Two-component all-electron methods for spin-orbit coupling

1 Two-component all-electron methods for spin-orbit coupling 

Since the many-electron wave function can be expanded in a linear combination of Slater determinants, its matrix element with a spin-orbit coupling operator of the form of Equation (3.6) can be expressed as a sum of matrix elements of the operator between Slater determinants. For a matrix element between Slater determinants which differ in exactly one spin orbital (i.e. which are singly excited from i — a with respect to each other), the matrix element is 

In an independent-particle model, Equation (3.7) defines a Fock operator describing valence electrons moving in a field generated by the electrons in orbitals k. Using this relationship, we define an approximate operator 

In general, a mean-field approximation is defined by any set of occupation numbers rt/t by means of a corresponding Fock operator matrix element, and the dependence of the results on the specific set of occupation numbers turned out to be very weak in practical calculations. This approximation has also been developed independently by Beming etal (2000). 

Hamiltonian resulting from the Douglas-Kroll transformation is particularly indicated for heavy elements and in variational calculations, because it is bounded from below (Samzow et al. 1992). The raw integrals are by now combined with AO and MO information from a variety of standard program packages (Molecule-Sweden, Columbus, Turbomole) and SOMF integrals are provided for BnSoc, Columbus, Molcas and LuciaRel. 

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