The Born-Oppenheimer approximation assumes that the wavefunctions of atomic nuclei and electrons in a molecule can be treated separately, primarily because the nuclei are much heavier than the electrons.
We begin with the molecular Hamiltonian operator :
where
-
and
refer to nuclei and electrons respectively.
- The first and second terms are the kinetic energy operator of nuclei and electrons respectively.
is the potential energy of interaction between two particles.
For example, the Hamiltonian for (see diagram above) is:
To solve the complicated Schrodinger equation for a molecule, Max Born and J. Oppenheimer proposed separating the electronic and nuclear motions. The first assumption the two scientists made is that the molecular wavefunction
is a product of the wavefunction of nuclear motion
and the wavefunction of electron motion
, i.e.
Question
Why is a function of nuclear coordinates
and electronic coordinates
?
Answer
The electron density of a molecule changes when the molecule vibrates and therefore the form of the wavefunction of electron motion changes based on the positions of the electrons and the nuclei.
With regard to eq51 and eq52, the Schrodinger equation is
Using the product rule, we have , which when substituted in eq53 gives
The second assumption Born and Oppenheimer made was that is a slowly varying function of
. This is because the heavier nuclei are considered relatively stationary as they move much more slowly than electrons. Therefore, the change in the distribution of the electrons with respect to the positions of the nuclei
is very small and eq54 becomes
Substituting the purely electronic Hamiltonian and its corresponding Schrodinger equation
in eq55, we have
Question
How do we interpret the term in eq56?
Answer
is the eigenvalue of the purely electronic Schrodinger equation
. Using the second assumption, we regard the nuclei as fixed while the electrons move.
in
then becomes a constant for a particular nuclear configuation. In other words,
is now a function of
and at the same time parametrically dependent on
. For each electronic state , the purely electronic Schrodinger equation is solved repeatedly using different fixed values of
, giving a set of
and a corresponding set of
. Since
, we can then easily calculate a value
for every
and obtain a curve (see diagram below). In effect,
is a function of
, making it the potential energy for nuclear motion:
We have therefore transformed eq53, the molecular Schrodinger equation, to eq57, which is an eigenvalue equation of a single variable of nuclear coordinates, with the eigenfunction as the wavefunction of nuclear motion, the eigenvalue
representing the total energy of the molecule, and the term
as the potential energy for nuclear motion.
Question
How does the above molecular potential energy curve relate to the quadratic potential energy curve derived using the harmonic oscillator model?
Answer
The two curves coincide at low vibration energy levels, i.e around the equilibrium internuclear distance (see diagram below). Nuclear motion includes translational, rotational and vibrational motions. As translational and rotational motions are purely kinetic, in eq57 is associated only with molecular vibrational motions. Consider a diatomic molecule with one of the nuclei at the origin and the other on the
-axis. The potential energy term becomes
, where
is the internuclear distance. Using a Taylor series, we can approximate the potential function by expanding it around its minimum at the equilibrium internuclear distance
:
At , the slope of
is zero, i.e.
. For small vibrations, where
is close to
, we ignore all the higher power terms, giving
. The physical observables of a system, such as the transition frequencies
of vibrational modes of a molecule, are not affected by the
term because it cancels out when we take the difference of two energy levels. Therefore, we can set
to zero, resulting in
, which is equivalent to the potential energy term of the harmonic oscillator (see eq49), where
and
.
In conclusion, the Born-Oppenheimer approximation offers a different approach, compared to the quantum harmonic oscillator, in deriving the Schrodinger equation for a diatomic molecule (see the Hamiltonian of eq57 vs eq48)) and the potential energy term for the vibrational motion of a diatomic molecule.