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Physical Review C

Hartree-Fock ansatz as saddle points of the convoluted Green’s function

BERTRAND G. GIRAUD, Yukap Hahn

Published Dec 1, 1989• 0 citations
Publisher

Abstract

A matrix element of a many-body Green's function is estimated via a multiple integral over a basis of products of single-particle wave functions. Saddle points of this integral generate Hartree (-Fock) equations. The result is applied to a simple, correlated kinetic energy model.
DOI
10.1103/physrevc.40.2907

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Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function $\ensuremath{\Phi}$. For two particles, we exactly express $\ensuremath{\Phi}$ in terms of (single-particle) orbitals for all energy levels and any interparticle interaction. Especially for the ground state, $\ensuremath{\Phi}$ is given by the simple Hartree-Fock and Hartree Ans\"atze for fermions and bosons, respectively. Implications for several and many particles as well as a numerical example for interacting bosons are provided.
We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.
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We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.