Spin-orbital operator

We shall consider only the leading part of the spin-orbit operator assumed in the phenomenological fonn... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

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 ... 

At this stage we will take the spin-orbit operator to have the general form... 

Here the components of excited state J are expressed in a representation that diagonalizes the spin-orbit operator. In general, this will be a complex representation. The principle of spectroscopic stability can again be used to express the components of Jin a representation that we denote jM. This representation is made up of space and spin parts where the spin part diagonalizes the spin operator. 

Since the angular momentum operator and the spatial part of a spin-orbit operator have the same symmetry, from a computational point of view the process of calculation of CjS0,1 or Cj 2 is almost identical to that of obtaining Aj or Bj, respectively. 

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). 

Spin-orbit operator and the part of the spin-orbit operator that acts on the spatial part of the wave function. 

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... 

An example of an operator which does not commute with SjijF is the spin-orbit operator ns0. For instance, for a two-electron atom... 

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 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. 

The scalar product (L S) occurring in the spin-orbit operator can be manipulated with the help of the escalator operators... 

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... 

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