Electron Correlations in Molecules and Crystals

Abstract

Electrons in molecules and crystals repel each other according to Coulomb’s law, with the repulsion energy depending on the interelectron distance as r ⁻¹₁₂ . This interaction creates a correlation hole around any electron, i.e. the probability to find any pair of electrons at the same point of spin-coordinate space is zero. From this point of view only the Hartree product ψ_H of molecular or crystalline spin-orbitals Ψ_i(x):

$$ \Psi _{\rm H} (x_1 ,x_2 ,...,x_{N_e } ) = \psi _1 (x_1 )\psi _2 (x_2 )...\psi _{N_e } (x_{N_e } ) $$

((5.1))

is a completely uncorrelated function. The Hartree product (5.1) describes the system of N_e electrons in an independent particle model. This independence means that the probability of simultaneously finding electron 1 at x₁, electron 2 at x₂, etc. (x means the set of coordinate r and spin σ variables) is given by

$$ \begin{array}{*{20}c} {{\text{|}}\Psi _{\rm H} (x_1 ,x_2 ,...,x_{N_e } )|^2 dx_1 dx_2 ...dx_{N_e } } \\ {{\text{ = }}|\psi _1 (x_1 )|^2 dx_1 |\psi _2 (x_2 )|^2 dx_2 ...|\psi _{N_e } (x_{N_e } )|^2 dx_{N_e } } \\ \end{array} $$

((5.2))

which is the probability of finding electron 1 at x1 times the probability of finding electron 2 at x₂, etc., i.e. product of probabilities.