Editing Atomic orbital (section)

== Electron properties ==
With the development of [[quantum mechanics]] and experimental findings (such as the two slit diffraction of electrons), it was found that the electrons orbiting a nucleus could not be fully described as particles, but needed to be explained by [[wave–particle duality]]. In this sense, electrons have the following properties:

'''Wave-like properties:'''
# Electrons do not orbit a nucleus in the manner of a planet orbiting a star, but instead exist as  [[standing wave]]s. Thus the lowest possible energy an electron can take is similar to the [[fundamental frequency]] of a wave on a string. Higher energy states are similar to [[harmonics]] of that fundamental frequency.
# The electrons are never in a single point location, though the probability of interacting with the electron at a single point can be found from the electron's [[wave function]]. The electron's charge acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wave function.

'''Particle-like properties:'''
# The number of electrons orbiting a nucleus can be only an integer.
# Electrons jump between orbitals like particles. For example, if one [[photon]] strikes the electrons, only one electron changes state as a result.
# Electrons retain particle-like properties such as: each wave state has the same electric charge as its electron particle. Each wave state has a single discrete spin (spin up or spin down) depending on its [[Quantum superposition|superposition]].

Thus, electrons cannot be described simply as solid particles. An analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when one electron is present. When more electrons are added, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ("electron cloud"<ref>{{cite book| title=The Feynman Lectures on Physics – The Definitive Edition, Vol 1 lect 6| page=11| year=2006| publisher= Pearson PLC, Addison Wesley|isbn =978-0-8053-9046-9|last1= Feynman|first1= Richard|last2= Leighton|first2=Robert B.|first3=Matthew|last3=Sands}}</ref>) tends toward a generally spherical zone of probability describing the electron's location, because of the [[uncertainty principle]].

One should remember that these orbital 'states', as described here, are merely [[eigenstates]] of an electron in its orbit.  An actual electron exists in a superposition of states, which is like a [[weighted average]], but with [[complex number]] weights.  So, for instance, an electron could be in a pure eigenstate (2, 1, 0), or a mixed state {{sfrac|1|2}}(2, 1, 0) + {{sfrac|1|2}} <math>i</math> (2, 1, 1), or even the mixed state {{sfrac|2|5}}(2, 1, 0) + {{sfrac|3|5}} <math>i</math> (2, 1, 1).  For each eigenstate, a property has an [[eigenvalue]].  So, for the three states just mentioned, the value of <math>n</math> is 2, and the value of <math>l</math> is 1. For the second and third states, the value for <math>m_l</math> is a superposition of 0 and 1.  As a superposition of states, it is ambiguous—either exactly 0 or exactly 1—not an intermediate or average value like the fraction {{sfrac|1|2}}.  A superposition of [[Quantum state|eigenstates]] (2, 1, 1) and (3, 2, 1) would have an ambiguous <math>n</math> and <math>l</math>, but <math>m_l</math> would definitely be 1.  Eigenstates make it easier to deal with the math.  You can choose a different [[Basis (linear algebra)|basis]] of eigenstates by superimposing eigenstates from any other basis (see [[Atomic_orbital#Real orbitals|Real orbitals]] below).

=== 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]].

=== Types of orbital ===
[[File:Atomic-orbital-clouds spdf m0.png|thumb|upright=1.5|3D views of some [[Hydrogen-like atom|hydrogen-like]] atomic orbitals showing probability density and phase ('''g''' orbitals and higher not shown)]]
Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the [[Schrödinger equation]] for a [[Hydrogen-like atom|hydrogen-like "atom"]] (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The [[coordinate system]]s chosen for orbitals are usually [[spherical coordinates]] {{math|(''r'', ''θ'', ''φ'')}} in atoms and [[Cartesian coordinate system|Cartesian]] {{math|(''x'', ''y'', ''z'')}} in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate: {{math|1=''ψ''(''r'', ''θ'', ''φ'') = ''R''(''r'') Θ(''θ'') Φ(''φ'')}}. The angular factors of atomic orbitals {{math|1=Θ(''θ'') Φ(''φ'')}} generate s, p, d, etc. functions as [[Spherical harmonics#Real form|real combinations]] of [[spherical harmonics]] {{math|''Y''<sub>''ℓm''</sub>(''θ'', ''φ'')}} (where {{mvar|ℓ}} and {{mvar|m}} are quantum numbers). There are typically three mathematical forms for the radial functions&nbsp;{{math|''R''(''r'')}} which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons:

# The ''hydrogen-like orbitals'' are derived from the exact solutions of the Schrödinger equation for one electron and a nucleus, for a [[hydrogen-like atom]]. The part of the function that depends on distance ''r'' from the nucleus has radial [[node (physics)|nodes]] and decays as <math> e^{-\alpha r} </math>.
# The [[Slater-type orbital]] (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital.
# The form of the [[Gaussian orbital|Gaussian type orbital]] (Gaussians) has no radial nodes and decays as <math> e^{-\alpha r^2} </math>.

Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.