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