MORSE OSCILLATOR
The Morse oscillator is a model for a particle in a one-dimensional anharmonic potential energy
surface with a dissociative limit at infinite displacement.1 It is commonly used for describing the
spectroscopy of diatomic molecules and anharmonic vibrational dynamics, and most of its
properties can be expressed through analytical expressions.2 The Morse potential is
2
V ( x)  De 1  e x  (1) where x  (r  r0 ) . De sets the depth of the energy
minimum at r = r0 relative to the dissociation limit
as r → ∞, and α sets the curvature of the potential.
If we expand V in powers of x about x=0,
V ( x)  12  x 2  16 g x3  241 hx 4 
we find that the harmonic, cubic, and quartic
  2 De 2 ,
expansion
coefficients
are
g  6 De 3 , and h  14 De 4 .
The Morse oscillator Hamiltonian for a
diatomic molecule of reduced mass mR bound by
this potential is
H
p2
 V ( x)
2 mR
and has the eigenvalues
2
En  0  n  12   xe  n  12  


n  0,1, 2,3...
(2)
Here 0  2 De 2 mR is the fundamental frequency and xe  0 / 4 De is the anharmonic
constant. The frequency ω0 is equivalent to equating the harmonic force constant
  ( 2V / x 2 ) x 0 with mR02 . The anharmonic constant xe is commonly seen in the
spectroscopy expression for the anharmonic vibrational energy levels
G (v)  e (v  12 )  e xe (v  12 ) 2  e ye (v  12 )3  (3) 1
Morse, P. M., Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels. Physical Review
1929, 34 (1), 57-64.
2
Lefebvre-Brion, H. l. n.; Field, R. W., The spectra and dynamics of diatomic molecules. 2nd ed.; Academic Press:
Boston, 2004.
From eq. (1.1) the ground state (or zero-point) energy is
E0  12 0 1  12 xe  (4) So the dissociation energy for the Morse potential is given by D0  De  E0 . The transition
energies are
En  Em  0  n  m  1  xe  n  m  12   (5) The proper harmonic expressions are obtained from the above for the Morse oscillator by setting
De   or xe  0 .
The wavefunctions for the Morse oscillator can also be expressed analytically in terms of
associated Laguerre polynomials Lbn ( z ) :3
(6)  n  N ne  z /2 z b/2Lbn ( z ) where
N n  [  b  n !/ (k  n)]1/2
z  k exp[ q ]
b  k  2n  1
k  4 De / 0
These expressions and those for matrix
elements in q, q2, e‒αq, and qe‒αq have
been given by Vasan and Cross.4
3
Gallas, J. A. C., Some matrix elements for Morse oscillators. Physical Review A 1980, 21 (6), 1829-1834.
Vasan, V. S.; Cross, R. J., Matrix elements for Morse oscillators. The Journal of Chemical Physics 1983, 78 (6),
3869-3871.
4