The building-up principle

Result

The building-up principle

T-004Home CU-101Threads quantum · structure

Statement

The ground-state electron configuration of an atom is obtained by filling the atomic orbitals of the previous result one electron at a time, subject to three rules: (1) the Madelung rule: orbitals fill in order of increasing \(n+l\), and for orbitals sharing the same \(n+l\), in order of increasing \(n\); (2) the Pauli exclusion principle: no orbital holds more than two electrons, and those two must have opposite spin (\(m_s=+\tfrac12\) and \(m_s=-\tfrac12\)); (3) Hund's rule: within a set of degenerate orbitals (equal \(n\) and \(l\), differing only in \(m_l\)), electrons occupy separate orbitals with parallel spin before any orbital is doubly occupied.

Why it matters

This is the rule that turns the four abstract quantum-number labels of the previous result into an actual, checkable prediction: the specific ground-state electron configuration of every element, from which nearly every other result in this unit — periodic trends, ionisation energies, valence and bonding behaviour — is read off. Without it, quantum numbers would be a classification scheme with nothing to classify.

It also explains, rather than merely asserts, the shape of the periodic table itself: the table's rows (periods) are exactly the principal shells being filled, and its block structure (s, p, d, f) is exactly which subshell is being filled, both direct consequences of the Madelung filling order applied element by element.

Hypotheses

Orbital energies in a multi-electron atom depend on both \(n\) and \(l\), not \(n\) alone.For the hydrogen atom itself (one electron), every orbital with the same \(n\) is exactly degenerate regardless of \(l\) (the accidental \(SO(4)\) symmetry noted on the Bohr-model page), so there is no unique "filling order" among, say, \(2s\) and \(2p\) — the Madelung rule has no content for a one-electron system. The rule is specifically a statement about how electron–electron screening breaks that degeneracy in atoms with two or more electrons. Ground state (lowest total energy), not an excited configuration.The building-up principle predicts the lowest-energy arrangement of electrons. An atom can be excited (e.g. by absorbing a photon) into a configuration that violates the Madelung filling order — a valid, physically real, but higher-energy state, such as promoting a \(2s\) electron to \(2p\) in an excited helium atom. This result says nothing about which excited states exist, only about the one occupied at \(0\,\text{K}\) in isolation.

Proof

The Pauli and Hund parts are the more fundamental postulates (cited from earlier results); the substantive new content here is the Madelung \((n+l)\) filling order, justified below as an empirical rule with a physical rationale (screening and penetration), not derived from first principles — no closed-form derivation of the Madelung rule from the many-electron Schrödinger equation exists; it is confirmed by numerical solution (Hartree–Fock and beyond) atom by atom.

\text{Pauli (previous result, Step 6): each orbital } (n,l,m_l) \text{ holds at most 2 electrons, opposite spin.}

Cited directly from this unit's quantum-numbers-and-orbitals result. This fixes the maximum capacity of every orbital and subshell, but not the order in which they fill. A

\text{Screening: an outer electron feels an effective nuclear charge } Z_{\text{eff}} = Z - S < Z, \quad S=\text{shielding by inner electrons.}

In a multi-electron atom, an electron in an outer orbital does not feel the bare nuclear charge \(Z\); inner electrons partially cancel (screen) the nuclear attraction. Orbitals that bring the electron closer to the nucleus on average, or that spend more time at small \(r\) (\(s\) orbitals penetrate closer to the nucleus than \(p\), which penetrate closer than \(d\), for the same \(n\)), are screened less and bound more tightly. B

\text{Penetration lowers a low-}l\text{ orbital's energy below a higher-}l\text{ orbital of the same or even slightly larger } n.

Because \(s\) orbitals (\(l=0\)) penetrate the inner shielding electron cloud more than \(p\), \(d\), or \(f\) orbitals of the same \(n\), an \(ns\) orbital is bound more tightly than the naive "energy depends on \(n\) alone" picture from hydrogen would suggest — tightly enough, in many cases, that \(ns\) drops below \((n-1)d\) or \((n-2)f\) in energy. B

\text{Empirical rule (Madelung, 1936): orbitals fill in order of increasing } (n+l); \text{ ties broken by increasing } n.

Named lemma, confirmed by numerical atomic structure calculations across the periodic table, cited here rather than derived: computing \(n+l\) for every orbital and sorting gives the observed filling order almost without exception (see Fails Without for the two exceptions in the first transition row). For example, \(4s\) has \(n+l=4+0=4\); \(3d\) has \(n+l=3+2=5\); since \(4<5\), \(4s\) fills before \(3d\), despite \(3d\) having the smaller principal quantum number. B

\text{Hund's rule (empirical, exchange-energy argument): within one subshell, singly occupy every orbital (parallel spin) before pairing any.}

Two electrons in separate orbitals of the same subshell, with parallel spins, have lower repulsion energy than two electrons forced into the same orbital (necessarily opposite spin, by Pauli) — both because same-spin electrons are kept further apart on average by the Pauli principle itself (reducing direct Coulomb repulsion) and because of a genuinely quantum "exchange energy" that further favours parallel-spin configurations. The full quantitative justification belongs to many-electron perturbation theory and is cited, not derived, here. C

Result

\text{fill by increasing } (n+l),\text{ ties by increasing } n;\quad \text{Pauli caps each orbital at 2};\quad \text{Hund maximises unpaired spins}

Reading. Three independent rules, layered together, generate a single, essentially unambiguous ground-state electron configuration for every element — the specific list of orbitals and their occupancy that this unit's subsequent periodic-trends result is built on.

Scope. Correctly predicts the ground-state configuration of the overwhelming majority of elements. A small number of exceptions occur where an alternative configuration gains extra stability from a half-filled or fully-filled subshell (see Fails Without), and configurations become genuinely difficult to assign unambiguously for some heavy \(d\)- and \(f\)-block elements where multiple close-lying configurations compete.

Corollaries & converses

The periodic table's period lengths (\(2,8,8,18,18,32\)) are the cumulative electron counts at which a new principal shell first starts filling under the Madelung order — a direct, checkable consequence of applying Step 4 element by element.
Valence electron configuration (the outermost, highest-\((n+l)\) occupied orbitals) recurs periodically because the Madelung order itself is periodic in a specific, checkable sense — the basis for why elements in the same column of the periodic table share similar chemistry.
Converse: given a claimed ground-state configuration, one can check it against Steps 1, 4, and 5 independently — correct total electron count (Pauli), correct filling order (Madelung), and maximal unpaired spins within any partially filled degenerate subshell (Hund) are all separately verifiable necessary conditions.

Fails without

Drop the Madelung rule as an exact prediction — chromium (\(Z=24\)): the naive Madelung filling gives \([\text{Ar}]3d^44s^2\), but the true ground state is \([\text{Ar}]3d^54s^1\). Promoting one \(4s\) electron into \(3d\) produces a half-filled \(3d^5\) subshell, which gains extra exchange stabilisation (Step 5's exchange-energy effect, here strong enough to outweigh the small \(4s\)/\(3d\) energy gap) exceeding the cost of leaving \(4s\) singly occupied.
Same failure, copper (\(Z=29\)): naive filling gives \([\text{Ar}]3d^94s^2\); the true ground state is \([\text{Ar}]3d^{10}4s^1\), for the same reason — a fully filled \(3d^{10}\) subshell is exceptionally stable, again outweighing the cost of an unfilled \(4s\).

Common errors

Filling orbitals by increasing \(n\) alone (as in hydrogen), giving \(3d\) before \(4s\). The correct criterion is increasing \(n+l\), which reverses this pair specifically because \(3d\) has \(n+l=5\) and \(4s\) has \(n+l=4\).
Writing \(3d^44s^2\) for chromium or \(3d^94s^2\) for copper instead of the true \(3d^54s^1\) and \(3d^{10}4s^1\) — the two best-known and most frequently mis-stated exceptions to naive Madelung filling.
Applying Hund's rule to already-filled (not partially filled) subshells, or forgetting it applies to degenerate orbitals within one subshell, not across different subshells or different atoms.
Writing the configuration for an ion by simply truncating the neutral atom's configuration from the "end" of the Madelung order, rather than removing electrons from the orbital of highest \(n\) first (which, for transition metal cations, means removing \(4s\) electrons before \(3d\) electrons, even though \(4s\) filled first) — the filling order and the ionisation order are not the same order.

Discussion

The building-up principle in this compact form is a 20th-century synthesis of several separate discoveries: Pauli's exclusion principle (1925, originally an empirical rule to explain spectroscopic term counts, only later connected to spin-statistics and the fermionic nature of electrons), Hund's rules (1925, from analysing atomic spectral term energies), and the specific \((n+l)\) filling order, worked out systematically by Erwin Madelung around 1936 by fitting to the growing body of known atomic spectra and ionisation data (and independently noted by Vsevolod Klechkovsky, so the rule is sometimes called the Klechkovsky rule in the Russian-language literature).

The rule remains, at its core, an empirical regularity rather than an exact theorem: no closed-form proof from the many-electron Schrödinger equation exists, because the many-electron problem itself has no closed-form solution (unlike the hydrogen atom). What can be shown, and is the modern justification, is that self-consistent-field calculations (Hartree–Fock and its refinements) reproduce the Madelung order for nearly every element when run numerically — the rule is confirmed computationally atom by atom, not derived symbolically once and for all.

The genuine exceptions (roughly twenty elements across the \(d\)- and \(f\)-blocks, not just chromium and copper) arise because the energy gap between adjacent orbitals in the Madelung order can be extremely small — comparable to, or smaller than, the exchange-energy gain from a half-filled or filled subshell — so which configuration is actually lowest in energy becomes a genuinely close numerical competition rather than a clean qualitative call. This is why an exhaustive list of every "anomalous" configuration in the \(d\)- and \(f\)-blocks has to be checked case by case against experiment or high-level computation, rather than predicted from a simple extension of the rule stated here.

Common misconception: that Hund's rule and the Pauli principle are two versions of the same idea. They are logically independent: Pauli caps orbital occupancy at two electrons of opposite spin (a hard constraint, no exceptions); Hund's rule is a separate, softer energetic preference about how electrons distribute themselves among several orbitals that are individually still well within their Pauli limit.

Worked examples

\text{Oxygen, } Z=8: \quad 1s^2\,2s^2\,2p^4

Fill by increasing \(n+l\): \(1s\) (\(n+l=1\), capacity \(2\)) then \(2s\) (\(n+l=2\), capacity \(2\)) then \(2p\) (\(n+l=3\), capacity \(6\)); with \(8\) electrons total, \(1s^2+2s^2\) uses \(4\), leaving \(4\) for \(2p\). By Hund's rule, the \(4\) electrons in the three degenerate \(2p\) orbitals fill as \(\uparrow\downarrow,\ \uparrow,\ \uparrow\) — one orbital paired, two singly occupied — giving \(2\) unpaired electrons, matching oxygen's observed paramagnetism. A

\text{Potassium, } Z=19: \quad 1s^22s^22p^63s^23p^6\,4s^1 \quad (\text{not } 3d^1)

After argon's core (\(1s^22s^22p^63s^23p^6\), \(18\) electrons), the next orbital by increasing \(n+l\) is \(4s\) (\(n+l=4\)), not \(3d\) (\(n+l=5\)); the \(19\)th electron goes into \(4s\), giving potassium its well-known single valence electron and strong chemical resemblance to sodium and lithium, the elements directly above it. A

\text{Chromium, } Z=24: \quad [\text{Ar}]\,3d^5\,4s^1 \quad (\text{not the naive } 3d^44s^2)

The exception from Fails Without, shown as a worked configuration: half-filled \(3d^5\) plus singly occupied \(4s\) beats the naive \(3d^44s^2\) filling because of the extra exchange stabilisation of a half-filled \(d\) subshell. B

Z=8\!:1s^22s^22p^4 \quad Z=19\!:[\text{Ar}]4s^1 \quad Z=24\!:[\text{Ar}]3d^54s^1

Reading. The same three rules generate three qualitatively different-looking outcomes — an ordinary \(p\)-block filling, a case where the Madelung reversal of \(4s\) before \(3d\) matters, and a genuine exception to the naive Madelung order — from one consistent procedure.

Scope. The identical three-rule procedure generates the ground-state configuration of every element in the periodic table, exceptions included, once the exception is checked against experiment as noted in Discussion.

Problems

Write the full ground-state electron configuration of nitrogen (\(Z=7\)) and state how many unpaired electrons it has.
Solution
\(1s^22s^22p^3\). The three \(2p\) electrons occupy the three degenerate \(2p\) orbitals singly by Hund's rule (\(\uparrow,\uparrow,\uparrow\)), giving \(3\) unpaired electrons — the maximum possible for a \(p^3\) configuration, and the reason nitrogen's half-filled \(2p^3\) subshell is unusually stable (directly analogous to chromium's half-filled \(3d^5\)).
Predict the ground-state configuration of copper (\(Z=29\)) by naive Madelung filling, then state the true configuration and explain the discrepancy.
Solution
Naive filling: after \([\text{Ar}]\) (\(18\)) plus \(4s^2\) (\(20\)), the remaining \(9\) electrons fill \(3d\), giving \([\text{Ar}]3d^94s^2\). The true configuration is \([\text{Ar}]3d^{10}4s^1\): promoting the second \(4s\) electron into \(3d\) completes a fully filled \(3d^{10}\) subshell, whose exchange stabilisation exceeds the small \(4s\)/\(3d\) energy gap being paid — the same mechanism as chromium (Worked Example 3), but for a filled rather than half-filled subshell.
A student claims that the \(\text{Cu}^{2+}\) ion (formed by removing 2 electrons from neutral copper) has configuration \([\text{Ar}]3d^8\), reasoning that since copper is \([\text{Ar}]3d^{10}4s^1\), removing 2 electrons "from the end" removes one \(4s\) and one \(3d\) electron. Correct the error.
Solution
The error is in the removal order, not the arithmetic: for transition-metal cations, electrons are removed from the orbital of highest principal quantum number \(n\) first, regardless of the filling order that built the neutral atom — a flagged Common Error above. Copper's \(4s\) electron (\(n=4\)) is removed before any \(3d\) electron (\(n=3\)), since once ionisation begins, \(4s\) is (by this stage) the higher-energy, more loosely bound orbital. Removing the single \(4s^1\) electron first, then one \(3d\) electron, gives \(\text{Cu}^{2+}=[\text{Ar}]3d^9\), not \([\text{Ar}]3d^8\).
Using Hund's rule, determine the number of unpaired electrons in a \(d^6\) configuration (e.g. \(\text{Fe}^{2+}\)) distributed over the five degenerate \(d\) orbitals.
Solution
With \(5\) orbitals available, Hund's rule fills all \(5\) singly first (using \(5\) electrons, all parallel spin), then the \(6\)th electron must pair up in one of the now-occupied orbitals (opposite spin, by Pauli). Result: \(4\) orbitals with \(1\) electron each (unpaired) and \(1\) orbital with \(2\) electrons (paired) — \(4\) unpaired electrons total, matching the observed high-spin configuration of \(\text{Fe}^{2+}\) in many of its compounds.