Atomic Many-Body Problem. I. General Theory of Correlated Wave Functions

Abstract

The purpose of the paper is to present the general theory of atomic wave functions which explicitly depend on the distances between the electrons. The total wave function for the atom will be written as a composition of one-electron spin-orbitals, and 2-electron, 3-electron,... $N$-electron functions, respectively. The wave function has the following properties: (1) It becomes identical with the exact solution of the many-body problem if one expands the many-electron functions in terms of a complete set of Slater determinants; (2) it represents superposition of configurations if one writes the many-electron functions in the form of linear combination of a finite set of determinants; (3) one obtains the generalization of the Hylleraas ${r}_{12}$-method introducing the interelectronic distances explicitly into the many-electron functions.It will be shown that the many-electron functions can be orthogonalized with respect to the one-electron spin-orbitals, without restricting the generality of the total wave function. General formulas for the matrix components of the Hamiltonian with respect to correlated functions will be derived. The approximation containing only 2-electron correlations will be discussed, and it will be pointed out that in this approximation all matrix components can be reduced to 2-electron and 3-electron integrals, respectively; and the calculation of the 3-electron integrals may be simplified by introducing two interelectronic distances as integration variables.