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

摘要

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 _m (a 1/2 x _A )|g(or f)| _n (b 1/2 x _B ); _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.View full textDownload full textKeywordsmatrix elements, harmonic oscillator, recurrence relations, Gaussian operator, exponential operatorRelated var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00268976.2012.668225