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Born-Oppenheimer

Born-Oppenheimer approximation
Definition:The Born–Oppenheimer approximation is one of the most effective simplifications in quantum mechanics. By decoupling the motions of the electrons and the nuclei, it introduces the concept of electronic structure and allows to solve Schrödinger’s equation for molecules and solid-state materials.
Explanation:The fundamentals of quantum mechanics ascertain that the energy eigenvalue, E, of a system of nuclei and electrons can be calculated by applying the Hamiltonian operator, , onto the wave function, Ψ, as stated in the famous time-independent Schrödinger equation: ĤΨ=EΨ. Note that the Hamiltonian contains two kinetic energy operators: e of the electrons and N of the nuclei, as well as three potential energy operators: nuclei–electron attraction (Ne), electron–electron (ee) as well as nuclei–nuclei repulsion (NN). Because of a large mass difference of nuclei and electrons (note that the nucleus is about 10000 times heavier than the electron in a Fe case), the latter move more rapidly and, therefore, instantaneously adjust to the nucleus movement. Hence, the moving electrons may be treated separately from the slowly moving nuclei, and the Hamiltonian is simplified into:
Ĥ=T̂e+V̂Ne+V̂ee.
In other words, the kinetic energy of the nuclei is set to zero assuming that the nuclei are stationary; then, the repulsion between nuclei becomes a constant value which may be added once the electronic structure is solved.

In practice, a ground state of a system can be found iteratively by alternatively calculating the electronic wave function in a set of fixed nucleus, followed by another optimization of the nucleus configuration by calculating the Hellmann–Feynman forces acting on the nucleus from the wave function.
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Diagram:

Max Born (1882‒1970, left) and Julius Robert Oppenheimer (1904‒1967, right).
SFB-Link:Real solid state materials can be treated in quantum-mechanical terms only by allowing approximations such as Born–Oppenheimer among others. It is used in any ab initio method used in SFB 761.
References:Born, Max; Oppenheimer, J. Robert (1927). "Zur Quantentheorie der Molekeln", Annalen der Physik 389 (20): 457–484.