Spin double group

As is outlined in Appendix A, the U can straightforwardly be determined, making use of the determinantal form (10) of Hso- In particular, an associated unitary 2x2 matrix can be found for each of the 12 elements of D3. The resulting group of order 24, the so-called spin double group is the symmetry group of the SO operator (10). 

Using the methods discussed in the tutorial (Sect. 2), the group of symmetry operations of Hso can be constructed explicitly. As is shown in Appendix B, the symmetry group of Hso of (53) is T, the spin double group of the tetrahedral rotation group, is of order 48. T also is the symmetry group of the total Hamiltonian H = Hes + Hso-... 

In addition to the cylindrical symmetry, the Hamiltonian is time-reversal invariant, that is, it commutes with the time-reversal operator t of (15). The symmetry group of //so is the spin double group of Cooi . 

Table 1 The symmetry operations of the spin double group Abbreviation e =...

Species of Spin Functions for Some Important Double Groups... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

For the calculation of the g values only the latter second-order term and the first order spin term, which contain j3H, are significant, and in the former either OMg and nMg, or n Mg and 0 Ms must belong to the same double group irreducible representation for the contribution to be non-vanishing. 

However, if Q(Qea) does not involve nuclear spin interactions, another symmetry group for tf(Qea) may be found. We let be the point group which acts on electronic spin coordinates and which is isomorphic to sp. The double group of is denoted and is obtained from double group contains twice as many elements as The two groups sp and may be combined to give an inner subdirect product which is denoted sp < a-i and is defined to contain elements of the form... 

Selection rules arise on considering how the transition operator f(t) transforms under the double group SF El We note that if f(t) is spin free it transforms the same way under SF El ° a as it does under S SF. Selection rules follow from the Wigner-Eckart theorem, just as for the spin-free case discussed in Section III. Selection rules for operators which contain spin may also be derived on considering22 how such operators transform in S SF El... 

Electron spin may interact directly 22,67,123 with the oscillating electromagnetic field of a photon. In this case the transition operator,, /(t) of Section IV, does not commute with the symmetric group S f, so that direct transitions between pure singlet and pure triplet states are allowed. Selection rules for this type of transition may be derived, as has been done by Chiu,22 by noting that f(t) has components which transform as different representations of the double group SF 0 (see Sect. VIII). 

The correctness of the sequence of ionic states based on the CNDO ASCF calculations has been confirmed by studying the effect of spin—orbit coupling on the UPS of the rhenium analog HRe(CO)5 (87, 161, 174). To consider the effects of spin-orbit coupling it is necessary to employ the double group C . 

The involvement of the spin-orbit coupling provides a passage from the G point group to the G double group. The interaction matrix Hy =... 

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