# Aufbau principle The **Aufbau principle** is the rule for building an [[Atom|atom]]'s ground-state [[Electron_configuration|electron configuration]] one [[Electron|electron]] at a time: each added electron takes the lowest-energy [[Atomic_orbital|orbital]] still open to it, no orbital holds more than two electrons and those two have opposite spins (the [[Pauli_exclusion_principle|Pauli exclusion principle]]), and electrons entering a set of equal-energy orbitals spread out one per orbital with parallel spins before any pair up ([[Hund's_rules|Hund's rule]]).[^b054-aufbau][^openstax-config] The name is the German *Aufbauprinzip*, "building-up principle"; it belongs to no one scientist. The order in which the subshells open is the empirical sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, which the Madelung rule summarises as increasing n + ℓ, and increasing n within a tie.[^openstax-config] In the microsim below the reader slides the [[Atomic_number|atomic number]] Z from 1 to 36 and watches a box-and-arrow diagram fill in that order, one arrow per electron; the readouts give the configuration string and the number of unpaired electrons, and the one equation on the HUD is the capacity of a subshell, `capacity = 2(2l + 1)`: 2 for s, 6 for p, 10 for d, 14 for f.[^b054-capacity] On the Chemistry flagship this article serves Part II — Modern principles › Matter, at the section *Electron configuration* (row K6), between the energy levels of the [[Bohr_model|Bohr model]] and the [[Periodic_table|periodic table]] whose shape the filling order produces. The Z control built here is the one every element family root in the flagship's §5 inherits, so the same slider that fills oxygen's boxes later selects a tile on the periodic-table landscape. The principle is a recipe, not a law of nature. It reproduces the configurations of most of the first 36 elements, but [[Chromium|chromium]] and [[Copper|copper]] break it, more elements break it in the fifth and sixth periods, and the sequence it uses has never been derived from first principles. The article states the rule, shows what the microsim does with it, lists the exceptions, and traces the rule back to [[Niels_Bohr|Niels Bohr]]'s theory of the periodic system in the early 1920s. ## Madelung energy ordering rule The Madelung rule orders subshells by the sum n + ℓ of the principal and azimuthal [[Quantum_number|quantum numbers]]; when two subshells tie, the one with the smaller n fills first. Reading the diagonals of a grid of n against ℓ gives the sequence quoted in the lead, the "diagonal rule" that a chemistry text presents as the filling order, with the note that it is empirical.[^b054-aufbau] The sequence has two surprising features that the microsim makes visible. First, 4s opens before 3d, so [[Potassium|potassium]] (Z = 19) and [[Calcium|calcium]] (Z = 20) put their outermost electrons into 4s while the 3d subshell stays empty; the bonding textbook's exercise on potassium asks for exactly this half-filled 4s.[^b054-config-ex] Second, once 3d begins to fill at [[Scandium|scandium]] it takes ten electrons before 4p opens at [[Gallium|gallium]], which is why the fourth period is eighteen elements long. In the microsim the reader has one control, Z from 1 to 36 with the default at [[Oxygen|oxygen]] (Z = 8). The boxes are drawn in Madelung order, 1s at the bottom and 4p at the top, with the Pauli limit of two arrows per box and Hund's rule deciding where the next arrow lands: an empty box in the current subshell before a half-filled one, and every arrow in a half-filled set pointing the same way.[^b054-aufbau] The readout strings the boxes into the configuration and counts the unpaired arrows. The defaults read 1s¹ for [[Hydrogen|hydrogen]], 1s² for [[Helium|helium]] and 1s²2s²2p⁴ for oxygen, whose two unpaired p electrons come straight from Hund's rule. The book's own worked configurations are the sim's tests: [[Phosphorus|phosphorus]] is 1s²2s²2p⁶3s²3p³ with three unpaired electrons, and the configuration [Ar]4s²3d⁵ is [[Manganese|manganese]].[^b054-config-p] [[Silicon|Silicon]] and [[Sulfur|sulfur]] each have two unpaired electrons,[^b054-config-ex] and [[Cobalt|cobalt]] ends in 3d⁷4s².[^b054-config-co] The unpaired count is what a magnetic measurement sees. The rule describes the order in which subshells fill as Z increases, not the order of orbital energies inside a finished atom. Once 3d is occupied it lies below 4s, so a [[Transition_metal|transition-metal]] atom loses its 4s electrons first when it becomes an [[Ion|ion]]: Fe²⁺ is [Ar]3d⁶, Zn²⁺ is [Ar]3d¹⁰ and Cr³⁺ is [Ar]3d³, each with the 4s box empty.[^b054-ions] The sim fills neutral atoms only; its Madelung sequence is a lookup, which is why the article marks it ILLUSTRATIVE of the filling order rather than a calculation of energies. ### Exceptions in the d-block At Z = 24 the diagram would put a second arrow into 4s and four into 3d, but chromium's measured ground state is [Ar]3d⁵4s¹, and at Z = 29 copper is [Ar]3d¹⁰4s¹ rather than 3d⁹4s².[^b054-exceptions][^openstax-config] In the sim these are the two moments when a 4s arrow jumps into 3d and flashes; the exception table is stored inline as `{24: '3d5 4s1', 29: '3d10 4s1'}`, and the Z range stops at 36 because the source lists only chromium, copper and [[Niobium|niobium]] as exceptions.[^b054-exceptions] Extending past [[Krypton|krypton]] needs a full configuration table. The textbook explanation, that a half-filled or filled d subshell carries extra stability, is a mnemonic rather than a mechanism; the same book says that "no simple method" predicts which elements will be exceptions.[^b054-exceptions] The underlying cause is that the 3d and 4s energies are close and cross as Z increases, so that small differences in electron–electron repulsion and exchange decide the ground state. The fifth and sixth periods show how weak the rule becomes there. Niobium is [Kr]4d⁴5s¹, [[Molybdenum|molybdenum]] [Kr]4d⁵5s¹, [[Ruthenium|ruthenium]] [Kr]4d⁷5s¹, [[Rhodium|rhodium]] [Kr]4d⁸5s¹, [[Palladium|palladium]] [Kr]4d¹⁰ with no 5s electron at all, [[Silver|silver]] [Kr]4d¹⁰5s¹, [[Platinum|platinum]] [Xe]4f¹⁴5d⁹6s¹ and [[Gold|gold]] [Xe]4f¹⁴5d¹⁰6s¹, all taken from the standard reference table of ground levels.[^nist-srd111] Six of the ten period-5 transition metals are exceptions. None of this changes the chemistry much, because the ions lose the s electrons anyway; it changes what a box-and-arrow diagram may claim to predict. ### Exceptions in the f-block The [[Lanthanide|lanthanides]] and [[Actinide|actinides]] open with the same kind of exception. The Madelung sequence puts 4f before 5d, but [[Lanthanum|lanthanum]] is [Xe]5d¹6s², [[Cerium|cerium]] is [Xe]4f¹5d¹6s² and [[Gadolinium|gadolinium]] is [Xe]4f⁷5d¹6s², each with one electron in 5d that the rule would have placed in 4f.[^nist-srd111] The actinides are worse: [[Actinium|actinium]] is [Rn]6d¹7s², [[Thorium|thorium]] [Rn]6d²7s² with no 5f electron, [[Protactinium|protactinium]] [Rn]5f²6d¹7s², [[Uranium|uranium]] [Rn]5f³6d¹7s², [[Neptunium|neptunium]] [Rn]5f⁴6d¹7s² and [[Curium|curium]] [Rn]5f⁷6d¹7s², because the 5f, 6d and 7s energies lie close together in the early actinides.[^nist-srd111] At the end of the series [[Lawrencium|lawrencium]] is listed as [Rn]5f¹⁴7s²7p¹, with its last electron in 7p rather than 6d, a relativistic effect.[^nist-srd111] A box-and-arrow sim extended to these rows would need a stored table for every one of them; the rule alone would place a dozen electrons wrongly. The f-block exceptions feed the long argument over which elements belong in group 3 of the periodic table, [[Lutetium|lutetium]] and lawrencium or lanthanum and actinium, which the periodic-table article takes up. ## History The building-up idea is Bohr's. Between 1921 and 1923 he assigned the electrons of every element to groups of orbits of the old quantum theory, adding one electron at a time and asking which orbit the newcomer would take, and in his Nobel lecture of December 11, 1922, he showed how the lengths of the periods, 2, 8, 8, 18, 18 and 32, follow from the way the groups close.[^bohr-nobel-lecture][^kragh2012] The scheme was guided by spectra and chemistry rather than derived from theory, and its historian Helge Kragh describes it as a theory of the chemical elements built on Bohr's physical intuition as much as on his quantum rules.[^kragh2012] What Bohr lacked was a reason for the group sizes. Wolfgang Pauli supplied it in 1925 with the exclusion principle, which allows at most one electron per set of four quantum numbers and so limits a shell of principal quantum number n to 2n² electrons,[^pauli1925] and Friedrich Hund's analysis of the spectra of scandium to nickel in the same year gave the rule for the spin arrangement inside a partly filled subshell.[^hund1925] The n + ℓ rule that orders the subshells came later, in Erwin Madelung's 1936 handbook, and it is still an empirical statement.[^madelung1936] ### The aufbau principle in the new quantum theory Quantum mechanics replaced Bohr's orbits with orbitals in 1926, but it kept his procedure. In the new theory an electron in an atom is labelled by four quantum numbers, n, ℓ, m_ℓ and m_s; the exclusion principle forbids two electrons from sharing all four, which is the Pauli limit of two per orbital with opposite [[Spin_(physics)|spins]];[^pauli1925] and the subshell capacity 2(2ℓ + 1) counts the 2ℓ + 1 values of m_ℓ times the two spin states.[^b054-capacity] Pauli's own account, in his Nobel lecture of December 13, 1946, traces the principle to the problem of why the electron groups of Bohr's periodic system close where they do.[^pauli-nobel-lecture] Hund's rule became a statement about exchange energy: electrons with parallel spins keep apart and repel each other less, so the configuration with the most unpaired spins lies lowest.[^hund1925] What quantum mechanics did not deliver is the filling order itself. Solving the many-electron [[Schrödinger_equation|Schrödinger equation]] gives each atom's ground state one element at a time, and the 4s-before-3d order, the chromium and copper exceptions and the f-block irregularities all come out of such calculations, but no closed formula orders the subshells for every Z. ### The <i>n</i> + <i><span>ℓ</span></i> energy ordering rule The rule that the sim runs was stated by Madelung in 1936 as a mnemonic for the observed order.[^madelung1936] Charles Janet had already drawn the periodic table it implies: his left-step table of 1930 ends each row at an s² element, so that the periods have lengths 2, 2, 8, 8, 18, 18, 32, 32, which are the n + ℓ groups of the rule.[^scerri2007] In 1962 the Soviet chemist V. M. Klechkovsky published a justification of the rule based on the Thomas–Fermi statistical model of the atom, which is why Russian texts call it the Klechkovsky rule.[^klechkovskii1962] In 1969 Per-Olov Löwdin listed the derivation of the n + ℓ rule from first principles among the unsolved problems of quantum chemistry, and the historian of the periodic table Eric Scerri has argued that it remains unsolved: modern calculations reproduce each configuration, but they do not explain why the simple rule should hold as often as it does, nor predict where it fails.[^lowdin1969][^scerri2007] The rule also says nothing about the energy ordering of the orbitals once they are occupied, which is the point at which students meet the 4s/3d reversal for ions.[^b054-ions] For the microsim these limits are the honest reading of its readout: the configuration string is what the rule predicts, and the flashes at Z = 24 and Z = 29 are where the atoms disagree. ## See also - [[Electron_configuration]] - [[Hund's_rules]] - [[Pauli_exclusion_principle]] - [[Periodic_table_(electron_configurations)]] - [[Bohr_model]] - [[Periodic_table]] - [[Block_(periodic_table)]] ## References [^b054-aufbau]: Blackstock, Lindsay; Brewer, Sharon; Cinel, Bruno (2022). *Chemical Bonding and Organic Chemistry*. Chapter 2, "Quantum Theory and Electronic Structure of Atoms", pp. 166–169 (the filling order as an empirical "diagonal rule", pp. 166–167; Pauli, pp. 168–169; Hund, p. 169). Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/chemical-bonding-and-organic-chemistry [^b054-capacity]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, p. 171 (subshell capacities 2, 6, 10, 14). [^b054-exceptions]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, p. 171 (Cr, Cu and Nb as exceptions; "no simple method" predicts them). [^b054-config-p]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, p. 172 (P as 1s²2s²2p⁶3s²3p³; [Ar]4s²3d⁵ identified as Mn). [^b054-config-co]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, p. 189 (Co ends in 3d⁷4s²). [^b054-config-ex]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, p. 190 (Si and S with two unpaired electrons each; K with a half-filled 4s). [^b054-ions]: Blackstock, Brewer and Cinel (2022), *Chemical Bonding and Organic Chemistry*, Chapter 2, pp. 173–174 (Fe²⁺ as 3d⁶) and Chapter 4, pp. 218–220 (cations lose ns before (n − 1)d; Zn²⁺ as [Ar]3d¹⁰, Cr³⁺ as [Ar]3d³). [^openstax-config]: Flowers, Paul; Neth, Edward; Robinson, William, et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 3, §3.4 "Electronic Structure of Atoms (Electron Configurations)" (the Aufbau, Pauli and Hund statements; the filling order 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p; Cr, Cu and Nb as exceptions). https://openstax.org/books/chemistry-atoms-first-2e/pages/3-4-electronic-structure-of-atoms-electron-configurations (PDF pp. 115–184, page to pin) [^nist-srd111]: National Institute of Standards and Technology. *Ground Levels and Ionization Energies for the Neutral Atoms*, NIST Standard Reference Database 111 (ground-state configurations of the neutral atoms). https://www.nist.gov/pml/ground-levels-and-ionization-energies-neutral-atoms [^bohr-nobel-lecture]: Bohr, Niels (1922). "The Structure of the Atom." Nobel Lecture, December 11, 1922. NobelPrize.org. https://www.nobelprize.org/prizes/physics/1922/bohr/lecture/ [^kragh2012]: Kragh, Helge (2012). *Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925*. Oxford University Press. Chapter 7, "A Theory of the Chemical Elements." https://doi.org/10.1093/acprof:oso/9780199654987.001.0001 [^pauli1925]: Pauli, W. (1925). "Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren." *Zeitschrift für Physik*, 31: 765–783. https://doi.org/10.1007/BF02980631 [^pauli-nobel-lecture]: Pauli, Wolfgang (1946). "Exclusion Principle and Quantum Mechanics." Nobel Lecture, December 13, 1946. NobelPrize.org. https://www.nobelprize.org/prizes/physics/1945/pauli/lecture/ [^hund1925]: Hund, F. (1925). "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel." *Zeitschrift für Physik*, 33: 345–371. https://doi.org/10.1007/BF01328319 [^madelung1936]: Madelung, Erwin (1936). *Die mathematischen Hilfsmittel des Physikers*, 3rd edition. Berlin: Springer. [^klechkovskii1962]: Klechkovskii, V. M. (1962). "Justification of the rule for successive filling of (n + l) groups." *Soviet Physics JETP*, 14 (2): 334. [^lowdin1969]: Löwdin, Per-Olov (1969). "Some comments on the periodic system of the elements." *International Journal of Quantum Chemistry*, 3 (S3): 331–334. https://doi.org/10.1002/qua.560030737 [^scerri2007]: Scerri, Eric R. (2007). *The Periodic Table: Its Story and Its Significance*. Oxford University Press (Janet's left-step table; the status of the Madelung rule; page to pin). ## Further reading - Blackstock, Brewer and Cinel (2022). *Chemical Bonding and Organic Chemistry*. Chapter 2, "Quantum Theory and Electronic Structure of Atoms", pp. 165–192. Portal Book 054. - Flowers, Neth, Robinson et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 3, §3.4. Portal Book 051. - Kragh (2012). *Niels Bohr and the Quantum Atom*. Chapter 7, on Bohr's building-up theory of the periodic system. - Scerri (2007). *The Periodic Table: Its Story and Its Significance*. Oxford University Press. ## External links - [Electronic Structure of Atoms (Electron Configurations)](https://openstax.org/books/chemistry-atoms-first-2e/pages/3-4-electronic-structure-of-atoms-electron-configurations), *Chemistry: Atoms First 2e*, OpenStax - [Ground Levels and Ionization Energies for the Neutral Atoms](https://www.nist.gov/pml/ground-levels-and-ionization-energies-neutral-atoms), NIST Standard Reference Database 111 - [Wolfgang Pauli's Nobel Lecture, "Exclusion Principle and Quantum Mechanics"](https://www.nobelprize.org/prizes/physics/1945/pauli/lecture/), NobelPrize.org <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Aufbau_principle.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Aufbau principle* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Aufbau_principle.html" data-title="Aufbau principle"></div> *Built from `MICROSIM_GUIDE/specs/sims/Aufbau_principle.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Aufbau_principle) : [Wikitube](https://en.wikitube.io/wiki/Aufbau_principle) · pinned revision [1360487775](https://en.wikipedia.org/w/index.php?oldid=1360487775) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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