Physical Chemistry , 1st ed.

The rotational energies of symmetric tops are further divided into prolate

and oblate symmetric tops. We first look at the prolate symmetric top, which

has IaIbIc. There is a total angular momentum quantum number Jthat

determines the quantized totalangular momentum:

total angular momentum J(J   1 )

There is an additional quantized value. Recognize that a molecule that is not

spherically symmetric can rotate three different ways. However, symmetric

tops have one unique rotation and two equivalent rotations. The unique axis

is called the figure axisfor the molecule. It turns out that the component of the

total angular momentum in the figure axis is also quantized. Since the defini-

tion of the figure axis depends on the molecule (it is molecule dependent), it

can be considered a sort of molecule-defined z-axis. (This is different from an

absolute,spatiallydefined z-axis.) The letter Kis used to indicate the quantized

figure-axis component of the total angular momentum,which follows the same

rules as z-component angular momentum:

LˆzmolKmol K0,1,2,...J (14.15)

Note that Kis boundedby J(just like mis bounded by ). Figure 14.6 shows

the differences in the definitions of the figure-axis component and the zcom-

ponent of the angular momentum. Angular momenta about both axes are

quantized in symmetric top molecules. Equations 14.9 and 14.15 are both ap-

plicable to the rotations of symmetric top molecules.

It can be shown (we omit the detailed derivation here) that the total rota-

tional energy of a prolate symmetric top is quantized.That energy, in terms

of the quantum number Jfor the total angular momentum andthe quantum

number Kfor the figure-axis component of angular momentum, has the fol-

lowing expression:

ErotBJ(J 1) (AB)K^2 (14.16)

In this case, the energy of the rotating molecule depends on twoquantum

numbers. Just as for linear molecules or the original 3-D rigid rotor, the en-

ergy is not dependent on the zcomponent of the angular momentum. It isde-

pendent on the figure-axis component. Because Ais always greater than B, the

second term in Kis always positive (K^2 is always nonnegative) and represents

an increase in the energy of the rotating molecule relative to a diatomic or

spherical top molecule.

For an oblate symmetric top, a very similar expression for the quantized en-

ergy of rotation is obtained, except that now the lower two moments of iner-

tia are equal and the largest is the unique one. But a similar derivation (again,

omitted here) leads to a similar expression:

ErotBJ(J 1) (CB)K^2 (14.17)

In this case, since Cis always less than B, the second term in equation 14.17

will always be negative, so the second term contributes to an overall decrease

in the rotational energy of the oblate molecule relative to a diatomic or spher-

ical top molecule.

Example 14.6

Ammonia has two defined inertial moments:I4.413 

10 ^47 kg m^2 and

I2.806 

10 ^47 kg m^2.

a.Label these as Ia,Ib, and Ic.

470 CHAPTER 14 Rotational and Vibrational Spectroscopy

MJ K

J

K  0

J  MJ

H

H

H

H C CCC

z

H

H

H

H C CCC

z

(a)

(b)
Figure 14.6 Jrepresents the total angular mo-
mentum of the molecule.MJis the zcomponent
ofJ. This z-axis can have any direction, but the
figure axis depends on the structure of the mole-
cule itself. (a) A rotation about the figure axis,
which means the angular momentum quantum
number Khas a large value. (b) A rotation out of
the figure axis, which means Kequals 0 even
though MJis nonzero. In both cases, the value of
Jis the same.