Second quantization and the Hartree–Fock approximation

Eberhard K. U. Gross, E. Runge, Olle Heinonen

Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.

Larry F. Abbott

Adrian Stan

In this chapter we introduce the fundamental concepts behind the Green Function Theory. We start from a description of the second quantization method and then, after introducing the ensemble average of an operator, we define the main object of the theory, the Green function. Further on, after obtaining the equation of motion for the Green function, we introduce the self-energy operator and we discuss the Hartree-Fock approximation, the second-Born approximation and the GW approximation to the self-energy.

J. M. Cook

Thomas Kjærgaard

We present a pedagogical derivation of the Hartree--Fock equations using the second-quantization atomic-orbital density-matrix formalism developed by Kjærgaard, Jørgensen, Olsen, Coriani, and Helgaker for AO-based response theory. The purpose is to introduce an alternative derivation of the Hartree--Fock equation, showing that the standard AO Hartree--Fock stationarity condition follows naturally from the exponential parametrization of the one-particle density matrix in a nonorthogonal AO basis. This route provides a compact bridge between elementary Hartree--Fock theory and the density-matrix machinery used in modern response theory and linear-scaling formulations.

M. Iwasaki

A canonical quantization method for the time-dependent Hartree-Fock equation is presented with the use of the generalized Hamiltonian method by Dirac.It is shown that in the Lipkin model our quantized Hamiltonian reproduces the original one approximately.