Difference potentials, spin-orbit operators

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Interaction matrix elements between states of different symmetry involving the spin-orbit operator have also been evaluated in various instances. A typical example is the predissociation of the b Z state in Oj by the Z and f Ilg states. The interaction is very much dependent on the internuclear separation and varies from zero around 2.4 a.u. to approximately 60cm at 3.5 a.u., where the potential surface crossing occurs in the neighborhood of 3 a.u. Hence in a detailed treatment of the probability y for non-radiationless transition, the explicit form of this matrix element has to be taken into account as... 

Ross, Ermler and Christiansen have recently carried out abinitio EP calculations of spin-orbit coupling in the group III A and group VIIA atoms. The spin-orbit splittings are computed with the operator represented as the difference of / +1 and I — j effective potentials. Comparisons were also made with the all-electron Dirac-Fock results as well as the first-order perturbation calculations. These authors have shown that the first-order perturbation results could be in error by 9% in comparison to the Dirac-Fock results and that the EP spin-orbit operator yields accurate results. 

We have written the operator Fl(x) as a function of the combined space-spin coordinates X, because while the spin summations can be carried out in Jl(x) before calculating matrix elements, Kl(x) may connect spin-orbitals that are off-diagonal in the spin wavefunctions however in the special case of the density matrix p (xi, Xa) arising from a wavefunction that is a spin singlet (5 = 0) one can show that must also be diagonal. This leads to a useful simplification here since we can usually assume this property for Wlo, and it means that Vl(x) reduces to a (non-local) function of the space variable r only we can therefore consistently parameterize the matrix elements for the whole potential, (/bI Vl(x) j) without having to decompose them into different spin combinations for the Coulomb and exchange potentials. 

The action of the crystal field potential should be considered in relationship to other perturbations such as the interelectron repulsion F 6 and eventually the spin-orbit coupling operator IT0. According to these relationships, several types of crystal field are distinguished (Table 8.8). The principal difference between them is which basis set is generated prior to the application of the actual perturbation operator. 

Table 1 shows the spin-orbit splittings of the neutral mercury atom from ZORA calculations. Different potentials have been used to construct the ZORA kinetic energy operator. In the column labelled ZORA/NUC, only the nuclear potential has been used while the column ZORA/FULL reports results from fully self-consistent ZORA calculations, in which the ZORA Hamiltonian is reconstructed in each SCF iteration using the full effective potential including the Hartree and exchange-correlation term. Note that these results are identical to a model potential calculation since for a neutral atom, the model potential is just the full self-consistent effective potential. The entries in the column labelled ZORA/COUL have been calculated using only the Coulombic parts of the model potential, i. e. the nuclear and Hartree potentials. 

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