Spin-orbit integrals

Spin-orbit integrals and spin-orbit pseudopotentials 

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  

extract a semi-empirical one-electron spin-orbit operator, 

import pre-computed all-electron atomic spin-orbit integrals, 

define a specific one-electron spin-orbit pseudopotential. 

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Single excitations and I ] differ by one pair of spin orbitals i,j. Contributions come from the one-electron spin-orbit integral and three-index two-electron integrals... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

Structure Studies of Diatomic Molecules Two-Electron Spin-Spin and Spin-Orbit Integrals. 

Table 1. spin-orbit integrals between doublet-doublet, quartet-quartet, and doublet-quartet states of CC>2+. Between parentheses there are given the schematic representations used in Figure 5. 

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 projectors ljmj) ljmj and their smns P/,/ i/2 are then reduced to combinations of usual spin-free projectors P/ onto spherical harmonics. Due to the spin projectors oc) (oc, 1)3) (j31, oc) (/3 [ and /3) (a, the computation of spin-orbit integrals is carried out for separate (oc, a) and (a,)3) spin-orbit matrices on the basis of atomic pseudo-spin-orbitals it can be shown that (/3, )3) = (oc, oc) and ()3, oc) = — (oc, )3). Of coiirse both approaches give the same results. 

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