Born-Oppenheimer approximation
A molecule is an assembly of positively charged nuclei and negatively charged
electrons that form a stable entity through the electrostatic forces which hold it all
together. Since all the particles which make up the molecule are moving relative to
each other, a full quantum mechanical description of the molecule is very complicated
and can only be obtained approximately. Fortunately, the overall motion of the
molecule can be broken down into various types of motion, namely, translational,
rotational, vibrational, and electronic. To a good approximation, the so called Born-
Oppenheimer approximation, each of these motions can be considered on its own
(Born & Oppenheimer 1927).
Their main objective was the separation of electronic and nuclear motions in a
molecule. The physical basis of this separation is quite simple. Both electrons and
nuclei experience similar forces in a molecular system, since they arise from a mutual
electrostatic interaction. However, the mass of the electron, is about four orders of
magnitude smaller than the mass of the nucleus. Consequently, the electrons are
accelerated at a much greater rate and move much more quickly than the nuclei.
Therefore, as an approximation, we can regard the dynamics of the electrons and
nuclei as largely independent. When describing the electrons, the nuclei can be
considered as being fixed in space. On longer timescales, the electrons can
immediately follow the much slower nuclear motions. To a very good approximation,
the total energy of a molecule is given as the sum of three components, corresponding
to the three modes of motion mentioned above, namely
Where the three terms on the right hand side represent the electronic, vibrational, and
rotational energy. We disregard overall translational energy of the molecule because it
leads mainly to collisions and, thus, has only an indirect influence on the internal
structure of molecules. Furthermore, we have neglected in Eq. (3.1) the interaction
between the three different types of motion. These interaction terms have to be
considered in a consistent treatment, and we will come back to them in the course of
this chapter, but it is more illustrative to neglect them as a first approximation. In
molecular spectroscopy, Eq. (3.1) is often expressed in units of the wave number ύ =
v/c , using the notation
(3.2)
with
(3.3)
The energy difference between two states can then be denoted by
( 3.4)
where the units of ύ are cm
–1
Since electrons move more rapidly than the nuclei, and the vibrations of nuclei are
more rapid than the rotation of molecules, the following relation holds:
(3.5)
We have already argued that the electronic energy is largest. The relative energies of
vibrations and rotations follow from simple order of magnitude estimates. The valence
electrons spread over the whole molecule (of size a ≈ 1 Å) and thus have typical
energies
(3.6)
In this estimate we have used the Rydberg energy but with a instead of the Bohr
radius. The energy scale of vibrations about the equilibrium separation req of two
nuclei in a molecule is defined by ℏω . To find ℏω we consider the potential energy
of the vibrational modes given by Mnω2(r – req) 2 , cf. Eqs. (3.33) and (3.34). We
have req ~ a. For vibrations with (r – req) ~ a the electron configuration would be
modified strongly and “costs” roughly one electronic energy Ee. From Mnω 2 (r –
req)2 ~ 2 2 e ℏ m a we find
(3.7)
Rotational energies are estimated by J 2 /I (cf. Eq. (3.13)) where J is the angular
momentum and I is the moment of inertia. With J 2 ~ 2 2 ℏ ∼ ℏ j j ( ) +1 and I ~
Mna 2 we get
(3.8)
The corresponding length scales are λe ~ 100–1000 nm (i.e. in the order of visible
wavelengths), λvib ~ 10 µm (i.e. in the infrared region), and λ rot ~ 1 mm. Therefore,
spectral lines of molecules are observed in the visible (actually near UV to about 1
µm) in the case of electronic transitions, in the infrared for vibrational transitions, and
in the sub-mm to about 1 mm region (microwaves) for rotational transitions.
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