Biomedical Engineering Reference
In-Depth Information
v
=
4
v
=
3
v
=
2
v
=
1
v
=
0
Harmonic
Experimental
Figure 3.6
Harmonic versus experimental energy levels
Our conclusion from the experimental data is that vibrational energy levels get pro-
gressively closer together as the quantum number increases. This suggests that whilst the
harmonic model is a reasonable one, we need to look more carefully at the form of the
potential in order to get better agreement with experiment.
3.5 The Morse Potential
Professional spectroscopists would be unhappy with the idea of using Hooke's law as a
model for the vibrational motion. They would be more concerned with matching their
experimental energy levels to a more accurate potential. Many such potentials have been
used over the years, with that due to Morse being widely quoted in elementary chemistry
texts. The Morse potential is as follows:
2
U
=
D
e
{
1
−
exp [
−
β(
R
−
R
e
)]
}
(3.18)
where
D
e
is the depth of the potential well, i.e. the thermodynamic dissociation energy, and
d
2
U
d
R
2
β
=
2
D
e
2μ
D
e
ω
e
2
=
(3.19)
This potential contains three parameters
D
e
, ω
e
and
R
e
, and so should be capable of giving a
better representation to the potential energy curve than the simple harmonic, which contains
just the two parameters
k
s
and
R
e
.
In the case of
1
H
35
Cl, a simple calculation shows that the dissociation energy
1
2
h
(2πω
e
)
D
e
=
D
0
+
(3.20)
