Physical Chemistry , 1st ed.

14.7 Centrifugal Distortions

Although the above sections suggest that rotational spectra are very simple,

complications arise because molecules are real systems, not ideal ones. Although

application of the ideal 3-D rigid rotor system to molecules works very well, it

does not do a perfect job of describing rotational spectroscopy.

Perhaps the biggest single nonideal factor in rotational spectroscopy is the

fact that molecules are not rigid rotors. It is assumed, for example, that the

molecules have certain specific and unchanging bond distances. This is not

the case. As molecules have larger and larger Jquantum numbers (corre-

sponding to higher and higher energies), they distort slightly but enough to

change the energy levels noticeably away from the ideal values. This effect is

called centrifugal distortion.

Consider the diatomic molecule in Figure 14.16a. As it rotates with a relatively

low Jquantum number, the chemical bond is strong enough to keep a fairly con-

stant bond length. However, as Figure 14.16b shows, at large Jthe rotating

molecule is distorted as the spinning atoms experience a sort of centrifugal force,

like we would feel on a fast-spinning merry-go-round. This force contributes to

a slight lengthening of the bond distance. Bond distances appear in the denom-

inator of the rotational constants, used to define I, the moment of inertia. This

lengthening of the bond therefore serves to lower the total energy of the rota-

tional state, so instead of rotational spectral lines being exactly 2Bapart, they be-

gin to get less than 2Bapart. The centrifugal distortion is thus observed as a

shrinking of the distances between adjacent lines in a rotational spectrum.

The effect on the rotational energy of centrifugal distortion is proportional

to [J(J 1)]^2 instead ofJ(J 1). Indeed, if the centrifugal distortion depended

on J(J 1), it would be worked into the Brotational constant! What is usu-

ally done is to mathematically fit the rotational energies to a general equation

having a form like

ErotBJ(J 1) DJJ^2 (J 1)^2 (14.25)

where DJis the centrifugal distortion constant.It usually is expressed in the

same units as B, either MHz or GHz for frequency, or cm^1 for wavenumber.

The positions of the rotational spectral lines are therefore given as

EJ+1EJErot 2 B(J 1)  4 DJ(J 1)^3 (14.26)

where as usual the quantum number Jrefers to the quantum number of the

lower-energy rotational state. Because the centrifugal distortion is less if the di-

atomic molecule bond is stiffer and therefore has a higher vibrational fre-

quency (which will be introduced in the next section), the centrifugal distor-

tion constant is often approximated by the following expression:

DJ ^4

 ̃

B

2

3

 (14.27)

where  ̃ is the wavenumber of the vibration (in cm^1 ) and Bmust also be ex-

pressed in units of cm^1.

Equation 14.26 shows that the energy difference between rotational levels

(energy differences are what spectra measure, after all) deviate from ideality by

a factor based on the third power of the rotational quantum number J.Table

14.2 lists some experimental Bvalues and DJvalues for several diatomic mol-

ecules.DJis typically very small, so effects due to centrifugal distortion are

usually noticeable only if the spectra are for rotations of small molecules (like

hydrogen), or if very high-energy rotational states are being probed.

14.7 Centrifugal Distortions 479

(a) Low^ J

(b) High^ J
Figure 14.16 (a) Normal bond length in a di-
atomic molecule. (b) As the rotational quantum
number increases, a centrifugal distortion causes
the bond to stretch. This adds some nonideal
component to the predicted rotational spectrum.

Table 14.2 Rotational constants Band

centrifugal distortion constants DJfor

selected diatomic moleculesa

Molecule B DJ

H 2 60.80 4.64 

10 ^2

D 2 30.42 1.159 

10 ^2

HCl 10.59 5.32 

10 ^4

HBr 8.473 3.72 

10 ^4

N 2 2.010 5.8 

10 ^6

NO 1.7046 ~5 

10 ^6

O 2 1.446 4.95 

10 ^6

aAll numbers have units of cm (^1).