Recurrence relations for one-dimensional harmonic oscillator matrix elements of Gaussian and exponential operators

Abstract

This paper reports the development of several general recurrence relations that can be used to evaluate one-dimensional, three centre harmonic oscillator matrix elements of the operators and f = exp(−cx _C ). The matrix elements have the general form ⟨φ _m (a ^1/2 x _A )|g(or f)|φ _n (b ^1/2 x _B )⟩; φ _m is the harmonic oscillator basis function for an eigenstate m. The coordinates are x _A  = x − A _x , and so on, where A _x , B _x , and C _x are points of reference for the displacement of a common atom whose instantaneous coordinate is x. A typical case might be that of a hydrogen atom referred to two wells located at A _x and B _x , and a second atom located at C _x on the x axis. The recurrence relations apply to all cases including the two centre A _x  = B _x and overlap integrals, A _x  ≠ B _x , c = 0, and C _x  = 0. Moreover, the recurrence relations can generate matrix elements to any order. The applications of some of these recursions are illustrated with several examples: (1) the variational treatment of the Morse oscillator using one-dimensional harmonic oscillator basis functions; (2) the development of a model of the Morse oscillator in Gaussian coordinates together with (3) the variational analysis of that model. In addition, (4) the simplest version of a symmetric double potential well system is examined using both the Morse oscillator and the model potential.

Keywords:

• matrix elements
• harmonic oscillator
• recurrence relations
• Gaussian operator
• exponential operator

Acknowledgements

This paper is dedicated to Professor William H. Miller on the occasion of his 70th birthday in Berkeley, CA, January 2012.

I thank Brett Dunlap for several helpful discussions of recursion relations.

Notes

Notes

1. While a variational calculation using harmonic oscillator basis functions has apparently not been carried out before, it is nevertheless the case that an excellent variational calculation using oscillators inside a spherical box has been reported 46. Moreover, the calculation of the Franck–Condon factors is similar to that used in 10, namely, the FC factor is expressed as a summation of eigenvector-weighted oscillator Franck–Condon factors.

2. The following discussion is based on calculations that involve the Morse double potential well model. While ionic molecules M⁺Br–H–Br⁻, in which H is shared between the two bromine atoms, and where M⁺ is a cation 47–49 exist, the molecular model discussed here has only indirect connection to those systems. The work of Evans and Ault, cited, involves compounds generated in rare gas and other matrices that clearly possess ionic bonds. In particular, any possible charged species are ignored here.