Self-consistent field method averaging procedure

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

A more balanced description thus requires multiconfiguration self-consistent field (MCSCF)-based methods, where the orbitals are optimized for each particular state or optimized for a suitable average of the desired states (state-averaged MCSCF). In semiempirical methods, however, an MCSCF procedure is normally not required due to the limited flexibility of the minimal valence atomic orbital basis commonly used in these methods. Instead, a multireference Cl method including a limited number of suitably chosen configurations will be appropriate. 

In the independent-partlcle-model (IPM) originally due to Bohr [1], each particle moves under the Influence of the outer potential and the average potential of all the other particles in the system. In modem quantum theory, this model was first Implemented by Hartree 12], who solved the corresponding one-electron SchrSdlnger equation by means of an iterative numerical procedure, which was continued until there was no change in the slgniflcant figures associated with the electric fields involved so that these could be considered as self-consistent this approach was hence labelled the Self-Conslstent-Fleld (SCF) method. In order to take the Pauli exclusion principle into account. Waller and Hartree [3] approximated the total wave function for a N-electron system as a product of two determinants associated with the electrons of... 

A better approximation can be obtained by taking the average repulsion between the electrons into account when determining the orbitals, a procedure known as the Hartree-Fock approximation. If the orbital for one of the electrons were somehow known, the orbital for the second electron could be calculated in the electric field of the nucleus and the first electron, described by its orbital. This argument could just as well be used for the second electron with respect to first electron.The goal is therefore to calculate a set of self-consistent orbitals, and this can be done by iterative methods. 

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