Spin-orbit term

Relativistic effects are significant for the heavier metals. The method of choice is nearly always relativistically derived effective core potentials. Explicit spin-orbit terms can be included in ah initio calculations, but are seldom used because of the amount of computational effort necessary. Relativistic calculations are discussed in greater detail in Chapter 33. 

Fig. 2.3. At left, energy levels for a Woods-Saxon potential with Vo — 50 MeV, R = 1.25A1/3 fm and a = 0.524 fm, neglecting spin-orbit interaction. At right, the same with spin-orbit term included. Adapted from Krane (1987).

Interaction of an electron s spin magnetic moment with the magnetic field it experiences by virtue of its motion relative to the external electric field (spin-orbit term). 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

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. 

There are three terms which appears in the first order relativistic expression the mass-velocity tehn, the Darwin term and the spin-orbit term[12]. Out of these terms the first two are comparatively easy to calculate, while the spin-orbit interaction term is more complicated. Fortunately, the spin-orbit interaction is often not too important for chemical properties, at least for the second row transition elements. It is therefore usual to neglect it in quantum chemical calculations. 

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with... 

It is well known that two main contributions determine the overall isotropic hyperfine coupling of a given atom together with small spin-orbit terms, which are, however, negligible for organic free radicals ... 

The spin-orbit terms, including centrifugal distortion, are written in the form... 

The energy level diagram shown in figure 10.90 was drawn using the first-order contributions of these three terms. In addition, the higher-order spin-orbit term given in equation (10.187) was included in the original analysis ... 

The AREP has the advantage that it may be used in standard molecular calculations that are based on A-S coupling. The AREP may be interpreted as containing the relativistic effects included in the Dirac Hamiltonian, with the exception of spin-orbit coupling. This form is the same as that presented by Kahn et al. (33) which is based on the relativistic treatment of Cowan and Griffin (34). The Hamiltonian employed by Cowan and Griffin is based on the Pauli approximation to the Dirac Hamiltonian with the omission of the spin-orbit term. 

This method yields good qualitative and, in some cases, quantitative results. It is especially useful in cases in which Cowan-Griffin orbital-based REPs, such as in this case, are employed because ab initio spin-orbit operators are not appropriate, since spin-orbit terms are not considered in the derivation... 

Another important effect due to the spin-orbit coupling comes into play whether the upper ionic core is specifically involved or not. This is because the excitation dynamics is very sensitive not only to the ionization potential or binding energy of the active electron but also to m, the projection of the orbital angular momentum along the polarization axis. Since spin-orbit terms are not... 

Except for the efforts mentioned above, relativistic calculations of shielding evaluate the main relativistic effects using one or two component limits of the four-component formalism, quasi-relativistic approaches. These avoid the variational collapse in the calculation of the scalar relativistic terms by employing frozen cores, or effective core potentials. Some include the one-electron spin-orbit terms, and sometimes the higher order spin-orbit terms too. Others include both scalar and spin-orbit terms. Ziegler... 

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