Useful extremum principle for the variational calculation of matrix elements

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 φₙₜ, 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=(Wₙₙ-W)φₙ≡q(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 φₙₜ approaches φₙ. The equation (H-Eₙₜ)Lₜ=q(φₙₜ), where Eₙₜ=(φₙₜ,Hφₙₜ), is known not to provide such an Lₜ. A practical procedure for handling complicated systems given a reasonably accurate Rayleigh-Ritz trial function φₙₜ is called for. This paper provides such a procedure using techniques developed in the establishment of variational bounds on scattering lengths. Given H and φₙₜ, we define Lₜ by ALₜ=q(φₙₜ), 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(Lₜₜ) is constructed which is an extremum for Lₜₜ=Lₜ. Variational parameters contained in Lₜₜ can be determined by extremizing M(Lₜₜ), thereby providing an approximation to Lₜ. The method is analogous to the determination of parameters in φₙₜ by the minimization of (φₙₜ,Hφₙₜ)(φₙₜ,φₙₜ). The method is immediately applicable to the variational determination of off-diagonal matrix elements Wₙₘ and of diagonal matrix elements of normal and of modified Green's functions.