Spin-orbit effects, second-order

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

The effect of suUur participation on the orbital g -shifts in the EPR spectra, illustrated in Pig. 20, accounts for the qualitatively different spectra observed for tyrosyl phenoxyl and Tyr-Cys phenoxyl radicals (Gerfen et al., 1996). The rhombicity of the simple tyrosyl radical EPR spectrum is a consequence of the splitting between gx and gy principal g -values. These g -shifts deviate from the free electron g--value ge = 2.00023) as a result of orbital angular momentum contributions. While a nondegenerate electronic state (such as the A" ground state for ere) contains no hrst-order unquenched orbital momentum, second-order spin-orbit mixing between close-lying a and a" functions results... 

Calculation of Second-Order Spin-Orbit Effects. 199... 

This means that fine-structure levels are expected to be equally spaced (by AA) however, second-order spin-orbit effects can distort the equidistant fine-structure pattern. 

For A > 0 states, second-order spin-orbit effects cause two types of J-independent level shifts proportional to AE (the form of the diagonal spin-orbit matrix element) or to 3E2 — S(S+1) (the form of the diagonal spin-spin matrix element). Second-order spin-orbit effects are unobservable for E and 2E states because they result in no new level splittings or changes in existing separations. For E states with S > 3, second-order spin-orbit effects are observable only as shifts similar to the form of the spin-spin interaction. See Levy (1973) for a demonstration of this fact. 

Thus, an approximate value for the unknown 3E 1E+ interaction can be obtained from an observable diagonal spin-orbit constant. As will be discussed later (Sections 3.4.4 and 5.3.3), second-order spin-orbit effects of this type contribute significantly to the effective spin-spin interaction in 3E states. 

Second-order effects arising from the product of matrix elements involving J+ L and L+ S operators have the same form as 7J+S. In the case of H2, the second-order effect seems to be smaller than the first-order effect, but in other molecules this second-order effect will be more important than the first-order contribution to the spin-rotation constant. These second-order contributions can be shown to increase in proportion with spin-orbit effects, namely roughly as Z2, but the direct spin-rotation interaction is proportional to the rotational constant. For 2n states, 7 is strongly correlated with Ap, the spin-orbit centrifugal distortion constant [see definition, Eq. (5.6.6)], and direct evaluation from experimental data is difficult. On the other hand, the main second-order contribution to 7 is often due to a neighboring 2E+ state. Table 3.7 compares calculated with deperturbed values of 7 7eff of a 2II state may be deperturbed with respect to 2E+ by... 

Second-order spin-orbit effects result in matrix elements which have the same form as the spin-spin operator. Symbolically,... 

Experimentally, this second-order spin-orbit effect is indistinguishable from the direct spin-spin interaction. 

As has already been mentioned, second-order spin-orbit effects result in matrix elements that have the same form as the spin-spin operator. Consequently, Xeff = Xss + Xso, where Xss is the direct spin-spin parameter and Xs° is the second-order spin-orbit contribution. The main second-order contributions are due to the nearest states often these are the states that belong to the same configuration as the state under consideration. Moreover, as these nearby states are in general spectroscopically well characterized, it is often relatively easy to estimate semiempirically the contribution of these nearby states to observed spin-spin constants. These contributions are called isoconfigurational second-order spin-orbit effects. Some selected examples are given in the following (see Fig. 3.15). 

Similarly to the 2S+ 4E+ interaction, a spin-spin interaction is possible between 5E and 4n states as assumed, for example, in NH by Smith, et al., (1976). For the spin-orbit operator, recall again the AS < 1 selection rule however, second-order spin-orbit effects can mix 5E and Jn states as follows ... 

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

For comparison with the usual second-order perturbation in the spin-orbit coupling, we assume that the first order calculation has taken all first-order effects into account as in Eq.(l 1). The second-order perturbation due to the interaction operator W is given by... 

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