S>In previous work N. N. Bogolyabov (Doklady Akad. Nauk 1l9, 244, 1958) suggested a generalization for the Fock method in which the energy minimum is found in the individual particle quasi-independent functions. The suggested variation utilizes a wider span of functions which consider the wave functions of particle pairs in addition to the quasi-independent functions of individual particles. The new variation is used for deriving the equations for a wave function of N particle systems in the many-body problem. The new principle and mathematical method permit both particle pair and N particle system correlations. It is pointed out that while the number of particles in spatially uniform problems is conserved, in spatially non-uniform problems it is not. (R.V.J.)
S>In previous work N. N. Bogolyabov (Doklady Akad. Nauk 1l9, 244, 1958) suggested a generalization for the Fock method in which the energy minimum is found in the individual particle quasi-independent functions. The suggested variation utilizes a wider span of functions which consider the wave functions of particle pairs in addition to the quasi-independent functions of individual particles. The new variation is used for deriving the equations for a wave function of N particle systems in the many-body problem. The new principle and mathematical method permit both particle pair and N particle system correlations. It is pointed out that while the number of particles in spatially uniform problems is conserved, in spatially non-uniform problems it is not. (R.V.J.)
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.
The minimum parameterization of the wave function is derived for the time-independent many-body problem. In this parameterization all electrons and electron pairs are indistinguishable which eliminates the exponential scaling with the number of particles plaguing all other correlation methods. An exponential scaling shown to stem from the expansion of the wave function in one-particle basis sets. It is demonstrated that using a geminal basis with a skew-symmetric orthogonal coefficient matrix, which fulfill a Lie algebra, the parameterization of the exact wave function becomes independent of the number of particles and only scale quadratic with the number of basis functions in the optimized basis and quartic in an arbitrary basis. The resulting antisymmetrized geminal power wave function is shown to fulfill the necessary and sufficient conditions for the exact wave function, treat all electrons and electron pairs as indistinguishable and be the most compact representation of the exact wave function possible. The number of parameters in the antisymmetrized geminal power wave function is exactly the same number of parameters as in the two-particle reduced density matrix thereby showing that these representations are equivalent just like the representations for a one-particle Hamiltonian. A detailed discussion of how the most compact wave function can be derived in general is also presented along with different geminal ansatze and sufficient closure relations. Due to the breaking of the scaling wall for the exact wave function it is expected that, if a general set of constraints can be found, even systems of biological relevance can be treated exactly in the future.
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.
By means of a suitably chosen variational principle, Euler equations are derived which generalize the Hartree-Fock equations in such a way as to include two-particle correlations. For a system of N fermions these equations involve N single-particle functions and $\frac{1}{2}N(N\ensuremath{-}1)$ pair correlation functions. The Euler equations are optimal in the sense that a more accurate estimation of the total energy would require the introduction of three-particle correlation functions and the retention of terms nonlinear in the two-particle functions. Although intended primarily for use in the calculation of atomic and molecular electronic structure, the Euler equations are here used to determine the Coulomb correlation energy in the case of the high-density electron gas. The result obtained is shown to have the same leading logarithmic divergence with density as that found previously by Gell-Mann and Brueckner.
For describing the pair correlations of the electrons in the many- electron systems a new method of reducing the many-particle problem to the two- particle problem is developed. Instead of the wave function, one- and twoparticle density matrices are used. In the process of deriving an equation for the two-particle wave function, which describes a pair of electrons in the field of the others, a transition to the second quantization representation is accomplished. The solution of the obtained equation is carried out with the help of perturbation theory. (auth)
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .