Franck-Condon factors for multidimensional harmonic oscillators

We present a simple formula for the overlap integrals of two sets of multi-dimensional harmonic oscillators. The oscillators have in general different equilibrium points, force constants, and natural vibration modes. The formula expresses the overlap matrix in the one-dimensional case, 〈 m′| n′′〉, a...

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Bibliographic Details
Published in:Chemical physics 1998-03, Vol.228 (1), p.227-240
Main Authors: Malmqvist, Per-Åke, Forsberg, Niclas
Format: Article
Language:English
Subjects:
Franck-Condon
Harmonic oscillator
Multi-dimensional
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Summary:We present a simple formula for the overlap integrals of two sets of multi-dimensional harmonic oscillators. The oscillators have in general different equilibrium points, force constants, and natural vibration modes. The formula expresses the overlap matrix in the one-dimensional case, 〈 m′| n′′〉, as a so-called LU decomposition, 〈m′|n′′〉=〈0′|0′′〉 ∑ L mtU tn , where the summation index has a range 0≤ t≤min( m, n), i.e., it is the matrix product of a lower-triangular matrix L with an upper-triangular U. These matrices are obtained from simple recursion formulae. This form is essentially retained in the multi-dimensional case. General matrix elements are obtained by exact and finite expressions, relating them to matrix elements over a single set of harmonic oscillator wave functions. We present test calculations with error estimates, also comparing with literature examples.
ISSN:0301-0104
DOI:10.1016/S0301-0104(97)00347-9