Methane Symmetry Operations - Improper rotations

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 S₄(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 S₄(x), i.e., right-handed rotation of the molecule-fixed axis system through 270° about the x axis [i · S₄(x) = C₄³(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 CH₄ 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 CH₄ 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.