Hall-Roothaan equations matrix form

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

The Roothaan-Hall equations Eq. (3.9.2) can be rewritten in matrix form ... 

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. 

If the basis set expansion for the Kohn-Sham orbitals in Equation (3.57) is substituted into the Kohn-Sham equations then it is possible to express them in a matrix form, identical in form to the Roothaan-Hall equations ... 

This leads to the matrix form of the Hartree-Fock equations, i.e. the Roothaan-Hall equations ... 

These equations have to be solved iteratively to reach self-consistency. They can be cast in matrix form yielding Roothaan-Hall-like equations (Hall 1951 Roothaan 1951). 

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. 

To use the Roothaan-Hall equations we want them in standard eigenvalue-like form so that we can diagonalize the Fock matrix F of Eq. 5.57 to get the coefficients c and the energy levels e, just as we did in connection with the extended Hiickel method (Section 4.4.1). The procedure for diagonalizing F and extracting the c s and e s and is exactly the same as that explained for the extended Hiickel method (although here the cycle is iterative, i.e. repetitive, see below) ... 

The overlap matrix. SCF-type semiempirical methods take the overlap matrix as a unit matrix, S = 1, so S vanishes from the Roothaan-Hall equations FC = SCe without the necessity of using an orthogonalizing matrix to transform these equations into standard eigenvalue form FC = Ce so that the Fock matrix can be diagonalized to give the MO coefficients and energy levels (Sections 4.4.3 and 4.4.1 Section 5.2.3.6.2). 

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