Physical Chemistry , 1st ed.

Ib, and Ic(instead ofIx,Iy, and Iz). The definitions for the various types of non-
linear molecules are therefore

Spherical top: IaIbIc

Oblate symmetric top: IaIbIc

Prolate symmetric top: IaIbIc

Asymmetric top: IaIbIc

It also becomes convenient at this point to define the following rotational
constants:

A

2



I

2

a

 (14.11)

B

2



I

2

b

 (14.12)

C

2



I

2

c

 (14.13)

Example 14.5

Assuming standard SI units, what is the unit on the rotational constants A,

B,or C?

Solution

Using just the units for the variables, which are J s for Planck’s constant and

kg m^2 for the moment of inertia, we have



k

(

g

J

s

m

)^2

 2 

kg

s

2

m^2

J

You should satisfy yourself that the reduction of units from the second step

to the third, which is the crucial one, is indeed valid.

Wavenumbers are commonly used as units to express the positions of rota-
tional transitions. If expressions in terms of cm^1 are desired, equations
14.11–14.13 can be written as

A

2



I

2

ahc



8 

h

2 Iac

B

2



I

2

bhc



8 ^2

h

Ibc



C

2



I

2

chc



8 

h

2 Icc

where cis the speed of light in units ofcentimetersper second.
The rotational energy of the molecules can then be expressed in terms ofA,
B, and Cas well as the rotational quantum number. For spherical tops and lin-
ear molecules:

ErotBJ(J 1) (14.14)

where we have used the rotational constant Bas the only necessary constant.



kg

s

2

m^2



2

s^2



kg m^2

14.4 Rotations in Molecules 469