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The Journal of Chemical Physics

Oscillators Perturbed by Gaussian Barriers

Sunney I. Chan, David Stelman

Published Aug 1, 1963• 72 citations
Publisher

Abstract

A simple method has been developed to generate the matrix elements of the operator exp (—βξ2) in the representation of the harmonic oscillator. These matrix elements are useful in double-minimum problems where the potential energy function may conveniently be represented by a perturbed harmonic oscillator with a Gaussian barrier. Some representative calculations involving such potential wells will also be presented.

DOI
10.1063/1.1734291

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A simple method has been developed to generate the matrix elements of the operator exp (—βξ2) in the representation of the harmonic oscillator. These matrix elements are useful in double-minimum problems where the potential energy function may conveniently be represented by a perturbed harmonic oscillator with a Gaussian barrier. Some representative calculations involving such potential wells will also be presented.
We study the double-well anharmonic oscillator when the barrier between the two wells is broad and high. In this limit, the energy gap between even- and odd-parity eigenstate pairs is exponentially small. We develop equations for similar exponentially small tunneling contributions to arbitrary matrix elements of arbitrary operators. As an example, we calculate the matrix elements $〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l}|n\ifmmode\pm\else\textpm\fi{}〉$ and $〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l+1}|n\ensuremath{\mp}〉$.
A new family of one-dimensional quantum models is proposed in terms of new potentials with a Gaussian asymptotic behavior but approaching to the potential of the harmonic o scillator when x→0x\to 0x→0. It is shown that, in the energy basis of the harmonic oscillator, the matrix elements of the Hamiltonian operators of these new models can be derived from generating functionals.
Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.