Energy Levels of a Morse Oscillator

atom mass

m

1

(amu)

12

atom mass

m

2

(amu)

16

fundamental vibrational frequency

ω

e



-1

cm



2170

dissociation energy

D

e



-1

cm



89600

equilibrium internuclear distance

R

e

(Å)

1.128

The Morse function

V(R)=

D

e



-2aR-

R

e



e

-2

-aR-

R

e



e



, where

R

is the internuclear distance, provides a useful approximation for the potential energy of a diatomic molecule. It is superior to the harmonic oscillator model in that it can account for anharmonicity and bond dissociation. The relevant experimental parameters are the dissociation energy

D

e

and the fundamental vibrational frequency

ω

e

, both conventionally expressed in wavenumbers (

-1

cm

), the equilibrium internuclear distance

R

e

in Angstrom units (Å), and the reduced mass

μ=

m

1

m

2

/(

m

1

+

m

2

)

in atomic mass units (amu). The exponential parameter is given by

a=

ω

e

μ/2

D

e

in appropriate units. The Schrödinger equation for the Morse oscillator is exactly solvable, giving the vibrational eigenvalues

ϵ

v

=

ω

e

v+

1

2

-

2

ω

e

4

D

e

2

v+

1

2

, for

v=0,1,2,...,

v

max

. Unlike the harmonic oscillator, the Morse potential has a finite number of bound vibrational levels with

v

max

≈2

D

e

/

ω

e

.