Spin-orbit operator/term

While the Hamiltonian operator Hq for the hydrogen atom in the absence of the spin-orbit coupling term commutes with L and with S, the total Hamiltonian operator H in equation (7.33) does not commute with either L or S because of the presence of the scalar product L S. To illustrate this feature, we consider the commutators [L, L S] and [S, L S],... 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

The expression for the contribution to the spin-orbit induced MCD intensity from perturbation of the ground state is somewhat reminiscent of an expression for the Ag quantity of EPR spectroscopy. The similarity lies in the paramagnetic term, Agp. This term is composed of integrals of a spin-orbit operator over molecular orbitals similar to the expression for the perturbation of the ground state in the presence of spin-orbit coupling (Eqs. 52-56). The paramagnetic contribution to Ag dominates for blue copper proteins and it was suspected that the MCD parameters and Amay have some sort of relationship. It was found that many of the terms that make large contributions to AgP do play a role in the MCD intensity but no simple relationship was found (160). 

It comprises the non-relativistic Hamiltonian of the form pf/2me + V and the relativistic correction terms, such as the mass-velocity operator —pf/8m c2, the Darwin term proportional to Pi E and the spin-orbit coupling term proportional... 

The spin-orbit operator LS given in Eq. (67) is expressed in terms of the individual electron-orbital and spin-momentum operators rather than the total momentum operators L and S. It can be shown (/, 5) that when evaluating integrals involving only LS functions of the same configuration, ls can be replaced by... 

For example, the singlet-triplet transitions in ethylenic compounds generally have tmax <3C 1. The fact that spin-forbidden transitions can be observed at all shows that the transition moment, f electric dipole operator. This operator also contains small terms such as quadrupole operators and spin-orbit operators. The latter is the part of any dynamical operator which couples orbital and spin angular moments this term is responsible for the appearance of weak triplet — singlet absorption spectra. 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

The spin-orbit does not interact for the A-term or E-term manifolds as these kets do not involve the angular momentum (in the cubic groups). Consequently all the g-limes degenerate energy levels within the model space possess zero matrix elements of the spin-orbit operator. 

Restricting our discussion to the subspace spanned by the terms 6Aig and 4 Tig, the matrix element of the spin-orbit operator have been evaluated by Weissbluth [59] using the formalism pioneered by Griffith [56] and ending at the eigenvalue problem of the 18 x 18 dimension (which is partly factored— Table 34). Then the second-order perturbation theory yields the energies of the lowest multiplets as... 

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

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... 

Semiempirical spin-orbit operators play an important role in all-electron and in REP calculations based on Co wen- Griffin pseudoorbitals. These operators are based on rather severe approximations, but have been shown to give good results in many cases. An alternative is to employ the complete microscopic Breit-Pauli spin-orbit operator, which adds considerably to the complexity of the problem because of the necessity to include two-electron terms. However, it is also inappropriate in heavy-element molecules unless used in the presence of mass-velocity and Darwin terms. 

We are now in a position to express the Dirac Hamiltonian in terms of radial operators and a four-component spin—orbital operator K defined by... 

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