Hartree-Fock method Pauli exclusion principle

The Hiickel molecular orbital (HMO) model of pi electrons goes back to the early days of quantum mechanics [7], and is a standard tool of the organic chemist for predicting orbital symmetries and degeneracies, chemical reactivity, and rough energetics. It represents the ultimate uncorrelated picture of electrons in that electron-electron repulsion is not explicitly included at all, not even in an average way as in the Hartree Fock self consistent field method. As a result, each electron moves independently in a fully delocalized molecular orbital, subject only to the Pauli Exclusion Principle limitation to one electron of each spin in each molecular orbital. 

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. 

Within the Hartree method, the electronic spin does not appear explicitly except for the fact that no more than two electrons may go into a single orbital. The existence of the Pauli exclusion principle, however, needs to be accounted for in order to go beyond the Hartree method, and that is what the Hartree-Fock method [120] is all about. We first formulate an arbitrary three-dimensional orbital for electron i by writing it as the product of a purely space-dependent part and a spin function (spinor), a or characterizing spin-up or spin-down electron, for example (pi Xi) = i(ri)iXi here we use x to indicate a variable which includes both space (r) and spin (a). A Hartree-like product wave function between two one-electron wave functions and 2 could then be written as... 

---