Formulation of the Quantum-Mechanical Many-Body Problem in Terms of One- and Two-Particle Functions

Abstract

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, $\Psi ={\Psi }_0+\Sigma {\mathrm{ij}}^{}{f}^{\mathrm{}\left(2\right)\mathrm{}}\mathrm{}\left(\mathrm{ij}\right)\mathrm{}$, where ${\Psi }_0$ is a determinant of one-particle wave functions and $f^{\mathrm{}\left(2\right)\mathrm{}}\mathrm{}\left(\mathrm{ij}\right)\mathrm{}$ is an antisymmetrized product of ($N-2\mathrm{}$) 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 ${\phi }_{\mathrm{ij}}$ 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.