Nonlinear ‘‘self-interaction’’ Hamiltonians of the form H (0)+λ〈r p〉 r q and their Rayleigh–Schrödinger perturbation expansions

Abstract

Rayleigh–Schrödinger perturbation expansions for eigenvalues E(λ) of nonlinear Hamiltonians of the form Ĥ(0)+λ〈r p〉rq, p,q≥1 are calculated using hypervirial (HV) and Hellmann–Feynman (HF) theorems. Such Hamiltonians are similar in form to those employed in the study of ‘‘self-interacting’’ systems, e.g., solute–solvent interactions. The specific cases considered for H(0) are one-dimensional harmonic oscillators and hydrogen atoms. The eigenvalue expansions for the nonlinear problems are compared with those of the linear problems where p=0, whose large-order behavior and summability properties are well-known. Also examined are the perturbation expansions for the expectation values 〈rk〉, which are also products of the HVHF method.