Optimised mean fields for atoms. I. Mean-field methods for the description of N-fermion systems

Siegert's representation (1960) for the grand canonical partition function of a fermion system with two-particle interactions serves as a starting point for a perturbation expansion around stationary points. It is shown that, in general, the latter are given by solutions of the time-independent Hartree equation. Introducing more general random fields, exchange terms are obtained in the stationarity equation. The procedure is extended to relativistic systems. A perturbative expansion for e.g. the ground-state energy of the N-particle relativistic bound-state problem is obtained which allows, contrary to 'Hamiltonian methods', a straightforward application of standard regularisation and renormalisation procedures.