(eq. 50) |

for *s* = 3 or 4. Molecule-fixed components of this equation are
normally considered. It can be seen from
(eq. 23) and
(eq. 39) that all molecule-fixed
components of * J* commute with all molecule-fixed components of

The rotational Hamiltonian for a spherical top molecule with zero or one quantum of a triply degenerate vibration excited can, to a first approximation, be written [4] as

(eq. 51) |

corresponding qualitatively to the fact that the rotational energy of a
spherical top is given by the product of a rotational constant *B*, which
is inversely proportional to the moment-of-inertia of the molecule, and the
square of the purely rotational angular momentum
[* J* + ζ

(eq. 52) |

When *υ* = 0,
= 0 and
*R* = *J*. When *υ** _{s}* = 1,

If laboratory-fixed components of the electric dipole moment operator are
expanded as a power series in the vibrational coordinates *Q _{s}*,
and if only terms linear in these coordinates for one value of

(eq. 53) |

where *s* = 3 or 4, and where
(∂*µ*/∂*Q _{s}*)
represents any one of the three dipole derivatives
(∂

(eq. 54) |

These selection rules are valid for fundamental bands, but not for overtones and combination bands [59], whose intensity is governed by higher-order terms omitted in (eq. 53).

In Fig. 5, transitions indicated by solid lines
obey the selection rule Δ*R* = 0 and are strong. The five
*P*^{o}(7) transitions and the one *P*^{−}(7)
transition, indicated by dashed lines, violate this rule and are weaker. The
remaining weak vibration-rotation transition in Fig. 5, indicated by the
dashed line among the *P*^{+}(7) transitions, does not violate the
Δ*R* = 0 selection rule. It does, however, violate a
selection rule requiring no change in the numerical counter superscript
[14,15]. This latter "selection rule"
depends on the relative values of the various interactions giving rise to the
splittings in the ground and first excited vibrational state [14,15], but has
been found to hold rather well for the infrared-active vibrational fundamentals
in CH_{4} [43−46,
61]. It is necessary, however, to "count"
the *υ*_{3} = 1
rovibrational levels from highest energy to lowest when
*R* = *J* ± 1, and from lowest energy to highest
when *R* = *J*, to make this selection rule valid.
(Readers should be cautioned that slight variations of the numerical counting
scheme described here will be found in the methane literature, though all
schemes have in common the selection rule that the counting index does not
change for allowed transitions.)