Spin-orbit interaction in complexes the double group

7 Spin-orbit interaction in complexes the double group 

Previously we stated that, for a state with orbital angular momentum L, the character for a rotation of angle a is given by 

When we use this formula to derive crystal field states of a complex, we have made the assumption that spin-orbit interaction is weak and hence is ignored. 

We may now use Table 8.7.1 to determine the representations, under O symmetry, spanned by a state with half-integer quantum number J. These results are summarized in Table 8.7.2. Those for states with integral J are included for convenience and completeness. In other words, this table contains all the results listed in Table 8.4.2, in addition to those for half-integer J states. 

To illustrate how to obtain the characters for rotation operations listed in Table 8.7.2, we take x ( = 2n) for J = xh as an example  

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