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 |
+  π + χ |
+ ½ π + χ |
π + χ |
− χ |
π − χ |
+ 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 |
