Editing Atomic orbital (section)

=== Formal quantum mechanical definition ===
Atomic orbitals may be defined more precisely in formal [[quantum mechanics|quantum mechanical]] language. They are approximate solutions to the [[Schrödinger equation]] for the electrons bound to the atom by the [[electric field]] of the atom's [[Atomic nucleus|nucleus]]. Specifically, in quantum mechanics, the state of an atom, i.e., an [[eigenstate]] of the atomic [[Hamiltonian (quantum mechanics)|Hamiltonian]], is approximated by an expansion (see [[configuration interaction]] expansion and [[basis set (chemistry)|basis set]]) into [[linear combination]]s of anti-symmetrized products ([[Slater determinant]]s) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their [[Spin (physics)|spin]] component, one speaks of '''atomic spin orbitals'''.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this [[Nuclear structure#The independent-particle model|independent-particle model]] of products of single electron wave functions.<ref>[[Roger Penrose]], ''[[The Road to Reality]]''</ref> (The [[London dispersion force]], for example, depends on the correlations of the motion of the electrons.)

In [[atomic physics]], the [[atomic spectral line]]s correspond to transitions ([[Atomic electron transition|quantum leaps]]) between [[quantum state]]s of an atom. These states are labeled by a set of [[quantum number]]s summarized in the [[term symbol]] and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup> for the ground state of [[neon]]-term symbol: <sup>1</sup>S<sub>0</sub>).

This notation means that the corresponding Slater determinants have a clear higher weight in the [[configuration interaction]] expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given [[Atomic electron transition|transition]]. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are [[fermion]]s ruled by the [[Pauli exclusion principle]] and cannot be distinguished from each other.<ref>{{cite book |last1=Levine |first1=Ira N. |title=Quantum Chemistry |date=1991 |publisher=Prentice-Hall |isbn=0-205-12770-3 |page=262 |edition=4th |quote=Therefore, the wave function of a system of identical interacting particles must not distinguish among the particles.}}</ref> Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when [[electron correlation]] is large.

Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the [[Hartree–Fock]] approximation, which is one way to reduce the complexities of [[molecular orbital theory]].