Representation, degenerate regular

Figure 6. Formfactor K(r) and nearest neighbour spacing distribution P(s) for (a) the pi s form the regular representation of the cyclic group Z24 (b) pi s represent the symmetric group. S, (c) a random set of pi s without symmetries. The dashed curve in (b) labeled red. Poisson corresponds to a distribution of degenerate levels being Poisson distributed otherwise.

Exercise 12.6-1 D3h= 2C3 3C2 crh 2S3 3crv, where crh = /C2, aV = IC2", C2-L(C2, C2")- Thus the improper binary axis C2 is normal to the 3C2 proper binary axes and the three improper C2" binary axes. There are, therefore, three irregular classes crh, 3 C2, and 3oy. There are six classes in all and therefore six vector representations (Nv = 6). There are three regular classes and therefore three spinor representations, each of which is doubly degenerate since J]3=1 2 = 22 + 22 + 22 = 12 = g. 

Fig. 2 Schematic representation of d-state orbitals (a) i) regular octahedron T2g state is triply degenerate, Eg state is doubly degenerate, and ii) tetragonal z-axis distortion each of these has dO occupancy, consistent with the ionic limit, (b), (c) and (d) represent respectively high-spin d, d, and a low-spin d ...

The out-of-plane distortion of a five-membered ring can always be described in terms of only two coordinates, since, of the nine (3x5-6) independent coordinates, seven (2x5-3) may be chosen in an arbitrary plane, the mean plane of the ring. These two coordinates can be chosen as a pair of symmetry displacement coordinates of a regular pentagon that transform together as the doubly degenerate representation of Ds. ... 

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