Electron correlation calculations Pauli exclusion principle

Further, if the wave function depends also on the electron spins, spin variables over all electrons should also be integrated we will see this below, in the calculation of exchange hole. The expression in the curly brackets above is exactly the XC hole PxCM(r, r ) defined in Equation 7.17. A comparison with Equation 7.19a shows that adding the hole to the density is similar to subtracting the density of one electron p(r )/N from it. The hole thus represents a deficit of one electron from the density. This is easily verified by integrating p tM(V, r ) over the volume dr, which gives a value of — 1. However, the structure of the hole is not simple and this is because of the motion of different electrons correlated due to the Pauli exclusion principle and the Coulomb interaction between them. Finally we note that the product p(r)p cM(r, r ) is symmetric with respect to an exchange in the variables... 

One of the main reasons for the good results obtained with the Hartree-Fock SCF method in electronic structure calculations for atoms and molecules is that the electrons keep away from each other due to the Pauli exclusion principle. This reduces the correlation between them, and provides a basis for the validity of the independent-particle model. The question arises as to the mechanisms that account for the validity of the SCF approximation in the vibrational case, which are obviously quite unrelated to the Pauli principle. 

Electron correlation in the triplet is a consequence of the Pauli exclusion principle, which is embodied mathematically in the antisymmetrization of the electronic wave function. Anti-symmetrization results in zero probability for two electrons of the same spin simultaneously being in the same AO in an LCAO-MO wave function. This obviously reduces the Coulomb repulsion between these electrons from the value, Jxy, that is calculated by computing the Coulomb repulsion energy between a pair of uncorrelated electrons, one in f/x and the other in ffy. 

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