## Methane Symmetry Operations

#### 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 *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.