Spin matrix, elements, electron propagator

The limit with respect to rj is taken because of integration techniques required in a Fourier transform from the time-dependent representation. Indices r and s refer to general, orthonormal spin-orbitals, r x) and os(x), respectively, where x is a space-spin coordinate. Matrix elements of the corresponding field operators, al and as/ depend on the N-electron reference state, N), and final states with N 1 electrons, labeled by the indices m and n. The propagator matrix is energy-dependent poles occur when E equals a negative VDE, Eq(N) — En(N — 1), or a negative VAE, Em(N +1) — Eq(N). 

A one-electron, zeroth-order Hamiltonian defines a set of reference eigenfunctions (spin-orbitals) and eigenvalues (e) such that the matrix elements of the corresponding inverse propagator matrix read... 

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