Abstract

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 (x^l), 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:FRANCK

Catalogue identifier:ADXX_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXX_v1_0

Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland

Programming language:Mathematica 5.0

Computer:Pentium M 1.4 GHz

Operating system:Mathematica 5.0

RAM:512 MB

No. of lines in distributed program, including test data, etc.:825

No. of bytes in distributed program, including test data, etc.:16 344

Distribution format:tar.gz

Nature 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 (x^l), 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:None

Running time:As an example, for the value of Franck–Condon Overlap Integral I_nn^′(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^′.

Keywords: Franck–Condon factors; Harmonic oscillator wave functions; Overlap integrals; Binomial coefficients; Hermite polynomials

PACS classification codes: 33.70.Ca; 31.15.-p; 02.60.-x; 32.30.-r

Article Outline

1. Introduction

2. General definitions and basic formulas

3. Expression for the Franck–Condon factors

4. Expressions for two-center matrix elements of x^l, exp(−2cx) and exp(−cx²)

5. Numerical results and discussion

Acknowledgements

References

This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).