Quantum Chemistry of Solids pp 147-191 | Cite as

# Electron Correlations in Molecules and Crystals

Chapter

## Abstract

Electrons in molecules and crystals repel each other according to Coulomb’s law, with the repulsion energy depending on the interelectron distance as is a completely uncorrelated function. The Hartree product (5.1) describes the system of which is the probability of finding electron 1 at

*r*_{12}^{−1}. 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)

*N*_{e}electrons in an independent particle model. This independence means that the probability of simultaneously finding electron 1 at*x*_{1}, electron 2 at*x*_{2},*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)

*x*1 times the probability of finding electron 2 at*x*_{2},*etc., i.e.*product of probabilities.## Keywords

Electron Correlation Periodic System Pair Domain Slater Determinant Distant Pair
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2007