Effective spin-orbit operator

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

An alternative form of the effective spin-orbit operator is obtained by expressing (2.2.39) in terms of the shift operators... 

The second-quantization representation of the effective spin-orbit operator now becomes... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

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... 

Teichteil et al.41 fit a spin-orbit pseudo-operator such that its action on a pseudo-orbital optimally reproduces the effect of the true spin-orbit operator on the corresponding all-electron orbital. Ermler, Ross, Christiansen, and co-workers42 9 and Titov and Mosyagin50 define a spin-orbit operator as the difference between the and j dependent relativistic effective pseudopotentials (REPs)51... 

One of the most rigorous approaches toward the definition of an effective one-electron spin-orbit operator appears to be the recently proposed... 

For compounds containing heavy atoms, spin-orbit and electron correlation energies are approximately of the same size, and one cannot expect these effects to be independent of each other. A variational approach that treats both interactions at the same level is then preferable to a perturbation expansion. Special care is advisable in the choice of the spin-orbit operator in this case. The variational determination of spin-orbit coupling requires a spin-orbit... 

Effective Potentials with Spin-Orbit Operators. I. Li through Ar. 

J. Chem. Phys., 93, 6654 (1990). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. IV. Cs through Rn. 

Relativistic Effective Potentials with Spin—Orbit Operators. VII. Am through Element 118. 

When compared with the diagonal element of the spin orbit operator in the effective Hamiltonian, AAE in a Hund s case (a) representation, we see that... 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

An accurate procedure for performing calculations that incorporate spin-orbit and other relativistic effects, and that represents intermediate coupling states for molecules containing heavy atoms, is based on A-S coupling in conjunction with the use of the ab initio REP-based spin-orbit operator and extended configuration interaction. The coupling scheme is more familiar than co-co coupling to chemists and physicists. In addition, the complexity of calculations necessary to achieve reliable results is computationally tractable. 

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