Physical Chemistry Third Edition

22.2 The Nonelectronic States of Diatomic Molecules 923

so that Eq. (22.2-23) becomes

h ̄^2

2 μ

(

rd^2 S

dr^2

−

J(J+ 1 )rS

r^2

)

+

(

E−Ve−

kx^2

2

)

rS 0

h ̄

2

2 μ

(

d^2 S

dr^2

−

J(J+ 1 )S

r^2

)

+

(

E−Ve−

kx^2

2

)

S 0 (22.2-25)

We express 1/r^2 as a power series inx:

1

r^2



1

(re+x)^2



1

r^2 e

(

1 −

2 x

re

+

3 x^2

re^2

+ ···

)

(22.2-26)

Ifxis quite small compared withre, it is a fairly good approximation to keep only the

first term of this series:

1

r^2

≈

1

re^2

(22.2-27)

Using this approximation and the fact thatd^2 S/dx^2 d^2 S/dr^2 , Eq. (22.2-25) becomes

−

h ̄^2

2 μ

d^2 S

dx^2

+

kx^2

2

S

(

E−Ve−

h ̄^2

2 μre^2

J(J+ 1 )

)

S (22.2-28)

Equation (22.2-28) is the same as the harmonic oscillator Schrödinger equation

of Eq. (15.4-1), except for the presence of the two constant terms subtracted from

the energy eigenvalue. The functionSis the same as the harmonic oscillator energy

eigenfunction, given by Eqs. (15.4-10), (15.4-11), and so on. The energy eigenvalue

Efor the relative energy is the harmonic oscillator energy eigenvalue of Eq. (15.4-8)

plus the two constant terms. (See Problem 15.6 for the effect of adding a constant to a

potential energy function.)

EEvJhνe

(

v+

1

2

)

+

h ̄^2

2 μre

J(J+ 1 )+Ve (22.2-29a)

Evib,v+Erot,J+Ve (22.2-29b)

The energy eigenvalue is a harmonic oscillator energy eigenvalue plus a rigid rotor

energy eigenvalue, plus a constant,Ve.

The quantityνeis the oscillator frequency of Eq. (14.2-29) except that in our case

we have a reduced massμinstead of a massm:

νe

1

2 π

√

k

μ

(22.2-30)

We can express the rotational energy in terms of the equilibriummoment of inertiaof

the diatomic molecule:

Ieμre^2 (22.2-31)

The quantum numbervcan equal 0, 1, 2, ..., and the quantum numberJcan also equal

0, 1, 2, .... We refer to the approximation of Eq. (22.2-29) as therigid rotor–harmonic

oscillator approximation.