A relativistic pair equation projected onto positive energy states

Abstract

A method, correct to order alpha 2, is suggested for the relativistic treatment of pair correlation. Projection operators for positive energy states are applied to the two-particle equation which describes the interaction between the electrons. After elimination of two of the four spatial components-each with four spin components-of a relativistic pair function, two coupled equations for the 'large-large' and 'small-small' components, rho LL and rho SS, remain. By neglecting certain terms in the coupled equations, the occurrence of homogeneous solutions with one electron in the positive and one electron in the negative energy continuum is avoided, and an equation for rho LL alone is obtained, which can be solved using numerical methods developed for the non-relativistic problem. For the Coulomb interaction, this procedure is found to be correct to O( alpha 2) even without the use of projection operators, whereas for the lowest-order Breit interaction (exchange of a single virtual transverse photon), projection operators give corrections to the pair function already in O( alpha 2), thereby removing the well known alpha 2 errors that occur if the Breit interaction is used in a higher-order calculation neglecting projection operators. If projection operators are included, the Breit interaction can be treated to higher orders without causing spurious alpha 2 terms. By means of an iterative procedure, pair correlation effects, due to the instantaneous Coulomb interaction and the exchange of single virtual transverse photons, can be evaluated to all orders.