Spin-orbit coupling parameters

There is a nice point as to what we mean by the experimental energy. All the calculations so far have been based on non-relativistic quantum mechanics. A measure of the importance of relativistic effects for a given atom is afforded by its spin-orbit coupling parameter. This parameter can be easily determined from spectroscopic studies, and it is certainly not zero for first-row atoms. We should strictly compare the HF limit to an experimental energy that refers to a non-relativistic molecule. This is a moot point we can neither calculate molecular energies at the HF limit, nor can we easily make measurements that allow for these relativistic effects. 

Table 5.11. Comparison of the Trans Dimer Ratio with the Ratio of the Squares of the Spin-Orbit Coupling Parameters°...

These two salts were studied by the Oxford physics group in the early days of ESR spectroscopy.8 The Cr(m) and V(n) salts were doped into diamagnetic single crystals of K3Co(CN)6 and K4Fe(CN)6, respectively. The spin-orbit coupling parameters found from the spectroscopic study are listed, together with the g-components, in Table 6.2. 

The direct dipole-dipole interaction between electron spins given in Eq. (14) can also contribute to D and E in the spin Hamiltonian. Various estimates of its contribution have shown it to be much smaller than the spin-orbit terms for transition-metal ions. For systems in which the crystal field is greatly distorted, this term can become large, however, and it is found to be the major source of D in the spin Hamiltonian of organic triplet-state molecules, where the spin-orbit terms are small as a result of the small size of the spin-orbit coupling parameter. 

In the experimental UPS of HRe(CO)5 (87, 161, 174) the splitting of the first and second bands is 0.31 eV and that between the second and third bands is 0.34 eV (Fig. 9). These relative spacings are consistent only with Scheme III, thus confirming that the e MO is less stable than the b2 MO. The spin—orbit coupling parameter, f5d(Re), of 0.25 eV deduced from the spectra compares with the range 0.26-0.34 eV estimated for the Re2 cation (136, 156). The reduction of fsd(Re) in HRe(C0)4 is consistent with the delocalization of metal 5d electrons into the 2ir MOs of the CO ligands. 

For BrRe(CO)s spin -orbit coupling effects are evident on both the first and third bands. The near equality of these splittings (0.27 eV) is between the spin—orbit coupling parameters for Re (0.25 eV) and Br (0.31 eV) and suggests that the e MOs are approximately equal mixtures of Re(5d) and Br(4p) contributions. 

Table 3.4 Values of the spin-orbit coupling parameter, X, for selected first row transition metal ions...

Pr branching ratio spin-orbit coupling parameter... 

Similar to the rare-earth trichlorides, also different ternary MYX compounds have been studied thoroughly under high pressure. The results for the pressure-induced changes of the Slater parameter F2 and the spin-orbit coupling parameter of these and other compounds are presented in table 5. Due to the difficulties with the DS model, the evaluation of the parameter shifts has been performed only in terms of the two covalency models. Assuming small changes for the free-ion parameters, the relative changes were approximated by ... 

According to table 5, in all cases except for some Eu3+ compounds, the decrease for F2 is larger than for . Thus, the SRC model plays the dominant role. In the case of some Eu3+ compounds on the contrary, the spin-orbit coupling parameter decrease more strongly than the Slater parameter, which leads to the conclusion that the CFC model is dominating here. 

Figure 5. Calculated energy levels for the PtClf ion in D h and Td symmetry. In the center, the energy levels are shown for D h coordination with the inclusion of a spin-orbit coupling parameter = 3000 cm1. The states are designated with the T notation for the D/ double group, which applies with the inclusion of spin-orbit coupling. For D4h dx2 y2 — dxy = 26,100 cm 1 dx2 y2 — dxz>yz = 30,200 cm 1 dx2 y2 — dz2 = 38,300 cm 1, F2 = 1000 cm 1, F = 65 cm 1. For the Td coordination, At = 14,000 cm 1, F2 = 1000 cm 1, and F = 65 cm 1. Observed bands for the crystal are shown at the right side...

The energy of the helium atom calculated above is the first-order energy, which differs from the true energy by an amount called the correlation energy this is a measure of the tendency of the electrons to avoid each other. The simplest improvement to the trial wave function is to allow Z in (6.29) to be a variable parameter, which we call (not to be confused with the spin-orbit coupling parameter in equation (6.20)) Z in the Hamiltonian (6.23) remains the same. The expression for the calculated energy,... 

With this approximation, the microscopic spin orbit coupling parameter is... 

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