Chem 120A
Molecular states
04/17/06
Spring 2006
Lecture 33
READING:
Engel (QCS): Ch. 1213, section B.2
McQuarrie and Simon: Chapter 7.2, 9.39.6
Basic energies and spectra
In Lecture 30 we saw that the large difference in the masses of nuclei and electrons allows us to apply the
BornOppenheimer approximation in order to solve the Schr¨odinger equation for a molecule. This allows
us to separate the nuclear and electronic degrees of freedom. This separation is equivalent to a separation
of time scales, meaning that since the nuclei are so massive the move on a much slower time scale than the
electrons. This leads to a separation of the energies that characterize the spectrum associated with absorption
and emission of light by a molecule. The magnitudes of the energies for electronic and nuclear (vibrational)
excitations are also quite different.
Recall that 1 eV
=
8065 cm
−
1
, 1 Hartree
=
219
,
474 cm
−
1
≈
27 eV. Electronic motion is confined to a typical
molecular size of
∼
1
˚
A. Using the particle in a box model, we can approximate the energy for electronic
motion, where
m
is the electron mass:
E
e
∼
¯
h
2
ma
2
∼
few eV
.
(1)
The nuclei move in the field of the fast moving electrons. In other words, the electronic energies, which
depend on the positions of the nuclei, represent an effective external potential in which the nuclei reside.
This potential is shown in Fig. 1.
There are two types of nuclear motion, vibration and rotation.
We
have already studied the case of nuclear vibration, which can be modeled by the harmonic oscillator. The
potential energy for the harmonic oscillator is proportional to
M
ω
2
vib
x
2
, where
ω
vib
is the angular frequency
of vibration and
M
is the nuclear mass. At equilibrium, we can take
x
=
a
(Fig. 1), so that the potential
a
Figure 1: A BornOppenheimer potential for a diatomic molecule with vibrational states shown. A harmonic
potential is also shown for comparison.
a
is the approximate size of the molecule.
Chem 120A, Spring 2006, Lecture 33
1
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v=1
v=2
v=3
K=0
1
2
3
~10
2
eV
~10
4
eV
}
rotational levels
vibrational levels
Figure 2: A rovibrational sequence for a molecule within the BornOppenheimer approximation. Here we
have also assumed that the kinetic energy for the nuclei can be separated into vibrational and rotational parts.
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 Spring '07
 Whaley
 Physical chemistry, Atom, Mole, pH, Energy, Kinetic Energy, Schr¨ dinger equation, Simon 9.39.6

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