Useful extremum principle for the variational calculation of matrix elements

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Variational principles for the estimation of the matrix element Wnn(φn, Wφn) for an arbitrary operator W are of great interest. The variational estimates are constructed from a trial wave function φn t, an approximation to the nth normalized bound-state eigenfunction φn, and of a trial auxiliary function Lt, an approximation to L which satisfies (H-En)L=(Wn n-W)φnq(φn). Variational-principle applications have been limited by the difficulty of obtaining a reasonable Lt, among other things, one demands that Lt approach L as φn t approaches φn. The equation (H-En t)Lt=q(φn t), where En t=(φn t,Hφn t), is known not to provide such an Lt. A practical procedure for handling complicated systems given a reasonably accurate Rayleigh-Ritz trial function φn t is called for. This paper provides such a procedure using techniques developed in the establishment of variational bounds on scattering lengths. Given H and φn t, we define Lt by A Lt=q(φn t), where A differs from H-En in that the influence of states 1 through n has effectively been "subtracted out"; the operator A is non-negative. A functional M(Lt t) is constructed which is an extremum for Lt t=Lt. Variational parameters contained in Lt t can be determined by extremizing M(Lt t), thereby providing an approximation to Lt. The method is analogous to the determination of parameters in φn t by the minimization of (φn t,Hφn t)(φn t,φn t). The method is immediately applicable to the variational determination of off-diagonal matrix elements Wn m and of diagonal matrix elements of normal and of modified Green's functions. © 1974 The American Physical Society.

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Physical Review A

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