Relativistic corrections spin -orbit coupling

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

The other relativistic effect entirely neglected so far is the spin-orbit coupling. For systems in nondegenerate states, the only first-order contribution to TAE comes from the fine structures in the corresponding atoms. Their effects can trivially be obtained from the observed electronic spectra, and hence the computational cost of this correction is fundamentally zero. 

For all results in this paper, spin-orbit coupling corrections have been added to open-shell calculations from a compendium given elsewhere I0) we note that this consistent treatment sometimes differs from the original methods employed by other workers, e.g., standard G3 calculations include spin-orbit contributions only for atoms. In the SAC and MCCM calculations presented here, core correlation energy and relativistic effects are not explicitly included but are implicit in the parameters (i.e., we use parameters called versions 2s and 3s in the notation of previous papers 11,16,18)). 

It comprises the non-relativistic Hamiltonian of the form pf/2me + V and the relativistic correction terms, such as the mass-velocity operator —pf/8m c2, the Darwin term proportional to Pi E and the spin-orbit coupling term proportional... 

The inclusion of relativistic corrections in the GGA thus does not resolve the problems of the GGA with the 5d transition metals, suggesting that the nonlocal contributions to Exc n beyond the first density gradient are important in these systems. In addition, the spin-orbit coupling of the valence electrons, neglected in this work, could be partially responsible for this discrepancy [25,30]. 

Various approaches can be pursued to compute spin-orbit effects. Four-component ab initio methods automatically include scalar and magnetic relativistic corrections, but they put high demands on computer resources. (For reviews on this subject, see, e.g., Refs. 18,19,81,82.) The following discussion focuses on two-component methods treating SOC either perturbationally or variationally. Most of these procedures start off with orbitals optimized for a spin-free Hamiltonian. Spin-orbit coupling is added then at a later stage. The latter approaches can be divided again into so-called one-step or two-step procedures as explained below. 

There is a further relativistic correction to the energy of the one-electron atom, which competes in magnitude and importance with the spin-orbit coupling. It can be analyzed directly using the special relativistically correct... 

With only minor modifications, the spin-independent relativistic corrections can be put into this Schrodinger equation and so be absorbed in the definition of the basis orbitals ipi the spin orbit coupling potential, however, is treated explicitly in the effective Hamiltonian. 

In Eq. (2), the wave-number runcorr of the first intense Laporte-allowed) electron transfer band is directly inserted, whereas fcorr in Eq. (1) has been corrected for spin-pairing energy (and in the case of f-group complexes, also for other effects of interelectronic repulsion and for the first-order relativistic effect usually called spin-orbit coupling ). These relatively minor corrections (usually below 8 kK) are not of great importance for the present review because they vanish for central atoms having no d-electrons in the groundstate, and hence, Xopt(M) and i< uncorr(M) coincide. 

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