Roothan equations

The proposed SCF-MI ab initio variational method (Gianinetti, Raimondi and Tomaghi, 1996) avoids the BSSE in an a priori fashion. The method is based on a modification of Roothan equations for closed shell systems. 

All quantum chemical calculations are based on the self-consistent field (SCF) method of Hatree and Fock (1928-1930) and the MO theory of Hund, Lennard-Jones, and Mulliken (1927-1929). A method of obtaining SCF orbitals for closed shell systems was developed independently by Roothaan and Hall in 1951. In solving the so-called Roothan equations, ab initio calculations, in contrast to semiempirical treatments, do not use experimental data other than the values of the fundamental physical constants. 

In the 1950s, the Roothan equation was too complex and computational demanding to be of practical use due to the occurrence of the many two electron repulsion integrals < rs g tu >. Thus, the number of such integrals increases as M4 with the number of atomic orbitals M. As a consequence, application of the Roothan equation was in the 1950s and 1960s characterized by a number of approximations. 

Once the wavefunction is defined, the HF Roothan equation is set in terms of the Fock matrix ( °) at infinite separation and the molecular interaction matrix (S ) and coefficient matrix C as expressed below ... 

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

The HF equations of Section 6.1 were first solved, for atoms, by a self-consistent field (S(2F) method devised by Hartree. The same approach and terminology is used in dealing with the finite-basis equations of the present section, (6.2.19) and (6.2.20) often being referred to as the Roothan form of the SCF equations . To clarify the concept of the self-consistent field, we turn to the closed-shell case (Roothan, 1951), in which spin (implicit in the development so far) is eliminated and the problem is to determine the spatial factors in the occupied spin-orbitals. 

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