Physical Chemistry Third Edition

922 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

Exercise 22.2

Find the frequency and wavelength of the radiation absorbed if a carbon monoxide molecule

makes a transition from theJ0 state to one of theJ1 states. In which region of the

electromagnetic spectrum does this radiation lie?

The Rotation and Vibration of Diatomic Molecules

In order to discuss vibration as well as rotation, we abandon the rigid rotor model.

Using the eigenvalue of̂L^2 from Eq. (17.2-27) we can write Eq. (22.2-13) in the form

−

d

dr

r^2

dR

dr

+

2 μr^2

h ̄^2

(V−E)R+J(J+ 1 )R 0 (22.2-20)

where we now omit the subscript on the relative energyErel. To proceed we need a

representation of the potential energyV(r). Curves that schematically representV(r)

for two electronic states are depicted in Figure 22.2. These curves conform to the general

pattern that the state of higher energy has the longer equilibrium internuclear distance

and a weaker bond. We representV(r) as a power series in the variablexr−re,

wherereis the value ofrat the minimum inV:

V(r)V(x)+

(

dV

dr

)

re

x+

1

2!

(

d^2 V

dr^2

)

re

x^2 + ··· (22.2-21)

where the subscript 0 means that the derivatives are evaluated atx0. The func-

tionV(r) is at a minimum ifx0, so the first derivative vanishes. To a fairly good

approximation, we truncate the series at the quadratic term and write

V(r)Ve+

1

2!

(

d^2 V

dx^2

)

0

x^2 Ve+

1

2

kx^2 (22.2-22)

where we use the same symbol forV whether it is expressed as a function ofror

ofxand whereVeV(re). The quantitykis a force constant like the one that we

introduced for the harmonic oscillator in Chapter 15. The function of Eq. (22.2-22)

is called aharmonic potential. The parabolas representing the harmonic potential are

included in Figure 22.2.

V

re r ́e

r

Truncated

power series

representations Actual

potential

energies

Figure 22.2 Vibrational Potential
Energy for a Typical Diatomic Mole-
cule in Two Electronic States.

The radial Schrödinger equation is now

d

dr

r^2

dR

dr

−J(J+1)R+

2 μr^2

h ̄^2

(

E−Ve−

kx^2

2

)

R 0 (22.2-23)

We define a new dependent variable

S(r)rR(r) (22.2-24)

The derivative of Eq. (22.2-23) is

d

dr

r^2

dR

dr



d

dr

r^2

dS/r

dr



d

dr

r^2

(

1

r

dS

dr

−

S

r^2

)



d

dr

(

r

dS

dr

−S

)

r

d^2 S

dr^2

+

dS

dr

−

dS

dr

r

d^2 S

dr^2