The Hamiltonian of a molecule in free space is by hypothesis invariant with respect to the interchange of identical particles and to the laboratory-fixed inversion operation. Since all operations of the molecular symmetry group as defined above are equivalent to some permutation-inversion operation, the Hamiltonian is invariant under all symmetry-group operations and is therefore of species A_{1}.
The Hamiltonian is often written as a sum of products of vibrational, rotational and nuclear spin operators. The requirement that it be of species A_{1} 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 A_{1} 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 v_{2} fundamental band [16], pure rotational transitions in the ground state [33-38], and transitions between states of different total nuclear spin [23, 36, 39].
The laboratory-fixed components µ_{X}, µ_{Y}, µ_{Z} of the electric dipole moment operator are by definition sums over all particles i in the molecule of products of the charge q_{i} and laboratory-fixed position coordinates X_{i }, Y_{i }, Z_{i }, i.e., Σ_{i }q_{i }X_{i }, Σ_{i }q_{i }Y_{i }, Σ_{i }q_{i }Z_{i }. These sums are clearly invariant to a permutation of the coordinates of identical particles, but change sign under the laboratory-fixed inversion operation. From the characters for T_{d} given in Table 1, and the fact that permutation-inversions correspond to point-group sense-reversing operations, we see that each of the laboratory-fixed components of the electric dipole moment is of species A_{2}.
Symmetry species of the molecule-fixed components µ_{x }, µ_{y }, µ_{z} 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
$$ \left[ \begin{array}{c} \mu_x\\ \mu_y\\ \mu_z \end{array}\right] = \left[ \begin{array}{c} ~\\ S(\chi\theta\phi) \\ ~ \end{array}\right] \left[\begin{array}{c} \mu_X \\ \mu_Y \\ \mu_Z \end{array}\right] ~ .$$
(eq. 35)
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 µ_{x }, µ_{y }, µ_{z} are of species F_{2x }, F_{2y }, F_{2z }, respectively.
The laboratory-fixed components J_{X }, J_{Y }, J_{Z } of the total angular momentum operator can be written as the sums Σ_{i}(Y_{i} P_{Zi} − Z_{i} P_{Yi}), Σ_{i}(Z_{i} P_{Xi} − X_{i} P_{Zi}), and Σ_{i}(X_{i} P_{Yi} − Y_{i} P_{Xi}) over all particles. These operators are invariant to permutations of identical particles. They are also clearly invariant to a laboratory-fixed inversion. Consequently, each of the three operators J_{X }, J_{Y }, J_{Z } is of species A_{1}.
The alert reader will recall that explicit expressions [4] for J_{X }, J_{Y }, J_{Z } 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 J_{X }, J_{Y }, J_{Z } as defined in (eq. 25) are all invariant with respect to the Eulerian angle transformations of the full T_{d} point group. The demonstration for operations in the D_{2d} subgroup of Table 5, of course, is quite simple.
Arguments using the analog of (eq. 35) indicate that the molecule-fixed components J_{x }, J_{y }, J_{z } of the total angular momentum operator transform according to (eq. 5), i.e., they belong to the symmetry species F_{1x }, F_{1y }, F_{1z }, respectively.
By arguments similar to those above, the laboratory-fixed components I_{X }, I_{Y }, I_{Z } of the total proton spin operator belong to the species A_{1}, and the molecule-fixed components I_{x }, I_{y }, I_{z } belong to the species F_{1x }, F_{1y }, F_{1z}.
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 [17] and molecule-fixed components of these operators are given in Table 18 and Table 19.
Species | I_{1X} | I_{2X} | I_{3X} | I_{4X} | I_{1Y} | I_{2Y} | I_{3Y} | I_{4Y} | I_{1Z} | I_{2Z} | I_{3Z} | I_{4Z} |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{1} | + | + | + | + | ||||||||
A_{1} | + | + | + | + | ||||||||
A_{1} | + | + | + | + | ||||||||
F_{2x} | + | − | + | − | ||||||||
F_{2y} | − | + | + | − | ||||||||
F_{2z} | + | + | − | − | ||||||||
F_{2x} | + | − | + | − | ||||||||
F_{2y} | − | + | + | − | ||||||||
F_{2z} | + | + | − | − | ||||||||
F_{2x} | + | − | + | − | ||||||||
F_{2y} | − | + | + | − | ||||||||
F_{2z} | + | + | − | − |
Species | I_{1x} | I_{2x} | I_{3x} | I_{4x} | I_{1y} | I_{2y} | I_{3y} | I_{4y} | I_{1z} | I_{2z} | I_{3z} | I_{4z} |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{2} | +1 | −1 | +1 | −1 | −1 | +1 | +1 | −1 | +1 | +1 | −1 | −1 |
E_{a} | +1 | −1 | +1 | −1 | +1 | −1 | −1 | +1 | ||||
E_{b} | +1 | −1 | +1 | −1 | −1 | +1 | +1 | −1 | −2 | -2 | +2 | +2 |
F_{1x} | +1 | +1 | +1 | +1 | ||||||||
F_{1y} | +1 | +1 | +1 | +1 | ||||||||
F_{1z} | +1 | +1 | +1 | +1 | ||||||||
F_{1x} | +1 | +1 | −1 | −1 | −1 | +1 | +1 | −1 | ||||
F_{1y} | +1 | +1 | −1 | −1 | +1 | −1 | +1 | −1 | ||||
F_{1z} | −1 | +1 | +1 | −1 | +1 | −1 | +1 | −1 | ||||
F_{2x} | +1 | +1 | −1 | −1 | +1 | −1 | −1 | +1 | ||||
F_{2y} | −1 | −1 | +1 | +1 | +1 | −1 | +1 | −1 | ||||
F_{2z} | −1 | +1 | +1 | −1 | −1 | +1 | −1 | +1 |
$$ \left[ \begin{array}{c} L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] = \left[ \begin{array}{c} Q_{sy} P_{sz} - Q_{sz} P_{sy}\\ Q_{sz} P_{sx} - Q_{sx} P_{sz}\\ Q_{sx} P_{sy} - Q_{sy} P_{sx}\end{array}\right] ~ , $$
(eq. 36)
if the proportionality constant ζ_{s} is suppressed. The quantity [2] ζ_{s}, 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 Q_{sx }, Q_{sy }, Q_{sz} and conjugate linear momenta P_{sx }, P_{sy }, P_{sz} for s = 3, 4 belong to the symmetry species F_{2x }, F_{2y }, F_{2z }, it can be shown fairly easily that the components L_{sx }, L_{sy }, L_{sz} of the vibrational angular momentum belong to the symmetry species F_{1x }, F_{1y }, F_{1z }.
Laboratory-fixed components of the vibrational angular momentum are normally not considered.