Methane Symmetry Operations - Laboratory-Fixed Rotations

15.   Laboratory-Fixed Rotations of CH₄ Considered as an Arbitrary Three-Dimensional Body

We now discuss briefly overall rotations of the CH₄ molecule in the laboratory, i.e., those rotations which do not arise by virtue of the symmetry of the molecule, but which are possible in free space for any three-dimensional body whatever. The distinction between overall rotations and point-group rotations is particularly important for theoretical discussions of nuclear spin interactions. Such discussions are almost always carried out using nuclear spin basis functions characterized by projections quantized along the laboratory-fixed Z axis, and require both types of symmetry operations, as illustrated in Section 16.

Let us return to (eq. 9) and consider an arbitrary rotation

$$R_{\alpha\beta\gamma}$$

applied to the laboratory-fixed coordinates of all particles. For laboratory-fixed coordinates we require the new coordinates of each particle to be related to the old coordinates by the equation

Unfortunately, it is not possible simultaneously to make the sense of a laboratory-fixed rotation the same as the sense of a point-group rotation of the displacement vectors and the sense of a point-group rotation of the equilibrium positions, since these latter two senses are opposite to each other. The convention chosen in (eq. 55) makes the sense of the laboratory-fixed rotation the same as that of the point-group equilibrium-position rotation (i.e., right-handed), and leads to a transformation equation (eq. 57) identical to equation (17.8) of Wigner [22]. Note, however, that (eq. 55) thus differs from (eq. 28).

The result indicated in (eq. 55) can be achieved on the right side of (eq. 9), by requiring the center-of-mass coordinates R_X, R_Y, R_Z to transform according to (eq. 55) also, and the matrix S(χθφ) to transform as follows [compare (eq. 29)]

No changes are required in either the a_i or the d_i.

Taking the transpose of (eq. 56), applying the unitary transformation of (eq. 30) and (eq. 31), and using the definition in (eq. 26), we find that rotational functions |kJm⟩ transform as follows under the laboratory-fixed rotations

The transformations specified by the right side of (eq. 57) consist of linear combinations of functions with different m quantum numbers but the same k quantum number, which is just the opposite of the transformations specified by (eq. 33).

Loosely speaking, the differences between (eq. 33) and (eq. 57) arise mathematically because we wish to rotate the S⁻¹(χθφ) matrix in (eq. 9) from the left when carrying out laboratory-fixed rotations, but from the right when carrying out molecular symmetry operations.

Figure 6 illustrates the rotations C₃ and C₃² about the laboratory-fixed (1,1,1) direction when the laboratory-fixed and molecule-fixed axes coincide. These particular rotations are depicted because they are similar to, but not identical with, the point-group rotation C₃(111) illustrated in Figure 3. A comparison of Figures (3c) and (6b) shows that the numbered equilibrium positions and the orientation of the molecule-fixed axis systems coincide in the two diagrams, but that the arrangement of displacement vectors does not. The differences between Figures (3c) and (6b) arise because (3c) represents simply a permutation of the original atom labels, whereas (6b) represents a rotation in the laboratory of the original molecule.

It can easily be seen that the laboratory-fixed rotations described in this section leave invariant the molecule-fixed components of the electric dipole-moment operator, the total angular momentum operator, etc., by noting that: (a) laboratory-fixed components of vector operators transform by hypothesis according to (eq. 55) under laboratory-fixed rotations, (b) the molecule-fixed and laboratory-fixed components of a vector operator are related by an equation like (eq. 35), and (c) the matrix S⁻¹(χθφ) transforms as given in (eq. 56) under laboratory-fixed rotations.