c a l c u l a t i o n s , s i n c e the m u l t i p o l e moment i n t e r a c t i o n s a re not used, however, we d i d not r e a l i s e t h a t t h i s f a c t o r was m i s s i n g , and - 157 -co n s e q u e n t l y some o f the r e s u l t s p r e s e n t e d i n Chapters \ I and V l l a r e i n e r r o r . We w i l l d i s c u s s the major i m p l i c a t i o n s o f t h i s e r r o r i n the p o t e n t i a l r o u t i n e i n the r e l e v a n t c h a p t e r s . The f o l l o w i n g m u l t i p o l e moments are c o n s i d e r e d t o be the most a c c u r a t e i n the l i t e r a t u r e . The quadrupole moment f u n c t i o n o f used i s the f i t o f T r u h l a r (1972) t o the r e s u l t s o f Kolos and Wolniewicz (1965) - o.L55i%~3(n-nyes (*vA*y*-. v 2 8b For the d i p o l e moment o f CO, the e m p i r i c a l e x p r e s s i o n d e r i v e d by Chackerian (1976) i s employed s ^ V.2.9a where e ^ I s the CO e q u i l i b r i u m bond l e n g t h . - 1.58 -The quadrupolc of CO was taken t o be V.2.9b where the e q u i l i b r i u m v a l u e i s the e x p e r i m e n t a l v a l u e r e p o r t e d by Meerts et a l . (1977) and the d e r i v a t i v e s i n V.2.9bwere o b t a i n e d by f i t t i n g a t h r e e term moment f u n c t i o n t o the r e s u l t s o f Amos (1979). The d i s p e r s i o n energy i s g i v e n by V ^ = -11 IS « V fa'1 [I * o • IOS px (LCO &,-) •+ooi'iPz(u»oeOl v.2.10 where the i s o t r o p i c and P 2 ( c o s & 2 ' c o e f f i c i e n t s are r e p o r t e d by Parker and Pack (1976). The P2(cos£j ) a n i s o t r o p y c o e f f i c i e n t i s g i v e n by Meyer (1976). As w i t h the GG i n t e r a c t i o n p o t e n t i a l (Chapter IV.2) v e r y l i t t l e i n i t i a l i z a t i o n can be performed s i n c e i t i s i m p o s s i b l e t o separate the v a r i a b l e s R and r 2 t o any l a r g e e x t e n t . One can o n l y c a l c u l a t e the f u n c t i o n s -1 -2 ^14' ^2' ^34' ^4 a n c* ^5 ( a P a r t f r o m the R and R f a c t o r s ) , and the m u l t i p o l e moment f u n c t i o n s d u r i n g the p o t e n t i a l i n i t i a l i z a t i o n . The c o m p u t a t i o n a l l y expensive t a s k s such as e x p o n e n t i a t i o n must be e v a l u a t e d at each v a l u e of R, however t h i s i s o n l y necessary at the i n i t i a l p a r t i a l wave, and i n a d d i t i o n some e x p o n e n t i a t i o n can be done a n a l y t i c a l l y ( e . g . , k\, a| , (471)^, e t c . ) . - 159 2.2 The P o t e n t i a l o f van Hemert Van Hemert (1983) has employed the e l e c t r o n gas model o f Gordon and Kim (1972) t o c a l c u l a t e the ^ + CO i n t e r a c t i o n at 4158 p o i n t s . Van Hemert (1983) a l s o r e p o r t s an a n a l y t i c a l e x p r e s s i o n o f the H£ + CO i n t e r -a c t i o n p o t e n t i a l d e r i v e d from h i s e l e c t r o n gas c a l c u l a t i o n s and the c o r r e c t second o r d e r l o n g range ( i n d u c t i o n and d i s p e r s i o n ) c o n t r i b u t i o n s as g i v e n by p e r t u r b a t i o n t h e o r y . The p o t e n t i a l ( h e r e a f t e r r e f e r r e d t o as the VH p o t e n t i a l ) i s expressed e x p l i c i t l y i n terms of r a d i a l and a n g u l a r v a r i a b l e s , by the use o f the s p h e r i c a l ex-pansion method (van der A v o i r d et a l . ( 1 9 8 0 ) , Berns and van der A v o i r d ( 1 9 8 0 ) ) . For two l i n e a r ( d i a t o m i c ) molecules i n s t a t e s , the d i s t a n c e and o r i e n t a t i o n dependence o f the i n t e r a c t i o n i s w r i t t e n ( i n a . u . ) . L-0 LfO i s o V.2.11 where £ - 9", $ ) , /~ - (&n <{>, ) , a*d, f{ ^ (Gx , 4^) and where the expansion c o e f f i c i e n t s , V^i^L > a r e f u n c t i o n s of d i s t a n c e o n l y and c o m p l e t e l y d e t e r m i n e the o r i e n t a t i o n a l dependence o f the p o t e n t i a l . Since the i n t e r a c t i o n p o t e n t i a l i s i n v a r i a n t under r o t a t i o n o f the a r b i t r a r y c o o r d i n a t e system we may set = (j> = 0, and employ a c o o r d i n a t e system of the type g i v e n i n F i g u r e 1. The paper of van Hemert (1983) does n o t 9 h o w e v e r , c l e a r l y d e f i n e the c o o r d i n a t e system t o be used. For the H2 + CO system t h e r e are two a l t e r n a t i v e s . ^ = &~2 = 0 16C -can d e f i n e the l i n e a r c o n f i g u r a t i o n i n which t he 0 i s c l o s e s t t o the H2 molecule (as i n F i g u r e 1 ) , a l t e r n a t i v e l y = 0 may d e f i n e a l i n e a r c o n f i g u r a t i o n w i t h C c l o s e s t t o the H2 mol e c u l e . I n an e a r l i e r p u b l i c a t i o n , van Hemert ( 1 9 8 1 ) , the l a t t e r c o n f i g u r a t i o n i s t a k e n , however some p r e l i m i n a r y c a l c u l a t i o n s s t u d y i n g the R dependence of the i n t e r a c t i o n p o t e n t i a l w i t h r e s p e c t t o 9- 0 have i n d i c a t e d t h a t van Hemert (1983) has adopted a c o o r d i n a t e system as g i v e n i n F i g u r e 1. I f we set ty\ = 0 the a n g u l a r f u n c t i o n s Plm^f^ can be expressed as Pi, (COO &,) (ceo & ^ ) C6*M4IL V.2.12 where — M and are a s s o c i a t e d Legendre f u n c t i o n s a c c o r d i n g t o the d e f i n i t i o n i n B r i n k and S a t c h l e r ( 1 9 7 5 ) . Since the i n t e r a c t i o n p o t e n t i a l can be decomposed i n t o s h o r t range, f i r s t o r d e r ( m u l t i p o l e moment i n t e r a c t i o n s ) , and second o r d e r ( i n d u c t i o n and d i s p e r s i o n i n t e r a c t i o n s ) l o n g range c o n t r i b u t i o n s , the V ^ ^ ^ can be ana l y z e d i n t h e same terms V.2.13 I () I By e x a m i n a t i o n o f a pLot of the l o g a r i t h m s o f the s h o r t range expansion c o e f f i c i e n t s as a f u n c t i o n o f R, f o r d i f f e r e n t r ^ and v a l u e s , the f o l l o w i n g a n a l y t i c a l form f o r ( R , r ^ , ) was suggested. V.2.14a , ,MAX 0 . MAX , , , MAX n where = 2, = 4, and L - 7, where s g n j _ | U i k i n d i c a t e s t h a t the expansion c o e f f i c i e n t s s h o u l d be n e g a t i v e f o r L ^ L = (2,1,1) and ( 2 , 2 , 2 ) , and where V.2.14b where r 1 = - r 1 and r 0 = r 0 - r 0 1 1 1,eq 2 2 2 , eq , , (0) (1) (2) (12) . , and where a , a , a , a are n u m e r i c a l parameters. S i m i l a r e x p r e s s i o n s h o l d f o r (rLlC%u (Tnr^) , ^UL^L. (fa O > and oL»L\C2.l- ^ T>'0 • This form does not account f o r the p r o b a b l y s m a l l , n o n - l i n e a r terms i n and . The s m a l l cross term i s included^however, s i n c e i t may c o n t r i b u t e s i g n i f i c a n t l y i n r o v i b r a t i o n a l c a l c u l a t i o n s . The c o e f f i c i e n t s A^2*.> HflQ.L-> ^ / i j i o a n d 0^1)L%L* i n V.2.14a and t h e i r dependence upon r ^ and (V.2.14b) were de t e r m i n e d s i m u l t a n e o u s l y i n a p o l y n o m i a l l e a s t squares f i t o f the l o g a r i t h m s o f the s p h e r i c a l expansion c o e f f i c i e n t s . Only the i s o t r o p i c d i s p e r s i o n energy i s g i v e n by van Hemert ( 1 9 8 3 ) , and t h i s i s g i v e n by V.2.15 - 162 where C»£ » £ g > and are the d i s p e r s i o n c o e f f i c i e n t s f o r CO and H2 a t t h e i r e q u i l i b r i u m bond l e n g t h , and have the f o l l o w i n g v a l u e s ( i n a.u.) Ct = 376, C f - 7 90 , W . Cf0 - 2 2. *cro, and where C^ was o b t a i n e d from known p o l a r i z a b i l i t i e s and from t h e known C^ v a l u e s o f the pure components, and the h i g h e r d i s p e r s i o n c o e f f i c i e n t s , Cg, C^Q, were o b t a i n e d w i t h the h e l p o f sum r u l e r a t i o s where one assumes equal mean e x c i t a t i o n e n e r g i e s f o r a l l m u l t i p o l e t r a n s i t i o n s (Mulder et a l . ( 1 9 8 0 ) ) . I t i s assumed t h a t C g and C 1 Q depend on the int^gsf'molecular d i s t a n c e t o the same measure as C6. Equation V.2.15 can be extended f o r use at i n t e r m e d i a t e d i s t a n c e s by m u l t i p l y i n g the r i g h t - h a n d s i d e w i t h a damping f u n c t i o n xPj^^e,8- s e e Meath and A l l m a t t ( 1 9 7 9 ) ) V.2.16 where R m = 7.55 a.u., and ^ = 0.7. The l o n g range m u l t i p o l e i n t e r a c t i o n s are g i v e n V.2.17 - 1.6 3 -where L , « > L * * Lr > a / U C U > and where w i t h / + ( ' r f ^ i , , ~)Jy , where ^Ly(^) ^ s t n e permanent 2^-pole moment of molecule X w i t h bond l e n g t h r . The paper of van Hemert (1983) g i v e s f o r m a l e x p r e s s i o n s f o r Ct^+Li+I $t)ef,) ^-^f^ > f o r o t h e r i n t r a m o l e c u l a r d i s t a n c e s the c o e f f i c i e n t s , lfi>r%} m u s t be found by e v a l u a t i n g ^»^Ly /^Of a s s u m i n g t h.c ^ ^ c ^ _ ^ ( r yjl(n _ r J ) ^ where r + and r _ are the l a r g e s t and s m a l l e s t i n t r a m o l e c u l a r d i s t a n c e s c o n s i d e r e d . I n c o n t r a s t t o the P p o t e n t i a l , t h i s i s an e x t r e m e l y s i m p l e i n t e r a c t i o n p o t e n t i a l t o compute. I n these c a l -c u l a t i o n s o n l y the s h o r t range and d i s p e r s i o n terms are employed, and we no t e t h a t a g r e a t d e a l o f the work i n v o l v e d i n computing these terms can be done once a t each o r i e n t a t i o n a t the i n i t i a l v a l u e o f R, and used r e p e a t e d l y t h r o u g h o u t t h e i n t e g r a t i o n . The f u n c t i o n s PlLiUiL , &^L%L, , bltLzL,GL,c%L,,ctlti2i. y and (fi^ are a l l , as one can see, independent o f R. I t i s w o r t h adding t h a t the p o t e n t i a l s u r f a c e employed by Bac'ic, Schinke and D i e r c k s e n (1984a) (and a l s o 1984b, see Chapter VL1) s h o u l d i n t h e o r y , be the most a c c u r a t e des-c r i p t i o n of the H2+CO i n t e r a c t i o n . I t ( t h e 'DK' p o t e n t i a l ) was c o n s t r u c t e d from t he e l a b o r a t e SCF c a l c u l a t i o n s o f D i e r c k s e n and 164 Kraemer (1984) o b t a i n e d at t h r e e CO bond d i s t a n c e s ( r 2 = 1.898 a.u., 2.132 a.u., and 2.234 a . u . ) , and has been shown t o p r o v i d e the most s a t i s f a c t o r y d e s c r i p t i o n of the r o t a t i o n a l e x c i t a t i o n o f CO by D2 (see Schinke et a l . (1984) - t h i s paper c o n t a i ns a summary of the r e s u l t s o b t a i n e d u s i n g a l l the t h r e e p o t e n t i a l s ) . However as we s h a l l show, p r i n c i p a l l y i n S e c t i o n 5, but a l s o i n Chapter VI.4 we doubt t h a t r o t a t i o n a l e x c i t a t i o n i s a s u f f i c i e n t l y s t r i n g e n t t e s t of an i n t e r a c t i o n p o t e n t i a l and u s i n g our c a l c u l a t i o n s o f the r o v i b r a t i o n a l e x c i t a t i o n o f CO by H2 t o g e t h e r w i t h the r e s u l t s o b t a i n e d by B a c i c , Schinke and D i e r c k s e n (1984a,b) we are a b l e t o make a more 'complete' comparison o f the i n t e r a c t i o n p o t e n t i a l s . 3. D e s c r i p t i o n of the System 3.1 IPS C a l c u l a t i o n s As a l r e a d y mentioned these IOS c a l c u l a t i o n s are performed w i t h the H2 molecule c o n s t r a i n e d i n i t s ground r o t a t i o n a l ( = 0) s t a t e . For p a r a - ^ i n i t s ground r o t a t i o n a l s t a t e the i n t e r a c t i o n p o t e n t i a l takes the f o r m : v. 3.1 The i n t e r n u c l e a r s e p a r a t i o n of the H2 molecule i s f i x e d a t i t s e q u i l i b r i u m v a l u e r e f l e c t i n g the f a c t t h a t the p r o b a b i l i t y f o r t h e v i b r a t i o n a l e x c i t a t i o n o f H2 i s n e g l i g i b l y s m a l l . T h i s has been demonstrated i n t h e b r e a t h i n g sphere (BS) c a l c u l a t i o n s o f Drolshagen 165 and G i a n t u r c o ( 1 9 8 3 ) . The a n i s o t r o p y o f the i n t e r a c t i o n p o t e n t i a l , due t o the asymmetry of the l l . ^ m o lecule i s i m p l i c i t l y averaged out ( i . e . o n l y the l e a d i n g , P Q ( c o s ^ ) , term o f the i n t e r a c t i o n p o t e n t i a l i s r e t a i n e d ) . The p o t e n t i a l o f Poulsen ( 1 9 8 2 ) , employed i n these c a l c u l a t i o n s , may be w r i t t e n V. 3.2 and ydisp _ ^ /g^/A * ft* P-+0 o8V?i (c&o 6 > O j The p o t e n t i a l o f van Hemert (1983) takes t h e form V. 3 . 3 J-where AOLZL^ (#0 ~ YoL%L%pP> The s h o r t range c o n t r i b u t i o n i s g i v e n by V. 3.4 where &oi,TL% = ^ oi^t^^ and s i m i l a r ex-p r e s s i o n s h o l d f o r , . a n d ^OL%L^ • The parameters ^o2XL^^oZ^—' are t a b u l a t e d i n Table 2 f o r 0 4, and 1^ = 0. W i t h these parameters V.3.4 i s v a l i d i n the r e g i o n 3.5^R^8.5 a.u. The d i s p e r s i o n c o n t r i b u t i o n i s g i v e n TABLE 2 Parameters used i n V.3.3 f o r the d e s c r i p t i o n o f the i n t e r - and i n t r a m o l e c u l a r d i s t a n c e dependence o f the s p h e r i c a l expansion c o e f f i c i e n t s f o r the f i r s t o r d e r p o t e n t i a l . L l L 2 L a( o ) a ( 2 ) b ( o ) b ( 2 ) 0 0 0 0. 3389+1 -0. 1294+1 -0. 1843+1 0. 7871+0 0 1 1 -0. 3737+1 0. 3828+1 0. 3622+0 -0. 2514+1 0 2 2 0. 1235+1 -0. 1285+1 -0. 1143+1 0. 6835+0 0 3 3 -0. 2508+1 0. 8553+0 -0. 3741+0 -0. 6706+0 0 4 4 -0. 2687+0 0. 7675+0 -0. 1436+1 -0. 6719+0 L l L 2 L c( o ) c ( 2 ) d ( o ) d ( 2 ) 0 0 0 0. 9654-1 -0. 1253+0 -0. 9360- 2 0. 8422-2 0 1 1 -0. 1598+0 0. 5038+1 0. 1600- 2 -0. 3011-1 0 2 2 -0. 2105-1 -0. 1128+0 -0. 1504- 2 0. 8145-2 0 3 3 -0. 9517-1 0. 1304+0 0. 1417- 2 -0. 5859-2 0 4 4 0. 3009-1 0. 1405+0 -0. 3514- 2 -0. 7125-2 - 166 by V.3.5 Both Poulsen (1982) and van Hemert (1983) take the e q u i l -i b r i u m i n t r a n u c l e a r d i s t a n c e s o f CO and ^ t o be t h e e x p e r i m e n t a l v a l u e s g i v e n by P r i s s e t t e et a l . ( 1 9 7 8 ) , i . e . r 1 = 1.4 a.u., and r 0 = 2.132 a.u. The v i b r a t i o n a l b a s i s f u n c t i o n s o f CO were ta k e n t o be Morse o s c i l l a t o r w a v e f u n c t i o n s . The n o r m a l i s e d Morse e i g e n f u n c t i o n s and e i g e n v a l u e s are g i v e n by Mies (196 4 ) . See Chapter IV. 3 f o r d e t a i l s . where £ * < 4 C = 0.9726472 a.u. As f o r the H2 Morse o s c i l l a t o r w a v e f u n c t i o n s , (j) i s quoted t o so many f i g u r e s so t h a t the parameters - 167 -used i n l/' are s e l f - c o n s i s t e n t and m a i n t a i n the o r t h o -n o r m a l i t y o f the w a v e f u n c t i o n s . For CO t h i s i s e x t r e m e l y i m p o r t a n t s i n c e we d e a l w i t h v e r y l a r g e and v e r y s m a l l numbers The c a l c u l a t i o n of the w a v e f u n c t i o n s i s d i s c u s s e d i n Appendix 3 a t the back o f t h i s t h e s i s . Both t he reduced masses o f CO,JLL^Q, and o f the H 2 + CO s y s t e r r i y ^ , were c a l c u l a t e d u s i n g n u c l i d e masses (Weast ( 1 9 7 5 ) ) , H = 1.0079 amu C = 12 amu 0 = 15.9994 amu 12 where 1 amu i s d e f i n e d as 1/12 o f the mass o f C, and employing the f a c t t h a t 1 a.u. = e l e c t r o n r e s t mass energy = 5.4858026-4 amu we f i n d that^tA-^Q and JUts take the n u m e r i c a l v a l u e s : JU,CQ = 12499.6 a.u. JJL, = 3427. 79 a.u. 3.2 B r e a t h i n g Sphere C a l c u l a t i o n s W i t h i n t he BS a p p r o x i m a t i o n (Chapter 11. 4) i n a d d i t i o n t o n e g l e c t i n g the r o t a t i o n a l degree of freedom o f H 2 , we a l s o n e g l e c t the r o t a t i o n a l degree o f freedom o f CO. T h e r e f o r e we average the i n t e r a c t i o n p o t e n t i a l over b o t h the H 2 and CO o r i e n t a t i o n s , and the p o t e n t i a l ( i s o t r o p i c ) may be w r i t t e n - 16 8 -V 6 S (*>*> J i ft* n «f7L V. 3.8 Only the p o t e n t i a l o f van Hemert (1983) i s employed i n these c a l c u l a t i o n s , and w i t h i n the BS a p p r o x i m a t i o n takes t he e x t r e m e l y s i m p l e form C a l c u l a t i o n s are performed employing both harmonic and Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e the v i b r a -t i o n a l m o t i o n o f t h e CO molecule. The d e t a i l s o f the Morse o s c i l l a t o r w a v e f u n c t i o n s have a l r e a d y been p r e s e n t e d i n S e c t i o n 3.1. The harmonic b a s i s f u n c t i o n s are the s o l u t i o n s o f (see Chapter IV. 4 ) . VBsfc,/e) = V. 3.9 where fa &oo Ho ^ O V. 3.10 where the harmonic p o t e n t i a l i s g i v e n by and the harmonic b a s i s f u n c t i o n s may be w r i t t e n f<0 3= 3-where i s a Hermite p o l y n o m i a l , N v 2 i s a normal-i z a t i o n c o e f f i c i e n t , and - 169 -( ic J The e i g e n e n e r g i e s a r e g i v e n by These c a l c u l a t i o n s were p e r f o r m e d p r i m a r i l y t o make a c o m p a r i s o n w i t h t h e BS r e s u l t s o f D r o l s h a g e n and G i a n t u r c o ( 1 9 8 3 , 1 9 8 4 ) , and t h e r e f o r e we t a k e t h e n u m e r i c a l v a l u e o f "Rw em p l o y e d by t h o s e a u t h o r s . The v a l u e s o f t h e r e d u c e d massesy^,^ nndytC , and t h e e q u i l i b r i u m i n t r a n u c l e a r d i s t ances and /J g^are t a k e n t o be t h o s e e m p l o y e d i n t h e IOS c a l c u l a t i o n s . 4. N u m e r i c a l Methods 4.1 IOS c a l c u l a t i o n s IOS c a l c u l a t i o n s have been p e r f o r m e d e m p l o y i n g t h e P and VH i n t e r a c t i o n p o t e n t i a l s . The e n e r g y r a n g e s t u d i e d i n t h e s e c a l c u l a t i o n s i s 1.1 *ftw ^ E "^2.9 •Rw, were t h e u n i t o f e n e r g y , -fiw = 0.26898eV. The P p o t e n t i a l i s e m p l o y e d a t t e n c o l l i s i o n e n e r g i e s , E^ = ( 0 . 9 + 0.2nHlw, where n = 1,2,3...10, w h i l e t h e VH p o t e n t i a l i s e m p l o y e d a t j u s t two e n e r g i e s , 1 . rftw and 2 . 9tiw, so t h a t t h e s e p o t e n t i a l s u r f a c e s c a n be compared. The r e s u l t s o f i n t e r e s t a r e t h e c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l d e a c t i v a t i o n p r o c e s s e s , v^ = 1 t o v ^ = 0, and v ^ = ^ t o v£ = 1 and 0. 170 -4.1.1 E v a l u a t i o n o f t h e p o t e n t i a l m a t r i x e l e m e n t s As s t a t e d i n C h a p t e r IV.4 t h e e v a l u a t i o n o f t h e p o t e n t i a l m a t r i x e l e m e n t s i n an IOS c a l c u l a t i o n r e q u i r e t h e s o l u t i o n o f an i n t e g r a l o f t h e t y p e ( c f . I I . 3 . 2 2 ) . VU' (*>**'>'f T-I m oo T- I • L O >£> z CM CM CM CM vD s D LO T-l r-. r \ T- l T-( T - l OO 0 0 0 0 0 0 L O L O L O L O II T—1 T- l T - l L O i o L O L O CM CM CM CM 0 4-J • • • T - l T - l T - l T H L O uO uO L O O CO c O o o • II C 1 1 1 o o o o o o o o E *x >-i CO O -—- T- l 1 • 1 1 •u O — T- l T - l CM CO CJ • OO < t OO 3 • CO 1 i — CO CO II CM II m 00 T H z z z T-i ; ; z CM Z Z Z 0 0 1^-O T-I oo o • CM T H CO > u • 3 • • • QJ 3 • O o o > • CO 1 - o CO sO • • T- l c oo 0) CO c T- l T - l r - l >-< c •r-l Csl ~ z CO — — — CO z z z CU •1-1 E - J T- l CO c E S-i o CO CO X o >-i T- l •l-4 T-l • • • U i-> -—. X ) o O o j_) CO •—- 1 1 1 CO U •—-E 00 r> CD • o r - r 4 J • 3 T- l sD T H CO C 3 • 1 1 1 • CO CO r^- cr. (0 CO co oo C 4 J 0 0 T H r ~ - cx> z z z CM Z Z Z -i CSJ CM CM sD CL 0) II • • • 14-1 II o o o 14-4 O-l X O "H X CO CO E o sD c E U + 1 1 O 00 T- l sO 05 C o oo o •r-l T - l O o z co z z z m : : : >-4 10 • o o L O CO 3 3 CM • o o CO T- l CM — . CO 172 d e p e n d e n t upon t h e l i m i t s r and r . , f o r t h e v a l u e s ' 1 ma x m i n o f t h e l i m i t s used i n t h e s e c a l c u l a t i o n s . V a r y i n g r 7 to max o r r . by 0.2 a.u. has an i n s i g n i f i c a n t e f f e c t on t h e mm 3 ° r e s u l t s a t a l l t h e v a l u e s o f R shown. The v a l u e s o f R and R . w i l l be t a k e n t o be 1.4 a.u., and 2.8 a.u. max mm ' r e s p e c t i v e l y i n a l l t h e c a l c u l a t i o n s e m p l o y i n g o u r Morse w a v e f u n c t i o n s . E m p l o y i n g an i n t e g r a t i o n r a n g e , 1.4 ^ ^ " ^ 2 . 8 a.u., o v e r t h e v i b r a t i o n a l c o o r d i n a t e , , we e v a l u a t e d t h e p o t e n t i a l m a t r i x e l e m e n t s u s i n g d i f f e r e n t Gauss L e g e n d r e q u a d r a t u r e s . I n T a b l e 4 we g i v e a c o m p a r i s o n o f t h e n u m e r i c a l v a l u e s o f Vov^^l ^ o r ^ z / " ^ $ e v a l u a t e d u s i n g 96-, 64-, and 48- p o i n t q u a d r a t u r e s i n c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . R e s u l t s a r e shown a t <5^s and a t R = 3,10 20 a.u. The a g r e e m e n t b e t w e e n t h e r e s u l t s o b t a i n e d u s i n g 96- and 64- p o i n t q u a d r a t u r e s i s v e r y good f o r v ' ^ 5 . A 96- p o i n t q u a d r a t u r e w i l l be e m p l o y e d i n a l l o u r p r e l i m i n a r y c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l , and o n l y t h e d i r e c t l y r e l e v a n t ( o p e n ) channels w i l l be r e t a i n e d i n t h e c o u p l e d e q u a t i o n s . We m e n t i o n t h a t c l o s e d c h a n n e l s s h o u l d have no e f f e c t upon t h e s t e p s i z e and t h e i n t e g r a t i o n r a n g e . Once t h e s e p a r a m e t e r s have been d e t e r m i n e d , t h e c o n v e r g e n c e o f t h e r e s u l t s as a f u n c t i o n o f t h e number o f v i b r a t i o n a l c h a n n e l s r e t a i n e d i n t h e c a l c u l a t i o n s w i l l be e x a m i n e d , and T a b l e 4 w i l l be u s e d t o ' s e l e c t ' a s u i t a b l e q u a d r a t u r e f o r o u r p r o d u c t i o n c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . 1/) 3 0) • 1—1 CO — c/) . — . cn CM 3 oo CO < f fib 4-J — - c z * CM •r-( > o — O > 1 — * J < r i-l C vD c CD ii e CP -1 CO CO c CD o a c o •1-1 CO Cu e o CO •r-l Z 4-) C — CD CO J_J -—• o a r. oo c • r - l C/J 3 X I 01 i_l CO 3 i—i CO > CD o o o - ? - t r^> ^o c^> i i i , -H r H r H 1 1 1 H I H I 1 i i i r-l O . I 1 1 1 - O ' 1 1 1 . -1 1 1 -,r> «' l 1 -"I vCl ' 1 " 1 ' 1 .Ci i m •X) i • *r> 1 1 - 1 . 1 - i i i r 'Kl 'f) rjO m m m <-H r H r-H i n m m C \ l CM r-l i—1 .—1 i—1 T - l H T - l i n m m O O O 1 1 1 o o o O O O C O *£> O 3 - — - - — ' - — -• N C M o o > CO o • • • — - • o o cm C O T-H CM 173 I n T a b l e 5 we show a c o m p a r i s o n o f t h e p o t e n t i a l m a t r i x e l e m e n t s , Vo V^(.R\&%)'> f o r V j / ^ " S , e v a l u a t e d u s i n g t h e VH p o t e n t i a l , u s i n g 96-, 64- and 48- p o i n t Gauss L e g e n d r e q u a d r a t u r e s i n t h e r a n g e 1.4 ^ ^ "^2.8 a.u. R e s u l t s a r e shown f o r R = 3,10, and 20 a.u., and a t Q^— 0°. As f o r t h e P p o t e n t i a l , v e r y good a g r e e m e n t i s f o u n d b e t w e e n t h e r e s u l t s o b t a i n e d u s i n g 96- and 64- p o i n t q u a d r a t u r e s f o r v ^ 5. We m e n t i o n t h a t a l t h o u g h t h e dependence o f t h e 1 ^ + CO i n t e r a c t i o n s as d e s c r i b e d by t h e P and VH p o t e n t i a l s ( a t a g i v e n v a l u e o f R) a r e p r o b a b l y q u i t e d i f f e r e n t , b o t h p o t e n t i a l s w i l l v a r y s m o o t h l y w i t h , compared t o t h e o s c i l l a t o r y b e h a v i o u r o f t h e w a v e f u n c t i o n s . Thus one w o u l d e x p e c t t h e c o n v e r g e n c e o f t h e p o t e n t i a l m a t r i x e l e m e n t s , as a f u n c t i o n o f t h e number o f q u a d r a t u r e p o i n t s u s e d t o e v a l u a t e them, t o be s i m i l a r f o r t h e two p o t e n t i a l s , as i s t h e c a s e . I n a l l t h e p r e l i m i n a r y c a l c u l a t i o n s e m p l o y i n g t h e VH p o t e n t i a l we w i l l e mploy a 96- p o i n t q u a d r a t u r e . I t i s w o r t h m e n t i o n i n g t h a t i t was f o u n d t h a t t h e p o t e n t i a l m a t r i x e l e m e n t s , e v a l u a t e d u s i n g t h e VH p o t e n t i a l , d i v e r g e d f o r i n t e r m o l e c u l a r s e p a r a t i o n s , R ^ 20 a.u. The r e a s o n s f o r t h i s w i l l be d i s c u s s e d i n S e c t i o n 4 . 1 . 2 ( i i ) t o g e t h e r w i t h t h e method e m p l o y e d t o a l l e v i a t e t h i s p r o b l e m . 4 1 . 2 S o l u t i o n o f t h e c o u p l e d e q u a t i o n s f o r f i x e d o r i e n t a t i o n s I n t h i s s e c t i o n we w i l l d e t e r m i n e t h e p a r a m e t e r s n e c e s s a r y f o r t h e a c c u r a t e i n t e g r a t i o n o f t h e f i x e d •J < CM 3 CO ^ C M d> — C w d ) u 3 xj CO X) CO 3 cr 0) u •v C 01 00 0) in en CO O 00 5^ o > 4-1 c 0) e o a ' i . z o C 0) 0) UH 4H CO e X ) c CO ON CO CO O) a) o o a a U-t 3C O > e o •rH u CO a e o o 0) x: oo c •H w 3 X) 0) •u CO 3 i—i CO > 0) O O O CM CM CM \D \£) i i i rH rH rH f 1 1 rH rH rH 1 1 1 1 1 1 ON OO rH I 1 1 rH CM 00 , 1 1 1 CO ON CO 00 O 00 m M U N \C> co to ON O ON CM CM CM oo oo oo O H O o o o CO CO CO CNJ CN CM vO vO >C CO CO CO CM CM CM CM CM CM CO CO CO O O O 1 1 1 O O O 1 1 1 o o o 1 1 1 i n o> 1 T-H H 1 1 1 CM 1 m CM oo rH z z sD Z Z CO z z o> CM CM CO CM ON r-l CM CM • • • o O o o 00 1 rH 1 i 1 o 1 rH m 00 oo CO 00 z z rH Z Z r - z z ON 00 O o ON rH rH rH CO O 1 O I O 1 CM ON o^ 00 o O r - \D Z Z 00 ON CO oo • o • O O 1 o 1 II CO JO U II - J o o CO O • • > • — • • o o OH CO rH CM - 174 -o r i e n t a t i o n c o u p l e d e q u a t i o n s i n t h e IOS c a l c u l a t i o n s e m p l o y i n g b o t h t h e P and VH p o t e n t i a l s . The p a r a m e t e r s o f i n t e r e s t a r e t h e i n t e g r a t i o n r a n g e , R •>* R ( w h e r e ° b ' mm max R . and R a r e t h e l o w e r and u p p e r l i m i t s o f t h e m m max i n t e g r a t i o n r a n g e r e s p e c t i v e l y ) , and t h e number o f i n t e -g r a t i o n s t e p s i n t h e r a n g e R R w h i c h we w i l l b K ° m m max d e n o t e by N. T h e r e f o r e i t f o l l o w s t h a t t h e s t e p s i z e i s g i v e n by The d e t e r m i n a t i o n o f a l l t h e s e p a r a m e t e r s w i l l be d i s c u s s e d i n some d e t a i l , one w i l l r e c a l l t h a t t h e i n t e g r a t i o n o f t h e c o u p l e d e q u a t i o n s i s by f a r t h e most i m p o r t a n t a s p e c t o f a s c a t t e r i n g c a l c u l a t i o n ( C h a p t e r IV.4) s i n c e i t a c c o u n t s f o r a b o u t 957a o f t h e t o t a l c o m p u t e r t i m e . I n t h e p r e s e n t c a l c u l a t i o n s t h e f i x e d a n g l e c o u p l e d e q u a t i o n s a r e s o l v e d e m p l o y i n g t h e R - m a t r i x p r o p a g a t o r a l g o r i t h m u s i n g p r o p a g a t o r s c o r r e s p o n d i n g t o a c o n s t a n t r e f e r e n c e p o t e n t i a l ( C h a p t e r 111.4 A.). A f i x e d s t e p s i z e was u s e d i n a l l t h e c a l c u l a t i o n s , some o f t h e p r o b l e m s e n c o u n t e r e d i n t h e use o f t h e s t e p s i z e a l g o r i t h m were d i s c u s s e d i n C h a p t e r IV. 4. We f i r s t d e t e r m i n e t h e above p a r a m e t e r s f o r t h e P p o t e n t i a l , and t h e n r e p e a t t h e t e s t s f o r t h e VH p o t e n t i a l . 4 . 1 . 2 ( 1 ) The P p o t e n t i a l I t w i l l be r e c a l l e d t h a t X i s e n e r g y d e p e n d e n t ( a l t h o u g h w e a k l y ) , t h i s i s d i s c u s s e d i n C h a p t e r I I I . 4 . 5 T h e r e i s ? t h e r e f o r e , t h e p o s s i b i l i t y t h a t t h e s t e p s i z e d e t e r m i n e d a t t h e h i g h e s t c o l l i s i o n e n e r g y may n o t be 175 s u f f i c i e n t l y s m a l l t o m a i n t a i n a c c u r a c y a t some o f t h e l o w e r e n e r g i e s . As d i s c u s s e d i n C h a p t e r n i as t h e e n e r p y i n c r e a s e s , s i n c e t h e e n e r g y i s o n l y p r e s e n t i n t h e d i a g o n a l e l e m e n t s o f t h e c o u p l i n g m a t r i x , t h e o f f d i a g o n a l e l e m e n t s o f t h e c o u p l i n g m a t r i x become l e s s i m p o r t a n t . C o n s e q u e n t l y t h e s t e p s i z e r e q u i r e d t o m a i n t a i n a c c u r a c y c a n be i n c r e a s e d . We b e g i n by e x a m i n i n g t h e e n e r g y dependence o f t h e s t e p s i z e r e q u i r e d t o m a i n t a i n a c c u r a c y . S i n c e we do n o t know t h e v a l u e s o f t h e p a r a m e t e r s Rm-^n a n d ^max w e e m P l ° y w h a t we c o n s i d e r t o be a g e n e r o u s i n t e g r a t i o n r a n g e i n w h i c h R . = 1 a.u., and R = 4 0 a.u. At ° mm ' max 40 a.u. v i b r a t i o n a l e x c i t a t i o n s h o u l d n o t be i m p o r t a n t , i n a d d i t i o n a s t a r t i n g v a l u e o f r . - 1 a.u. s h o u l d " mm be s a t i s f a c t o r y f o r t h e i ^ + CO s y s t e m as t h i s i s w e l l i n t h e c l a s s i c a l l y f o r b i d d e n r e g i o n . I n T a b l e 6 we p r e s e n t a c o m p a r i s o n o f r e s u l t s o b t a i n e d e m p l o y i n g s e v e r a l s t e p s i z e s a t t h e l o w e s t and h i g h e s t c o l l i s i o n e n e r g i e s c o n s i d e r e d i n t h e s e c a l c u l a t i o n s (E = 1.1 -riw and E = 2.9 "ffw r e s p e c t i v e l y ) and a t L = 0. I t i s c l e a r t h a t a t b o t h e n e r g i e s t h e r e s u l t s c o n v e r g e m o n o t o n i c a l l y t o t h e c o r r e c t answer as t h e number o f s e c t o r s i s i n c r e a s e d . T h i s m o n o t o n i c c o n v e r g e n c e i s an e x t r e m e l y u s e f u l f e a t u r e , K i r k p a t r i c k ( 1 9 8 3 ) r e p o r t s t h a t i f one employs t h e s t e p s i z e a l g o r i t h m , t h e r e s u l t s t e n d t o o s c i l l a t e . As m e n t i o n e d p r e v i o u s l y ( C h a p t e r H I ) a p p r o x i m a t e p o t e n t i a l a l g o r i t h m s r e q u i r e a l o t o f n u m e r i c a l e f f o r t t o o b t a i n h i g h l y a c c u r a t e r e s u l t s , TABLE 6 C o m p a r i s o n o f o b t a i n e d u s i n g t h e P p o t e n t i a l , and d i f f e r e n t numbers o f s t e p s o v e r t h e i n t e g r a t i o n r a n g e a t L = 0. ( a ) 195 s t e p s , ( b ) 390 s t e p s , ( c ) 780 s t e p s , ( d ) 3900 s t e p &2 -0° 90° 180° E = 1.1 -hw v _ =l-»v ?=0 ( a ) 0 .543172-6 0. 792105-7 0. 506654-7 ( b ) 0 .549581-6 0. 800173-7 0. 512574-7 ( c ) 0 .550265-6 0. 801073-7 0. 513210-7 ( d ) 0 . 550392-6 0. 801254-7 0. 513327-7 E = 2.9 "ttw v o = l-*v£ = 0 ( a ) 0 .547403-2 0. 110918-2 0. 280446-2 z z ( b ) 0 .553915-2 0. 111796-2 0. 283291-2 ( c ) 0 .554369-2 0. 111832-2 0. 283506-2 ( d ) 0 .554371-2 0. 111828-2 0. 283534-2 v o=2-»v~=0 0 . 100165-5 0. 405892-7 0. 137423-6 z z 0 .102429-5 0. 412326-7 0. 140279-6 0 .102593-5 0. 412679-7 0. 140513-6 0 .102612-5 0. 412682-7 0. 140541-6 v 0 = 2->v< = l 0 . 726507-3 0. 146264-3 0. 186629-3 z z 0 .734216-3 0. 147415-3 0. 188675-3 0 .734834-3 0. 147494-3 0. 188849-3 0 .734923-3 0. 147500-3 0. 188861-3 I(Vx-y^i I&z)l i n u n i t s o f (a-u' R . = 1 a . u, 40 a.u mm ' ' max N.B. The f o l l o w i n g i n f o r m a t i o n may be u s e f u l f o r C h a p t e r V I . I f E^ , d e n o t e s t h e i n i t i a l k i n e t i c e n e r g y r e l a t i v e t o t h e ^ 2 = 1 j j _ = 0 t h r e s h o l d t h e n a t 1.1 hw, Ej = 233.5549 c m ' \ and a t 2.9 hw, E T = 4138.5631 c m - 1 . 1.76 a l t h o u g h t h e y can o b t a i n r e a s o n a b l e a c c u r a c y w i t h much l e s s e f f o r t . T h i s i s r e f l e c t e d i n I he r e s u l t s o f T a b l e 6 where t h e use o f 195 s t e p s g i v e s r e a s o n a b l e r e s u l t s . I t i s c l e a r f r o m T a b l e 6 t h a t t h e c o n v e r g e n c e o f t h e r e s u l t s i s s i m i l a r a t b o t h e n e r g i e s ( i . e . t h e s t e p s i z e r e q u i r e d t o m a i n t a i n a c c u r a c y i s n o t n o t i c e a b l y e n e r g y d e p e n d e n t i n t h e e n e r g y r a n g e we s t u d y ) . The s t e p s i z e i s d e p e n d e n t on t h e p a r t i a l wave p a r a m e t e r . As a l r e a d y m e n t i o n e d , i n t h e s e c a l c u l a t i o n s t h e s e c t o r R - m a t r i c e s u s e d were a p p r o p r i a t e t o a p p r o x -i m a t i n g t h e e l e m e n t s o f t h e l o c a l l y d i a g o n a l m a t r i x as c o n s t a n t ( l i J . 4 . 4 6 ) . T h e r e f o r e t h e s t e p s i z e s h o u l d be s u f f i c i e n t l y s m a l l so t h a t t h e c o u p l i n g m a t r i x ( i n c l u d i n g 2 t h e L ( L + 1)/R t e r m ) c a n be a c c u r a t e l y a p p r o x i m a t e d as c o n s t a n t w i t h i n t h e s e c t o r . As d i s c u s s e d i n C h a p t e r 2 I I I t h e r a t e o f change o f t h e t e r m L ( L + 1)/R i n c r e a s e s w i t h L and t h e r e f o r e , a s t e p s i z e w h i c h i s s u f f i c i e n t l y s m a l l t o a c c u r a t e l y c a l c u l a t e r e s u l t s f o r t h e h i g h e s t L v a l u e w i l l a l s o be s u f f i c i e n t f o r t h e l o w e r v a l u e s . I n T a b l e 7 we show a c o m p a r i s o n o f r e s u l t s o b t a i n e d e m p l o y i n g d i f f e r e n t s t e p s i z e s a t E = 2. 9tiw and a t L = 30, 50, 70, 90. R e s u l t s a r e shown f o r 90°. At t h e two h i g h e s t v a l u e s o f L shown we have o m i t t e d and 1 s i n c e t h e y a r e e x t r e m e l y s m a l l , and i t i s i m p o s s i b l e t o m a i n t a i n a c c u r a c y ( i . e . we a r e d e a l i n g w i t h n u m e r i c a l t h e v a l u e s o f f o r v 2 2 t o v / = 0 z e r o s ) . N o t i c e t h a t t h e v a l u e s o f f o r w l -J < 3 1—1 O N CO • •r-l C M 4_J c II CD 4-J UJ o a x-t CO a . CD CD 00 x : c 4-> CO rH O O C c •i-i o co •1-1 3 4-J CO X> VH CD 00 c CD •r-l 4_) CO C 4_J •r-l X) o CD JC 4 J . 3 CD * > O CO a CD 4-1 CO > O CO 4 r J CD X) E 3 14-1 C O 4.) c C o CD U) r4 •r-l CD U-l CO 14-1 a •rH E X) O c_> X) C CO « o c r to a CD O O C ^ co CO a CD 4_> CO O 00 co a CD o co co a CD 4-> CO LPl O N CO O c r o O O on II rJ < t < r < f CM > > CM > \D VJO S O o o o o m • m m i n i i i T H • T-H T - l T - l 1 1 1 r l H H r l 1 l l l i T H i i i m r^- oo 1 1 1 1 CM uO 00 I I I ! T-i m vo CM c r c r oo r — m m m CM 00 r-l C O O C O T H o rH c r c r CM T- l C M CM C M m CM C O O sO \£> \0 r-- r-. >H vO vD C O C O C O C O oo O O 00 00 c r c r c r c r o o o o O o o o O O O O C O C O C O C O c r c r c r c r m m m m 1 O 1 1 1 c r oo o 1 I I I C M CM I I I 1 O oo - O C O C O C O T-H T H T-l T-l r-^ r>» r - CM CM CM C M C M CM CM C M CNJ C N CNI C N m m m m O o o o O o o o O O o o CO X> O X) o o rH 3 CO O 3 co X CO E C •rH E P5 * - C M > CM II CM > - 177 -t h e s e t r a n s i t i o n s a r e a l r e a d y e x t r e m e l y s m a l l a t L = 50. I n v i e w o f t h e r e s u l t s p r e s e n t e d i n T a b l e 7 we c l a i m t h a t t h e s t e p s i z e r e q u i r e d t o m a i n t a i n a c c u r a c y i s w e a k l y d e p e n d e n t upon L, f o r t h e r a n g e o f L v a l u e s c o n -s i d e r e d i n t h e s e c a l c u l a t i o n s . At E = 2.9 "nw, i f one c o n s i d e r s a l l p a r t i a l waves w i t h L ^ 80 , t h i s i s s u f f i c i e n t t o e v a l u a t e i n t e g r a l r o t a t i o n a l l y summed c r o s s s e c t i o n s c o n v e r g e d t o much l e s s t h a n 1 % f o r a l l t h e v i b r a t i o n a l t r a n s i t i o n s s t u d i e d . As m e n t i o n e d i n C h a p t e r 111 we b e g i n t h e i n t e g r a t i o n i n t h e c l a s s i c a l l y f o r b i d d e n r e g i o n ( i . e . we s t a r t t h e i n t e g r a t i o n a t a p o i n t where t h e i n t e r a c t i o n p o t e n t i a l i s much g r e a t e r t h a n t h e c o l l i s i o n e n e r g y ) . Thus a v a l u e o f R . d e t e r m i n e d a t E = 2.9"ftw and L = 0 s h o u l d m m be s a t i s f a c t o r y a t a l l o t h e r p a r t i a l waves, and a t a l l e n e r g i e s l e s s t h a n 2.9 -fiw. The c o n v e r g e n c e o f t h e r e s u l t s as a f u n c t i o n o f R . was e x a m i n e d k e e p i n g R c o n s t a n t m m v & m a x a t 40 a.u., and e m p l o y i n g a s t e p s i z e o f 0.01 a.u. ( c o r r e s p o n d i n g t o N = 3900 i n T a b l e 6 ) . The r e s u l t s o f o u r c a l c u l a t i o n s a r e shown i n T a b l e 8. As e x p e c t e d ^ m i n = ^ a ' u ' i - s m o r e t h a n s u f f i c i e n t t o m a i n t a i n a c c u r a c y , t h e r e s u l t s o b t a i n e d w i t h R . = 1,2 and 3 a.u. a r e mm ' i n e x c e l l e n t a g r e e m e n t f o r t h e t r a n s i t i o n s shown. D e t e r m i n i n g s a t i s f a c t o r y v a l u e s f o r R i s r a t h e r ° J max more d i f f i c u l t . E s s e n t i a l l y a t a g i v e n c o l l i s i o n e n e r g y R m a x i s d e t e r m i n e d by two f a c t o r s . I f t h e a s y m p t o t i c b o u n d a r y c o n d i t i o n i s f i t t e d i n t e r m s o f S p h e r i c a l B e s s e l f u n c t i o n s ( C h a p t e r H I ) t h e e f f e c t o f t h e c e n t r i f u g a l 'I AMU' 8 C o m p a r i s o n o f I $&=(v-a.->V& I © t ) I ( a . u . o b t a i n e d u s i n g t h e P p o t e n t i a l , and d i f f e r e n t v a l u e s o f R . a t E = 2.9 -ffw and L = 0. m i n ( a ) R . = m i n 1 a . u. 5 ( b ) R . = m m 2 a. u . , ( c ) R . mm 3 a . u . = 0° 90° 180° v 2=l-*v£ 0 (a ) ( b ) ( c ) 0 . V)4 V/ 1 -2 11 11 0 .11182H-11 11 2 0 .283534-• i 11 2 v 2 = 2 = ^ v ^ = 0 0 .203623-5 11 0 .412682-11 11 i 0 .140541-• i 11 6 v 2=2-^v£=l 0 .734923-3 i i 0 .147500-11 11 3 0 .188871-11 11 3 R = 40 a.u., X = 0.01 a.u. max ' 178 p o t e n t i a l i s i m p l i c i t l y i n c l u d e d i n t h e B e s s e l f u n c t i o n s . Thus we need t o i n t e g r a t e t h e c o u p l e d e q u a t i o n s t o a v a l u e o f R a t w h i c h t h e i n t e r a c t i o n p o t e n t i a l has max r become n e g l i g i b l e . T h e r e f o r e one c o u l d d e t e r m i n e R ° b max a t E = 1.1 "fiw and L = 0, however t h e r e i s a n o t h e r f a c t o r , as L I n c r e a s e s one f i n d s t h a t R i n c r e a s e s ( e . g . see max b T a b l e 4, C h a p t e r IV. 4 ) . When L i n c r e a s e s t h e c l a s s i c a l t u r n i n g p o i n t moves t o a l a r g e r R v a l u e . One must f i t t h e b o u n d a r y c o n d i t i o n i n t h e c l a s s i c a l l y a l l o w e d r e g i o n where t h e s o l u t i o n t o t h e S c h r o d i n g e r e q u a t i o n i s a l i n e a r c o m b i n a t i o n o f o s c i l l a t o r y f u n c t i o n s ( e . g . I I I . 4 . 2 4 ) and n o t an e x p o n e n t i a l l y d e c a y i n g f u n c t i o n . I n T a b l e 9 we show t h e c o n v e r g e n c e o f f^Vj,)/&x)/ as a f u n c t i o n o f R a t E = 1.1 -tfw and L = 0, 10, and max ' ' 20. The r e s u l t s were o b t a i n e d m a i n t a i n i n g K . - 1 a.u., b min ' and X = 0.01 a.u. I t i s c l e a r t h a t i f one m a i n t a i n s a v a l u e o f R = 2 5 a.u. a t a l l t h e v a l u e s o f L shown t h e n max one may o b t a i n r e s u l t s c o n v e r g e d t o w i t h i n a few p e r c e n t . We n o t e t h a t a l t h o u g h we m a i n t a i n t h i s v a l u e o f R f o r p a r t i a l waves w i t h L ^ 20, i t becomes p r o -max p ' K g r e s s i v e l y more d i f f i c u l t t o m a i n t a i n a c c u r a c y s i n c e t h e v a l u e s o f / $L {^.y Vz. I@z)l become e x t r e m e l y s m a l l . ( I n d e e d , a t t h i s e n e r g y we f o u n d t h a t t o e v a l u a t e an i n t e g r a l r o t a t i o n a l l y summed c r o s s s e c t i o n c o n v e r g e d t o w i t h i n 17c we o n l y n e e ded t o c o n s i d e r a l l L 3 0 ) . I n T a b l e 10 we show t h e r e s u l t s o f c a l c u l a t i o n s a t E =2.9fiw and L = 0, 50, 70, and 90 a t one CO o r i e n t a t i o n , TABLIi 9 Comparison o f | $(£ \fa.->%, \ ©a) I (a.u.) o b t a i n e d u s i n g the P p o t e n t i a l , and d i f f e r e n t v a l u e s o f R a t E = l . i -trw. m a x (a) R max = 20 a.u., (b) R max = 25 a.u R max 30 a.u. 90 o 180 o L 0 (a) 0.550275-6 (b) 0.550410-6 (c) 0.550405-6 0.801278-7 0.801220-7 0.801263-7 0.513012-7 0.513555-7 0.513326-7 10 0.188512-6 0.188535-6 0.188527-6 0.224245-7 0.224223-7 0.224235-7 0.206638-7 0.206622-7 0.206617-7 20 0.119863-8 0.119793-8 0.119787-8 0.531074-10 0.530723-10 0.530704-10 0.563807-9 0.566084-9 0.566042-9 R . = 1 min a.u., 0.01 a.u, TABLE 10 Comparison o f .u.) o b t a i n e d u s i n g the P p o t e n t i a l , and d i f f e r e n t v a l u e s o f R at r ' max E = 1.9 and 2.9 -ftw at % = o! (a) R max = 20 a.u., (b) R = 25 a max u. , (c) R max 30 a.u L = 0 50 70 90 K - 2.9 t»w V 2 = l — ^ 2 = 0 (a) (b) (c) 0.554382-2 0.554372-2 0.554371-2 0.109423-3 0.109421-3 0.109421-3 0. 0. 0. 656646-7 656648-7 656651-7 0 0 0 .279149-.276843-.276696--12 -12 -12 v2=2-*v^=0 0.102618-5 0.102612-5 0.102612-5 0.558360-12 0.558315-12 0.558320-12 v 2=2=*^=0 0.734951-3 0.734922-3 0.734924-3 0.204087-7 0.204074-7 0.204076-7 E - 1.9 -Rw v 2 = 1 " * v 2 = 0 (a) (b) (c) 0.367333-3 0.367322-3 0.367326-3 0.675062-8 0.674968-8 0.674968-8 R . = 1 a.u., X = 0.01 a.u. mi R\ 7 179 £t = 0°. The r e s u l t s i l l u s t r a t e t h a t as th e p a r t i a l wave parameter i n c r e a s e s the i n t e g r a t i o n range must be extended, however n o t i c e t h a t f o r the range o f L valu e s c o n s i d e r e d t h e i n c r e a s e i n R i s not s i g n i f i c a n t . max ° A v a l u e o f R = 25 a.u. i s a b l e t o converge the r e s u l t s max ° t o w i t h i n a few per cent at a l l the L val u e s shown (and f o r a l l the t r a n s i t i o n s shown). In a d d i t i o n one may see t h a t the v a l u e o f k does not i n c r e a s e s i g n i f i c a n t l y max J w i t h d e c r e a s i n g energy (e.g. compare E = 2.9 fiw and L = 0 w i t h the r e s u l t s f o r E = 1.1 iTw and L = 0 i n Table 9 ) . I n Table 10 we a l s o show r e s u l t s f o r E = 1.9 'fiw and L = 0, and 50 ( t h e v a l u e o f j SL(VZ)V%, I decreases r a p i d l y f o r L ^ 5 0 ) . Again a v a l u e o f R m a x = 25 a.u. i s s a t i s f a c t o r y at t h i s energy. We note t h a t the convergence o f the r e s u l t s i s s i m i l a r i n a l l o r i e n t a t i o n s , as i s i l l u s t r a t e d i n Table 9. In p r a c t i c e we s t u d i e d the dependence o f the r e s u l t s on l < m . J X ro co i - : co i n i n co ro co CM CO ro o c \ rH co VC VO o o I I W W O VO ro o rH rH CN rH ro C N rs er> ro O rH ro in r - co CM CN o o o o o , CO XI O XI OJ o r -ro C N rH rH VO o d o o o o co m CN l l w w vo o f » rH C N VD ro rH «^ C N r -r— ro vo C N C N ro rH i n rH o o o o o CNJ rH O j r H W P o g o c l o CO ro I i n o CO C N I w rH ON CO ro vo c o vo C N ro vc r -O ,CN r-CN I W ro CM CO CN CO U 1 vo o rH ro ro a W fN co i n ro n rH VD n o o CN o I W CO CN CO vo C N ro C N o ro o o o o | o CN CN I W j W ro |ro * r C N J C N I T . r> IT. i n o CO vc i n a o I i [ i W i r m r> cr. i r m r - O o a o d o C N CO H r H | C I I k id w o VD ro n co CN O CO CN r - o r -i n co ro CN CO rH r~ i n CS C N K U ro a o ro O r CO vo C N o ON vo ro i n D CO . CN O o J-J o 0) C L w CD J_J • • H 3 II • CN - J CO — — r> m H ' - 4 II CN r J , w X) E •—• )-< a) R-CN OJ II 60 c CN CO r-J l-i o •H -—-4-1 I-I u CO -—-o r-l x ; CO w a . t H OJ OJ w II r-l CN CO • J 14H i—1 O D ^—-CJ XI c a> o rH • H o 1_) E o CO r-l •H 0) II U< i-J CO c CN) > •H 0) 0) x : QJ rH X) CO f—1 CO X) OJ c 1 5 a.u. The c a l c u l a t i o n s were performed employing a s m a l l s t e p s i z e (0.01 a . u . ) , and m a i n t a i n i n g R m^ n = 1 a.u. and R m a x = 20 a.u R e s u l t s are shown f o r E = 1.1 and 2.9 -fiw a t L = 0 and a t $ 2 = 0 ° ' ^ s m a y b e s e e n t n e r e s t r i c t i o n p l a c e d on V 0 ) / y i f o r R > 1 5 a.u. has no s i g n i f i c a n t e f f e c t on the r e s u l t s . I n a d d i t i o n i t i s apparent t h a t no more than one c l o s e d channel need be i n c l u d e d a t the e n e r g i e s s t u d i e d i n these c a l c u l a t i o n s , the convergence o f the r e s u l t s as a f u n c t i o n o f the v i b r a t i o n a l b a s i s w i l l be d i s c u s s e d i n more d e t a i l i n S e c t i o n 4.1.3, however we m e n t i o n at CM W -) CO < CO e CD j_) O a x > x : C •r- l O x ) CO > CM 3 CO V c o C/5 •H U CO a e o O CNI Gb C CO O II i-J oo oo oo iXD oo oo • i II 1 i n 1 uO 1 1 ON i i rv oo CNI CM v t v f ON ON \D \D 00 00 T ON 00 OO rv rv CNI rv rv ON ON O O m 00 oo ON ON II rH rH rH rH rH rH CM O o o o O O > r-l vf 1 v t 1 v t 1 A CO oo 00 oo rH rH rH rH O o oo oo ON l o o CNI CM CNI \£> rv rv oo oo CNI vf oo oo oo 1 oo 1 CO 0O 1 3 o 1 l/O 1 m 1 O 1 O 1 rv 1 m rH rH rv \D rH CO II v f v f O O rH rH • rH rH i n m m rH oo O ^ C N I CO OO 00 00 rH • • • • • • o o o O o o by V II .—. CNI CO X) c m > — —' CD i-i > J_I oo O 11 i i—i oo CO rv rv rv rv CO O I 1 1 1 1 1 •r-l o ON O rH oo oo II CM v£5 rv m m oo C II rH ON rH rH o O 0) ,—- CO CM 00 00 rv rv i J v. CM CM CM ON ON ON o > CO OO O o vO a CNJ A . rv iv. rH rH rH i-H T ON rH rH rH rH to — • rH • • • • • • o r-l O o o o o o r—1 II •—• i—l i—l CNI > CO X) 0 W o U* > CO X> r-l — — II MH CO O C w O r-l -—• r-l 'r-l QJ OJ 4J c X> CO C CN oo v t m E 4J cO D o x : 182 t h i s stage t h a t convergence i s found t o be worst at the ' c o l l i n e a r ' o r i e n t a t i o n s , ^ £ = 0° a r*d 180° (see Table 20, S e c t i o n 4.1.3). I n c o n c l u s i o n we c o n s i d e r t h a t r e a sonable s o l u t i o n t o the above problem. I t i s p o s s i b l e t h a t t h e r e are a number o f t y p o g r a p h i c a l e r r o r s i n the paper o f van Hernert ( 1 9 8 3 ) , however we doubt t h i s . One w i l l r e c a l l t h a t u s i n g t he parameters o f Table 2, the s h o r t range p o t e n t i a l o f van Hemert (1983) i s v a l i d i n the range 3.5^.R^8.5 a.u., and t h e r e -f o r e i t i s more l i k e l y t h a t t h e p o t e n t i a l f i t i s ' b r e a k i n g down' f o r R ,>8 . 5 a.u. I n Table 13 we examine the energy dependence o f th e s e c t o r w i d t h , a comparison o f r e s u l t s o b t a i n e d employing s e v e r a l s t e p s i z e s at E = 1.1 and 2.9 'ftw and at L = 0 i s shown. G e n e r a l l y speaking the convergence o f t h e r e s u l t s as a f u n c t i o n o f the number of steps i n the i n t e g r a t i o n range i s l e s s s a t i s f a c t o r y t h a n f o r those e v a l u a t e d employing the P p o t e n t i a l ( c f . Table 6 ) , however n o t i c e t h a t t h e m a t r i x elements are much s m a l l e r i n the pr e s e n t c a l -c u l a t i o n s . Indeed, i t i s e x t r e m e l y d i f f i c u l t t o m a i n t a i n accuracy a t = 180^ where the m a t r i x elements are should be so s t r o n g l y d i m i n i s h e d at t h i s o r i e n t a t i o n , as mentioned above we b e l i e v e t h a t t h i s p o t e n t i a l i s programmed c o r r e c t l y . N e v e r t h e l e s s , such s m a l l elements sho u l d not be p a r t i c u l a r l y i m p o r t a n t , the c o n t r i b u t i o n s s e t t i n g V 0 , .* O f o r R>15 a.u. would appear t o a v e r y s m a l l . I t i s not c l e a r why the magnitude o f TABLE 13 Comparison of I S L , ( I & z ? ) l ^ ^ a - u-' ^ e v a l u a t e d u s i n g the VH p o t e n t i a l , a n d d i f f e r e n t numbers of steps over the i n t e g r a t i o n range at L = 0. (a) 390 «- t eps (b) 780 s t e p s , ( c ) 3900 s t e p s . 0° 90° 1.80° E = 1.1 fiw v 9 = l-»v'=0 (a) 0 .966552- 8 0. 491236-7 0 .446622- 11 (b) 0 .968689- 8 0. 492512-7 0 .482391- 11 (c) 0 .969240- 8 0. 492872-7 0 .492373- 11 E = 2.9 -fiw v 9 = l-»v^ = 0 (a) 0 .243481- 3 0. 424361-3 0 .129801- 5 L (b) 0 .244149- 3 0. 425246-3 0 .130127- 5 (c) 0 .244307- 3 0. 425435-3 0 .130174- 5 v 9 = 2->v'=0 0 .1568 3 3-8 0. 586880-8 0 .411421 - 13 L 0 .15 7 716-8 0. 5892V/-H 0 .41544 3- 1 3 0 .157926- 8 0. 589720-H 0 .4 1 582 3- 1 3 2-*v' = l 0 .244994- 4 0. 548479-4 0 .408167- 7 I z 0 .245715- 4 0. 549541-4 0 .409302- 7 0 .245887- 4 0. 549747-4 0 .409416- 7 1 a . u. , R = 4 0 a.u. - 183 -at & 2 ^ 0 ° and & 2 2*90° are a f a c t o r of'vLO 3 l a r g e r and are t h e r e f o r e the major c o n t r i b u t o r s (even d e s p i t e t he s m a l l s i n 9^ weight f a c t o r at Q 2 ^ 0°' • A s f ° r t n e p p o t e n t i a l , we conclude t h a t the s t e p s i z e i s not h i g h l y dependent on the energy. I n Table 14 we study the L dependence of the s t e p s i z e , r e s u l t s are shown at E = 2.9 -tiw and L = 30, 50, and 70 at $ 2 ~ 90°. At L >70 the r e s u l t s become e x t r e m e l y s m a l l , and i t i s i m p o s s i b l e t o m a i n t a i n accuracy. The r e s u l t s i n d i c a t e t h a t the VH p o t e n t i a l does not o n l y produce r e s u l t s which are very d i f f e r e n t i n magnitude t o these e v a l u a t e d u s i n g the P p o t e n t i a l , but a l s o t h e L d i s t r i b u t i o n o f r e s u l t s i s p r o b a b l y q u i t e d i f f e r e n t f o r t h e two p o t e n t i a l s , indeed we mention t h a t the range o f L v a l u e s needed t o be c o n s i d e r e d at a g i v e n energy i s s l i g h t l y s m a l l e r f o r the VH p o t e n t i a l . For example, at E = 2.9 -fiw we r e q u i r e a l l 70, compared w i t h L 80 needed i n the same c a l c u l a t i o n employing the P p o t e n t i a l ( t o ensure t h a t the i n t e g r a l c r o s s s e c t i o n s are converged t o w i t h i n 17J. From the r e s u l t s p r e s e n t e d i n Tables 13 and 14 we conclude t h a t a s t e p s i z e o f 0.05 a.u. ( c o r r e s p o n d i n g t o 780 steps i n Table 14) w i l l be s a t i s -f a c t o r y t o e v a l u a t e the r e s u l t s converged t o w i t h i n a few p e r c e n t at the en e r g i e s s t u d i e d (E = 1.1 and 2.9"Rw) and at the v a l u e s o f L s t u d i e d ( L i 70). As f o r t h e P p o t e n t i a l we de t e r m i n e a s u i t a b l e v a l u e f o r R . at the h i g h e s t c o l l i s i o n energy, and lowest p a r t i a l wave. The r e s u l t s o f our c a l c u l a t i o n s TABLE 14 2 .1 i nod Comparison of ] SL (v2y l&z^)l^~ ( a . i i . ) usLng the VII potent La L, and d i f f e r e n t numbers o f steps over the i n t e g r a t i o n range at E = 2.9-tiw. (a) 390 s t e p s , (b) 780 s t e p s , ( c ) 3900 s t e p s . ©. 2 90 o 30 50 70 V 0 = l-»v£ = 0 (a) 0.939152-4 (b) 0.940646-4 (c) 0.940927-4 0.160521-5 0.160591-5 0.160586-5 0.134979-9 0.134829-9 0.134526-9 v 2 = 2-»v 2 = 0 0.478953-10 0.480228-10 0.480441-10 0.216384-16 0.216324-16 0.359366-16 v 2 = 2=>v 2 = 1 0.202576-5 0.202800-5 0.202830-5 0.194849-10 0.194594-10 0.193257-10 - 184 -at E = 2.9 "Aw and L 0 are shown i n Table 15. The r e s u l t s e v a l u a t e d m a i n t a i n i n g R . = 1,2 and 3 a.u. to mm ' are i n e x c e l l e n t a c c o r d . A v a l u e of R . = 2 a.u. w i l l mm be employed i n a l l our c a l c u l a t i o n s . I n Table 16 we examine the convergence of the r e s u l t s as a f u n c t i o n o f R a t E = 1.1 -tfw and L = 0, and 10. max ' We mention t h a t a t t h i s energy we o n l y r e q u i r e p a r t i a l waves w i t h L-^20, the r e s u l t s shown i n Table 16 i n d i c a t e t h a t a v a l u e o f R = 2 5 a.u. i s s a t i s f a c t o r y at a l l max J the v a l u e s of L c o n s i d e r e d at t h i s energy. N o t i c e t h a t we have o m i t t e d the r e s u l t s a t = 180°, because of the s m a l l v a l ues of \ % J y - ^ \ f ^ a t t h i s o r i e n t a t i o n i t i s d i f f i c u l t t o m a i n t a i n accuracy. I n Table 17 we show the r e s u l t s of c a l c u l a t i o n s at E = 2 . 9'ft"w and a t L = 0, 50,and 70 at = °°- W e n o t i c e t h a t , a s f o r the r e s u l t s e v a l u a t e d u s i n g the P p o t e n t i a l , the v a l u e o f R r e q u i r e d t o m a i n t a i n accuracy does not v a r y max M J J s i g n i f i c a n t l y w i t h L. A l t h o u g h E = 1.1 and 2.9'ft w are the o n l y c o l l i s i o n e n e r g i e s s t u d i e d i n these c a l -c u l a t i o n s , the n u m e r i c a l methods developed i n t h i s s e c t i o n are used f o r our b r e a t h i n g sphere c a l c u l a t i o n s ( S e c t i o n 4.2). Thus we a l s o show r e s u l t s e v a l u a t e d at E = 1.9-rIw and L = 0, 50 i n Table 17, t o g i v e a b e t t e r idea o f the energy dependence of R m a x- As can be seen from Tables 16 and 17, the v a l u e of R r e q u i r e d t o e v a l u a t e ' max ^ s a t i s f a c t o r y r e s u l t s does not v a r y s i g n i f i c a n t l y w i t h e i t h e r the c o l l i s i o n energy or the p a r t i a l wave parameter. TABLE 15 Comparison o f [ Sj_^ V^ ' | I ^ ~ ( a . u . ) ^ o b t a i n e d u s i n g the VH p o t e n t i a l , and d i f f e r e n t v a l u e s o f R . at r min E = 2.9 -ftw and L = 0. (a) R . = mm 1 a . u. , (b) R . mm = 2 a.u., (c ) R . = 3 mm a.u. 0° 90° 180° v 2 = l->v' 2 = 0 (a) (b) ( c ) 0 .244307-3 0.425435 ! 1 -3 0.130173-11 5 v 2 = 2-^v^=0 0 .157926-8 11 11 0. 589720 11 -8 0.415623-11 13 v 2 = 2 - ^ v 2 = l 0 .245887-4 • i 0. 549747 ? i 11 -4 0.409416-11 11 7 R = 40 a.u., X = 0.01 a.u. TABLE 16 Comparison o f /S^ (vL^ V^/Q^)/( a . u . ) ^ o b t a i n e d u s i n g the VH p o t e n t i a l , and d i f f e r e n t v a l u e s o f R . a t K ' max E = 1.1 fiw. ) R = 2 0 a.u., (b) R = 2 5 a.u., ( c ) R = 3 0 a.u max ' max ' max 8 2 = 0° 90° L 0 (a) 0. 973249-8 0. 498876- 7 (b) 0. 976812-8 0. 491184- 7 (c ) 0. 969240-8 0. 492872- 7 10 0. 319256-8 0. 136018- 7 0. 337400-8 0. 136823- 7 0. 334605-8 0. 135893- 7 R . = 1 a.u., X = 0.01 a.u. Comparison of ISL(^LIY^&2.~)l^~ ( a . u . ) ^ o b t a i n e d u s i n g the VH p o t e n t i a l , and d i f f e r e n t v a l u e s o f R at max E = 1.9 and 2.9 -Kw at G>2 = 0 . (a) R max 20 a.u. , (b) R ' max = 25 a.u., ( c ) R max 30 a L = 0 50 70 E = 2.9 -hw v r l-*v£ = 0 (a) 0 (b) 0 (c ) 0 .244316-3 .244305-3 .244307-3 0.702591-5 0. 70 3090-5 0. 703027-5 0 0 0 . 109478 .106089 .106144 -7 -7 -7 v 2 = 2->v 2 = 0 0 0 0 .157998-8 .157962-8 .157926-8 0. 332031-14 0. 326795-14 0. 317196-14 v 2 = 2-»v£ = l 0 0 0 .245012-4 .245971-4 .245887-4 0.168242-8 0.172874-8 0.169934-8 E = 1.9 -hw v 2 = l-*v^=0 (a) 0 (b) 0 (c) 0 .125501-4 .125585-4 .125570-4 0.522292-9 0.583155-9 0.598745-9 185 I n t h i s r e s p e c t the c o n c l u s i o n s are q u a l i t a t i v e l y s i m i l a r t o those drawn f o r the P p o t e n t i a l , a l t h o u g h as may be seen the VH p o t e n t i a l w i l l p r o b a b l y produce r a t h e r d i f f e r e n t n u m e r i c a l v a l u e s f o r the cross s e c t i o n s . I t is,however, d i f f i c u l t t o be c o n c l u s i v e a t t h i s stage s i n c e not a l l the L v a l u e s have been examined, and the L d i s t r i b u t i o n s of the p a r t i a l c r o s s s e c t i o n s appear t o be v e r y d i f f e r e n t f o r the two p o t e n t i a l s ( c f . Tables 16 and 17 w i t h Tables 9 and 1 0 ) . Also i t i s not c l e a r ( d e s p i t e the s t r o n g $ 2 dependence o f the S-matrices as computed u s i n g the VH p o t e n t i a l ) t h a t when the a n g u l a r i n t e g r a t i o n (over ?)" 2) has been c a r r i e d o u t , t h a t the p o t e n t i a l s w i l l produce s i g n i f i c a n t l y d i f f e r e n t r e s u l t s at a g i v e n v a l u e of L ( a t a g i v e n c o l l i s i o n e n e r g y ) . For example, the two p o t e n t i a l s produce r a t h e r d i f f e r e n t r e s u l t s at 5^2 A? 180°, however the s i n ^ w e i g h t i n g a t such o r i e n t a t i o n s i s e x t r e m e l y s m a l l (and i s zero a t 180°), and t h e r e f o r e i t i s u n l i k e l y t h a t t h i s s i g n i f i c a n t d i f -f e r e n c e i n the s i n g l e o r i e n t a t i o n r e s u l t s w i l l have such a p r o f o u n d e f f e c t upon the f i n a l r e s u l t s . The v a l u e s o f the parameters R . , R , and X used i n these c a l -v mm' max' c u l a t i o n s are i d e n t i c a l t o those used f o r the P p o t e n t i a l ( i . e . R . = 2 a.u., R = 25 a.u., and X = 0.05 a . u . ) . mm ' max ' 4.1.3 D e s c r i p t i o n of the CO molecule I n t h i s s e c t i o n we examine the convergence of the r e s u l t s w i t h r e s p e c t t o the number o f v i b r a t i o n a l channels r e t a i n e d i n t h e f i x e d angle coupled e q u a t i o n s . - 186 -4 . 1 . 3 ( i ) The P p o t e n t i a l I n Table 18 we p r e s e n t the r e s u l t s of c a l c u l a t i o n s performed at two c o l l i s i o n e n e r g i e s , E = 1 .9 -h"w and 2 . 9 -ffw . One would expect convergence t o be worst at these c o l l i s i o n e n e r g i e s , s i n c e 1 .9 "ffw and 2 . 9 "n"w are just below the 2 a n d 3 t h r e s h o l d s r e s p e c t i v e l y . R e s u l t s are shown at t h r e e CO o r i e n t a t i o n s , = 0", 90°, 180°. One w i l l r e c a l l t h a t c l o s e d channels are r e q u i r e d t o a c c u r a t e l y d e s c r i b e the d e f o r m a t i o n o f the t a r g e t (Chapter m.4.6). Since i n t h e present study the H2 molecule i s a s p h e r i c a l l y symmetric p a r t i c l e , one may t h i n k o f the c o l l i s i o n as one i n v o l v i n g an atom and a d i a t o m i c m o l e c u l e , r a t h e r t h a n two d i a t o m i c molecules Thus the c o l l i n e a r approaches are g i v e n by = 0° and 180'', p h y s i c a l l y one would expect t h e d e f o r m a t i o n of the t a r g e t t o be g r e a t e s t at these col I i n c u r ( 'head on') c o l l i s i o n s , c e r t a i n l y t h i s would appear t o be r e f -l e c t e d i n the r e s u l t s shown i n Table 18. I t i s c l e a r from the r e s u l t s p r e s e n t e d i n Table 18 t h a t t o o b t a i n s a t i s f a c t o r y r e s u l t s , converged t o w i t h i n 170, f o r a l l the t r a n s i t i o n shown, and at a l l t h e e n e r g i e s s t u d i e d i n these c a l c u l a t i o n s , i t i s necessary t o i n c l u d e a l l the open, and one c l o s e d v i b r a t i o n a l c h a n n e l . T h i s i s c o n f i r m e d f o r the M^ = 1 t o v^ = 0 process by BSD i n t h e i r study of the CO + H 2 ( = 0) system u s i n g the DK p o t e n t i a l . T A B U ' IK C o m p a r i s o n o f ISU( Vk^Vju'/^O/ o b t a i n e d u s i n g t h e P p o t e n t i a l , and r e t a i n i n g d i f f e r e n t numbers o f v i b r a t i o n a l c h a n n e l s i n t h e c o u p l e d e q u a t i o n s a t L = 0. ( a ) 2 c h a n n e l s , ( b ) 3 c h a n n e l s , ( c ) 4 c h a n n e l s , ( d ) 5 c h a n n e l s . 0° 90° 180° E = 1.9 -ttw ( a ) 0. 367282- 3 0. 659697- 4 0. 102243- 3 v 9 = l ~ * v ' = 0 ( b ) 0. 4 2 4 8 7 1 - 3 0. 672928- 4 0. 141248- 3 z z (c ) 0. 425085- 3 0. 672930- 4 0. 141544- 3 E = 2.9 -n w ( b ) 0. 554327- 2 0. 1 1 1 8 3 1 - 2 0. 283510- 2 v 9 = l ~ * v 9 = 0 ( c ) 0. 555726- 2 0. 111832- 2 0. 285922- 2 Z z ( d ) 0. 555693- 2 0. 111825- 2 0. 285917- 2 v 9 = 2 - > v 9 = 0 0. 102593- 5 0. 412676- 7 0. 140512- 6 z z 0. 126618- 5 0. 420444- 7 0. 218819- 6 0. 1 2 7 1 6 1 - 5 0. 420880- 7 0. 221177- 6 v 9 = 2-^v^ = l 0. 734842- 3 0. 147493- 3 0. 188848- 3 z z 0. 908000- 3 0. 150288- 3 0. 305620- 3 0. 911883- 3 0. 150453- 3 0. 308867-3 jSL(^^l9%)l i n u n i t s o f ( a . u . ) - 187 -I t w i l l be r e c a l l e d t h a t i n Sect i o n 4.1.1 we compared t h e v a l u e s o f t h e p o t e n t i a l m a t r i x e l e m e n t s e v a l u a t e d u s i n g d i f f e r e n t Gauss L e g e n d r e q u a d r a t u r e s . I n T a b l e 4 r e s u l t s a r e shown c a l c u l a t e d u s i n g 9 6 - , 64-,and 48-p o l n t q u a d r a t u r e s . I t may be seen t h a t f o r t h e v i b r a t i o n a l , s t a t e s r e q u i r e d i n t h e s e c a l c u l a t i o n s , ^ 3, t h e a g r e e -ment b e t w e e n t h e r e s u l t s e v a l u a t e d u s i n g t h e s e q u a d r a t u r e s i s e x c e l l e n t ( e v e n a t R = 20 a.u. where t h e i n t e r a c t i o n p o t e n t i a l i s e x t r e m e l y s m a l l and i t i s d i f f i c u l t t o m a i n t a i n a c c u r a c y ) . S i n c e we employ an a p p r o x i m a t e p o t e n t i a l a l g o r i t h m , no e x p l i c i t r e f e r e n c e t o t h e p o t e n t i a l m a t r i x e l e m e n t s i s r e q u i r e d a f t e r t h e c a l c u l a t i o n a t t h e i n i t i a l p a r t i a l wave, t h e r e f o r e we a r c a b l e t o g e n e r a t e a l a r g e number o f r e s u l t s w i t h v e r y l i t t l e n u m e r i c a l e f f o r t . N o t i c e t h a t we have a l r e a d y shown t h a t t h e s t e p s i z e r e q u i r e d t o m a i n t a i n a c c u r a c y does n o t v a r y s i g n i f i c a n t l y w i t h L. I n t h e p r e s e n t c a l c u l a t i o n s we chose t o employ a 96- p o i n t Gauss L e g e n d r e q u a d r a t u r e , t h i s means t h a t t h e c a l c u l a t i o n a t t h e i n i t i a l p a r t i a l wave i s q u i t e e x p e n s i v e , i n T a b l e 19 we g i v e t h e a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d f o r t h e e v a l u a t i o n o f K^ ,!/^ (B"^) a t 460 v a l u e s o f R f o r V ^ / t f / ) ^ 3 u s i n g t h e IBM 370/168 c o m p u t e r a t NUMAC, however s u c h a q u a d r a t u r e e n s u r e s e x t r e m e l y h i g h n u m e r i c a l a c c u r a c y w i t h no i n c r e a s e i n c o m p u t e r t i m e a t s u b s e q u e n t p a r t i a l waves. TABLE 19 C o m p a r i s o n o f a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d t o e v a l u a t e V^,^ f ^ < 9 z ) f ° r v 2 ( v ^ ) ^ 3 a t 460 v a l u e s o f R u s i n g t h e IBM 370/168 a t NUMAC e m p l o y i n g d i f f e r e n t N - p o i n t Gauss L e g e n d r e q u a d r a t u r e s . ( a ) P P o t e n t i a l , ( b ) VH p o t e n t i a l • N 4H 64 96 T i m e ( s ) (a ) 13.27 19 . 24 27 .24 ( b ) 9.41 11.35 15.22 188 4 . 1 . 3 ( 1 1 ) The VH P o t e n t i a 1. I n T a b l e 20 we examine t h e c o n v e r g e n c e of I S^^VDVI I w i t h r e s p e c t t o t h e number o f v i b r a t i o n a l c h a n n e l s r e t a i n e d i n t h e c o u p l e d e q u a t i o n s . R e s u l t s a r e shown a t E = 1 . 1 , 1.9, and 2 . 9-+lw . The c o n v e r g e n c e o f t h e r e s u l t s as a f u n c t i o n o f i n c r e a s i n g i s i n q u a l i t a t i v e a g r e e m e n t w i t h t h e r e s u l t s o b t a i n e d w i t h t h e P p o t e n t i a l , n o t i c e i t i s e x t r e m e l y d i f f i c u l t t o m a i n t a i n a c c u r a c y a t E = 1.1-hw, and &2 = 180° be c a u s e o f t h e s m a l l m a t r i x e l e m e n t s a t t h i s o r i e n t a t i o n , as m e n t i o n e d p r e v i o u s l y , t h e e l e m e n t s a t 0 ° , and 90° a r e however a f a c t o r o f U 10 l a r g e r , and w o u l d be e x p e c t e d t o be t h e m a j o r c o n t r i b u t i o n s t o t h e f i n a l r e s u l t s . Thus we o n l y need c o n s i d e r v i b r a t i o n a l c h a n n e l s w i t h Vz.(Vji)^3 i n t h e s e c a l c u l a t i o n s , and as f o r t h e P p o t e n t i a l , e x c e l l e n t a g r e e m e n t i s f o u n d b e t w e e n t h e v a l u e s o f Vv-^v^ (^) e v a l u a t e d e m p l o y i n g t h e t h r e e Gauss L e g e n d r e q u a d r a t u r e s ( 9 6 - , 64-,and 48- p o i n t s ) . A g a i n we t a k e a d v a n t a g e o f t h e a p p r o x i m a t e p o t e n t i a l a l g o r i t h m and employ a 96-p o i n t q u a d r a t u r e . A l s o g i v e n i n T a b l e 19 a r e t h e a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d f o r t h e e v a l u a t i o n o f ( & ) ^-"^ a t 460 v a l u e s o f R f o r V^(V) ^ 3 u s i n g t h e IBM 370/168 c o m p u t e r a t NUMAC i n c a l c u l a t i o n s e m p l o y i n g t h e VH p o t e n t i a l . N o t i c e t h a t t h e s i m p l e f o r m o f t h e VH p o t e n t i a l i s r e f l e c t e d i n t h e r e l a t i v e m a g n i t u d e o f t h e c o m p u t e r t i m e s r e q u i r e d by t h e two p o t e n t i a l s . I n T a b l e 21 we g i v e t h e a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d a t t h e i n i t i a l and s u b s e q u e n t p a r t i a l waves, t o i n t e g r a t e t h e f i x e d a n g l e c o u p l e d d i f f e r e n t i a l e q u a t i o n s TABLE 20 C o m p a r i s o n o f JS^ (v^ Vjf f o b t a i n e d u s i n g t h e VH p o t e n t i a l , and r e t a i n i n g d i f f e r e n t numbers o f v i b r a t i o n a l c h a n n e l s i n t h e c o u p l e d e q u a t i o n s a t L = 0« ( a ) 2 c h a n n e l s , ( b ) 3 c h a n n e l s , (c ) 4 c h a n n e l s , (d channe1 e 90 o 180 o E = 1.1 -fiw v 9 = l - > v ^ = 0 ( a ) 0.966436-8 0.490959-7 0.329766-11 1 1 ( b ) 0.111314-7 0.552493-7 0.422946-11 ( c ) 0.111222-7 0.553562-7 0.512090-11 E = 1 . 9 -fiw v 9 = l - > v ^ = 0 ( a ) 0.125488-4 0.283992-4 0.299236-7 L ( b ) 0.146779-4 0.323814-4 0.376952-7 ( c ) 0.1468 32-4 0.32 3900-4 0.3 76H94-7 K 2 . 9 -fiw V 2 = l - > v ^ = 0 v 2 = 2 - > v ^ = 0 v 2 = 2->v^ = l ( b ) 0.244146-3 ( c ) 0.244344-3 ( d ) 0.244317-3 0.157751-8 0.189566-8 0.190486-8 0.245799-4 0.306938-4 0.308410-4 0.425232-3 0.425471-3 0.425437-3 0.589139-8 0.704364-8 0.706919-8 0.549464-4 0.661795-4 0.664163-4 0.130173-5 0.130173-5 0.130121-5 0.408489-13 0.499670-13 0.593993-13 0.410020-7 0.553685-7 0.560729-7 TABU- : 21 A c o m p a r i s o n o f a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d t o s o l v e t h e f i x e d a n g l e c o u p l e d e q u a t i o n s at one o r i e n t a t i o n u s i n g t h e C r a y - 1 c o m p u t e r . Times a r e shown f o r b o t h t h e P and VH p o t e n t i a l s a t t h e i n i t i a l and s u b s e q u e n t p a r t i a l waves. ( a ) 3 v i b r a t i o n a l c h a n n e l s , ( b ) 4 c h a n n e l s T i m e ( s ) I n i t i a l p a r t i a l S u b s equent p a r t i a l wave M r waves P p o t e n t i a l ( a ) ( b ) 3.29 3.40 0 .19 0.25 VH p o t e n t i a l (a ) ( b ) 1 . 74 1 .90 0.19 0.25 460 s t e p s i n t h e r a n g e 2 . 0 ^ R ^ 2 5 . 0 a.u. - 189 -a t one o r i e n t a t i o n u s i n g t h e C r a y - 1 c o m p u t e r . T h r e e and f o u r v i b r a t i o n a l c h a n n e l s a r e r e t a i n e d , and a p p r o x i m a t e l y 460 s t e p s o f f i x e d l e n g t h a r e t a k e n . As can be seen i t i s p o s s i b l e t o g e n e r a t e r e s u l t s a t s u b s e q u e n t p a r t i a l waves w i t h v e r y l i t t l e n u m e r i c a l e f f o r t . N o t i c e , i n a d d i t i o n , t h a t t h e c o m p u t e r t i m e r e q u i r e d a t s u b s e q u e n t p a r t i a l waves i s i n d e p e n d e n t o f t h e f o r m o f t h e i n t e r -a c t i o n p o t e n t i a l . 4.2 B r e a t h i n g Sphere C a l c u l a t i o n s I n t h e s e c a l c u l a t i o n s t h e CO m o l e c u l e i s a p p r o x i m a t e d as a b r e a t h i n g s p h e r e , i t i s c o n s i d e r e d t o be a s p h e r i c a l l y s y m m e t r i c s c a t t e r i n g p a r t n e r w h i c h i n t e r a c t s w i t h t h e H2 m o l e c u l e v i a an e f f e c t i v e , a v e r a g e d p o t e n t i a l ( V . 3 . 8 ) w h i c h i s i n d e p e n d e n t o f t h e i r r e l a t i v e o r i e n t a t i o n s . E s s e n t i a l l y t h e BS a p p r o x i m a t i o n i s a v e r y s i m p l e (one o r i e n t a t i o n ) v e r s i o n o f t h e IOS a p p r o x i m a t i o n . S i n c e a l l CO o r i e n t a t i o n s a r e e q u i v a l e n t , one o n l y needs t o s o l v e t h e f i x e d a n g l e c o u p l e d e q u a t i o n s a t one o r i e n t a t i o n . One w i l l r e c a l l t h a t i n t h e IOS a p p r o x i m a t i o n r o t a t i o n a l l y summed c r o s s s e c t i o n s a r e e v a l u a t e d u s i n g ( c f . I I . 3 . 3 0 ) . ^ty'0 J' v.4.2 S i n c e i n t h e BS a p p r o x i m a t i o n t h e T m a t r i x i s i n d e p e n d e n t o f t h e CO o r i e n t a t i o n , we may e v a l u a t e t h e i n t e g r a l i n V.4.2 a n a l y t i c a l l y t o g i v e ( c f . I I . 4 . 3 ) . 190 V.4. 3 These BS c a l c u l a t i o n s a r e p e r f o r m e d e m p l o y i n g t h e VH p o t e n t i a l , and b o t h h a r m o n i c and Morse o s c i l l a t o r w a v e f u n c t i o n s a r e u s e d t o d e s c r i b e t h e v i b r a t i o n a l m o t i o n o f t h e CO m o l e c u l e . BS c a l c u l a t i o n s e m p l o y i n g h a r m o n i c o s c i l l a t o r wavef u n c t i oris were p e r f o r m e d p r i m a r i l y i n o r d e r I n make a c o m p a r i s o n w i t h I he r e s u l t s o f D r o l s h a g e n and G i a n t u r c o ( 1 9 8 3 , 1 9 8 4 ) , we n o t e t h a t i t i s n o t w o r t h g o i n g i n t o t h e same d e g r e e o f d e t a i l r e g a r d i n g t h e c a l -c u l a t i o n o f t h e h a r m o n i c w a v e f u n c t i c n s as i s done f o r t h e Morse w a v e f u n c t i o n s . They a r e e x t r e m e l y s i m p l e f u n c t i o n s ( s e e S e c t i o n 3.2) and r e l a t i v e l y e a s y t o c a l -c u l a t e , q u a l i t a t i v e l y t h e y a r e s i m i l a r t o Morse w a v e f u n c t i o n s f o r r e a s o n a b l y low v i b r a t i o n a l s t a t e s ( v 2 ^ 5 ) . They a r e s i g n i f i c a n t o v e r t h e same ra n g e o f , t h e v i b r a t i o n a l c o o r d i n a t e , 1 . 4 ^ r-, ^ 2 . 8 a.u., and t h e e i genetic r g i es a r e r o u g h l y 2% h i g h e r t h a n t h e c o r r e s p o n d i n g Morse e i g e n -e n e r g i e s . We add t h a t h a r m o n i c o s c i l l a t o r w a v e f u n c t i o n a r e e m p l o y e d o n l y i n t h e s e c a l c u l a t i o n s , whereas t h e Morse w a v e f u n c t i o n s a r e e m p l o y e d i n t h e c a l c u l a t i o n s p r e s -e n t e d i n t h i s c h a p t e r , and i n C h a p t e r s V I and V I I . As a l r e a d y s t a t e d i n S e c t i o n 3.2 we assume a v a l u e o f *n"w = 0.269 eV f o r t h e h a r m o n i c o s c i l l a t o r b a s i s f u n c t i o n s , s i n c e t h i s v a l u e i s q u o t e d by D r o l s h a g e n and G i a n t u r c o ( 1 9 8 3 , 1 9 8 4 ) . C a l c u l a t i o n s a r e p e r f o r m e d 191 a t t e n b a r y c e n t r i c c o l l i s i o n e n e r g i e s , E , i n t h e r a n g e 1.1 «^ E < 2 . 9 - n w , where E n = (0.9 + 0.2n) -fiw ( n = 1,2,3... The v a l u e o f - f i w e m p l o y e d i n o u r IOS c a l c u l a t i o n s i s s l i g h t l y d i f f e r e n t (ftw = 0.26898 e V ) . The s m a l l d i f f e r e n c e s b e t w e e n t h e n u m e r i c a l v a l u e s o f ' f f w , and t h e Morse and h a r m o n i c e i g e n e n e r g i e s s h o u l d n o t s e r i o u s l y a f f e c t any c o m p a r i s o n s b e t w e e n t h e c a l c u l a t i o n s . I n a d d i t i o n we p e r f o r m c a l c u l a t i o n s a t two c o l l i s i o n e n e r g i e s (E = 1 . 1 , and 2.9-fiw, where «ftw = 0.26898 eV) e m p l o y i n g o u r Morse o s c i l l a t o r w a v e f u n c t i o n s . S i n c e t h e s e BS c a l c u l a t i o n s a r e e s s e n t i a l l y s i m p l e IOS c a l c u l a t i o n s , t h e n u m e r i c a l d e t a i l s d e v e l o p e d f o r o u r IOS c a l c u l a t i o n s s h o u l d be s a t i s f a c t o r y f o r t h e s e c a l c u l a t i o n s . As s t a t e d i n S e c t i o n 3.2 t h e i n t e r a c t i o n p o t e n t i a l i s j u s t t h e i s o t r o p i c (CO and o r i e n t a t i o n i n d e p e n d e n t ) t e r m . I g n o r i n g t h e CO a n i s o t r o p y t e r m s s h o u l d n o t s i g n i f i c a n t l y a f f e c t t h e v a r i a t i o n o f t h e i n t e r a c t i o n p o t e n t i a l w i t h r e s p e c t t o t h e i n t e r m o l e c u l a r s e p a r a t i o n , R, hence we w i l l e m p loy t h e v a l u e s o f R m £ n R , and X d e t e r m i n e d i n t h e p r e v i o u s s e c t i o n . I n max' 1 a d d i t i o n t h e n e g l e c t o f t h e CO a n i s o t r o p y t e r m s s h o u l d n o t s e r i o u s l y a f f e c t t h e d e s c r i p t i o n o f t h e CO m o l e c u l e ( i . e . we i n c l u d e one c l o s e d v i b r a t i o n a l c h a n n e l , and a l l open c h a n n e l s i n o u r IOS c a l c u l a t i o n s ) . T h i s l a t t e r a s s u m p t i o n i s easy t o c h e c k , i n T a b l e 22 we p r e s e n t t h e c o n v e r g e n c e o f t h e r e s u l t s w i t h r e s p e c t t o t h e number o f v i b r a t i o n a l s t a t e s i n c l u d e d i n t h e c o u p l e d e q u a t i o n s . The r e s u l t s a r e shown a t E = 1.9 -tiw and E = 2.9 *nw, TABLE 22 The c o n v e r g e n c e o f / S t£K , y V j y l w i t h r e s p e c t t o t h e number o f v i b r a t i o n a l c h a n n e l s a t L = 0. The r e s u l t s a r e e v a l u a t e d t r e a t i n g t h e r o t a t i o n a l d e g r e e o f f r e e d o m o f t h e CO w i t h i n t h e BS a p p r o x i m a t i o n , and u s i n g h a r m o n i c o s c i l l a t o r w a v e f u n c t i o n s t o a p p r o x i m a t e t h e v i b r a t i o n a l m o t i o n ( a ) 2 c h a n n e l s , ( b ) 3 c h a n n e l s , ( c ) 4 c h a n n e l s , ( d ) 5 c h a n n e l s v 2=l=»v^=0 v 2 = 2 ~ * v 2 = 0 v 2 = 2 - > v 2 = l E (-11 w) 1.9 ( a ) 0.941781-5 ( b ) 0.947348-5 ( c ) 0.947238-5 2.9 ( b ) 0.208947-3 ( c ) 0.208941-3 ( d ) 0.208910-3 0.764802-9 0.905397-9 0.909486-9 0.144718-4 0.173181-4 0.173994-4 / ^Va/>V^,0/ i n u n i t s o f ( a . u . ) 19 2 and a t L = 0, and were e v a l u a t e d u s i n g t h e h a r m o n i c wavet u n c t i o n s . As can he seen (and as e x p e c t c.d ) t h e i n c l u s i o n o f one c l o s e d v i b r a t i o n a l , c h a n n e l i n t h e c o u p l e d e q u a t i o n s i s s u f f i c i e n t t o e n s u r e t h a t t h e v a l u e s o f f o r a l l t h e t r a n s i t i o n s s t u d i e d i n t h e s e c a l c u l a t i o n s . I t i s c l e a r t h a t t o c o n v e r g e t h e r e s u l t s f o r t h e t r a n s i t i o n = 1 t o v ' = 0 one need o n l y r e t a i n t h e open v i b r a t i o n a l c h a n n e l s i n t h e r a n g e o f c o l l i s i o n e n e r g i e s s t u d i e d (E l e V ) . I t i s w o r t h r e p e a t i n g t h a t t h e s e BS c a l c u l a t i o n s a r e c o m p u t a t i o n a l l y v e r y s i m p l e , t h e i n t e r a c t i o n p o t e n t i a l t a k e s an e x t r e m e l y s i m p l e f o r m , and t h e c o u p l e d e q u a t i o n s need o n l y be s o l v e d a t one o r i e n t a t i o n a t each p a r t i a l wave. The c a l c u l a t i o n s were p e r f o r m e d on t h e IBM 370/168 c o m p u t e r a t NUMAC. Even w i t h o u t t a k i n g a d v a n t a g e o f t h e e f f i c i e n t g e n e r a t i o n o f r e s u l t s a t s u b s e q u e n t p a r t i a l w aves, u s i n g s t o r e d i n f o r m a t i o n g e n e r a t e d a t t h e i n i t i a l v a l u e o f L, t h e a p p r o x i m a t e c o m p u t e r t i m e r e q u i r e d f o r one p a r t i a l wave was o n l y 6 seconds ( r e t a i n i n g f o u r v i b r a -t i o n a l c h a n n e l s i n t h e c o u p l e d e q u a t i o n s ) . 4.3 A n g u l a r Q u a d r a t u r e I n t h e s e IOS c a l c u l a t i o n s we e v a l u a t e two t y p e s o f c r o s s s e c t i o n s . Those f o r t h e i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s i n CO f o r j 2 = 0 t o , e v a l u a t e d u s i n g a r e c o n v e r g e d t o w i t h i n a few p e r c e n t ( c f . I I . 3. 2 9 ) , or • < -1.0 -O.Q -0.* -0.4 -0.2 0.0 0.2 0.4 0.4 0.0 1.0 F i g u r e 2 The v a r i a t i o n w i t h Q-2 of t h e r e a l p a r t ( a ) and i m a g i n a r y p a r t (b) of i n u n i t s of ( a . u . ) a t E = 2 .9 -fiw and L = 0. A l s o shown ( c ) i s the ^ 2 v a r i a t i o n -1.0 -4.0 -0.4 -0.4 -4.2 0.0 0.2 0.4 0.6 0.B 1.0 0.0O4S 0.0020 O.OO 5 0.0010 o.ooos 1.0 -0.0 -0.1 2 0.0 0. J 0.4 O.A 0.0 C O S B o - 194 -a r e r e q u i r e d t o i n t e g r a t e e i t h e r t h e r e a l o r i m a g i n a r y p a r t s o f ^tCh^/^x) t h a n / ( f ^ 0 > / ^ ° C o n s e q u e n t l y we w i l l d e v o t e t h e m a j o r i t y o f t h i s s e c t i o n t o t h e e v a l u a t i o n o f t h e i n d i v i d u a l r o t a t i o n a l c r o s s s e c t i o n s , &*(Vlhr&Y'i!fz!} u s i n g t h e P p o t e n t i a l , f o r t h e v i b r a t i o n a l t r a n s i t i o n v 2 = 1 t o v 2 = 0 ( n o t e t h a t s u c h c r o s s s e c t i o n s a r e n o t p r e s e n t e d f o r v 2 = 2 t o v 2 = 1 and 0 ) . The m a j o r d i f f e r e n c e s b e t w e e n t h e s e c a l c u l a t i o n s , and t h o s e d e s c r i b e d i n C h a p t e r I V f o r t h e H 2 + H + s y s t e m a r e t h e number o f r o t a t i o n a l s t a t e s ( j ^ ) t h a t must be i n c l u d e d i n t h e c a l c u l a t i o n s , and t h e number o f o r i e n t a t i o n s a t w h i c h t h e c o u p l e d e q u a t i o n s must be s o l v e d . I'or a d i a t o m i c m o l e c u l e such as H 2, t h e r e i s a r i g o r o u s s e l e c t i o n r u l e w h i c h a l l o w s o n l y t r a n s i t i o n s forAj^ = e v e n t o o c c u r , however i n t h e case o f a h e t e r o n u c l e a r d i a t o m i c m o l e c u l e t h e r e i s no c o r r e s p o n d i n g s e l e c t i o n r u l e and c o n s e q u e n t l y odd t r a n s i t i o n s a r e a l s o p o s s i b l e . S e c o n d l y t h e r e l a t i v e l y l a r g e e n e r g y s p a c i n g o f t h e H 2 r o t a t i o n a l s t a t e s means t h a t e ven a t a c o l l i s i o n e n e r g y o f E = 4.67 eV r e l a t i v e l y few r o t a t i o n a l s t a t e s a r e e n e r g e t i c a l l y a c c e s s i b l e a f t e r t h e c o l l i s i o n has t a k e n p l a c e , and c o n s e q u e n t l y i t was o n l y n e c e s s a r y t o c o n s i d e r t h e e v a l u a t i o n o f c r o s s s e c t i o n s , S ^ ^ v ^ O * ^ ^A^ '^ 5 i n v o l v i n g r o t a t i o n a l s t a t e s up t o ^ ' = 10. I n C h a p t e r I V . 5 , T a b l e 6 c l e a r l y i l l u s t r a t e s t h a t s t a t e s w i t h J 2'^> 10 need n o t be c o n s i d e r e d f o r a ' c o m p l e t e s u m m a t i o n ' i n I V . 4 . 6 . 195 V.4.6 CO, h o w e v e r , has a f a r s m a l l e r r o t a t i o n a l c o n s t a n t t h a n H 2 ( B C Q = 1.9313 c m - 1 c f = 59.3125 c m - 1 ) ( H e r z b ^ r g ( 1 9 5 0 ) ) , and c o n s e q u e n t l y t h e r e a r e many c l o s e l y packed r o t a t i o n a l l e v e l s . Even a t t h e r e l a t i v e l y l o w c o l l i s i o n e n e r g i e s s t u d i e d i n t h e s e c a l c u l a t i o n s (E -^-leV) a c o n s i d e r a b l y l a r g e r number o f r o t a t i o n a l s t a t e s a r e a c c e s s i b l e a f t e r v i b r a t i o n a l r e l a x a t i o n has t a k e n p l a c e . F o r e xample a t E = 1.1-fiw t h e r e a r e • ^ v 34 s t a t e s a c c e s s i b l e ( a t E = 1.9 -ftw, )!f ^ 4 6 , and a t E = ' J2max ' 2.9 ^ftw, j'^max —' ^ 6 ) . ^ e m e n t i o n t h e s e a r e o n l y e s t i m a t e s , t h e e n e r g y o f t h e l e v e l s was c a l c u l a t e d u s i n g t h e s i m p l e e x p r e s s i o n ^ j ^ ' = ^QQ^2^2 + ^ ^ w n i - c n becomes p r o g r e s s i v e l y i n a c c u r a t e as i n c r e a s e s . As n o t e d i n C h a p t e r I V . 4 when one d e a l s w i t h a h e t e r o n u c l e a r t a r g e t m o l e c u l e we have no c h o i c e b u t t o e m p l o y an N - p o i n t q u a d r a t u r e i n t h e r a n g e as t h e S - m a t r i x i s n o t s y m m e t r i c a b o u t 9~2 = 7£/2, as i n t h e c a s e o f a h o m o n u c l e a r d i a t o m i c t a r g e t m o l e c u l e . T h i s i n i t s e l f f o r c e s a l a r g e i n c r e a s e i n t h e number o f o r i e n t a t i o n s a t w h i c h t h e c o u p l e d e q u a t i o n s must be s o l v e d . The q u a d r a t u r e t h a t we employ w i l l a u t o m a t i c a l l y f o r c e an u p p e r bound on t h e number o f r o t a t i o n a l ( s t a t e s t h a n c a n be i n c l u d e d i n t h e c a l c u l a t i o n s . By 196 d e f i n i t i o n i f one i s d e a l i n g w i t h an N - p o i n t q u a d r a t u r e , t h e n t h o s e p o i n t s a r e t h e z e r o s o f V/y^ o (S^yO^ t h e r e f o r e C fvzO ^ V-[j^- N) ~ O • For )2)> N t h e number o f osc i L l a t i o n s o f >° ) i s g r e a t e r t h a n N, and t h e q u a d r a t u r e g i v e s u n s a t i s f a c t o r y r e s u l t s . E v i d e n t l y i f we e mploy an N - p o i n t q u a d r a t u r e , t h e e v a l u a t i o n o f c r o s s s e c t i o n s , (TC/iO-^l^/x ) i s o n l y p o s s i b l e // f o r r o t a t i o n a l s t a t e s up t o j 2 = N - 1 . I t i s n o t w o r t h e m p l o y i n g more t h a n ~ 3 0 q u a d r a t u r e p o i n t s s i n c e t h i s e n s u r e s e x c e l l e n t c o n v e r g e n c e f o r r o t a t i o n a l l y summed c r o s s s e c t i o n s ( s e e T a b l e 2 5 ) . Thus a t t h e m a j o r i t y o f t h e c o l l i s i o n e n e r g i e s s t u d i e d i n t h e s e c a l c u l a t i o n s t h e r o t a t i o n a l b a s i s , e m p l o y e d f o r CO, w i l l be i n a d e q u a t e ( a n d c o n s e q u e n t l y t h e s u m m a t i o n o v e r j 2 i n V.4.6 w i l l be i n c o m p l e t e ) . E v i d e n t l y we must r u l e o u t t h e use o f V.4.6 t o c a l c u l a t e c r o s s s e c t i o n s w i t h i n i t i a l j 2 ^ 0 u s i n g c r o s s s e c t i o n s e v a l u a t e d f o r i n i t i a l j 2 = 0 ( a t a l l b u t p o s s i b l y che l o w e s t e n e r g y , E = 1 . 1 * f t w ) . I n T a b l e 23 we show t h e v a l u e s o f -i O X ) V- to U 3 c r to O in to 3 Q. co E O O U O a o in O a CM C I O a 0 0 CM CO o N ^ -J o o m >—1 t-J *x> 1 1 1 vO NO ^O 1 i I 1 1 1 v£! NO 1 NP IN. I N I N . i n m i n 1 1 1 O O 0 0 i I I VD O J < r 1 1 1 1 1 ..-) CO 1 1 1 ND v f c 1 1 a - . 1 , H 1 m 1 1 1 CNI l O K I I f 0 0 NO NO CM . - 1 (X) O m c o r-,i i - l *r> \£> vO NO < t I - - < t CM i H CO X) NO NO •JD IN. NO CO CO CO fN- ON o o H H \ D CM CM CM CO CO CO c o c o c o CM CsJ CNJ i — i T-l T-l CO CO CO o > v o m o o o o o o O O O o o o o o o O O O o o o < r vO 0 0 o CM NT X 5 1—1 i - l CM CM CM CM CM CM ND 1 1 | m m m i i i NO 1 ND ND I i NO NO VO I i i NO NO NO I i i OO 1 o o o o 1 1 NO vO • i i 1 ND 1 1 CO NO 1 1 1 NO CO 0 0 1 i 1 0 0 CO CM 1 o o 1 1 CO TH i i i c o T—i o o fN. P-. 0 0 o o c o o NO I N . 0 0 CX> O TH o o m o fN. < t o ON I N . CM CO CO ON O i n CM H ON NO c o i — i i n < J CO CO ON CJN T-l N t C M O N ON ON < t o o NO < t CM i n c o NO O 0 0 ON ON (N- ON O c o ON IN. ON CO CO I N . ON NO TH CM (N. CO O CO CO m IN. ON < t < t CO o o m c o ON 0 0 ND m m i n N£> < f CM 1-J CO IN. T-H i n c o NO u o ON N t m i I N . CNJ CM CM O O O i—1 TH i - l |N- | N p.. i — l i - l CM I N c o m CM CM CM 1-1 1—1 1—1 T—1 1—1 T-1 < t < * < t I N i - ^ CM CM CM NO m T—J 1—J 1 1 o o o o o o o o o O O O o a o o o o o o o CO X l O CM NO o o CM CM CM NO 1 1 NO t NO 1 >X) NO 1 1 i IN. I fN. 1 NO I N . I i IN. • 0 0 • fN. 1 fN. 1 NO NO r N i i i NO I fN. fN. 1 1 1 0 0 1 < t I o 1 0 0 1 1 CO v£> 1 o o 1 CM 1 CM 1 1 CM CM i m i m 1 CO 1 CM i i i o o fN. ON 1 m 1 1 c o m < t o o CM O ON CM m CO I-J NO vO m NO o o I N . o i - l ND CM m NO IN. CM < t ON CO CO m o o O i - i IN. CM fN. ON (N. o o IN. < t ON ON CO ON TH ON < t NO ON ND CM m • i - l •<—) - 19 7 e v L d e n t t h a t a t a g i v e n e n e r g y c o n v e r g e n c e i m p r o v e s w i t h I , . P h y s i c a l l y as I, d e c r e a s e s L he i s a b l e t o a p p r o a c h c l o s e r t o t h e CO where t h e a n i s o t r o p y o f t h e i n t e r a c t i o n p o t e n t i a l i s g r e a t e r and t h e r e f o r e xSt,C'4-/)^ 0/^.)becomes a s t r o n g e r f u n c t i o n o f . Con-s e q u e n t l y as L i n c r e a s e s f e w e r q u a d r a t u r e p o i n t s a r e r e q u i r e d t o i n t e g r a t e IyO jo^i o v e r ^ • As t h e c o l l i s i o n e n e r g y , E, i n c r e a s e s t h e k i n e t i c e n e r g y a v a i l a b l e f o r t h e c o l l i s i o n i n c r e a s e s , c o n -s e q u e n t l y t h e H 2 i s a b l e t o a p p r o a c h c l o s e r t o t h e CO, we w o u l d e x p e c t t h e r e f o r e t h a t d e c r e a s i n g E has a s i m i l a r e f f e c t on t h e c o n v e r g e n c e o f t h e c r o s s s e c t i o n s as i n c r e a s i n g L. T h i s i s i l l u s t r a t e d i n T a b l e 24 where we p r e s e n t t h e v a l u e s o f Ojjl) f o r E = 1.9-ttw and L = 0, c a l c u l a t e d u s i n g 28- and 32- p o i n t Gauss L e g e n d r e q u a d -r a t u r e s . C l e a r l y a t t h i s e n e r g y t h e c o n v e r g e n c e o f t h e r e s u l t s f o r a g i v e n v a l u e o f i s f a r s u p e r i o r t h a n f o r t h e c o r r e s p o n d i n g r e s u l t s e v a l u a t e d a t E = 2.9 "nw. F i n a l l y i n T a b l e 25 we compare t h e v a l u e s o f r o t a t i o n a l l y summed c r o s s s e c t i o n s , & ( Vz^Vg. f ° r t h e t r a n s i t i o n s V 2 = 1 t o v ^ = 0, and V 2 = 2 t o v£ = 1 and 0, o b t a i n e d u s i n g 28- and 32- p o i n t Gauss L e g e n d r e q u a d r a t u r e s . The r e s u l t s shown a r e e v a l u a t e d a t E = 2.9 -fiw and L = 0, e m p l o y i n g b o t h t h e P and VH i n t e r a c t i o n p o t e n t i a l s . N o t i c e t h a t f o r t h e P p o t e n t i a l t h e c o n v e r g e n c e o f t h e s e r o t a t i o n a l l y summed c r o s s s e c t i o n s i s f a r b e t t e r t h a n TABLE 24 C o m p a r i s o n o f IOS c r o s s s e c t i o n s c a l c u l a t e d u s i n g d i f f e r e n t Gauss L e g e n d r e q u a d r a t u r e s a t E = 1.9 +iw and L = 0 : ( a ) 28 p o i n t s ; ( b ) 32 p o i n t s . C r oss s e c t i o n s i n u n i t s o f L = 0 C~L(lO~* Oj£) L = 0 11 h = 0 ( a ) 0.27724758-7 ( b ) 0.27722092-7 h -= 14 0. 24216229-7 0. 24336987-7 h - 2 0. 18740944-7 0.18743008-7 h -^ 16 0.62067915-8 0. 58336099-8 h = 4 0.23523323-7 0.23530474-7 h -= 18 0.19206660-8 0.25222250-8 h = 6 0.40410774-7 0.40388637-7 h -= 20 0. 26893630-8 0.21472692-8 h = 8 0.61689808-8 0.61883541-8 h -= 22 0.46625192-8 0.28491694-8 h = 10 0.2 572429 7-7 0.25784212-7 h -- 24 0.15124 308-8 0.44086015-8 ! 1 h = 12 0.44926814-7 0.44883217-7 h -= 26 0.16927551-7 0.70263780-8 TABLE 2 5 C o m p a r i s o n o f I OS r o ! i on.i 1 l y summed c r o s s s e c t i o n s , CPiv,^—5^2'' c v''' U l 1 ^ e (' u s ' n & d i f f e r e n t Gauss L e g e n d r e q u a d r a t u r e s : ( a ) 28; ( b ) 32. Cro s s s e c t i o n s i n c a l c u l a t e d a t E = 2.9 -fiw and L = 0. P p o t e n t i a l VH p o t e n t i a l V 2 = = 0 ( a ) ( b ) 0. 0. 46418014-5 46487479-5 0.10024156-5 0.10100988-5 v 2 = =0 0. 0. 34516585-9 34637290-9 0.89766816-11 0.90089657-11 V 2 " * v 2 = 1 0. 0. 18626094-5 18857920-5 0.16397943-6 0.16 556 7 50-6 | - 198 -f o r t h e c o r r e s p o n d i n g i n d i v i d u a l r o t a t i o n a l c r o s s s e c t i o n s I t i s w o r t h m e n t i o n i n g t h a t s i n c e t h e summat i o n o v e r j 2 ' i s p e r f o r m e d a n a l y t i c a l l y i n V.4.5, t h e s e r o t a t i o n a l l y summed c r o s s s e c t i o n s a r e n o t r e s t r i c t e d by t h e q u a d r a t u r e u s e d t o e v a l u a t e them ( a s s u m i n g o f c o u r s e , t h a t t h e number o f q u a d r a t u r e p o i n t s i s s u f f i c i e n t t o p r o v i d e c o n v e r g e d r e s u l t s ) . C l e a r l y f o r b o t h t h e P and VH p o t e n t i a l s t h e a g r e e m e n t o f t h e r e s u l t s e v a l u a t e d u s i n g 28- and 32- p o i n t q u a d r a t u r e s i s e x c e l l e n t . I n c o n c l u s i o n we w i l l e mploy t h e f o l l o w i n g Gauss L e g e n d r e q u a d r a t u r e s i n a l l o u r IOS c a l c u l a t i o n s : 1 . 1 ^ E ^ 1 . 9 - f f w 28- p o i n t s . 2 . K E < 2.9 -tfw 32- p o i n t s , L ^ 50. 28- p o i n t s , L > 50. 5. R e s u l t s and d i s c u s s i o n I n t h i s s e c t i o n we w i l l p r e s e n t c r o s s s e c t i o n s and r a t e c o e f f i c i e n t s f o r t h e p r o c e s s o f c o l l i s i o n a l d e - e x c i t a t i o n e v a l u a t e d i n c a l c u l a t i o n s e m p l o y i n g t h e P and VH p o t e n t i a l s u r f a c e s . T a b l e 26 c o n t a i n s t h e n u m e r i c a l v a l u e s o f t h e r o t a t i o n a l l y summed c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l d e - e x c i t a t i o n p r o c e s s e s v 2 = 1 t o v£ = 0, v 2 = 2 t o v ^ = 0, and v 2 = 2 t o v 2 = 1, e v a l u a t e d i n t h e IOS c a l -c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . C r o s s s e c t i o n s f o r t h e c o r r e s p o n d i n g e x c i t a t i o n p r o c e s s e s may be d e t e r -m i n e d u s i n g t h e f o l l o w i n g d e t a i l e d b a l a n c e r e l a t i o n s h i p 2, 2-v ° v.5.1 R o t a t i o n a l l y summed c r o s s s e c t i o n s , <3"*( v9-^> vi. )( % ), 1 2 1 o f o r t h e v i b r a t i o n a l d e - e x c i t a t i o n o f C 0 by p a r a - i - ^ i n i t s g r o u n d s t a t e . The b a r y c e n t r i c e n e r g y , E, i s e x p r e s s e d i n u n i t s o f f i w = 0.26898 eV and i s r e l a t i v e t o t h e r o v i b r a t i o n a l g r o u n d s t a t e . Numbers i n b r a c k e t s a r e powers o f 10. E(-hw) 0" ( l - * 0 ) 1) 1.1 0 . 9 5 2 ( - 6 ) 1.3 0 . 1 0 3 ( - 4 ) 1.5 0 . 4 8 8 ( - 4 ) 1.7 0 . 1 5 1 ( - 3 ) 1 .9 0. 3 6 5 ( - 3 ) 2.1 0. 7 4 6 ( - 3 ) 0 . 2 2 0 ( - 9 ) 0 . 2 6 0 ( - 5 ) 2.3 0 . 1 3 5 ( - 2 ) 0 . 3 6 2 ( - 8 ) 0 . 2 5 7 ( - 4 ) 2.5 0 . 2 2 5 ( - 2 ) 0 . 2 5 2 ( - 7 ) 0 . 1 1 6 ( - 3 ) 2.7 0 . 3 4 9 ( - 2 ) 0 . 1 1 2 ( - 6 ) 0. 3 5 K - 3 ) 2.9 0 . 5 1 5 ( - 2 ) 0. 3 7 K - 6 ) 0 . 8 3 0 ( - 3 ) 199 These a r e n o t r e c i p r o c a l p r o c e s s e s and t h i s d e t a i l e d b a l a n c e t y p e c o n d i t i o n i s imposed by t h e IOS a p p r o x i -m a t i o n . Such a r e l a t i o n s h i p i s n o t u n r e a s o n a b l e i n t h e ICS a p p r o x i m a t i o n , where t h e r o t o r s l a t e s a r e c o n s i d e r e d d e g e n e r a t e . The c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l d e - e x c i t a t i o n o f CO a r e e x t r e m e l y s m a l l . An e x p l a n a t i o n can be p r o v i d e d w i t h i n t h e d i s t o r t e d wave a p p r o x i m a t i o n ( e . g . see G i a n t u r c o ( 1 9 7 9 ) ) . I n t h i s a p p r o x i m a t i o n t h e c r o s s s e c t i o n s a r e p r o p o r t i o n a l t o a r a d i a l i n t e g r a l o v e r t h e c o r r e s p o n d i n g c o u p l i n g p o t e n t i a l w e i g h t e d by t h e r a d i a l f u n c t i o n s o f t h e i n i t i a l , , and f i n a l v ' , s t a t e s . Because o f t h e low c o l l i s i o n e n e r g i e s t h e i n i t i a l w a v e f u n c t i o n i s m o d e r a t e l y o s c i l l a t i n g w h i l e t h e f i n a l w a v e f u n c t i o n i s much more r a p i d l y o s c i l l a t i n g because o f t h e r e l e a s e d e n e r g y ( e . g . f o r = 1-^0, t h e e n e r g y r e l e a s e d i s 2152 cm ^) and t h u s t h e m a g n i t u d e o f t h e o v e r l a p i n t e g r a l i s s m a l l . I n a d d i t i o n , i t i s w o r t h n o t i n g t h a t i t s m a g n i t u d e depends on a d e l i c a t e i n t e r p l a y b e t w e e n t h e two w a v e f u n c t i o n s a r o u n d t h e c l a s s i c a l t u r n i n g p o i n t . F i g u r e 3 shows t h e d i s t r i b u t i o n o f t h e M ^ = 1 t o = 0 v i b r a t i o n a l d e - e x c i t a t i o n c r o s s s e c t i o n s , E = 1 . 1 , 1.9,and 2.9"tTw, e v a l u a t e d i n t h e 10S c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . I t i s c l e a r t h a t as t h e t o t a l c o l l i s i o n e n e r g y i n c r e a s e s t h e r e i s a t e n d e n c y t o p o p u l a t e h i g h e r r o t a t i o n a l s t a t e s o f t h e g r o u n d amongst t h e f i n a l r o t a t i o n a l s t a t e s f o r x 1 0 " 7 1.8 1.6 1 .4 1.2 1.0 0.8 0.6 0.4 0.2 F i g u r e 3 0 The d i s t r i b u t i o n of the v i b r a t i o n a l d e - e x c i t a t i o n i o ~ ^ c r o s s s e c t i o n ^'(lo-^ojx.)^^) amongst t h e f i n a l r o t a t i o n a l s t a t e s , ^2 • ( a ) E = 1.1 -hw; s j 2. 5 (b) E = 1. 9 -hw; ( c ) E = 2.9 -hw. A 2 - ° t 1.5 1.0 0.5 0 x 1 0 ~ 4 4.0 3.5 3.0 2.5 2.0 1.0 0.5 0 _ 200 _ v i b r a t i o n a l s t a t e o f t h e CO m o l e c u l e . At t h e two h i g h e r c o l l i s i o n e n e r g i e s (E = 1 . 9 -ffw and 2 . 9-tfw) t h e c r o s s s e c t i o n s a r e o b v i o u s l y n o t c o n v e r g e d w i t h r e s p e c t t o t h e f i n a l r o t a t i o n a l s t a t e e v i d e n t l y s t a t e s w i t h J2^>26 s h o u l d be c o n s i d e r e d . The o s c i l l a t o r y b e h a v i o u r o f (P ( / O ) w i t h / r e s p e c t t o i s due t o t h e a n i s o t r o p y o f t h e i n t e r a c t i o n p o t e n t i a l . C o n s e q u e n t l y i t changes w i t h b o t h t h e r a n g e o f ^2 a n c * t h e c o l l i s i o n e n e r g y . I t w i l l be r e c a l l e d t h a t f o r t h e r o v i b r a t i o n a l e x c i t a t i o n o f a h o m o n u c l e a r d i a t o m i c m o l e c u l e , s u c h as 1 ^ ( e . g . see t h e r e s u l t s f o r t h e r o v i b r a t i o n a l e x c i t a t i o n o f l ^ by H + i n C h a p t e r I V ) t h e r e i s a r i g o r o u s s e l e c t i o n r u l e w h i c h o n l y a l l o w s t r a n s i t i o n s f o r w h i c h = even t o o c c u r . I n t h e case o f an " a l m o s t h o m o n u c l e a r " d i a t o m i c m o l e c u l e ( i . e . a h e t e r o n u c l e a r d i a t o m i c f o r w h i c h t h e odd a n i s o t r o p y i n t h e i n t e r a c t i o n p o t e n t i a l i s much s m a l l e r t h a n t h e even a n i s o t r o p y , as i n t h e case o f CO) t h e odd A.^ t r a n s i t i o n s a r e d i m i n i s h e d by d e s t r u c t i v e i n t e r f e r e n c e , b u t y e t have f i n i t e c r o s s s e c t i o n s . T h i s e f f e c t has been demon-s t r a t e d by McCurdy and M i l l e r ( 1 9 7 7 ) u s i n g s e m i c l a s s i c a l m e t h o d s , and i t i s w o r t h n o t i n g t h a t s i m i l a r e f f e c t s h a v e been o b s e r v e d by An d r e s e t a l . ( 1 9 8 2 ) , and S c h i n k e e t a l . ( 1 9 8 4 ) i n s t u d i e s o f t h e r o t a t i o n a l e x c i t a t i o n o f CO by D 2. I n F i g u r e 4 we show a c o m p a r i s o n b e t w e e n t h e IOS r o t a t i o n a l l y summed c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l - 7 I I L 0 403 807 1210 1613 2016 E^,( cra"^ ) F i g u r e 4 A c o m p a r i s o n of the IOS c a l c u l a t i o n s of r o t a t i o n a l l y summed c r o s s s e c t i o n s f o r the v i b r a t i o n a l d e - e x c i t a t i o n p r o c e s s = 1 t o v£ = 0,ci 1—¥• 0) , i n u n i t s of . =« e & : B S D ; = 0 — 0 - = : P r e s e n t c a l c u l a t i o n s . E j i s measured r e l a t i v e t o t h e v 2 = 1 j - ^ = 0 t h r e s h o l d . - 201 -r e l a x a t i o n p r o c e s s V 2 = 1 t o v2 = 0 c a l c u l a t e d by B a c i c , S c h i n k e and D i e r c k s e n ( 1 9 8 4 a ) ( h e r e a f t e r r e f e r r e d t o as BSD) w i t h t h e c o r r e s p o n d i n g r e s u l t s e v a l u a t e d i n t h e p r e s e n t IOS c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . I t may be s e e n f r o m F i g u r e 4 t h a t t h e c r o s s s e c t i o n s e v a l u a t e d by BSD a r e r o u g h l y a f a c t o r o f 2.5 - 3.5 l a r g e r t h a n t h e r e s u l t s o f t h e p r e s e n t c a l c u l a t i o n s , i n t h e e n e r g y r a n g e shown. BSD employ t h e DK p o t e n t i a l , use Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e t h e v i b r a t i o n a l m o t i o n o f t h e CO m o l e c u l e , and H 2 i s c o n s t r a i n e d i n i t s g r o u n d r o t a t i o n a l s t a t e . V i b r a t i o n a l t r a n s i t i o n s a r e i n d u c e d by t h e r ^ -dependence o f t h e i n t e r a c t i o n p o t e n t i a l . I f we e x p r e s s t h e i n t e r a c t i o n p o t e n t i a l i n t h e f o r m V.5.2. we may e x p a nd i n a T a y l o r s e r i e s a b o u t t h e e q u i l i b r i u m CO i n t e r n u c l e a r s e p a r a t i o n , i*i,e4 , and o b t a i n V. 5. 3 The s u c c e s s i v e t e r m s o f t h e e x p a n s i o n V.5.3 a r e r e s p o n s i b l f o r t h e t r a n s i t i o n s w i t h ^ V ^ = 0,1,2... I t w i l l be r e c a l l e d t h a t t h e r ^ - d e p e n d e n c e o f t h e P and DK p o t e n t i a l s 202 were o b t a i n e d i n r a t h e r d i f f e r e n t ways. The P p o t e n t i a l was c o n s t r u c t e d by u s i n g t h e ab i n i t i o SCF p o t e n t i a l e n e r g y p o i n t s o f P r i s s e t t e e t a l . ( 1 9 7 8 ) and F l o w e r e t a l . ( 1 9 7 9 ) . The o r i g i n a l SCF d a t a were a v a i l a b l e o n l y f o r t h e CO e q u i l i b r i u m bond d i s t a n c e , / i ^ e ^ . j a n < 3 t h e dependence on r 2 was o b t a i n e d by r e p r e s e n t i n g t h e i n t e r a c t i o n p o t e n t i a l as a s u p e r p o s i t i o n o f atom-atom i n t e r a c t i o n s . On t h e o t h e r hand t h e DK p o t e n t i a l was c o n s t r u c t e d f r o m t h e e l a b o r a t e SCF c a l c u l a t i o n s o f D i e r c k s e n and Kraemer ( 1 9 8 4 ) . These SCF p o t e n t i a l e n e r g i e s where c a l c u l a t e d a t t h r e e CO bond d i s t a n c e s ( r 2 = 1.898a.u., 2.132a.u., 2 . 2 3 4 a . u . ) . I n v i e w o f t h e d i f f e r e n t p r o c e d u r e s o f o b t a i n i n g t h e ^ - d e p e n d e n c e o f t h e i n t e r a c t i o n , t h e d i s c r e p a n c i e s seen i n F i g u r e 4 a r e p e r h a p s n o t so s u r p r i s i n g , What i s r a t h e r s u r p r i s i n g i s t h a t BSD f i n d t h a t t h e m a g n i t u d e o f t h e v i b r a t i o n a l d e - e x c i t a t i o n c r o s s s e c t i o n s a r e e x t r e m e l y d e p e n d e n t upon t h e v i b r a t i o n a l l y e l a s t i c p o t e n t i a l , V p ^ ^ e ^ ) ^ , ) , t h i s t e r m c a n n o t d i r e c t l y i n f l u e n c e t h e v i b r a t i o n a l i n e l a s t i c c r o s s s e c t i o n s ( t h e o v e r l a p i n t e g r a l i n v o l v i n g t h i s t e r m w i l l be z e r o ) . The DK p o t e n t i a l e m p l o y e d by BSD i n v o l v e s a damping f u n c t i o n o f t h e f o r m >/K'f) *J f o r V.5.4 where t h e p a r a m e t e r s and D were d e t e r m i n e d by s i m u l t a n -e o u s l y f i t t i n g r o t a t i o n a l l y i n e l a s t i c and t o t a l d i f f e r e n t i a l 203 c r o s s s e c t i o n s f o r D 2 + CO ( S c h i n k e e t a l . ( 1 9 8 4 ) ) . The f u n c t i o n V.5.4 m u l t i p l i e s t h e d i s p e r s i o n t e r m o f t h e p o t e n t i a l and any change o f t h e s e p a r a m e t e r s must be c o n s i s t e n t w i t h t h e beam d a t a . BSD f i n d t h a t c h a n g i n g D f r o m 4.46A* t o 4.70ft r e s u l t s i n a s h i f t i n t h e z e r o p o i n t o f t h e v i b r a t i o n a l e l a s t i c p o t e n t i a l , V , o f o n l y 0.08A1, however t h i s i s s u f f i c i e n t t o d e c r e a s e t h e c r o s s s e c t i o n f o r E ^ 80 cm""^ by a f a c t o r o f 1.5. The s e n s i t i v i t y d e c r e a s e s w i t h t h e c o l l i s i o n e n e r g y , t h e e x p l a n a t i o n o f t h i s b e h a v i o u r i s easy i f one r e c a l l s t h a t t h e c r o s s s e c t i o n i s m a i n l y d e t e r m i n e d by t h e t u r n i n g p o i n t r e g i o n ( a t E pz> 2000 cm t h e c r o s s s e c t i o n o n l y d e c r e a s e s by a f a c t o r o f ^ 1.1) . The z e r o p o i n t o f t h e i s o t r o p i c p a r t o f t h e P p o t e n t i a l i s r o u g h l y 0. 16A* s m a l l e r t h a n t h e DK p o t e n t i a l ( w i t h D = 4.46A*). I n o r d e r t o f i t t h e d i f f r a c t i o n o s c i l l a t i o n s o f t h e + CO t o t a l d i f f e r e n t i a l c r o s s s e c t i o n ( S c h i n k e e t a l . ( 1 9 8 4 ) ) i t s h o u l d be s h i f t e d t o l a r g e r d i s t a n c e s w h i c h w o u l d p r o b a b l y d e c r e a s e t h e c r o s s s e c t i o n s w h i c h a r e a l r e a d y a f a c t o r 2.5 - 3.5 s m a l l e r t h a n t h o s e c a l c u l a t e d by BSD. As BSD p o i n t o u t t h e s e n s i t i v i t y on t h e a b s o l u t e l o c a t i o n o f t h e e l a s t i c p o t e n t i a l p u t s a s t r o n g r e s t r i c t i o n on t h e a c c u r a c y o f ab i n i t i o o r e m p i r i c a l p o t e n t i a l s u r f a c e s . I n v i e w o f t h e p r e s e n t r e s u l t s i t i s q u e s t i o n a b l e w h e t h e r t h e e x t r a p o l a t i o n f r o m t h e e q u i l i b r i u m s e p a r a t i o n t o a r b i t r a r y CO i n t e r -n u c l e a r d i s t a n c e s , as p e r f o r m e d by P o u l s e n ( 1 9 8 2 ) , i s 204 s u f f i c i e n t l y a c c u r a t e . However, d e s p i t e t h e d i f f e r e n c e s i n t h e d e - e x c i t a t i o n c r o s s s e c t i o n s we w i l l show l a t e r i n t h i s s e c t i o n t h a t b o t h t h e P and DK p o t e n t i a l s p r o v i d e a r e a s o n a b l e d e s c r i p t i o n o f t h e e x p e r i m e n t a l r e s u l t s f o r t h e d e - e x c i t a t i o n o f CO ( v 2 = 1) by o r t h o - ^ . The e x p e r i m e n t a l d a t a a v a i l a b l e ( e . g . Andrews and Simpson ( 1 9 7 6 ) , and S t a r r e t a l . ( 1 9 7 4 ) ) i s i n t h e f o r m o f v i b r a t i o n a l r e l a x a t i o n r a t e c o e f f i c i e n t s . F o r a gas i n t r a n s l a t i o n a l e q u i l i b r i u m , t h e r a l e c o e f f i c i e n t s f o r i n d i v i d u a l p r o c e s s e s v2J2'—^ v2^2 a r e r e ^ - a t e < ^ t o t h e c o r r e s p o n d i n g c r o s s s e c t i o n s by a v e r a g i n g o v e r a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n ( s e e C h a p t e r I ) . i s t h e r e d u c e d mass o f t h e s y s t e m , U~ Iis t h e I n i t i a l r e l a t i v e v e l o c i t y o f t h e two m o l e c u l e s , k i s Boltzmann's c o n s t a n t , and T i s t h e t e m p e r a t u r e . We can o b t a i n t h e e x p r e s s i o n i n t e r m s o f an a v e r a g e o v e r , t h e i n i t i a l c o l l i s i o n e n e r g y o f t h e s y s t e m i n m o l e c u l a r s t a t e ( v 7 j 9 ) Ob v. 5.5 where 7 T V 2 by u s i n g E = -^JA-Ob - 205 -Pure r o t a t i o n a l r e l a x a t i o n ( 4 v 2 = 0) ^ s e x t r e m e l y r a p i d ( A l e x a n d e r ( 1 9 7 5 ) ) and c o n s e q u e n t l y t h e r e l a x a t i o n p r o c e s s o b s e r v e d e x p e r i m e n t a l l y i s t h e o v e r a l l r e l a x a t i o n of t h e r o t a t i o n a l , s t a t e s o f t h e v 2 = 1 s t a t e t o t h e v 2 = 0 s t a t e . W i t h i n t h e IOS a p p r o x i m a t i o n V.5.6 i s i n d e p e n d e n t o f t h e i n i t i a l r o t a t i o n a l s t a t e j 2 and t h u s i t i s e q u a l t o t h e r e l a x a t i o n r a t e a v e r a g e d o v e r j 2 i n d e p e n d e n t of t h e p a r t i c u l a r f o r m o f t h e d i s t r i b u t i o n f u n c t i o n . ( I n t h e BS a p p r o x i m a t i o n a l l r o t a t i o n a l s t a t e s a r e o m i t t e d , w i t h t h e e x c e p t i o n o f j 2 = 0 ) . We t h e r e f o r e c h ose j 2 = 0 and w r i t e t h e v i b r a t i o n a l r a t e (summed o v e r f i n a l r o t a t i o n a l s t a t e j 2 ) J*-V. 5. 7 The i n t e r p o l a t i o n o f t h e a v a i l a b l e r e s u l t s ( r o t a t i o n a l l y summed c r o s s s e c t i o n s , ^ * ( v 2 — ^ v 2 ) , f o r t h e p r o c e s s v 2 = 1 — ^ 0 ) r e q u i r e d t o e v a l u a t e t h e i n t e g r a l i n V.5.6 was p e r f o r m e d o v e r l o g ( ^ M v 2 = 1 v ' = 0 ) ) as a f u n c t i o n of l o g E^. A s p l i n e i n t e r p o l a t i o n p r o c e d u r e was e m p l o y e d t o o b t a i n t h e v a l u e s o f t h e c r o s s s e c t i o n s t o e v a l u a t e t h e i n t e g r a l by a Gauss L e g e n d r e q u a d r a t u r e . I n F i g u r e 5, we p r e s e n t t h e v a l u e s o f t h e v i b r a t i o n a l r e l a x a t i o n r a t e e v a l u a t e d u s i n g t h e r e s u l t s o f t h e I0S c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . We compare o u r r e s u l t s w i t h t h e 'V-T' s e m i c l a s s i c a l r e s u l t s o f P o u l s e n and B i l l i n g ( 1 9 8 2 ) , i n w h i c h i t i s s u p p o s e d T - K . 0 - H . 5 --15.0 -•15.5 --16.0 --16.5 1000 751 579 455 204 0.09 0.10 0.11 0.12 0.13 0. H 0. 15 0. 16 t e m p e r a t u r e , T ( K ) , :oef f i c i e n t , oi (cm->s F i g u r e 5. The v a r i a t i o n w i t h o t t h e v i b r a t i o n a l r e l a x a t i o n c : F u l l c u r v e : p r e s e n t c a l c u l a t i o n s ; b r o k e n c u r v e : P o u l s e n and B i l l i n g ( 1 9 8 2 ) ; + :Andrews and Simpson ( 1 9 7 6 ) ; j j f : S t a r r e t a l . ( 1 9 7 4 ) . 206 t h a t a l l the v i b r a t i o n a l e x c i t a t i o n energy of the CO molecule i s c o n v e r t e d i n t o r e l a t i v e t r a n s l a t i o n a l energy ( i . e . t h e molecule does not become r o t a t i o n a l l y e x c i t e d i n the c o l l i s i o n ) . I t i s w o r t h n o t i n g t h a t Poulsen and B i l l i n g (1982) a l s o employ the P p o t e n t i a l and Morse o s c i l l a t o r v i b r a t i o n a l e i g e n f u n c t i o n s . A l s o shown are the e x p e r i m e n t a l p o i n t s o f Andrews and Simpson (1976) f o r ortho-'ri^ and o f S t a r r et a l . (1974) f o r normal l ^ . I t i s more a p p r o p r i a t e t o compare the t h e o r e t i c a l r e s u l t s i n F i g u r e 5 w i t h the e x p e r i m e n t a l r e s u l t s f o r o r t h o - l ^ t han w i t h the measurements i n normal U-2' E x p e r i m e n t a l l y i t i s found t h a t the r e l a x a t i o n of CO i n p a r a - ^ i s much f a s t e r t h a n i n o r t h o - ' r ^ . The d i f f e r e n c e i s due t o the near-resonance process (V.2.4) V.5.8a which i s not p r e s e n t f o r o r t h o - l ^ , but which enhances the r e s u l t s o b t a i n e d i n normal 1 ^ . A c c o r d i n g t o Poulsen and B i l l i n g (1982) such a comparison between experiment and t h e o r y i s not unreasonable, these a u t h o r s f i n d t h a t pure v i b r a t i o n a l - t r a n s l a t i o n a l processes Co(«i-0-tH^fy) Co(vz*o} -f Hi Q,) V. 5.8b are much more e f f i c i e n t than processes i n which ^ j , ^ 0 except f o r t h e near=resonance process above. I n a d d i t i o n , they f i n d t h a t the 'V-T' cross s e c t i o n s are e s s e n t i a l l y - 207 -independent o f the i n i t i a l H2 r o t a t i o n a l s t a t e . While i n t he next c h a p t e r we show the l a t t e r statement t o be t r u e , the former c o n c l u s i o n i s shown t o be erroneous (Chapter V I ) . I n the c a l c u l a t i o n o f Poulsen and B i l l i n g ( 1 9 8 2 ) , c l a s s i c a l mechanics were used t o d e s c r i b e the r o t a t i o n o f the CO molecule and the r e l a t i v e m o t i o n of the CO and 1 ^ m o l e c u l e s . The v i b r a t i o n a l m otion of the CO was t r e a t e d quantum m e c h a n i c a l l y , t h i s s e m i c l a s s i c a l a p p r o x i m a t i o n w i l l be h e r e a f t e r r e f e r r e d t o as t h e SCA. S i m i l a r r e s u l t s t o those shown i n F i g u r e 5 have been o b t a i n e d by P r i c e et a l . (1983) f o r the He + CO system. These a u t h o r s compare the r e s u l t s of q u a n t a l c a l c u l a t i o n s , i n which the c l o s e - c o u p l i n g a p p r o x i m a t i o n i s a p p l i e d t o the v i b r a t i o n a l m o t i o n and the I0S a p p r o x i m a t i o n t o the r o t a t i o n a l m o t i o n , w i t h c a l c u l a t i o n s employing the SCA. When the sudden a p p r o x i m a t i o n i s a l s o a p p l i e d i n the s e m i c l a s s i c a l c a l c u l a t i o n s , these a u t h o r s f i n d e x c e l l e n t agreement between the two sets of c a l c u l a t i o n s o f the r a t e c o e f f i c i e n t f o r v i b r a t i o n a l r e l a x a t i o n , however when t h i s a p p r o x i m a t i o n i s removed, d i s c r e p a n c i e s are found which are i n the same sense and s i m i l a r i n magnitude t o those shown i n F i g u r e 5. I t would appear fr o m these r e s u l t s t h a t the energy-sudden comoonent of the IOS a p p r o x i m a t i o n i s r e s p o n s i b l e f o r the d i s c r e p a n c i e s between the t h e o r e t i c a l c u r v e s . R e c e n t l y many such s p e c u l a t i o n s about the a p p l i c a b i l i t y 208 o f the IOS a p p r o x i m a t i o n t o d e s c r i b e v i b r a t i o n a l r e l a x a t i o n have a r i s e n i n the l i t e r a t u r e ( e . g . iMaricq et a l . ( 1 9 3 3 ) , P r i c e et a l . ( 1 9 8 3 ) , B i l l i n g and C l a r y (1982), ana J o i i c a r d and B i l l i n g ( 1 9 8 4 ) ) . However BSD cast doubt on t h i s i n t e r p r e t a t i o n , they p o i n t out t h a t the t e s t o f the IOS a p p r o x i m a t i o n a g a i n s t the SCA i s not unambiguous s i n c e b o t h methods i n v o l v e d i f f e r e n t a p p r o x i m a t i o n s . These a u t h o r s have c a l c u l a t e d r o t a t i o n a l l y summed cross s e c t i o n s t r e a t i n g the CO r o t a t i o n a l m o t i o n w i t h i n t he CS and IOS a p p r o x i m a t i o n s . The CS cross s e c t i o n s are r o u g h l y a f a c t o r 1.5 l a r g e r than the IOS cross s e c t i o n s , and independent o f the i n i t i a l r o t a t i o n a l s t a t e o f the CO molecule f o r low j ^ ( _)2& (~>) • As '^ D p o i n t o u t , i f the IOS a p p r o x i m a t i o n Is a c c u r a t e f o r 0 i t i s d i f f i c u l t t o imagine t h a t i t i s i n a c c u r a t e by a f a c t o r 2 t o 4 f o r i n i t i a l CO r o t a t i o n a l s t a t e s lower than e i g h t o r so. I t appears, t h e r e f o r e t h a t t he d i s c r e p a n c y between the c a l c u l a t i o n s i s more l i k e l y due t o the use of c l a s s i c a l mechanics. W i t h i n the SCA the t r a n s l a t i o n a l energy due t o the quantum t r a n s i t i o n i s not r i g o r o u s l y d e f i n e d . I t i s normal t o take a symmetrized energy a c c o r d i n g t o the a r i t h m e t i c v e l o c i t y average f o r the i n i t i a l and f i n a l channels ( e .g. see P r i c e et a l . ( 1 9 8 3 ) ) . T h i s i s s a t i s f a c t o r y i f the energy due t o the quantum t r a n s i t i o n i s s m a l l compared t o the t o t a l energy, however, the energy t r a n s f e r f o r the V2 = 1 t o v ' = 0 t r a n s i t i o n ( ^ 2 1 5 2 cm~^ ) i s s i g n i f i c a n t compared w i t h the i n i t i a l 209 -t r a n s l a t i o n a l e n e r g i e s which c o n t r i b u t e most at the lower temperatures i n F i g u r e 5 ( T ^ 500 K ) . I t i s c l e a r t h a t the d i s c r e p a n c y between the two c a l c u l a t i o n s decreases w i t h i n c r e a s i n g t e m p e r a t u r e . Both the IOS a p p r o x i m a t i o n and the SCA would be expected t o improve w i t h t e m p e r a t u r e . Secondly, as BSD have shown, the t u r n i n g p o i n t r e g i o n i s i m p o r t a n t i n the c a l c u l a t i o n o f v i b r a t i o n a l l y i n e l a s t i c c r o s s s e c t i o n s . C l a s s i c a l mechanics i s u s u a l l y not a p p r o p r i a t e t o p r o p e r l y d e s c r i b e the t u r n i n g p o i n t r e g i o n , e s p e c i a l l y i f the t r a n s i t i o n p r o b a b i l i t i e s are o f the o r d e r of 1 0 - ^ - 10~^; the c l a s s i c a l l y f o r b i d d e n r e g i o n i s not probed a t a l l by the c l a s s i c a l t r a j e c t o r i e s . I n F i g u r e 6 we compare the v i b r a t i o n a l r e l a x a t i o n r a t e e v a l u a t e d u s i n g the r e s u l t s o f the IOS c a l c u l a t i o n s employing t h e P p o t e n t i a l w i t h the c o r r e s p o n d i n g r e s u l t s o b t a i n e d by BSD. The r e s u l t s o f BSD are shown f o r two v a l u e s o f the parameter D (4.46 & and 4.70 K). For r e f e r e n c e the e x p e r i m e n t a l r e s u l t s f o r o r t h o - l ^ (Andrews and Simpson ( 1 9 7 6 ) ) a r e shown. As can be seen the r e s u l t s e v a l u a t e d by BSD agree e x t r e m e l y w e l l w i t h the e x p e r i m e n t a l d a t a , e s p e c i a l l y i f a v a l u e of D = 4.70 K i s t a k e n . However i t w i l l be r e c a l l e d t h a t the CS cross s e c t i o n s e v a l u a t e d by BSD are a f a c t o r o f 1.5 l a r g e r than the c o r r e s p o n d i n g IOS c r o s s s e c t i o n s . T h e r e f o r e i f the r e s u l t s shown i n F i g u r e 6 are i n c r e a s e d by a f a c t o r of 1.5 the r e s u l t s of BSD move above the e x p e r i m e n t a l data ( n e v e r t h e l e s s the agreement w i l l s t i l l be r e a s o n a b l e ) , • ( .|» ± 4. ) UOSdllJTS pUB SM9J.pUV j o s t j u i o d x ^ u a u i T a a d x a ^Ml uMoqs o s^v (^9V'V = a ) a s g : • •—•— ' suo-t:iB-[noT.BD q u a s a a d : - • — o = a - • ( j - S g ^ - j - O I } ^ ° ' q u a i D i j j a o o u o i q e x e x a - i "tBuoxqe^qiA aq^ j o a-inqBaadaia^ q^tM U O ^ B U B A aqq j o suoTie-[nD"[BD SOI 3 ° uosf-iBdiiioo v '9 a - m S f j ooe 1 ooz 001 — i — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 1 1 J I 1 L J l _ O 00 OL 001 210 however the present r e s u l t s move c l o s e r t o the e x p e r i m e n t a l p o i n t s . There i s another f a c t o r , the s t r e n g t h o f A j ( = 2 t r a n s i t i o n s , t h i s w i l l be d i s c u s s e d i n d e t a i l i n Chapter V I . For the present d i s c u s s i o n we w i l l assume t h a t Poulsen and B i l l i n g (1982) are c o r r e c t i n t h e i r o b s e r v a t i o n t h a t t r a n s i t i o n s w i t h ^ j ^ / 0 are not i m p o r t a n t (except f o r the near-resonance process above). Thus d e s p i t e t h e d i s c r e p a n c i e s i n the cross s e c t i o n s e v a l u a t e d u s i n g th e P and DK p o t e n t i a l , and the r e s u l t i n g c r i t i c i s m o f the P p o t e n t i a l , b o t h p o t e n t i a l s p r o v i d e a r e a s o n a b l e d e s c r i p t i o n o f the e x p e r i m e n t a l r e s u l t s f o r o r t h o - l - ^ . I n Table 27 (columns 3 and 4) we compare the r e s u l t s o f IOS c a l c u l a t i o n s employing the P p o t e n t i a l and the VH p o t e n t i a l . The e l e c t r o n gas model p o t e n t i a l of van Hemert (1983) e x p l i c i t l y g i v e s the v a r i a t i o n o f the i n t e r a c t i o n p o t e n t i a l w i t h r^. However, i t was found by B i l l i n g and Poulsen (1983) t h a t t h i s p o t e n t i a l s u r f a c e g i v e s r e l a t i v e l y poor agreement w i t h the e x p e r i m e n t a l measurements f o r r o t a t i o n a l l y i n e l a s t i c s c a t t e r i n g i n t h e D2 +C0 system. Since r o t a t i o n and v i b r a t i o n are s t r o n g l y coupled i n the H2 + CO system one would not expect the VH p o t e n t i a l t o reproduce r o v i b r a t i o n a l c r oss s e c t i o n s p a r t i c u l a r l y w e l l . As can be seen the agreement o f t h e r e s u l t s , e v a l u a t e d u s i n g the VH p o t e n t i a l , w i t h t h e e x p e r i m e n t a l p o i n t s shown i n F i g u r e 5 would be u n s a t i s f a c t o r y . c o C M o W 0 0 C CT> O 1-1 o C QJ QJ Cfl c/> I — I in 3 in O O O c O r - l •r-l CO 4 J T-l CO 4-J 4-J C QJ CJ 4~l X O QJ C X I OJ T3 o •H r- l 4-1 CO o C TO < o y> H •r-i QJ 4-1 4-1 co C rJ -H rO , •r-l f* > Pd | X > O CU O e 0 £. (/> CO • H e •r-l Q> / , r-l O J D U U C X CO QJ x: QJ 4_i o e w o o QJ XI LO 4-1 o r-l O QJ x: C 4-J o -a 4 J c o co E in C O • H 4.1 u c: 3 QJ rJ X ) QJ in in QJ 1-1 a ~ ^ QJ Q J ^ W r-in QJ O a. QJ rJ CO CO 1-1 X) u-i •rJ QJ > CO 3 in QJ 'r-l 00 QJ H X> E 1-1 o • 4-1 T J CO X ) r - l QJ r - l > ^ o u r - l m a o E QJ O C QJ QJ X3 u 14-1 o QJ 4-1 CO 4_> 1/5 O — •r-l — 4_>N. OJ CO > C 4 J o o E E ~ U co i — i i/) X I co CO C £ rW o O -rl o I T CNI > C O QJ CO 0 0 V) 4-J r4 O O U o 4-J > , C J rJ rO QJ QJ E T I X QJ QJ 4-J c o C QJ 4-i co in > QJ r4 a OJ CO c o •r-l 4-1 CO u X) •r-l > o - 211 -I t I s c l e a r from Table 27 t h a t the agreement w i t h the r e s u l t s o b t a i n e d u s i n g the P p o t e n t i a l d e t e r i o r a t e s w i t h i n c r e a s i n g A\fz • For ilvf^ = 1 the r e s u l t s d i f f e r by a f a c t o r 2-2*5, however f o r 2 the agreement i s e x t r e m e l y poor. As a l r e a d y mentioned v i b r a t i o n a l t r a n s i t i o n s are induced by the ^-dependence o f the i n t e r a c t i o n p o t e n t i a l . The s u c c e s s i v e terms o f the expansion V.5.3 are r e s p o n s i b l e f o r the t r a n s i t i o n s A V j . ^ 0,1,2... While i t i s p o s s i b l e t h a t reasonable agreement i n the f i r s t o r d e r term , _ ^ ^ > a . / , might be p r e d i c t e d by two independent d e t e r m i n a t i o n s o f the i n t e r a c t i o n p o t e n t i a l , agreement i n second o r d e r and h i g h e r o r d e r d e r i v a t i v e s becomes p r o g r e s s i v e l y l e s s l i k e l y . Consequently we are not s u r p r i s e d by t h e d e t e r -i o r a t i o n of the agreement w i t h . I t i s w o r t h n o t i n g t h a t the v i b r a t i o n a l l y e l a s t i c p o t e n t i a l c a l c u l a t e d by van Hemert (1983) has a zero p o i n t which i s 0.13& g r e a t e r than t h a t of the P p o t e n t i a l . Consequently the i s o t r o p i c p a r t s o f the VH and DK p o t e n t i a l s are q u i t e s i m i l a r , and we mention t h a t t h e i s o t r o p i c p a r t s o f the two i n t e r a c t i o n p o t e n t i a l s were e s s e n t i a l l y d e r i v e d from s i m i l a r p r i n c i p a l s (as f a r as the d i s p e r s i o n energy i s c o n c e r n e d ) . N e v e r t h e l e s s th e v ^ = 1 — ^ 0 d e - e x c i t a t i o n c r o ss s e c t i o n s c a l c u l a t e d u s i n g the two p o t e n t i a l s are v e r y d i f f e r e n t . E v i d e n t l y when comparing d i f f e r e n t p o t e n t i a l s u r f a c e s the terms o f major i n t e r e s t s h o u l d be the terms t h a t are d i r e c t l y - 212 -r e s p o n s i b l e f o r the t r a n s i t i o n b e i n g s t u d i e d . I n o r d e r t o compare w i t h the c a l c u l a t i o n s of Drolshagen and G i a n t u r c o (1983, 1984), we performed c a l c u l a t i o n s i n which the r o t a t i o n a l m o t i o n of the CO was t r e a t e d w i t h i n t he b r e a t h i n g sphere (BS) a p p r o x i m a t i o n . The i n t e r a c t i o n p o t e n t i a l o f van Hemert (1983) was employed, and harmonic o s c i l l a t o r w a v e f u n c t i o n s were used t o d e s c r i b e t h e v i b r a t i o n a l m otion o f the CO m o l e c u l e . I n Table 27 (column 6) we p r e s e n t some o f the r e s u l t s o f these c a l c u l a t i o n s . Comparison o f columns 4 and 5 i n Table 27 shows t h a t c r oss s e c t i o n s computed u s i n g the BS a p p r o x i m a t i o n are r o u g h l y an o r d e r of magnitude s m a l l e r than the c o r r e s p o n d i n g IOS r e s u l t s ; the l a r g e s t d i s c r e p a n c i e s b e i n g found a t l o w e r energy. I n t h i s r e s p e c t , our c o n c l u s i o n s are i n q u a l i t a t i v e agreement w i t h these o f G i a n t u r c o e t a l . (1982) f o r the CO + He system. I n a d d i t i o n note t h a t BSD f i n d s i m i l a r agreement between r e s u l t s o b t a i n e d u s i n g t h e BS and IOS a p p r o x i m a t i o n s . As a l r e a d y mentioned th e r o t a t i o n a l - v i b r a t i o n a l c o u p l i n g i s l a r g e f o r H2 + CO, and s i n c e the BS a p p r o x i m a t i o n n e g l e c t s t h i s c o u p l i n g i t must be d i s m i s s e d as a r e a l i s t i c dynamical approach t o s t u d y the H2 + CO system. I n column 5 o f Table 27 we p r e s e n t the r e s u l t s of c a l c u l a t i o n s employing the BS a p p r o x i m a t i o n , and u s i n g Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e the v i b r a t i o n a l m o t i o n o f the CO m o l e c u l e . A comparison o f the r e s u l t s computed u s i n g the BS a p p r o x i m a t i o n , but u s i n g d i f f e r e n t d e s c r i p t i o n s o f the v i b r a t i o n a l - 213 -m o t i o n o f the CO molecule i n d i c a t e s t h a t the c a l c u l a t i o n o f r o v i b r a t i o n a l c r o s s s e c t i o n s i s l e s s s e n s i t i v e t o th e form of the v i b r a t i o n a l e i g e n f u n c t i o n s than t o the dynamical a p p r o x i m a t i o n employed or t o the form o f the i n t e r a c t i o n p o t e n t i a l . I n F i g u r e 7, we compare v a l u e s o f the v i b r a t i o n a l r e l a x a t i o n c o e f f i c i e n t , Oi^j^Q , as computed by Drolshagen and G i a n t u r c o ( 1 9 8 4 ) , w i t h our own r e s u l t s , employing th e VH p o t e n t i a l , harmonic o s c i l l a t o r w a v e f u n c t i o n s , and the BS a p p r o x i m a t i o n . The paper o f Drolshagen and G i a n t u r c o (1983) c o n t a i n s graphs which d i s p l a y t he v a r i a t i o n w i t h energy o f r o t a t i o n a l l y summed cross s e c t i o n s f o r th e v i b r a t i o n a l d e - e x c i t a t i o n processes {v^ = 1 t o vj£ = 0, and V2 = 2 t o v£ = 0 ) . A comparison of the r e s u l t s o f Drolshagen and G i a n t u r c o (1983) w i t h our own r e s u l t s f o r the t r a n s i t i o n (v£ = 1 t o v ' = 0 ) , by v i s u a l i n s p e c t i o n of the r e l e v a n t graph , shows t h a t t h e v a l u e s computed by these a u t h o r s are s y s t e m a t i c a l l y s m a l l e r than our own. We have no s a t i s f a c t o r y e x p l a n a t i o n f o r the d i s c r e p a n c i e s i n e i t h e r the d e - e x c i t a t i o n c r o s s s e c t i o n s or i n the v a l u e s o f the v i b r a t i o n a l r a t e c o e f f i c i e n t s shown i n F i g u r e 7. Since t h e cross s e c t i o n s e v a l u a t e d by Drolshagen and G i a n t u r c o (1984) are s y s t e m a t i c a l l y lower than our own, we would expect a c o r r e s p o n d i n g l o w e r i n g of the r e l a x a t i o n r a t e c o e f f i c i e n t i n the e n t i r e t emperature range shown i n F i g u r e 7. The lowest energy id 4 lOuO 7 S I )7') 244 2 04 15.0 15.5 16.0 16.5 17.0 - 1 7 . 5 -_18.5 I 1 1 1 1 1 1 1 1 0 .09 0.10 0.11 0 .12 0.13 0.14 0 .15 0 .16 o H F i g u r e 7 A comparison o f BS c a l c u l a t i o n s o f the v a r i a t i o n w i t h t e m p e r a t u r e , T(K) o f the v i b r a t i o n a l r e l a x a t i o n c o e f f i c i e n t , e M c n H s - ! ) . F u l l curve : pr e s e n t c a l c u l a t i o n s ; broken curve : Drolshagen and G i a n t u r c o ( 1 9 8 4 ) . - 214 -s t u d i e d i n t h i s c a l c u l a t i o n i s 1.1 "ftw ( c o r r e s p o n d i n g t o T * 6. Summary We have performed quantum mechanical c a l c u l a t i o n s of the r o v i b r a t i o n a l d e - e x c i t a t i o n o f CO by h^. I n the p r e s e n t s t u d y t he 1 ^ molecule i s c o n s t r a i n e d i n i t s ground r o t a t i o n a l s t a t e ( j , = 0 ) . C a l c u l a t i o n s are performed u s i n g b o t h the P p o t e n t i a l , and the VH p o t e n t i a l i n which the r o t a t i o n a l degree o f freedom o f the CO molecule i s t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n , and i n which the v i b r a t i o n a l motion i s approximated by Morse o s c i l l a t o r w a v e f u n c t i o n s . We f i n d t h a t our IOS c r o s s s e c t i o n s f o r the v i b r a t i o n a l d e - e x c i t a t i o n process = 1 t o v ' = 0 are r o u g h l y a f a c t o r o f 3 s m a l l e r t h a n t h e c o r r e s p o n d i n g r e s u l t s e v a l u a t e d by BSD employing the DK p o t e n t i a l . We are not a b l e t o g i v e a s i m p l e e x p l a n a t i o n o f t h i s o b s e r v a t i o n , however we f i n d t h e computed v a l u e s o f the r a t e c o e f f i c i e n t f o r the v i b r a t i o n a l d e - e x c i t a t i o n o f CO(v2 = 1) by o r t h o - l ^ are i n reasonable agreement w i t h the e x p e r i m e n t a l p o i n t s o f Andrews and Simpsons (1 9 7 6 ) . The r e s u l t s o b t a i n e d by BSD are i n b e t t e r agreement w i t h e x p e r i m e n t , however we note t h a t i n view o f t h e CS and IOS r e s u l t s o b t a i n e d by BSD the use o f the CS a p p r o x i m a t i o n should i n c r e a s e our IOS r a t e s by a f a c t o r o f 1.5, and thus the agreement w i t h experiment would improve. The s e m i - c l a s s i c a l r e s u l t s o f Poulsen and B i l l i n g (1982) u s i n g t he P p o t e n t i a l are i n e x c e l l e n t agreement w i t h t he e x p e r i m e n t a l r e s u l t s , 21? however, i n view o f the r e s u l t s o f BSD we suspect t h a t the s e m i - c l a s s i c a l a p p r o x i m a t i o n employed by these a u t h o r s i s not s u i t a b l e to study the vibr a t i o n a l e x c i t a t i o n o f CO by H2, and thus we c o n s i d e r t h i s agreement p u r e l y acc1 d e n t a l . The agreement between the IOS r e s u l t s o b t a i n e d u s i n g t h e P and VH p o t e n t i a l d e t e r i o r a t e s w i t h i n c r e a s i n g . We a t t r i b u t e t h i s t o the d i f f i c u l t y o f o b t a i n i n g c o n s i s t e n t h i g h e r o r d e r terms i n two independent d e r i v a t i o n s o f an i n t e r a c t i o n p o t e n t i a l . We have performed BS c a l c u l a t i o n s employing t he VH p o t e n t i a l , and u s i n g b o t h harmonic and Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e the v i b r a t i o n a l m o t i o n o f CO. The BS a p p r o x i m a t i o n g i v e s cross s e c t i o n s which are r o u g h l y an o r d e r o f magnitude s m a l l e r than the c o r r e s p o n d i n g IOS cross s e c t i o n s e v a l u a t e d u s i n g t h e same p o t e n t i a l s u r f a c e . We c o n c l u d e , i n view o f t h i s , and t he r e s u l t s o b t a i n e d by BSD, t h a t the BS a p p r o x i m a t i o n s h o u l d be r u l e d out as a p o s s i b l e means of s t u d y i n g th e v i b r a t i o n a l d e - e x c i t a t i o n o f CO by H2. I n a d d i t i o n , we c o n c l u d e , by comparing t he BS r e s u l t s o b t a i n e d u s i n g d i f f e r e n t v i b r a t i o n a l w a v e f u n c t i o n s , t h a t the c a l c u l a t i o n of r o v i b r a t i o n a l c r o s s s e c t i o n s i s l e s s s e n s i t i v e t o the form o f the v i b r a t i o n a l e i g e n f u n c t i o n s than t o the dynamical a p p r o x i m a t i o n s employed or t o the form o f the i n t e r a c t i o n p o t e n t i a l . The agreement w i t h t h e BS r e s u l t s of Drolshagen 218 and G i a n t u r c o (1983, 1984), i s u n s a t i s f a c t o r y . The reasons f o r the d i s c r e p a n c i e s between both the computed v a l u e s o f t h e cross s e c t i o n s and the r a t e c o e f f i c i e n t s f o r t he v i b r a t i o n a l d e - e x c i t a t i o n o f C0(v2 = 1) by o r t h o -H2 are u n r e s o l v e d . We co n c l u d e , i n view o f the r e s u l t s o f G i a n t u r c o e t a l . (1980) and BSD.that the r e s u l t s p r e s e n t e d by Drolshagen and G i a n t u r c o (1983, 1984) are s e r i o u s l y i n e r r o r . 219 CHAPTER VI NEAR-RESONANCE VIBRATIONAL RELAXATION OF 12ClbO IN COLLISIONS WITH PARA-H-1. I n t r o d u c t i o n As s t a t e d i n Chapter V . l , s i n c e the e a r l y experiment of Hooker and M i l l i k a n (1963) t h e r e has been c o n s i d e r a b l e i n t e r e s t i n the H2 + CO system. Measurements have been made i n o r t h o - ^ ( = 1,3,5 ) , and para-H2 ( j - ^ = 0,2,4 ) as w e l l as normal-H2 ? over a wide range o f k i n e t i c temp-e r a t u r e s . I t has been observed e x p e r i m e n t a l l y t h a t para-H2 i s more e f f i c i e n t than o r t h o - ^ i n the v i b r a t i o n a l d e a c t i v a t i o n o f C 0 ( v 2 = 1) ( M i l l i k a n and Osburg ( 1 9 6 4 ) ) . These a u t h o r s a n t i c i p a t e d t h a t t h e d i f f e r e n c e must be due t o the p o p u l a t i o n s of d i f f e r e n t r o t a t i o n a l l e v e l s , and they suggested t h a t the f o l l o w i n g near-resonance t r a n s i t i o n makes the d e a c t i v a t i o n f a s t e r i n p a r a - ^ (V.2.4). v i . 1.1 More r e c e n t l y Andrews and Simpson (1976) have p u b l i s h e d e x p e r i m e n t a l measurements f o r normal -H2, p a r a - ^ and ortho-H2 i n the temp e r a t u r e range 7 7 ^ T ^ 340K. A s a t i s f a c t o r y t h e o r e t i c a l i n t e r p r e t a t i o n o f t h i s e f f e c t has been needed f o r s e v e r a l y e a r s . The e a r l i e s t 220 a t t e m p t s t o e x p l a i n the e x p e r i m e n t a l o b s e r v a t i o n are those o f Sharma and Kern ( 1 9 7 1 ) ) , and Nelson and Roberts ( 1 9 7 3 ) . Both s t u d i e s c l o s e w i t h a r e a s o n a b l e agreement between t h e o r y and experiment a f t e r making numerous assumptions and s i m p l i f i c a t i o n s c o n c e r n i n g the dynamics. Some of the a p p r o x i m a t i o n s have been shown t o be i n a p p r o p r i a t e f o r the H2 + CO system, f o r example, the b r e a t h i n g sphere a p p r o x i m a t i o n f o r the CO r o t a t i o n a l degree o f freedom used by Nelson and Roberts (1973) and the use of s t r a i g h t -p a t h c l a s s i c a l t r a j e c t o r i e s employed by Sharma and Kern (1971) (see Chapter V.5). The l a c k o f d e t a i l e d i n f o r m a t i o n about the i n t e r a c t i o n p o t e n t i a l a t t h a t time p r o h i b i t e d any more r e a l i s t i c dynamical s t u d y . While Sharma and Kern (1971) c o n s i d e r o n l y the l o n g range m u l t i p o l e moments w i t h a hard sphere r a d i u s of 3.3A, Nelson and Roberts (1973) t r e a t the v i b r a t i o n - r o t a t i o n c o u p l i n g as a f i t parameter from the b e g i n n i n g . As d i s c u s s e d i n Chapter V . l i t i s o n l y r e c e n t l y t h a t p o t e n t i a l s u r f a c e s s u i t a b l e t o study the v i b r a t i o n a l d e - e x c i t a t i o n o f CO by H2 have become a v a i l a b l e i n the l i t e r a t u r e (Poulsen ( 1 9 8 2 ) , van Hemert ( 1 9 8 3 ) , and Bacic, Schinke, and D i e r c k s e n (1984a;). The f i r s t r e a l i s t i c d ynamical study o f the near-resonance process (VI.1.1) was c a r r i e d out by Poulsen and B i l l i n g ( 1 9 8 2 ) . These a u t h o r s c a l c u l a t e d r e l a x a t i o n r a t e s w i t h i n a s e m i - c l a s s i c a l ( c l a s s i c a l p a t h ) a p p r o x i m a t i o n employing the P p o t e n t i a l . The agreement w i t h the e x p e r i m e n t a l d a t a f o r both p a r a -221 and o r t h o - I ^ i s e x c e l l e n t . I t i s shown t h a t the d i f f e r e n c e between the o r t h o - and para- r a t e i s due t o the near-resonance process which the a u t h o r s conclude i s p r i m a r i l y induced by the l o n g range m u l t i p o l e i n t e r a c t i o n between the hexadecapole moment o f and the d i p o l e moment of CO. I t w i l l be r e c a l l e d t h a t i n the p r e v i o u s c h a p t e r we s t u d i e d t he d e - e x c i t a t i o n o f C0(v2 = 1) by c o n s t r a i n e d i n i t s ground r o t a t i o n a l s t a t e ( j ^ = 0 ) . I n the pr e s e n t c h a p t e r we extend the c a l c u l a t i o n s p r e s e n t e d i n Chapter V by t a k i n g e x p l i c i t account o f the 1 ^ r o t a t i o n a l degree o f freedom. The sudden a p p r o x i m a t i o n i s i n a p p l i c a b l e t o H2, owing t o t h e l a r g e v a l u e o f i t s r o t a t i o n a l c o n s t a n t , and t h e r e f o r e we employ the more a c c u r a t e c o u p l e d s t a t e s (CS) a p p r o x i m a t i o n (Chapter I I ) . However, i n r e l a t i v e l y h i g h energy c o l l i s i o n s i n v o l v i n g l i g h t e r c o l l i s i o n p a r t n e r s , f o r example H +, s a t i s f a c t o r y r e s u l t s can be o b t a i n e d employing t he IOS a p p r o x i m a t i o n (e.g. see Schinke and McGuire ( 1 9 7 8 a ) ) . The r o t a t i o n a l degree o f freedom of the CO i s t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n . I n Chapter V.5 i t was shown t h a t the P p o t e n t i a l p r o v i d e s a b e t t e r d e s c r i p t i o n o f r o v i b r a t i o n a l e x c i t a t i o n o f CO by H2 t h a n t h e VH p o t e n t i a l , c o n s e q u e n t l y o n l y the P p o t e n t i a l w i l l be employed i n these c a l c u l a t i o n s . Since we now take e x p l i c i t account o f the r o t a t i o n a l m o t i o n o f the H2, the H2 a n i s o t r o p y terms o f the i n t e r a c t i o n must be i n c l u d e d . I n t h i s c h a p t e r we are p r i m a r i l y concerned w i t h t h e c a l c u l a t i o n o f cross s e c t i o n s and t h e r a t e - 222 c o e f f i c i e n t f o r t he near-resonance p r o c e s s , and the subsequent c a l c u l a t i o n o f the r a t e o f d e a c t i v a t i o n o f CO by p a r a - l ^ . However, the dependence o f the r e s u l t s on the r o t a t i o n a l s t a t e o f the H2 w i l l a l s o be s t u d i e d i n some d e t a i l . Very r e c e n t l y a quantum mechanical study o f the near-resonance process was c a r r i e d out by Bacic, Schinke, and D i e r c k s e n (1984b). These a u t h o r s employed the same d e c o u p l i n g a p p r o x i m a t i o n s as used i n the present s t u d y , but employed t he DK p o t e n t i a l . The r a t e f o r p a r a - l ^ agrees t o w i t h i n a f a c t o r o f two w i t h the e x p e r i m e n t a l r a t e i n the t e m p e r a t u r e range 1 0 0 ^ T 3 0 0 K , h o w e v e r „ a l t h o u g h the r e l a x a t i o n r a t e f o r the near-resonance process i s found t o be l a r g e i t s c o n t r i b u t i o n t o the t o t a l p ara-H2 r a t e i s o n l y o f minor importance u n l i k e i n the st u d y o f Poulsen and B i l l i n g (1982). 2. D e s c r i p t i o n o f the system I n Chapter V.5 i t was shown t h a t t h e P p o t e n t i a l p r o v i d e s a b e t t e r d e s c r i p t i o n o f r o v i b r a t i o n a l e x c i t a t i o n o f CO by H2 tha n t he VH p o t e n t i a l , c o n s e q u e n t l y o n l y t h e P p o t e n t i a l w i l l be employed i n these c a l c u l a t i o n s . Since we now take e x p l i c i t account o f the r o t a t i o n a l degree o f freedom o f l - ^ , the H2 a n i s o t r o p y terms i n the i n t e r a c t i o n p o t e n t i a l must be i n c l u d e d i n the c a l c u l a t i o n s . The i n t e r a c t i o n p o t e n t i a l may be w r i t t e n i i V I . 2.1 where 223 I n k e e p i n g w i t h t he dynamical a p p r o x i m a t i o n s employed i n these c a l c u l a t i o n s ( i . e . , the r o t a t i o n a l motion o f CO i s t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n , the r o t a t i o n a l m o t i o n o f 1 ^ i s t r e a t e d w i t h i n the CS a p p r o x i m a t i o n ) , we c o n s i d e r o n l y those terms w i t h = 0 (Chapter I I . 3 . 3 ) . As i n the p r e v i o u s 1 ^ + CO c a l c u l a t i o n s the i n t e r n u c l e a r s e p a r a t i o n o f the 1 ^ molecule i s c o n s t r a i n e d a t i t s e q u i l i b r i u m v a l u e fj^z^ where The p o t e n t i a l may now be w r i t t e n VI .2.2 where 31 — ^ 2 - * The c o o r d i n a t e system i s t h a t d e f i n e d by f i g u r e 1 of Chapter V.2, see Chapter V.2 f o r d e t a i l s o f the P p o t e n t i a l . As mentioned i n Chapter V.2 t h e r e i s an e r r o r i n the l o n g range m i i t i p o l e moment i n t e r a c t i o n s g i v e n i n the paper o f Poulsen (1 9 8 2 ) . These i n t e r a c t i o n s are g i v e n by ( e . g . see Flower et a l . ( 1 9 7 9 ) , L e a v i t t (1980)) ( c f . V.2.7) V = -i* Hy° k*" r,T") V I . 2 . 3 - 224 -where (S^ and (J^y are the quadrupole and hexadecapole Co oo moments of the molecule r e s p e c t i v e l y , a n d ^ and cy are the d i p o l e and quadrupole moments of the CO molecule r e s p e c t i v e l y and where VI .2.4 I n t he paper o f Poulsen (1982) t he f a c t o r o f 1/2 i s o m i t t e d f r o m VI.2.4 ( B a c i c , Schinke, and D i e r c k s e n (1984b)). This e r r o r was pr e s e n t i n our computer programme, and t h e r e f o r e some o f the cross s e c t i o n s p r e s e n t e d i n t h i s c h a p t e r a r e i n e r r o r . The e f f e c t s o f the above e r r o r are d i s c u s s e d i n d e t a i l i n Appendix 4 a t the end o f t h i s t h e s i s . The v i b r a t i o n a l b a s i s f u n c t i o n s o f the CO were taken t o be the Morse o s c i l l a t o r w a v e f u n c t i o n s employed i n th e IOS c a l c u l a t i o n s p r e s e n t e d i n Chapter V. The d e t a i l s o f these w a v e f u n c t i o n s are g i v e n i n Chapter V.3, and t h e i r c a l c u l a t i o n i s d i s c u s s e d i n Appendix 3 at the back o f t h i s t h e s i s . The major o b j e c t o f these c a l c u l a t i o n s i s t o e v a l u a t e c r o s s s e c t i o n s and r a t e c o e f f i c i e n t s f o r the near-resonance p r o c e s s . VI.2.5 The energy d e f e c t , A E , a s s o c i a t e d w i t h a near-resonance process i s e x t r e m e l y i m p o r t a n t . Small changes i n can l e a d t o s i g n i f i c a n t changes i n the magnitude o f the - 22 5 -near-resonance c r o s s s e c t i o n s (e.g. see Bacic, Schinke, and D i e r c k s e n ( 1 9 8 4 b ) ) . I t was t h e r e f o r e , c o n s i d e r e d i m p o r t a n t t h a t A E was as c l o s e as p o s s i b l e t o va l u e s quoted i n the l i t e r a t u r e ( e . g . , Andrews and Simpson (19 7 6 ) , Poulsen and B i l l i n g ( 1 9 8 2 ) ) . B a c i c , Schinke, and Di e r c k s e n (1984b) ( h e r e a f t e r r e f e r r e d t o as BSD2) employed a v a l u e f o r the d e f e c t which i s r o u g h l y a f a c t o r o f 2 too h i g h , and on the o p p o s i t e s i d e o f resonance. I n Appendix 4, i n a d d i t i o n t o d i s c u s s i n g the e r r o r i n our p o t e n t i a l r o u t i n e we w i l l p resent the r e s u l t s o f BSD2 o b t a i n e d employing the P p o t e n t i a l . I n p a r t i c u l a r i t w i l l be shown t h a t t h e erroneous energy d e f e c t used by BSD2 leads t o an o v e r e s t i m a t i o n o f the cross s e c t i o n s f o r the near-resonance process by a f a c t o r o f ^ 2 . The v a l u e s o f the energy d e f e c t s employed by the above a u t h o r s are g i v e n i n Table 1. There are a number o f sources i n the l i t e r a t u r e from which one may o b t a i n values f o r the v i b r a t i o n a l e i g e n e n e r g i e s ( e . g . , Herzhtfrg ( 1 9 5 0 ) , Mies ( 1 9 6 4 ) , Mizushima (1975)) and v a l u e s f o r the r o t a t i o n a l e i g e n e n e r g i e s ( e . g . , Herzb^rg ( 1 9 5 7 ) , Mizushima (19 7 5 ) , Flower and Launay ( 1 9 7 7 ) ) . I n Table 2 we pre s e n t the v a l u e s o f the energy d e f e c t , A E , r e s u l t i n g from a number o f co m b i n a t i o n s o f th e above v i b r a t i o n a l and r o t a t i o n a l e n e r g i e s . I t was dec i d e d t o employ the e i g e n e n e r g i e s c a l c u l a t e d u s i n g the data from Herzb^rg ( 1 9 5 0 ) , s i n c e these ensure a reasonable v a l u e f o r the d e f e c t . I t w i l l be r e c a l l e d t h a t i n our p r e v i o u s s t u d y o f H 2 + CO i n Chapter V we employed the TABLE 1 Values of the Energy D e f e c t , i & E , employed by v a r i o u s a u t h o r s . A E(cm ^) Andrews and Simpson (1976) 82.0 Poulsen and B i l l i n g (1982) 83.3 Bacic, Schinke, and D i e r c k s e n (1984b) -43.3 TABLE 2 Values o f the Energy D e f e c t ,A E ( c m ^ ) , c a l c u l a t e d u s i n g v i b r a t i o n a l and r o t a t i o n a l e i g e n e n e r g i e s t a k e n from the l i t e r a t u r e . V i b r a t i o n a l Herzh*rrg Mizushima Mies Mies Eige n e n e r g i e s Rotat i o n a l Herzbjdrg Mizushima Herzberg Mizushima A E (cm 87 .03 91.41 95.98 100.36 226 -Morse v i b r a t i o n a l e i g e n e n e r g i e s g i v e n by Mies (1964). We c o n s i d e r t h a t t h i s s l i g h t i n c o n s i s t e n c y should have an i n s i g n i f i c a n t e f f e c t on the 'V-T" cross s e c t i o n s ( i . e . , any comparisons t h a t are made w i t h t he r e s u l t s o f Chapter V s h o u l d be v a l i d ) . I n c o n c l u s i o n t he v i b r a t i o n a l and r o t a t i o n a l e i g e n e n e r g i e s are c a l c u l a t e d u s i n g : -V i b r a t i o n a l E i g e n e n e r g i e s VI.2.6 where UTQ = 2156. 7721cm" 1, UT€^ = 13.4148 cm - 1, and M-Y = 0.0303 cm - 1. R o t a t i o n a l E i g e n e n e r g i e s V I . 2. 7 where B = 59.3125 cm 1 and D = 0.0456552 cm - 1, o o The e q u i l i b r i u m CO i n t e r n u c l e a r s e p a r a t i o n i s ta k e n t o be The reduced masses o f the CO and + CO system are taken t o be JU/co = 12499.6 a.u., and JbL = 3427.79 a.u. r e s p e c t i v e l y . See Chapter V.3 f o r more d e t a i l s . 3. Numerical Methods We exte n d t h e study o f the v i b r a t i o n a l d e - e x c i t a t i o n o f C0(v2 = 1) by p a r a - ^ t j ^ = 0) p r e s e n t e d i n Chapter V by p e r f o r m i n g c a l c u l a t i o n s i n which t he r o t a t i o n a l degree o f freedom o f the ^ molecule i s t r e a t e d e x p l i c i t l y 227 We are p r i m a r i l y i n t e r e s t e d i n the c a l c u l a t i o n o f cross s e c t i o n s f o r the near-resonance process ( V I . 2 . 5 ) and f o r the 'V-T' process i n which = = 2. We note t h a t we examine the near-resonance process i n a model c a l c u l a t i o n i n which o n l y the d i r e c t l y r e l e v a n t r o t a t i o n a l s t a t e s = 2 and 6 are i n c l u d e d i n the coupled e q u a t i o n s . T h i s i s d i s c u s s e d i n S e c t i o n 4.2.1. R e l a x a t i o n r a t e c o e f f i c i e n t s e v a l u a t e d u s i n g cross s e c t i o n s c a l c u l a t e d f o r these processes are used t o g e t h e r w i t h t he r a t e c o e f f i c i e n t e v a l u a t e d i n Chapter V f o r the 'V-T' process i n which = j | = 0 t o c a l c u l a t e t h e r a t e c o e f f i c i e n t f o r t h e d e - e x c i t a t i o n o f C0(v2 = 1) by p a r a - H 2 . The above processes are s t u d i e d a t s i x e n e r g i e s , E^ , = 64.8322, 83.0, 129.915, 208.0, 2299.36, and 3167.14 cm - 1, where Ej denotes the i n i t i a l k i n e t i c energy f o r the c o l l i s i o n ( i . e . r e l a t i v e t o the v 2 = 1 = 2 t h r e s h o l d ) . I n a d d i t i o n we s t u d y the dependence o f the v i b r a t i o n a l d e - e x c i t a t i o n o f CO ( v 2 = 1) on the r o t a t i o n a l s t a t e s o f H 2. The ' V - T ' ( j ^ = 2) cross s e c t i o n s and r a t e s are used t o g e t h e r w i t h the 'V-T' c a l c u l a t i o n s e v a l u a t e d i n Chapter V t o study the dependence o f the v i b r a t i o n a l d e - e x c i t a t i o n process v 2 = 1=#>0 on the i n i t i a l r o t a t i o n a l s t a t e o f H 2. Also we e v a l u a t e cross s e c t i o n s f o r the Aj-^ = 0 and 2 r o t a t i o n a l t r a n s i t i o n s i n the same c a l c u l a t i o n s t o examine the r e l a t i v e magnitude o f = 2 c r o s s s e c t i o n s compared t o A j ^ = 0 cro s s s e c t i o n s , and a l s o t o examine the e f f e c t o f the = 2 t r a n s i t i o n s on the 'V-T' r e s u l t s . 228 I n t h i s s e c t i o n we s h a l l present the n u m e r i c a l methods r e q u i r e d f o r the e v a l u a t i o n of the cross s e c t i o n s f o r th e above processes. Where p o s s i b l e the n u m e r i c a l methods developed f o r the c a l c u l a t i o n s p r e s e n t e d i n Chapter V w i l l be employed. 3.1 S o l u t i o n of the coupled e q u a t i o n s at f i x e d o r i e n t a t i o n I n these c a l c u l a t i o n s we a g a i n employ the R-matrix p r o p a g a t o r method t o s o l v e the f i x e d angle coupled e q u a t i o n s , u s i n g p r o p a g a t o r s c o r r e s p o n d i n g t o a c o n s t a n t r e f e r e n c e p o t e n t i a l (Chapter I I I . 4 . 4 ) . A f i x e d s t e p s i z e i s employed i n a l l of the c a l c u l a t i o n s . I n these c a l c u l a t i o n s we s h a l l work i n terms o f k i n e t i c b a r y c e n t r i c e n e r g i e s r a t h e r t h a n t o t a l c o l l i s i o n e n e r g i e s , s i n c e f o r two d i f f e r e n t processes the t o t a l e n e r g i e s may be v e r y d i f f e r e n t , w h i l s t the i n i t i a l (and p o s s i b l y f i n a l ) k i n e t i c e n e r g i e s are i d e n t i c a l . For example c o n s i d e r the f o l l o w i n g p r o c e s s , where o n l y one r o t a t i o n a l s t a t e i s r e t a i n e d i n the c o u p l e d th e i n i t i a l k i n e t i c energy by , t h e n t h e c o r r e s p o n d i n g t o t a l energy i s g i v e n by V I . 3 . 1 e q u a t i o n s ( i . e . = ) . I f we l e t j * = 0, and denote VI.3. 2 - 22.9 -where E ( v 2 = 1) i s the energy o f the f i r s t e x c i t e d v i b r a t i o n a l s t a t e o f CO ( r e l a t i v e t o v 2 = 0 ) . I f we l e t j 1 = 2 and m a i n t a i n the same i n i t i a l k i n e t i c energy, , t h e n the c o r r e s p o n d i n g t o t a l energy i s now g i v e n by V VI.3.3 where E ( j ^ = 2) i s the energy of the second e x c i t e d r o t a t i o n a l s t a t e o f H 2 above = 0. A l t h o u g h the k i n e t i c e n e r g i e s o f the two processes are i d e n t i c a l , the t o t a l e n e r g i e s d i f f e r by E ( j ^ = 2 ) . Given these s i m p l e processes, and g i v e n t h a t t he i n i t i a l k i n e t i c e n e r g i e s are i d e n t i c a l , the s t e p s i z e , and i n t e g r a t i o n range d e t e r m i n e d f o r the process I n which j-j = 0 should be adequate t o study the process w i t h = 2. The a d d i t i o n o f the H 2 a n i s o t r o p y terms i n t o t h e p o t e n t i a l s u r f a c e s h o u l d have an i n s i g n i f i c a n t e f f e c t on t h e v a r i a t i o n of the i n t e r a c t i o n p o t e n t i a l as a f u n c t i o n o f R. Below we s h a l l g e n e r a l i s e our arguments t o more c o m p l i c a t e d s i t u a t i o n s ( i . e . where t h e r e i s more t h a n one r o t a t i o n a l c h a n n e l ) , and t h e r e f o r e use n u m e r i c a l i n t e g r a t i o n parameters determined f o r n o n - r o t a t i n g H 2 ( = 0) i n c a l c u l a t i o n s where the r o t a t i o n a l degree o f freedom o f the H 2 molecule i s e x p l i c i t l y t r e a t e d . Consequently where p o s s i b l e we s h a l l use the s t e p s i z e and i n t e g r a t i o n range d e t e r m i n e d i n Chapter V.4.1. I t w i l l be r e c a l l e d t h a t i n Chapter V.4 the l o w e r l i m i t o f t h e i n t e g r a t i o n range, R . , was d e t e r m i n e d - 2 30 -a t t h e h i g h e s t t o t a l c o l l i s i o n energy, E = 6291.4021 cm-"'" (which i s a l s o the h i g h e s t t r a n s l a t i o n a l e n e r g y ) . As s t a t e d i n Chapter V.4 the v a l u e o f R . = 2 a.u. s h o u l d be s a t i s -v mm f a c t o r y a t a l l e n e r g i e s ^ 6291.4021 cm-"'" s i n c e we are r e q u i r e d t o s t a r t t h e i n t e g r a t i o n i n the c l a s s i c a l l y f o r b i d d e n r e g i o n where the i n t e r a c t i o n p o t e n t i a l i s much g r e a t e r than t h e k i n e t i c energy. I t may be shown t h a t t h e maximum k i n e t i c energy i n these c a l c u l a t i o n s i s 5310.5314 cm-"'", and co n s e q u e n t l y t he v a l u e o f R . = 2 a.u. ' M 7 min s h o u l d be s a t i s f a c t o r y f o r these c a l c u l a t i o n s , and w i l l be used t h r o u g h o u t t h i s s t u d y . The energy ( o r more p r e c i s e l y the t r a n s l a t i o n a l e n e r g i e s ) o n l y e n t e r i n t o the c a l c u l a t i o n s i n t h e d i a g o n a l elements o f the p o t e n t i a l m a t r i x , t h e r e f o r e as s t a t e d i n Chapter V.4, as these d i a g o n a l elements decrease the o f f d i a g o n a l elements w i l l be become more i m p o r t a n t . Consequently the s t e p s i z e r e q u i r e d t o m a i n t a i n accuracy may have t o be decreased. However, t h i s dependence o f the s t e p s i z e on the energy i s weak, as shown i n Chapter V.4, the s t e p s i z e r e q u i r e d t o m a i n t a i n accuracy d i d not v a r y s i g n i f i c a n t l y w i t h the i n i t i a l t r a n s l a t i o n a l energy, E T, f o r 233.5549 ^ E T ^ 4138.5631cm - 1 (where E^ i s r e l a t i v e t o ^2 = l j - ^ = 0 ) . The minimum k i n e t i c energy c o n s i d e r e d i n these c a l c u l a t i o n s (and t h e r e f o r e the minimum d i a g o n a l element i n a p o t e n t i a l m a t r i x i n any of these c a l c u l a t i o n s ) i s 64.8322 cm - 1. I n Table 3 we show the values o f / $ L (^%"^\ e v a l u a t e d , r e t a i n i n g two v i b r a t i o n a l channels i n the coup l e d e q u a t i o n s , employing s e v e r a l d i f f e r e n t s t e p s i z e s TABLE 3 Compari son o b t a i n e d u s i n g d i f f e r e n t numbers of steps over the i n t e g r a t i o n range at E^ , = 64.8322 cm f o r the v i b r a t i o n a l d e - e x c i t a t i o n p r o c e s s , = 1 t o v ' = 0, i n CO. (a) 160 s t e p s , (b) 380 s t e p s , ( c ) 760 s t e p s , (d) 3800 steps a 2 = 0° 90° 180° L 0 (a) 0. 73965840- 7 0. 12582501-7 0 .46712753- 8 (b) 0. 74905092- 7 0. 12724302-7 0 .47296381- 8 (c) 0. 75016444- 7 0. 12742139-7 0 .47365258- 8 (d) 0. 75040162- 7 0. 12746161-7 0 .47379490- 8 10 0. 10910313- 7 0. 16531459-8 0 .92584605- 9 0. 11045749- 7 0. 16717943-8 0 .93725740- 9 0. 11061607- 7 0. 16741672-8 0 .93859710- 9 0. 11064945- 7 0. 16747069-8 0 .93887056- 9 ISL faivj ley* i n u n i t s o f (a.u.) R . - 2 a.u., R = 40 a.u. mm ' max - 231 at E-y = 64.8322 cm ^ at t h r e e CO o r i e n t a t i o n s ( 0 ^ = 0°, 90°, 180°) and at L = 0, and 10. The r e s u l t s were o b t a i n e d m a i n t a i n i n g R . = 2 a.u., and R = 4 0 a.u. ° rain ' max ( w h i c h , as s t a t e d i n Chapter V.4, should be more than adequate t o converge v i b r a t i o n a l l y i n e l a s t i c c r o ss s e c t i o n s ) . As may be seen the agreement between the r e s u l t s o b t a i n e d employing 3800, 760, 380 and 190 steps i n t he range 2 . 0 ^ R ^ 4 0 . 0 a.u. i s e x t r e m e l y good at each of the v a l u e s of L shown. I t appears t h a t l o w e r i n g - 1 -1 the k i n e t i c energy from 233.5549 cm t o 64.8322 cm has no s i g n i f i c a n t e f f e c t on the convergence o f the r e s u l t s (see Table 9, Chapter V.4). Since we found the energy dependence of the step s i z e t o be weak i n our p r e v i o u s H2 * CO c a l c u l a t i o n s i t would have been r a t h e r s u r p r i s i n g i f we had concluded o t h e r w i s e . I n a d d i t i o n we found t h a t t he s t e p s i z e r e q u i r e d t o m a i n t a i n accuracy d i d not v a r y s i g n i f i c a n t l y w i t h the p a r t i a l wave parameter, L, f o r 100. The range o f e n e r g i e s , E^, c o n s i d e r e d i n the present c a l c u l a t i o n s i s lower than t h a t s t u d i e d i n Chapter V.4, however we found t h a t the near-resonance p a r t i a l cross s e c t i o n s decrease more s l o w l y w i t h L than the "V-T" r e s u l t s . (Indeed,we found t h a t at Ej = 3167.14 cm ^, we needed t o c o n s i d e r L ^ 80 t o converge the i n t e g r a l cross s e c t i o n s t o w i t h i n 1% f o r b o t h the ' V - T ' ( j ^ = 2) and near-resonance ( j ^ = 2, j j _ = 6) processes. As mentioned above we expect t h a t the R v a r i a t i o n of the i n t e r a c t i o n p o t e n t i a l w i l l not be s i g n i f i c a n t l y a f f e c t e d by the a d d i t i o n of the H 9 232 a n i s o t r o p y terms i n the p o t e n t i a l . By employing a s t e p s i z e which i s e x t r e m e l y e f f i c i e n t f o r n o n - r o t a t i n g l ^ we s h o u l d be a b l e t o accommodate any changes i n the R v a r i a t i o n of the p o t e n t i a l t h a t are p r e s e n t i n r o t a t i n g H2 c a l c u l a t i o n s . I n c o n c l u s i o n we w i l l employ a s t e p s i z e o f 0.05 a.u. ( c o r r e s p o n d i n g t o 760 steps i n Table 3) i n a l l of our c a l c u l a t i o n s , which as one w i l l r e c a l l was used i n our p r e v i o u s H2 + CO c a l c u l a t i o n s . As noted i n Chapter V.4 i t i s necessary t o i n t e g r a t e the coupled e q u a t i o n s s u f f i c i e n t l y f a r i n t o the a s y m p t o t i c r e g i o n t o where the i n t e r a c t i o n p o t e n t i a l has become n e g l i g i b l e b e f o r e the boundary c o n d i t i o n may be f i t t e d , a l s o i t i s necessary t h a t the boundary c o n d i t i o n be f i t t e d i n the c l a s s i c a l l y a l l o w e d r e g i o n (where the s o l u t i o n i s a l i n e a r c o m b i n a t i o n of o s c i l l a t o r y f u n c t i o n s ) . I n as a f u n c t i o n of the upper l i m i t o f t h e i n t e g r a t i o n range, R . a t E T = 64.8322 cm . The r e s u l t s are shown f o r t h r e e CO o r i e n t a t i o n s ( & x = 0°, 90°,and 180°) at L = 0, 10, and 20. These r e s u l t s are e v a l u a t e d m a i n t a i n i n g ^min = ^ a.u., a s t e p s i z e of 0.01 a.u. ( c o r r e s p o n d i n g t o 3800 steps i n Table 3 ) , and r e t a i n i n g two v i b r a t i o n a l channels i n the coupled e q u a t i o n s . As may be seen,at L = 20 i t i s i m p o s s i b l e t o m a i n t a i n accuracy as we are d e a l i n g w i t h n u m e r i c a l z e r o s . The r e s u l t s i n d i c a t e t h a t r e d u c i n g the k i n e t i c energy from 233.5549 c m - 1 t o 64.8322 cm 1 does not s i g n i f i c a n t l y i n c r e a s e the r e q u i r e d v a l u e o f R „.In Table 4 a v a l u e R = 2 5 a.u. i s Table 4 we show the convergence o f max T max max "IAHI.K l-s C o m p a r i s o n o f / Sl tvz~V' ~°> ^ ' Cy/sol&z)J o b t a i n e d u s i n g d i f f e r e n t v a l u e s o f R ( a . u . ) a t E„ = 64.8322 cm „ & max T ( a ) 20 .0, ( b ) 25.0, ( c ) 30 0 . = 0° 90° 180° L 0 ( a ) 0. 74975379- 7 0 .12752299- 7 0 .47262148- 8 ( b ) 0. 75026976- 7 0 . 12744359-7 0 .47376643- 8 ( c ) 0. 7 502110 3-7 0 . 1 2745729-7 0 .47376178- 8 i o 0. 1106 5894- 7 0 . 1 6 7 344 2 5-8 0 .94681785- 9 0. 11061700- 7 0 . 16745149-8 0 .93894562- 9 0. 11060899- 7 0 .16747638- 8 0 .93890258- 9 20 0. 14782977- 15 0 .13784626- 17 0 .67046550- 14 0. 83126358- 18 0 .11468204- 19 0 .49776530- 16 0. 13198875- 17 0 .11565855- 19 0 .24848210- 16 /5^(V J V» -who I n f o r m e d u s I. h,-i( i\ j 'I I r . - m s i H o n s .1 r e I m p o r t a n t , and c o n s e q u e n t l y c l a i m e d t h a t o u r 'model c a l c u l a t i o n ' of t h e n e a r - r e s o n a n c e p r o c e s s was i n a d e q u a t e ( N o t i c e t h a t t h e i n c l u s i o n o f j u s t = 2 and = 6 i n t h e r o t a t i o n a l b a s i s e x p l i c i t l y i g n o r e s A j ^ = 2 t r a n s i t i o n s ) . T h i s i s i n s h a r p c o n t r a s t t o P o u l s e n and B i l l i n g ( 1 9 8 2 ) who f i n d t h a t & j ^ = 2 t r a n s i t i o n s a r e i n s i g n i f i c a n t ( c o m p a r e d to t h e c o r r e s p o n d i n g A j ^ = 0 t r a n s i t i o n s ) . We s t u d y c r o s s s e c t i o n s f o r t h e r o t a t i o n a l t r a n s i t i o n s = 0-^0 ^ 0-V2, 1-^1 , and 1.-^ 3 ( w h i l e t h e CO u n d e r g o e s t h e v i b r a t i o n a l t r a n s i t i o n , ~ 1—^0) i n c l u d i n g o n l y t h e d i r e c t l y r e l e v a n t r o t a t i o n a l c h a n n e l s ( = 0 and 2, and = 1 and 3 r e s p e c t i v e l y ) t o e x a m i n e t h e c l a i m s made by S c h i n k e . I n S e c t i o n 4 we p r e s e n t o u r c a l c u l a t i o n s o f Aj-^ = 0 and A j ^ = 2 c r o s s s e c t i o n s and r a t e s ^ a n d as m e n t i o n e d a b o v e , d i s c u s s t h e v a l i d i t y o f t h e r o t a t i o n a l b a s i s s e t e m p l o y e d t o examine t h e n e a r - r e s o n a n c e p r o c e s s , i n t h e l i g h t o f o u r c a l c u l a t i o n s . 3.3 D e s c r i p t i o n o f t h e CO m o l e c u l e I n t h e p r e s e n t c a l c u l a t i o n s t h e i n t e r a c t i o n p o t e n t i a l i s more a n i s o t r o p i c t h a n t h a t e m p l o y e d i n o u r p r e v i o u s H2 + CO c a l c u l a t i o n s . T h i s i s due e n t i r e l y t o t h e i n t r o d u c t i o n o f t h e r o t a t i o n a l d e g r e e o f f r e e d o m o f t h e m o l e c u l e , and t h e r e a r e no a d d i t i o n a l a n i s o t r o p y t e r m s d e s c r i b i n g t h e CO m o l e c u l e . I n a d d i t i o n t h e r a n g e o f t o t a l b a r y c e n t r i c c o l l i s i o n e n e r g i e s s t u d i e d i n t h e p r e s e n t c a l c u l a t i o n s i s l o w e r t h a n t h a t s t u d i e d i n C h a p t e r V(E >S 5664.76 cm compared w i t h 6291.4021 c m - ^ ) . We t h e r e f o r e c o n c l u d e - 21 r j -t h a t t h e t r e a t m e n t o f t h e CO m o l e c u l e i n o u r p r e v i o u s c a l c u l a t i o n s s h o u l d be s a t i s f a c t o r y , and a c c o r d i n g l y we r e t a i n t h r e e v i b r a t i o n a l c h a n n e l s i n a l l t h e s e c a l -c u l a t i o n s . I t w i l l be r e c a l l e d t h a t t h i s b a s i s e n s u r e s t h a t t h e c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l d e - e x c i t a t i o n p r o c e s s , V2 = 1 t o v ^ = 0 , a r e c o n v e r g e d t o w i t h i n a few p e r c e n t (see T a b l e 18, C h a p t e r V . 4 ) . F o r t h e s e d i a t o m - d i a t o m c a l c u l a t i o n s t h e p o t e n t i a l m a t r i x e l e m e n t s may be w r i t t e n ( c f . I I . 3 . 2 2 ) $ V I . 3.4 where V I . 3 . 4 a and where 1 / \ / / s. I V . 3 . 4 b To e v a l u a t e c r o s s s e c t i o n s f o r t h e n e a r - r e s o n a n c e p r o c e s s one must p e r f o r m t h r e e s e p a r a t e c a l c u l a t i o n s a t each CO o r i e n t a t i o n , I9^ > • The c o u p l e d e q u a t i o n s must be s o l v e d a t t h r e e v a l u e s o f JLj ( i . e . -=/2-/ ^ 2, s i n c e J2~f ^ m i n ( J l = = 6 ) ) . The v i b r a t i o n a l p o t e n t i a l m a t r i x e l e m e n t s , ^>*" C&)&%^) ' a r e n o w e v e r i n d e p e n d e n t o f _ / i y , and c o n s e q u e n t l y one o n l y need e v a l u a t e t h e v a l u e s o f 236 ] / ^ Y ^ ^ . ) ' 1 1 l h G i n i t i a l v a l u e o f J2,, , and t h e y may be use d at s u b s e q u e n t v a l u e s o f _/2^ . However i t was f o u n d t h a t t h e r e was i n s u f f i c i e n t s t o r a g e a v a i l a b l e t o do t h i s , and c o n s e q u e n t l y t h e ma L r i x e l e m e n t s ^7^f % (ftf^S*) had t o be e v a l u a t e d a t each v a l u e o f ^^-j I n o u r p r e v i o u s f ^ + CO c a l c u l a t i o n s we e m p l o y e d a 96- p o i n t Gauss L e g e n d r e q u a d r a t u r e t o e v a l u a t e t h e v i b r a t i o n a l p o t e n t i a l m a t r i x e l e m e n t s , V^""*'^ &zS^) . E m p l o y i n g t h i s q u a d r a t u r e , and r e t a i n i n g t h e same v i b r a t i o n a l b a s i s i t was f o u n d t h a t t h e c a l c u l a t i o n o f * " ^ y was r o u g h l y a f a c t o r o f 3 more e x p e n s i v e i n t h e p r e s e n t c a l c u l a t i o n s , due t o t h e a d d i t i o n a l t e r m s i n t h e i n t e r a c t i o n p o t e n t i a l . S i n c e we employ an a p p r o x i m a t e p o t e n t i a l a l g o r i t h m t o s o l v e t h e c o u p l e d d i f f e r e n t i a l e q u a t i o n s , t h e c a l c u l a t i o n s a t s u b s e q u e n t p a r t i a l waves r e q u i r e no e x p l i c i t r e f e r e n c e t o t h e p o t e n t i a l m a t r i x e l e m e n t s , h owever we c o n s i d e r e d t h e i n c r e a s e i n c o m p u t e r t i m e a t t h e i n i t i a l p a r t i a l wave u n a c c e p t a b l y h i g h . N o te t h a t t h i s i s f u r t h e r i n c r e a s e d by t h e a d d i t i o n o f r o t a t i o n a l s t a t e s i n t o t h e c a l c u l a t i o n , and i n a d d i t i o n as t h e number o f H2 r o t a t i o n a l s t a t e s i s i n c r e a s e d we f o u n d t h a t t h e a d v a n t a g e g a i n e d a t s u b s e q u e n t p a r t i a l waves d e c r e a s e s . I t i s w o r t h m e n t i o n i n g t h a t t h i s i s one o f t h e m a j o r r e a s o n s f o r e m p l o y i n g o u r model c a l -c u l a t i o n t o e v a l u a t e c r o s s s e c t i o n s f o r t h e n e a r - r e s o n a n c e p r o c e s s . E s s e n t i a l l y we c o u l d n o t a f f o r d t o r e t a i n more t h a n two r o t a t i o n a l c h a n n e l s . T h i s i s d i s c u s s e d i n S e c t i o n - 2 37 -4 . 2 . 1 . One w i l l r e c a l l t h a t a 9 6 - p o i n t q u a d r a t u r e e n s u r e s e x t r e m e l y h i g h n u m e r i c a l a c c u r a c y . The a g r e e m e n t b e t w e e n t h e r e s u l t s e v a l u a t e d e m p l o y i n g 4 8 - , 64-, and 96- p o i n t Gauss L e g e n d r e q u a d r a t u r e s i s e x c e l l e n t ( s e e T a b l e 4, C h a p t e r V . 4 ) . I t s h o u l d t h e r e f o r e be p o s s i b l e t o r e d u c e t h e number o f q u a d r a t u r e p o i n t s u s e d t o e v a l u a t e Vj^*" (ffip&z^ c o n s i d e r a b l y w i t h o u t any s i g n i f i c a n t l o s s o f a c c u r a c y . I n T a b l e 5 we compare t h e v a l u e s o f /Si_ (^x^i-^Hy^z^i e v a l u a t e d a t E.j, = 64.8322 cm ^ and L = 0 e m p l o y i n g 64-48 - , and 28- p o i n t Gauss L e g e n d r e q u a d r a t u r e s t o e v a l u a t e t h e v i b r a t i o n a l p o t e n t i a l m a t r i x e l e m e n t s . T h r e e v i b r a t i o n a l c h a n n e l s , and two r o t a t i o n a l c h a n n e l s ( j ^ = 2 and 6) a r e r e t a i n e d i n t h e c o u p l e d e q u a t i o n s . The r e s u l t s a r e shown f o r JZf = 0 and 6£ = 0°, 90°, and 180° f o r t h e t r a n s i t i o n s V2 = 1 = 2-^v^ = 0 = 2, and ^2 = 1 = 2-^v^ = 0 j ' = 6. At t h i s e n e r g y , w h i c h i s t h e l o w e s t s t u d i e d , any e r r o r s i n t h e e v a l u a t i o n o f t h e p o t e n t i a l m a t r i x e l e m e n t s s h o u l d have maximum e f f e c t on t h e v a l u e s o f l&L(v1.fiSti->V3!f.t,n1lG^>l'3- • As can be seen t h e a g r e e m e n t b e t w e e n t h e r e s u l t s e v a l u a t e d e m p l o y i n g 28-, 48- , and 64- p o i n t q u a d r a t u r e s i s v e r y good, i n a d d i t i o n i t i s w o r t h n o t i n g t h a t a 2 8 - p o i n t q u a d r a t u r e r e s u l t s i n a r e d u c t i o n o f a f a c t o r o f 3 i n t h e c o m p u t e r t i m e r e q u i r e d by a 9 6 - p o i n t q u a d r a t u r e f o r t h e c a l c u l a t i o n o f l ^ f * , ^ ) f o r vs.Ai/>4 3 . I n c o n c l u s i o n a 2 8 - p o i n t Gauss L e g e n d r e q u a d r a t u r e w i l l be e m p l o y e d t h r o u g h o u t t h e s e c a l c u l a t i o n s . TAIil.K 5 C o m p a r i s o n o f / S t . i^^i^r0} o b t a i n e d u s i n g d i f f e r e n t N - p o i n t Gauss L e g e n d r e q u a d r a t u r e s t o evaluate_£he v i b r a t i o n a l p o t e n t i a l m a t r i x e l e m e n t s a t = 64.8322 cm and L = 0. ( a ) N = 64, ( b ) N = 48, (c ) N - 28 • = 0° 90° 180° = 2 t o i\ -• 2 (a ) 0. 18974629-6 0. 78139596- 8 0. 36535433- 8 1 ( b ) 0. 18974629-6 0. 78139 596-8 0. 36535433- 8 f c ) 0. 19038483-6 0. 78146348- 8 0. 36776693- 8 j { = 6 (a) 0. 29757500-6 0. 13217708- 5 0. 15669486- 5 1 ( b ) 0. 29757500-6 0. 13217708- 5 0. 15669486- 5 ( c ) 0. 29867127-6 0. 13201466- 5 0. 15660986- 5 /Si, (v^hi-al = 0J\/2=ojl/Jli/&2)/ i n u n i t s o f ( a . u . ) 2 238 3.4 Q u a d r a t u r e o v e r t h e CO o r i e n t a t i o n s I n t h e p r e s e n t s t u d y we s h a l l be p r i n c i p a l l y i n t e r e s t e d i n t h e c a l c u l a t i o n o f r o t a t i o n a l l y summed c r o s s s e c t i o n s , C*(vx-/ji — O ) > s u c h c r o s s s e c t i o n s may be e m p l o y e d t o e v a l u a t e v i b r a t i o n a l r e l a x a t i o n r a t e c o e f f i c i e n t s . I n C h a p t e r V.4 t h e q u a d r a t u r e o v e r t h e CO o r i e n t a t i o n s , &^ , was e x a m i n e d i n some d e t a i l . I n t h e e n e r g y r a n g e c o n s i d e r e d i n t h o s e c a l c u l a t i o n s , ^ 4138.5631 cm \ a 28- p o i n t Gauss L e g e n d r e q u a d r a t u r e was a b l e t o d e t e r m i n e r o t a t i o n a l l y summed c r o s s s e c t i o n s f o r t h e v i b r a t i o n a l d e - e x c i t a t i o n p r o c e s s \'^ - 1 t o - 0 t o w i t h i n a few p e r c e n t ( s e e T a b l e 2 5 > C h a p t e r V . 4 ) . On t h e o t h e r hand t h e c o n v e r g e n c e w i t h r e s p e c t t o t h e number o f o r i e n t a t i o n s , » i n c l u d e d i n t h e c a l c u l a t i o n s , o f t h e c o r r e s p o n d i n g c r o s s s e c t i o n s f o r t h e i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s i n CO was l e s s s a t i s f a c t o r y . One w i l l r e c a l l t h a t t o e v a l u a t e c r o s s s e c t i o n s f o r t h e t r a n s i t i o n V2 = 1 j - ^ = 0 ^ = v 2 = 0 = o n e must e v a l u a t e an i n t e g r a l o f t h e t y p e ( V . 4 . 4 ) t h a t i n v o l v e s t h e f u n c t i o n YJio C^®® w h i c h becomes more o s c i l l a t o r y as a f u n c t i o n o f ^ a s i n c r e a s e s . I n t h e p r e s e n t c a l c u l a t i o n s h o w e v e r , we o n l y e v a l u a t e s u c h c r o s s s e c t i o n s , f o r t h e r o t a t i o n a l t r a n s i t i o n ^ = 0-^0 i n CO. For t h i s t r a n s i t i o n t h e s p h e r i c a l h a r m o n i c Yj^^ ((M2>B^) i s a c o n s t a n t , and t h e r e f o r e t h e c o n v e r g e n c e as a f u n c t i o n o f t h e number o f 9^0 v a l u e s i n c l u d e d i n t h e c a l c u l a t i o n o f t h e s e c r o s s s e c t i o n s i s s i m i l a r t o t h a t o f t h e r o t a t i o n a l l y summed c r o s s s e c t i o n s . However i t i s n o t i d e n t i c a l s i n c e t h e - 239 -i n t e g r a l one must p e r f o r m t o e v a l u a t e £3"° ^ V ^ ^ i ~~^ > / ^ i n v o l v e s / fv^/-^i > W^'-fe-l w h i c h i s a l e s s o s c i l l a t o r y f u n c t i o n o f t h a n t h e r e a l and i m a g i n a r y p a r t s o f t h e S - m a t r i x , i n c l u d e d i n t h e i n t e g r a l s f o r ^^^^) • See C h a p t e r V.4 f o r a more d e t a i l e d d i s c u s s i o n . As m e n t i o n e d p r e v i o u s l y i n t h i s s e c t i o n , t h e i n t e r a c t i o n p o t e n t i a l u s e d i n t h e p r e s e n t c a l c u l a t i o n s i s more a n i s o t r o p i c t h a n t h a t e m p l o y e d i n o u r p r e v i o u s 1 ^ + CO c a l c u l a t i o n s , a l t h o u g h as a l s o m e n t i o n e d t h i s a d d i t i o n a l a n i s o t r o p y i s e n t i r e l y due t o t h e l ^ m o l e c u l e , c o n s e q u e n t l y a q u a d r a t u r e o v e r t h e CO o r i e n t a t i o n s , (9*2 > s h o u l d n o t be s e r i o u s l y a f f e c t e d . I t was f o u n d ( s e e C h a p t e r V.4) t h a t t h e c o n v e r g e n c e o f t h e r e s u l t s as a f u n c t i o n o f t h e number o f o r i e n t a t i o n s c o n s i d e r e d i n t h e c a l c u l a t i o n i m p r o v e d w i t h b o t h d e c r e a s i n g K.j., and i n c r e a s i n g L. T h e r e f o r e i n t h e e n e r g y r a n g e c o n s i d e r e d i n t h e s e c a l c u l a t i o n s , E.j, ^ 31.67.14 cm~^ a 2 8 - p o i n t Gauss L e g e n d r e q u a d r a t u r e s h o u l d be c a p a b l e o f e v a l u a t i n g r o t a t i o n a l l y summed c r o s s s e c t i o n s , and i n d i v i d u a l r o t a t i o n a l c r o s s s e c t i o n s f o r ^ - 0 —5" 0 t o w i t h i n a few p e r c e n t . I n c o n c l u s i o n a 2 8 - p o i n t Gauss L e g e n d r e q u a d r a t u r e i s e m p l o y e d i n a l l t h e s e c a l c u l a t i o n s . 4. R e s u l t s and d i s c u s s i o n The r e s u l t s p r e s e n t e d i n t h i s s e c t i o n a r e t h o s e e v a l u a t e d e m p l o y i n g t h e P p o t e n t i a l as p r e s e n t e d by P o u l s e n ( 1 9 8 2 ) ( i . e . t h e f a c t o r o f 0.5 i s o m i t t e d f r o m V I . 2 . 4 ) . The e f f e c t s o f t h i s e r r o r a r e d i s c u s s e d i n d e t a i l i n A p p e n d i x 4 a t t h e back o f t h i s t h e s i s . E s s e n t i a l l y we 'MO show t h a t o n l y t h e r e s u l t s f o r t h e n e a r - r e s o n a n c e p r o c e s s have been s e r i o u s l y a f f e c t e d , t h e c r o s s s e c t i o n s a r e o v e r e s t i m a t e d by a f a c t o r o f 4. The r e s u l t s f o r t h e Aj-^ = 0 a r i d 2 t r a n s i t i o n s a r e changed by a few p e r c e n t . 4.1 The dependence o f t h e v i b r a t i o n a l d e - e x c i t a t i o n o f CO on t h e r o t a t i o n a l s t a t e o f 1 ^ I n C h a p t e r V we p r e s e n t e d t h e r e s u l t s o f a c a l c u l a t i o n I n w h i c h we s t u d i e d ( h e r o v i b r a t i o n a I e x c i t n l i o n o f CO by n o n - r o t a t i n g ^ 2 ^ 1 ^ * '° s t u c ' y t n e dependence o f t h e r e s u l t s on t h e r o t a t i o n a l s t a t e o f t h e m o l e c u l e , we have p e r f o r m e d a number o f c a l c u l a t i o n s i n w h i c h t h e H2 was c o n s t r a i n e d i n i t s s e c o n d e x c i t e d r o t a t i o n a l s t a t e ( = 2 ) . I n F i g u r e 1 we show a c o m p a r i s o n o f p a r t i a l c r o s s s e c t i o n s , 0*^ (V2 = 1 , j^, ^ = U — ^ v 2 = ^' ^ 1 ' ^2 = ^' f o r j - ^ = 0 and 2. The c o m p a r i s o n i s made a t t h e same v a l u e o f t h e i n i t i a l k i n e t i c e n e r g y E,p ( i . e . t h e c o l l i s i o n e n e r g i e s d i f f e r by t h e e n e r g y s e p a r a t i o n o f t h e j j = 2 , and j.j 0 r o t a t i o n a l l e v e l s o f 11^ ) . R e s u l t s a r e shown f o r Ej - 66 7.5 cm ^, 15 35.3 cm \ and 28 3 7.0 cm -'. I t i s e v i d e n t t h a t a t a l l t h e e n e r g i e s s t u d i e d t h e v a r i a t i o n o f 0^" w i t h r e s p e c t t o L i s v e r y s i m i l a r f o r b o t h 0 and 2. The p a r t i a l c r o s s s e c t i o n s a r e o n l y shown f o r e v e r y t e n t h p a r t i a l wave a t t h e two l o w e s t e n e r g i e s , and a t e v e r y t w e n t y p a r t i a l waves a t t h e h i g h e s t e n e r g y . Hence t h e s h a r p changes i n g r a d i e n t t h a t a r e seen i n t h e p l o t s . A more ' c o m p l e t e ' c o m p a r i s o n w o u l d i n v o l v e t h e c o m p a r i s o n F i g u r e 1 A c o m p a r i s o n o f p a r t i a l wave c r o s s s e c t i o n ^ , i n f o r = 0 ( s o l i d l i n e s ) and - 2 (dashed l i n e s ) . The i n i t i a l k i n e t i c e n e r g i e s a r e ( a ) 667.5 (b) 1535.3 cm and -1 -1 > ( c ) 2837.0 cm -1 • 10.8 -I0.B 7 ( l O ^ c m V 1 ) , c a l c u l a t e d T o r j 0 ( C h a p t e r V) and j , 2. T ( K ) Vr 200 4.4 4.0 220 5.9 5.2 240 7.7 6.6 260 10.1 8.5 280 13.0 11.0 300 16.6 14.0 400 4« 44 500 I I 7 1 1 (> 600 2 30 26 7 242 -pot (M)( I . i I . One w i l l r r r . i l l ( h . i t I he p o l o n l i ; i I in.i L r i x e l e m e n t s i n v o l v e a w e i g h t f a c t o r X g i v e n by I V . 4 . 1 Because o f t h e t r i a n g u l a r c o n d i t i o n f o r t h e a n g u l a r momentum c o u p l i n g c o e f f i c i e n t s o n l y t h e Yoo ( c o s ^ ) t e r m can d i r e c t l y i n f l u e n c e t h e = 0 >0 c r o s s s e c t i o n , h o wever f o r = 2 — > 2 i t e r m s w i t h * ^ - j = 2 c o n t r i b u t e . S i n c e t h e above r a t e s a r e v e r y s i m i l a r one w o u l d e x p e c t t h e Y2Q(COS ©"^) p o t e n t i a l t e r m s t o be s m a l l , i n a d d i t i o n P o u l s e n and B i l l i n g ( 1 9 8 2 ) f i n d t h a t t h e A j 1 = 2 r a t e s a r e i n s i g n i f i c a n t compared t o t h e c o r r e s p o n d i n g A j ^ = 0 r a t e s . T h i s h o w e v e r , i s n o t t r u e , S c h i n k e ( p r i v a t e c o m m u n i c a t i o n ) p o i n t e d o u t t h a t f r o m i n t e r m e d i a t e t o l a r g e i n t e r m o l e c u l a r s e p a r a t i o n s ( R ^ 6 a.u.) t h e Y2Q(COS$*^) t e r m i s t h e d o m i n a n t t e r m o f t h e P p o t e n t i a l . T h i s has been c o n f i r m e d by o u r own e x a m i n a t i o n o f t h e P p o t e n t i a l , and a l s o i n t h e c a l c u l a t i o n o f A j x = 2 c r o s s s e c t i o n s . I n T a b l e 7 we show r o t a t i o n a l l y summed c r o s s s e c t i o n s , (J* ( v ^ = 1 j i = ^ ' v 2 = f ° r J i = 0„and j ' = 0 and 2 a t Ej = 233.5549and 1101.3346 cm . These r e s u l t s were o b t a i n e d e m p l o y i n g a r o t a t i o n a l b a s i s s e t i n c l u d i n g = 0 and 2. As can be seen t h e = 2 c r o s s s e c t i o n s a r e r o u g h l y 507o o f t h e Aj^ = 0 c r o s s s e c t i o n s . A l s o shown i n T a b l e 7 a r e t h e c o r r e s p o n d i n g = 0 c r o s s s e c t i o n s w l -J OQ < CM > f « * U3 0C o X ) c CO c j CD 0 ) 03 03 o O *o 0) E E o u c — QJ U 0) o U-l LM II CO X ) 00 C o 1—1 a E CD n —' .a X ) c CO c o c/3 •r-i u CO CL E O o co O t CM O r o II o o t I CM 0 0 O ^0 I C O I I Q> C O O T -H o o U X) I I oo CNi T — i m T - i ON o o CO I oo 0 0 CO i i 0 0 0 0 CNI 0 0 m a n d (-=fi,e£,^ ( S e c t i o n 2 ) , however t h e m a j o r c o n t r i b u t i o n t o t h e i n t e g r a l w i l l come f r o m t e r m s i n v o l v i n g (f^, — /z,e<£~) • T n e l e a d i n g l o n g r a n g e t e r m s a r e t h e r e f o r e ( c f . V I . 2 . 3 ) . Vt^ ' - tfr,(c0>8C) VI.4.2a V I . 4 . 2 b V I . 4 . 2 c V I . 4 . 2 d where 0.4876, titfjf0y 0 . 0 8 4 0 8 , = 2.4790, and d^/^ = 1.879. For t h e d i s c u s s i o n o f A j | = 2 c r o s s s e c t i o n s i t i s o n l y n e c e s s a r y t o n o t i c e t h a t \/» a n d V^rn a r e - 244 -l a r g e . What i s t h e r e f o r e , s u r p r i s i n g i s t h e s i m i l a r i t y o f t h e 'V-T' c a l c u l a t i o n s f o r = 0 and 2, i n v i e w o f t h e r e s u l t s shown i n T a b l e 7. One w i l l r e c a l l t h a t f o r = 2=^2 we must a l s o c o n s i d e r t e r m s w i t h = 2 as w e l l as t e r m s w i t h = 0. However, t h e w e i g h t f a c t o r s i n v o l v i n g = 2 a r e r a t h e r d i f f e r e n t t o t h o s e i n v o l v i n g = 0 . I n T a b l e 8 we show t h e n u m e r i c a l v a l u e s o f t h e w e i g h t f a c t o r s , X, f o r a number o f c o m b i n a t i o n s o f 3l = j { ) and=f2,^. As can be s e e n t h e w e i g h t f a c t o r s f o r ^ ^ = 2 a r e much s m a l l e r t h a n f o r ^ ^ = 0 w h i c h p r o b a b l y e x p l a i n s t h e s i m i l a r i t y o f t h e 'V-T' c a l c u l a t i o n s . I n a d d i t i o n we have e v a l u a t e d r e s u l t s f o r o r t h o -^2 ^ 1 ~ ^' R o t a t i o n a l l y summed c r o s s s e c t i o n s , ( V 2 = 1 j f ^ V 2 = 0 j { ) f o r = 1 and = 1 and 3 a t E^ = 233.5549 cm -^ a r e a l s o shown i n T a b l e 7. I t i s a p p a r e n t t h a t a t a g i v e n e n e r g y ( ) t h e = 1-^3 c r o s s s e c t i o n i s l a r g e r t h a n t h e = 0«^2 c r o s s s e c t i o n . I t a p p e a r s t h a t = 2 c r o s s s e c t i o n s a r e d e p e n d e n t u p o n t h e i n i t i a l r o t a t i o n a l s t a t e a l t h o u g h a l l o f them a r e i n d u c e d by t h e same p o t e n t i a l t e r m , and i n a d d i t i o n t h e c o r r e s p o n d i n g w e i g h t f a c t o r s , X, shown i n T a b l e 9 a r e r o u g h l y t h e same f o r c o m b i n a t i o n s o f j ^ , , J2y,and c o n s i d e r e d . The d i f f e r e n c e i s due t o t h e d i f f e r e n c e o f t h e e n e r g y d e f e c t s f o r t h e two r o t a t i o n a l t r a n s i t i o n s , i . e . TABLE 8 Weigh t f a c t o r s , X, f o r = 0 0 0 1 2 0 0 1 2 0 2 0.286 2 1 2 0.143 2 2 2 0.286 TABI.K 9 Weight f a c t o r s , X, f o r j - ^ + 0 2 0 2 0.447 1 3 0 2 0.393 1 3 1 2 0.321 - 245 -V I . 4 . 3 a V I . 4 . 3 b Any t r a n s i t i o n w h i c h r e d u c e s t h e e n e r g y d e f e c t i n c r e a s e s t h e r e l a x a t i o n c r o s s s e c t i o n , c o n s e q u e n t l y t h e c r o s s s e c t i o n f o r = 1—>3 i s l a r g e r t h a n t h a t f o r = 0 - > 2 . As m e n t i o n e d i n A p p e n d i x 4 t h e above o b s e r v a t i o n s c o n c e r n i n g A = 2 t r a n s i t i o n s a r e i n q u a n t i t a t i v e a g r eement w i t h t h e r e s u l t s o b t a i n e d by BSD2 i n t h e i r c a l c u l a t i o n s e m p l o y i n g t h e P p o t e n t i a l . However, BSD2 f i n d t h a t when t h e DK p o t e n t i a l i s u s e d t h e = 0 — ^ 2 c r o s s s e c t i o n s a r e o f t h e same o r d e r as t h e = 0 * — ^ 0 c r o s s s e c t i o n s , and t h e e f f e c t i s more p r o n o u n c e d f o r j j •- 1 ( a i d f o r = 2 ) . At l a r g e d i s t a n c e s t h e Y ^ Q ( C O S ^ ) p o t e n t i a l t e r m i s p r o -Co p o r t i o n a l t o yW-fn , t h e f i r s t d e r i v a t i v e o f t h e d i p o l e moment o f CO ( i . e . V I . 4 . 2 a ) . I n t h e DK p o t e n t i a l t h e l o n g r a n g e Y 2 Q ( C O S $ ^ ) i n t e r a c t i o n i s i n c l u d e d i n t h e CO SCF c a l c u l a t i o n . On t h e SCF l e v e l i - s p r o b a b l y o v e r -e s t i m a t e d by a f a c t o r o f 1.5-2 ( e . g . see J a q u e t e t a l . ( 1 9 8 0 ) ) . Hence t h e o v e r e s t i m a t i o n o f A j ^ = 2 t r a n s i t i o n s o b t a i n e d w i t h t h e DK p o t e n t i a l . I t w i l l be r e c a l l e d t h a t i n C h a p t e r V t h e 'V-T' r e l a x a t i o n r a t e ( = 0) was compared w i t h t h e e x p e r i m e n t a l r a t e f o r ortho-H2« I n d o i n g so we were g u i d e d by t h e s t u d y o f P o u l s e n and B i l l i n g ( 1 9 8 2 ) . These a u t h o r s f o u n d t h a t t h e r e l a x a t i o n r a t e f o r A j - ^ = 2 t r a n s i t i o n s a r e much s m a l l e r t h a n t h e r a t e f o r A j ^ = 0. T h e r e f o r e we - 246 -c o n c u r w i t h BSD2 i n t h a t we c o n s i d e r t h e i n t e r p r e t a t i o n o f t h e 1V-T' r a t e as an o r t h o r a t e i s w r o n g . The r a t e , 0 ( = 1 j | = = 0 , = 0) i s r o u g h l y a f a c t o r o f 2 t o 3 l o w e r t h a n t h e e x p e r i m e n t a l r a t e f o r o r t h o -H 2 • BSD have shown t h a t r e l a x a t i o n c r o s s s e c t i o n s e v a l u a t e d w i t h t h e CO m o l e c u l e t r e a t e d w i t h i n t h e CS a p p r o x i m a t i o n a r e r o u g h l y a f a c t o r o f 1.5 l a r g e r t h a n t h e c o r r e s p o n d i n g I O S c r o s s s e c t i o n s ( C h a p t e r V . 5 ) . We t h e r e f o r e a r g u e t h a t t h e P p o t e n t i a l i s c a p a b l e o f p r o v i d i n g a b e t t e r d e s c r i p t i o n o f t h e e x p e r i m e n t a l r a t e f o r o r t h o - I - ^ t h a n t h e DK p o t e n t i a l i f t h e r a t e c o e f f i c i e n t s f o r t h e A j ^ = 0 and 2 p r o c e s s e s a r e e v a l u a t e d u s i n g c r o s s s e c t i o n s c a l c u l a t e d w i t h i n t h e CS a p p r o x i m a t i o n . On t h e o t h e r h a n d , t h e j^j] = 0 r e l a x a t i o n r a t e c a l c u l a t e d u s i n g JOS c r o s s s e c t i o n s e v a l u a t e d e m p l o y i n g t h e DK p o t e n t i a l a g r e e w e l l w i t h e x p e r i m e n t . The i n c l u s i o n o f t h e A j ^ = 2 r a t e , and t h e use o f t h e CS a p p r o x i m a t i o n w o u l d r u i n t h i s a g r e e m e n t . We m u s t , o f c o u r s e , assume t h a t t h e above r a t i o o f CS a p p r o x i m a t i o n r a t e s t o IOS a p p r o x i m a t i o n r a t e s f o r n o n -r o t a t i n g H 2 i s v a l i d f o r r o t a t i n g H 2 . I n v i e w o f t h e l a r g e Y 2 Q ( C O S © ^ ) t e r m i t i s n o t c l e a r why P o u l s e n and B i l l i n g ( 1 9 8 2 ) o b t a i n e d such i n s i g n i f i c a n t = 2 r e l a x a t i o n r a t e s . T h i s i s p r o b a b l y a n o t h e r i n d i c a t i o n t h a t t h e s e m i c l a s s i c a l a p p r o x i m a t i o n e m p l o y e d by t h e s e a u t h o r s i s i n a p p r o p r i a t e f o r t h e 1 ^ + CO s y s t e m ( s e e C h a p t e r V.5 f o r a more d e t a i l e d d i s c u s s i o n ) . - 247 -4.2 The n e a r - r e s o n a n c e p r o c e s s 4.2.1 D i s c u s s i o n o f t h e model c a l c u l a t i o n As s t a t e d i n S e c t i o n 3 when i n v e s t i g a t i n g t h e n e a r -r e s o n a n c e p r o c e s s ( V I . 2 . 5 ) we i n c l u d e o n l y t h e d i r e c t l y r e l e v a n t r o t a t i o n a l s t a t e s o f t h e m o l e c u l e , = 2 and 6. T h i s i s h o w e v e r , c o n t r a r y t o t h e u s u a l p r o c e d u r e o f r e t a i n i n g a l l t h e c h a n n e l s up t o and i n c l u d i n g t h e c h a n n e l s o f I n t e r e s t , and a d d i n g e x t r a c h a n n e l s ( i f n e c e s s a r y ) t o c o n v e r g e t h e t r a n s i t i o n s b e i n g s t u d i e d . To s t u d y t h e n e a r - r e s o n a n c e p r o c e s s one w o u l d need t o i n c l u d e a l l t h e r o t a t i o n a l c h a n n e l s up t o = 6 ( a t l e a s t ) , as i s d o n e b y B S D 2 . Our 'model c a l c u l a t i o n ' e x p l i c i t l y n e g l e c t s t h e A j - ^ = 2 t r a n s i t i o n s , w h i c h a r e s i g n i f i c a n t c ompared t o t h e c o r r e s p o n d i n g Aj^ = 0 c r o s s s e c t i o n s . When we p e r f o r m e d t h e s e c a l c u l a t i o n s , one o f o u r r e a s o n s f o r s u c h a model c a l c u l a t i o n was t h e s i m i l a r i t y o f t h e A j | = 0 r e s u l t s f o r - 0 and 2. As e x p l a i n e d i n S e c t i o n 4.1 one w o u l d e x p e c t t h e Y 2 Q ( C O S & - ^ ) t e r m s o f t h e i n t e r a c t i o n p o t e n t i a l t o be s m a l l , i n a d d i t i o n P o u l s e n and B i l l i n g ( 1 9 8 2 ) f i n d t h a t A j ^ = 2 r a t e s a r e i n s i g n i f i c a n t compared t o t h e c o r r e s p o n d i n g A j ^ = 0 r a t e s , i n t h e i r s e m i - c l a s s i c a l s t u d y e m p l o y i n g t h e P p o t e n t i a l . However, a t l o w t e m p e r a t u r e s , E.p j& 300 K, where we e x p e c t e d t h e n e a r - r e s o n a n c e p r o c e s s t o be d o m i n a n t ( e . g . see Andrews and Simpson ( 1 9 7 6 ) ) , as i s c o n f i r m e d by o u r c a l c u l a t i o n s ( s e e T a b l e 11) b o t h A j ^ = 0 and 2 c r o s s s e c t i o n s a r e i n s i g n i f i c a n t c ompared t o t h e c o r r e s p o n d i n g n e a r - r e s o n a n c e c r o s s s e c t i o n s , e ven - 248 -when one a c c o u n t s f o r t h e e r r o r i n o u r programme and we b e l i e v e ( i n t h e r e g i m e o f t h e r m a l e n e r g i e s ) t h a t t h e n u m e r i c a l v a l u e s o f t h e n e a r - r e s o n a n c e c r o s s s e c t i o n s have n o t been s e r i o u s l y a f f e c t e d by t h e t r u n c a t i o n o f t h e r o t a t i o n a l b a s i s s e t . T h i s i s c o n f i r m e d a t E^. = 208 cm-''' by t h e c a l c u l a t i o n o f BSD2 e m p l o y i n g t h e P p o t e n t i a l , and e m p l o y i n g t h e = 0,2,4, and 6 r o t a t i o n a l s t a t e s o f t h e H 2 m o l e c u l e . Good a g r e e m e n t i s f o u n d b e t w e e n t h e r e s u l t s o f o u r c a l c u l a t i o n , and t h a t o f BSD2 f o r t h e n e a r - r e s o n a n c e c r o s s s e c t i o n as i s shown i n A p p e n d i x 4. I t i s w o r t h m e n t i o n i n g t h a t , as BSD2 c o n c l u d e , such a model c a l c u l a t i o n i s n o t f e a s i b l e i f one emp l o y s t h e D K i n t e r a c t i o n p o t e n t i a l due t o t h e p o o r d e s c r i p t i o n o f t h e Y 2 Q ( C O S ^ ^ ) t e r m ( a n d t h e r e s u l t i n g o v e r e s t i m a t e d A j ^ = 2 c r o s s s e c t i o n s ) . I n S e c t i o n 4.2.2 we show t h a t i n t h e r e g i m e o f t h e r m a l e n e r g i e s t h e c r o s s s e c t i o n s f o r t h e t r a n s i t i o n = 2=^4 a r e l a r g e r t h a n t h e c o r r e s p o n d i n g n e a r - r e s o n a n c e c r o s s s e c t i o n s . At t h e two h i g h e s t e n e r g i e s s t u d i e d i n t h e s e c a l c u l a t i o n s we w o u l d e x p e c t o u r model c a l c u l a t i o n t o be u n s a t i s f a c t o r y . At such h i g h k i n e t i c e n e r g i e s , Ej = 2299.36 cm and 3167.14 cm ^, t h e 'V-T' p r o c e s s e s a r e d o m i n a n t (see T a b l e 11) and c o n s e q u e n t l y t h e n e g l e c t o f = 2 t r a n s i t i o n s w i l l p r o b a b l y n o t be j u s t i f i e d . However, one w i l l r e c a l l t h a t t h e m a g n i t u d e o f t h e A j - ^ = 2 c r o s s s e c t i o n s r e l a t i v e t o t h e c o r r e s p o n d i n g A j ] ^ = 0 c r o s s s e c t i o n s d e c r e a s e w i t h E,p, and t h e r e f o r e a t t h e s e h i g h e n e r g i e s w i l l n o t be p a r t i c u l a r l y i m p o r t a n t , n e v e r t h e l e s s t h e y a r e p r o b a b l y 7 4 { ) s t i l l c o m p a r a b l e i n m a g n i t u d e t o t h e c o r r e s p o n d i n g n e a r -r e s o n a n c e c r o s s s e c t i o n s . I n c o n c l u s i o n we w o u l d e x p e c t t h e n e a r - r e s o n a n c e r e s u l t s t o be o v e r e s t i m a t e d a t t h e s e h i g h e n e r g i e s , due t o t h e n e g l e c t o f c o u p l i n g t o t h e = 4 and 0 r o t a t i o n a l s t a t e s , h o w e v e r , n o t i c e t h a t w h i l e t h e d e s c r i p t i o n o f t h e n e a r - r e s o n a n c e p r o c e s s may n o t be c o m p l e t e l y s a t i s f a c t o r y i t w i l l f o r m an i n s i g n i f i c a n t c o n t r i b u t i o n t o t h e r a t e o f d e a c t i v a t i o n o f CO (v,^ = 1) by p a r a - H , , a I such h i g h e n e r g i e s . The n e a r - r e s o n a n c e p r o c e s s i s o n l y i m p o r t a n t i n t h e e n e r g y r a n g e I 0 0 ^ ^ 500 K ( e . g . see Andrews and Simpson ( 1 9 7 6 ) ) . A n o t h e r r e a s o n f o r e m p l o y i n g t h i s model c a l c u l a t i o n , i s t h a t i t was t h e l a r g e s t f e a s i b l e . We d i d n o t have s u f f i c i e n t c o m p u t e r t i m e t o p e r f o r m c a l c u l a t i o n s i n c l u d i n g t h e = 0 and = 4 r o t a t i o n a l s t a t e s , and s t u d y s u f f i c i e n t e n e r g i e s t o a c c u r a t e l y e v a l u a t e v i b r a t i o n a l r e l a x a t i o n r a t e c o e f f i c i e n t s . As m e n t i o n e d i n S e c t i o n 3 t o o b t a i n c r o s s s e c t i o n s f o r t h e n e a r - r e s o n a n c e p r o c e s s i I. i s n e c e s s a r y t o s o l v e t h e c o u p l e d d i f f e r e n t i a l , e q u a t i o n s a t t h r e e va l u e s o f Jlf X ^ ) a t each o f t h e 28 CO o r i e n t a t i o n s . I n T a b l e 10 we g i v e t h e a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d t o s o l v e t h e c o u p l e d d i f f e r e n t i a l e q u a t i o n s a t one o r i e n t a t i o n u s i n g t h e C r a y - 1 c o m p u t e r . T h r e e v i b r a t i o n a l c h a n n e l s a r e r e t a i n e d i n a l l t h e c a l c u l a t i o n s , and t h e e q u a t i o n s a r e s o l v e d a t 460 v a l u e s o f R. The s i z e of r o t a t i o n a l b a s i s i s i n c r e a s e d as shown i n T a b l e 10. Times f o r t h e i n i t i a l and s u b s e q u e n t p a r t i a l waves a r e g i v e n f o r each v a l u e o f J l | t h a t must be c o n s i d e r e d . At t h e i n i t i a l p a r t i a l TABLE 10 The a p p r o x i m a t e c o m p u t e r t i m e s r e q u i r e d t o s o l v e t h e f i x e d a n g l e c o u p l e d e q u a t i o n s a t one o r i e n t a t i o n u s i n g t h e C r a y - 1 c o m p u t e r . T h r e e v i b r a t i o n a l c h a n n e l s a r e i n c l u d e d i n a l l t h e c a l c u l a t i o n s , and t h e r o t a t i o n a l c h a n n e l s r e t a i n e d a r e ; -( a ) = 2, and 6, ( b ) = 0,2 and 6, ( c ) = 0,2,4, and 6. Times a r e g i v e n f o r t h e i n i t i a l and s u b s e q u e n t p a r t i a l waves at J l t = 0 1 , ana 2 . 0 L 2 T i m e ( s ) I n i t i a l ( a ) 3 . 70 70 3 70 P a r t i a l ( b ) 5 .00 ii Wave ( c ) 6 . 50 5 00 5 00 Subsequent 0 .50 0 50 0 50 P a r t i a l 1 .00 11 Waves 2 .00 1 00 1 .00 - 250 -wave i n c l u d i n g Che = 0 and 4 r o t a t i o n a l s t a t e s r e s u l t s i n a A, 50% i n c r e a s e i n t h e c o m p u t e r t i m e n e c e s s a r y f o r o u r model c a l c u l a t i o n , however a t s u b s e q u e n t v a l u e s o f L we f i n d t h a t t h e i n c l u s i o n o f t h e s e s t a t e s r e s u l t s i n an i n c r e a s e i n t h e t i m e by 200%. A l t h o u g h a t s u b s e q u e n t p a r t i a l waves t h e r e i s no e x p l i c i t r e f e r e n c e t o t h e p o t e n t i a l m a t r i x e l e m e n t s , I t I s n e c e s s a r y t o p e r f o r m v a r i o u s m a t r i x o p e r a t i o n s ( e . g . , i n v e r s i o n , m u l t i p l i c a t i o n , e t c . ) . I n p a r t i c u l a r we have f o u n d t h a t as t h e d i m e n s i o n s o f t h e m a t r i c e s i n c r e a s e , m a t r i x m u l t i p l i c a t i o n becomes a r a t h e r c o s t l y o p e r a t i o n , and e v e n t u a l l y b e g i n s t o f o r m a s i g n i f i c a n t p e r c e n t a g e o f t h e t o t a l c o m p u t e r t i m e r e q u i r e d to s o l v e t h e c o u p l e d e q u a t i o n s . F o r example i f we r e t a i n f o u r r o t a t i o n a l s t a t e s , i t a c c o u n t s f o r ' v 307o o f t h e t o t a l t i m e . I n c o n c l u s i o n as t h e s i z e o f t h e r o t a t i o n a l b a s i s i s i n c r e a s e d t h e a d v a n t a g e g a i n e d at s u b s e q u e n t p a r t i a l waves d e c r e a s e s as i s shown i n T a b l e 10. I n o u r H D + CO c a l c u l a t i o n s ( C h a p t e r V I I ) t h i s p r o b l e m was a l l e v i a t e d by o p t i m i z i n g t h e m a t r i x m u l t i p l i c a t i o n r o u t i n e f o r t h e C r a y - 1 c o m p u t e r , however n o t e t h a t i n t h e H D + CO s y s t e m a model c a l c u l a t i o n o f t h e t y p e p e r f o r m e d i n t h e s e c a l c u l a t i o n s i s n o t f e a s i b l e . I n t h e p r e s e n t c a l c u l a t i o n we b e l i e v e t h a t o u r model c a l c u l a t i o n i s a s u f f i c i e n t l y a c c u r a t e d e s c r i p t i o n o f t h e m o l e c u l e i n t h e r e g i m e o f t h e r m a l e n e r g i e s , where t h e n e a r - r e s o n a n c e p r o c e s s i s d o m i n a n t . - 251 -4.2-2 R e s u l t s I n T a b l e 11 we compare r o t a t i o n a l l y summed c r o s s s e c t i o n s , 0*Mv2 = 1 j — ^ = 0 ) f o r t h e ' V-T' p r o c e s s , i n w h i c h j-j •=• 2, and f o r t h e n e a r - r e s o n a n c e p r o c e s s , i n w h i c h = 2 and = 6. I t may be seen t h a t t h e c r o s s s e c t i o n s f o r t h e n e a r - r e s o n a n c e p r o c e s s a r e a l m o s t i n d e p e n d e n t o f t h e i n i t i a l k i n e t i c e n e r g y whereas t h e 1 V-T' c r o s s s e c t i o n s i n c r e a s e r a p i d l y w i t h E^ ,. The e n e r g y d i f f e r e n c e b e t w e e n t h e i n i t i a l and f i n a l c h a n n e l s becomes l e s s i m p o r t a n t a t Ej. i n c r e a s e s . T h e r e f o r e t h e m a g n i t u d e o f t h e n e a r - r e s o n a n c e c r o s s s e c t i o n s r e l a t i v e t o t h e A>3i = 0 c r o s s s e c t i o n s d e c r e a s e s w i t h Ej as i s c l e a r l y o b s e r v e d i n Table 1 1 . Due t o t h e e r r o r i n t h e p o t e n t i a l r o u t i n e t h e n e a r - r e s o n a n c e c r o s s s e c t i o n s a r e r o u g h l y a f a c t o r o f 4 t o o l a r g e , a s i s shown i n A p p e n d i x 4. Even when t h e c r o s s s e c t i o n s shown a r e r e d u c e d by a f a c t o r o f ' V 4, i t i s e v i d e n t f r o m T a b l e 11 t h a t i n t h e r e g i o n o f t h e r m a l e n e r g i e s , E^ ^ 300 K,one w o u l d c e r t a i n l y e x p e c t t h e n e a r - r e s o n a n c e p r o c e s s t o s i g n i f i c a n t l y i n f l u e n c e t h e r a t e o f v i b r a t i o n a l d e a c t i v a t i o n o f C0(v2 = 1) by p a r a - H 2 , a l t h o u g h as we w i l l show l a t e r i n t h i s S e c t i o n t h e n e a r - r e s o n a n c e r e l a x a t i o n r a t e i s i n s u f f i c i e n t t o a c c o u n t f o r t h e d i f f e r e n c e b e t w e e n t h e e x p e r i m e n t a l r a t e s f o r o r t h o - and p a r a - t ^ . I t i s c l e a r f r o m T a b l e 11 t h a t t h e n e a r - r e s o n a n c e c r o s s s e c t i o n s r i s e as t h e = 1 = 2 t h r e s h o l d i s a p p r o a c h e d . We a r e u n a b l e t o o f f e r a s a t i s f a c t o r y e x -p l a n a t i o n f o r t h i s u n p h y s i c a l b e h a v i o u r . We b e l i e v e TABLE 11 A c o m p a r i s o n o f r o t a t i o n a l l y summed c r o s s s e c t i o n s , n e a r - r e s o n a n c e v i b r a t i o n a l r e l a x a t i o n p r o c e s s . Note t h a t 1 cm ^ = 1.4388K. E.j. (cm ^ ) ( 1 2 = * 02) ( 1 2 - * 06) 64.8322 0.359(-6 ) 0 . 5 9 6 ( - 4 ) 83.0 0 . 3 8 9 ( - 6 ) 0 . 5 5 6 ( - 4 ) 129.915 0 . 5 3 4 ( - 6 ) 0 . 5 1 5 ( - 4 ) 208.0 0 . 9 1 4 ( - 6 ) 0 . 5 0 3 ( - 4 ) 2299.36 0 . 6 6 8 ( - 3 ) 0 . 5 4 8 ( - 4 ) 3167.14 0 . 2 0 9 ( - 2 ) 0 . 5 6 3 ( - 4 ) Numbers i n b r a c k e t s a r e powers o f 10. - 2r>2 -t h a i o u r n u m e r i c a l met I K i d s d e s c r i b e d I n .Seel I o n U a n d i n C h a p t e r V.4 a r e an a c c u r a t e d e s c r i p t i o n o f t h i s s y s t e m and some a d d i t i o n a l c a l c u l a t i o n s have i n d i c a t e d t h a t t h e e r r o r i n o u r programme i s n o t r e s p o n s i b l e f o r t h i s b e h a v i o u r . I t i s w o r t h n o t i n g t h a t t h e 'V-T' c r o s s s e c t i o n s were e v a l u a t e d e m p l o y i n g i d e n t i c a l n u m e r i c a l methods and s i n c e t h e e n e r g y v a r i a t i o n o f t h e s e c r o s s s e c t i o n s a p p e a r s t o be s a t i s f a c t o r y , i t i s p r o b a b l e t h a t t h e t r u n c a t i o n o f t h e r o t a t i o n a l b a s i s s e t m i g h t be t h e cause o f t h e u n p h y s t c a l e n e r g y v a r i a t i o n o f t h e n e a r - r e s o n a n c e c r o s s -s e c t i o n s . However a t v e r y low e n e r g i e s , \iy^ 100 K, where t h e r i s e o f t h e c r o s s s e c t i o n s i s most n o t i c e a b l e , t h e = 2 r o t a t i o n a l l e v e l i s n o t s i g n i f i c a n t l y p o p u l a t e d , and c o n s e q u e n t l y s u c h c r o s s s e c t i o n s w i l l have an i n -s i g n i f i c a n t e f f e c t on t h e r a t e o f v i b r a t i o n a l r e l a x a t i o n o f C O ( v 2 = 1) by p a r a - H 2 . I n T a b l e 12 we compare t h e v a l u e s o f t h e n e a r - r e s o n a n c e r e l a x a t i o n r a t e c o e f f i c i e n t w i t h t h e c o r r e s p o n d i n g r e s u l t s e v a l u a t e d by P o u l s e n and B i l l i n g ( 1 9 8 2 ) . I t i s w o r t h r e p e a t i n g t h a t b o t h c a l c u l a t i o n s employ t h e P p o t e n t i a l , and use Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e t h e v i b r a t i o n a l m o t i o n o f t h e CO m o l e c u l e . The e n e r g y d e f e c t s f o r t h e n e a r - r e s o n a n c e p r o c e s s , e m p l o y e d i n t h e c a l c u l a t i o n s , d i f f e r by 3.73 cm~^ ( s e e T a b l e 2 ) , t h i s s m a l l d i f f e r e n c e i s i n s i g n i f i c a n t compared t o t h e d i f f e r e n t d y n a m i c a l a p p r o a c h e s u s e d i n t h e c a l c u l a t i o n s . I f t h e r e l a x a t i o n r a t e f r o m t h e p r e s e n t c a l c u l a t i o n s i s d i v i d e d by a f a c t o r TABLE 12 A c o m p a r i s o n o f t h e r a t e c o e f f i c i e n t f o r v i b r a t i o n a l r e l a x a t i o n by t h e n e a r - r e s o n a n c e p r o c e s s ; ( a ) p r e s e n t IOS c a l c u l a t i o n ; ( b ) s e m i -c l a s s i c a l c a l c u l a t i o n o f P o u l s e n and B i l l i n g ( 1 9 3 2 ) . T ( K ) 0 < ( v ? = 1 -= 2 - * v 2 ' , ^ 0 j f = ^ 6) (.•i ) ( b ) 80 65 35 100 67 36 120 70 37 140 73 38 160 76 39 180 79 40 200 82 4 1 220 85 42 240 88 43 260 91 44 280 94 45 300 97 4 5 400 111 48 500 124 51 600 136 53 0 ( {^2 = 1 = 2 - ? > V 2 = 0 j ' = 6) i n 10 ^ c m ^s 1 - 2 r> 3 -o f ^ 4 t o a c c o u n t f o r t h e e r r o r i n t h e p o t e n t i a l r o u t i n e , t h e y a r e r o u g h l y a f a c t o r o f 2 l o w e r t h a n t h e r e s u l t s o b t a i n e d by P o u l s e n and B i l l i n g . I t i s w o r t h r e p e a t i n g t h a t when we had a c c o u n t e d f o r t h i s e r r o r , and t h e ' i n c o r r e c t ' r o t a t i o n a l e n e r g y l e v e l s e m p l o y e d by BSD2, o u r r e s u l t s w ere i n r e a s o n a b l e a g reement w i t h t h e r e s u l t s o b t a i n e d by BSD2 i n a c a l c u l a t i o n e m p l o y i n g t h e P p o t e n t i a l , and w h i c h i n c l u d e d t h e = 0 and 4 s t a t e s i n t h e r o t a t i o n a l b a s i s ( s e e A p p e n d i x 4 ) . The d i s c r e p a n c i e s b e t w e e n t h e p r e s e n t c a l c u l a t i o n s and t h o s e o f P o u l s e n and B i l l i n g ( 1 9 8 2 ) i s p e r h a p s y e t a n o t h e r i n d i c a t i o n t h a t t h e SCA e m p l o y e d by t h e s e a u t h o r s i s n o t s u i t a b l e t o s t u d y t h e r o v i b r a t i o n a l e x c i t a t i o n o f CO by . I n F i g u r e 2 we compare t h e r e s u l t s o f BSD2 f o r t h e n e a r - r e s o n a n c e p r o c e s s e v a l u a t e d i n c a l c u l a t i o n s e m p l o y i n g t h e DK p o t e n t i a l , and i n c l u d i n g t h e = 0 and 4 s t a t e s i n t h e i r r o t a t i o n a l b a s i s s e t , w i t h t h e c o r r e s p o n d i n g r e s u l t s e v a l u a t e d i n t h e p r e s e n t s t u d y . We r e c a l l t h a t BSD2 employ Morse o s c i l l a t o r w a v e f u n c t i o n s t o d e s c r i b e t h e v i b r a t i o n a l m o t i o n o f t h e CO m o l e c u l e . A l t h o u g h t h e m a g n i t u d e o f t h e c r o s s s e c t i o n s o b t a i n e d u s i n g t h e two p o t e n t i a l s a r e somewhat d i f f e r e n t , b o t h t h e P and DK p o t e n t i a l s p r o d u c e n e a r - r e s o n a n c e c r o s s s e c t i o n s w h i c h a r e s l o w l y v a r y i n g w i t h e n e r g y . B o t h c a l c u l a t i o n s g i v e o v e r e s t i m a t e d n e a r - r e s o n a n c e c r o s s s e c t i o n s . I n t h e p r e s e n t c a l c u l a t i o n s t h e c r o s s s e c t i o n s a r e r o u g h l y a f a c t o r o f 4 t o o l a r g e , a l s o BSD2 o v e r e s t i m a t e t h e p r o c e s s by 0 . 0 0 0 1 8 0 . 0 0 0 1 6 0 . 0 0 0 1 4 0 . 0 0 0 1 2 0 . 0 0 0 1 0 0 . 0 0 0 0 8 0 . 0 0 0 0 6 0 . 0 0 0 0 4 0 . 0 0 0 0 2 -0.00000 2 0 0 4 0 0 6 0 0 8 0 0 cm ) F i g u r e 2. A c o m p a r i s o n o f r o t a t i o n a l l y summed c r o s s s e c t i o n s f o r t h e n e a r - r e s o n a n c e p r o c e s s , e-(ii,-9> 0 6) A*. -) : P r e s e n t c a l c u l a t i o n s ; : BSD2 . A l s o shown a r e t h e c r o s s s e c t i o n s e v a l u a t e d by BSD2 f o r t h e 'V-T' p r o c e s s i n w h i c h J, = 2 ( - - - ) , and f o r t h e Aj't = 2 p r o c e s s i n whch j = 2 , and /' = 4 - 254 -a f a c t o r o f ^ . 2 . T h i s d i s c r e p a n c y i n t h e c a l c u l a t i o n s o f BSD2 i s due t o t h e use o f i n a c c u r a t e r o t a t i o n a l e n e r g y l e v e l s , as i s shown i n A p p e n d i x 4. The DK p o t e n t i a l has no Y ^ Q ( C O S ) t e r m ( w h i c h c o u p l e s d i r e c t l y A = 4 t r a n s i t i o n s ) . BSD2 employed t h e l o n g - r a n g e Y ^ Q ( C O S ©-^) t e r m s p r e s e n t e d by P o u l s e n ( 1 9 8 2 ) ( i . e . e q u a t i o n s V I . 4 . 2 c , d ) . I n d o i n g so t h e y were g u i d e d by t h e o b s e r v a t i o n o f P o u l s e n and B i l l i n g ( 1 9 8 2 ) , t h e s e a u t h o r s s t a t e t h a t t h e n e a r - r e s o n a n c e p r o c e s s i s p r i m a r i l y d e t e r m i n e d by t h e l o n g r a n g e i n t e r a c t i o n b e t w e e n t h e h e x a d e c a p o l e moment o f l - ^ , and t h e d i p o l e ( a n d q u a d r u p o l e ) moment o f CO. I n v i e w o f t h e dependence o f t h e n e a r - r e s o n a n c e p r o c e s s on t h e s e i n t e r a c t i o n s ( A p p e n d i x 4 ) , t h e y a r e c l e a r l y v e r y i m p o r t a n t . However, e v e n when one a c c o u n t s f o r t h e d i s c r e p a n c i e s d i s c u s s e d a b o v e , t h e DK and P p o t e n t i a l s g i v e d i f f e r e n t n e a r - r e s o n a n c e c r o s s s e c t i o n s . C l e a r l y t h e l o n g r a n g e Y^Qtcos© 1^) t e r m s a r e n o t t h e o n l y c o n t r i b u t o r s t o t h e n e a r - r e s o n a n c e p r o c e s s . I n p a r t i c u l a r i t i s i n t e r e s t i n g t o n o t e t h a t t h e s o - c a l l e d l o n g r a n g e m u l t i p o l e moment ( Y ^ Q ( C O S © ^ ) ) i n t e r a c t i o n s a r e i n c l u d e d i n b o t h t h e p o t e n t i a l s u r f a c e s a t a l l i n t e r -m o l e c u l a r d i s t a n c e s i n t h e i n t e g r a t i o n r a n g e . As m e n t i o n e d above P o u l s e n and B i l l i n g ( 1 9 8 2 ) c o n c l u d e t h a t t h e n e a r -r e s o n a n c e p r o c e s s i s p r i m a r i l y d e t e r m i n e d by t h e l o n g r a n g e i n t e r a c t i o n s . However, i t i s n o t o b v i o u s ( a t l e a s t t o u s ) t h a t t h e n e a r - r e s o n a n c e p r o c e s s i s o n l y d e t e r m i n e d by t h e m u l t i p o l e i n t e r a c t i o n s a t l o n g r a n g e . The p r o c e s s 255 may be determined ( p a r t i a l l y ) by these i n t e r a c t i o n s at s h o r t range (R j£ 6 a . u . ) . This by i t s e l f does not e x p l a i n the d i s c r e p a n c i e s between the near-resonance cross s e c t i o n s d e t e r m i n e d by the P and DK p o t e n t i a l s . However, i t i s w o r t h n o t i n g t h a t a t s h o r t range the m u l t i p o l e moment i n t e r a c t i o n s g i v e n by Poulsen (1982) w i l l be i n a c c u r a t e , s i n c e the forms o f i n t e r a c t i o n s g i v e n by VI.4.2a,b,c,d are a s y m p t o t i c forms. I f these i n t e r a c t i o n s are i m p o r t a n t at s h o r t range, t h e n we c o n s i d e r t h a t an a c c u r a t e d e t e r -m i n a t i o n o f the near-resonance process would be e x t r e m e l y d i f f i c u l t , s i n c e one would r e q u i r e an a c c u r a t e ab i n i t i o d e t e r m i n a t i o n o f the m u l t i p o l e moment i n t e r a c t i o n s at R J& 6 a.u. Also shown i n F i g u r e 2 are the cross s e c t i o n s f o r the = 2-^2 and = 2=>4 t r a n s i t i o n s e v a l u a t e d by BSD2. C l e a r l y t he near-resonance r a t e w i l l not form a s i g n i f i c a n t c o n t r i b u t i o n t o the r a t e o f d e - e x c i t a t i o n of CO by para-H2. The near-resonance cross s e c t i o n s are much s m a l l e r than theAj-^ = 2 cross s e c t i o n s . Note t h a t the = 2,=^4 cross s e c t i o n s are s t r o n g l y d i m i n i s h e d by the P p o t e n t i a l , s i n c e t h i s p o t e n t i a l employs the more a c c u r a t e v a l u e f o r the d i p o l e moment of CO. I n a d d i t i o n the near-resonance cross s e c t i o n s are c l e a r l y not s i g n i f i c a n t l y l a r g e r than the fa.]^ = 0 cross s e c t i o n s at t h e r m a l e n e r g i e s ( c f . Table 1 1 ) . I n c o n c l u s i o n i t i s apparent t h a t the P p o t e n t i a l p r o v i d e s a b e t t e r d e s c r i p t i o n o f the near-resonance process '/Mi than the DK p o t e n t i a l , a l s o i t would appear t h a t the P p o t e n t i a l p r o v i d e s a b e t t e r d e s c r i p t i o n o f the e x p e r i -mental r e s u l t s f o r o r t h o - H ^ . We d i s c u s s e d the e f f e c t o f Aj-j^ = 2 t r a n s i t i o n s , and the CS a p p r o x i m a t i o n i n S e c t i o n 4.1. I n view o f the d i f f e r e n t d e t e r m i n a t i o n s of the dependence of the i n t e r a c t i o n s i t i s s u r p r i s i n g ( a t l e a s t t o us) t h a t the P p o t e n t i a l s hould p r o v i d e a b e t t e r d e s c r i p t i o n of the v i b r a t i o n a l e x c i t a t i o n o f CO by (see Ch/ipl or V. r)) . He f o r e the present s t u d i e s , and the c a l c u l a t i o n s of BSD and BSD2 had been c a r r i e d o u t , the DK p o t e n t i a l was c o n s i d e r e d t o be the most a c c u r a t e d e s c r i p t i o n o f the 1 ^ + CO i n t e r a c t i o n ( e . g . see Schinke et a l . ( 1 9 8 4 ) ) . Indeed the DK p o t e n t i a l was c o n s t r u c t e d employing h i g h l y a c c u r a t e SCF c a l c u l a t i o n s , and Schinke et a l . (1984) have shown t h a t the DK p o t e n t i a l p r o v i d e s the best o v e r a l l d e s c r i p t i o n of r o t a t i o n a l e x c i t a t i o n i n the D2 1 CO system. The i n t e r a c t i o n potent i . i l Is expressed i n the form ( V I . 2 . 1 ) *l VI.4.4 we may expand i n a T a y l o r s e r i e s about the e q u i l i b r i u m CO i n t e r n u c l e a r s e p a r a t i o n , 2^.jC£_ ' a n <^ o b t a i n ( c f . V.5.3). VI .4.5 4± t l - k e O j i vr 257 As noted i n Chapter V, the su c c e s s i v e terms o f the expansion VI.4.5 are r e s p o n s i b l e f o r the t r a n s i t i o n s w i t h o n l y i n v o l v e the f i r s t term (and i n the c a l c u l a t i o n s T2 i s c o n s t r a i n e d at i t s e q u i l i b r i u m v a l u e , f ^ e * ^ , and t h e r e f o r e i t cannot be c o n s i d e r e d as a s u f f i c i e n t exam-i n a t i o n o f an i n t e r a c t i o n p o t e n t i a l . However, c a l c u l a t i o n s i n v o l v e the second ( c h i e f l y ) , and h i g h e r o r d e r terms i n the expansion. Such terms are more d i f f i c u l t t o d e t e r m i n e , and c o n s e q u e n t l y v i b r a t i o n a l e x c i t a t i o n i s a more s t r i n g e n t t e s t o f an i n t e r a c t i o n p o t e n t i a l . I n o r d e r t o compare w i t h the measurements of the r a t e o f v i b r a t i o n a l d e a c t i v a t i o n o f C0(v2 = 1) i n pa r a -H2, the r a t e c o e f f i c i e n t s must be we i g h t e d by the r e l a t i v e p o p u l a t i o n s o f the H2 r o t a t i o n a l c h a n n e l s , g i v e n by a Boltzmann d i s t r i b u t i o n . At low te m p e r a t u r e s o n l y the = 0 and 2 l e v e l s are s i g n i f i c a n t l y p o p u l a t e d , t h e r e f o r e we may w r i t e : ^ Vy = 0,1,2 . T h e r e f o r e r o t a t i o n a l e x c i t a t i o n w i l l s t u d y i n g the v i b r a t i o n a l e x c i t a t i o n process A v 2 1 w i l l ur0 IV.4.6 where *0 o 0C0 (-r) = 0 v7 + «< ( V i - / f,' 2. -=? oJ= t>) - 2 58 -and 53 ~f / „ E(0,2) i s the energy o f = 2 l e v e l above the = 0 ground s t a t e o f , and k i s Boltzmann's c o n s t a n t s . I n F i g u r e 3 we compare t he r e s u l t s o f the pr e s e n t c a l c u l a t i o n s w i t h the e x p e r i m e n t a l r e s u l t s o f Andrews and Simpson (1 9 7 6 ) . Also shown are the r e s u l t s o f Poulsen and B i l l i n g (1982) and the r e s u l t s o f BSD2. C l e a r l y t h e best agree-ment w i t h experiment i s o b t a i n e d by BSD2, however i n view o f the o v e r e s t i m a t e d Aj^ - 2 t r a n s i t i o n s we c o n s i d e r t h i s l e v e l o f agreement a c c i d e n t a l . The near-resonance process i s o v e r e s t i m a t e d i n our own c a l c u l a t i o n s , c o n s e q u e n t l y the P p o t e n t i a l p r o v i d e s a r e l a t i v e l y poor d e s c r i p t i o n o f the e x p e r i m e n t a l r e s u l t s f o r p a r a - l ^ . Most n o t a b l y , n e i t h e r t h e P or DK p o t e n t i a l s produce a s u f f i c i e n t l y l a r g e near-resonance r a t e t o account f o r the d i f f e r e n c e between t he r a t e s f o r para- and o r t h o -H2• These r e s u l t s are i n sharp c o n t r a s t t o the c o n c l u s i o n o f Poulsen and B i l l i n g ( 1 9 8 2 ) , however i n view o f the d i s c r e p a n c i e s between the r e s u l t s o f Poulsen and B i l l i n g and our own r e s u l t s ( e . g. f o r Aj-^ = 0 and 2 t r a n s i t i o n s ) the r a t e o b t a i n e d f o r p a r a - r ^ must be t r e a t e d w i t h a degree o f c a u t i o n . I n the l i g h t o f the pr e s e n t study (and a l s o the study o f BSD2 employing the P p o t e n t i a l ) we c o n s i d e r t h i s good agreement w i t h experiment a c c i d e n t a l . The p r e s e n t c a l c u l a t i o n s were performed t r e a t i n g t h e CO w i t h i n the IOS a p p r o x i m a t i o n , i f the CS a p p r o x i m a t i o n had been employed one might expect an i n c r e a s e of a f a c t o r 2.0 \-T(K) F i g u r e 3. The v a r i a t i o n w i t h t e m p e r a t u r e of the v i b r a t i o n a l r e l a x a t i o n c o e f f i c i e n t , o< (T) ( i n u n i t s o f 10"!^cm^s -!). S o l i d curve : p r e s e n t c a l c u l a t i o n s : dashed curve : Poulsen and B i l l i n g ( 1 9 8 2 ) ; + : BSD2;^ : Andrews and Simpson ( 1 9 7 6 ) . 259 o f i n the r e l a x a t i o n r a t e s . However even then the agreement w i t h experiment would s t i l l be f a r from s a t i s f a c t o r y . As shown i n t h i s study the magnitude o f the near-resonance process i s h i g h l y dependent on the l o n g range m u l t i p o l e i n t e r a c t i o n s , the i n t e r a c t i o n s between HH the hexadecapole moment of 1 ^ , Q^^y and the f i r s t d e r i v a t i v e CO of the d i p o l e moment of CO, JU^f^ ( c h i e f l y ) and the f i r s t CO d e r i v a t i v e o f the quadrupole moment of CO, ( VI . 4 . 2c , d ) . —^y^%/4o * ~kp> °& VI.4.7a VI.4.7b As mentioned i n Chapter V.2 Poulsen (1982) employed the most a c c u r a t e m u l t i p o l e moment f u n c t i o n s a v a i l a b l e i n the l i t e r a t u r e . I t i s wo r t h r e p e a t i n g t h a t the i n t e r a c t i o n s VI.4.7a,b were a l s o employed i n the DK p o t e n t i a l . However, i n view o f the poor d e s c r i p t i o n o f the d i f f e r e n c e between t he e x p e r i m e n t a l r a t e s f o r o r t h o - and para-H2, i t i s necessary t o b r i n g i n t o q u e s t i o n the accuracy o f the hexadecapole moment of ^ and the f i r s t d e r i v a t i v e o f t h e d i p o l e moment of CO. HH Q^^was determined by K a r l et a l . (1975) who employed the e l e c t r o n i c w a v e f u n c t i o n s o f Kolos and Wolniewicz (1965, 1975). For 2.2 a.u. these a u t h o r s employed the 54 term wave f u n c t i o n s o f Kolos and Wolniewicz ( 1 9 6 5 ) , and f o r l a r g e r i n t e r n u c l e a r s e p a r a t i o n s , ^ 2.2 a.u., they used the f i r s t 44 ( o r 30) terms o f the f u n c t i o n 260 used by Kolos and WolnLewicz (197 5 ) . I t i s apparent from the paper of Karl et a l . (1975) t h a t above computations ensure e x c e l l e n t convergence, and the a u t h o r s c o n s i d e r HH the v a l u e o f Q ^ ^ t o be a c c u r a t e t o a few p e r c e n t . CO jU{Q w a s d e t e r m i n e d by Chackerian (1976) by combining n u m e r i c a l s o l u t i o n s o f the Sc h r o d i n g e r e q u a t i o n w i t h a b s o l u t e i n t e n s i t y d a ta o f v i b r a t i o n - r o t a t i o n bands. I t i s e v i d e n t from the paper o f Chackerian (1976) t h a t CO t h e r e i s some doubt c o n c e r n i n g the accuracy ofJLL(,y • However i n view o f the good agreement between the r e s u l t s o f Chackerian (1976) w i t h the e x p e r i m e n t a l d e t e r m i n a t i o n (Weisback and Chackerian ( 1 9 7 3 ) ) , and the t h e o r e t i c a l d e t e r m i n a t i o n ( T i p p i n g (1976)) o f the m a t r i x elements f o r v i b r a t i o n - r o t a t i o n t r a n s i t i o n s ( i n v o l v i n g A v 2 = 1^ i n CO, i t i s not p o s s i b l e t h a t the f i r s t d e r i v a t i v e o f the d i p o l e moment i s i n e r r o r by as much as a f a c t o r o f 2. Indeed t h i s i s the l e v e l o f e r r o r t h a t one CO HH would have t o f i n d i n b o t h ^ t . ^ and Q ^ ^ t o p r o v i d e a s a t i s -f a c t o r y e x p l a n a t i o n f o r the poor d e s c r i p t i o n o f the e x p e r i -mental r a t e s f o r p a r a - ^ . I t i s i n t e r e s t i n g t h a t Poulsen and B i l l i n g (1982) p o s t u l a t e t h a t i f t he h i g h e r m u l t i p o l e moments o f CO were t r e a t e d e x p l i c i t l y l i k e the d i p o l e and the quadrupole ( t h e i n t e r a c t i o n between the h i g h e r m u l t i p o l e moments o f H2 and CO are accounted f o r i n the s h o r t range p o t e n t i a l ) t h a t l a r g e r near-resonance cross s e c t i o n s might be produced. Very r e c e n t l y the above a u t h o r s p u b l i s h e d the r e s u l t s o f a c a l c u l a t i o n o f the r a t e c o e f f i c i e n t f o r the v i b r a t i o n a l 261 de-exc I t n t I on of (!()(v^ I ) by p.'ir.i II., emp I <>y i ng the I' p o t e n t i a l i n which the o c t o p o l e and hexadecapole o f the CO molecule were t r e a t e d e x p l i c i t l y (Poulsen and B i l l i n g ( 1 9 8 4 ) ) . They f i n d t h a t the r a t e c o e f f i c i e n t f o r the near-resonance process are i n c r e a s e d by sx, 507o. However, i n view o f the d i s c r e p a n c i e s between the r e s u l t s o f Poulsen and B i l l i n g (1982) and the present c a l c u l a t i o n s (and a l s o those o f BSD2) we c o n s i d e r t h a t the r a t e c o e f f i c i e n t f o r the d e - e x c i t a t i o n o f C0(v2 = 1) by para-H2 as e v a l u a t e d by Poulsen and B i l l i n g (1984) , which agree e x t r e m e l y w e l l w i t h the e x p e r i m e n t a l r e s u l t s , s hould perhaps be t r e a t e d w i t h a degree o f c a u t i o n . As a l r e a d y d i s c u s s e d i t i s p r o b a b l e t h a t t he s e m i - c l a s s i c a l a p p r o x i m a t i o n employed by Poulsen and B i l l i n g (1984) i s not a p p r o p r i a t e t o s t u dy the v i b r a t i o n a l e x c i t a t i o n o f CO by H2, and o n l y a f t e r a f u l l y quantum mechanical c a l c u l a t i o n o f the near-resonance process has been performed employing the new P p o t e n t i a l (e.g. the P p o t e n t i a l employing the e x p l i c i t e x p r e s s i o n s f o r the o c t o p o l e and hexadecapole moments of CO) can the s i g n i f i c a n c e o f these h i g h e r m u l t i p o l e moments be p r o p e r l y assessed. 5. Summary We have performed quantum mechanical c a l c u l a t i o n s o f t he v i b r a t i o n a l d e - e x c i t a t i o n o f CO by H2, where the r o t a t i o n a l degree o f freedom of the H2 molecule has been t r e a t e d e x p l i c i t l y w i t h i n the CS a p p r o x i m a t i o n . The r o t a t i o n a l degree o f freedom o f the CO molecule was t r e a t e d 262 w i t h i n the IOS a p p r o x i m a t i o n , and the v i b r a t i o n a l m otion wasdescribed u s i n g Morse o s c i l l a t o r w a v e f u n c t i o n s . The i n t e r a c t i o n p o t e n t i a l s u r f a c e o f Poulsen (1982) has been employed. We have been p r i m a r i l y i n t e r e s t e d i n the c a l -c u l a t i o n o f cross s e c t i o n s and r a t e s f o r the near-resonance p r o c e s s , a l t h o u g h we have a l s o s t u d i e d the f o l l o w i n g processes CO (v±-0+H% ( A ) Co(Vz'°) + «z. QiO f o r 2\ = 3\ a n <3 = j - ^ ^ 2 where 2. The m u l t i p o l e moment i n t e r a c t i o n s ( V I . 4 . 2 ) g i v e n by Poulsen (1982) are a f a c t o r o f 2 too l a r g e , these e r r o r s were present i n our p o t e n t i a l r o u t i n e . Only the cro s s s e c t i o n s e v a l u a t e d f o r the near-resonance process have been s e r i o u s l y a f f e c t e d , they are a f a c t o r o f 4 too h i g h . We have found t h a t the cross s e c t i o n s and r a t e s f o r the 'V-T1 processes i n which = are independent o f the r o t a t i o n a l s t a t e o f the 1 ^ m o l e c u l e , f o r s m a l l j ^ . I n a d d i t i o n we f i n d t h a t the cross s e c t i o n s f o r the process i n which = + 2 are r o u g h l y 507o o f the c o r r e s p o n d i n g Aj-^ = 0 cro s s s e c t i o n s f o r = 0, and t h a t t he e f f e c t i s more pronounced f o r i n i t i a l r o t a t i o n a l s t a t e , j - ^ = 1. Our r e s u l t s are i n q u a n t i t a t i v e agreement w i t h t h e c a l c u l a t i o n s o f BSD2 employing the P p o t e n t i a l , however when the DK p o t e n t i a l i s employed by these a u t h o r s , w h i l e t he 'V-T' cr o s s s e c t i o n s are independent o f - 263 -f o r small, j j , they are- roughly a f a c t o r of 3 h i g h e r than the present r e s u l t s , a l s o Aj^ 2 cross sect Ions are found t o be the same o r d e r as the c o r r e s p o n d i n g Aj^ = 0 cross s e c t i o n s f o r = 0, and the e f f e c t i s more pronounced f o r = 1 and 2. The l a t t e r e f f e c t i s a t t r i b u t e d t o the o v e r e s t i m a t i o n o f the Y2Q(COS ) term o f the DK p o t e n t i a l . Poulsen and B i l l i n g (1982) a l s o f i n d t h a t the 1V-T 1 r a t e i s i n s e n s i t i v e t o , f o r s m a l l j ^ , a l t h o u g h the 'V-T' r a t e i s r o u g h l y a f a c t o r o f ^ 2 . 5 g r e a t e r than t h a t found i n the present c a l c u l a t i o n s , and i n a d d i t i o n these a u t h o r s f i n d t h a t A) = 2 r a t e s arc i n s i g n i f i c a n t compared t o t h e c o r r e s p o n d i n g Aj-^ = 0 r a t e s . I t i s not c l e a r t o us why they come t o t h i s c o n c l u s i o n s i n c e they a l s o employ the P p o t e n t i a l . I n the l i g h t o f our r e s u l t s , we concur w i t h BSD2 t h a t the i n t e r p r e t a t i o n o f the Aj^ = 0 r a t e as an o r t h o - f ^ r a t e i s wrong. I n a d d i t i o n we conclude t h a t the P p o t e n t i a l i s capable o f p r o v i d i n g a more s a t i s -f a c t o r y d e s c r i p t i o n o f the e x p e r i m e n t a l r a t e f o r the d e a c t i v a t i o n o f C0(v2 = 1) by o r t h o - f ^ than the DK p o t e n t i a l i f one i n c l u d e s the e f f e c t o f A j ^ = 2 r a t e s , and t r e a t s the r o t a t i o n a l degree o f freedom o f the CO molecule i n the CS a p p r o x i m a t i o n . We f i n d t h a t the cross s e c t i o n s f o r the near-resonance process are almost independent o f the i n i t i a l k i n e t i c energy, i n c o n t r a s t the Aj^ = 0 cross s e c t i o n s are r a p i d l y v a r y i n g w i t h energy. At th e r m a l e n e r g i e s the near-resonance process i s the dominant p r o c e s s , however, the computed - 264 -r a t e i s i n s u f f i c i e n t t o account f o r the d i f f e r e n c e between the e x p e r i m e n t a l r a t e s of the d e - e x c i t a t i o n o f CO by o r t h o - and para-H^ as determined by Andrews and Simpson (197 6 ) . When the e r r o r i n our p o t e n t i a l r o u t i n e i s accounted f o r , the agreement w i t h the e x p e r i m e n t a l r e s u l t s i s d i m i n i s h e d . Our r e s u l t s f o r the near-resonance process d i s a g r e e w i t h b o t h t h e r e s u l t s o f Poulsen and B i l l i n g (1982) and of BSD2. Even when the present r e s u l t s are c o r r e c t e d t o account f o r the e r r o r i n our programme, t he agreement w i t h the above c a l c u l a t i o n s i s s t i l l u n s a t i s f a c t o r y . The r e s u l t s o f BSD2 f o r t h i s process are o v e r e s t i m a t e d by a f a c t o r o f ^ 2 due t o the use o f i n a c c u r a t e r o t a t i o n a l energy l e v e l s , even when t h i s c o r r e c t i o n i s made d i s -c r e p a n c i e s remain. However, g i v e n t h e same p o t e n t i a l , energy d e f e c t , and r o t a t i o n a l b a s i s s e t , we f i n d good agreement between our r e s u l t s and those o f BSD2 f o r t h i s p r o c ess. I n p a r t i c u l a r we conclude t h a t the P p o t e n t i a l p r o v i d e s the best d e s c r i p t i o n o f the near-resonance p r o c e s s , but we are unable t o o f f e r a s a t i s f a c t o r y e x p l a n a t i o n f o r the poor agreement w i t h e x p e r i m e n t . We c o n s i d e r t h a t the d i s c r e p a n c i e s o f the present c a l c u l a t i o n s w i t h those o f Poulsen and B i l l i n g (1982) are p r o b a b l y due t o the f a i l u r e o f the s e m i - c l a s s i c a l a p p r o x i m a t i o n employed by these a u t h o r s . I n c o n c l u s i o n t h e near-resonance process i s s e n s i t i v e t o h i g h o r d e r terms i n the i n t e r a c t i o n p o t e n t i a l , t h e energy d e f e c t , and the H 9 r o t a t i o n a l b a s i s s e t . We agree - 265 _ w i t h BSD2 i n t h a t we c o n s i d e r an a c c u r a t e ab i n i t i o study o f the near-resonance process ( V I . 2 . 5 ) i n a complex system l i k e H~ + CO i s not f e a s i b l e at the p r e s e n t t i m e . 7 (> f> CHAI'TKK V I I ROVIBRATTONAL EXCITATION OF 1 2C 1 6Q BY HP 1. I n t r o d u c t i o n I n t h i s c h a p t e r we w i l l p r e s e n t c a l c u l a t i o n s o f the r o v i b r a t i o n a l e x c i t a t i o n o f CO by HD. The m o t i v a t i o n s f o r such a st u d y are t h r e e f o l d . There are no e x i s t i n g t h e o r e t i c a l s t u d i e s o f t h i s process i n the l i t e r a t u r e , s econdly t h e r e i s a p o t e n t i a l energy s u r f a c e a v a i l a b l e , and t h i r d l y t h e r e are e x p e r i m e n t a l r e s u l t s w i t h which we can compare our r e s u l t s . The p o t e n t i a l energy s u r f a c e f o r HD + CO can be o b t a i n e d from a knowledge o f the ^ + CO p o t e n t i a l by means o f a c o - o r d i n a t e t r a n s f o r m a t i o n . W i t h i n the Born-Oppenheimer (BO) a p p r o x i m a t i o n (see Chapter I ) , i s o t o p i c s u b s t i t u t i o n w i t h i n one o f the c o n s t i t u e n t molecules merely causes a s h i f t o f the c e n t r e o f mass o f the mo l e c u l e , l e a v i n g the n a t u r e o f the i n t e r m o l e c u l a r i n t e r a c t i o n u n a f f e c t e d . Thus w i t h i n the v a l i d i t y o f the BO a p p r o x i m a t i o n , the i n t e r a c t i o n p o t e n t i a l f o r HD +C0 expressed i n c o o r d i n a t e s about the c e n t r e o f mass of the HD molecule can be e x t r a c t e d from a H2 + CO i n t e r a c t i o n p o t e n t i a l by a s u i t a b l e c o o r d i n a t e t r a n s f o r m a t i o n . An a c c u r a t e n u m e r i c a l procedure f o r c a r r y i n g out such a t r a n s f o r m a t i o n i s a v a i l a b l e i n the l i t e r a t u r e ( L i u et a l . ( 1 9 7 8 ) ) , and w i l l be employed i n these c a l c u l a t i o n s . The e x p e r i m e n t a l data a v a i l a b l e i s i n the form o f r e l a x a t i o n r a t e c o e f f i c i e n t s . Andrews and Simpson - 267 -(1976) have o b t a i n e d the r a t e c o e f f i c i e n t f o r the v i b r a t i o n a l d e - e x c i t a t i o n o f C0(v2 = 1) by HD i n the tempe r a t u r e range, 75&T-S&340 K. Altho u g h we w i l l compare the r a t e c o e f f i c i e n t e v a l u a t e d u s i n g our r e s u l t s w i t h the e x p e r i -mental d a t a , the main o b j e c t o f these c a l c u l a t i o n s i s t o d e t e r m i n e the e f f e c t s o f the i s o t o p i c s u b s t i t u t i o n on d i f f e r e n t aspects o f the s c a t t e r i n g c a l c u l a t i o n s . For example we w i l l d i s c u s s i t s e f f e c t on the d e s c r i p t i o n o f the c o l l i s i o n p a r t n e r s , and on the parameters such as t he s t e p s i z e and the i n t e g r a t i o n range. Thus t h r o u g h o u t the course o f t h i s study we w i l l make numerous comparisons w i t h r e s u l t s o b t a i n e d f o r the 1^ i CO system. To f a c i l i t a t e these comparisons the same dynamical a p p r o x i m a t i o n s used i n our s t u d i e s o f the l - ^ + CO system are employed i n the pr e s e n t c a l c u l a t i o n s . Thus the r o t a t i o n a l degrees o f freedom o f the CO and HD molecules are t r e a t e d w i t h i n the IOS and CS a p p r o x i m a t i o n s r e s p e c t i v e l y . Since HD i s a h e a v i e r molecule than H2, the IOS a p p r o x i m a t i o n w i l l be l e s s v a l i d f o r the CO molecule (Chapter I I . 3 . 3 ) . The v i b r a t i o n a l motion o f the CO molecule w i l l be t r e a t e d w i t h i n t h e c l o s e - c o u p l i n g a p p r o x i m a t i o n . Only the i n t e r a c t i o n p o t e n t i a l o f Poulsen (1982) i s used i n these c a l c u l a t i o n s , the t r a n s f o r m a t i o n o f t h i s p o t e n t i a l i s d e s c r i b e d i n S e c t i o n 2, i n S e c t i o n 3 we p r e s e n t the n u m e r i c a l d e t a i l s and make comparisons w i t h t h e H^ + CO system, and f i n a l l y i n S e c t i o n 4 we - 268 -compare our r e s u l t s w i t h the e x p e r i m e n t a l data and draw our f i n a l c o n c l u s i o n s r e g a r d i n g t h i s system. 2. The I n t e r a c t i o n P o t e n t i a l As mentioned i n S e c t i o n 1, w i t h i n t h e BO a p p r o x i m a t i o n , i s o t o p i c s u b s t i t u t i o n does not change the n a t u r e o f the i n t e r a c t i o n p o t e n t i a l , hence f o r example the 1 ^ + CO i n t e r a c t i o n p o t e n t i a l o f Poulsen (1982) has been employed i n T>2 + CO c a l c u l a t i o n s w i t h o u t any m o d i f i c a t i o n s ( B t l l i n g and Poulsen ( 1 9 8 2 ) ) . However, i f o n l y one o f the atoms i s i s o t o p i c a l l y s u b s t i t u t e d , t h i s causes a s h i f t o f the c e n t r e o f mass of the molecule towards the h e a v i e r n u c l e u s . Hence i f one of t h e H atoms i s r e p l a c e d by a D atom, t h i s causes the c e n t r e o f mass t o s h i f t a d i s t a n c e ^ towards the D atom, as shown i n F i g u r e 1. H 6L 9 o F i g u r e 1. The H2 + CO i n t e r a c t i o n p o t e n t i a l may be expressed ( c f . V I . 2 . 2 ) . 269 and i n terms of the new c o o r d i n a t e s ( f%. ? fe* ) the HD + CO i n t e r a c t i o n may be w r i t t e n V I I . 2 . 2 where R , , and 6^ are s i m p l y r e l a t e d t o R, ^ , and &^ by g e o m e t r i c a l r e l a t i o n s K -K/(l+irk-2fG*y&/ vn. 2.3a V I I . 2 . 3 b and 9-^ - «•/ ^ a VI 1.2.3c where Since as mentioned above the i s o t o p i c change does not s i g n i f i c a n t l y a l t e r t h e p o t e n t i a l s u r f a c e , we have V a,*)* 6 6 ^ 9 , > ^ , ^ 0 and u s i n g the o r t h o g o n a l i t y p r o p e r t i e s o f the Legendre p o l y n o m i a l s (e.g. see L i u e t a l . ( 1 9 7 8 ) , and Kreek and Le Roy ( 1 9 7 5 ) ) . *J I U T T O /. 270 where 3C * C&G &( . | N D x,'^ LOU &f. The f u n c t i o n s Usfc (&%~> &*~) are determined n u m e r i c a l l y from the exact F^ + CO p o t e n t i a l , (^ifi-a.) 1~g} by e v a l u a t i n g the i n t e g r a l V I I . 2 . 4 u s i n g Gauss Legendre q u a d r a t u r e . T h i s t e c h n i q u e has been employed i n s e v e r a l c a l c u l a t i o n s ( e . g. see L i u e t a l . ( 1 9 7 8 ) , and Buck e t a l . (1980)) b o t h these a u t h o r s have shown the above method t o be more a c c u r a t e than an a n a l y t i c a l method f o r d e t e r m i n i n g the f u n c t i o n s i n terms o f the f u n c t i o n s , d e r i v e d by Kreek and Le Roy ( 1 9 7 r j ) . Furthermore the method p r e s e n t e d by Kreek and Le Roy (1975) i n v o l v e s much c o m p l i c a t e d a l g e b r a , and would be e x t r e m e l y t e d i o u s t o employ, e s p e c i a l l y i n view o f t h e complex form o f th e P p o t e n t i a l . To date o n l y atom-diatom p o t e n t i a l s have been t r a n s -formed u s i n g V I I . 2 . 4 ( L i u et a l . ( 1 9 7 8 ) , Buck et a l . ( 1 9 8 0 ) , and L i u and McCourt ( 1 9 7 9 ) ) . For d i a t o m - d i a t o m p o t e n t i a l s r e l a t i o n s VI I . 2. 3a, V I I . 2. 3b, .incl V I I . 2.4 are s t i l l v a l i d , however one must ensure t h a t the r e l a t i v e c o n f i g u r a t i o n o f the two diatoms remains f i x e d as the i n t e r m o l e c u l a r v e c t o r moves from A t o A^. Thus one should t r a n s f o r m from t o u s i n g r e l a t i o n V I I . 2 . 3 c , however i n the p r e s e n t c a l c u l a t i o n s we o m i t t e d t h i s t r a n s f o r m a t i o n . For an i n t e r m o l e c u l a r s e p a r a t i o n o f R = 3 a.u. which i s the minimum i n these c a l c u l a t i o n s , i t s o m i s s i o n r e s u l t s i n an e r r o r i n 6^ o f £ 3° which i s u n i m p o r t a n t when one r e c a l l s t h a t t he CO molecule 27] i s t r e a t e d w i t h i n the IOS a p p r o x i m a t i o n which i g n o r e s some of the p o t e n t i a l a n i s o t r o p i c s . To t e s t the t r a n s f o r m a t i o n V I I . 2 . 4 f o r the p r e s e n t c a l c u l a t i o n s we e v a l u a t e d the exact HD + CO p o t e n t i a l by s u b s t i t u t i n g V I I . 2 . 3 a and V I I . 2 . 3 b d i r e c t l y i n t o V I I . 2 . 1 , and compared these 'exact' r e s u l t s w i t h the n u m e r i c a l values o b t a i n e d by employing V I I . 2 . 4 and V I I . 2 . 2 Such a comparison i s g i v e n i n Table 1, r e s u l t s are shown f o r &l = 0°, and S^f = 90° a t s e v e r a l i n t e r m o l e c u l a r s e p a r a t i o n s , R, and at t h r e e CO i n t e r n u c l e a r d i s t a n c e s , r r^. The v a l u e o f the s h i f t , S, was taken t o be 0.1665 /7^^ (Kreek and Le Roy ( 1 9 7 5 ) ) , where H^ip ^ s t n e e q u i l i b r i u m s e p a r a t i o n g i v e n i n Chapter V. 3 , = 1.4 a.u. L i u et a l . (1978) c l a i m t h a t f o r an i n t e g r a l o f the ty p e V I I . 2 . 4 an N-point Gaussian q u a d r a t u r e , w i t h V I I . 2 . 5 s h o u l d be h i g h l y a c c u r a t e , hence f o r the r e s u l t s p r e s e n t e d the 24- p o i n t Gauss Legendre q u a d r a t u r e employed s h o u l d be s a t i s f a c t o r y . Table 1 c l e a r l y i l l u s t r a t e s t h a t as more terms are r e t a i n e d i n the expansion V I I . 2 . 2 , t he r e s u l t s t e n d t o t he exact v a l u e s . As R i n c r e a s e s t he number of terms t h a t must be r e t a i n e d t o achieve a g i v e n accuracy decreases; as expected t he p o t e n t i a l i s more a n i s o t r o p i c a t c l o s e approach. For the range o f R v a l u e s c o n s i d e r e d i n these c a l c u l a t i o n s ( 3 . 0 ^ R ^ 3 5 . 0 a.u.) i t i s c l e a r OQ < H e u QJ CO \-> QJ - O e c QJ CO 3 r-* CO > QJ CO 3 H-l o c o CO •r-l CO a E o u c o • H O C 4-i • QJ O oo CO T - l fe er a-J O 4 -1 i-> CO •1-1 c X T3 •r- i ua 00 c •r - l CO X ) c + o •r - l • CO Q 0 o to l-Tl et e r - l u O N H QJ X OJ on .-t CM CM vO •£> c o c o 00 00 d O I I O CM o o ; s t m t O o o o sO vO CO CO CO o d d cs L O i n O N O N S f CNl CNl CO CO CO o o o < t I I C O ' " O c o c o O O I I 00 T - I C O CNI T - I T - I : oo oo T H T H T-l T - l d d O d m i n • 1 N O i i O N i T - I 1 va- i CO CO CO i n CNI CO CM CM CNI m m vO v£> O N O N m m O O O O CNI CM m un d d O I d I i n i n m m I 6 CO 1 CO m O O N T—1 C O m CO CM vO T - I O N T-l T-l C r~- r - a> d> ci d d d d d d O O o o o o o o o o o o + + + + + + + + + + + + 0) CD CNI w CQ < O a -> CO a e o o 05 05 CO o 05 CU X ) e c CNI (0 »-l 00 c • H 05 c •r-l O a CO o • TH CNI o 00 T-l 05 c •r-l O a CM r-l o m TH in 4-1 C •r-l 0 a o O rH CN rH o 05 c •r-l O a o 00 ON x o i£> O CO 3 CO "Si < u I - ON O oo «—I CNI , cO vD CO CO 00 O d m U 0 I I J I N CNJ QN ON S o : CO CO d d CNJ < } • CO C O O > C T > CO IN IN O N T -vf \D CO CO ; O O O ON ON o d d 999 m oo o> CO CO CO CNI CNJ CNJ o d d co £> o -v >A ON n , CNI CNJ CO CO d d m m I I ON CO VJD CJN I N rH I N 00 VJD I N o o I I CO <}" i m i n I N I N m O ON vo m O O i n m d d I i m m vD c o v£> rH 00 m r— CO CNI . O O CNI CNI oo oo d d i i NT I N j - ON CNJ CNI ON O \D m CNJ CNJ 00 00 d d i i c~> cO c o ^ rH CO x^> oS 3 O m i n : CNI 3 3 ON ON ON C\l CNI CNI d d d d d d o + o + o + 9 I N m CO CNJ 00 m m ON 00 ON TH CNI 00 00 tN oo CNI CO CO o rH fN I N ; 1 o O r rH rH rH CO NJ" CD U1 27 3 -a l l o w s f o r c o l l i s i o n a l t r a n s i t i o n s between the a d j a c e n t r o t a t i o n a l m a n i f o l d s of HD ( i . e . a&j^ = - 1 c o l l i s i o n a l s e l e c t i o n r u l e i s o p e r a t i v e i n these c a l c u l a t i o n s ) . As a l r e a d y mentioned a number o f times i n t h i s t h e s i s the l o n g range m u l t i p o l a r i n t e r a c t i o n s as g i v e n by Poulsen (1982) are a f a c t o r o f 2 too l a r g e (Chapter V I ) . The e f f e c t s o f these e r r o r s on the p r e s e n t c a l c u l a t i o n s are d i s c u s s e d i n Appendix 5 at the end of t h i s t h e s i s . I t i s w o r t h m e n t i o n i n g t h a t the l e a d i n g l o n g range ( m u l t i p o l a r ) term i s now t h e i n t e r a c t i o n between t h e d i p o l e moment of HD and the d e r i v a t i v e s ( w i t h r e s p e c t _ 3 t o ^ ) o f the d i p o l e moment o f CO,which v a r i e s as R (e. g . see L e a v i t t ( 1 9 8 0 ) ) . 3. Numerical d e t a i l s As s t a t e d i n S e c t i o n ^ o n e of the o b j e c t s o f these c a l c u l a t i o n s i s t o make a comparison w i t h t he e x p e r i m e n t a l l y d e t e r m i n e d r a t e c o e f f i c i e n t f o r the v i b r a t i o n a l r e l a x a t i o n of CO ( v 2 = 1) by HD i n the temp e r a t u r e range, 7 5 ^ T ^ 3 4 0 K . Thus we are concerned w i t h the c a l c u l a t i o n o f r o t a t i o n a l l y summed cross s e c t i o n s , CT (Wi.^ — ^ ^3!^-/°^ , f o r the process * V I I . 3 . 1 i n which the HD molecule undergoes the r o t a t i o n a l t r a n s i t i o n from j ^ = 0 and 1 t o j ^ ^ 5 . We p e r f o r m c a l c u l a t i o n s a t s i x c o l l i s i o n e n e r g i e s , E = 1.03, 1.04, 1.06, 1.07, 1.09 and 1.1 'fi'w, where the u n i t o f energy, fiw = 0.26989 eV. 274 -T h i s range o f e n e r g i e s enables us t o d e t e r m i n e a reasonable v a l u e f o r the r a t e c o e f f i c i e n t i n the t e m p e r a t u r e range, 140^-T^240K. Above 240K the j 1 = 2 r o t a t i o n a l s t a t e o f HD i s s i g n i f i c a n t l y p o p u l a t e d ( ^ > 20% of HD i s i n the = 2 l e v e l ) , t o o b t a i n the r a t e c o e f f i c i e n t f o r T ^ 240 K we would have t o study e n e r g i e s at which the y^*rl^l^'2. channel i s open. Such c a l c u l a t i o n s are not f e a s i b l e w i t h the computer time a v a i l a b l e f o r t h i s s t u d y . The n u m e r i c a l d e t a i l s o f the CO molecule are those g i v e n i n Chapter V, the v i b r a t i o n a l b a s i s r e q u i r e d i n these c a l c u l a t i o n s i s d i s c u s s e d l a t e r i n t h i s s e c t i o n . The mass of the HD molecule i s found by adding the n e u t r o n r e s t mass g i v e n by Weast (1975) t o the mass o f the H2 m o l e c u l e g i v e n i n Chapter V. 3, the e q u i l i b r i u m sep-a r a t i o n o f HD i s the v a l u e o f ^ e ^ , g i v e n f o r H£ i n t h a t s e c t i o n . The reduced mass o f the system i s found t o be 4975.83 a.u. a f a c t o r o f M . 5 g r e a t e r t h a n t h a t f o r t he H2 + CO system. The r o t a t i o n a l e i g e n e n e r g i e s o f the HD molecule are t a k e n t o be those c a l c u l a t e d u s i n g th e parameters g i v e n by Herzburg ( 1 9 5 0 ) , and are g i v e n b y ^ ' ^ ^ O - v ^ * 0 " where B = 44.6586 cm" 1 and D = 0.0257873 c m - 1 o o As i n our p r e v i o u s c a l c u l a t i o n s the coupled e q u a t i o n s are s o l v e d u s i n g the R-matrix p r o p a g a t o r method employing p r o p a g a t o r s c o r r e s p o n d i n g t o a c o n s t a n t r e f e r e n c e p o t e n t i a l (see Chapter I I I ) . The i s o t o p i c s u b s t i t u t i o n does not s i g n i f i c a n t l y a f f e c t t he v a r i a t i o n o f the i n t e r a c t i o n 275 p o t e n t i a l as a f u n c t i o n o f R , t h e r e f o r e the s t e p s i z e , X, and t h e s t a r t i n g p o i n t , R ^ n > r e q u i r e d t o m a i n t a i n accuracy s h o u l d be s i m i l a r t o those d e t e r m i n e d f o r our H2 + CO c a l c u l a t i o n s (see Chapter V . 4 . 1 ) . We began our convergence t e s t s w i t h HD c o n s t r a i n e d t o i t s ground r o t a t i o n a l s t a t e , and found t h a t a v a l u e o f X = 0.05 a.u. i s a b l e t o converge the r e s u l t s t o w i t h i n a few per cent at a l l the e n e r g i e s s t u d i e d i n these c a l c u l a t i o n s , i n c r e a s i n g X t o 0.1 a.u. changes t h e r e s u l t s by 170. I n a d d i t i o n we found t h a t a v a l u e o f R = 3 a.u. was mm s a t i s f a c t o r y . At a g i v e n c o l l i s i o n energy the range o f p a r t i a l wave parameters t h a t must be c o n s i d e r e d t o converge the i n t e g r a l c r oss s e c t i o n s t o w i t h i n 1% i s r o u g h l y a f a c t o r o f 1*5 h i g h e r t h a n t h a t r e q u i r e d i n a c o r r e s p o n d i n g H2 + CO c a l c u l a t i o n . For example, a t E = 1.1-ttw we r e q u i r e d a l l p a r t i a l cross s e c t i o n s w i t h L < 30 t o e v a l u a t e G* (v*. =ljfi-0 vz!~<^//-0) i n our H2 + CO c a l c u l a t i o n s , i n the HD + CO system we must c o n s i d e r a l l p a r t i a l c r o ss s e c t i o n s w i t h L ^ 4 5 . T h i s i s due t o the h i g h e r reduced mass o f the HD + CO system, which makes t h e i n t e r a c t i o n p o t e n t i a l more i m p o r t a n t a t a g i v e n v a l u e o f L (see Chapter I V . A ) . D e s p i t e the h i g h e r reduced mass we found no s i g n i f i c a n t i n c r e a s e i n the upper l i m i t o f the i n t e g r a t i o n range employed / i n our H_ + CO c a l c u l a t i o n s , a v a l u e o f R = 2 5 a.u. 2 max i s a b l e t o m a i n t a i n convergence t o w i t h i n a few per cent a t a l l t he c o l l i s i o n e n e r g i e s s t u d i e d . I t i s w o r t h - 2 7b -n o t i n g t h a t employing a s t e p s i z e o f 0.1 a.u. and i n c r e a s i n g the v a l u e o f t o 40 a.u. changes the r e s u l t s by max & J £ 1%. As expected v i b r a t i o n a l e x c i t a t i o n i s not t i m p o r t a n t f o r R ^-25 a.u. I n Table 3 we show the v a l u e s of fSL (Vt.=/f,Jli ] &£)J e v a l u a t e d at E = 1.1 f[w and L= 0, 10 at &r = 0° u s i n g e. s t e p s i z e o f 0.1 a.u. R e s u l t s are shown f o r = 0 / / a t two va l u e s o f R , R = 25.0, 30.0 a.u. Also max' max ' shown are the c o r r e s p o n d i n g 1V-T' r e s u l t s e v a l u a t e d f o r the H 2 + CO system. This comparison w i t h the H_2 + CO r e s u l t s g i v e s the f i r s t i n d i c a t i o n o f the e f f e c t s o f the i s o t o p i c s u b s t i t u t i o n on the f i n a l r e s u l t s . P h y s i c a l l y s i n c e t he HD molecule i s h e a v i e r t h a n H 2 i t i s l e s s e f f e c t i v e i n v i b r a t i o n a l l y e x c i t i n g CO than H 2 ( i . e . a t a g i v e n k i n e t i c energy the HD moves a t a slower speed). When we i n t r o d u c e d the HD r o t a t i o n a l degree o f freedom i n t o the c a l c u l a t i o n we found t h a t i t was not p o s s i b l e t o converge the va l u e s o f (v%.~lrft&i p Vjis0^t &f/P^,} f*~ f o r the i n d i v i d u a l r o t a t i o n a l processes t o the same l e v e l o f accuracy as shown i n Table 3. As d i s c u s s e d l a t e r i n t h i s s e c t i o n we c l a i m t h a t t h i s i s due t o the t r a n s f e r o f f l u x t o and from the HD r o t a t i o n a l s t a t e s , which are c l o s e l y spaced and s t r o n g l y c oupled by the p o t e n t i a l . Only when we sum the v a l u e s o f iStfe-t^t^H) V&'OjfSif/fy?*)} over t he f i n a l HD r o t a t i o n a l s t a t e s , j ^ , do we f i n d r e a s o n a b l e agreement. TABLE Comparison o f jSL (^i^vj* ol&£)t[ a •u d i f f e r e n t v a l u e s o f R' at E max e v a l u a t e d u s i n g 1.1 -hw and = 0°. Also shown are r e s u l t s o b t a i n e d f o r the H~ + CO system (a) R ; - 25 a.u., (b) R 7 0 max max 2 30 a u. HD CO (a) (b) 0.24441874-7 0.24441238-7 H 2 + CO 0.55028077-6 0.55028163-6 10 (a) 0.10376396-7 (b) 0.10376931-7 0.18848531-6 0.18848694-6 TABLE 4 Comparison o f r e s u l t s o b t a i n e d r e t a i n i n g d i f f e r e n t numbers of v i b r a t i o n a l channels i n the coupled e q u a t i o n s at E = 1.1 *fiw. Also shown are r e s u l t s f o r the H^ + CO system. (a) 2 c h a n n e l s , (b) 3 cha n n e l s , ( c ) 4 cha n n e l s . e' 2 . 0° 90° 180° HD + CO (a) 0. 24441874- 7 0. 36120377- 8 0. 96716586- 9 (b) 0. 28660642- 7 0. 36762079- 8 0. 14648386- 8 (c ) 0. 28662355- 7 0. 36762540- 8 0. 14649408- 8 H 2 + CO 0. 55028077- 6 0. 80104030- 7 0. 51323274- 7 0. 59879974- 6 0. 80773173- 7 0. 62622036- 7 0. 59880180- 6 0. 80772145- 7 0. 62641953- 7 Values o f / &L. (V^ o/&^ ) / 2 ( a . u. ) 2 o b t a i n e d a t L = 0 • 277 Due t o the enhanced reduced mass of the HD + CO system the v i b r a t i o n a l channels o f CO a r e more s t r o n g l y c oupled by the p o t e n t i a l . I n Table 4 we show the convergence f o r b o t h the H 2 + CO and HD + CO systems. As one can see i t i s s u f f i c i e n t t o i n c l u d e a l l the open and one c l o s e d v i b r a t i o n a l channel t o o b t a i n s a t i s f a c t o r y r e s u l t s f o r b oth systems at t h i s c o l l i s i o n energy, however n o t i c e t h a t as expected the c l o s e d channels are much more i m p o r t a n t i n the HD + CO system. For example a t 9 2 =&L = 180° the a d d i t i o n o f the second e x c i t e d v i b r a t i o n a l s t a t e , = 2, changes the r e s u l t by r o u g h l y 50% (even a t t h i s energy, E = 1.1-fiw, which i s w e l l below t h e = 2 t h r e s h o l d ) , whereas the c o r r e s p o n d i n g r e s u l t i s changed by s\s 20% i n the H 2 + CO system. Even at E = 1.9 -ttw, at ^ = 180°, the a d d i t i o n o f = 2 o n l y changes the r e s u l t by ^ 40% i n the H 2 t CO system (see Table 18 , Chapter V ) . This imposes a s t r o n g r e s t r i c t i o n upon the energy range we can study,as we s h a l l show l a t e r i n t h i s s e c t i o n i t i s necessary t o i n c l u d e a t l e a s t t h e f i r s t s i x r o t a t i o n a l l e v e l s of HD t o o b t a i n s a t i s f a c t o r y r e s u l t s , w i t h such a l a r g e r o t a t i o n a l b a s i s we found t h a t the a d d i t i o n o f each v i b r a t i o n a l channel i n c r e a s e d the computer time r e q u i r e d t o s o l v e t he coupled e q u a t i o n s by ^v50%,. E v i d e n t l y as the c o l l i s i o n energy i n c r e a s e s beyond E = 1.1 -fiw, i t w i l l soon be necessary o f at L = 0, a t t h r e e CO o r i e n t a t i o n s , g^= 0°, 90°, 180 E = 1.1 -^Tw and 278 t o i n c l u d e at l e a s t two c l o s e d v i b r a t i o n a l channels t o o b t a i n a s a t i s f a c t o r y d e s c r i p t i o n of the CO molecule and such c a l c u l a t i o n s are not f e a s i b l e , i f s u f f i c i e n t e n e r g i e s are t o be s t u d i e d t o e v a l u a t e a r e a s o n a b l y a c c u r a t e v a l u e f o r the r a t e c o e f f i c i e n t . We employ a v i b r a t i o n a l b a s i s c o n s i s t i n g of v 2 ^ 2 i n a l l these c a l c u l a t i o n s , the v i b r a t i o n a l p o t e n t i a l m a t r i x elements are e v a l u a t e d u s i n g a 28- p o i n t Gauss Legendre q u a d r a t u r e , t h i s ensures the r e s u l t s are converged t o w i t h i n a few per cent at a l l the e n e r g i e s s t u d i e d i n t h i s c a l c u l a t i o n . G e n e r a l l y speaking the HD molecule was much more d i f f i c u l t t o d e a l w i t h than the CO molecule. We r e c a l l t h a t i n our H2 + CO c a l c u l a t i o n s we d e termined the s t e p s i z e and i n t e g r a t i o n range r e q u i r e d t o m a i n t a i n accuracy at a g i v e n i n i t i a l k i n e t i c energy, E T, i n c a l c u l a t i o n s i n v o l v i n g H 2 c o n s t r a i n e d t o i t s ground r o t a t i o n a l s t a t e . These parameters were then employed i n c a l c u l a t i o n s s t u d y i n g r o t a t i n g H 2 at the same i n i t i a l k i n e t i c energy (see Chapter V I ) . I n the present c a l c u l a t i o n s , however, we do not take account o f the i n i t i a l r o t a t i o n a l s t a t e o f HD ( i . e . i f we study HD ( j - ^ = 0) at a t o t a l c o l l i s i o n energy o f E, we do not i n c r e a s e the c o l l i s i o n energy t o E + £>j^ t o study H D ( j 1 / 0 ) ) . T h i s appears t o be a reasonable approach s i n c e the r o t a t i o n a l energy l e v e l s o f HD are v e r y c l o s e l y spaced compared w i t h the energy s p a c i n g o f the v i b r a t i o n a l energy l e v e l s i n CO ( c f . £ V j a _ j ^ 2000 cm - 1, 90 cm - 1, *~ 1300 cm 279 The s l i g h t change i n the i n i t i a l k i n e t i c energy at a g i v e n c o l l i s i o n energy s h o u l d not have any s i g n i f i c a n t e f f e c t upon the i n t e g r a t i o n parameters. I n Table 5 we show the convergence o f the r e s u l t s as a f u n c t i o n o f the number o f r o t a t i o n a l s t a t e s i n c l u d e d i n the coupled e q u a t i o n s f o r both the HD + CO and H 2 + CO systems. The r e s u l t s are shown f o r the t r a n s i t i o n v 2 = 1 j 1 = 0 - ^ v 2 / = 0 j ' = 0 a t E = l . H w and L = 0 and at B^= 180°. We were e x t r e m e l y s u r p r i s e d by the number of r o t a t i o n a l s t a t e s t h a t had t o be i n c l u d e d i n the HD + CO c a l c u l a t i o n s t o achieve convergence t o w i t h i n /v /57 t ), as can be seen we r e q u i r e at l e a s t s i x s t a t e s . I n c o n t r a s t the convergence o f the r e s u l t s i n t h e H 2 + CO system i s e x t r e m e l y r a p i d , o n l y two channels are needed t o achieve the same degree o f convergence. I t i s w o r t h m e n t i o n i n g t h a t the same computer programme was used t o study each system. The HD +- CO c a l c u l a t i o n s o n l y r e q u i r e d t h a t t r i v i a l m o d i f i c a t i o n s be made t o the e x i s t i n g r o u t i n e s (e.g.,we must implement the p o t e n t i a l t r a n s f o r m a t i o n V I I . 2 . 4 , which appears t o be w o r k i n g c o r r e c t l y , as i s shown i n S e c t i o n 2, and we must account f o r t h e Z k j ^ = 1 c o l l i s i o n a l s e l e c t i o n r u l e i n HD). Al t h o u g h channels denoted by v^ = 1 are open o n l y f o r ^ 1 a t t h i s energy, channels denoted by = 0 are open f o r ^ 6 . The o n l y r e asonable e x p l a n a t i o n we are a b l e t o o f f e r i s t h a t these open channels (which are c l o s e l y spaced, and s t r o n g l y coupled by the p o t e n t i a l ) TABLE 5 The convergence of / SL (v** = 0 yV^~ Ojf- o/&l)j ( a . u. ) 2 as a f u n c t i o n o f the number o f r o t a t i o n a l s t a t e s r e t a i n e d i n the coupled e q u a t i o n s at E = 1.1 'ftw and L = 0. R e s u l t s a r e shown f o r bo t h the HD + CO and H^ + CO system a t »»' = ^ = 1 8 0 ° -Number of R o t a t i o n a l HD t CO H., i CO State? -1 0.96713045-9 0.51323744-7 2 0.70576707-8 0.60346349-7 3 0.67240391-8 0.60381697-7 4 0.11571327-8 0.60378485-7 5 0.19664733-9 6 0.15790107-9 7 0.15804647-9 8 0.15773195-9 2 80 are h a v i n g a s i g n i f i c a n t e f f e c t upon the convergence of the r e s u l t s . N o t i c e t h a t i n the H 2 + CO system, at the same c o l l i s i o n energy r e l a t i v e l y few r o t a t i o n a l channels are open, the channels t h a t are open are more w i d e l y spaced (and the p o t e n t i a l i s l e s s i m p o r t a n t due t o the s m a l l e r reduced mass o f the H 2 + CO system). We have c a r r i e d out s i m i l a r n u m e r i c a l t e s t s a t o t h e r CO o r i e n t a t i o n s a t t h i s energy, and at lower c o l l i s i o n e n e r g i e s . We conclude t h a t i f we r e t a i n the f i r s t s i x r o t a t i o n a l channels i n the c a l c u l a t i o n s , t h i s i s s u f f i c i e n t t o converge t he r e s u l t s f o r the t r a n s i t i o n s from v 2 = 1 j | = 0 and 1 t o v 2 = O j ' , f o r ^ 4 t o w i t h i n •^107o at a l l the e n e r g i e s s t u d i e d . However, the cross s e c t i o n s f o r t r a n s i t i o n s t o j | = 5 are u n s a t i s f a c t o r y , they are o v e r e s t i m a t e d by a f a c t o r o f 2. As a l r e a d y mentioned we found t h a t the convergence (as a f u n c t i o n o f R' ) o f the r e s u l t s f o r the i n d i v i d u a l max r o t a t i o n a l t r a n s i t i o n s i n HD i s e x t r e m e l y poor, however we found r e a s o n a b l e agreement between the va l u e s o f StUv^l^^ol&l) where, where j - ^ = 0 and 1 i e v a l u a t e d at d i f f e r e n t v a l u e s of R' ( f o r R^ ^ 2 5 a . u . ) . max max I n Table 6 we show the convergence o f the r e s u l t s w i t h r e s p e c t t o R ' a t E = 1.03 and 1.1 -Rw, at Jta = 0, max ' f ' and = 180° f o r s e v e r a l values o f the p a r t i a l wave parameter L. As may be seen the agreement between the r o t a t i o n a l l y summed S-matrices, N •—' •—• CNI O CT> v O O O i A O L O CTN ON H vD (fl CO •a VJD H CM H H CO < ! " CNl CNl o o* - - — ' a • a • • • • • • • • • o o o o o o o o o O O O CJ o • 00 0 + H CO -Q ll X L O o o o C-s! CNl r-l r-l r H o> o> o> CJN cr> ON OO 00 00 4-J 1 1 1 r~» ON v£) 1 1 1 oo 00 I I I < f ON s O 1 1 1 ON L O CNl 4-J 4_> cO O O ro ( N ON CNI ^ I N | N CO L O CO O oo cr> co O L O r - o • L O H L O CNJ L O NJ" O X r-l L O CO 00 00 o L O CO L O CO H U- l CO-* •* CNJ i— 1 r - l r - l r-l CNJ H \0 CO CNl CO CO ^ E tt> n . . . T3 OS — . o o o o o o o o o o o o 0) 4-J 4-1 CO O o II >-0 i—i > co d n H H H oo oo oo O I H O ^ oo oo oo QJ r—1 CO 1 1 1 s D r -1 1 1 CNJ *£ ) CO 1 1 1 vD 00 I s m uo o I N co o 4-J 1—1 v O CM LO 00 CO r - < t ON i r i H v C 1—1 O CjN CJ\ CM O L O vO co oo CNl ^ J - CO u o >^ O O ^ N CO r H r H H ON L O CNl CO O w II " CO CNJ CO H H H H 00 H \£> vaO vO 4) B . • • • • • • * • • • U ii O O O o o o o o o O O O , , . , , _ , — , , . o v — - Q / ) CO rO U T J QJ U H V — - — - — • - — ' — o •r-l 14-4 <4H O o CO ^ — * a V) w II 3 CO O 6 Q) CD Jp O H o • < r-l cO > 'cO > r J ta r H H 281 i s f a r s u p e r i o r t o the agreement between the S-matrices f o r the process = = 0 ( g i v e n f o r comparison, note t h a t the S-matrices f o r a l l the r o t a t i o n a l processes are s i m i l a r i n ma g n i t u d e ) . As I, i n c r e a s e s the agreement between t he r o t a t i o n a l l y summed r e s u l t s does however, d e t e r i o r a t e , t h e o n l y reasonable e x p l a n a t i o n we can o f f e r , i s t h a t s i n c e the n u m e r i c a l v a l u e s o f the S-m a t r i c e s are s m a l l , the computer code i s h a v i n g d i f f i c u l t y m a i n t a i n i n g accuracy. The p a r t i a l cross s e c t i o n s w i t h these h i g h L va l u e s ( i . e . L 20 at; E = 1.03-tTw, and L 30 at E = 1.1 -ffw) do however, form a r e l a t i v e l y s m a l l c o n t r i b u t i o n t o the f i n a l i n t e g r a l cross s e c t i o n . Wo have c a r r i e d out s i m i l a r t e s t s f o r the ^2 ' ^ system at E = 1.1 f i w , and L = 0, 10. Again we conclude t h a t the convergence o f the r e s u l t s f o r t h e i n d i v i d u a l r o t a t i o n a l process = j | = 0 i s l e s s s a t i s f a c t o r y t h a n f o r t he r o t a t i o n a l l y summed r e s u l t s . (However, the r e s u l t s e v a l u a t e d a t R ; = 25.0 and 30.0 a.u. max f o r the t r a n s i t i o n = j | = 0 do agree t o w i t h i n 57 0). The o n l y c o n c l u s i o n we can draw r e g a r d i n g the r e s u l t s e v a l u a t e d f o r t h i s system , i s t h a t the l o n g range i n t e r -a c t i o n s are c a u s i n g f l u x t o be t r a n s f e r r e d from one r o t a t i o n a l s t a t e t o a n o t h e r , c o n s e q u e n t l y we are unable t o de t e r m i n e i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s , but as a l r e a d y shown i n t h i s s e c t i o n v i b r a t i o n a l e x c i t a t i o n i s not i m p o r t a n t f o r R^ a x 2> 25 a.u., and thus we f i n d r e asonable agreement between t h e v a l u e s o f ^LJ (^2. S ^ ^ S ® /®&> ^ 282 e v a l u a t e d at R = 25.0, 30.0 and 35.0 a.u. ( i n d e e d max ' i t i s h i g h l y u n l i k e l y t h a t t h e l ^ or HD c o u l d v i b r a t i o n a l l y e x c i t e the CO molecule at such l a r g e i n t e r m o l e c u l a r s e p a r a t i o n s ) . The l e a d i n g l o n g range i n t e r a c t i o n i n the HD + CO i n t e r a c t i o n p o t e n t i a l i s , as mentioned i n S e c t i o n 2, t h e i n t e r a c t i o n between the d i p o l e moment of HD, and the d e r i v a t i v e s ( w i t h respect t o r,^) of the d i p o l e moment o f CO,which v a r i e s as K . However t h i s i n t e r a c t i o n s h o u l d o n l y have a s i g n i f i c a n t i n f l u e n c e on the convergence o f the v i b r a t i o n a l l y e l a s t i c r e s u l t s (as was seen i n the ^ + H + system, Chapter IV.A). As shown i n S e c t i o n 2, the i n t e r a c t i o n p o t e n t i a l i s t e n d i n g t o zero as R 0 0 5 i n a d d i t i o n we are c o n f i d e n t , i n t h e l i g h t o f our ^ + CO c a l c u l a t i o n s , t h a t t he a s y m p t o t i c boundary c o n d i t i o n i s f i t t e d c o r r e c t l y i n our programme. One w i l l r e c a l l t h a t our r e s u l t s f o r 1^ 1 CO are I n good agreement w i t h the r e s u l t s o f BSD2 e v a l u a t e d u s i n g the P p o t e n t i a l , as i s shown i n Appendix 4. We are r e a s o n a b l y sure t h a t t h e convergence problems seen i n HD + CO are a l l caused by the l o n g range m u l t i -p o l a r i n t e r a c t i o n s (as we c l a i m above), when we employ j u s t the s h o r t range and d i s p e r s i o n terms of the p o t e n t i a l i n t h e s c a t t e r i n g c a l c u l a t i o n s we f i n d e x c e l l e n t agreement between the r e s u l t s f o r the i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s ( t h e y agree t o w i t h i n a / 1 % ) e v a l u a t e d at th e v a l u e s o f R' g i v e n i n Table 6. 283 We have c a r r i e d out numerous t e s t s on the p o t e n t i a l t r a n s f o r m a t i o n g i v e n i n S e c t i o n 2, i n p a r t i c u l a r we are c o n f i d e n t t h a t the t r a n s f o r m e d m u l t i p o l a r i n t e r -a c t i o n s are s u f f i c i e n t l y a c c u r a t e t o study r o v i b r a t i o n a l e x c i t a t i o n ( p r i o r t o these c a l c u l a t i o n s , the t r a n s f o r m a t i o n V I I . 2 . A g i v e n by L i u et a l . (1978) had o n l y been employed i n c a l c u l a t i o n s s t u d y i n g pure r o t a t i o n a l e x c i t a t i o n ) . For example, a t f i x e d v a l u e s o f , and 6^ we e v a l u a t e d the r a t i o (employing j u s t the m u l t i p o l a r i n t e r -a c t i o n s ) . X ^ U OY/s.^ -//)) - r ^ v o VI1'3-2 where (rzLii) = (JL (P/> \ ) > * 0 . At b o t h R = 20.0 and 30 a.u. we f i n d e x c e l l e n t aeree-max b ment between t h e va l u e s of X ( t h e y agree t o e i g h t s i g n i f i c a n t f i g u r e s ) f o r i = 1, , 28, as one w i l l r e c a l l we use a 28- p o i n t Gauss Legendre q u a d r a t u r e t o e v a l u a t e the v i b r a t i o n a l p o t e n t i a l m a t r i x elements. E v i d e n t l y one c o u l d i n c l u d e the m u l t i p o l a r i n t e r a c t i o n s i n these c a l -c u l a t i o n s e x p l i c i t l y u s i n g n u m e r i c a l v a l ues f o r the HD d i p o l e and quadrupole moments taken from the l i t e r a t u r e , however a number o f a d d i t i o n a l c a l c u l a t i o n s i n d i c a t e t h a t t h i s procedure would r e s u l t i n q u a l i t a t i v e l y s i m i l a r r e s u l t s . Since we f i n d good agreement between the r o t a t i o n a l l y summed r e s u l t s , the c o n c l u s i o n t h a t one should draw - 284 -i s t h a t the 'net' e f f e c t o f the Long range f o r c e s are u n i m p o r t a n t , and t h e r e f o r e w h i l e t h e e r r o r s i n the p o t e n t i a l r o u t i n e might change i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s , i t i s q u e s t i o n a b l e t h a t the r o t a t i o n a l l y summed ( v i b r a t i o n a l ) r e s u l t s s h o u l d be s e r i o u s l y a f f e c t e d . However, as we show i n Appendix 5, the e r r o r s i n our p o t e n t i a l r o u t i n e have s i g n i f i c a n t l y affected both the cr o s s s e c t i o n s f o r i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s and f o r the v i b r a t i o n a l t r a n s i t i o n s . As i s di s c u s s e d i n Appendix 5, i t i s apparent t h a t t h e l o n g range i n t e r a c t i o n s p l a y a s i g n i f i c a n t r o l e i n d e t e r m i n i n g the n u m e r i c a l v a l u e s o f the v i b r a t i o n a l d e - e x c i t a t i o n cross s e c t i o n s , which would appear t o be i n c o n f l i c t w i t h our c o n c l u s i o n drawn above. We have, u n f o r t u n a t e l y , no s a t i s f a c t o r y e x p l a n a t i o n f o r When the s c a t t e r i n g c a l c u l a t i o n s were performed we employed 320 steps o f f i x e d l e n g t h i n the range 3 R S^,35 a.u. which i s more than s a t i s f a c t o r y t o o b t a i n converged r o t a t i o n a l l y summed ( v i b r a t i o n a l ) cross s e c t i o n s . Because o f the u n c e r t a i n t i e s i n the cross s e c t i o n s f o r the i n d i v i d u a l r o t a t i o n a l t r a n s i t i o n s , i n S e c t i o n 4 we present o n l y the r o t a t i o n a l l y summed cross s e c t i o n s , t h i s . V I I . 3 . 3 where = 0 and 1. 285 M a i n t a i n i n g the- same i n t e g r a t i o n range, and s t e p s i z e the computer time r e q u i r e d t o s o l v e the coupled e q u a t i o n s , r e t a i n i n g the same number o f v i b r a t i o n a l s t a t e s and no r o t a t i o n a l s t a t e s ( i . e . j u s t j ^ = 0 ) , i s r o u g h l y a f a c t o r o f 5 g r e a t e r t h a n t h a t r e q u i r e d f o r the H2 + CO system. Using e q u a t i o n V I I . 2 . 4 t o t r a n s f o r m the i n t e r a c t i o n p o t e n t i a l i s e x t r e m e l y e x p e n s i v e , even r e t a i n i n g j u s t seven terms i n the expansion V I I . 2 . 2 , and employing an 8- p o i n t Gauss hegendre q u a d r a t u r e t o e v a l u a t e the i n t e g r a l . We t h e r e f o r e compute the p o t e n t i a l u s i n g V I I . 2 . 4 b e f o r e p e r f o r m i n g the s c a t t e r i n g c a l c u l a t i o n s and use a s p l i n e i n t e r p o l a t i o n procedure t o e v a l u a t e the p o t e n t i a l at the r e q u i r e d v a l u e s o f rV ( t h e p o t e n t i a l i s 1pre-computed' at a l l the v a l u e s o f , the CO o r i e n t a t i o n s , and , the v i b r a t i o n a l c o o r d i n a t e s r e q u i r e d ) . Routines w r i t t e n i n FORTRAN IV capable o f i n t e r p o l a t i n g data by means of a c u b i c s p l i n e are a v a i l a b l e and have been used numerous times i n work pr e s e n t e d i n t h i s t h e s i s (e.g. see Chapters V.5 and VI.4) . I n Table 7 we show a comparison of the v a l u e s of Sly*0 6 f B ° ; t ^ s ° / < % ) e v a l u a t e d u s i n g 137-, 86-, and 69- p o i n t s p l i n e s i n the range 2 . 0 ^ R ^ 35 a.u. The p o t e n t i a l was computed r e t a i n i n g 9 terms i n the expansion V I I . 2 . 2 and employing a 10- p o i n t Gauss Legendre q u a d r a t u r e t o e v a l u a t e the i n t e g r a l V I I . 2 . 4 . Also shown i s the n u m e r i c a l v a l u e o f the S-matrix e v a l u a t e d TABLE 7 comparison o f r e s u l t s o b t a i n e d u s i n g d i f f e r e n t N-point s p l i n e s i n the range 2 ^ R^ 35 a.u. t o gener a t e t he p o t e n t i a l . Also shown i s the r e s u l t o b t a i n e d . i n a c a l c u l a t i o n i n which the p o t e n t i a l was e v a l u a t e d at each v a l u e o f R i n the i n t e g r a t i o n range u s i n g V I I . 2 . 4 . (a) N = 69 p o i n t s , (b) N = 86 p o i n t s , ( c ) N = 137 p o i n t s , (d) R e sult o b t a i n e d u s i n g V I I . 2 . 4 at each v a l u e o f R . (a) 0.12964002 - 8 (b) 0.11887458 - 8 (c ) 0.11943994 - 8 (d) 0.11630300 - 8 Re s u l t s e v a l u a t e d at E = 1.03-+fw and L = 0, and at 9^ 180°. j x = 0 o n l y , j{£ 5. TABLE 8 The convergence o f (\?2 = l - ^ v , ^ = 0) w i t h r e s p e c t t o the number o f CO o r i e n t a t i o n s , Bj, , i n c l u d e d i n the c a l c u l a t i o n s at E = 1.1 -ftw and L = 0 . (a) 16 o r i e n t a t i o n s , (b) 20 o r i e n t a t i o n s , ( c ) 28 o r i e n t a t i o n s o~y ( v 0 = l - > v ' = 0) ( K 2 ) h = o j t = 1 (a) 0.20560412-8 0.17204594-8 (b) 0.20598170-8 0.16401752-8 (c) 0.20569189-8 0.16563214-8 286 i n a c a l c u l a t i o n i n which the p o t e n t i a l i s computed u s i n g the i n t e g r a l V I I . 2 . 4 at each s t e p i n the i n t e g r a t i o n range. R e s u l t s are shown f o r E = 1.03 -fiw and L = 0 a t S^= 180° and = 0. At t h i s energy, which i s t h e lowest s t u d i e d i n these c a l c u l a t i o n s s i f we employ a 137 p o i n t s p l i n e ( i . e . i n which the p o t e n t i a l i s p r e -computed every 0.25 a.u. i n the range 2.0-^ R 35.0 a.u., r e c a l l t h a t the c oupled e q u a t i o n s are s o l v e d i n the range 3.0 & R £s 35 a.u., t h e r e i s t h e r e f o r e no e x t r a -p o l a t i o n of d a t a ) , t h i s ensures t h a t the r e s u l t s are d e t e r m i n e d t o w i t h i n /'^ 57o at t h i s energy . Note t h a t as the c o l l i s i o n energy i n c r e a s e s , e r r o r s i n the p o t e n t i a l m a t r i x elements s h o u l d be l e s s i m p o r t a n t . I n c o n c l u s i o n the 137- p o i n t s p l i n e w i l l be employed i n a l l our c a l -c u l a t i o n s . We m ention t h a t i t r e s u l t s i n a r e d u c t i o n i n t he computer time needed t o s o l v e the c oupled e q u a t i o n s at one o r i e n t a t i o n by a f a c t o r of ^ 4 ( i f t h r e e v i b r a t i o n a l and s i x r o t a t i o n a l channels are i n c l u d e d i n the c a l c u l a t i o n s ) Even w i t h t h i s vast r e d u c t i o n i n computer time at the i n i t i a l p a r t i a l wave, the c a l c u l a t i o n s are s t i l l e x t r e m e l y e x p e n s i v e . Using the IBM 370/168 computer at NUMAC the approximate computer time r e q u i r e d a t the i n i t i a l p a r t i a l wave i s 110 s (70s at JLp = 0 and 40s a t J Z j = 1) and at subsequent p a r t i a l waves i s ^ 60s (40s at JZf = 0 and 20s at JLj = 1 ) . Using the s p l i n e t o g e n e r a t e the p o t e n t i a l has no e f f e c t upon the computer time r e q u i r e d a t subsequent p a r t i a l waves, s i n c e t h e r e i s no e x p l i c i t r e f e r e n c e t o the p o t e n t i a l m a t r i x elements. As a l r e a d y 287 mentioned (Chapter VI.4 ) , the m a j o r i t y o f the computer t i m e at subsequent p a r t i a l waves (when we are d e a l i n g w i t h ' l a r g e ' m a t r i c e s ) i s accounted f o r by m a t r i x m u l t i p l i c a t i o n . By v e c t o r i z i n g the m a t r i x m u l t i p l i c a t i o n , and R-matrix p r o p a g a t o r r o u t i n e s we are a b l e t o reduce the computer time g i v e n above f o r the c a l c u l a t i o n s a t subsequent v a l u e s o f L by a f a c t o r o f ^ 1 5 u s i n g the Cray-1 computer ( r a t h e r than the more u s u a l f a c t o r o f ^ 8) . I t i s w o r t h r e p e a t i n g t h a t due t o the enhanced reduced mass o f the HD + CO system (a f a c t o r o f 1.5 l a r g e r than t h a t f o r H 9 +• CO) we need t o c a l c u l a t e ^ 5 0 % more p a r t i a l waves at a g i v e n energy. Since t h e coupled e q u a t i o n s are t o be s o l v e d at 16 o r i e n t a t i o n s we c o n s i d e r s i x r o t a t i o n a l channels the maximum f e a s i b l e . I n Table 8 we compare r o t a t i o n a l l y summed ( w i t h r e s p e c t t o b o t h f i n a l CO and HD r o t o r s t a t e s ) cross 20-, and 28- p o i n t Gauss Legendre q u a d r a t u r e s at E = 1.1 "Hw and L = 0. We r e c a l l t h a t the convergence o f the r e s u l t s (as a f u n c t i o n o f the number of CO o r i e n t a t i o n s i n c l u d e d i n t h e c a l c u l a t i o n s ) improves w i t h b o t h i n c r e a s i n g L and d e c r e a s i n g E (Chapter V.4.3). As can be seen a 1 6 - p o i n t Gauss Legendre q u a d r a t u r e ensures good r e s u l t s f o r b o t h j ^ = 0 and 1. Employing t h i s q u a d r a t u r e we c o n s i d e r t h a t the r e s u l t s w i l l be e v a l u a t e d t o w i t h i n «%^570 a t a l l t he e n e r g i e s s t u d i e d i n these c a l c u l a t i o n s . 4. R e s u l t s and d i s c u s s i o n I n t h i s s e c t i o n we w i l l p r e s e n t and d i s c u s s t he s e c t i o n s , ) , e v a l u a t e d u s i n g 16-288 r e s u l t s of our c a l c u l a t i o n s of the v i b r a t i o n a l d e - e x c i t a t i o n o f C0(v2 = 1) by HD. I n Table 9 we show the n u m e r i c a l v a l u e s o f the r o t a t i o n a l l y summed cross s e c t i o n s , (Tj (^%~ / 4^, i . e . S ^ ° V I I . 4 . 1 where = 0 and 1. Both the c o l l i s i o n energy, E, and the i n i t i a l t r a n s l a t i o n a l energy, E T, are g i v e n i n Table 9. As b o t h t he V2 = i = 0 and V2 = 1 j ^ = 1 t h r e s h o l d s are approached the cr o s s s e c t i o n s b e g i n t o r i s e . We are r e a s o n a b l y sure t h a t t h i s t h r e s h o l d b e h a v iour i s due t o o r b i t i n g resonances. The e f f e c t J v f pot cut 1.11 f o r the system, at ,1 p a r t i a l wave parameter L, i s g i v e n by y £ u (k. I*') •*• LOzlll " 2/*(f!f VI1-4-2 where JA* i s the reduced mass of the system and where V^ n (_(R ) i s the s p h e r i c a l l y symmetric component of the HD + CO i n t e r a c t i o n p o t e n t i a l . Resonances can occur o n l y when the e f f e c t i v e p o t e n t i a l possesses an a t t r a c t i v e w e l l capable o f s u p p o r t i n g one or more bound s t a t e s . These resonances are s a i d t o be of shape or Feshbach type a c c o r d i n g t o whether they occur i n an open or a c l o s e d channel ( e . g. see C h i l d ( 1 9 7 4 ) ) . At E = 1.03°nw we have l o c a t e d a Feshbach resonance i n the V2 = 1 = 1 c h a n n e l . The k i n e t i c energy of the system i n t h i s s t a t e i s -3.43 lO - ^ a.u., compared w i t h TABLE 9 R o t a t i o n a l l y s u m m e d cross s e c t i o n s ^ ( i n A ) f o r the v i b r a t i o n a l r e l a x a t i o n processes i n which = 0 and = 1. E i s the t o t a l c o l l i s i o n energy, E^(cm i s the i n i t i a l k i n e t i c energy, r e l a t i v e t o the V^*/ j \ t h r e s h o l d ( j 1 = 0 or 1 ) . * O E (t>w) E., 0 ~ , Q ( v 2 -1 -*v£ = 0) E, the comparison o f (T) 291 w i t h the experimentally d e t e rmined r a t e c o e f f i c i e n t (Andrews and Simpson (1976)) i s g i v e n i n F i g u r e 4 f o r the t e m p e r a t u r e range, 1 4 0 ^ T ^ 240 K. Above T > 240 K, the agreement w i t h the e x p e r i m e n t a l p o i n t s i s e x t r e m e l y poor, as mentioned i n S e c t i o n 3, we need t o c o n s i d e r c o l l i s i o n e n e r g i e s ^ 1.1*ffw ( i n i t i a l k i n e t i c e n e r g i e s ^> 300 K) t o a c c u r a t e l y e v a l u a t e the r a t e c o e f f i c i e n t f o r T ^ 240 K. I n the t e m p e r a t u r e range shown the t h e o r e t i c a l l y d e t e r m i n e d r a t e i s a f a c t o r /v 2-3 lower than the e x p e r i m e n t a l p o i n t s . I n view of our p r e v i o u s comparisons w i t h experiment (Chapter V,5, and Chapter V I . 4 ) we c o n s i d e r t h i s l e v e l of agreement t o be s a t i s f a c t o r y , indeed we would have been s u r p r i s e d I f the agreement had been b e t t e r . The summation over j ' , t he f i n a l r o t a t i o n a l s t a t e , i n e q u a t i o n V I I . 4 . 1 i n c l u d e s = 5, d e s p i t e the cross s e c t i o n s f o r the r o t a t i o n a l t r a n s i t i o n s t o j | = 5 b e i n g o v e r e s t i m a t e d by a f a c t o r of ^ 2 . As mentioned i n S e c t i o n 3, the s i x r o t a t i o n a l s t a t e s ( j ^ ^ 5) employed i n t h e c a l c u l a t i o n s i s i n s u f f i c i e n t t o a c c u r a t e l y determine the c r oss s e c t i o n s w i t h = 5. At the e n e r g i e s we s t u d y i n these c a l c u l a t i o n s , E ^ 1 .1 -flw, the = 6 s t a t e i s open, the cross s e c t i o n s w i t h = 5 a l s o account f o r the f l u x t h a t s h o u l d have gone i n t o the = 6 channel. When one accounts f o r the e r r o r i n t h e p o t e n t i a l r o u t i n e the agreement w i t h experiment w i l l be d i m i n i s h e d , as i s d i s c u s s e d i n Appendix 5. Indeed i t appears t h a t the r e s u l t i n g agreement w i t h experiment would be e x t r e m e l y 17.0 17.2 H O !60 180 200 220 T F i g u r e 4. The v a r i a t i o n w i t h t e m p e r a t u r e , T v i b r a t i o n a l r e l a x a t i o n c o e f f i c i e n t , 0{(T) ( i n cm Js -1 K) , o f the u n i t s o f S o l i d curve : presen t c a l c u l a t i o n s ; " : e x p e r i m e n t a l r e s u l t s o f Andrews and Simpson (19 7 6 ) . 292 p o o r . Some improvement would r e s u l t i f the r o t a t i o n a l degree o f freedom o f CO was t r e a t e d w i t h i n the CS a p p r o x i -m a t i o n (assuming t h a t t h e r a t i o o f CS t o IOS r a t e s , as d e t e r m i n e d by BSD f o r ^ ( j - ^ = 0) + CO, i s v a l i d f o r HD ( j 1 4 0) + CO). The IOS r a t e would be i n c r e a s e d by a f a c t o r o f 'W. rj, i n s u f f i c i e n t t o account f o r the d i s c r e p a n c i e s between experiment and t h e o r y . However, i n view o f the expense o f these IOS c a l c u l a t i o n s , we c o n s i d e r t h a t a s i m i l a r c a l c u l a t i o n employing the CS a p p r o x i m a t i o n would not be f e a s i b l e . As we have mentioned a l r e a d y i n t h i s c h a p t e r , i t i s apparent from our d i s c u s s i o n i n Appendix 5 c o n c e r n i n g the e r r o r s i n the P p o t e n t i a l , t h a t the l o n g range i n t e r -a c t i o n s p l a y a s i g n i f i c a n t r o l e i n d e t e r m i n i n g the n u m e r i c a l values o f the v i b r a t i o n a l d e - e x c i t a t i o n c r oss s e c t i o n s . None o f the t r a n s i t i o n s s t u d i e d i n these c a l c u l a t i o n s are near-resonance, so we f i n d i t e x t r e m e l y d i f f i c u l t t o u n d e r s t a n d why we a r r i v e at the above c o n c l u s i o n , e s p e c i a l l y i n view o f the r e s u l t s o b t a i n e d f o r t he H2 + CO system. One w i l l r e c a l l t h a t t h e A j ^ = 0 and 2 cross s e c t i o n s were changed by o n l y a few per cent when we accounted f o r t he e r r o r i n our p o t e n t i a l r o u t i n e (Appendix 4 ) . I n p a r t i c u l a r , i t i s not c l e a r , at l e a s t t o us, why t h e agreement w i t h experiment i s so poor. F i n a l l y we conclude t h a t we would have t o spend a g r e a t d e a l more time i n v e s t i g a t i n g t h i s system b e f o r e we f u l l y u n d e r s t a n d a l l the r e s u l t s p r e s e n t e d i n t h i s c h a p t e r . However, u s i n g the p r e s e n t r e s u l t s we are 29} ab l e t o dr/iw .1 number of c o n c l u s i o n s r c g . i r d i n g the e f f e c t s of the i s o t o p i c s u b s t i t u t i o n on the c a l c u l a t i o n s , and have some i n d i c a t i o n o f the l e v e l o f agreement between experiment and t h e o r y . 294 CHAPTER VI11 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK The main r e s u l t s p r e s e n t e d i n t h i s t h e s i s are the c a l c u l a t i o n s o f the r o v i b r a t i o n a l e x c i t a t i o n o f CO by para-H^ (Chapters V and V I ) . The work on H 2 + CO i s e s s e n t i a l l y complete i n the sense t h a t we have a rea s o n a b l e u n d e r s t a n d i n g of the s e n s i t i v i t y o f the v i b r a t i o n a l r e l a x a t i o n cross s e c t i o n s upon t he p o t e n t i a l energy s u r f a c e , the dynamical a p p r o x i m a t i o n s , n u m e r i c a l p r o c e d u r e s , and parameters. The most c r u c i a l p o i n t i s c e r t a i n l y t h e i n t e r a c t i o n p o t e n t i a l , and s i n c e a l l the a v a i l a b l e H 2 + CO i n t e r a c t i o n p o t e n t i a l s h a v e now been t e s t e d ( t h e p o t e n t i a l s o f Poulsen (1982) and van Hemert (1983) i n t h i s work, and the p o t e n t i a l c o n s t r u c t e d from t h e SCF data o f D i e r c k s e n and Kraemer (1984) by BSD and BSD2) i t seems p o i n t l e s s c o n t i n u i n g these c a l -c u l a t i o n s . The p o t e n t i a l o f van Hemert (1983) i s un-s a t i s f a c t o r y , o f the two r e m a i n i n g s u r f a c e s , n e i t h e r "stands o u t " as the best o v e r a l l d e s c r i p t i o n o f the H 2 + CO i n t e r a c t i o n . However, t o g e t h e r t h e P and DK p o t e n t i a l s p r o b a b l y r e p r e s e n t the best we can do f o r a complex system l i k e H 2 + CO, g i v e n presert ab i n i t i o t e c h n i q u e s . Of s p e c i a l i n t e r e s t i n t h i s work i s the near r e -sonance - process ( V I . 1 . 1 ) . I t i s e v i d e n t from our r e s e a r c h t h a t t h i s i s an e x t r e m e l y complex p r o c e s s , and i t w i l l be some time b e f o r e a s a t i s f a c t o r y 2')') t h e o r e t i c a l d e s c r i p t i o n o f t h i s process i s a c h i e v e d , e s p e c i a l l y , i f as p o s t u l a t e d i n Chapter V I , t h e m u l t i p o l e moment i n t e r a c t i o n s are i m p o r t a n t at s h o r t range. Poulsen and B i l l i n g (1982) suggest t h a t the cross s e c t i o n s f o r t h i s process would be enhanced i f h i g h e r m u l t i p o l e moments o f CO were t r e a t e d e x p l i c i t l y i n t h e p o t e n t i a l s u r f a c e . Indeed, such a c a l c u l a t i o n has been performed (Poulsen and B i l l i n g ( 1 9 8 4 ) ) , the P p o t e n t i a l was employed and the o c t o p o l e and hexadecapole moments o f CO were t r e a t e d e x p l i c i t l y . However, i t i s p o s s i b l e t h a t the semi-c l a s s i c a l a p p r o x i m a t i o n employed by these a u t h o r s i s i n a p p l i c a b l e t o the 1 ^ + CO system, and t h e r e f o r e the r e s u l t s p r e s e n t e d by Poulsen and B i l l i n g (1984) s h o u l d be t r e a t e d w i t h a degree o f c a u t i o n . Only when a quantum mechanical c a l c u l a t i o n has been performed can the importance o f the h i g h e r m u l t i p o l e s o f CO be p r o p e r l y assessed. Due t o the e r r o r i n our p o t e n t i a l r o u t i n e , t h e HD -t CO c a l c u l a t i o n s p r e s e n t e d i n Chapter V I I are s e r i o u s l y i n e r r o r , however i t I s not w o r t h r e p e a t i n g these c a l -c u l a t i o n s . The c a l c u l a t i o n s are e x t r e m e l y expensive c o m p u t a t i o n a l l y , and the p r e s e n t r e s u l t s do at l e a s t g i v e us a reasonable u n d e r s t a n d i n g o f the e f f e c t s o f i s o t o p i c s u b s t i t u t i o n . N e v e r t h e l e s s i t i s p r o b a b l y w o r t h c o n t i n u i n g work on t h i s system t o d e t e r m i n e more p r e c i s e l y the r o l e p l a y e d by the l o n g range m u l t i p o l a r i n t e r a c t i o n s . I n Chapter IV we used the 1^ + H + system t o examine th e a p p l i c a b i l i t y o f the a s y m p t o t i c s e r i e s expansions 296 of t he type i n t r o d u c e d by G a i l i t i s (1976) i n heavy p a r t i c l e s c a t t e r i n g c a l c u l a t i o n s . 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I n the IOS a p p r o x i m a t i o n the a s y m p t o t i c form o f the coupled d i f f e r e n t i a l e q u a t i o n s takes t he f o r m , i f we r e t a i n t h e f i r s t two l e a d i n g l o n g range terms o f the i n t e r -a c t i o n p o t e n t i a l ( c f . I I I . A . 3 0 ) . A l . 3 Th i s e q u a t i o n must be s o l v e d , the s o l u t i o n w i l l be i n terms o f the a s y m p t o t i c s e r i e s expansions A(R) and B(R) To achieve t h i s we f i r s t c o n s i d e r the a s y m p t o t i c form o f the coupled e q u a t i o n s , i f no i n t e r a c t i o n p o t e n t i a l terms are r e t a i n e d , ( c f . I I I . 4 . 2 6 ) . Al .4 - A2 -T h i s e q u a t i o n i s t h e R i c c a t i - B e s s e l e q u a t i o n (Abramowitz and S t e g u n ( 1 9 6 5 ) ) , which has r e a l l i n e a r l y independent s o l u t i o n s which may be w r i t t e n R e w r i t i n g A1.3 and Al.A i n m a t r i x n o t a t i o n , we have 6t" + ULQr - o / * -t - o . A l . 5a A l . 5b We w r i t e the s o l u t i o n o f A1.5a ( o r A1.3) i n terms o f the s o l u t i o n s o f Al.5b ( o r A l . 4 ) . S o - A / + B// Then £'V fc* - P £ * 4?/' Whence p , ^ ^ _ _ g ^ ^ 2 8 ^ Put ^ Also put ^ s ^ ^ ^ T h e n V W A -/4*r + M-B*r - 2 8.^- O 3*4(08 -Bw +yB +*A'=O As R =^^> , A 1 and B 0 T h e r e f o r e put ~ l*f &- ^ Ofad> B ~ &~ We then f i n d A1.6a Al .6b - A3 -At l a r g e enough R the e q u a t i o n s are uncoupled, so we may w r i t e ( c f . I I I . 4 . 3 3 a , b ) A1.7a A1.7b Put U T = £ - % £ C Al.Sa *L ~ 4 j o(mR~™-P ?>f A1.8c and 6" - /G^K'^'V • A1'8d S u b s t i t u t i n g A1.8a,b,c,d, i n t o A1.7a,b we f i n d ^ 6 A1.9a A1.9b C o n s i d e r i n g l e a d i n g terms i n A1.9a By d e f i n i t i o n ©< 0 ^O^^^O hence A4 C o n s i d e r i n g l e a d i n g terms i n A1.9b o^s J oast p-ft from Al.9b fto ^(^-t 0 ~ <*o f> from 1.9a E ^ ^ — D R e w r i t i n g A1.9a m+\e(&m (m+«-0 r-*~«~ Q^fSrn ( m ^ O ^ ^ J Al.lOa R e w r i t i n g A1.9b Al . l O b Consider A l . l O a w i t h m = 1 I f we o n l y r e t a i n t h e l e a d i n g l o n g range term A5 Also one can e a s i l y v e r i f y Chat Consider A.10b and m = n - 2 ( £ 1) on LHS: /3„_ z (2,r\ - "20 (in - 3 ) - a C^-n - ^ " ) * O 1 A l . l l a From Al.lOa ^ A l . l l b A l . l l a r e l a t e s ^3/)--^ and n—l and ckr\ — l A l . l 2 b r e l a t e s ^^^.j and 3^ /} - / e.g. i f n = 3, A.12a r e l a t e s rH i n CO fH i n i n CO 00 i n fH o SO ON op o 00 i n 00 00 ON op ON fH 00 co or 00 tH 00 SO CO CO CM r - ao fH ON CO fH 00 l ~ - o r ON i n o 00 t o •H o so so 00 or fH fH o r - so 00 so co rH O so o OS CN so 00 CN 00 CO r~ i n fH r -co CM 00 so CN CM so ON CN o ON CO SO fH r~ o o r» or 00 o ON o so SO so i n fH fH o p~ m i n i n co o i n o ON or so r - ON SO so r- so so CO so i n ON • V i n 00 CN r o p~ o fH m fH p- or CM 00 rH co so ON r - OS so o r co CN iH fH 00 so o o o o o o o o o o o o o o o o o 1 . 1 I I 1 I I CM fH H H O fH fH rH fH fH f H CM CN CM CN CM CM O O o o o o o o O O O O o o o O O 1 1 1 1 ± 1 1 1 1 1 1 1 1 1 1 1 1 a a QQO a a q a QQO Q Q Q Q Q ON ON m o so rH oo r- 00 CO ^ ON Of SO SO r- rH tH co oo oo r- p» o m r~ oo m o i n o o fH SO CO OT r- ON CN i n or r - ON i n O 00 or or CM ON r - i n 00 rH CO or ON or co or o r~ co CO CM rH CM CO ON CO CO P > O rH ON 00 o so rH CM o ON CO r* so o rH P - ON i n rH co r~ or rH o co i n ON so o P - o CM CN CM ON 00 co i n fH so CO CM or CM O rH r - m op O CM P - CO ao so p - o r - , H so ON r - rH ON fH CO 00 rH CO CM m o r ON CM or so m so o ao co i n so rH O so P--rH CN ON or or CM r-t ON SO CO CN rH ON SO or CM rH e • 0 * 0 • • • • e • e • o o o o o o o o o o o o o o o o o I I I I I I I I CN rH fH O fH rH rH rH fH CM CM CM CM CM CN co co O O O o o o o O O i O O O 1 1 o O 1 • O o O i i i Q | | Q Q T 1 1 Q Q Q Q Q i Q Q 1 1 Q Q 1 1 Q Q Q a a m P - P~ oo or so or 00 CM or or CM or P - CM rH i n i n i n r - CO CO CM so so ON CO m so CO CO ON i n o iH O CO or CM CO CO i n P - r o ON P - or or co i n CM rH 00 or so rH rH rH i n so r - or o i n rH SO ON ON CM rH O CM rH CM 00 00 ON co or o P - ON o o ON rH i n O CM or i n p - or rH CN CM co r - rH o or CM rH r - i n o fH CO r » rH i n O SO fH P - rH 00 ON P^ o o fH SO P - rH CN i n op 00 CM ON CO o r rH CM CO or so or r » O CO or CO o CO CM ON so 00 ON O o CM rH ON o rH CO i n i n so i n p - P - CO oo o r p~ o 00 ON CN so r-l rH CO so ON co i n CN ON or CM o r rH rH r - rH P - CN 00 CM i n CO o r o 00 ON rH SO CO P^ O ON SO CM ON or ON or r - CO so CO ON CO O ON CN rH rH or ON rH so so 00 00 i n r~ P - o CO O rH SO CTN SO ON CO i n r-> CO o o o P~ ON CO » r i n ON CM CO ON i n i n CO CM rH ON so or CM fH fH ON SO o O O o o o o O O o o o o o o o o I I I I I I I I O rH rH CM CM CM CM O O O O O O O + I I I I I I Q Q Q Q Q Q Q 00 SO CM o r f H P- P» p~ so co o co or o r oo m co o r m oo ^ O rH i n CO fH CO or so CM ao so ON m so CM fH O P* or rH 00 ON co oo oo or oo so m ON CM rH o r CM o r so so CM so ON m ON CM CO fH so so r» ON SO fH fH O 00 oo or CO CM P- SO 00 SO rH r - co CM co ON f » oo ON SO C I 00 CM O 00 ON rH SO CM CM ON i n ON fH «H ON m CM rH CO CO co co or or so i n O O o o o o o o 1 1 1 1 1 1 1 l Q Q Q Q Q Q Q Q p» i n ON m CM rH CM or or i n SO rH O CN o rH CO 00 CO P* ON rH CO or so oo i n SO o rH or ON rH CM 00 so r- CO CO fH »H SO rH or i n 00 ON CO fH 00 ON ON o r CM oo r- CO 00 P- 00 00 P* m tH 00 ON 00 i n ON or SO O so or or CO 00 00 ON CO o r CM co fH fH so p - or m ON so P- ON r-so O CM O rH co r~ r-CO rH CM o r oo i n so co i n i n CO fH O P*- fH o co or CM tH i n fH r-i n i n rH o o o o o I I I + Q Q Q Q CO CM rH CO vo or so o O CM P* ON SO rH O ON * r r - so i n ^ r oo ON co U~l P - V ON or p»- oo r -i n tH t-4 ON co 00 O CJ CM CO r- i n r~ o rH P - o oo ON 00 SO 00 ON o r r~- oo SO CM O ON ON ON rH ON f H r H i H r H C N C M C M C M O O O O O O O O I I I I I I I I Q Q Q Q Q Q Q Q s o o c o o s o o o o r s o fOfHr~orrHfosoor soincNcocosocoo CMONCor-CMinino oosoooinosooON P ^ O N r H i n C M r H O i H OOONCMCNOOONSOSO ON « in o co ON M O C M O N C M i n o r o O N i n O O P ^ S O O N C M C M P - i n o o o o s o c o ^ i n i n o c M o c o o o r o r o r o r c N s o P - o c o s o o r C O C N C O r H C O i n p - O r - © p - o o i n o c M C M C O f H r H S O C O C M r H CO CO CO CO o r o r or o o o I I I o I o I o o • I I I Q Q Q 1 Q 1 Q Q i Q ON rH CO so > , < S ^ , *«dL 2-/3T. a.u. ^1 and-fiw are taken from Mies ( 1 9 6 4 ) : -= 261.22 ( d i m e n s i o n l e s s ) •fiw = 0.26898 eV y / ^ o i s the reduced mass of CO, and takes the v a l u e jUCo= 12499.6 a.u. Using these parameters the r e m a i n i n g parameters D, and ^ A10 -are found t o be D = 0.64556 a.u. ^ = 0.9726472 a.u. As mentioned i n Chapter V.3, i s quoted t o so many f i g u r e s t o ensure t h a t the parameters i n a r e s e l f - c o n s i s t e n t , and m a i n t a i n the o r t h o n o r m a l i t y o f the w a v e f u n c t i o n s . For CO t h i s i s e x t r e m e l y i m p o r t a n t s i n c e we must d e a l w i t h v e r y l a r g e and v e r y s m a l l numbers The c a l c u l a t i o n o f CO Morse w a v e f u n c t i o n s i s t h e r e f o r e r a t h e r l e s s s t r a i g h t f o r w a r d than f o r example, the c a l -c u l a t i o n o f H2 Morse wavef u n c t i o n , f o r which f ^ g ^ i s 38 l a r g e ( ^ 10 ) , but can n e v e r t h e l e s s be handled by a computer. The W h i t t a k e r f u n c t i o n , &/j, — —Vs. , can be w r i t t e n i n terms of a g e n e r a l i s e d Leguerre p o l y -nomial (Chapter I V ) . Wto, ^ _ 00 = e- H % (-1) Z j " a " So the n o r m a l i z a t i o n a c t u a l l y used i s M i . = Ptv^vJ . Hence tf\ * ( € - / - / vj/ P - y j ) , the n o r m a l i z a t i o n c o e f f i c i e n t s , AA, are e v a l u a t e d r e c u r s i v e l y employing s t a r t i n g w i t h f\f2 s $ -However, as one w i l l r e c a l l i t i s not p o s s i b l e t o e v a l u a t e A/j^in one s t e p due t o the l a r g e v a l u e of / 7 ^ ^ — , never-t h e l e s s t he c a l c u l a t i o n o f the w a v e f u n c t i o n s can be - A l l made r e l a t i v e l y s i m p l e by t a k i n g j u d i c i a l c o m b i n a t i o n s o f l a r g e and s m a l l numbers. The w a v e f u n c t i o n s are c a l c u l a t e d once i n a s c a t t e r i n g c a l c u l a t i o n , d u r i n g the i n i t i a l i z a t i o n o f the p o t e n t i a l . T h e i r c a l c u l a t i o n i s shared between two r o u t i n e s , LAGR which i s c a l l e d once at each v a l u e o f , and e v a l u a t e s A3. 3 f o r a l l the v i b r a t i o n a l s t a t e s c o n s i d e r e d i n t h e s c a t t e r i n g c a l c u l a t i o n s s t a r t i n g w i t h /V0 s—: = , and WFN which i s c a l l e d f o r each c o m b i n a t i o n o f V2 and t o e v a l u a t e t h e p o t e n t i a l m a t r i x elements. WFN e v a l u a t e s A3.4 N o t i c e t h a t A3.3 i n v o l v e s e w i t h i s manageable u s i n g the computer ( w h i l e e - ^ ^ i s n o t ) . 1 8 i-The n u m e r i c a l v a l u e o f N = (1943.3674 10 ) 2 o «= = 1 Q 2 5 0 LAGR e v a l u a t e s A(y) i n the f o l l o w i n g manner:-1. The q u a n t i t y P = e - ^ ^ i s c a l c u l a t e d . 2. A do loop i s e n t e r e d and the f o l l o w i n g c a l c u l a t i o n i s performed t e n t i m e s 9 to 2 5 3. The do lo o p i s l e f t . 4. LAGR e v a l u a t e s N Q = P x (1943.3674 10 ° ) 2 . 5. LAGR then c a l c u l a t e s A(y) f o r each v a l u e o f ^2^2^ needed. A12 Al t h o u g h no more than t h r e e open v i b r a t i o n a l channels are r e q u i r e d i n any of our c a l c u l a t i o n s , however so t h a t t h e CO molecule i s p r o p e r l y d e s c r i b e d i t i s necessary t o i n c l u d e c l o s e d v i b r a t i o n a l c h a n nels. When these s t u d i e s were begun we d i d not know how many channels would have t o be i n c l u d e d i n the coup l e d e q u a t i o n s t o o b t a i n s a t i s f a c t o r y r e s u l t s . T h e r e f o r e , so t h a t the convergence o f the r e s u l t s as a f u n c t i o n o f the s i z e o f the v i b r a t i o n a l b a s i s c o u l d be p r o p e r l y t e s t e d , we i n c l u d e d the f i r s t s i x ( v 2 ^ 5) Morse w a v e f u n c t i o n s i n our p r e l i m i n a r y c a l c u l a t i o n s s t u d y i n g the e v a l u a t i o n o f the p o t e n t i a l m a t r i x elements (Chapter V.4.1). Some of our CO Morse o s c i l l a t o r w a v e f u n c t i o n s are shown i n F i g u r e 1. Q u a l i t a t i v e l y they appear s a t i s f a c t o r y , we mention t h a t u s i n g our t e c h n i q u e o f e v a l u a t i n g N , o u t l i n e d above, we were a b l e t o reproduce t he MOEX wa v e f u n c t i o n s employed i n our H2 + H + c a l c u l a t i o n s . I n Table 1 we show the va l u e s o f t h e o v e r l a p i n t e g r a l , I , f o r V 2 ( v ^ ) ^ 5, where I i s g i v e n by A3.5 was e v a l u a t e d u s i n g a 9 6 - p o i n t Gauss Legendre q u a d r a t u r e i n the range, 1.4^^^:2.8 a.u. As expected o r t h o n o r m a l i t y d e t e r i o r a t e s w i t h V2 s i n c e we e v a l u a t e the i n t e g r a l n u m e r i c a l l y . I t i s w o r t h m e n t i o n i n g t h a t i n Chapter V.4.1 we show t h a t the v a l u e s o f t h e p o t e n t i a l OS A3. 5 r2 (a . u.) F i g u r e 1. CO Morse o s c i l l a t o r wavefunct i o n s . TABLE 1 ^ Values of the overlap i n t e g r a l , K^,*') = / U>^° wJ/°J/z f o r v 2 ( v ^ ) ^ 5 . J ° *• % 0 0 1.0 1 0.1408(-15) 1.0 2 -0.3058(-14) 0.6277(--14) 1.0 3 -0.1214(-13) -0.1354(--12) -0.3205( -13) 1.0 4 0.1689(-11) -0.9326(--12) -0.6397( -12) 0. 3891 ( -12) 1.0 5 -0.2567(-11) -0.210K--10) -0.205H -10) 0. 1259( -11) 0.1022( 1.0 Numbers i n b r a c k e t s are powers o f 10. - A13 -m a t r i x elements are i n s e n s i t i v e t o changes of ^ 0.2 a.u. i n the v a l u e s of the i n t e g r a t i o n l i m i t s . I n view o f the r e s u l t s p r e s e n t e d i n F i g u r e 1 and Table 1 we c l a i m t h a t our CO Morse w a v e f u n c t i o n s are s a t i s f a c t o r y . - A I 4 -APPENDIX 4 I n t h i s appendix we w i l l d i s c u s s t h e e f f e c t o f the e r r o r i n our P p o t e n t i a l r o u t i n e on the 1 ^ + CO r e s u l t s p r e s e n t e d i n Chapter VI. I n a d d i t i o n we w i l l compare our c a l c u l a t i o n s w i t h the cr o s s s e c t i o n s e v a l u a t e d by BSD2 i n a c a l c u l a t i o n employing t h e P p o t e n t i a l . N.B. BSD2 = Bacic, Schinke, and D i e r c k s e n (1984b). - A 1 5 -As d i s c u s s e d i n Chapter VL 2, t h e r e i s an e r r o r i n t h e l o n g range m u l t i p o l e i n t e r a c t i o n s g i v e n i n t h e paper o f Poulsen ( 1 9 8 2 ) . These i n t e r a c t i o n s a re g i v e n by ( c f . V I . 2 . 3 ) . ' ' A4.1a Va«i- ™ (sy^"X^R-S (12-Yzo (u« fi>,) Yz& (t&o©O) A4 . 1 h -~¥TL (2T?y^ $»\c*R-(> (to Y«0 (un&D Y,» (c*i&3) ' A4.1c A4.1d ' /T>HH s* where U( and ( J ^ are the quadrupole and hexadecapole moments o f the H2 molecule r e s p e c t i v e l y , andjU^° and are t he d i p o l e and quadrupole moment of t h e CO molecule r e s p e c t i v e l y . However, the i n t e r a c t i o n s g i v e n by A4.1 are a f a c t o r o f 2 t o o l a r g e . T h i s e r r o r was d e t e c t e d by BSD2, but was p r e s e n t i n our p o t e n t i a l r o u t i n e . We have s i n c e c o r r e c t e d t h i s e r r o r i n our computer programme, and have r e p e a t e d the s c a t t e r i n g c a l c u l a t i o n a t an i n i t i a l k i n e t i c energy of Ej, = 208 cm . The r e s u l t s o f these c a l c u l a t i o n s f o r the 1V-T' process i n which = V-J ^ 0~y 9 o n the form o f the m u l t i p o l a r i n t e r -a c t i o n s . As i s d i s c u s s e d i n Chapter V I I . 3 , we are r e a s o n a b l y sure t h a t the t r a n s f o r m a t i o n V I I . 2 . 4 i s s u f f i c i e n t l y a c c u r a t e t o s t u d y v i b r a t i o n a l e x c i t a t i o n , and we are - A 22 -t h e r e f o r e , unable t o p r o v i d e a s a t i s f a c t o r y e x p l a n a t i o n f o r the r e s u l t s p r e s e n t e d i n t h i s appendix. I n p a r t i c u l a r the agreement between the e x p e r i m e n t a l r e s u l t s o f Andrews and Simpson (1976) and the t h e o r e t i c a l r a t e w i l l be d i m i n i s h e d (Chapter V I I . 4 ) , and i n view o f the r e s u l t s p r e s e n t e d i n t h i s appendix i t appears t h a t t he r e s u l t i n g agreement w i l l be e x t r e m e l y poor.

Molecular collisions. 12 - Generalized phase shifts

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Molecular collisions. 12 - Generalized phase shifts

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