#### 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

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

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

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

is satisfied.

New Eulerian angles are chosen such that

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

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

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

This is consistent with a left-hand side obtained by replacing **R**_{i} by −**R**_{j}. 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}.