Relativistic Effects in Pseudopotentials

So far we have made no mention of relativistic effects, apart from a development in terms of pseudospinors rather than pseudoorbitals. Since the intended use of most pseudopotentials is in code based on a nonrelativistic formulation of quantum chemistry, the separation of the spin-free and spin-orbit terms is essential. Two approaches are again possible. 

What we must not do is to add an all-electron spin-orbit operator, such as the Breit-Pauli or Douglas-Kroll operator, because the effect of the core is not included in these operators, nor is the removal of the core tail. The all-electron spin-orbit operators behave as 1/r, and since the pseudospinors have minimal core amplitude, the spin-orbit effect will be grossly underestimated. 

The other approach is to use a spin-dependent equation, such as the Dirac-Hartree-Fock or Wood-Boring equation, to obtain spin-dependent pseudopotentials, and take the appropriate averages and differences to obtain a spin-free relativistic pseudopotential and a spin-orbit pseudopotential. The formalism for the latter approach is as follows. 

A pseudopotential obtained from a spin-dependent equation may be written as an expansion in I and j. 

Here we are only concerned with the angular expansion, so we have dropped the local potential U r). The angular spinors can be expanded into orbital and spin parts  

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