Mean-Field Approximations for Spin-Orbit Interaction

Mean-Field Approximations for Spin-Orbit Interaction 

The two-electron spin-orbit term can amount to 20 to 50% of the total spin-orbit splitting, and has the opposite sign to the one-electron contribution. This two-electron interaction should therefore be included in any calculation that aims at a quantitative description of spin-orbit effects. However, the number of two-electron spin-orbit integrals is larger than the number of two-electron Coulomb integrals by almost an order of magnitude. It would therefore be useful if we could find an approximation in which we only needed to evaluate a set of one-electron spin-orbit integrals. 

The spin-orbit operators for the model potential and pseudopotential approximations are one-electron operators. These operators include the effect of the two-electron spin-orbit interaction used in the mean-field approximation to derive the model potential or pseudopotential. Molecular calculations with these potentials therefore include, at least at the atomic level, the two-electron spin-orbit terms. This is just the kind of approximation we are looking for. 

It would be very convenient if we could do the same for all-electron methods. One simple approach would be to just scale the nuclear charge of the one-eleetron term such that 

An alternative approach is based on the application of these spin-orbit terms in a perturbation theory (PT) or configuration interaction (Cl) step following a scalar mean-field calculation. For such an application we will need to evaluate matrix elements over wave functions, expressed as linear combinations of Slater determinants. To do this, we apply the usual Slater-Condon rules. We start by looking at the one-electron term, where we may examine the elements of the operator 

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