Exact evaluation and recursion relations of two-center harmonic oscillator matrix elements

Using vibrational wave functions of two relatively displaced harmonic oscillators of arbitrary frequencies, Franck–Condon overlap integrals and matrix elements of xl, exp(−2cx), and exp(−cx2) (x is the internuclear separation) are obtained. Useful three-term, four-term, and five-term recursion relations among these matrix elements are derived. It is shown that all of the relevant matrix elements can be obtained from a mere knowledge of the lowest two Franck–Condon overlap integrals. Results are illustrated by computation of Franck–Condon factors for the A 1∑+u–X 1∑+g and the B 1Πu–X 1∑+g systems of 7Li2.