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Fig. 6.9 Dependence of the
potential energy of the
nuclei of a diatomic
molecule as a function of
the distance between the
nuclei. Differences between
the harmonic (
U
0
) and
anharmonic (
U
) functions
(
top
)
Let us turn to the simplest physical example—the description of the dynamics of
an arbitrary diatomic molecule. Potential energy of the interaction between the
nuclei of the molecule is a function with a rather complicated form, which increases
sharply at short distances between the nuclei, passes through a minimum at the
equilibrium point of the internuclear distance, and asymptotically approaches zero
with increasing internuclear distance (Fig.
6.9
). It is easy to see that near the
minimum, this function can be rather precisely approximated by a parabola,
which corresponds to the so-called harmonic approximation of the potential energy
of the nuclei of the molecule
2
k
e
r
ð
r
e
Þ
U
¼
:
2
This, in turn, implies that the force acting between the nuclei of a molecule
depends linearly on the change of the distance between them
F
¼
k
e
r
ð
r
e
Þ:
This approximation is important, since the value
k
e
determines the oscillation
frequency of the nuclei
r
2
k
e
ʽ ¼
2
ˀ
,
is the reduced mass of the nuclei of the molecule.
Thus, three quantities: the equilibrium distance between the nuclei
r
e
, the depth
of the minimum—the dissociation energy—
D
e
, and the second derivative of the
energy of the nuclei of the molecule at the equilibrium point
k
e
allow for describing
the basic properties of a diatomic molecule.
The minimum of the potential energy of the nuclei corresponds to a bound state
of the molecule. Therefore, the solution of the Schr¨dinger equation shows that the
nuclei of the molecule are mainly in the vicinity of
r
e
, and the simple model
where
ʼ
