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Methane Symmetry Operations

4.4   Explicit Eulerian-Angle transformations

For 16 of the 24 operations in the point group Td, the explicit transformations of the Eulerian angles take rather complicated forms, e.g., θnew = arccos(−cosχ sinθ), and are not really useful as substitution transformations. For the other eight operations, however, the explicit transformations are quite simple, as shown in Table 5. As it happens, the eight point-group operations occurring in Table 5 comprise a subgroup D2d of Td in which the z axis is the S4 axis.

Table 5. Transformations of the Eulerian angles for operations of one of the D2d subgroups of Td

  E S4(z) S43(z) C2(z) C2(x) C2(y) σd(110) σd(−110)
χnew χ 3/2 π + χ + ½ π + χ π + χ − χ π − χ 3/2 π − χ + ½ π − χ
θnew θ θ θ θ π − θ π − &theta π − θ π − θ
φnew φ φ φ φ π + φ π + φ π + φ π + φ

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