Dirac-Fock equations construction

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

An alternative to the operator approach is to start from the matrix equations (Filatov 2002). Then the elimination the small-component, the construction of the transformation and the transformed Fock matrix are all straightforward. There is no difficulty with interpretation because the inverse of a matrix is well defined. The matrix to be inverted is positive definite so it presents no numerical problems. The drawback of a matrix method is that the basis set for the small component must be used, at least to construct the potentials that appear in the inverse. In that case, the same number of integrals is required as in the full Dirac-Hartree-Fock method, and there is no reduction in the integral work or the construction of the Fock matrix. 

The answer lies in the interative way in which these ostensibly many body equations are solved. One really solves the one electron Dirac equation for each individual electron moving in a field produced by the electron distribution found in the previous iteration. The solutions to the one electron Dirac equation are, of course, well behaved, and only the positive energy solutions are kept in preparing the next iteration. When convergence is finally obtained, one has, in effect, solved the many body equation using projection operators constructed with the solutions of the many body equation. Not only does this intuitively seem to be a good way to define the A > but Mittleman has shown that using projection operators obtained from a Hartree-Fock... 

---