Spin-orbit contributions

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

Although it is unfortunate that spin-orbit coupling and the electron dipolar interaction give fine structure terms of the same form, it is possible to separate the effects. Since the spin-orbit contribution to D is related to the g-tensor ... 

For vanadocene second-order perturbation calculations of the spin-orbit contribution to the zero field splitting, DLS, showed the II ( 2) level to lie below A ( - ), but the estimated value, 1.4 to 1.5 cm-1, constitutes only about half of the observed result. How-... 

We have not as yet however treated the charge-transfer data available for complexes of the 5 d series. For these latter species though the effective spin-orbit coupling constants are often of the order of 3 kK. or more, as compared with only about 1 kK. for Ad systems, and smaller values still for the 3d elements. Consequently, as for the d—d transitions it is often necessary explicitly to consider relativistic effects in the interpretation of charge-transfer spectra, and in particular to make allowance for the changes in spin-orbit contributions which may accompany a given di transition. In fact one of us has shown (18) that these changes are... 

Table 29. Spin-orbit contributions and corrections for use in treatment of charge-transfer data...

As deduced from their magnetic moments [Con(salen)] /zeff=2.38 Mb [Con(saloph)] Meff 2.74 mb], 27 they are low spin complexes. Even if the spin-orbit contribution makes such value slightly greater than expected, it indicates an electronic configuration with a single unpaired electron (d7-dyz2dxz2dz22dxy ). 

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)). 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

Table II compares these empirical estimates with those obtained from the DSW calculation. Relativistic contributions have little effect on the spin-dipolar interactions, and both calculations are in reasonably good agreement with the empirical estimates. The spin-orbit contributions are also in moderately good agreement with the empirical estimates, showing that electron currents about the -axis are considerably more important than those about axes in the plane of the ligand. Indeed, in view of the approximations that enter into Equation 7, (, ), one might have as much confidence in the DSW result as in the empirical estimate given in the final column.

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

What is the contribution to the total energy of the 3p level of Carbon made by the two 2p orbitals alone Of course, the two Is and two 2s spin-orbitals contribute to the total energy, but we artificially ignore all such contributions in this example to simplify the problem. 

The spin-orbit contribution simply equals the sum of the relevant interactions in each shell of equivalent electrons... 

It is interesting to notice that the numerical coefficients at F2 rapidly decrease with growth of the power of this integral. For atoms with small Z values t]p electrostatic interaction. In the case of neutral atoms the spin-orbit contribution together with the mixed term exceeds the electrostatic contribution to the variance and excess of the atomic spectrum of the configuration njF only for n> 6. 

Scalar Pauli computations and ZORA calculations with spin-orbit yield quite similar results, Table III, although the latter includes spin-orbit effects whereas the former does not. However, we note from a direct inspection of the ZORA calculations that the spin-orbit contributions are small (not shown in the table). The U-F bond is mostly po- and type and should only have very little s character at the fluorine center. Hence, Fermi contact contributions are small, and spin-orbit is not relevant here, according to the qualitative discussion above. 

It follows from Table V that spin-orbit effects are relevant for the heavy metal shieldings and, since the spin-orbit contribution does not always have the same sign, for the relative chemical shifts. In this connection, it is interesting to note that the ZORA spin-orbit numbers are shifted as compared to their Pauli spin-orbit counterparts. This effect can be attributed, at least partly, to core contributions at the metal while scalar contributions of the core orbitals are approximately accounted for by the frozen core approximation (6,7), spin-orbit contributions of the core orbitals are neglected. Hence, the more positive (diamagnetic) shieldings from the ZORA method are due to spin-orbit/Fermi contact contributions of the s orbitals in the uranium core. 

Calculations of the fine-structure parameters by Veseth,258 the spin-orbit contribution to the zero-field splitting,259 and the Verdet constant for O2 have also been reported.260... 

A few other studies on these molecules have appeared. BCN was the subject of a calculation with a small GTO basis set,427 and the spin-spin interaction in the ground states of NCN, CNN, and CCO has been computed by Williams, using a STO-4G and an extended 4-31G basis set.428 Results were not in very good agreement with experiment. Although spin-orbit contributions were believed to be the main reason... 

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