## Methane Symmetry Operations

#### 4.3 Improper rotations

Consider now a sense-reversing point-group operation, represented by the
rotation-reflection symbol *S*_{n}. (Formally,
*n* = 1 gives a planar reflection, *n* = 2 gives
the inversion, and *n* > 3 gives the higher-order
rotation-reflections.) Again [4], the vibrational
displacement vectors **d**_{i} must be replaced by the new
displacement vectors

where *N* is the 3 × 3 improper rotation matrix
*D*(*S*_{n}) associated with the operation
*S*_{n} in (eq. 1). The index
*j* is chosen such that

is satisfied.

New Eulerian angles are chosen such that

is satisfied. The negative sign in (eq. 18) has been introduced of
necessity, in order to make a solution of that equation possible. Since the
matrix *N* represents an improper rotation, with a determinant of
−1, the product of *N* and *S*(χ, θ, φ)
*cannot* be represented as another proper rotation matrix. However, the
matrix −*N* represents a proper rotation, and the product of
−*N* and *S*(χ, θ, φ) *can* be represented
as another proper rotation matrix. Formally, the matrix −*N*
corresponds to the proper rotation *i* · *S*_{n},
i.e., to the proper rotation obtained by multiplying the sense-reversing
operation *S*_{n} under consideration by the molecule-fixed
inversion operation *i*. This formal equivalence arises from the presence
of the minus sign in (eq. 18), and is true regardless of whether or not
*i* or *i* · *S*_{n} is contained in the point
group of the molecule.

**R**_{new } is set equal to −**R** for
sense-reversing 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}. Improper
rotations thus correspond to permutation-*inversion* operations, with the
permuted indices related by (eq. 17).

Figure 4 illustrates: (a) an arbitrary
instantaneous configuration of the methane molecule; (b) the
transformation of vibrational displacement vectors required for the point group
operation *S*_{4}(*x*), i.e., left-handed rotation through
90° about the *x* axis, followed by reflection in the *yz*
plane; and (c) the transformation of rotational angles required for
*S*_{4}(*x*), i.e., right-handed rotation of the
molecule-fixed axis system through 270° about the *x* axis
[*i* · *S*_{4}(*x*) =
*C*_{4}^{3}(*x*)]. The transformation of
center-of-mass coordinates is not illustrated. Nevertheless, it can be seen
that the final result corresponds to the permutation (1432)* as defined in
Section 3.

It should be stressed that even though the point group *T*_{d}
contains sense-reversing operations, none of these operations actually reverses
the sense of the CH_{4} framework (see
Fig. 4). This result, at first surprising, arises
because the sense-reversing effect of the permutation part of the operation is
counteracted by the sense-reversing effect of the inversion part of the
operation. Permutation-inversion operations do exist, of course, which reverse
the sense of the CH_{4} framework [23], but
these are not feasible, and are excluded from the permutation-inversion
molecular symmetry group and from the isomorphic point group
*T*_{d}.