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

(eq. 55) |

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}*,

(eq. 56) |

No changes are required in either the * a_{i}* or the

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

(eq. 57) |

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
*C*_{3} and *C*_{3}^{2} 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 *C*_{3}(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.