Roothaan-Hall equations example

The first shortcoming of a minimal basis set.. . bias toward atoms with spherical environments. . . may be addressed by providing two sets of valence basis functions ( inner and outer functions). For example, proper linear combinations determined in the solution of the Roothaan-Hall equations allow for the fact that the p orbitals which make up a tight o bond need to be more contracted than the p orbitals which make up a looser k bond. 

In the m sets of equations 5.54-1-5.54-m each set itself contains m equations (the subscript of , for example, runs from 1 to m), for a total ofmxm equations. These equations are the Roothaan-Hall version of the Hartree-Fock equations they were obtained by substituting for the MO s j/ in the HF equations a linear combination of basis functions ( s weighted by c s). The Roothaan-Hall equations are usually written more compactly, as... 

Using the Roothaan-Hall Equations to do ab initio Calculations - an Example... 

Fig. 5.11 STO-1G energy versus bond length r for H-He+. The calculation for r = 0.800 A was done largely by hand (see Section Using the Roothaan-Hall Equations to do Ab initio Calculations - an Example ) the others were done with the program Gaussian 92 [29]...

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). 

The Hall-Roothaan equations for the case of the helium ground state appear, when equation 5.39 is written out in an appropriate linear combination of functions upon which the variation principle procedure can be applied to return the best energy in a calculation. For example, for the double-zeta basis of Slater functions used in the previous section, we have... 

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