Improving an algebraic approach to calculate approximate Franck–Condon factors of diatomic molecules

Abstract

Using the BCH theorem, we express the Hamiltonian of a Morse oscillator as a complete series of powers of the creation and annihilation operators for the harmonic oscillator. In this way, we improve the results of a previous work that uses a Bogoliubov-Tyablikov tranformation to calculate the Franck-Condon factors by means of equivalent harmonic oscillators potentials. 0 1992John Wiley & Sons, Inc.

Simple evaluation of Franck–Condon factors and non‐Condon effects in the Morse potential

“…However, this approach may present a similar disadvantage because the calculation involves a sum over all positive and negative nodes of a polynomial. (20) (which agree exactly with the calculations of Ref. [11] making the integration of the Morse wave functions), the calculation using harmonic oscillator wave functions with oscillator length a 0 ϭ [(j ϩ 1 2 ) 1/2 ␤] Ϫ1 , and the calculation with modified harmonic oscillator wave functions as given in Ref.…”

“… The parameters of this transition are j 1 = 69.494, β 1 = 0.326a, R 1 = 5.876a 0 , j 2 = 48.106, β 2 = 0.459a, and R 2 = 5.0535a 0 . Successive entries correspond to Kusch and Hessel 22, Drallos and Wadehra 27, Rivas‐Silva et al 20, Ley‐Koo et al 14, and our results using formula (20) with the scaled integral (22). …”

“…, and R 2 ϭ 5.0535a 0 . Successive entries correspond to Kusch and Hessel [22], Drallos and Wadehra [27], Rivas-Silva et al [20], Ley-Koo et al [14], and our results using formula (20) with the scaled integral (22). The parameters of this transition are j 1 ϭ 62.05,…”

“…As mentioned in the introduction, several approaches have been proposed to find FC factors (20) in analytic form, including the use of different versions of the harmonic oscillator approximation of the Morse wave functions 4–6, 19, 20, the expansion of the integral (22) into series containing polygamma functions 21, the use of a Laguerre quadrature 11, and others. We now show that the integral (22), although not tabulated, is an analytic well‐behaved function that satisfies relations that can be used for its reliable evaluation using a personal computer.…”

“…The term ϭ 0 corresponds to the FC part. The only difference with (20) is that instead of a single integral J a (k 1 , k 2 , y 1 , y 2 ) we have a sum of integrals J a (k 1 ϩ , k 2 , y 1 , y 2 ), each of them multiplied by the corresponding coefficient in the expansion (32). That means that we only have to calculate some more integrals and multiply them by some coefficients that should be chosen in a suitable way.…”