Editing Electron configuration (section)

== 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}}</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>