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Fix Matter see-also invoke

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Fix Matter see-also invoke

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{{Short description|Function describing an electron in an atom}}

{{Quantum matter backlink|Atoms}}

An '''atomic orbital''' is a quantum-mechanical function describing the location, wave-like behavior, and probability distribution of an [[Physics:Quantum atoms/electron|electron]] in an [[Physics:Quantum atoms/atom|atom]].<ref>{{cite book |last1=Orchin |first1=Milton |last2=MacOmber |first2=Roger S. |last3=Pinhas |first3=Allan R. |last4=Wilson |first4=R. Marshall |title=The Vocabulary and Concepts of Organic Chemistry |chapter=Atomic Orbital Theory |date=2005 |pages=1–24 |doi=10.1002/0471713740.ch1 |isbn=978-0-471-68028-4 }}</ref> Orbitals are central to [[Physics:Quantum atoms/electron configuration|electron configuration]], chemical bonding, and atomic spectra.
[[File:Electron_configuration.png|thumb|450px|right|Atomic orbitals describe the probability clouds in which electrons are likely to be found around a nucleus.]]
== Description ==
In [[Physics:Quantum mechanics|quantum mechanics]], an orbital is not a fixed path followed by an electron. Instead, it is a wave function whose squared magnitude gives the probability density for finding the electron in a region around the nucleus.<ref>{{cite book |last=Daintith |first=J. |url=https://archive.org/details/dictionaryofchem0000unse_r3p4/page/408/mode/2up?view=theater |title=Oxford Dictionary of Chemistry |publisher=Oxford University Press |year=2004 |isbn=978-0-19-860918-6 |location=New York |pages=407–409}}</ref>

Atomic orbitals are characterized by quantum numbers. The principal quantum number {{mvar|n}} is related to the shell and energy scale. The azimuthal quantum number {{mvar|ℓ}} determines the orbital type and angular momentum. The magnetic quantum number {{mvar|mℓ}} describes orientation. A fourth quantum number, the spin projection {{mvar|ms}}, describes electron spin.

An orbital can contain at most two electrons, and those two electrons must have opposite spin according to the Pauli exclusion principle.<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}}</ref>

== Electron properties ==
Electrons in atoms show both wave-like and particle-like behavior.

'''Wave-like properties include:'''

* electrons form standing-wave states around the nucleus
* the electron position is described probabilistically
* orbital shapes arise from wave interference and boundary conditions

'''Particle-like properties include:'''

* electrons carry a definite electric charge
* electrons occupy discrete quantum states
* transitions between orbitals involve absorption or emission of photons

This wave-particle behavior is why orbitals are often visualized as electron clouds rather than planetary orbits.

== Quantum numbers ==
Atomic orbitals are labeled by quantum numbers:

* {{mvar|n}} — principal quantum number; shell, size, and energy scale
* {{mvar|ℓ}} — azimuthal quantum number; orbital shape and angular momentum
* {{mvar|mℓ}} — magnetic quantum number; orbital orientation
* {{mvar|ms}} — spin quantum number; electron spin projection

The allowed values are restricted:

* {{mvar|n}} = 1, 2, 3, ...
* {{mvar|ℓ}} = 0 to {{mvar|n}} − 1
* {{mvar|mℓ}} = −{{mvar|ℓ}} to +{{mvar|ℓ}}
* {{mvar|ms}} = +1/2 or −1/2

The Pauli exclusion principle states that no two electrons in the same atom can have the same set of all four quantum numbers.

== Orbital names ==
Orbitals are named using the principal quantum number followed by a letter representing {{mvar|ℓ}}:

* {{mvar|ℓ}} = 0: s orbital
* {{mvar|ℓ}} = 1: p orbital
* {{mvar|ℓ}} = 2: d orbital
* {{mvar|ℓ}} = 3: f orbital

The labels s, p, d, and f come from early spectroscopic descriptions: sharp, principal, diffuse, and fundamental. For {{mvar|ℓ}} values above 3, the letters continue alphabetically as g, h, i, k, and so on.<ref>{{cite book|first=David|last=Griffiths|year=1995|title=Introduction to Quantum Mechanics|pages=190–191|publisher=Prentice Hall|isbn=978-0-13-124405-4}}</ref><ref>{{cite book|first=Ira|last=Levine|year=2000|title=Quantum Chemistry|edition=5|pages=[https://archive.org/details/quantumchemistry00levi_0/page/144 144–145]|publisher=Prentice Hall|isbn=978-0-13-685512-5|url=https://archive.org/details/quantumchemistry00levi_0/page/144}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=QbQJAgAAQBAJ&pg=PA106|title=Quanta, Matter, and Change: A Molecular Approach to Physical Chemistry|publisher=Oxford University Press|isbn=978-0-19-920606-3|year=2009|page=106|first1=Peter |last1=Atkins |first2=Julio |last2=de Paula |first3=Ronald |last3=Friedman}}</ref>

== Shapes of orbitals ==
The simplest orbital is the s orbital, which is spherically symmetric. The p orbitals have two lobes and are oriented along different axes. The d and f orbitals have more complex lobed shapes.

Orbital images usually show surfaces enclosing regions where the electron has a high probability of being found. These diagrams do not show fixed paths; they show probability distributions.

The number and arrangement of nodes are determined by quantum numbers. Nodal surfaces are regions where the probability density is zero.

== Hydrogen-like orbitals ==
The simplest exact orbitals are those of hydrogen-like atoms, which contain one electron. In such systems, the Schrödinger equation can be solved analytically.

Hydrogen-like orbitals are described using a radial part and an angular part:

:<math>\psi(r,\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi)</math>

For atoms with more than one electron, exact analytical solutions are not generally possible. Approximation methods such as Hartree–Fock theory and molecular orbital theory are used.

== Orbital energy ==
In one-electron atoms, orbital energy depends mainly on {{mvar|n}}. In multi-electron atoms, electron-electron interactions make the energy depend also on {{mvar|ℓ}}.

This explains why the filling order of orbitals in many-electron atoms is not simply 1s, 2s, 2p, 3s, 3p, 3d, and so on. Instead, the approximate filling sequence begins:

:1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p

This sequence is closely related to the structure of the periodic table and to [[Physics:Quantum atoms/electron configuration|electron configuration]].

== Electron configuration ==
Orbitals provide the framework for electron configuration. Electrons fill orbitals according to quantum rules such as:

* the Pauli exclusion principle
* Hund’s rule
* the Aufbau principle

The repeating structure of the periodic table arises because complete sets of s, p, d, and f orbitals can hold 2, 6, 10, and 14 electrons respectively.

== Transitions between orbitals ==
Electrons can move between orbital states by absorbing or emitting photons. Such transitions occur only when the photon energy matches the energy difference between the two states.

These transitions produce spectral lines and are central to atomic spectroscopy.

== History ==
Early atomic models treated electrons as orbiting particles. J. J. Thomson discovered the electron in 1897.<ref name="referenceC">{{cite journal |last1=Thomson |first1=J. J. |title=XL. Cathode Rays |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=October 1897 |volume=44 |issue=269 |pages=293–316 |doi=10.1080/14786449708621070 }}</ref> Hantaro Nagaoka proposed an orbit-based model in 1904.<ref name="Nagaoka 1904 445–455">{{cite journal |last1=Nagaoka |first1=H. |title=LV. Kinetics of a system of particles illustrating the line and the band spectrum and the phenomena of radioactivity |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=May 1904 |volume=7 |issue=41 |pages=445–455 |doi=10.1080/14786440409463141 }}</ref> Niels Bohr later introduced quantized electron orbits in 1913.<ref name="Bohr 1913 476">{{cite journal |last1=Nicholson |first1=J. W. |title=The Constitution of Atoms and Molecules |journal=Nature |date=May 1914 |volume=93 |issue=2324 |pages=268–269 |doi=10.1038/093268a0 |bibcode=1914Natur..93..268N |doi-access=free }}</ref>

The modern orbital concept emerged after the development of quantum mechanics, de Broglie matter waves, the Schrödinger equation, and Heisenberg’s uncertainty principle.<ref>{{cite journal |last1=Heisenberg |first1=W. |title=Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik |journal=Zeitschrift für Physik |date=March 1927 |volume=43 |issue=3–4 |pages=172–198 |doi=10.1007/BF01397280 |bibcode=1927ZPhy...43..172H }}</ref><ref>{{cite journal|last=Bohr | first=Niels| title=The Quantum Postulate and the Recent Development of Atomic Theory|journal=Nature |date=April 1928| volume=121 | pages=580–590|doi=10.1038/121580a0 |bibcode = 1928Natur.121..580B| issue=3050 |doi-access=free}}</ref>

The term ''orbital'' was introduced by Robert S. Mulliken in 1932 as a shortened form of ''one-electron orbital wave function''.<ref>{{cite journal |last1=Mulliken |first1=Robert S. |title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations |journal=Physical Review |date=July 1932 |volume=41 |issue=1 |pages=49–71 |doi=10.1103/PhysRev.41.49 |bibcode=1932PhRv...41...49M }}</ref><ref>{{cite journal |last1=Murrell |first1=John N |title=The origins and later developments of molecular orbital theory |journal=International Journal of Quantum Chemistry |date=5 September 2012 |volume=112 |issue=17 |pages=2875–2879 |doi=10.1002/qua.23293 }}</ref>

== Properties ==

* describes electron probability density
* linked to [[Physics:Quantum atoms/energy level|energy levels]]
* determines chemical behavior
* provides the basis for [[Physics:Quantum atoms/electron configuration|electron configuration]]
* classified as s, p, d, f, and higher orbitals
* each orbital holds at most two electrons
* transitions between orbitals produce spectral lines

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Atomic orbital|1}}

Physics:Quantum atoms/orbital - Revision history

imported>WikiHarold: Fix Matter see-also invoke

imported>WikiHarold: Fix Matter see-also invoke