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Physical Review

Generalized Self-Consistent Field Theory and the Dielectric Formulation of the Many-Body Problem

Morrel H. Cohen

Published May 15, 1963• 35 citations
Publisher

Abstract

A generalized self-consistent field (SCF) theory of many-particle systems is developed by modifying the usual functional relationship between energy and wave function. A parameter $\ensuremath{\xi}$, $0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1$, is thus introduced into the energy so that for $\ensuremath{\xi}=0$, the particles interact only with a self-consistent field, for $\ensuremath{\xi}=1$ the particles interact dynamically among themselves with the correct field, and for intermediate values of $\ensuremath{\xi}$ both SCF and dynamical interactions occur. A systematic development of the theory is given for time-independent and time-dependent probelms, for finite temperatures, and for both uniform and nonuniform systems. The derivative with respect to $\ensuremath{\xi}$ of the total energy, of free energy, is expressed in terms of the dielectric function and an improved version of the dielectric formulation of the many-body problem thereby obtained. A brief discussion of the advantages of the method, possible applications, and further generalizations or extensions is included.

DOI
10.1103/physrev.130.1301

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Physical review. B, Condensed matter•Feb 15, 1979•18 citations
A generalized self-consistent field (SCF) theory of many-particle systems is developed by modifying the usual functional relationship between energy and wave function. A parameter \ensuremathξ\ensuremath{\xi}\ensuremathξ, 0\ensuremath≤\ensuremathξ\ensuremath≤10\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}10\ensuremath≤\ensuremathξ\ensuremath≤1, is thus introduced into the energy so that for \ensuremathξ=0\ensuremath{\xi}=0\ensuremathξ=0, the particles interact only with a self-consistent field, for \ensuremathξ=1\ensuremath{\xi}=1\ensuremathξ=1 the particles interact dynamically among themselves with the correct field, and for intermediate values of \ensuremathξ\ensuremath{\xi}\ensuremathξ both SCF and dynamical interactions occur. A systematic development of the theory is given for time-independent and time-dependent probelms, for finite temperatures, and for both uniform and nonuniform systems. The derivative with respect to \ensuremathξ\ensuremath{\xi}\ensuremathξ of the total energy, of free energy, is expressed in terms of the dielectric function and an improved version of the dielectric formulation of the many-body problem thereby obtained. A brief discussion of the advantages of the method, possible applications, and further generalizations or extensions is included.
A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the MMM occupied orbitals ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ of an $N(\ensuremath{\le}M)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\ensuremath{\varrho}}=\ensuremath{\Sigma}{\ensuremath{\mu}=1}^{M}|{\ensuremath{\psi}}_{\ensuremath{\mu}}〉〈{\ensuremath{\psi}}_{\ensuremath{\mu}}|$. The condition placed on $\mathbf{\ensuremath{\varrho}}$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting M=NM=NM=N, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For M>NM>NM>N there are three freedoms one may take with the fundamental condition on $\mathbf{\ensuremath{\varrho}}$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite M>NM>NM>N, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of NNN identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.
The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.
Gor'kov factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)〉〈\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉$, is made the basis of a generalized self-consistent-field (SCF) theory precisely analogous to that previously developed for Hartree factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1)\ensuremath{\psi}(4)〉〈({\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)〉$. The generalized SCF method is reviewed in the context of Gor'kov factorization. The corresponding simple SCF theory is developed to illustrate the method and is shown to give a simple Hamiltonian version of the Bardeen-Cooper-Schrieffer theory of super-conductivity and, more generally, of coherent pairing. The notion of an external pairing field is introduced and the corresponding response functions developed via a formalism like that of Kubo. General fluctuation-dissipation theorems are proved for the response functions. An equation of state is then obtained by a formulation analogous to the dielectric formulation. The insertion of the simple SCF approximation to the response function into the equation of state yields a generalization of the usual low-density (or short-range) evaluation of the grand potential to arbitrary temperatures. Screening the bare interaction with this approximate response function converts the former into the ttt matrix of the independent-pair approximation. The entire analysis gives an underlying unity to the currently disparate treatments of superconductivity, of long-range correlations, and of short-range correlations.
A self-consistent field Hamiltonian is constructed, for arbitrary pair interactions, which yields (in principle) an exact value for the energy of a chosen state of a system of NNN particles, as well as estimates for other states. The method for extending the formalism to interactions involving arbitrary numbers of particles is indicated and is straightforward. The method is carried out within the bound-state framework, and while motivated by the recent work of Bethe, results in a Hamiltonian somewhat different from his. An argument is presented which indicates that the independent-particle eigenfunctions of this Hamiltonian are "good" representations of the original NNN-body system. A discussion of the practicality of the computations is included.
In this paper we derive a formula for the time-dependent Hartree-Fock dielectric function by using Green's-function theory. We develop a set of self-consistent equations for the self energy, polarization, and Green's function which can be iterated to obtain successively more accurate approximations for the polarization. In this procedure the Hartree and Hartree-Fock approximations occur naturally as the first and second steps in the iterations. As the iterations are carried out, we obtain not only the expressions for the polarization, but also the one-electron Hartree-Fock equations, so that there is no question as to which polarization formula corresponds to which equation. In this way we obtain an expression for the dielectric function which clearly and directly corresponds to the Hartree-Fock equation and which is therefore called the time-dependent Hartree-Fock (TDHF) dielectric function. In this TDHF formula for the dielectric function, the first term is the time-dependent Hartree or random-phase approximation dielectric function. We show that in order to be consistent with the theory, one should use Hartree wave functions and energies to compute random-phase approximation dielectric functions and Hartree-Fock wave functions and energies to compute TDHF dielectric functions. Numerical results are presented for the free-electron gas.