Symmetric rotors
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص444-445
2025-12-04
71
Symmetric rotors
In symmetric rotors, two moments of inertia are equal but different from the third (as in CH3Cl, NH3, and C6H6); the unique axis of the molecule is its principal axis (or figure axis). We shall write the unique moment of inertia (that about the principal axis) as I|| and the other two as I⊥. If I|| > I⊥, the rotor is classified as oblate (like a pancake, and C6H6); if I|| < I⊥ it is classified as prolate (like a cigar, and CH3Cl). The classical expression for the energy, eqn 13.22, becomes
Again, this expression can be written in terms of J2 = Ja2 + Jb2 + Jc2:
Now we generate the quantum expression by replacing J2 by J(J + 1)$2, where J is the angular momentum quantum number. We also know from the quantum theory of angular momentum (Section 9.7b) that the component of angular momentum about any axis is restricted to the values K$, with K = 0, ±1,..., ± J. (K is the quantum number used to signify a component on the principal axis; MJ is reserved for a component on an externally defined axis.) Therefore, we also replace Ja 2 by K2h2. It follows that the rotational terms are
F(J,K) = BJ(J + 1) +(A− B)K2 J =0, 1, 2,... K=0,±1,..., ± J (13.29)
With
Equation 13.29 matches what we should expect for the dependence of the energy levels on the two distinct moments of inertia of the molecule. When K = 0, there is no component of angular momentum about the principal axis, and the energy levels depend only on I⊥ (Fig. 13.13). When K =±J, almost all the angular momentum arises from rotation around the principal axis, and the energy levels are determined largely byI||. The sign of K does not affect the energy because opposite values of K correspond to opposite senses of rotation, and the energy does not depend on the sense of rotation.
Fig. 13.13 The significance of the quantum number K. (a) When |K| is close to its maximum value, J, most of the molecular rotation is around the figure axis. (b) When K=0 the molecule has no angular momentum about its principal axis: it is undergoing end-over-end rotation.
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