Hartree-Fock theory. The independent-particle model

1 HARTREE-FOCK THEORY. THE INDEPENDENT-PARTICLE MODEL 

In Chapters 1 and 3 we introduced the idea of a molecular orbital (MO), as a 1-electron wavefunction, generally delocalized, describing an electron free to move over the whole molecule in some appropriate effective field and in Section 1.3 a simple example was used to show how such orbitals could be constructed. In this chapter we develop various forms of the self-consistent field (SCF) method, whose aim is to optimize the MOs in order to obtain a best many-electron function of given form. This given form will usually be a single determinant in other words we shall consider the representation of the wavefunction by a single term of the complete set expansion (3.1.4), choosing the MOs so as to obtain the best 1-term approximation to the exact molecular wavefunction. 

To derive (6.1.2), we start from the variational energy approximation (3.3.7a) associated with the function (6.1.1), namely, 

With the change we obtain a corresponding first-order variation of the energy 

To proceed, it is convenient to introduce two new operators. The first of these is a coulomb operator J,- associated with an electron in V l- working on an arbitrary function of the variable jci, it has the effect 

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