The evaluation of condon overlap integrals and computation of spectral intensities for anharmonic vibrational states

It was proposed to obtain a method for introducing some correction for vibrational anharmonicity in the calculation of intensities of vibronic spectral bands of polyatomic molecules, which would still permit the rabid computation of these intensities. The spectral intensities of interest were calculable from Condon overlap integrals in accordance with the Franck Condon principle. The problem studied was essentially that of evaluating Condon overlap integrals for anharmonic vibrational states. The study was limited to spectral series for which the vibrational states could be satisfactorily approximated as one dimensional vibrations. Anharmonic vibrational wave functions were obtained by application of the quantum mechanical perturbation method to a vibrational potential expressed in a series form containing the usual harmonic quadratic term and an additional higher order anharmonic term. The wave function solutions were expressed as linear combinations of the exact wave functions of a corresponding harmonic oscillator. The work was carried out entirely in the normal coordinate notation of polyatomic molecules. Specific wave functions were obtained for both cubic and quartic anharmonic vibrational states. The constant coefficients in the anharmonic potentials were evaluated in terms of spectroscopically measureable quantities by requiring solutions for the energy states to agree with the accepted spectroscopic expression for these energy states. This resulted in the expression of the wave function dependence on the vibrational system parameters in terms of the same spectroscopic quantities.