The method of superposition of configurations, which provides a general solution of the quantum-mechanical many-particle problem (for fermions), is reformulated so that equations may be compared with those used in the Brueckner theory. Important differences occur in application to finite or nonuniform systems for which the Hartree-Fock or Brueckner self-consistent orbitals are not plane waves. In such cases nonvanishing single-particle matrix elements occur which cannot be described by the Brueckner formalism based on a two-particle operator. Equations for an effective two-particle operator, equivalent to the variational equations of the method of superposition of configurations, are derived for a basis of Hartree-Fock orbitals. At the expense of making the orbital basis dependent upon the effective two-particle operator, the orbital basis can be determined by a condition which is essentially that of the Brueckner method. This condition removes a class of matrix elements which do not necessarily vanish in the Hartree-Fock basis although they would be neglected to second order in a perturbation calculation starting from the Hartree-Fock wave function. The equations for the effective two-particle operator are formally the same in both cases but lead to different operators because of the different choice of basis. In neither case can the equations be written in terms of products of such operators, a formalism assumed in the Brueckner theory. It is shown that the Brueckner condition is not equivalent to the condition which would determine the best possible orbital basis for the form of wave function implied by the use of an effective operator dependent on two particles only. The argument of the present paper is limited to systems with a finite number of particles, since the variational principle used here is not applicable without modification to systems with an infinite number of particles.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix
〈m1m2∣A∣p1p2〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix
A. Hole-state energies are given self-consistently in terms of the matrix
A, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of
N antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order
k has two sets of indices of each
k variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order
N over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete
k-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly
N spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order
M, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the
M 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=N, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For
M>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>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
N 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.
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.
A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of
N fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and
f(2)(ij) is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained
N one-particle equations for the
N one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.