Vibrational Transition Moments and Dipole Derivatives

Oscillator strengths and integrated absorption intensities of simple vibrational absorption (infrared) spec-tra are related to the squares of electric-dipole transition moments,1 Dnvv ′ = 〈Ψnv|µ̂|Ψnv′ 〉 · 〈Ψnv′|µ̂|Ψnv〉, (1) where n denotes the electronic state, v and v ′ denote vibrational states, and |Ψnv 〉 and |Ψnv′ 〉 denote initial and final vibronic states, respectively. We may compute the electric-dipole vibrational transition moment beginning from the Born-Oppenheimer approximation, in which we assume that the total vibronic wave function, Ψnv(r,R), may be written as a product of an electronic wave function, ψn(r;R) and a vibrational wave function, χnv(R), where r and R denote the collective electronic and nuclear coordinates, respectively. Then the electric-dipole transition matrix element may be written as 〈Ψnv(r,R)|µ̂|Ψnv′(r,R)〉 = 〈χnv(R)|〈ψn(r;R)|µ̂|ψn(r;R)〉|χnv′(R)〉 = 〈χnv|〈µ̂〉n|χnv′ 〉 (2) where 〈µ̂〉n denotes the expectation value of the electric-dipole operator in the n-th Born-Oppenheimer electronic state. The dependence of the 〈µ̂〉n on the nuclear coordinates is usually approximated by the first term of its Taylor expansion about a reference geometry R0 (i.e., the electrical harmonic approxima-tion): 〈µ̂〉n ≈ 〈µ̂〉0+∑