VARIATIONAL SOLUTIONS TO THE BRILLOUIN-WIGNER PERTURBATION DIFFERENTIAL EQUATIONS

摘要

The usualbrillouin-Wigner (BW) perturbation theory series expansions for the energy and wave function of a perturbed system are replaced by a set of perturbation differential equations. Thus it seems probable that many of the developments. of Rayleigh-Schrödinger (RS) perturbation theory which depend largely on the RS perturbation differential-equations should carry over to BW theory. A variational method, analogous to the hylleraas principle in RS theory, is derived which can be used to obtain approximate solutions to the n-th order BW perturbation equation for systems in the lowest energy state of a given symmetry. The BW energy to: (2n)-th order obtained in this manner is an upper bound to the exact BW energy to (2n)-th order if the (n-l)-th order wave function is known exactly. This efwcourse is usually true for n = 1 only. It is shown, however, that these variational techniques give an upper bound to the total energy even if the (n-l)-th order BW wave function is unknown Finally a convenient matrix method of applying the variational principles is suggested and a method of using this formulation of BW perturbation theory is discussed formally.