Calculation of Second-Order Spin-Orbit Effects

2 Calculation of Second-Order Spin-Orbit Effects 

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). 

Configuration Order of states0 State Direct Hss Isoconfigurational second-order Hso SS xalc. AlS,c.(iso) eff /vexD. 

The 7r2 configuration gives rise to three states, 3E (the lowest state), 1 A, and 1S+ states (see Section 3.2.3). Spin-orbit interaction is possible only between the 3Eq and basis functions that have the same value of fi and the same e/f symmetry (both are e). The other selection rules, AS = 1 and E+ E, are also satisfied. Since AA = AE = 0, only the l2jS2j part of the spin-orbit operator is relevant and, as has been shown in Eq. (3.4.16), 

For valence states of homonuclear molecules, the it orbital is the antibonding counterpart, ng, of the bonding nu orbital. One then finds that aWu a g (aw aw ). Thus the interaction between the 3E+ and 3E states will dominate the Xs ° values for the 7t37t configuration and is given in Table 3.8. For example, in the P2 molecule, it has been observed that A (a3E+) = —3.26 cm-1 and A (b3E ) = +3.20 cm-1 (Brion, et al., 1974 Brion and Malicet, 1976). For this heavy (third-row) molecule, the direct spin-spin parameter is negligible. 

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