Exact analytical expressions and numerical analysis of two-center Franck-Condon factors and matrix elements over displaced harmonic oscillator wave functions

A detailed study is undertaken, using various techniques, in deriving analytical formula of Franck-Condon overlap integrals and matrix elements of various functions of power (xl), exponential (exp(−2cx)) and Gaussian (exp(−cx²)) over displaced harmonic oscillator wave functions with arbitrary frequencies. The results suggested by previous experience with various algorithms are presented in mathematically compact form and consist of generalization. The relationships obtained are valid for the arbitrary values of parameters and the computation results are in good agreement with the literature. The numerical results illustrate clearly a further reduction in calculation times.

Program summary

Program name:FRANCKCatalogue identifier:ADXX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXX_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandProgramming language:Mathematica 5.0Computer:Pentium M 1.4 GHzOperating system:Mathematica 5.0RAM:512 MBNo. of lines in distributed program, including test data, etc.:825No. of bytes in distributed program, including test data, etc.:16 344Distribution format:tar.gzNature of problem:The programs calculate the Franck-Condon factors and matrix elements over displaced harmonic oscillator wave functions with arbitrary quantum numbers (n,n₁), frequencies (a,a₁) and displacement (d) for the various functions of power (xl), exponential (exp(−2cx)) and Gaussian (exp(−cx²)).Solution method:The Franck-Condon factors and matrix elements are evaluated using binomial coefficients and basic integrals.Restrictions:The results obtained by the present programs show great numerical stability for arbitrary quantum numbers (n,n₁), frequencies (a,a₁) and displacement (d).Unusual features:NoneRunning time:As an example, for the value of Franck-Condon Overlap Integral Inn^′(d;α,α^′)=0.004405001887372332 with n=3, n₁=2, a=4, a₁=3, d=2, the compilation time in a Pentium M 1.4 GHz computer is 0.18 s. Execution time depends on the values of integral parameters n, n^′, d, α, α^′.