The Stability of Matter: From Atoms to Stars pp 253-255 | Cite as

# Variational Principle for Many-Fermion Systems

## Abstract

If ψ is a determinantal variational trial function for the *N*-fermion Hamiltonian, *H*, with one- and two-body terms, then *e* _{0} ⩽ 〈 ψ,*H*ψ〉= *E*(*K*), where *e* _{0} is the ground-state energy, *K* is the one-body reduced density matrix of ψ, and *E*(*K*) is the well-known expression in terms of direct and exchange energies. If an *arbitrary* one-body *K* is given, which does not come from a determinantal ψ, then *E*(*K*) ⩾ *e* _{0} does not necessarily hold. It is shown, however, that if the two-body part of *H* is positive, then in fact *e* _{0}⩽*e* _{HF} ⩽ *E*(*K*), where *e* _{HF} is the Hartree-Fock ground-state energy.

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