Editing Born–Oppenheimer approximation (section)

== Detailed description ==
The BO approximation recognizes the large difference between the [[electron]] mass and the masses of atomic nuclei, and correspondingly the time scales of their motion. Given the same amount of momentum, the nuclei move much more slowly than the electrons. In mathematical terms, the BO approximation consists of expressing the [[wavefunction]] (<math>\Psi_\mathrm{total}</math>) of a molecule as the product of an electronic wavefunction and a nuclear ([[Molecular vibration|vibrational]], [[Rotational spectroscopy|rotational]]) wavefunction. <math> \Psi_\mathrm{total} = \psi_\mathrm{electronic} \psi_\mathrm{nuclear} </math>. This enables a separation of the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]] into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently.

In the first step, the nuclear [[kinetic energy]] is neglected,<ref group="note">Authors often justify this step by stating that "the heavy nuclei move more slowly than the light [[Electron|electrons]]". Classically, this statement makes sense only if the [[momentum]] ''p'' of electrons and nuclei is of the same order of magnitude. In that case ''m''<sub>n</sub> ≫ ''m''<sub>e</sub> implies ''p''<sup>2</sup>/(2''m''<sub>n</sub>) ≪ ''p''<sup>2</sup>/(2''m''<sub>e</sub>).  It is easy to show that for two bodies in circular orbits around their center of mass (regardless of individual masses), the momenta of the two bodies are equal and opposite, and that for any collection of particles in the center-of-mass frame, the net momentum is zero.  Given that the center-of-mass frame is the lab frame (where the molecule is stationary), the momentum of the nuclei must be equal and opposite to that of the electrons.  A hand-waving justification can be derived from quantum mechanics as well. The corresponding operators do not contain mass and the molecule can be treated as a [[Particle in a box|box containing the electrons and nuclei]]. Since the kinetic energy is ''p''<sup>2</sup>/(2''m''), it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 10<sup>4</sup>.{{Citation needed|date=January 2015}}</ref> that is, the corresponding operator ''T''<sub>n</sub> is subtracted from the total [[molecular Hamiltonian]]. In the remaining electronic Hamiltonian ''H''<sub>e</sub> the nuclear positions are no longer variable, but are constant parameters (they enter the equation "parametrically"). The electron&ndash;nucleus interactions are ''not'' removed, i.e., the electrons still "feel" the [[Coulomb potential]] of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the ''clamped-nuclei'' approximation.)

The electronic [[Schrödinger equation]]
: <math> H_\text{e}(\mathbf r, \mathbf R) \chi(\mathbf r, \mathbf R) = E_\text{e} \chi(\mathbf r, \mathbf R) </math>
where <math> \chi(\mathbf r, \mathbf R)
 </math>, the electronic wavefunction for given positions of nuclei (fixed '''R'''), is solved approximately.<ref group=note>Typically, the electronic Schrödinger equation for molecules cannot be solved exactly. Approximation methods include the [[Hartree-Fock method]]</ref> The quantity '''r''' stands for all electronic coordinates and '''R''' for all nuclear coordinates. The electronic energy [[eigenvalue]] ''E''<sub>e</sub> depends on the chosen positions '''R''' of the nuclei. Varying these positions '''R''' in small steps and repeatedly solving the electronic [[Schrödinger equation]], one obtains ''E''<sub>e</sub> as a function of '''R'''. This is the [[potential energy surface]] (PES): <math> E_e(\mathbf R)</math>. Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the [[adiabatic theorem]], this manner of obtaining a PES is often referred to as the ''adiabatic approximation'' and the PES itself is called an ''adiabatic surface''.<ref group=note>It is assumed, in accordance with the [[adiabatic theorem]], that the same electronic state (for instance, the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a discontinuity (jump) in the PES if electronic state switching would occur.{{Citation needed|date=January 2015}}</ref>

In the second step of the BO approximation, the nuclear kinetic energy ''T''<sub>n</sub> (containing partial derivatives with respect to the components of '''R''') is reintroduced, and the Schrödinger equation for the nuclear motion<ref group=note>This equation is time-independent, and stationary wavefunctions for the nuclei are obtained; nevertheless, it is traditional to use the word "motion" in this context, although classically motion implies time dependence.{{Citation needed|date=January 2015}}
</ref>
: <math> [T_\text{n} + E_\text{e}(\mathbf R)] \phi(\mathbf R) = E \phi(\mathbf R) </math>
is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the [[Eckart conditions]]. The eigenvalue ''E'' is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule.{{clarify |date=July 2019 |reason=This text was moved from introduction where context didn't seem sufficient. Averaging over electronic configurations may perhaps be relevant for defining a molecular equilibrium, but seems wrong to define a potential energy surface associated with a particular electronic state}} In accord with the [[Hellmann–Feynman theorem]], the nuclear potential is taken to be an average over electron configurations of the sum of the electron&ndash;nuclear and internuclear electric potentials.