Four-component Dirac-Hartree-Fock equations for solids; generalization of the relativistic Hartree-Fock equations

Abstract

Relativistic Hartree-Fock equations will be presented for 1D-, 2D- and 3D-infinite systems. 4-Component spinors with Gaussian basis sets will be used and the so-called kinetic balance beween the small and large components of the spinors is introduced. The resulting somewhat complicated generalized matrix eigenvalue equation for solids is described. It is also shown for 1D- and 2D-systems how the MP2 and MP2- r 12 methods could be applied in their relativistic form. With the help of them, on the one hand, the total energy per unit cell (including correlation effects) can be computed. On the other hand, applying the inverse Dyson equation, the relativistic band structure can also be corrected for correlation. Furthermore, it is argued that to obtain more reliable one-electron functions and energies for the correlation calculations, one has to include into the relativistic Hartree-Fock equations the frequency-dependent Breit operator, the first-order Lamb shift terms (the difference of the self energy and mass correction terms) and the vacuum polarization term in the Uehling potential approximation (all the other quantum electrodynamical effects are order(s) of magnitude smaller). Including all these terms into the expression of the relativistic total energy, the corresponding generalized Dirac-Hartree-Fock equations obtained after performing the variational calculations are presented. Finally a scheme is proposed in which most of the electrons are treated in the standard way (Dirac-Hartree-Fock equations with only Coulomb interactions and calculation of all the other terms with the aid of first-order perturbation theory), while for the core electrons of large Z atoms or ions, the generalized relativistic HF equations are used. Therefore, one applies the solutions of the generalized relativistic HF equations for the construction of the relativistic Slater determinant in the case of core electrons, while for the rest of the electrons the one-electron functions are obtained from the standard Dirac-Hartree-Fock equations.