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

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Abstract
For transitions between two offset oscillators with equal force constants, the optical band shapes are described by the distribution Wpu giving the probability that the optical transition is accompanied by the generation of pu phonons. For the Condon approximation and for thermal equilibrium among the initial‐oscillator vibrational levels, Wpu is shown through the Manneback recursion formulas for the overlap integrals to satisfy the recursion formula S0<m>vWpu+1+ puWpu −S0vWpu−1 =0. S0 and <m>v are offset and temperature parameters, respectively. From this Wpu recursion formula, the moments of the Wpu distribution, its approximate forms, its exact expression, and numerical values are obtained in simple ways.