View source for Electron configuration

{{Short description|Mode of arrangement of electrons in different shells of an atom}}
[[File:Electron orbitals.svg|right|thumb|upright=1.5|[[Electron]] [[Atomic orbital|atomic]] and [[molecular orbital]]s]]
[[File:Electron shell 003 Lithium - no label.svg|thumb|120px|A [[Bohr model|Bohr diagram]] of [[lithium]]]]

In [[atomic physics]] and [[quantum chemistry]], the '''electron configuration''' is the [[Distribution (mathematics)|distribution]] of [[electron]]s of an [[atom]] or [[molecule]] (or other physical structure) in [[Atomic orbital|atomic]] or [[molecular orbital]]s.<ref name="IUPAC1">{{GoldBookRef|file=C01248|title=configuration (electronic)}}</ref> For example, the electron configuration of the [[neon]] atom is {{nowrap|1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup>}}, meaning that the 1s, 2s, and 2p [[Electron shell#Subshells|subshells]] are occupied by two, two, and six electrons, respectively.

Electronic configurations describe each electron as moving independently in an [[Atomic orbital|orbital]], in an average [[Field (physics)|field]] created by the [[Atomic nucleus|nuclei]] and all the other electrons. Mathematically, configurations are described by [[Slater determinants]] or [[configuration state function]]s.

According to the laws of [[Introduction to quantum mechanics|quantum mechanics]], a [[Energy level|level of energy]] is associated with each electron configuration. In certain conditions, electrons are able to move from one configuration to another by the emission or absorption of a [[quantum]] of energy, in the form of a [[photon]].

Knowledge of the electron configuration of different atoms is useful in understanding the structure of the [[periodic table|periodic table of elements]], for describing the [[chemical bond]]s that hold atoms together, and in understanding the [[chemical formula]]s of compounds and the [[Molecular geometry|geometries of molecules]]. In bulk materials, this same idea helps explain the peculiar properties of [[lasers]] and [[semiconductors]].

== Shells and subshells ==
{{Main|Electron shell}}
{| class="wikitable" align=right
|-
!
! s  ({{mvar|l}} = 0)
! colspan="3" |p ({{mvar|l}} = 1)
|-
!
! ''m'' = 0
! ''m'' = 0
! colspan="2" |''m'' = ±1
|-
!
! s
! p<sub>''z''</sub>
! p<sub>''x''</sub>
! p<sub>''y''</sub>
|- align=center
!''n'' = 1
| [[File:Atomic-orbital-cloud n1 l0 m0.png|32px]]
|
|
|
|- align=center
!''n'' = 2
| [[File:Atomic-orbital-cloud n2 l0 m0.png|48px]]
| [[File:Atomic-orbital-cloud n2 l1 m0.png|48px]]
| [[File:Atomic-orbital-cloud n2 px.png|48px]]
| [[File:Atomic-orbital-cloud n2 py.png|48px]]
|}

Electron configuration was first conceived under the [[Bohr model]] of the [[atom]], and it is still common to speak of [[Electron shell|shells and subshells]] despite the advances in understanding of the [[Quantum mechanics|quantum-mechanical]] nature of [[Electron|electrons]].

An [[electron shell]] is the [[Set (mathematics)|set]] of [[Quantum state|allowed states]] that share the same [[principal quantum number]], ''n'', that electrons may occupy. In each [[Term (logic)|term]] of an electron configuration, ''n'' is the [[Natural number|positive integer]] that precedes each [[Atomic orbital#Shapes of orbitals|orbital letter]] (e.g. [[helium]]'s electron configuration is 1s<sup>2</sup>, therefore ''n'' = 1, and the orbital contains two electrons). An atom's ''n''th electron shell can accommodate 2''n''<sup>2</sup> electrons. For example, the first shell can accommodate two&nbsp;electrons, the second shell eight&nbsp;electrons, the third shell eighteen, and so on. The factor of two arises because the number of allowed states doubles with each successive shell due to [[Spin quantum number|electron spin]]—each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +{{1/2}} (usually denoted by an up-arrow) and one with a spin of −{{1/2}} (with a down-arrow).

A [[Electron shell#Subshells|subshell]] is the set of states defined by a common [[azimuthal quantum number]], {{mvar|l}}, within a shell. The value of {{mvar|l}} is in the range from 0 to ''n''&nbsp;−&nbsp;1. The values {{mvar|l}}&nbsp;= 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively. For example, the 3d subshell has ''n''&nbsp;=&nbsp;3 and {{mvar|l}}&nbsp;=&nbsp;2. The maximum number of electrons that can be placed in a subshell is given by 2(2{{mvar|l}} + 1). This gives two electrons in an s&nbsp;subshell, six electrons in a p&nbsp;subshell, and ten electrons in a d&nbsp;subshell.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,{{efn|name="SchrodNote"|In formal terms, the [[quantum number]]s ''n'', {{mvar|l}} and ''m''{{sub|{{mvar|l}}}} arise from the fact that the solutions to the time-independent [[Schrödinger equation]] for [[hydrogen-like atom]]s are based on [[spherical harmonics]].}} in particular the [[Pauli exclusion principle]], which states that no two electrons in the same atom can have the same values of the four [[quantum number]]s.<ref>{{GoldBookRef|file=PT07089|title=Pauli exclusion principle}}</ref>

Exhaustive technical details about the complete quantum mechanical theory of atomic spectra and structure can be found and studied in the basic book of Robert D. Cowan.<ref>{{Cite book |last=Cowan |first=Robert D. |title=The Theory of Atomic Structure and Spectra |date=2020 |publisher=University of California Press |isbn=9780520906150}}</ref>

== Notation ==
{{See also|Atomic orbital}}
Physicists and chemists use a standard notation to indicate the electron configurations of atoms and molecules. For atoms, the notation consists of a sequence of atomic [[Electron shell#Subshells|subshell]] labels (e.g. for [[phosphorus]] the sequence 1s, 2s, 2p, 3s, 3p) with the number of electrons assigned to each subshell placed as a superscript. For example, [[hydrogen]] has one electron in the s-orbital of the first shell, so its configuration is written 1s<sup>1</sup>. [[Lithium]] has two electrons in the 1s-subshell and one in the (higher-energy) 2s-subshell, so its configuration is written 1s<sup>2</sup>&nbsp;2s<sup>1</sup> (pronounced "one-s-two, two-s-one"). [[Phosphorus]] ([[atomic number]] 15) is as follows: 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup>&nbsp;3s<sup>2</sup>&nbsp;3p<sup>3</sup>.

For atoms with many electrons, this notation can become lengthy and so an abbreviated notation is used. The electron configuration can be visualized as the [[core electron]]s, equivalent to the [[noble gas]] of the preceding [[Period (periodic table)|period]], and the [[valence electron]]s: each element in a period differs only by the last few subshells. Phosphorus, for instance, is in the third period. It differs from the second-period [[neon]], whose configuration is 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup>, only by the presence of a third shell. The portion of its configuration that is equivalent to neon is abbreviated as [Ne], allowing the configuration of phosphorus to be written as [Ne]&nbsp;3s<sup>2</sup>&nbsp;3p<sup>3</sup> rather than writing out the details of the configuration of neon explicitly. This convention is useful as it is the electrons in the outermost shell that most determine the chemistry of the element.

For a given configuration, the order of writing the orbitals is not completely fixed since only the orbital occupancies have physical significance. For example, the electron configuration of the [[titanium]] ground state can be written as either [Ar]&nbsp;4s<sup>2</sup>&nbsp;3d<sup>2</sup> or [Ar]&nbsp;3d<sup>2</sup>&nbsp;4s<sup>2</sup>. The first notation follows the order based on the [[Aufbau principle#Madelung energy ordering rule|Madelung rule]] for the configurations of neutral atoms; 4s is filled before 3d in the sequence Ar, K, Ca, Sc, Ti. The second notation groups all orbitals with the same value of ''n'' together, corresponding to the "spectroscopic" order of orbital energies that is the reverse of the order in which electrons are removed from a given atom to form positive ions; 3d is filled before 4s in the sequence Ti<sup>4+</sup>, Ti<sup>3+</sup>, Ti<sup>2+</sup>, Ti<sup>+</sup>, Ti.

The superscript 1 for a singly occupied subshell is not compulsory; for example [[aluminium]] may be written as either [Ne]&nbsp;3s<sup>2</sup>&nbsp;3p<sup>1</sup> or [Ne]&nbsp;3s<sup>2</sup>&nbsp;3p. In atoms where a subshell is unoccupied despite higher subshells being occupied (as is the case in some ions, as well as certain neutral atoms shown to deviate from the [[Aufbau principle#Madelung energy ordering rule|Madelung rule]]), the empty subshell is either denoted with a superscript 0 or left out altogether. For example, neutral [[palladium]] may be written as either {{nowrap|[Kr] 4d<sup>10</sup> 5s<sup>0</sup>}} or simply {{nowrap|[Kr] 4d<sup>10</sup>}}, and the [[Lanthanum|lanthanum(III)]] ion may be written as either {{nowrap|[Xe] 4f<sup>0</sup>}} or simply [Xe].<ref>{{Cite book|last1=Rayner-Canham|first1=Geoff|title=Descriptive Inorganic Chemistry|last2=Overton|first2=Tina|publisher=Macmillan Education|year=2014|isbn=978-1-319-15411-0|edition=6|location=|pages=13–15}}</ref>

It is quite common to see the letters of the orbital labels (s, p, d, f) written in an italic or slanting typeface, although the [[International Union of Pure and Applied Chemistry]] (IUPAC) recommends a normal typeface (as used here). The choice of letters originates from a now-obsolete system of categorizing [[spectral lines]] as "[[Sharp series|'''s'''harp]]", "[[Principal series (spectroscopy)|'''p'''rincipal]]", "[[Diffuse series|'''d'''iffuse]]" and "[[Fundamental series|'''f'''undamental]]" (or "'''f'''ine"), based on their observed [[fine structure]]: their modern usage indicates orbitals with an [[azimuthal quantum number]], {{mvar|l}}, of 0, 1, 2 or 3 respectively. After f, the sequence continues alphabetically g, h, i... ({{mvar|l}}&nbsp;= 4, 5, 6...), skipping j, although orbitals of these types are rarely required.<ref>{{cite web| url=http://scienceworld.wolfram.com/physics/ElectronOrbital.html|year=2007 |first=Eric W.|last= Weisstein|title=Electron Orbital|work=wolfram}}</ref><ref>{{cite book|title=General Chemistry
|first1=Darrell D. |last1=Ebbing|first2= Steven D. |last2=Gammon|url=https://books.google.com/books?id=_vRm5tiUJcsC&pg=PA284 |page=284|isbn=978-0-618-73879-3|date=2007-01-12|publisher=Cengage Learning }}</ref>

The electron configurations of molecules are written in a similar way, except that [[molecular orbital]] labels are used instead of atomic orbital labels (see below).

== Energy of ground state and excited states ==
The energy associated to an electron is that of its orbital. The energy of a configuration is often approximated as the sum of the energy of each electron, neglecting the electron-electron interactions. The configuration that corresponds to the lowest electronic energy is called the [[stationary state|ground state]]. Any other configuration is an [[excited state]].

As an example, the ground state configuration of the [[sodium]] atom is 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup>&nbsp;3s<sup>1</sup>, as deduced from the Aufbau principle (see below). The first excited state is obtained by promoting a 3s electron to the 3p subshell, to obtain the
1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup>&nbsp;3p<sup>1</sup> configuration, abbreviated as the 3p level. Atoms can move from one configuration to another by absorbing or emitting energy. In a [[sodium-vapor lamp]] for example, sodium atoms are excited to the 3p level by an electrical discharge, and return to the ground state by emitting yellow light of wavelength 589&nbsp;nm.

Usually, the excitation of [[valence electron]]s (such as 3s for sodium) involves energies corresponding to [[photon]]s of visible or [[ultraviolet]] light. The excitation of [[core electron]]s is possible, but requires much higher energies, generally corresponding to [[X-ray]] photons. This would be the case for example to excite a 2p electron of sodium to the 3s level and form the excited 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>5</sup>&nbsp;3s<sup>2</sup> configuration.

The remainder of this article deals only with the ground-state configuration, often referred to as "the" configuration of an atom or molecule.

== History ==
[[Irving Langmuir]] was the first to propose in his 1919 article "The Arrangement of Electrons in Atoms and Molecules" in which, building on [[Gilbert N. Lewis]]'s [[cubical atom]] theory and [[Walther Kossel]]'s chemical bonding theory, he outlined his "concentric theory of atomic structure".<ref>{{cite journal |last1=Langmuir |first1=Irving |author1-link=Irving Langmuir |date=June 1919 |title=The Arrangement of Electrons in Atoms and Molecules |journal=Journal of the American Chemical Society |volume=41 |issue=6 |pages=868–934 |doi=10.1021/ja02227a002|bibcode=1919JAChS..41..868L |url=https://zenodo.org/record/1429026 }}</ref> Langmuir had developed his work on electron atomic structure from other chemists as is shown in the development of the [[History of the periodic table]] and the [[Octet rule]].
[[Niels Bohr]] (1923) incorporated Langmuir's model that the [[Periodic table|periodicity]] in the properties of the elements might be explained by the electronic structure of the atom.<ref name="Bohr">{{cite journal | last = Bohr | first = Niels | s2cid = 123582460 | author-link = Niels Bohr | title = Über die Anwendung der Quantumtheorie auf den Atombau. I | journal = Zeitschrift für Physik| year = 1923 | volume = 13 | issue = 1 | page = 117|bibcode = 1923ZPhy...13..117B |doi = 10.1007/BF01328209 }}</ref> His proposals were based on the then current [[Bohr model]] of the atom, in which the electron shells were orbits at a fixed distance from the nucleus. Bohr's original configurations would seem strange to a present-day chemist: [[sulfur]] was given as 2.4.4.6 instead of 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup>&nbsp;3s<sup>2</sup>&nbsp;3p<sup>4</sup> (2.8.6). Bohr used 4 and 6 following [[Alfred Werner]]'s 1893 paper. In fact, the chemists accepted the concept of atoms long before the physicists. Langmuir began his paper referenced above by saying,<blockquote>«…The problem of the structure of atoms has been attacked mainly by physicists who have given little consideration to the chemical properties which must ultimately be explained by a theory of atomic structure. The vast store of knowledge of chemical properties and relationships, such as is summarized by the Periodic Table, should serve as a better foundation for a theory of atomic structure than the relatively meager experimental data along purely physical lines... These electrons arrange themselves in a series of concentric shells, the first shell containing two electrons, while all other shells tend to [[Octet rule|hold eight]].…»</blockquote>The valence electrons in the atom were described by [[Richard Abegg]] in 1904.<ref>{{cite journal
| doi = 10.1002/zaac.19040390125
| volume = 39
| issue = 1
| pages = 330–380
| last = Abegg
| first = R.
| title = Die Valenz und das periodische System. Versuch einer Theorie der Molekularverbindungen
| trans-title = Valency and the periodic system. Attempt at a theory of molecular compounds
| journal = Zeitschrift für Anorganische Chemie
| year = 1904
| url = https://zenodo.org/record/1428102
}}</ref>

In 1924, [[Edmund Clifton Stoner|E.&nbsp;C.&nbsp;Stoner]] incorporated [[Arnold Sommerfeld|Sommerfeld's]] third quantum number into the description of electron shells, and correctly predicted the shell structure of sulfur to be 2.8.6.<ref>{{cite journal | doi = 10.1080/14786442408634535 | last = Stoner | first = E.C. | author-link = Edmund Clifton Stoner | title = The distribution of electrons among atomic levels | journal = Philosophical Magazine |series=6th Series| volume = 48 | year = 1924 | pages = 719–36 | issue = 286}}</ref> However neither Bohr's system nor Stoner's could correctly describe the changes in [[Emission spectrum|atomic spectra]] in a [[magnetic field]] (the [[Zeeman effect]]).

Bohr was well aware of this shortcoming (and others), and had written to his friend [[Wolfgang Pauli]] in 1923 to ask for his help in saving quantum theory (the system now known as "[[old quantum theory]]"). Pauli hypothesized successfully that the Zeeman effect can be explained as depending only on the response of the outermost (i.e., valence) electrons of the atom. Pauli was able to reproduce Stoner's shell structure, but with the correct structure of subshells, by his inclusion of a fourth quantum number and his [[Pauli exclusion principle|exclusion principle]] (1925):<ref>{{cite journal | last = Pauli | first = Wolfgang | s2cid = 122477612 | author-link=Wolfgang Pauli | title = Über den Einfluss der Geschwindigkeitsabhändigkeit der elektronmasse auf den Zeemaneffekt | journal = Zeitschrift für Physik| year = 1925 | volume = 31 | issue = 1 | pages = 373 | doi = 10.1007/BF02980592|bibcode = 1925ZPhy...31..373P }} English translation from {{cite journal | last = Scerri | first = Eric R. | url = http://www.chem.ucla.edu/dept/Faculty/scerri/pdf/BJPS.pdf | title = The Electron Configuration Model, Quantum Mechanics and Reduction | journal = The British Journal for the Philosophy of Science| year = 1991 | volume = 42 | issue = 3 | pages = 309–25 | doi = 10.1093/bjps/42.3.309}}</ref>
{{Blockquote|
It should be forbidden for more than one electron with the same value of the main quantum number ''n'' to have the same value for the other three quantum numbers ''k'' [{{mvar|l}}], ''j'' [''m<sub>{{mvar|l}}</sub>''] and ''m'' [''m<sub>s</sub>''].
}}
The [[Schrödinger equation]], published in 1926, gave three of the four quantum numbers as a direct consequence of its solution for the hydrogen atom:{{efn|name="SchrodNote"}} this solution yields the atomic orbitals that are shown today in textbooks of chemistry (and above). The examination of atomic spectra allowed the electron configurations of atoms to be determined experimentally, and led to an empirical rule (known as Madelung's rule (1936),<ref name="Madelung">{{cite book | last = Madelung | first = Erwin | author-link = Erwin Madelung | title = Mathematische Hilfsmittel des Physikers | location = Berlin | publisher = Springer | year = 1936}}</ref> see below) for the order in which atomic orbitals are filled with electrons.

{{anchor|Madelung rule}}

== Atoms: Aufbau principle and Madelung rule ==
<!-- This section is linked from Electron configuration -->
{{See also|Electron configurations of the elements (data page)}}
The [[aufbau principle]] (from the [[German language|German]] ''Aufbau'', "building up, construction") was an important part of [[Niels Bohr|Bohr's]] original concept of electron configuration. It may be stated as:<ref>{{GoldBookRef|file=AT06996|title=aufbau principle}}</ref>
:''a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy subshells are filled before electrons are placed in higher-energy orbitals.''

[[File:Klechkowski rule 2.svg|upright=1.35|thumb|The approximate order of filling of atomic orbitals, following the arrows from 1s to 7p. (After 7p the order includes subshells outside the range of the diagram, starting with 8s.)]]The principle works very well (for the ground states of the atoms) for the known 118&nbsp;elements, although it is sometimes slightly wrong. The modern form of the aufbau principle describes an order of [[Specific orbital energy|orbital energies]] given by [[Aufbau principle#Madelung energy ordering rule|Madelung's rule (or Klechkowski's rule)]]. This rule was first stated by [[Charles Janet]] in 1929, rediscovered by [[Erwin Madelung]] in 1936,<ref name="Madelung" /> and later given a theoretical justification by [[V. M. Klechkowski]]:<ref>{{cite journal | title = Theoretical justification of Madelung's rule | journal = Journal of Chemical Education | last = Wong | first = D. Pan | year = 1979 | issue = 11 | pages = 714–18 | volume = 56 | doi = 10.1021/ed056p714|bibcode = 1979JChEd..56..714W }}</ref>
# [[Electron shell#Subshells|Subshells]] are filled in the order of increasing ''n''&nbsp;+&nbsp;{{mvar|l}}.
# Where two subshells have the same value of ''n''&nbsp;+&nbsp;{{mvar|l}}, they are filled in order of increasing ''n''.
This gives the following order for filling the orbitals:
:1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, (8s, {{Not a typo|5g}}, 6f, 7d, 8p, and 9s)

In this list the subshells in parentheses are not occupied in the ground state of the heaviest atom now known ([[Oganesson|Og]], ''Z''&nbsp;=&nbsp;118).

The aufbau principle can be applied, in a modified form, to the [[proton]]s and [[neutron]]s in the [[atomic nucleus]], as in the [[Nuclear shell model|shell model]] of [[nuclear physics]] and [[nuclear chemistry]].

=== Periodic table ===
{{Main article|Block (periodic table)}}
[[File:Periodic table blocks spdf (32 column).svg|thumb|upright=1.5|Electron configuration table showing [[Block (periodic table)|blocks]].]]
The form of the [[periodic table]] is closely related to the atomic electron configuration for each element. For example, all the elements of [[alkaline earth metal|group 2]] (the table's second column) have an electron configuration of [E]&nbsp;''n''s{{sup|2}} (where [E] is a [[noble gas]] configuration), and have notable similarities in their chemical properties. The periodicity of the periodic table in terms of [[periodic table block]]s is due to the number of electrons (2, 6, 10, and 14) needed to fill s, p, d, and f subshells. These blocks appear as the rectangular sections of the periodic table. The single exception is [[helium]], which despite being an s-block atom is conventionally placed with the other [[noble gas]]ses in the p-block due to its chemical inertness, a consequence of its full outer shell (though there is discussion in the contemporary literature on whether this exception should be retained).

The electrons in the [[Valence electron|valence (outermost) shell]] largely determine each element's [[Chemical property|chemical properties]]. The similarities in the chemical properties were remarked on more than a century before the idea of electron configuration.{{efn|The similarities in chemical properties and the numerical relationship between the [[atomic weight]]s of [[calcium]], [[strontium]] and [[barium]] was first noted by [[Johann Wolfgang Döbereiner]] in 1817.}}

=== Shortcomings of the aufbau principle ===
The aufbau principle rests on a fundamental postulate that the order of orbital energies is fixed, both for a given element and between different elements; in both cases this is only approximately true. It considers atomic orbitals as "boxes" of fixed energy into which can be placed two electrons and no more. However, the energy of an electron "in" an atomic orbital depends on the energies of all the other electrons of the atom (or ion, or molecule, etc.). There are no "one-electron solutions" for systems of more than one electron, only a set of many-electron solutions that cannot be calculated exactly{{efn|Electrons are [[identical particle]]s, a fact that is sometimes referred to as "indistinguishability of electrons". A one-electron solution to a many-electron system would imply that the electrons could be distinguished from one another, and there is strong experimental evidence that they can't be. The exact solution of a many-electron system is a [[N-body problem|''n''-body problem]] with ''n''&nbsp;≥&nbsp;3 (the nucleus counts as one of the "bodies"): such problems have evaded [[Mathematical analysis|analytical solution]] since at least the time of [[Leonhard Euler|Euler]].}} (although there are mathematical approximations available, such as the [[Hartree–Fock method]]).

The fact that the aufbau principle is based on an approximation can be seen from the fact that there is an almost-fixed filling order at all, that, within a given shell, the s-orbital is always filled before the p-orbitals. In a [[hydrogen-like atom]], which only has one electron, the s-orbital and the p-orbitals of the same shell have exactly the same energy, to a very good approximation in the absence of external electromagnetic fields. (However, in a real hydrogen atom, the [[energy level]]s are slightly split by the magnetic field of the nucleus, and by the [[quantum electrodynamic]] effects of the [[Lamb shift]].)

=== Ionization of the transition metals ===
The naïve application of the aufbau principle leads to a well-known [[paradox]] (or apparent paradox) in the basic chemistry of the [[transition metal]]s. [[Potassium]] and [[calcium]] appear in the periodic table before the transition metals, and have electron configurations [Ar]&nbsp;4s{{sup|1}} and [Ar]&nbsp;4s{{sup|2}} respectively, i.e. the 4s-orbital is filled before the 3d-orbital. This is in line with Madelung's rule, as the 4s-orbital has ''n''&nbsp;+&nbsp;{{mvar|l}}&nbsp;=&nbsp;4 (''n''&nbsp;=&nbsp;4, {{mvar|l}}&nbsp;=&nbsp;0) while the 3d-orbital has ''n''&nbsp;+&nbsp;{{mvar|l}}&nbsp;=&nbsp;5 (''n''&nbsp;=&nbsp;3, {{mvar|l}}&nbsp;=&nbsp;2). After calcium, most neutral atoms in the first series of transition metals ([[scandium]] through [[zinc]]) have configurations with two 4s electrons, but there are two exceptions. [[Chromium]] and [[copper]] have electron configurations [Ar]&nbsp;3d{{sup|5}}&nbsp;4s{{sup|1}} and [Ar]&nbsp;3d{{sup|10}}&nbsp;4s{{sup|1}} respectively, i.e. one electron has passed from the 4s-orbital to a 3d-orbital to generate a half-filled or filled subshell. In this case, the usual explanation is that "half-filled or completely filled subshells are particularly stable arrangements of electrons". However, this is not supported by the facts, as [[tungsten]] (W) has a Madelung-following d{{sup|4}}&nbsp;s{{sup|2}} configuration and not d{{sup|5}}&nbsp;s{{sup|1}}, and [[niobium]] (Nb) has an anomalous d{{sup|4}}&nbsp;s{{sup|1}} configuration that does not give it a half-filled or completely filled subshell.<ref name=mustdie>{{cite journal |last1=Scerri |first1=Eric |date=2019 |title=Five ideas in chemical education that must die |journal=Foundations of Chemistry |volume=21 |pages=61–69 |doi=10.1007/s10698-018-09327-y|s2cid=104311030 }}</ref>

The apparent paradox arises when electrons are ''removed'' from the transition metal atoms to form [[ion]]s. The first electrons to be ionized come not from the 3d-orbital, as one would expect if it were "higher in energy", but from the 4s-orbital. This interchange of electrons between 4s and 3d is found for all atoms of the first series of transition metals.{{efn|There are some cases in the second and third series where the electron remains in an s-orbital.}} The configurations of the neutral atoms (K, Ca, Sc, Ti, V, Cr, ...) usually follow the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, ...; however the successive stages of ionization of a given atom (such as Fe<sup>4+</sup>, Fe<sup>3+</sup>, Fe<sup>2+</sup>, Fe<sup>+</sup>, Fe) usually follow the order 1s, 2s, 2p, 3s, 3p, 3d, 4s, ...

This phenomenon is only paradoxical if it is assumed that the energy order of atomic orbitals is fixed and unaffected by the nuclear charge or by the presence of electrons in other orbitals. If that were the case, the 3d-orbital would have the same energy as the 3p-orbital, as it does in hydrogen, yet it clearly does not. There is no special reason why the Fe{{sup|2+}} ion should have the same electron configuration as the chromium atom, given that [[iron]] has two more protons in its nucleus than chromium, and that the chemistry of the two species is very different. Melrose and [[Eric Scerri]] have analyzed the changes of orbital energy with orbital occupations in terms of the two-electron repulsion integrals of the [[Hartree–Fock method]] of atomic structure calculation.<ref>{{cite journal | last = Melrose | first = Melvyn P. |author2=Scerri, Eric R. | title = Why the 4s Orbital is Occupied before the 3d | journal = Journal of Chemical Education | volume = 73 | issue = 6 | pages = 498–503 | year = 1996 | doi = 10.1021/ed073p498|bibcode = 1996JChEd..73..498M }}</ref> More recently Scerri has argued that contrary to what is stated in the vast majority of sources including the title of his previous article on the subject, 3d orbitals rather than 4s are in fact preferentially occupied.<ref>{{cite magazine |last=Scerri |first=Eric |author-link=Eric Scerri |date=7 November 2013 |title=The trouble with the aufbau principle |url=https://eic.rsc.org/feature/the-trouble-with-the-aufbau-principle/2000133.article |url-status=live |magazine=[[Education in Chemistry]] |volume=50 |issue=6 |pages=24–26 |publisher=[[Royal Society of Chemistry]] |archive-url=https://web.archive.org/web/20180121061346/https://eic.rsc.org/feature/the-trouble-with-the-aufbau-principle/2000133.article |archive-date=21 January 2018 |access-date=12 June 2018}}</ref>

In chemical environments, configurations can change even more: Th<sup>3+</sup> as a bare ion has a configuration of [Rn]&nbsp;5f<sup>1</sup>, yet in most Th<sup>III</sup> compounds the thorium atom has a 6d<sup>1</sup> configuration instead.<ref>{{cite journal |first1=Ryan R. |last1=Langeslay |first2=Megan E. |last2=Fieser |first3=Joseph W. |last3=Ziller |first4=Philip |last4=Furche |first5=William J. |last5=Evans |title=Synthesis, structure, and reactivity of crystalline molecular complexes of the {[C<sub>5</sub>H<sub>3</sub>(SiMe<sub>3</sub>)<sub>2</sub>]<sub>3</sub>Th}<sup>1−</sup> anion containing thorium in the formal +2 oxidation state |journal=Chem. Sci. |year=2015 |volume=6 |pages=517–521 |doi=10.1039/C4SC03033H|pmc=5811171 |pmid=29560172 |issue=1 }}</ref><ref>{{cite book|last1 = Wickleder|first1 = Mathias S.|first2 = Blandine|last2 = Fourest|first3 = Peter K.|last3 = Dorhout|ref = Wickleder et al.|contribution = Thorium|title = The Chemistry of the Actinide and Transactinide Elements|editor1-first = Lester R.|editor1-last = Morss|editor2-first = Norman M.|editor2-last = Edelstein|editor3-first = Jean|editor3-last = Fuger|edition = 3rd|date = 2006|volume = 3|publisher = Springer|location = Dordrecht, the Netherlands|pages = 52–160|url = http://radchem.nevada.edu/classes/rdch710/files/thorium.pdf|doi = 10.1007/1-4020-3598-5_3| isbn=978-1-4020-3555-5 |url-status = dead|archive-url = https://web.archive.org/web/20160307160941/http://radchem.nevada.edu/classes/rdch710/files/Thorium.pdf|archive-date = 2016-03-07}}</ref> Mostly, what is present is rather a superposition of various configurations.<ref name=mustdie /> For instance, copper metal is poorly described by either an [Ar]&nbsp;3d{{sup|10}}&nbsp;4s{{sup|1}} or an [Ar]&nbsp;3d{{sup|9}}&nbsp;4s{{sup|2}} configuration, but is rather well described as a 90% contribution of the first and a 10% contribution of the second. Indeed, visible light is already enough to excite electrons in most transition metals, and they often continuously "flow" through different configurations when that happens (copper and its group are an exception).<ref>{{Cite journal|doi = 10.26434/chemrxiv.11860941|title = The Chemical Bond Across the Periodic Table: Part 1 – First Row and Simple Metals|last1 = Ferrão|first1 = Luiz|last2 = Machado|first2 = Francisco Bolivar Correto|last3 = Cunha|first3 = Leonardo dos Anjos|last4 = Fernandes|first4 = Gabriel Freire Sanzovo|url = https://figshare.com/articles/The_Chemical_Bond_Across_the_Periodic_Table_Part_1_First_Row_and_Simple_Metals/11860941|journal =[[ChemRxiv]] | s2cid=226121612 |access-date = 23 August 2020|archive-date = 1 December 2020|archive-url = https://web.archive.org/web/20201201001121/https://figshare.com/articles/The_Chemical_Bond_Across_the_Periodic_Table_Part_1_First_Row_and_Simple_Metals/11860941|url-status = dead|url-access = subscription|doi-access = free}}</ref>

Similar ion-like 3d{{sup|''x''}}&nbsp;4s{{sup|0}} configurations occur in [[transition metal complex]]es as described by the simple [[crystal field theory]], even if the metal has [[oxidation state]]&nbsp;0. For example, [[chromium hexacarbonyl]] can be described as a chromium atom (not ion) surrounded by six [[carbon monoxide]] [[ligand]]s. The electron configuration of the central chromium atom is described as 3d{{sup|6}} with the six electrons filling the three lower-energy d orbitals between the ligands. The other two d orbitals are at higher energy due to the crystal field of the ligands. This picture is consistent with the experimental fact that the complex is [[diamagnetic]], meaning that it has no unpaired electrons. However, in a more accurate description using [[molecular orbital theory]], the d-like orbitals occupied by the six electrons are no longer identical with the d orbitals of the free atom.

=== Other exceptions to Madelung's rule ===
There are several more exceptions to [[Aufbau principle#Madelung energy ordering rule|Madelung's rule]] among the heavier elements, and as atomic number increases it becomes more and more difficult to find simple explanations such as the stability of half-filled subshells. It is possible to predict most of the exceptions by Hartree–Fock calculations,<ref>{{cite journal | last = Meek | first = Terry L. |author2=Allen, Leland C. | title = Configuration irregularities: deviations from the Madelung rule and inversion of orbital energy levels | journal = [[Chemical Physics Letters]] | volume = 362 | issue = 5–6 | pages = 362–64 | doi=10.1016/S0009-2614(02)00919-3 | year = 2002|bibcode = 2002CPL...362..362M }}</ref> which are an approximate method for taking account of the effect of the other electrons on orbital energies. Qualitatively, for example, the 4d elements have the greatest concentration of Madelung anomalies, because the 4d–5s gap is larger than the 3d–4s and 5d–6s gaps.<ref name=primefan>{{cite web |url=http://www.primefan.ru/stuff/personal/ptable.pdf |title=Периодическая система химических элементов Д. И. Менделеева |trans-title=D. I. Mendeleev's periodic system of the chemical elements |last=Kulsha |first=Andrey |date=2004 |website=primefan.ru |access-date=17 May 2020 |language=ru}}</ref>

For the heavier elements, it is also necessary to take account of the [[Relativistic quantum chemistry|effects of special relativity]] on the energies of the atomic orbitals, as the inner-shell electrons are moving at speeds approaching the [[speed of light]]. In general, these relativistic effects<ref>{{GoldBookRef|file=RT07093|title=relativistic effects}}</ref> tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals.<ref>{{cite journal | first = Pekka | last = Pyykkö | title = Relativistic effects in structural chemistry | journal = [[Chemical Reviews]] |year = 1988 | volume = 88 | pages = 563–94 | doi = 10.1021/cr00085a006 | issue = 3}}</ref> This is the reason why the 6d elements are predicted to have no Madelung anomalies apart from lawrencium (for which relativistic effects stabilise the p<sub>1/2</sub> orbital as well and cause its occupancy in the ground state), as relativity intervenes to make the 7s orbitals lower in energy than the 6d ones.

The table below shows the configurations of the f-block (green) and d-block (blue) atoms. It shows the ground state configuration in terms of orbital occupancy, but it does not show the ground state in terms of the sequence of orbital energies as determined spectroscopically. For example, in the transition metals, the 4s orbital is of a higher energy than the 3d orbitals; and in the lanthanides, the 6s is higher than the 4f and 5d. The ground states can be seen in the [[Electron configurations of the elements (data page)]]. However this also depends on the charge: a [[calcium]] atom has 4s lower in energy than 3d, but a Ca<sup>2+</sup> cation has 3d lower in energy than 4s. In practice the configurations predicted by the Madelung rule are at least close to the ground state even in these anomalous cases.<ref>See the [https://physics.nist.gov/PhysRefData/Handbook/periodictable.htm NIST tables]</ref> The empty f orbitals in lanthanum, actinium, and thorium contribute to chemical bonding,<ref name=Glotzel>{{cite journal |last1=Glotzel |first1=D. |date=1978 |title=Ground-state properties of f band metals: lanthanum, cerium and thorium |journal=Journal of Physics F: Metal Physics |volume=8 |issue=7 |pages=L163–L168 |doi=10.1088/0305-4608/8/7/004|bibcode=1978JPhF....8L.163G }}</ref><ref name=LaF3>{{cite journal |last1=Xu |first1=Wei |last2=Ji |first2=Wen-Xin |first3=Yi-Xiang |last3=Qiu |first4=W. H. Eugen |last4=Schwarz |first5=Shu-Guang |last5=Wang |date=2013 |title=On structure and bonding of lanthanoid trifluorides LnF<sub>3</sub> (Ln = La to Lu) |journal=Physical Chemistry Chemical Physics |volume=2013 |issue=15 |pages=7839–47 |doi=10.1039/C3CP50717C|pmid=23598823 |bibcode=2013PCCP...15.7839X }}</ref> as do the empty p orbitals in transition metals.<ref>[https://pubs.rsc.org/en/content/articlehtml/2015/sc/c5sc02776d Example for platinum]</ref>

Vacant s, d, and f orbitals have been shown explicitly, as is occasionally done,<ref>See for example [http://www.primefan.ru/stuff/chem/front2019.png this Russian periodic table poster] by A. V. Kulsha and T. A. Kolevich</ref> to emphasise the filling order and to clarify that even orbitals unoccupied in the ground state (e.g. [[lanthanum]] 4f or [[palladium]] 5s) may be occupied and bonding in chemical compounds. (The same is also true for the p-orbitals, which are not explicitly shown because they are only actually occupied for lawrencium in gas-phase ground states.)
{| class="wikitable"
|+Electron shells filled in violation of Madelung's rule<ref>{{cite book|first1= G. L.|last1= Miessler |first2= D. A.|last2= Tarr|title=Inorganic Chemistry|edition=2nd |publisher=Prentice-Hall |year=1999|page=38}}</ref> (red)<br />Predictions for elements 109–112<ref name=Haire />
|-
! colspan=3 | Period 4 || &nbsp; || colspan=3 | Period 5 || &nbsp; || colspan=3 | Period 6 || &nbsp; || colspan=3 | Period 7
|-
! Element || Z || Electron Configuration || &nbsp; || Element || Z || Electron Configuration || &nbsp; || Element || Z || Electron Configuration || &nbsp; || Element || Z || Electron Configuration
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Lanthanum]] || 57 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> <span style="color:red;">4f<sup>0</sup> 5d<sup>1</sup></span> || &nbsp; || [[Actinium]] || 89 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>0</sup> 6d<sup>1</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Cerium]] || 58 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> <span style="color:red;">4f<sup>1</sup> 5d<sup>1</sup></span> || &nbsp; || [[Thorium]] || 90 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>0</sup> 6d<sup>2</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Praseodymium]] || 59 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>3</sup> 5d<sup>0</sup> || &nbsp; || [[Protactinium]] || 91 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>2</sup> 6d<sup>1</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Neodymium]] || 60 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>4</sup> 5d<sup>0</sup> || &nbsp; || [[Uranium]] || 92 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>3</sup> 6d<sup>1</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Promethium]] || 61 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>5</sup> 5d<sup>0</sup> || &nbsp; || [[Neptunium]] || 93 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>4</sup> 6d<sup>1</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Samarium]] || 62 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>6</sup> 5d<sup>0</sup> || &nbsp; || [[Plutonium]] || 94 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>6</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Europium]] || 63 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>7</sup> 5d<sup>0</sup> || &nbsp; || [[Americium]] || 95 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>7</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Gadolinium]] || 64 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> <span style="color:red;">4f<sup>7</sup> 5d<sup>1</sup></span> || &nbsp; || [[Curium]] || 96 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> <span style="color:red;">5f<sup>7</sup> 6d<sup>1</sup></span>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Terbium]] || 65 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>9</sup> 5d<sup>0</sup> || &nbsp; || [[Berkelium]] || 97 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>9</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Dysprosium]] || 66 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>10</sup> 5d<sup>0</sup> || &nbsp; || [[Californium]] || 98 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>10</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Holmium]] || 67 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>11</sup> 5d<sup>0</sup> || &nbsp; || [[Einsteinium]] || 99 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>11</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Erbium]] || 68 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>12</sup> 5d<sup>0</sup> || &nbsp; || [[Fermium]] || 100 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>12</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Thulium]] || 69 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>13</sup> 5d<sup>0</sup> || &nbsp; || [[Mendelevium]] || 101 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>13</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|f-block}}"
| colspan=3 | &nbsp; || &nbsp; || colspan=3 | &nbsp; || &nbsp; || [[Ytterbium]] || 70 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>0</sup> || &nbsp; || [[Nobelium]] || 102 || &#91;[[radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>0</sup>
|- bgcolor="{{element color|d-block}}"
| [[Scandium]] || 21 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>1</sup> || &nbsp; || [[Yttrium]] || 39 || &#91;[[krypton|Kr]]&#93; 5s<sup>2</sup> 4d<sup>1</sup> || &nbsp; || [[Lutetium]] || 71 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>1</sup> || &nbsp; || [[Lawrencium]] || 103 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> <span style="color:red;">6d<sup>0</sup> 7p<sup>1</sup></span>
|- bgcolor="{{element color|d-block}}"
| [[Titanium]] || 22 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>2</sup> || &nbsp; || [[Zirconium]] || 40 || &#91;[[krypton|Kr]]&#93; 5s<sup>2</sup> 4d<sup>2</sup> || &nbsp; || [[Hafnium]] || 72 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>2</sup> || &nbsp; || [[Rutherfordium]] || 104 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>2</sup>
|- bgcolor="{{element color|d-block}}"
| [[Vanadium]] || 23 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>3</sup> || &nbsp; || [[Niobium]] || 41 || &#91;[[krypton|Kr]]&#93; <span style="color:red;">5s<sup>1</sup> 4d<sup>4</sup></span> || &nbsp; || [[Tantalum]] || 73 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>3</sup> || &nbsp; || [[Dubnium]] || 105 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>3</sup>
|- bgcolor="{{element color|d-block}}"
| [[Chromium]] || 24 || &#91;[[argon|Ar]]&#93; <span style="color:red;">4s<sup>1</sup> 3d<sup>5</sup></span> || &nbsp; || [[Molybdenum]] || 42 || &#91;[[krypton|Kr]]&#93; <span style="color:red;">5s<sup>1</sup> 4d<sup>5</sup></span> || &nbsp; || [[Tungsten]] || 74 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>4</sup> || &nbsp; || [[Seaborgium]] || 106 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>4</sup>
|- bgcolor="{{element color|d-block}}"
| [[Manganese]] || 25 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>5</sup> || &nbsp; || [[Technetium]] || 43 || &#91;[[krypton|Kr]]&#93; 5s<sup>2</sup> 4d<sup>5</sup> || &nbsp; || [[Rhenium]] || 75 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>5</sup> || &nbsp; || [[Bohrium]] || 107 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>5</sup>
|- bgcolor="{{element color|d-block}}"
| [[Iron]] || 26 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>6</sup> || &nbsp; || [[Ruthenium]] || 44 || &#91;[[krypton|Kr]]&#93; {{red|5s<sup>1</sup> 4d<sup>7</sup>}} || &nbsp; || [[Osmium]] || 76 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>6</sup> || &nbsp; || [[Hassium]] || 108 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>6</sup>
|- bgcolor="{{element color|d-block}}"
| [[Cobalt]] || 27 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>7</sup> || &nbsp; || [[Rhodium]] || 45 || &#91;[[krypton|Kr]]&#93; <span style="color:red;">5s<sup>1</sup> 4d<sup>8</sup></span> || &nbsp; || [[Iridium]] || 77
|| &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>7</sup> || &nbsp; || [[Meitnerium]] || 109
|| &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>7</sup>
|- bgcolor="{{element color|d-block}}"
| [[Nickel]] || 28 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>8</sup> or <br /> &#91;[[argon|Ar]]&#93; <span style="color:red;">4s<sup>1</sup> 3d<sup>9</sup></span> ([[Nickel#Electron configuration dispute|disputed]])<ref>{{cite book |url=https://archive.org/details/periodictableits0000scer |url-access=registration |pages=[https://archive.org/details/periodictableits0000scer/page/239 239]–240 |title=The periodic table: its story and its significance |author=Scerri, Eric R. |publisher=Oxford University Press|year=2007 |isbn=978-0-19-530573-9}}</ref>|| &nbsp; || [[Palladium]] || 46 || &#91;[[krypton|Kr]]&#93; <span style="color:red;">5s<sup>0</sup> 4d<sup>10</sup></span> || &nbsp; || [[Platinum]] || 78 || &#91;[[xenon|Xe]]&#93; <span style="color:red;">6s<sup>1</sup></span> 4f<sup>14</sup> <span style="color:red;">5d<sup>9</sup></span> || &nbsp; || [[Darmstadtium]] || 110
|| &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>8</sup>
|- bgcolor="{{element color|d-block}}"
| [[Copper]] || 29 || &#91;[[argon|Ar]]&#93; <span style="color:red;">4s<sup>1</sup> 3d<sup>10</sup></span> || &nbsp; || [[Silver]] || 47 || &#91;[[krypton|Kr]]&#93; <span style="color:red;">5s<sup>1</sup> 4d<sup>10</sup></span> || &nbsp; || [[Gold]] || 79 || &#91;[[xenon|Xe]]&#93; <span style="color:red;">6s<sup>1</sup></span> 4f<sup>14</sup> <span style="color:red;">5d<sup>10</sup></span> || &nbsp; || [[Roentgenium]] || 111 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>9</sup>
|- bgcolor="{{element color|d-block}}"
| [[Zinc]] || 30 || &#91;[[argon|Ar]]&#93; 4s<sup>2</sup> 3d<sup>10</sup> || &nbsp; || [[Cadmium]] || 48 || &#91;[[krypton|Kr]]&#93; 5s<sup>2</sup> 4d<sup>10</sup> || &nbsp; || [[Mercury (element)|Mercury]] || 80 || &#91;[[xenon|Xe]]&#93; 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>10</sup> || &nbsp; || [[Copernicium]] || 112 || &#91;[[Radon|Rn]]&#93; 7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>10</sup>
|}

The various anomalies describe the free atoms and do not necessarily predict chemical behavior. Thus for example neodymium typically forms the +3 oxidation state, despite its configuration {{nowrap|[Xe] 4f<sup>4</sup> 5d<sup>0</sup> 6s<sup>2</sup>}} that if interpreted naïvely would suggest a more stable +2 oxidation state corresponding to losing only the 6s electrons. Contrariwise, uranium as {{nowrap|[Rn] 5f<sup>3</sup> 6d<sup>1</sup> 7s<sup>2</sup>}} is not very stable in the +3 oxidation state either, preferring +4 and +6.<ref name=Jorgensen>{{cite book |last=Jørgensen |first=Christian K. |date=1988 |title=Handbook on the Physics and Chemistry of Rare Earths |volume=11 |chapter=Influence of rare earths on chemical understanding and classification |pages=197–292 |doi=10.1016/S0168-1273(88)11007-6|isbn=978-0-444-87080-3 }}</ref>

The electron-shell configuration of elements beyond [[hassium]] has not yet been empirically verified, but they are expected to follow Madelung's rule without exceptions until [[unbinilium|element 120]]. [[Unbiunium|Element 121]] should have the anomalous configuration {{nowrap|<nowiki>[</nowiki>[[Oganesson|Og]]<nowiki>]</nowiki> 8s<sup>2</sup> {{color|red|5g<sup>0</sup>}} 6f<sup>0</sup> 7d<sup>0</sup> {{color|red|8p<sup>1</sup>}}}}, having a p rather than a g electron. Electron configurations beyond this are tentative and predictions differ between models,<ref>{{cite journal |last1=Umemoto |first1=Koichiro |last2=Saito |first2=Susumu |date=1996 |title=Electronic Configurations of Superheavy Elements |url=https://journals.jps.jp/doi/pdf/10.1143/JPSJ.65.3175 |journal=Journal of the Physical Society of Japan |volume=65 |issue=10 |pages=3175–9 |doi=10.1143/JPSJ.65.3175 |bibcode=1996JPSJ...65.3175U |access-date=31 January 2021|url-access=subscription }}</ref> but Madelung's rule is expected to break down due to the closeness in energy of the {{Not a typo|5g}}, 6f, 7d, and 8p<sub>1/2</sub> orbitals.<ref name=Haire>{{cite book| title=The Chemistry of the Actinide and Transactinide Elements| editor1-last=Morss|editor2-first=Norman M.| editor2-last=Edelstein| editor3-last=Fuger|editor3-first=Jean| last1=Hoffman|first1=Darleane C. |last2=Lee |first2=Diana M. |last3=Pershina |first3=Valeria |chapter=Transactinides and the future elements| publisher= [[Springer Science+Business Media]]| year=2006| isbn=978-1-4020-3555-5| location=Dordrecht, The Netherlands| edition=3rd}}</ref> That said, the filling sequence 8s, {{Not a typo|5g}}, 6f, 7d, 8p is predicted to hold approximately, with perturbations due to the huge spin-orbit splitting of the 8p and 9p shells, and the huge relativistic stabilisation of the 9s shell.<ref>{{cite conference |url=https://www.epj-conferences.org/articles/epjconf/pdf/2016/26/epjconf-NS160-01001.pdf |title=Is the Periodic Table all right ("PT OK")? |last1=Pyykkö |first1=Pekka |date=2016 |conference=Nobel Symposium NS160 – Chemistry and Physics of Heavy and Superheavy Elements}}</ref>

== Open and closed shells ==
{{About|the concept in physics|the software|Open Shell|section=TRUE}}
In the context of [[atomic orbital]]s, an '''open shell''' is a [[valence shell]] which is not completely filled with [[electron]]s or that has not given all of its valence electrons through [[chemical bond]]s with other [[atom]]s or [[molecule]]s during a [[chemical reaction]]. Conversely a '''closed shell''' is obtained with a completely filled valence shell. This configuration is very [[Stable nuclide|stable]].<ref>{{cite web|url=http://www.newi.ac.uk/buckleyc/periodic.htm|title=Periodic table|access-date=2007-11-01|archive-url=https://web.archive.org/web/20071103074338/http://www.newi.ac.uk/buckleyc/periodic.htm|archive-date=2007-11-03|url-status=dead}}</ref>

For molecules, "open shell" signifies that there are [[Unpaired electron|unpaired electrons]]. In [[molecular orbital]] theory, this leads to molecular orbitals that are singly occupied. In [[computational chemistry]] implementations of molecular orbital theory, open-shell molecules have to be handled by either the [[restricted open-shell Hartree–Fock]] method or the [[unrestricted Hartree–Fock]] method. Conversely a closed-shell configuration corresponds to a state where all [[molecular orbital]]s are either doubly occupied or empty (a [[diradical|singlet state]]).<ref>{{cite book|chapter-url=http://www.semichem.com/ampacmanual/ci.html |url=http://www.semichem.com/ampacmanual/ |publisher=Semichem, Inc. |chapter=Chapter 11. Configuration Interaction|title=AMPAC™ 10 User Guide}}</ref>  Open shell molecules are more difficult to study computationally.<ref>{{cite web|url=http://iopenshell.usc.edu/|title=Laboratory for Theoretical Studies of Electronic Structure and Spectroscopy of Open-Shell and Electronically Excited Species – iOpenShell|website=iopenshell.usc.edu}}</ref>

== Noble gas configuration ==
{{Further|Noble gas}}
'''Noble gas configuration''' is the electron configuration of [[Noble gas|noble gases]]. The basis of all [[chemical reaction]]s is the tendency of [[chemical elements]] to acquire [[Stable nuclide|stability]]. [[Main-group element|Main-group atoms]] generally obey the [[octet rule]], while [[transition metal]]s generally obey the [[18-electron rule]]. The [[noble gas]]es ([[Helium|He]], [[Neon|Ne]], [[Argon|Ar]], [[Krypton|Kr]], [[Xenon|Xe]], [[Radon|Rn]]) are less [[Electrical reactance|reactive]] than other [[Chemical element|elements]] because they already have a noble gas configuration. [[Oganesson#Predicted compounds|Oganesson is predicted]] to be more reactive due to [[Relativistic quantum chemistry|relativistic effects]] for heavy atoms.
:{|class=wikitable
! Period
! Element
! colspan="7"| Configuration
|-
| 1 || [[Helium|He]] || 1s<sup>2</sup>|| || || || || ||
|-
| 2 || [[Neon|Ne]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>|| || || || ||
|-
| 3 || [[Argon|Ar]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>||3s<sup>2</sup> 3p<sup>6</sup>|| || || ||
|-
| 4 || [[Krypton|Kr]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>||3s<sup>2</sup> 3p<sup>6</sup>||4s<sup>2</sup> 3d<sup>10</sup> 4p<sup>6</sup>|| || ||
|-
| 5 || [[Xenon|Xe]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>||3s<sup>2</sup> 3p<sup>6</sup>||4s<sup>2</sup> 3d<sup>10</sup> 4p<sup>6</sup>||5s<sup>2</sup> 4d<sup>10</sup> 5p<sup>6</sup>|| ||
|-
| 6 || [[Radon|Rn]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>||3s<sup>2</sup> 3p<sup>6</sup>||4s<sup>2</sup> 3d<sup>10</sup> 4p<sup>6</sup>||5s<sup>2</sup> 4d<sup>10</sup> 5p<sup>6</sup>||6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>10</sup> 6p<sup>6</sup>||
|-
| 7 || [[Oganesson|Og]] || 1s<sup>2</sup>||2s<sup>2</sup> 2p<sup>6</sup>||3s<sup>2</sup> 3p<sup>6</sup>||4s<sup>2</sup> 3d<sup>10</sup> 4p<sup>6</sup>||5s<sup>2</sup> 4d<sup>10</sup> 5p<sup>6</sup>||6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>10</sup> 6p<sup>6</sup>||7s<sup>2</sup> 5f<sup>14</sup> 6d<sup>10</sup> 7p<sup>6</sup>
|}

Every system has the tendency to acquire the state of stability or a state of minimum energy, and so chemical elements take part in chemical reactions to acquire a stable electronic configuration similar to that of its nearest [[noble gas]]. An example of this tendency is two [[hydrogen]] (H) atoms reacting with one [[oxygen]] (O) atom to form [[water]] (H<sub>2</sub>O). Neutral atomic hydrogen has one electron in its [[Valence electron|valence shell]], and on formation of water it acquires a share of a second electron coming from oxygen, so that its configuration is similar to that of its nearest noble gas [[helium]] (He) with two electrons in its valence shell. Similarly, neutral atomic oxygen has six electrons in its valence shell, and acquires a share of two electrons from the two hydrogen atoms, so that its configuration is similar to that of its nearest noble gas [[neon]] with eight electrons in its valence shell.

== Electron configuration in molecules ==
'''Electron configuration in molecules''' is more complex than the electron configuration of atoms, as each [[molecule]] has a different [[Molecular orbital|orbital structure]]. The [[molecular orbital]]s are labelled according to their [[Molecular symmetry|symmetry]],{{efn|The labels are written in lowercase to indicate that they correspond to one-electron functions. They are numbered consecutively for each symmetry type ([[irreducible representation]] in the [[character table]] of the [[point group]] for the molecule), starting from the orbital of lowest energy for that type.}} rather than the [[atomic orbital]] labels used for [[Atom|atoms]] and [[Monatomic ion|monatomic ions]]; hence, the electron configuration of the [[Oxygen#Allotropes|dioxygen]] molecule, O{{sub|2}}, is written 1σ{{sub|g}}{{sup|2}}&nbsp;1σ{{sub|u}}{{sup|2}}&nbsp;2σ{{sub|g}}{{sup|2}}&nbsp;2σ{{sub|u}}{{sup|2}}&nbsp;3σ{{sub|g}}{{sup|2}}&nbsp;1π{{sub|u}}{{sup|4}}&nbsp;1π{{sub|g}}{{sup|2}},<ref>Levine I.N. ''Quantum Chemistry'' (4th ed., Prentice Hall 1991) p.376 {{ISBN|0-205-12770-3}}</ref><ref>Miessler G.L. and Tarr D.A. ''Inorganic Chemistry'' (2nd ed., Prentice Hall 1999) p.118 {{ISBN|0-13-841891-8}}</ref> or equivalently 1σ{{sub|g}}{{sup|2}}&nbsp;1σ{{sub|u}}{{sup|2}}&nbsp;2σ{{sub|g}}{{sup|2}}&nbsp;2σ{{sub|u}}{{sup|2}}&nbsp;1π{{sub|u}}{{sup|4}}&nbsp;3σ{{sub|g}}{{sup|2}}&nbsp;1π{{sub|g}}{{sup|2}}.<ref name="IUPAC1" /> The term 1π{{sub|g}}{{sup|2}} represents the two [[Electron|electrons]] in the two [[Degenerate energy levels|degenerate]] π*-orbitals ([[Antibonding molecular orbital|antibonding]]). From [[Hund's rules]], these electrons have parallel [[Electron magnetic moment|spins]] in the [[ground state]], and so dioxygen has a net [[magnetic moment]] (it is [[paramagnetic]]). The explanation of the paramagnetism of dioxygen was a major success for [[molecular orbital theory]].

The electronic configuration of polyatomic molecules can change without absorption or emission of a [[photon]] through [[vibronic coupling]]s.

=== Electron configuration in solids ===
In a [[solid]], the electron states become very numerous. They cease to be discrete, and effectively blend into continuous ranges of possible states (an [[electron band]]). The notion of electron configuration ceases to be relevant, and yields to [[band theory]].

== Applications ==
The most widespread application of electron configurations is in the rationalization of [[Chemical property|chemical properties]], in both [[Inorganic chemistry|inorganic]] and [[organic chemistry]]. In effect, electron configurations, along with some simplified forms of [[molecular orbital theory]], have become the modern equivalent of the [[valence (chemistry)|valence]] concept, describing the number and type of [[Chemical bond|chemical bonds]] that an [[atom]] can be expected to form.

This approach is taken further in [[computational chemistry]], which typically attempts to make [[Quantitative analysis (chemistry)|quantitative estimates]] of chemical properties. For many years, most such calculations relied upon the "[[linear combination of atomic orbitals]]" (LCAO) approximation, using an ever-larger and more complex [[basis set (chemistry)|basis set]] of [[Atomic orbital|atomic orbitals]] as the starting point. The last step in such a calculation is the assignment of electrons among the molecular orbitals according to the aufbau principle. Not all [[Computational chemistry#Methods|methods in computational chemistry]] rely on electron configuration: [[density functional theory]] (DFT) is an important example of a method that discards the model.

For [[Atom|atoms]] or [[Molecule|molecules]] with more than one [[electron]], the motion of electrons are [[Electron correlation|correlated]] and such a picture is no longer exact. A very large number of electronic configurations are needed to exactly describe any multi-electron system, and precisely associating a certain energy level with any single configuration is not possible. However, the electronic [[wave function]] is usually dominated by a very small number of configurations and therefore the notion of electronic configuration remains essential for multi-electron systems.

A fundamental application of electron configurations is in the interpretation of [[Emission spectrum|atomic spectra]]. In this case, it is necessary to supplement the electron configuration with one or more [[term symbol]]s, which describe the different [[Energy level|energy levels]] available to an atom. Term symbols can be calculated for any electron configuration, not just the [[Ground state|ground-state]] configuration listed in tables, although not all the energy levels are observed in practice. It is through the analysis of atomic spectra that the ground-state electron configurations of the elements were experimentally determined.

== See also ==
* [[Born–Oppenheimer approximation]]
* [[d electron count]]
* [[Electron configurations of the elements (data page)]]
* [[Extended periodic table]]&nbsp;– discusses the limits of the periodic table
* [[Group (periodic table)]]
* [[HOMO/LUMO]]
* [[Molecular term symbol]]
* [[Octet rule]]
* [[Periodic table (electron configurations)]]
* [[Spherical harmonics]]
* [[Unpaired electron]]
* [[Valence shell]]

== Notes ==
{{notelist}}
==References==
{{Reflist|colwidth=30em}}

== External links ==
{{Commons category|Electron configurations}}
* [http://www.hydrogenlab.de/elektronium/HTML/einleitung_hauptseite_uk.html What does an atom look like? Configuration in 3D]

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