Spin-orbit coupling numerical calculations

As described in Sections II.A. 1 and II.A.4, the numerical procedures required to calculate term parameters and C terms induced by the perturbation of the transition moment by spin-orbit coupling are nearly identical. Almost all of the comments concerning the calculation of terms made in Section III.A.1 apply equally well to the spin-orbit-induced C terms. The major difference between the two types of calculation is that the spin-orbit interaction is often significantly larger than the influence of a magnetic field. 

The correlation diagram that correlates the intermediate- (or medium-) field states with the weak-field states is shown in Figure 8.1. The same states must arise independently of the order in which the crystal-field and spin-orbit coupling perturbations are applied. The numbers in parentheses are the degeneracies of the states they provide a useftd check on the accuracy of numerical calculations. 

Numerical calculations of (Sz) at 300 K obtained with eq. (22) for all /((III) free ions are collected in table 3 (Golding and Halton, 1972). Pinkerton et al. (1985) have demonstrated that (Sz) are relatively insensitive to the choice of various sets of reported spin-orbit coupling constants. Moreover, except for R = Sm and Eu, (Sz) displays a minor dependence on the temperature and the data calculated at 300 K (table 3) are amenable for a reliable treatment of contact shifts in solution around room temperature. The close proximity of excited states possessing different J manifolds for Sm(III) requires a precise calculation of (Sz) at each temperature as is the case for R = Eu. However, the latter metal brings some specific complications associated with the absence of a well-defined gj value for its 7 = 0 ground state. Golding... 

Attention may now therefore be turned away from the manner in which numerical values of the splitting parameters may be calculated and directed instead towards the application of the ligand field model to systems of pseudo-axial symmetry, and the results of such a treatment, the f1 configuration being conveniently treated as a first example. In this case the single f-electron produces only a 2F free ion spectroscopic state which under the influence of spin-orbit coupling yields the two J levels, 2F5/2 and 2F7/2, the former lying lower by 7/2 %. On application of a pseudo-axial,... 

Numerical calculations have been performed, so far, only for ordinary spin-orbit coupling in superconductors (Capelle etal. 1997, 1998). Figure 5.18 shows the difference A Ps in the superconducting phase divided by the corresponding quantity APN in the normal state as a function of (T/ Tc) for a simple model superconductor. The figure shows that, below the critical temperature Tc, the dichroic response is dramatically modified compared with the normal state. Without spin-orbit coupling both numerator and denominator would be zero, while without superconducting coherence... 

Fig. 5.5 Calculated values of the reduction factor F for different degrees of tetragonal distortion of an octahedral Fe environment. The 10 Dq tgg — eg separation = 10,000 cm and the spin-orbit coupling / = — 80 cm. The numerical figures refer to a convenient splitting factor Ds derived from the operator form of tv = Dsifs — 2), splitting the /i levels by 3Z>5. The low values of Fat low temperatures and small distortions are caused by the spin-orbit coupling. [Ref. 30, Fig. 1]...

Numerical ab initio calculations for selected examples with polarized basis sets and Cl of reasonably size confirmed that the size of the matrix elements within the active space matrix is negligible. In contrast, the elements of that involve both the active and inner shells are large, since is primeuily due to the shielding of nuclei by inner-shell electrons [11]. It is therefore common practice in many semiquantitative applications, to account for the effect of the fixed-core electrons by replacing the factor gPgZ r in by the empirical value of the atomic spin-orbit coupling constant valence p orbitals on... 

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

It is evident that, from the comparison of calculated curves with experimental values of the effective magnetic moment as a function of temperature, the magnitude of certain parameters may be obtained which characterize the lowest electronic state such as low-symmetry splitting A, the orbital reduction factor k and the spin-orbit coupling parameter A. A considerable activity has been noted in this particular area within the years covered by this volume and, for numerous transition metal complexes, the parameters A, k and A have been determined from magnetic data. In the tables, these results have been refered to where applicable and the detailed values of the parameters were listed. 

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