Upper and Lower Bounds for Ground-State Second-Order Perturbation Energy

Abstract

The first‐order perturbation equation, using the Dalgarno—Lewis formulation, is arranged in a form analogous to the Poisson equation for the electrostatic potential produced by a charge distribution in a medium of variable dielectric constant. Thus, the Thomson and Dirichlet variational principles of electrostatics can be used to obtain approximate solutions to the first‐order perturbation equation for systems in either the ground state or the lowest energy state of a given symmetry. The Thomson principle provides a useful lower bound to the second‐order perturbation energy. The Dirichlet principle is derivable from the Rayleigh—Ritz or Hylleraas principles and gives an upper bound to the second‐order energy. For excited states, the Sinanoğlu principle provides the upper bound. By optimizing the scaling of the trial perturbed wavefunction, the Hylleraas principle is presented in a somewhat improved form. As an example, the polarizability of atomic hydrogen is used to illustrate both the Thomson and Dirichlet principles and to place upper and lower bounds on the polarizability.