Variational Methods for Many-Body Problems

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Abstract
A variational approach to the approximate solution of a wide variety of self-consistency problems is described. Attention is focused on the variational formulation of the nonlinear integral equation, which determines the single-particle Green's function of a many-particle system; such an equation arises from a particular truncation of the hierarchy of equations which dynamically couple the complete set of Green's functions. To illustrate the method, we have applied it, with encouraging results, to the numerical determination of the self-energy function for the helium atom in the Hartree-Fock approximation. The applicability of this approach to the general problem of electron-atom scattering is pointed out.