  We present a variational solution of the T-matrix integral equation within a
local approximation. This solution provides a simple form for the T matrix
similar to Hubbard models but with the local interaction depending on momentum
and frequency. By examining the ladder diagrams for irreducible polarizability,
a connection between this interaction and the local-field factor is
established. Based on the obtained solution, a form for the T-matrix
contribution to the electron self-energy in addition to the GW term is
proposed. In the case of the electron-hole multiple scattering, this form
allows one to avoid double counting.
