Construction of resonance projection operators: Application to two-electron targets

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Abstract
A derivation of resonance projection operators, in the Feshbach sense, is presented in detail, with inclusion of angular momentum and spin variables. The resultant operators, P and Q, are written in a transparent and manifestly symmetric form. The operators depend on eigensolutions of a homogeneous integral equation. The equation is solved and hence the operators are explicitly constructed for the two-electron (i.e., He-like) targets in a nonseparable (the open-shell) approximation; this procedure can readily be extended to any configuration-interaction-type target function. The kernel of the auxiliary integral equation is also explicitly derived for a Hylleraas-type target function; a Rayleigh-Ritz variational principle is written down for its eigenfunctions. It is pointed out that these (Feshbach) P and Q operators are not formally idempotent, but it is shown that their matrix elements between antisymmetric functions are. This makes them completely suitable for calculation of autoionization states via diagonalization of QHQ. A basic problem in using the concomitant optical potential for scattering calculations remains and is briefly discussed. The generalization of these operators to the inelastic domain is also given (in an appendix).