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In-Depth Information
6
4
2
n
= 1
J
= 0
6
4
2
n
= 0
Electronic
excited state
J
= 0
Zero-point energy
6
4
2
n
= 3
J
= 0
6
4
Electronic
transition
2
n
= 2
J
= 0
6
Rotational
transitions
4
2
n
= 1
J
= 0
6
Vibrational
transition
4
2
n
= 0
Electronic
ground state
J
= 0
Zero-point energy
Figure 3.1
The different energy levels of a compound. The separation of the rotational energy levels (
J
is the
number of rotational quanta) is very small with respect to the separation of the vibrational levels
and that of electronic levels between different orbitals. Note the zero-point energy, a
consequence of the uncertainty principle, which only appears for the vibrational energy levels.
attached by a spring: this is the model of the harmonic oscillator. College physics tells us
that if such a system does not lose energy by friction or radiation, the sum of the potential
and kinetic energies remains constant. The potential energy
V
(
r
) of a diatomic molecule
is a function of the separation distance
r
between the nuclei and goes through a minimum
for
r
=
r
0
. The potential energy
V
(
r
) can be expanded about the value at the minimum as
d
V
d
r
r
0
(
d
2
V
d
r
2
1
2
2
V
(
r
)
=
V
(
r
0
)
+
r
−
r
0
)
+
r
0
(
r
−
r
0
)
+···
(3.3)
which assumes that terms with degrees
2, known as anharmonic, are small enough to be
neglected: this is the parabolic or harmonic approximation of the potential well. The first
term on the right-hand side is the energy at the minimum and the second is zero because the
derivative is evaluated at the minimum. Let us call
k
the second-order derivative calculated
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