Breit-Pauli spin-orbit integrals

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

The use of Breit-Pauli or no-pair spin-orbit operators to compute the spin-orbit integrals implies to deal with the full nodal structure of the orbitals [2]. When AREPs are used at the SCF step the pseudoorbitals, as eigenfunctions of the valence Fock-operators, have lost their nodal structure in the core region, exactly where spin-orbit operators essentially act, making it impossible to apply such operators in pseudopotential schemes. Three solutions can be employed to evaluate the spin-orbit integrals on nodeless pseudoorbitals ... 

The presence of two-electron operators in the Breit-Pauli and similar expressions for makes their use computationally quite demanding, because such operators have nonvanishing matrix elements even between Slater determinants that differ in two spin-orbital occupancies and because there are many two-electron integrals. This is especially true in studies of photochemical reaction paths, where information about spin-orbit coupling is needed at many geometries. Several simplifications have been quite popular. 

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