Semi-empirical methods Roothaan equations

HyperChem s ab initio calculations solve the Roothaan equations (59) on page 225 without any further approximation apart from the use of a specific finite basis set. Therefore, ab initio calculations are generally more accurate than semi-empirical calculations. They certainly involve a more fundamental approach to solving the Schrodinger equation than do semi-empirical methods. 

A feature common to the semi-empirical methods is that the overlap matrix, S (in Equation (2.225)), is set equal to the identity matrix I. Thus all diagonal elements of the overlap matrix are equal to 1 and all off-diagonal elements are zero. Some of the off-diagonal elements would naturally be zero due to the use of orthogonal basis sets on each atom, but in addition the elements that correspond to the overlap between two atomic orbitals on different atoms are also set to zero. The main implication of this is that the Roothaan-Hall equations are simplified FC = SCE becomes FC = CE and so is immediately in standard matrix form. It is important to note that setting S equal to the identity matrix does not mean that aU overlap integrals are set to zero in the calculation of Fock matrix elements. Indeed, it is important specifically to include some of the overlaps in even the simplest of the semi-empirical models. 

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