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{{Short description|Principle of atomic physics}}
{{redirect|Atomic build-up|the spread of nuclear weapons|Nuclear proliferation}}
{{sidebar periodic table|structure|image=Aufbau animated.png|caption=Electrons occupy the shells and sub-shells of an atom in approximate accordance with the Aufbau principle.}}

In [[atomic physics]] and [[quantum chemistry]], the '''Aufbau principle''' ({{IPAc-en|'|au|f|b|au}}, from {{langx|de|Aufbauprinzip|lit='''building-up principle'''}}), also called the '''Aufbau rule''', states that in the [[ground state]] of an [[atom]] or [[ion]], [[electron]]s first fill [[Electron shell#Subshells|subshells]] of the lowest available [[energy]], then fill subshells of higher energy. For example, the 1s subshell is filled before the 2s subshell is occupied. In this way, the electrons of an atom or ion form the most stable [[electron configuration]] possible. An example is the configuration {{nowrap|1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> 3s<sup>2</sup> 3p<sup>3</sup>}} for the [[phosphorus]] atom, meaning that the 1s subshell has 2 electrons, the 2s subshell has 2 electrons, the 2p subshell has 6 electrons, and so on.

The configuration is often abbreviated by writing only the [[valence electron]]s explicitly, while the [[core electron]]s are replaced by the symbol for the last previous [[noble gas]] in the [[periodic table]], placed in square brackets. For phosphorus, the last previous noble gas is neon, so the configuration is abbreviated to [Ne] 3s<sup>2</sup> 3p<sup>3</sup>, where [Ne] signifies the core electrons whose configuration in phosphorus is identical to that of neon.

Electron behavior is elaborated by other principles of [[atomic physics]], such as [[Hund's rule]] and the [[Pauli exclusion principle]]. Hund's rule asserts that if [[Degenerate orbitals|multiple orbitals of the same energy]] are available, electrons will occupy different [[Atomic orbital|orbitals]] singly and with the same [[Spin (physics)|spin]] before any are occupied doubly. If double occupation does occur, the Pauli exclusion principle requires that electrons that occupy the same orbital must have different [[Spin (physics)|spin]]s (+{{1/2}} and −{{1/2}}).

Passing from one [[Chemical element|element]] to another of the next higher [[atomic number]], one [[proton]] and one electron are added each time to the neutral atom.
The maximum number of electrons in any [[Electron shell|shell]] is 2''n''<sup>2</sup>, where ''n'' is the [[principal quantum number]].
The maximum number of electrons in a subshell is equal to 2(2{{mvar|l}}&nbsp;+&nbsp;1), where the [[azimuthal quantum number]] {{mvar|l}} is equal to 0, 1, 2, and 3 for s, p, d, and f subshells, so that the maximum numbers of electrons are 2, 6, 10, and 14 respectively. In the [[ground state]], the electronic configuration can be built up by placing electrons in the lowest available subshell until the total number of electrons added is equal to the atomic number. Thus subshells are filled in the order of increasing energy, using two general rules to help predict electronic configurations:
# Electrons are assigned to subshells in order of increasing value of ''n''&nbsp;+&nbsp;{{mvar|l}}.
# For subshells with the same value of ''n''&nbsp;+&nbsp;{{mvar|l}}, electrons are assigned first to the subshell with lower ''n''.

A version of the aufbau principle known as the [[nuclear shell model]] is used to predict the configuration of protons and [[neutron]]s in an [[atomic nucleus]].<ref>{{cite book |last1=Cottingham |first1=W. N. |last2=Greenwood |first2=D. A. |title=An introduction to nuclear physics |publisher=Cambridge University Press |date=1986 |isbn=0-521-31960-9 |chapter=Chapter 5: Ground state properties of nuclei: the shell model |chapter-url-access=registration |chapter-url=https://archive.org/details/introductiontonu0000cott }}</ref>

==Madelung energy ordering rule==
[[File:Aufbau Principle-en.svg|thumb|left|The states crossed by same red arrow have same ''n''&nbsp;+&nbsp;{{mvar|l}} value. The direction of the red arrow indicates the order of state filling.]]
[[Image:Atomic_orbitals_as_triangles.svg|thumb|left|For multielectron atoms the energy spectra of shells interleave resulting in the ''n''&nbsp;+&nbsp;{{mvar|l}} rule]]
In neutral atoms, the approximate order in which subshells are filled is given by the ''n''&nbsp;+&nbsp;{{mvar|l}} rule, also known as the:
* Madelung rule (after [[Erwin Madelung]])
* Janet rule (after [[Charles Janet]])
* Klechkowsky rule (after [[Vsevolod Klechkovsky]])
* Wiswesser's rule (after [[William Wiswesser]])
* Moeller's rubric<ref>{{Cite book |last=Pfennig |first=Brian W. |url=https://www.google.fr/books/edition/Principles_of_Inorganic_Chemistry/TctVEAAAQBAJ?hl=en&gbpv=1&dq=moeller+diagram+electrons&pg=PA100&printsec=frontcover |title=Principles of Inorganic Chemistry |date=2022-02-02 |publisher=John Wiley & Sons |isbn=978-1-119-65032-4 |language=en}}</ref>
* ''aufbau'' (building-up) rule or
* diagonal rule<ref>{{cite web |title=Electron configuration |date=19 September 2013 |website=[[WyzAnt]] |url=http://www.wyzant.com/resources/lessons/science/chemistry/electron_configuration }}</ref>

Here ''n'' represents the principal quantum number and {{mvar|l}} the azimuthal quantum number; the values {{mvar|l}}&nbsp;= 0, 1, 2, 3 correspond to the s, p, d, and f subshells, respectively. Subshells with a lower ''n''&nbsp;+&nbsp;{{mvar|l}} value are filled before those with higher ''n''&nbsp;+&nbsp;{{mvar|l}} values. In the many cases of equal ''n''&nbsp;+&nbsp;{{mvar|l}} values, the subshell with a lower ''n'' value is filled first.  The subshell ordering by this rule is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, 8s, 5g, ... For example, thallium (''Z''&nbsp;=&nbsp;81) has the ground-state configuration {{nowrap|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>1</sup>}}<ref>{{cite book |last1=Miessler |first1=Gary L. |last2=Tarr |first2=Donald A. |title=Inorganic Chemistry |edition=2nd |publisher=Prentice Hall |date=1998  |page=38 |isbn=0-13-841891-8}}</ref> or in condensed form, [Xe] 6s<sup>2</sup> 4f<sup>14</sup> 5d<sup>10</sup> 6p<sup>1</sup>.

Other authors write the subshells outside of the noble gas core in order of increasing ''n'', or if equal, increasing ''n'' + ''l'', such as Tl (''Z''&nbsp;=&nbsp;81) {{nowrap|[Xe]4f<sup>14</sup> 5d<sup>10</sup> 6s<sup>2</sup> 6p<sup>1</sup>}}.<ref name=Jolly>{{cite book |last1=Jolly |first1=William L. |title=Modern Inorganic Chemistry |edition=1st |publisher=McGraw-Hill |date=1984 |pages=[https://archive.org/details/trent_0116300649799/page/10 10–12] |isbn=0-07-032760-2 |url=https://archive.org/details/trent_0116300649799/page/10 }}</ref> They do so to emphasize that if this atom is [[Ionization|ionized]], electrons leave approximately in the order 6p, 6s, 5d, 4f, etc. On a related note, writing configurations in this way emphasizes the outermost electrons and their involvement in chemical bonding. 

In general, subshells with the same ''n''&nbsp;+&nbsp;{{mvar|l}} value have similar energies, but the s-orbitals (with {{mvar|l}} = 0) are exceptional: their energy levels are appreciably far from those of their ''n''&nbsp;+&nbsp;{{mvar|l}} group and are closer to those of the next ''n''&nbsp;+&nbsp;{{mvar|l}} group. This is why the [[periodic table]] is usually drawn to begin with the s-block elements.<ref>{{cite journal |last1=Ostrovsky |first1=V. N. |date=1981 |title=Dynamic symmetry of atomic potential |url= |journal=Journal of Physics B: Atomic and Molecular Physics |volume=14 |issue=23 |pages=4425–4439 (4429) |doi=10.1088/0022-3700/14/23/008 |bibcode=1981JPhB...14.4425O |access-date=}}</ref>

The Madelung energy ordering rule applies only to neutral atoms in their ground state. There are twenty elements (eleven in the d-block and nine in the f-block) for which the Madelung rule predicts an electron configuration that differs from that determined experimentally, although the Madelung-predicted electron configurations are at least close to the ground state even in those cases.

One inorganic chemistry textbook describes the Madelung rule as essentially an approximate empirical rule although with some theoretical justification, based on the [[Thomas–Fermi model]] of the atom as a many-electron quantum-mechanical system.<ref name=Jolly/>

===Exceptions in the d-block===

The [[Valence electron|valence]] d-subshell "borrows" one electron (in the case of palladium two electrons) from the valence s-subshell.

{| class="wikitable"
|-
! Atom !! [[Chromium|<sub>24</sub>Cr]] !! [[Copper|<sub>29</sub>Cu]] !! [[Niobium|<sub>41</sub>Nb]] !! [[Molybdenum|<sub>42</sub>Mo]] !! [[Ruthenium|<sub>44</sub>Ru]] !! [[Rhodium|<sub>45</sub>Rh]] !! [[Palladium|<sub>46</sub>Pd]] !! [[Silver|<sub>47</sub>Ag]] !! [[Platinum|<sub>78</sub>Pt]] !! [[Gold|<sub>79</sub>Au]] !! [[Lawrencium|<sub>103</sub>Lr]]
|-
! Core electrons
| [Ar] || [Ar] || [Kr] || [Kr] || [Kr] || [Kr] || [Kr] || [Kr] || [Xe]&nbsp;4f<sup>14</sup> || [Xe]&nbsp;4f<sup>14</sup> || [Rn]&nbsp;5f<sup>14</sup>
|-
! Madelung rule 
| 3d<sup>4</sup>&nbsp;4s<sup>2</sup>|| 3d<sup>9</sup>&nbsp;4s<sup>2</sup> || 4d<sup>3</sup>&nbsp;5s<sup>2</sup> || 4d<sup>4</sup>&nbsp;5s<sup>2</sup> || 4d<sup>6</sup>&nbsp;5s<sup>2</sup> || 4d<sup>7</sup>&nbsp;5s<sup>2</sup> || 4d<sup>8</sup>&nbsp;5s<sup>2</sup> || 4d<sup>9</sup>&nbsp;5s<sup>2</sup> || 5d<sup>8</sup>&nbsp;6s<sup>2</sup> || 5d<sup>9</sup>&nbsp;6s<sup>2</sup> || 6d<sup>1</sup>&nbsp;7s<sup>2</sup> 
|-
! Experimental 
| 3d<sup>5</sup>&nbsp;4s<sup>1</sup> || 3d<sup>10</sup>&nbsp;4s<sup>1</sup> || 4d<sup>4</sup>&nbsp;5s<sup>1</sup> || 4d<sup>5</sup>&nbsp;5s<sup>1</sup> || 4d<sup>7</sup>&nbsp;5s<sup>1</sup> || 4d<sup>8</sup>&nbsp;5s<sup>1</sup> || 4d<sup>10</sup> || 4d<sup>10</sup>&nbsp;5s<sup>1</sup> || 5d<sup>9</sup>&nbsp;6s<sup>1</sup>|| 5d<sup>10</sup>&nbsp;6s<sup>1</sup> || 7s<sup>2</sup>&nbsp;7p<sup>1</sup>
|}

For example, in [[copper]] <sub>29</sub>Cu, according to the Madelung rule, the 4s subshell (''n''&nbsp;+&nbsp;{{mvar|l}} = 4&nbsp;+&nbsp;0 = 4) is occupied before the 3d subshell (''n''&nbsp;+&nbsp;{{mvar|l}} = 3&nbsp;+&nbsp;2 = 5). The rule then predicts the electron configuration {{nowrap|1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> 3s<sup>2</sup>  3p<sup>6</sup> 3d<sup>9</sup> 4s<sup>2</sup>}}, abbreviated {{nowrap|[Ar] 3d<sup>9</sup> 4s<sup>2</sup>}} where [Ar] denotes the configuration of [[argon]], the preceding noble gas. However, the measured electron configuration of the copper atom is {{nowrap|[Ar] 3d<sup>10</sup> 4s<sup>1</sup>}}. By filling the 3d subshell, copper can be in a lower [[Energy level|energy state]].

A special exception is [[lawrencium]] <sub>103</sub>Lr, where the 6d electron predicted by the Madelung rule is replaced by a 7p electron: the rule predicts {{nowrap|[Rn] 5f<sup>14</sup> 6d<sup>1</sup> 7s<sup>2</sup>}}, but the measured configuration is {{nowrap|[Rn] 5f<sup>14</sup> 7s<sup>2</sup> 7p<sup>1</sup>}}.

===Exceptions in the f-block===

The valence d-subshell often "borrows" one electron (in the case of thorium two electrons) from the valence f-subshell. For example, in [[uranium]] <sub>92</sub>U, according to the Madelung rule, the 5f subshell (''n''&nbsp;+&nbsp;{{mvar|l}} = 5&nbsp;+&nbsp;3 = 8) is occupied before the 6d subshell (''n''&nbsp;+&nbsp;{{mvar|l}} = 6&nbsp;+&nbsp;2 = 8). The rule then predicts the electron configuration {{nowrap|[Rn] 5f<sup>4</sup> 7s<sup>2</sup>}} where [Rn] denotes the configuration of [[radon]], the preceding noble gas. However, the measured electron configuration of the uranium atom is {{nowrap|[Rn] 5f<sup>3</sup> 6d<sup>1</sup> 7s<sup>2</sup>}}.
{| class="wikitable"
|-
! Atom !! [[Lanthanum|<sub>57</sub>La]] !! [[Cerium|<sub>58</sub>Ce]] !! [[Gadolinium|<sub>64</sub>Gd]] !! [[Actinium|<sub>89</sub>Ac]] !! [[Thorium|<sub>90</sub>Th]] !! [[Protactinium|<sub>91</sub>Pa]] !! [[Uranium|<sub>92</sub>U]] !! [[Neptunium|<sub>93</sub>Np]] !! [[Curium|<sub>96</sub>Cm]]
|-
! Core electrons  
| [Xe] || [Xe] || [Xe] || [Rn] || [Rn] || [Rn] || [Rn] || [Rn] || [Rn]
|-
! Madelung rule 
| 4f<sup>1</sup>&nbsp;6s<sup>2</sup> || 4f<sup>2</sup>&nbsp;6s<sup>2</sup> || 4f<sup>8</sup>&nbsp;6s<sup>2</sup> || 5f<sup>1</sup>&nbsp;7s<sup>2</sup> || 5f<sup>2</sup>&nbsp;7s<sup>2</sup> || 5f<sup>3</sup>&nbsp;7s<sup>2</sup> || 5f<sup>4</sup>&nbsp;7s<sup>2</sup> || 5f<sup>5</sup>&nbsp;7s<sup>2</sup> || 5f<sup>8</sup>&nbsp;7s<sup>2</sup>
|-
! Experimental 
| 5d<sup>1</sup>&nbsp;6s<sup>2</sup> || 4f<sup>1</sup>&nbsp;5d<sup>1</sup>&nbsp;6s<sup>2</sup> || 4f<sup>7</sup>&nbsp;5d<sup>1</sup>&nbsp;6s<sup>2</sup> || 6d<sup>1</sup>&nbsp;7s<sup>2</sup> || 6d<sup>2</sup>&nbsp;7s<sup>2</sup> || 5f<sup>2</sup>&nbsp;6d<sup>1</sup>&nbsp;7s<sup>2</sup> || 5f<sup>3</sup>&nbsp;6d<sup>1</sup>&nbsp;7s<sup>2</sup> || 5f<sup>4</sup>&nbsp;6d<sup>1</sup>&nbsp;7s<sup>2</sup> || 5f<sup>7&nbsp;</sup>6d<sup>1</sup>&nbsp;7s<sup>2</sup>
|}

All these exceptions are not very relevant for chemistry, as the energy differences are quite small<ref name=Jorgensen>{{cite journal |last1=Jørgensen |first1=Christian |date=1973 |title=The Loose Connection between Electron Configuration and the Chemical Behavior of the Heavy Elements (Transuranics) |journal=Angewandte Chemie International Edition |volume=12 |issue=1 |pages=12–19 |doi=10.1002/anie.197300121}}</ref> and the presence of a nearby atom can change the preferred configuration.<ref name=FIII19>{{cite book |last1=Feynman |first1=Richard |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |date=1964 |title=The Feynman Lectures on Physics |url=https://feynmanlectures.caltech.edu/III_19.html |publisher=Addison–Wesley |volume=3 |chapter=19. The Hydrogen Atom and The Periodic Table |isbn=0-201-02115-3}}</ref> The periodic table ignores them and follows idealised configurations.<ref name=Jensen2009>{{cite journal |author1-link=William B. Jensen |last1=Jensen |first1=William B. |date=2009 |title=Misapplying the Periodic Law |journal=Journal of Chemical Education |volume=86 |issue=10 |pages=1186 |doi=10.1021/ed086p1186 |bibcode=2009JChEd..86.1186J |doi-access=free }}</ref> They occur as the result of interelectronic repulsion effects;<ref name=Jorgensen/><ref name=FIII19/> when atoms are positively ionised, most of the anomalies vanish.<ref name=Jorgensen/>

The above exceptions are predicted to be the only ones until [[unbinilium|element 120]], where the 8s shell is completed. [[Unbiunium|Element 121]], starting the g-block, should be an exception in which the expected 5g electron is transferred to 8p (similarly to lawrencium). After this, sources do not agree on the predicted configurations, but due to very strong [[relativistic quantum chemistry|relativistic effects]] there are not expected to be many more elements that show the expected configuration from Madelung's rule beyond 120.<ref name="BFricke">{{Cite journal |last1=Fricke |first1=Burkhard |year=1975 |title=Superheavy elements: a prediction of their chemical and physical properties |journal=Recent Impact of Physics on Inorganic Chemistry |volume=21 |pages=[https://archive.org/details/recentimpactofph0000unse/page/89 89–144] |doi=10.1007/BFb0116498 |url=https://archive.org/details/recentimpactofph0000unse/page/89 |access-date=4 October 2013 |series=Structure and Bonding |isbn=978-3-540-07109-9 }}</ref> The general idea that after the two 8s elements, there come regions of chemical activity of 5g, followed by 6f, followed by 7d, and then 8p, does however mostly seem to hold true, except that relativity "splits" the 8p shell into a stabilized part (8p<sub>1/2</sub>, which acts like an extra covering shell together with 8s and is slowly drowned into the core across the 5g and 6f series) and a destabilized part (8p<sub>3/2</sub>, which has nearly the same energy as 9p<sub>1/2</sub>), and that the 8s shell gets replaced by the 9s shell as the covering s-shell for the 7d elements.<ref name=BFricke/><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>


==History==

===The aufbau principle in the new quantum theory===
[[File:Sommerfeld ellipses.svg|thumb|In the old quantum theory, orbits with low angular momentum (s- and p-subshell) get closer to the nucleus.]]
The principle takes its name from German, ''{{lang|de|Aufbauprinzip}}'', "building-up principle", rather than being named for a scientist. It was formulated by [[Niels Bohr]] in the early 1920s.<ref name=Kragh>Kragh, Helge, '7 A Theory of the Chemical Elements', Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 (Oxford, 2012; online edn, Oxford Academic, 24 May 2012), https://doi.org/10.1093/acprof:oso/9780199654987.003.0007, accessed 23 Feb. 2024.</ref> This was an early application of [[quantum mechanics]] to the properties of electrons and explained chemical properties in physical terms. Each added electron is subject to the electric field created by the positive [[Electric charge|charge]] of the atomic nucleus ''and'' the negative charge of other electrons that are bound to the nucleus. Although in [[hydrogen]] there is no energy difference between subshells with the same principal quantum number ''n'', this is not true for the outer electrons of other atoms.

In the [[old quantum theory]] prior to quantum mechanics, electrons were supposed to occupy classical elliptical orbits. The orbits with the highest [[angular momentum]] are "circular orbits" outside the inner electrons, but orbits with low angular momentum (s- and p-subshell) have high [[orbital eccentricity|subshell eccentricity]], so that they get closer to the nucleus and feel on average a less strongly screened [[Effective nuclear charge|nuclear charge]].

[[Wolfgang Pauli]]'s model of the atom, including the effects of electron spin, provided a more complete explanation of the empirical aufbau rules.<ref name=Kragh/>

===The ''n''&nbsp;+&nbsp;''{{mvar|l}}'' energy ordering rule===
A [[Alternative periodic tables#Left-step periodic table (Janet, 1928)|periodic table in which each row corresponds to one value of ''n''&nbsp;+&nbsp;{{mvar|l}}]] (where the values of ''n'' and {{mvar|l}} correspond to the principal and azimuthal quantum numbers respectively) was suggested by [[Charles Janet]] in 1928, and in 1930 he made explicit the quantum basis of this pattern, based on knowledge of atomic ground states determined by the analysis of [[Emission spectrum|atomic spectra]]. This table came to be referred to as the left-step table. Janet "adjusted" some of the actual ''n''&nbsp;+&nbsp;{{mvar|l}} values of the elements, since they did not accord with his energy ordering rule, and he considered that the discrepancies involved must have arisen from measurement errors. As it happens, the actual values were correct and the ''n''&nbsp;+&nbsp;{{mvar|l}} energy ordering rule turned out to be an approximation rather than a perfect fit, although for all elements that are exceptions the regularised configuration is a low-energy excited state, well within reach of chemical bond energies.

In 1936, the German physicist [[Erwin Madelung]] proposed this as an empirical rule for the order of filling atomic subshells, and most English-language sources therefore refer to the Madelung rule. Madelung may have been aware of this pattern as early as 1926.<ref>{{cite journal |title=The Order of Electron Shells in Ionized Atoms |last1=Goudsmit |first1=S. A. |last2=Richards |first2=Paul I. |journal=[[Proceedings of the National Academy of Sciences of the United States of America|Proc. Natl. Acad. Sci.]] |pages=664–671 (with correction in [https://doi.org/10.1073/pnas.51.5.906 issue 5, p&nbsp;906]) |volume=51 |issue=4 |date=1964 |url=http://www.pnas.org/content/51/4/664.full.pdf |bibcode=1964PNAS...51..664G |doi=10.1073/pnas.51.4.664 |pmid=16591167 |doi-access=free|pmc=300183 }}</ref> The Russian-American engineer [[Vladimir Karapetoff]] was the first to publish the rule in 1930,<ref>{{cite journal |last1=Karapetoff |first1=Vladimir |date=1930 |title=A chart of consecutive sets of electronic orbits within atoms of chemical elements |url= |journal=Journal of the Franklin Institute |volume=210 |issue=5 |pages=609–624 |doi=10.1016/S0016-0032(30)91131-3 |access-date=}}</ref><ref>{{cite journal |last1=Ostrovsky |first1=Valentin N. |date=2003 |title=Physical Explanation of the Periodic Table |url= |journal=Annals of the New York Academy of Sciences |volume=988 |issue=1 |pages=182–192 |doi=10.1111/j.1749-6632.2003.tb06097.x |pmid=12796101 |bibcode=2003NYASA.988..182O |s2cid=21629328 |access-date=}}</ref> though Janet also published an illustration of it the same year.

In 1945, American chemist [[William Wiswesser]] proposed that the subshells are filled in order of increasing values of the function<ref>{{cite journal |last1=Wiswesser |first1=William J. |title=The Periodic System and Atomic Structure I. An Elementary Physical Approach |journal=Journal of Chemical Education |date=July 1945 |volume=22 |issue=7 |pages=314–322 |doi=10.1021/ed022p314 |bibcode=1945JChEd..22..314W |url=https://pubs.acs.org/doi/pdf/10.1021/ed022p314 |access-date=5 September 2020}}</ref>
:<math>W(n,l) = n + l - \frac{l}{l + 1}. </math> 
This formula correctly predicts both the first and second parts of the Madelung rule (the second part being that for two subshells with the same value of ''n''&nbsp;+&nbsp;{{mvar|l}}, the one with the smaller value of ''n'' fills first). Wiswesser argued for this formula based on the pattern of both angular and radial nodes, the concept now known as [[Effective nuclear charge|orbital penetration]], and the influence of the core electrons on the valence orbitals.

In 1961 the Russian agricultural chemist [[V.M. Klechkovsky|V.M. Klechkowski]] proposed a theoretical explanation for the importance of the sum ''n''&nbsp;+&nbsp;{{mvar|l}}, based on the Thomas–Fermi model of the atom.<ref>{{cite journal |last1=Klechkovskii |first1=V.M. |title=Justification of the Rule for Successive Filling of (n+l) Groups |journal=Journal of Experimental and Theoretical Physics |date=1962 |volume=14 |issue=2 |page=334 |url=http://jetp.ras.ru/cgi-bin/e/index?t=&au=+Klechkovskii&yf=2022&yt=2022&se=1&a=s |access-date=23 June 2022}}</ref> Many French- and Russian-language sources therefore refer to the Klechkowski rule.<ref>{{cite book |last1=Sakho |first1=Ibrahima |title=Introduction to Quantum Mechanics 1: Thermal Radiation and Experimental Facts Regarding the Quantization of Matter |date=2019 |publisher=Wiley |isbn=978-1786304872 |page=115 |url=https://books.google.com/books?id=n9W0DwAAQBAJ&q=madelung+rule&pg=PA116 |access-date=11 April 2021}}</ref> '

The full Madelung rule was derived from a similar potential in 1971 by Yury N. Demkov and Valentin N. Ostrovsky.<ref name=DO>{{cite journal |last1=Demkov |first1=Yury N. |last2=Ostrovsky |first2=Valentin N. |date=1972 |title=n+l Filling Rule in the Periodic System and Focusing Potentials |url=http://jetp.ras.ru/cgi-bin/e/index/e/35/1/p66?a=list |journal=Journal of Experimental and Theoretical Physics |volume=35 |issue=1 |pages=66–69 |doi= |bibcode=1972JETP...35...66D |access-date=25 November 2022}}</ref> They considered the potential 
:<math>U_{1/2}(r) = -\frac{2v}{rR(r+R)^2}</math> 
where <math>R</math> and <math>v</math> are constant parameters; this approaches a [[Coulomb potential]] for small <math>r</math>. When <math>v</math> satisfies the condition 
:<math>v=v_N=\frac{1}{4}R^2 N(N+1)</math>, 
where <math>N=n+l</math>, the zero-energy solutions to the [[Schrödinger equation]] for this potential can be described analytically with [[Gegenbauer polynomials]]. As <math>v</math> passes through each of these values, a manifold containing all states with that value of <math>N</math> arises at zero energy and then becomes bound, recovering the Madelung order. The application of [[Perturbation theory (quantum mechanics)|perturbation-theory]] show that states with smaller <math>n</math> have lower energy, and that the s-orbitals (with <math>l=0</math>) have their energies approaching the next <math>n+l</math> group.<ref name=DO/><ref name=shattered>{{cite book |last1=Thyssen |first1=Pieter |last2=Ceulemans |first2=Arnout |author-link= |date=2017 |title=Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table |url= |location= |publisher=Oxford University Press |pages=360–381 |isbn=9780190611392}}</ref>

In recent years it has been noted that the order of filling subshells in neutral atoms does not always correspond to the order of adding or removing electrons for a given atom. For example, in the fourth row of the periodic table, the Madelung rule indicates that the 4s subshell is occupied before the 3d. Therefore, the neutral atom ground state configuration for K is {{nowrap|[Ar] 4s<sup>1</sup>}}, Ca is {{nowrap|[Ar] 4s<sup>2</sup>}}, Sc is {{nowrap|[Ar] 4s<sup>2</sup> 3d<sup>1</sup>}} and so on. However, if a [[scandium]] atom is ionized by removing electrons (only), the configurations differ: Sc is {{nowrap|[Ar] 4s<sup>2</sup> 3d<sup>1</sup>}}, Sc<sup>+</sup> is {{nowrap|[Ar] 4s<sup>1</sup> 3d<sup>1</sup>}}, and Sc<sup>2+</sup> is {{nowrap|[Ar] 3d<sup>1</sup>}}. The subshell energies and their order depend on the nuclear charge; 4s is lower than 3d as per the Madelung rule in K with 19 protons, but 3d is lower in Sc<sup>2+</sup> with 21 protons. In addition to there being ample experimental evidence to support this view, it makes the explanation of the order of ionization of electrons in this and other [[transition metal]]s more intelligible, given that 4s electrons are invariably preferentially ionized.<ref name="Scerri EiC">{{cite magazine |title=The Trouble With the Aufbau Principle |magazine=[[Education in Chemistry]] |url=https://eic.rsc.org/feature/the-trouble-with-the-aufbau-principle/2000133.article |last=Scerri |first=Eric |date=7 November 2013 |issue=6 |pages=24–26 |volume=50 |publisher=[[Royal Society of Chemistry]] }}</ref> Generally the Madelung rule should only be used for neutral atoms; however, even for neutral atoms there are exceptions in the d-block and f-block (as shown above).

==See also==
* [[Ionization energy]]

==References==
{{Reflist}}

==Further reading==
* [http://www.iun.edu/~cpanhd/C101webnotes/modern-atomic-theory/images/energy-levels.jpg Image: Understanding order of shell filling] {{Webarchive|url=https://web.archive.org/web/20141115122916/http://www.iun.edu/~cpanhd/C101webnotes/modern-atomic-theory/images/energy-levels.jpg |date=2014-11-15 }}
* [[Jan C. A. Boeyens|Boeyens, J. C. A.]]: ''Chemistry from First Principles''. Berlin: Springer Science 2008, {{ISBN|978-1-4020-8546-8}}
* {{cite journal |doi=10.1007/s10698-005-2141-y |last=Ostrovsky |first=V.N. |journal=Foundations of Chemistry |volume=7 |issue=3 |pages=235–39 |title=On Recent Discussion Concerning Quantum Justification of the Periodic Table of the Elements  |year=2005 |s2cid=93589189 }}
* {{cite journal |title=On the dynamical symmetry of the periodic table. II. Modified Demkov-Ostrovsky atomic model |first1=Y. |last1=Kitagawara |year=1984 |journal=J. Phys. B |volume=17 |issue=21 |pages=4251–59 |doi=10.1088/0022-3700/17/21/013|last2=Barut |first2=A.O. |bibcode=1984JPhB...17.4251K }}
* {{cite journal |title=Transition Metals and the Aufbau Principle |first=L. G. |last=Vanquickenborne |year=1994 |journal=Journal of Chemical Education |volume=71 |issue=6 |pages=469–471 |url=http://depa.fquim.unam.mx/amyd/archivero/4svs3d_26264.pdf |bibcode=1994JChEd..71..469V |doi=10.1021/ed071p469 }}
* {{cite journal |title=On the Madelung Rule |first=E.R. |last=Scerri |year=2017 |journal=Inference |volume=1 |issue=3 |url=http://inference-review.com/article/on-the-madelung-rule |access-date=2018-04-15 |archive-date=2017-04-12 |archive-url=https://web.archive.org/web/20170412121031/http://inference-review.com/article/on-the-madelung-rule |url-status=dead }}

==External links==
* [http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html#aufbau Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund's Rule] from [[Purdue University]]

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