Useful extremum principle for the variational calculation of matrix elements

Variational principles for the estimation of the matrix element $W_{\mathrm{nn}}\equiv ({\phi }_n,W{\phi }_n)$ for an arbitrary operator $W$ are of great interest. The variational estimates are constructed from a trial wave function ${\phi }_{nt}$, an approximation to the nth normalized bound-state eigenfunction ${\phi }_n$, and of a trial auxiliary function $L_t$, an approximation to $L$ which satisfies $(H-E_n)L=(W_{nn}-W){\phi }_n\equiv q\left({\phi }_n\right)$. Variational-principle applications have been limited by the difficulty of obtaining a reasonable $L_t$, among other things, one demands that $L_t$ approach $L$ as ${\phi }_{nt}$ approaches ${\phi }_n$. The equation $(H-E_{nt})L_t=q\left({\phi }_{nt}\right)$, where $E_{nt}=({\phi }_{nt},H{\phi }_{nt})$, is known not to provide such an $L_t$. A practical procedure for handling complicated systems given a reasonably accurate Rayleigh-Ritz trial function ${\phi }_{nt}$ is called for. This paper provides such a procedure using techniques developed in the establishment of variational bounds on scattering lengths. Given $H$ and ${\phi }_{nt}$, we define $L_t$ by $A{L}_t=q\left({\phi }_{nt}\right)$, where $A$ differs from $H-E_n$ in that the influence of states 1 through $n$ has effectively been "subtracted out"; the operator $A$ is non-negative. A functional $M\left(L_{tt}\right)$ is constructed which is an extremum for $L_{tt}=L_t$. Variational parameters contained in