Spin orbit coupling constant

The spin of an electron gives rise to a magnetic moment which can interact with effective magnetic field set up by its orbital motion about the nucleus. 

Although this spin-orbit interaction is essentially a relativistic effect it may be approached classically. For hydrogen-like atoms the spin-orbit hamiltonian is 

When this expression is extended to many-electron systems, two related problems arise. Firstly, what is the effective spin-orbit hamiltonian for the electron in open shells Secondly, what is the potential in which they move For a hydrogen-like atom the field would be written 

Blume and Watson,97- 98 using spherical tensor methods, were able to reduce the second part of this expression to a form suitable for calculation. They derived expectation values of this operator from non-relativistic atomic Hartree-Fock wavefunctions and hence the spin-orbit coupling constant, , for many atoms and ions. Consistent results for atoms were also obtained by Hinkley using wavefunctions both from exponential basis sets100 and from gaussian basis sets.101 Agreement with experiment is good. 

Lo et a/,102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanis103 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin orbit coupling 

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Figure 5, Low-eriergy vibronic spectrum in a electronic state of a linear triatomic molecule. The parameter c determines the magnitude of splitting of adiabatic bending potential curves, is the spin-orbit coupling constant, which is assumed to be positive. The zero on the...

Self-inductance L Spin orbit coupling constant A... 

Figure A The variation with temperature and spin-orbit coupling constant, of the magnetic moments of octahedral, low-spin, d" ions, (The values of at 300 K are marked for individual ions).

Fig. 2. Temperature dependence of the HS fraction % according to the Ising model. The employed parameter values are = 150 K, Aj = Aj = 500 cm and X = — 100 em h Here, Aj and Aj are the orbital energy differences between the and levels and between the Bj and levels, respectively, X being the spin-orbit coupling constant. The model parameters A, Aj, and X determine the value of AG. The levels result from the HS iron(II) ground state in orthorhombic symmetry according to Bj -1- B2 + B. The figures on the curves specify the values of...

Here, the orbital-splitting parameters Aj = Aj = 500 cm", the spin-orbit coupling constant X = — 100 cm" and = 150 K have been assumed [35]. 

The only term which has non vanishing matrix elements with both A g and T2g terms is the arising from the electron configuration t gSg- If the spin-orbit coupling constant is denoted with the matrix elements are [126] ... 

The spin—orbit coupling constants f4f and fsd, which represent the relativistic spin—orbit interaction in the 4f and 5d shells, also determined by means of the radial wave functions Rrd of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23... 

Table 3. Calculated Slater—Condon Integrals, Spin—Orbit Coupling Constants, and Ligand Field Parameters (in cm-1) for CsMgBr, Eu2+ Considering the Ground (GC) and Excited (EC) Configurations Local Structures of the Eu2+ Impurity...

The amount of trans dimer formed upon the photolysis of acenaphthylene in heavy-atom solvents could be correlated with the square of the spin-orbit coupling constant for the heaviest atom in the solvent (see Table 5.11) ... 

From measurements of this type Thomaz and Stevens found a linear relationship for a graph of log(kJrPC,2) vs. where n is the number of halogen atoms in the molecule, is the spin-orbit coupling constant, and Em is the polarographic half-wave reduction potential of the heavy-atom quencher (Figure 5.16). This correlation suggests that an exciplex is formed by partial... 

In ruthenocene it also proved possible at 77 K to resolve the broad band containing the 12+ -> 1n, and 1Z+ -+ x4> transitions, but as for Fe(Cp)2 only one spin-forbidden d-d band, assigned as shown in Table 11, could be found, despite a spin-orbit coupling constant of the order of 1 kK. Note that although the 3I1 (on S4), 34 (o2 tr 5 3), and 3II (o2 n 6 3) levels are predicted (81) to be split by , %, and 31, respectively, abnormally large band widths would not be anticipated since only one component of each triplet level is capable of mixing with the nearby singlet states. 

Here, L is the angular momentum operator, ge is the g value for the free electron (ge = 2.0023), and X is the spin-orbit coupling constant. Considering only the z direction, this equation becomes... 

In Equation 1.5, A is the spin-orbit coupling within a given Russell-Saunders multiplet, which is related to the spin-orbit coupling constant of the ion, C, by the relation A = f/2.S, the + sign applying for n <7 and the - sign for n > 7 [ 15]. The effect of spin-orbit coupling is to split the terms in multiplets with same L,... 

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