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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. 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 formidable, consecutive steps. This approximation was proposed in the early days of quantum mechanics by Born and Oppenheimer (1927) and is still indispensable in quantum chemistry.$Psi\_\{\; total\}\; =\; psi\_\{\; electronic\}\; times\; psi\_\{\; nuclear\}$ ## Short description

The Born-Oppenheimer (BO) approximation is ubiquitous in quantum chemical calculations of molecular wavefunctions. It consists of two steps. ### Footnotes

## Derivation of the Born-Oppenheimer approximation

It will be discussed how the BO approximation may be derived and under which conditions it is applicable. At the same time we will show how the BO approximation may be improved by including vibronic coupling. To that end the second step of the BO approximation is generalized to a set of coupled eigenvalue equations depending on nuclear coordinates only. Off-diagonal elements in these equations are shown to be nuclear kinetic energy terms.
It will be shown that the BO approximation can be trusted whenever the PESs, obtained from the solution of the electronic Schrödinger equation, are well separated:

_{A} and M_{A}—the atomic number and mass of nucleus A.### Historical note

The Born-Oppenheimer approximation is named after M. Born and R. Oppenheimer who wrote a paper
[Annalen der Physik, vol. 84, pp. 457-484 (1927)] entitled: Zur Quantentheorie der Moleküle (On the Quantum Theory of Molecules). This paper describes the separation of electronic motion, nuclear vibrations, and molecular rotation. Somebody who expects to find in this paper the BO approximation—as it is explained above and in most modern textboooks—will be in for a surprise. The reason being that the presentation of the BO approximation is well hidden in Taylor expansions (in terms of internal and external nuclear coordinates) of (i) electronic wave functions, (ii) potential energy surfaces and (iii) nuclear kinetic energy terms. Internal coordinates are the relative positions of the nuclei in the molecular equilibrium and their displacements (vibrations) from equilibrium. External coordinates are the position of the center of mass and the orientation of the molecule. The Taylor expansions complicate the theory and make the derivations
very hard to follow. Moreover, knowing that the proper separation of vibrations and rotations was not achieved in this paper, but only 8 years later [by C. Eckart, Physical Review, vol. 46, pp. 383-387 (1935)] (see Eckart conditions), one is not very much motivated to invest much effort into understanding the work by Born and Oppenheimer, however famous it may be. Although the article still collects many citations each year, it is safe to say that it is not read anymore (except perhaps by historians of science).
### External links

Resources related to the Born-Oppenheimer approximation:## See also

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 $psi\_\{electronic\}$ 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 repeated for many different nuclear configurations. The set of electronic energies thus computed becomes a function of the 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; 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 an averaged electron-nuclear attraction.
The BO approximation follows from the inertia of electrons to be negligible in comparison to the atom to which they are bound.

In the first step the nuclear kinetic energy is neglected, that is, the corresponding operator T_{n} is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian H_{e} the nuclear positions enter as parameters. The electron-nucleus interactions are not removed and 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

- $H\_mathrm\{e\}(mathbf\{r,R\}\; );\; chi(mathbf\{r,R\})\; =\; E\_mathrm\{e\}\; ;\; chi(mathbf\{r,R\})$

is solved (out of necessity approximately). The quantity r stands for all electronic coordinates and R for all nuclear coordinates. Obviously, the electronic energy eigenvalue E_{e} 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_{e} as a function of R. This is the potential energy surface (PES): E_{e}(R) . 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.

In the second step of the BO approximation the nuclear kinetic energy T_{n} (containing partial derivatives with respect to the components of R) is reintroduced and the Schrödinger equation for the nuclear motion

- $$

- $E\_0(mathbf\{R\})\; ll\; E\_1(mathbf\{R\})\; ll\; E\_2(mathbf\{R\}),\; cdots$ for all :$mathbf\{R\}$.

We start from the exact non-relativistic, time-independent molecular Hamiltonian:

- $$

- $$

It is useful to introduce the total nuclear momentum and to rewrite the nuclear kinetic energy operator as follows:

- $T\_mathrm\{n\}\; =\; sum\_\{A\}\; sum\_\{alpha=x,y,z\}\; frac\{P\_\{Aalpha\}\; P\_\{Aalpha\}\}\{2M\_A\}$

Suppose we have K electronic eigenfunctions $chi\_k\; (mathbf\{r\};\; mathbf\{R\})$ of $H\_mathrm\{e\},$, that is, we have solved

- $$

- $$

The total wave function $Psi(mathbf\{R\},mathbf\{r\})$ is expanded in terms of $chi\_k\; (mathbf\{r\};\; mathbf\{R\})$:

- $$

- $$

- $big(mathbb\{H\}\_mathrm\{e\}(mathbf\{R\})big)\_\{k\text{'}k\}\; equiv\; langle\; chi\_\{k\text{'}\}(mathbf\{r\};mathbf\{R\})$

- $$

- $left[mathbb\{H\}\_mathrm\{n\}(mathbf\{R\})\; +\; mathbb\{H\}\_mathrm\{e\}(mathbf\{R\})\; right]$

The column vector $boldsymbol\{phi\}(mathbf\{R\})$ has elements $phi\_k(mathbf\{R\}),;\; k=1,ldots,K$. The matrix $mathbb\{H\}\_mathrm\{e\}(mathbf\{R\})$ is diagonal and the nuclear Hamilton matrix is non-diagonal with the following off-diagonal (vibronic coupling) terms,

- $$

The vibronic coupling in this approach is through nuclear kinetic energy terms. Solution of these coupled equations gives an approximation for energy and wavefunction that goes beyond the Born-Oppenheimer approximation. Unfortunately, the off-diagonal kinetic energy terms are usually difficult to handle. This is why often a diabatic transformation is applied, which retains part of the nuclear kinetic energy terms on the diagonal, removes the kinetic energy terms from the off-diagonal and creates coupling terms between the adiabatic PESs on the off-diagonal.

If we can neglect the off-diagonal elements the equations will uncouple and simplify drastically. In order to show when this neglect is justified, we suppress the coordinates in the notation and write, by applying the Leibniz rule for differentiation, the matrix elements of $T\_\{textrm\{n\}\}$ as

- $$

The diagonal ($k\text{'}=k$) matrix elements $langlechi\_\{k\}|big(P\_\{Aalpha\}chi\_kbig)rangle\_\{(mathbf\{r\})\}$ of the operator $P\_\{Aalpha\},$ vanish, because this operator is Hermitian and purely imaginary. The off-diagonal matrix elements satisfy

- $$

- $$

Conversely, if all surfaces are well separated, all off-diagonal terms can be neglected and hence the whole matrix of $P^\{A\}\_alpha$ is effectively zero. The third term on the right hand side of the expression for the matrix element of T_{n} (the Born-Oppenheimer diagonal correction) can approximately be written as the matrix of $P^\{A\}\_alpha$ squared and, accordingly, is then negligible also. Only the first (diagonal) kinetic energy term in this equation survives in the case of well-separated surfaces and a diagonal, uncoupled, set of nuclear motion equations results,

- $$

We reiterate that when two or more potential energy surfaces approach each other, or even cross, the Born-Oppenheimer approximation breaks down and one must fall back on the coupled equations. Usually one invokes then the diabatic approximation.

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