#### 4.2 Proper rotations

Consider first a proper-rotation point-group operation *C*_{n }, since proper rotations represent a slightly easier case than sense-reversing point-group operations. Following the commonly accepted prescription [4], we must rotate the vibrational displacement vectors **d**_{i}, but leave the equilibrium position labels unchanged. This geometrical operation can be represented algebraically as

$$(\mbox{$d$}_i)_{\rm new} = M ~\mbox{$d$}_j ~ , $$

where *M* is the 3 x 3 proper rotation matrix *D*(*C*_{n}) associated with the operation *C*_{n} in (eq. 1). The index *j* is chosen for given *i* such that the equation

$$\mbox{$a$}_j = M^{-1} ~\mbox{$a$}_i ~ , $$

involving the equilibrium positions, is satisfied.

New Eulerian angles are chosen such that

$$S(\chi_{\rm new} , \theta_{\rm new} , \phi_{\rm new})= M~S(\chi , \theta , \phi) ~ , $$

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.

**R**_{new} is set equal to **R** for proper-rotation point-group operations.

Replacing **d**_{i} by (**d**_{i})_{new} , etc., on the right-hand side of (eq. 9), we obtain the new expression

$$\mbox{$R$}+S^{-1} (\chi\theta\phi) M^{-1} (\mbox{$a$}_i + M\mbox{$d$}_j) = \mbox{$R$}+S^{-1} (\chi\theta\phi) (\mbox{$a$}_j + \mbox{$d$}_j) ~ .$$

This is consistent with a left-hand side obtained by replacing **R**_{i} by +**R**_{j}. 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 *C*_{3}(111), and (c) the transformation of rotational angles required for *C*_{3}(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.