NIST: Methane Symmetry Operations

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

(eq. 16)

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

(eq. 17)

is satisfied.

New Eulerian angles are chosen such that

(eq. 18)

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

(eq. 19)

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.