Physical Chemistry Third Edition

22.4 The Rotation and Vibration of Polyatomic Molecules 937

symmetry number of BF 3 is 6 (three positions with one side of the molecule upward,

and three more positions with the other side up). For this molecule, only one-sixth

of the conceivable sets of values ofJ,K, andMcan occur. The symmetry number

of methane is 12 (three orientations with each of the four hydrogens upward). Only

one-twelfth of the conceivable sets of values ofJ,M, andKcan occur for methane.

Exercise 22.8

Find the symmetry numbers of the molecules in their equilibrium conformations:

a.Chloroform, CHCl 3

b.Water, H 2 O

c.Benzene, C 6 H 6

Vibrations of Polyatomic Molecules

In a polyatomic molecule there are several bond lengths and bond angles that can

oscillate about their equilibrium values. However, each bond length or bond angle

does not oscillate independently of the others. It turns out that there are collective

motions of some or all of the nuclei that can oscillate independently, callednormal

modes. Determining the normal modes is a complicated process, and we give only a

brief summary.^6 Some software packages such as CAChe and Spartan carry out the

process automatically. We assume that the molecule cannot rotate and is in a fixed

orientation. For the first nucleus, letq 1 be the displacement of the nucleus from its

equilibrium position in thexdirection, letq 2 be its displacement in theydirection,

andq 3 be its displacement in thezdirection. For the second nucleus, letq 4 ,q 5 , and

q 6 be similar displacements, and so on. There are 3nsuch variables if there aren

nuclei.

The Born–Oppenheimer energy acts as the vibrational potential energy. We assume

aharmonic potential energy. That is, we assume that the potential energy depends on

theqvariables in the following way:

EBOV(q 1 ,q 2 ,...,q 3 n)Ve+

1

2

∑^3 n

i 1

∑^3 n

j 1

bijqiqj (22.4-13)

where theb’s are constants. In addition to terms withij, “cross terms” occur in

whichij. The presence of the cross terms make theqcoordinates interfere with

each other so that they do not vibrate independently. Equation (22.4-13) is analogous

to Eq. (22.2-22) and will be a good approximation for small values of theq’s.

z

y

x





Figure 22.5 Three Angles to Spec-

ify the Orientation of a Nonlinear

Polyatomic Molecule.The anglesθ

andφare used to specify the orienta-

tion of one molecular axis. The angleψ

is used to specify the angle of rotation

about this axis.

We now find a transformation to a new set of coordinates such that each new coordi-

nate can oscillate independently from the other new coordinates. The potential energy

function must be given by a formula like that of Eq. (22.4-3) without cross terms.

The number of such coordinates turns out to be smaller than 3n, because the Born–

Oppenheimer energy of the molecule is independent of its location and orientation.

Three coordinates can be used to specify the location of the center of mass of the

molecule. For a linear or diatomic molecule, two angular coordinates specify the orien-

tation of the molecule. For a nonlinear molecule, three angular coordinates are required

to specify the orientation of the molecule, as shown in Figure 22.5. The anglesθand

φspecify the direction of an axis (as in spherical polar coordinates) and the angle

ψspecifies the angle of rotation about this axis. The Born–Oppenheimer energy can

(^6) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross,Molecular Vibrations, McGraw-Hill, New York, 1955.