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 x^l, 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 ¹Π_u–X ¹∑⁺_g systems of ⁷Li₂.