Spin-orbit coupling corrections

Substituting the parameters, we have ab = 0.058. (The upper sign applies if the components are listed in the order x, y, z in Table 4.3, the lower sign if the order is y, x, z.) Finally, we get b2 = 0.660, a2 = 0.005. The dz2 component is not really significant, given the accuracy of the data and the theory, i.e., most of the departure from axial symmetry can be explained by the spin-orbit coupling correction. 

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)). 

Aev is a core-valence correction obtained as the difference between ae-CCSD(T)/cc-pCVQZ and fc-CCSD(T)/cc-pCVQZ energies. Azpve is the harmonic zero-point vibrational correction obtained at the ae-CCSD(T)/cc-pCVTZ level, AAnh. is the correction due to anharmonic effects, calculated at the fc-MP2/cc-pVDZ level. Amvd is the correction for scalar-relativistic effects (one electron Darwin and mass-velocity terms) obtained at the ae-CCSD(T)/cc-pCVTZ level [101, 102], Aso is a spin-orbit coupling correction, which may be non-zero only for open-shell species. For the C, O and F atoms, Aso amounts to —0.35599, —0.93278 and —1.61153 kJ/mol, respectively [103]. The remaining contributions take care of the correction to the full triple excitations and perturbative treatment of quadruples Ax = ccsdt/cc-pvtz - ccsd(t)/cc-pvtz, A(q) = E CCSDT(Q)/cc-pVDZ—-E ccsDT/cc-pVDz- The final atomization energies are obtained by adding all the incremental contributions... 

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,... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

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. 

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