The Hamiltonian is often written as a sum of products of vibrational, rotational and nuclear spin operators. The requirement that it be of species A1 is then used to exclude mathematically possible but physically inadmissible terms. This application of group theory is particularly useful in methane, where interaction terms between vibrational, rotational and nuclear spin motions are relatively complicated to set up correctly. These interaction operators, which are not in general of species A1 with respect to transformations of only the vibrational, only the rotational or only the nuclear spin variables, give rise to many interesting physical effects, e.g., non-zero intensity in the v2 fundamental band , pure rotational transitions in the ground state [33-38], and transitions between states of different total nuclear spin [23, 36, 39].
Symmetry species of the molecule-fixed components µx , µy , µz of the electric dipole moment operator can be determined correctly by intuition. But, as in Sec. 6, it is safer to proceed algebraically, recalling that transformation from laboratory-fixed to molecule-fixed vector components takes place via the direction cosine matrix, and that transformation properties of this matrix under the molecular symmetry group operations were fixed earlier (Sec. 7.2). We have
We must now apply symmetry operations to the right side of (eq. 35), and then determine transformations for the left side consistent with the changes which occur on the right. It is not difficult to show that the left side of (eq. 35) transforms according to (eq. 1). Thus, as expected intuitively, the molecule-fixed dipole-moment components µx , µy , µz are of species F2x , F2y , F2z , respectively.
The alert reader will recall that explicit expressions  for JX , JY , JZ were given in (eq. 25) as functions only of the Eulerian angles and partial derivatives with respect to these angles, and that transformations of the Eulerian angles have already been specified. A somewhat involved mathematical argument shows that indeed JX , JY , JZ as defined in (eq. 25) are all invariant with respect to the Eulerian angle transformations of the full Td point group. The demonstration for operations in the D2d subgroup of Table 5, of course, is quite simple.
Arguments using the analog of (eq. 35) indicate that the molecule-fixed components Jx , Jy , Jz of the total angular momentum operator transform according to (eq. 5), i.e., they belong to the symmetry species F1x , F1y , F1z , respectively.
In fact, however, there are twelve proton spin operators, one three-component vector for each of the four protons. The species of twelve linearly independent combinations of both laboratory-fixed  and molecule-fixed components of these operators are given in Table 18 and Table 19.
if the proportionality constant ζs is suppressed. The quantity  ζs, by which (eq. 36) must be multiplied to obtain the true vibrational angular momentum operator, lies between −1 and +1. Its precise value depends on the geometry and force field of the molecule [40,41]. Since the vibrational coordinates Qsx , Qsy , Qsz and conjugate linear momenta Psx , Psy , Psz for s = 3, 4 belong to the symmetry species F2x , F2y , F2z , it can be shown fairly easily that the components Lsx , Lsy , Lsz of the vibrational angular momentum belong to the symmetry species F1x , F1y , F1z .
Laboratory-fixed components of the vibrational angular momentum are normally not considered.