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].

Symmetry species of the molecule-fixed components *µ _{x }*,

(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 }*,

The alert reader will recall that explicit expressions
[4] for *J _{X }*,

Arguments using the analog of (eq. 35) indicate that
the molecule-fixed components *J _{x }*,

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 |

(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 }*,

Laboratory-fixed components of the vibrational angular momentum are normally not considered.