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Methane Symmetry Operations - Proper rotations

4.2   Proper rotations

Consider first a proper-rotation point-group operation Cn , 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 di, but leave the equilibrium position labels unchanged. This geometrical operation can be represented algebraically as

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

(eq. 12)

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

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

(eq. 13)

involving the equilibrium positions, is satisfied.

New Eulerian angles are chosen such that

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

(eq. 14)

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

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

(eq. 15)

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.


Created September 20, 2016, Updated June 2, 2021