Methane Symmetry Operations
4.4 Explicit EulerianAngle transformations
For 16 of the 24 operations in the point group T_{d}, the
explicit transformations of the Eulerian angles take rather complicated forms,
e.g., θ_{new} = 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 pointgroup operations occurring in
Table 5 comprise a subgroup D_{2d} of
T_{d} in which the z axis is the
S_{4} axis.
Table 5. Transformations of the Eulerian angles for operations of one of
the D_{2d} subgroups of T_{d}

E 
S_{4}(z) 
S_{4}^{3}(z) 
C_{2}(z) 
C_{2}(x) 
C_{2}(y) 
σ_{d}(110) 
σ_{d}(−110) 
χ_{new} 
χ 
+ π + χ 
+ ½ π + χ 
π + χ 
− χ 
π − χ 
+ π − χ 
+ ½ π − χ 
θ_{new} 
θ 
θ 
θ 
θ 
π − θ 
π − &theta 
π − θ 
π − θ 
φ_{new} 
φ 
φ 
φ 
φ 
π + φ 
π + φ 
π + φ 
π + φ 