# Born-Oppenheimer approximation

The Born-Oppenheimer approximation, also known as the adiabatic approximation, is a technique used in quantum chemistry and condensed matter physics in order to de-couple the motion of nuclei and electrons. It is based upon the fact that typical electronic velocities far exceed those of nuclei.

## Hand-waving derivation of the approximation

Since the mass of atomic nuclei are far greater than the mass of those electrons orbiting them (by a factor of about 1000), for a given energy, the electrons move much faster than the nuclei. To get an idea of what kinds of numbers we are talking, it is instructive to note that a typical electron velocity is about [itex]10^6 ms^{-1}[itex] (the Fermi velocity), while those of nuclei are about [itex]10^3ms^{-1}[itex] (the speed of sound). The much swifter electronic system can always respond quickly to changes in the configuration of nuclei, thus allowing the electronic system to remain in its ground state (for that particular configuration of nuclei).

The motion of the electrons can therefore be considered decoupled from the motion of the nuclei, which leads to the elimination of several terms from the Schrödinger equation - in practice one goes ahead and solves the quantum mechanical problem only for the system of electrons, and treating the nuclei either as entities fixed in a lattice, or perhaps as having some phononic degrees of freedom. The Schrödinger equation is then written out for discrete sets of fixed nuclear position:

[itex]\hat{H}_{e}\Psi_{e}(\mathbf{r} {;} \mathbf{R}) = \left\lbrace -\frac{1}{2}\sum_{i}\lambda_{i}^{2} - \sum_{Ai}\frac{Z_{A}}{R_{Ai}} + \sum_{i>j}\frac{1}{r_{ij}} \right\rbrace \Psi_{e}(\mathbf{r} {;} \mathbf{R}) = E_{e}(\mathbf{R})\Psi_{e}(\mathbf{r} {;} \mathbf{R}) [itex],

where i and j denote electrons and A denotes nuclei, with the distances rij = |ri-rj| and RAi = |RA-ri|.

The Born-Oppenheimer approximation is a good one, and has become a routine foundation stone for the physical study of solid and molecular systems. It is implicitly used in most computational calculations.

## Beyond the Born-Oppenheimer Approximation

The explicit consideration of the coupling of electronic and nuclear (vibrational) movement is known as electron-phonon coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as vibronic coupling.

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