Roothaan Hall matrix method

The fact that the Roothaan-Hall equations Eqs. 5.56 are actually a total of mzm equations suggests that they might be expressible as a single matrix equation, since the single matrix equation AB = 0, where A and B are m x m matrices, represents m x m simple equations, one for each element of the product matrix AB (work it out for two 2x2 matrices). A single matrix equation would be easier to work with than m1 equations and might allow us to invoke matrix diagonalization as in the case of the simple and extended Hiickel methods (Sections 4.3.4 and 4.4.1). To subsume the sets of equations 5.54-1-5.54-m, i.e. Eqs. 5.56, into one matrix... 

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). 

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

Usually, the parent molecules M are confined to some limited size that allows rapid determination of the parent molecule density matrices within a conventional ab initio Hartree-Fock-Roothaan-Hall scheme, followed by the determination of the fragment density matrices and the assembly of the macro-molecular density matrix using the method described above. The entire iterative procedure depends linearly on the number of fragments, that is, on the size of the target macromolecule M. When compared to the conventional ab initio type methods of computer time requirements growing with the third or fourth power of the number of electrons, the linear scaling property of the ADMA method is advantageous. 

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. 

Hall and, independently, Roothaan showed that by introducing a set of known spatial functions the integro-differential equation can be transformed into a set of algebraic equations and thus can be solved using standard matrix methods. That is, if the molecular orbitals, ilj(r), are expressed as a linear combination of M one-electron functions known as basis functions... 

Semiempirical methods, like ab initio methods, define the elements of the Fock matrix, using the Roothaan-Hall equation, as shown in equation (I),... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

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