Matrix Approximations to the Dirac Hamiltonian for Molecular Calculations

Abstract

This chapter develops several approximate methodologies for the solution of the Dirac equation in a finite basis set. The approximations are made in the matrix formalism rather than in the operator formalism. The operator equation that underlies the developments is the modified Dirac equation, which enables a spin separation similar to that found in the Breit-Pauli or the Douglas-Kroll formalism. The small component is eliminated by a non-unitary transformation that generates a nonunit metric, but one on which the large component is normalized. The transformation depends on the one-particle eigenstates. In the first approximation this dependence is frozen at the atomic level and subsumed into the contraction coefficients of the atomic basis sets. The second approximation classifies atoms as “relativistic” or “nonrelativistic”, allowing relativistic effects to be incorporated into an otherwise nonrelativistic calculation. To both of these approximations is added a further approximation for scalar relativistic effects in which the two-electron terms are treated approximately, and only the one-electron integrals need be modified. The errors in the first approximation are negligible for most chemical purposes, and the errors in the last approximation are also mostly negligible. The errors made in treating atoms nonrelativistically has a large Z dependence, but is sufficiently accurate for most chemical purposes for elements up to Ar.