In quantum chemistry

?, the computation of the energy and

wavefunction of an average-size

molecule is a formidable task that is alleviated by the

**Born–Oppenheimer (BO) approximation**, named after

Max Born and J. Robert Oppenheimer

?. For instance the benzene

? molecule consists of 12 nuclei and 42 electrons. The time independent

Schrödinger equation, which must be solved to obtain the energy and molecular

wavefunction of this

molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei. The

**BO approximation** makes it possible to compute the

wavefunction in two less complicated consecutive steps. This approximation was proposed in 1927, in the early period of

quantum mechanics, by Born and Oppenheimer and is still indispensable in quantum chemistry

?.

In basic terms, it allows the

wavefunction of a

molecule to be broken into its electronic and nuclear (vibrational, rotational) components.

In the first step of the

**BO approximation** the electronic

Schrödinger equation is solved, yielding the

wavefunction ψelectronic depending on electrons only. For benzene

? this

wavefunction depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the

equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates. In the second step of the

**BO approximation** this function serves as a potential in a

Schrödinger equation containing only the nuclei—for benzene

? an equation in 36 variables.

The success of the

**BO approximation** is due to the high

ratio between nuclear and electronic masses. The approximation is an important tool of quantum chemistry

?; without it only the lightest molecule, H

_{2}, could be handled, and all computations of molecular wavefunctions for larger molecules make use of it. Even in the cases where the

**BO approximation** breaks down, it is used as a point of departure for the computations.

The electronic energies, constituting the nuclear potential, consist of kinetic energies,

interelectronic repulsions and

electron–nuclear attractions. In a handwaving manner the nuclear potential is taken to be an averaged electron–nuclear attraction. The BO approximation

? follows from the

inertia of electrons being considered to be negligible in comparison to the

atom to which they are bound.

wikipedia - Born Oppenheimer approximation underline added, see

repulsion,

attraction
See Also

**12.11 - Eighteen Attributes or Dimensions**
**Angular Momentum coupling**
**magnetic moment**
**Quantum coupling**
**Renner-Teller Effect**
**Rotational-vibrational coupling**
**rovibronic coupling**
**spin-orbit coupling**
**spintronics**