Physical Chemistry Third Edition

924 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

The wave function for the relative motion of the nuclei is now

ψrΘJM(θ)ΦM(φ)

Sv(r−re)

r

ψrot,JM(θ,Φ)ψvib,v(x) (22.2-32)

The rotational wave function is the same spherical harmonic function that occurred

with the hydrogen atom or the rigid rotor:

ψrot,JMΘJM(θ)ΦM(φ) (22.2-33)

The vibrational wave function is equal to a harmonic oscillator wave function divided

byr, the internuclear distance.

ψvib,vR(r)

Sv

r



ΨHO,v

r

(22.2-34)

The wave function is given in the Born–Oppenheimer approximation by the wave

function for relative nuclear motion times the translational wave function times the

electronic wave function:

ψtotψtrψrelψelψtrψrotψvibψel (22.2-35)

The energy is the translational energy plus the relative energy in Eq. (22.2-29).

EEtr+EvJEtr+hνe

(

v+

1

2

)

+ ̄

h^2

2 Ie

J(J+1)+Ve (22.2-36)

We write this as

EtotEtr+Evib+Erot+Eel (22.2-37)

where

EtrEnxnynz

h^2

8 M

[

n^2 x

a^2

+

n^2 y

b^2

+

n^2 z

c^2

]

(22.2-38)

Evibhνe

(

v+

1

2

)

(22.2-39)

Erot

h ̄^2

2 Ie

J(J+ 1 ) (22.2-40)

EelVeV(re)EBO(re) (22.2-41)

The electronic energyEelis equal to a constant. The rest of the Born–Oppenheimer

energy was taken as the potential energy of vibration. Sometimes the zero-point vibra-

tional energy is included in the constant electronic energy, so that

EelVe+

hve

2

(alternate version) (22.2-42)

and

Evibhνev (alternate version) (22.2-43)