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In this section, we will discuss one of the most important
and fundamental approximations in molecular quantum mechanics.
This approximation was developed by Max Born and J. Robert Oppenheimer
in 1927. We will consider a very general molecule with
nuclei and electrons. The coordinates of the nuclei are
. The coordinates of the electrons are
, and their spin variables are
.
For shorthand, we will denote the
complete set of nuclear coordinates as and the
set of electrons coordinates as and
the complete set of electron coordinates and spin variables. The total molecular
wave function depends on variables,
which makes it a very cumbersome object to deal with.
The BornOppenheimer approximation leads to a very important
simplifaction of the wave function.
If we could neglect the electronnuclear interaction,
then the wave function would be a simple
product
.
However, we cannot neglect this term, but it might still
be possible to write the wave function as a product.
We note, first, that most nuclei are 34 orders of magnitude heaver
than an electron. For the lightest nucleus, the proton,
This mass difference is large enough to have important physical
consequences. Let us think classically about this mass difference first.
If two particles interact in some way, and one is much heavier
than the other, the light particle will move essentially as a
``slave'' of the heavy particle. That is, it will simply follow
the heavy particle wherever it goes, and, it will move
rapidly in reponse to the heavy particle motion. As an illustration
of this phenomenon, consider the simple mechanical system pictured
below:
Considering this as a classical system, we expect that the motion will
be dominated by the large heavy particle, which is attached to a fixed
wall by a spring. The small, light particle, which is attached to the
heavy particle by a spring will simply follow the heavy particle and execute
rapid oscillations around it. The figure below (bottom panel) illustrates
this:
In this illustration, when the heavy particle moves even a tiny amount, the
light particle executes many oscillations around the heavy particle. Thus,
we see that the light particle moves quite a bit, and we can think of the
light particle as generating ``on the fly'' an effective potential
for the heavy particle, at least at the present position
of the heavy particle.
What we can conclude, therefore, is that we can approximately fix the
position of the heavy particle and just allow the light particle to move
around before we advance the heavy particle to its next position.
How do the different mass scales of electrons and nuclei manifest themselves
in the timeindependent Schrödinger equation? Light particles, such as electrons,
tend to have very diffuse, delocalized wave functions, while heavy particles,
such as the nuclei, tend to have wave functions that are very localized about
the classical positions. This is illustrated in the figure below:
Figure 1:
Crude picture of matter. Nuclear wave functions are shown
above the axis, electronic wave functions are shown below.

Thus, taking the idea of solving the electronic problem for fixed nuclear
positions seriously, we can write the molecular wave function as
which suggests that the electronic wave function
is solved using the nuclear positions
simply as parameters that characterize the wave function in the
same way that the electron mass and electron charge do.
This is not an exact wave function for the molecule but an
approximate one. Hence, we call this the BornOppenheimer
approximation to the wave function. The total classical
energy of the molecule is
where and are the electronic and nuclear
kinetic energies. Therefore, the total Hamiltonian of the
molecule is
where and are the kinetic energy operators
that result from substituting momenta for derivatives. Thus,
and contain second derivatives with
respect to electronic and nuclear coordinates, respectively.
The details of these operators are not that important for
this discussion.
We now write the total molecular Hamiltonian as
where we assign
If we now substitute this into the Schrödinger equation, we obtain
Let us consider how the electronic and nuclear kinetic energy operators
act on the wave function. contains derivatives with respect
only to electronic coordinates. Thus, it has no effect on the nuclear
wave function, and we can write
The operator contains derivatives with respect to
nuclear coordinates, thus it affects both parts of the wave function:
The last term arises from an application of the product rule using the
fact that
. What is important here is that
if the nuclear wave function is highly localized, then its curvature is
high in the vicnity of the nucleus, and hence,
is
very large compared to
since
changes much less dramatically spatially due to the delocalized nature of the
electronic wave function. Thus, we need only keep the term in the above
equation that contains
:
Using this fact in the Schrödinger equation, we can write the equation as
or
We now divide both sides by
, which
gives
Note that the right side depends only on the nuclear coordinates , which
means that we can write compactly as some function
. Hence,
we obtain
or
which is called the electronic Schrödinger equation. it yields
a set of wave functions
and energies
characterized by a set of
quantum numbers , which is the full set of quantum numbers
needed to characterize the wave function, e.g. for hydrogen.
Thus, the electronic Schrödinger equation can be written as
Note that the potential energy depends on the electronic coordinates
and the fixed nuclear coordinates . Thus, the nuclear
coordinates can be thought of as additional parameters in the
potential energy. Alternatively, we can think of the nuclear
as providing a fixed background potential energy in addition to the
internal electronelectron repulsion that governs the
shape of the electronic wave function.
Now the nuclear part comes from setting the right side of the
full Schrödinger equation equal to
, which yields:
and if we multiply by
, we obtain the nuclear Schrödinger equation
The Schrödinger equations for the electronic and nuclear part of the
wave function are not exact but approximate based on the
separation of electronic and nuclear degrees of freedom we
assume can be made based on the mass differential.
Two interesting facts about the approximate Schrödinger equations
should be noted. First, the electronic Schrödinger equation
yields a set of energy levels
that
depend on the nuclear configuration. Thus, at each nuclear
configuration we choose, we get a different set of
energy levels. These energy levels change continuously
with the nuclear configuration, in fact. At the same
time, the nuclear Schrödinger equation involves a potential
energy with two terms
,
i.e. the nuclearnuclear repulsion plus the electronic
energy level
as a function of .
Therefore, there is a different nuclear Schrödinger equation
for each electronic energy level. The
potential energies
are called BornOppenheimer potential energies
or BornOppenheimer surfaces. They are surfaces
(or hypersurfaces) because they must be viewed as
continuous functions of (see figure below):
The potential energy
determines the geometry of the molecule by minimization, and the
nuclear probability distribution by solution of the nuclear
Schrödinger equation. It also determines the nuclear dynamics if
we solve the nuclear timedependent Schrödinger equation. Thus,
, the electronic energy levels, are
of utmost importance in molecular quantum mechanics.
For this reason, we need to understand the distributions
of electrons in molecules.
Incidentally, we could now introduce a further approximation in
which we treat the nuclei as classical particles with
a potential energy
.
The nuclei would then move as classical point particles
via Newton's Second Law, which would now take the form
Next: The hydrogen molecule ion
Up: lecture_13
Previous: Overview of molecular quantum
Mark E. Tuckerman
20111110