Roothaan-Hall equations illustrated

As seen in Section 3.5, the Roothaan-Hall (or Pople-Nesbet for the UHF case) equations must be solved iteratively since the Fock matrix depends on its own solutions. The procedure illustrated in Figure 3.3 involves the following steps. 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

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