Born–Oppenheimer approximation

The Born- Oppenheimer approximation or Born-Oppenheimer approximation or adiabatic approximation is an approximation to simplify the Schrödinger equation for a system with more severe and at least one light particles. It takes place in the quantum mechanical treatment of molecules and solids application, as these consist of at least two atomic nuclei and a large number of much lighter electrons. The approximation is in physical chemistry widely used as an analytically exact solution of the Schrödinger equation is known only for the simplest systems, such as the hydrogen atom, or ion. The Born -Oppenheimer approximation is named after Max Born and J. Robert Oppenheimer, and was first published in 1927 in the Annals of Physics.

Motivation

The quantum- mechanical state of function of a molecule or solid is a function of the degrees of freedom of all the electrons and atomic nuclei. In the following is omitted, the spin degree of freedom of the particles, the positions of all the electrons are combined in the vector, the positions of all the atoms in the vector. The state function is determined as usual from the corresponding Schrödinger equation

The molecular Hamiltonian

In addition to the kinetic operators

And

And repulsion between the electrons

The repulsion between the cores

And the attraction between nuclei and electrons

As mentioned in the introduction, the molecular Schrödinger equation can be solved analytically only for the simplest systems. A numerical solution of the whole system is not feasible due to the high dimensionality. In order to make the molecular Schrödinger equation releasably, so an approximation is needed.

Principle

The Born- Oppenheimer approximation separates the molecular Schrödinger equation into an equation for the electrons and one for the cores. The two sub-problems are then solved much easier by taking advantage of symmetries. Separation of the electronic and nuclear degrees of freedom is carried out based on the large difference in mass, resulting in a higher inertia of the cores. Because all of the particles with one another to interact primarily by Coulomb forces are equally substantially the electrons are much more accelerated than the cores.

The essence of the Born- Oppenheimer approximation can be represented as follows:

• From the perspective of the electrons, the nuclei stand still practical. So first the kinetic operator of the nuclei is neglected. This results in a Schrodinger equation for the electrons, wherein the position of the cores included as a parameter in the attractive and repulsive potential energy potential. This results in electronic eigenstates and associated self- energies, which depend parametrically on the positions of the nuclei.
• In contrast, the movement of the nuclei of the instantaneous position of the electron is almost unaffected. However, the nuclei feel the natural energy of the electronic state. Thus, each electronic state generates its own potential in which the nuclei move then.

Mathematical formulation

The requirement for this approach is the assumption that the movement of electrons, and the core movement can be separated. This assumption leads to a molecular wave equation, which consists of a product of the electron wave function and the core wave function:

Next you meet the assumptions that:

• Depends on the positions of the core, but not its speed. That is, the core movement is much less than the movement of electrons, that they can be assumed to be fixed and only flows in as a parameter
• And hence the core shaft function depends only on the nuclear coordinate.

Applying now the Hamiltonian () on the entire wave function, so you get two separate expressions:

• One for the movement of the electron

Electron Schrödinger equation:

With

• And one for the core movement

In sum, the large difference in the relative masses between electrons and nuclei thus makes it possible to separate the wave function into an electron - wave equation and a nuclear wave equation.

Method

For various core distances, the Schrödinger equation is solved successively. Is finally obtained a relationship between the length and the binding energy of the molecule. This is expressed by the potential curve. From the potential curve, the equilibrium distance and dissociation energy of the bond can be determined.

The Born- Oppenheimer approximation leads to good results for molecules in the ground state, particularly those with heavy nuclei. However, they can lead to very poor results for excited molecules and cations, which is particularly important when the photoelectron spectroscopy.