Physical Chemistry Third Edition

926 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

potential energy can be used.^1 The most commonly used representation is theMorse

function

V(r)V(re)+De

(

1 −e−a(r−re)

) 2

(22.2-44)

whereDeis thedissociation energy, which is equal to the energy required to dissociate

the molecule from the state of minimumV. The parameteradetermines the curvature of

the function and is equal tok/(2De). The values of these parameters must be determined

for each substance. Figure 22.3a depicts the Morse function for the CO molecule. The

Schrödinger equation for the Morse function has been solved.^2 It has also been treated

(^2080) in second-order perturbation theory.^3
20
10
100 120 140 160
V
/10
219
J
r/10^212 m
(a)
~1082 kJ mol^21
~1.80 310218 J
80
20
20
10
5
15
100 120 140 160
V
/10
219
J
r/10^212 m
(b)
Morse potential
energy function
First 20 vibrational
levels
Figure 22.3 The Vibrational Poten-
tial Energy and Energy Levels of
the CO Molecule.(a) The vibrational
potential energy represented by the
Morse function. (b) The first 20 vibra-
tional energy levels for the Morse poten-
tial. The Morse potential parameters for
CO areDe 11 .2eV 1. 80 × 10 −^18 J
anda2.2994× 1010 m−^1. The force
constant for the harmonic potential is
k1900 N m−^1.
Exercise 22.3
a.Using the general relation between potential energy and force, Eq. (E-6) of Appendix E,
obtain a formula for the force on a nucleus in a diatomic molecule described by the Morse
potential function.
b.Show thatDeis equal to the difference in potential energy between the minimum and the
value for larger.
c.Show thatreis the value ofrat the minimum and show that there is no force ifrre.
Corrections for anharmonicity, for centrifugal stretching, and for interaction between
vibration and rotation can be added to the energy level expression, giving for the energy
of vibration and rotation
EvJhνe

(

v+

1

2

)

−hνexe

(

v+

1

2

) 2

+hBeJ(J+1)

(22.2-45)

−hDJ^2 (J+1)^2 −hα

(

v+

1

2

)

J(J+1)

The constant parameters in this expression are all positive and are given for the Morse

potential by^4

α

3 h^2 νe

16 π^2 μr^2 eDe

(

1

are

−

1

a^2 r^2 e

)

(22.2-46)

xe

hνe

4 De

(22.2-47)

Be

h ̄

4 πIe



h

8 π^2 Ie

(22.2-48)

D

4 Be

νe^2

(22.2-49)

(^1) J. C. Davis, Jr.,Advanced Physical Chemistry, The Ronald Press, New York, 1965, p. 285.
(^2) P. M. Morse,Phys. Rev., 34 , 57 (1929).
(^3) B. A. Pettit,J. Chem. Educ., 75 , 1170 (1998).
(^4) J. C. Davis,op. cit., p. 351 (note 1).