where M is the 3 x 3 proper rotation matrix D(Cn) associated with the operation Cn in (eq. 1). The index j is chosen for given i such that the equation
involving the equilibrium positions, is satisfied.
New Eulerian angles are chosen such that
is satisfied. It is always possible to do this, since the product of two rotations, e.g., M and S(χ , θ , φ), can always be represented as a third rotation.
Rnew is set equal to R for proper-rotation point-group operations.
Replacing di by (di)new , etc., on the right-hand side of (eq. 9), we obtain the new expression
This is consistent with a left-hand side obtained by replacing Ri by +Rj. Thus, proper rotations correspond to pure permutation operations, with the permuted indices related by equation (eq. 13).
Figure 3 illustrates: (a) an arbitrary instantaneous configuration of the methane molecule, (b) the transformation of vibrational displacement vectors required for the point group operation C3(111), and (c) the transformation of rotational angles required for C3(111). It can be seen that the complete transformation consists of a rotation of the vibrational displacement vectors through 120° in a left-handed sense about the (1,1,1) direction, followed by a rotation of the molecule-fixed axis system (containing the equilibrium positions and attached displacement vectors) through 120° in a right-handed sense about the (1,1,1) direction. The final result corresponds to the permutation (132) as defined in Section 3.