Perturbational-Variational Approach to the Calculation of Variational Wave Functions. I. Theory

Abstract

Certain aspects of Rayleigh-Schr\odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.