# The Born-Oppenheimer Approximation

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schrödinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei.

The full Hamiltonian for the molecular system can then be written as:

$\mathbf{H} = \mathbf{T}^{elec}(\vec{r}) + \mathbf{T}^{nucl}(\vec{r}) + \mathbf{V}^{nucl-elec}(\vec{R},\vec{r}) + \mathbf{V}^{elec}(\vec{r}) + \mathbf{V}^{nucl}(\vec{r})$

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei:

\begin{aligned} \mathbf{H}^{elec} &= -\frac{1}{2}\sum\limits_i^{elecs}(\frac{\partial^2}{\partial x_i^2}+\frac{\partial^2}{\partial y_i^2}+\frac{\partial^2}{\partial z_i^2}) - \sum\limits_i^{elecs}\sum\limits_I^{nucl}(\frac{\mathrm{Z}_I}{\lvert \vec{R_I}-\vec{r_i} \rvert}) &+ \overset{elecs}{\sum\limits_i \sum\limits_{j \textless i}}(\frac{1}{\lvert \vec{r_i}-\vec{r_j} \rvert}) + \overset{nucl}{\sum\limits_I \sum\limits_{J \textless I}}(\frac{Z_I Z_J}{\lvert \vec{R_I}-\vec{R_J} \rvert}) \end{aligned}

Note that the fundamental physical constants drop out with the use of atomic units.

This Hamiltonian is then used in the Schrödinger equation describing the motion of electrons in the field of fixed nuclei:

$\mathbf{H}^{elec}\psi^{elec}(\vec{r},\vec{R}) = E^{\textit{eff}}(\vec{R})\psi^{elec}(\vec{r},\vec{R})$

Solving this equation for the electronic wavefunction will produce the effective nuclear potential function . It depends on the nuclear coordinates and describes the potential energy surface for the system.

Accordingly, Eeff is also used as the effective potential for the nuclear Hamiltonian:

$\mathbf{H}^{nucl} = \mathbf{T}^{nucl}(\vec{R})+ E^{\textit{eff}}(\vec{R})$

This Hamiltonian is used in the Schrödinger equation for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schrödinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules.