Algebraic Hartree-Fock finite basis expansions

4 Algebraic Hartree-Fock finite basis expansions 

This is the stationary condition for the energy functional, when orbital functions are expanded in the specified basis set. RHF equations are related to UHF equations exactly as they are in the theory based on integrodifferential OEL equations. 

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Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree-Fock equations - the ubiquitous independent particle model - in their matrix form. The Hartree-Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. 

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