Methane Symmetry Operations - Proper rotations

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

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

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.

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

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 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.