The Born-Oppenheimer Approximation

Perhaps the great disappointment of quantum chemistry is that, whilst the Schrödinger equation is powerful enough to describe almost all chemistry, it is too complex to solve for all but the simplest of systems. Even the simplest molecule, H₂⁺, consists of three particles, thus producing a Schrödinger equation that is impossible to solve analytically. To overcome this difficulty a variety of approximations are made, the most common of which is the Born-Oppenheimer approximation [6].

The masses of the nuclei are much greater than the electrons, hence the electrons can respond almost instantaneously to any change in the nuclear positions. Thus, to a good approximation, we can think of the electrons as moving in a field of fixed nuclei. Within this approximation, the nuclear kinetic energy term can be neglected and the nuclear-nuclear repulsion term can be considered a constant. These two terms can therefore be removed^5.1 to form the electronic Hamiltonian, $\hat{H}_{elec}$.

What remains is termed the electronic Schrödinger equation,
$$\begin{align}\hat{H}_{elec} \, \Psi_{elec}({\bf r}_{i};{\bf R}_{A}) = E_{elec}({\bf R}_{A}) \, \Psi_{elec}({\bf r}_{i};{\bf R}_{A}). \end{align}$$
The notation $\Psi_{elec}({\bf r}_{i};{\bf R}_{A})$ implies that the electronic wavefunction depends on the nuclear positions only parametrically -- a different wavefunction is defined for each nuclear configuration.

$E_{elec}({\bf R}_{A})$ is only the electronic energy; to regain the total energy (for fixed nuclei) we must add the nuclear-nuclear repulsion constant.
$$\begin{align}E_{tot} = E_{elec} + \sum_{A}^{M}\sum_{B>A}^{M}\frac{Z_{A}Z_{B}}{\vert{\bf R}_{A}-{\bf R}_{B}\vert} \end{align}$$

Repeating the calculation with a different nuclear arrangement allows the potential energy surface to be mapped out and the equilibrium geometry to be found. The work of this thesis is entirely within the Born-Oppenheimer approximation, so for clarity, the `tot' and `elec' subscripts will be dropped and only electronic Hamiltonians and wavefunctions will be considered.