Effective-potential expansion method for the many-body problem at finite temperatures. I. Basic formalism

The effective-potential expansion method, namely, a variational approach to the many-body problem at zero temperature, is extended to nonzero temperatures by the introduction of a new trial density matrix in the Gibbs variational principle for the grand potential Ω. The trial density matrix contains an effective potential as a variational parameter. We have given an expansion form for Ω in terms of the effective potential. Special care is taken to evaluate the entropy term correctly.