Sum rules and other properties involving resonance projection operators

Abstract

Projection operators describing electron scattering from multielectron atoms (and ions) have recently been derived in detail [A. Temkin and A. K. Bhatia, Phys. Rev. A 31, 1259 (1985)]. The operators (P and Q) involve the eigenspectrum of an auxiliary eigenvalue equation in addition to the eigenfunctions of the target system. In the present paper, a sum rule for these auxiliary eigenvalues is derived. It is verified for the two-electron target with use of an open-shell approximation both in the elastic and inelastic domains. For Hylleraas target functions the auxiliary eigenspectrum is infinite; nevertheless, a variational solution of the auxiliary problem corresponding to a simple but judiciously chosen Hylleraas target approximation is shown to exhaust the sum rule to better than five significant figures. The above sum rule applies to individual angular momentum states; if one sums over all angular momentum states, one obtains a super sum rule in which the right-hand side is a simple integer. Super sum rules for inelastic scattering from the two-electron target are also treated (in an appendix).