Recursion analysis of the configurational-coordinate model for equal force constants

Abstract

For transitions between two offset oscillators with equal force constants, the optical band shapes are described by the distribution W_pu giving the probability that the optical transition is accompanied by the generation of p_u phonons. For the Condon approximation and for thermal equilibrium among the initial‐oscillator vibrational levels, W_pu is shown through the Manneback recursion formulas for the overlap integrals to satisfy the recursion formula S₀<m>_vW_pu+1+ p_uW_pu −S_0vW_pu−1 =0. S₀ and <m>_v are offset and temperature parameters, respectively. From this W_pu recursion formula, the moments of the W_pu distribution, its approximate forms, its exact expression, and numerical values are obtained in simple ways.