Aufbau Principle and Electron Configurations

Every element in the periodic table is an atom built the same way: drop electrons, one at a time, into the lowest single-particle orbital still available. That single instruction — the Aufbau ("building-up") principle — plus one veto and one tie-breaker generates the entire architecture of chemistry. The veto is Pauli Exclusion Principle: each spatial orbital sleeps at most two electrons, one spin up, one spin down. The tie-breaker, for spins inside a partly-filled subshell, is Hund's Rules. This article owns the filling engine that sits between them. The engine has one nontrivial gear: the order of the orbital energies. In a bare Coulomb field energies depend only on the principal number $n$ , and every subshell of a shell is degenerate. But an electron in a real atom does not see the bare nucleus — it sees the nucleus screened by the cloud of the others, and screening lifts the $\ell$ -degeneracy. Low- $\ell$ orbitals penetrate the screen, feel a larger effective charge, and sink. The upshot is the empirical Madelung rule: orbitals fill in order of increasing $n+\ell$ , ties broken by smaller $n$ . Its most famous consequence is the outlier that makes the transition metals possible — $4s$ ( $n+\ell=4$ ) fills before $3d$ ( $n+\ell=5$ ), even though $3d$ has the smaller principal number. We take the Madelung sequence, prove it minimizes the independent-electron energy, count electrons into it to read off ground configurations, and then confront the honest failures. Chromium and copper break the naive filling, and they break it for a reason the same physics predicts: a half-filled or filled $d$ -subshell is stabilized by exchange energy — the mechanism owned by Exchange Forces and Exchange Energy — enough to steal an electron from $4s$ . Understanding both the rule and its exceptions is the difference between memorizing the periodic table and deriving it.

Key takeaways

The Aufbau principle is a useful rule for constructing ground-state electron configurations by filling orbitals according to their usual energy ordering, while respecting the Pauli exclusion principle — two opposite-spin electrons per spatial orbital.
The filling order is the Madelung $(n+\ell)$ rule: lower $n+\ell$ first, ties broken by smaller $n$ — giving $1s\,2s\,2p\,3s\,3p\,4s\,3d\,4p\,5s\,4d\dots$
The signature outlier: the Madelung rule places $4s$ before $3d$ because $4s$ has the lower $n+\ell$ value, while its greater penetration also helps make it lower in energy in neutral atoms; as the $3d$ subshell fills, however, the relative $4s$ and $3d$ energies can change.
Subshell capacity is $2(2\ell+1)$ , so $s,p,d,f$ hold $2,6,10,14$ ; shell capacity is $2n^2 = 2,8,18,32$ — the block widths and period lengths of the periodic table.
The rule is a model: it minimizes the sum of orbital energies in an independent-electron picture, but the true state is correlated and the level ordering shifts as subshells fill.
The famous exceptions are chromium, $[\mathrm{Ar}]\,3d^5 4s^1$ , and copper, $[\mathrm{Ar}]\,3d^{10}4s^1$ : the small energy difference between the $4s$ and $3d$ subshells allows electron rearrangement, with exchange and other many-electron effects contributing to the stability of the observed configurations.
Key idea: build the atom by dropping electrons into the lowest available orbital, two per spatial orbital, in Madelung $(n+\ell)$ order — which minimizes the independent-electron energy — and read the exceptions as exchange energy overruling a near-degenerate filling.

The core insight

The ground configuration is built by filling single-particle orbitals lowest-energy-first (Madelung $n+\ell$ order, smaller $n$ breaking ties), two opposite-spin electrons per spatial orbital. This minimizes the independent-electron energy $\sum_a \varepsilon_a$ ; where two orbitals are nearly degenerate, exchange energy can flip the order, producing the Cr and Cu exceptions.

Where this lives today

The Aufbau engine is the backbone of every introductory account of chemistry and the starting point of every quantitative one. Modern electronic-structure methods — Hartree-Fock and, dominantly, density-functional theory — do not assume the Madelung order; they compute self-consistent orbital energies and let the filling fall out, precisely so they can capture the near-degeneracies where the naive rule fails. In that sense Aufbau is the zeroth approximation these methods refine. Its exceptions remain an active check on those methods: reproducing the correct ground configurations of the transition metals, lanthanides, and actinides — where $s$ , $d$ , and $f$ levels crowd together and relativistic effects intrude for the heavy elements — is a standard benchmark. At the frontier of the periodic table, the superheavy elements test whether the $(n+\ell)$ ordering survives strong relativistic reordering at all: predicting the configurations of elements beyond $Z\approx118$ is an open problem where relativistic quantum chemistry, not the Madelung rule, has the last word.

The physical picture

Picture the atom as an apartment building. Floors are shells (the principal number $n$ ); each floor has rooms (the orbitals). Exclusion is the house rule: every room sleeps at most two, one spin-up and one spin-down. Aufbau is the tenants' preference: take the cheapest available room first. If that were the whole story, you would fill floor by floor and the shell number would be all that mattered. The twist is that some upper-floor rooms are cheaper than lower-floor ones. The cost of a room is its orbital energy, and that energy is set by how much of the nuclear charge the electron actually feels. An electron with low angular momentum has a weaker centrifugal barrier near the nucleus, so its wavefunction can penetrate strongly through the screening cloud of the other electrons and sample a more nearly unscreened nuclear charge. It is cheap. A high- $\ell$ electron is kept farther from the nucleus by its larger centrifugal barrier, spends more of its probability density in the screened region, and therefore feels a smaller effective nuclear charge. It is dear. So the $4s$ room, despite having the higher principal quantum number, is placed before $3d$ by the Madelung $(n+\ell)$ rule: $4s$ has $n+\ell=4$ , while $3d$ has $n+\ell=5$ . Penetration and screening also influence the actual orbital energies, so the ordering is a useful filling rule rather than an exact universal energy hierarchy.

Figure

The Madelung staircase: filling by increasing n+ℓ, and the outlier where 4s beats 3d

Why fill lowest-first at all? In the independent-electron approximation, the ground state is obtained by occupying the available one-electron orbitals so as to minimize the total orbital energy, subject to the Pauli exclusion principle. Thus, when the orbital energies are known, the lowest-energy orbitals are occupied first — the basic idea behind the Aufbau rule. That is a short argument, and we make it precise below.

The filling engine: Hamiltonian, screening, and the sum rule

The exact stationary states of an $N$ -electron atom are eigenstates of the full Hamiltonian,

\hat{H}=\sum_{i=1}^{N}\left(-\frac{\hbar^2}{2m}\nabla_i^2-\frac{Ze^2}{4\pi\varepsilon_0 r_i}\right)+\sum_{i<j}\frac{e^2}{4\pi\varepsilon_0\,|\mathbf{r}_i-\mathbf{r}_j|} \qquad (1)

where: $N$ — number of electrons; $Z$ — nuclear charge; $\mathbf{r}_i$ — position of electron $i$ ; $m$ — electron mass; $\varepsilon_0$ — vacuum permittivity. The physical read: the last term, the mutual repulsion, couples every coordinate to every other, so no closed-form solution exists for $N\ge 2$ . The Aufbau picture buys tractability by replacing that repulsion with a spherically averaged screen.

V_{\mathrm{eff}}(r)=-\frac{Z_{\mathrm{eff}}(r)\,e^2}{4\pi\varepsilon_0\,r} \qquad (2)

where: $Z_{\mathrm{eff}}(r)$ — the effective nuclear charge after screening, running from nearly $Z$ deep inside to a small residual charge far out. Because $V_{\mathrm{eff}}$ is central, its eigenstates still carry $(n,\ell,m_\ell)$ and separate into a radial function times a spherical harmonic — the radial machinery of the hydrogenic problem is owned by The Hydrogen Atom: Coulomb Radial Equation, recalled here by name. What is new relative to bare hydrogen is that $V_{\mathrm{eff}}$ makes the energy depend on both $n$ and $\ell$ : the $\ell$ -degeneracy is lifted, and low- $\ell$ orbitals sink.

\varepsilon_{n\ell}\ \text{filled in order of increasing}\ (n+\ell),\ \text{ties broken by smaller}\ n \qquad (3)

The Madelung rule is the compact statement of that ordering. It is empirical — it summarizes the outcome of the penetration-and-screening physics rather than being derived from a formula — but it reproduces the observed filling of the ground states across the table. The resulting sequence,

1s\;2s\;2p\;3s\;3p\;4s\;3d\;4p\;5s\;4d\;5p\;6s\;4f\;5d\;6p\;7s\;5f\;6d \qquad (4)

is exactly what a $(n+\ell,\,n)$ sort of the orbitals produces, as the first derivation verifies. Each orbital's occupancy cap follows from exclusion:

g_{n\ell}=2(2\ell+1),\qquad 2\sum_{\ell=0}^{n-1}(2\ell+1)=2n^2 \qquad (5)

where: $g_{n\ell}$ — the maximum electrons in a subshell; the factor $2\ell+1$ counts the $m_\ell$ values (spatial orbitals) and the factor $2$ the spin states. These give $s,p,d,f = 2,6,10,14$ and shell totals $2,8,18,32$ — the block widths and period lengths of the periodic table. The capacity rule itself is owned by Pauli Exclusion Principle, which states it as $2(2\ell+1)=2,6,10,14$ ; we consume it here to fill the sequence.

Reading off configurations

With the sequence in hand and the caps fixed, a ground configuration is a counting exercise: pour $Z$ electrons into (4), filling each subshell to capacity before moving on. Carbon ( $Z=6$ ) fills $1s^2 2s^2 2p^2$ ; argon ( $Z=18$ ) closes at $1s^2 2s^2 2p^6 3s^2 3p^6$ ; potassium ( $Z=19$ ) then adds an electron not to $3d$ but to $4s$ , because $4s$ comes first in the sequence — the direct fingerprint of the $(n+\ell)$ outlier. The independent-electron total energy that the filling minimizes is

E_{\text{tot}}\approx\sum_{a\in\text{occ}}\varepsilon_a \qquad (6)

and the antisymmetric state the configuration actually names is a Slater determinant of the occupied spin-orbitals — the construction owned by Slater Determinants, recalled by name.

Derivation AThe Madelung sequence from an (n+ℓ, n) sort, and why lowest-first minimizes the energy

We establish the two structural facts the whole scheme rests on: that sorting the orbitals by $(n+\ell)$ with smaller $n$ breaking ties reproduces the empirical filling sequence (4), and that filling the lowest orbitals first is the energy-minimizing choice under the exclusion constraint. The goal is to show Aufbau is not a mnemonic but a consequence of two things — an ordering rule and a variational argument — each of which can be checked mechanically. The payoff is a filling engine you can turn by hand.

Step 1. Generate and sort the orbitals. List the allowed orbitals $(n,\ell)$ with $\ell=0,\ldots,3$ up through $n=7$ . Sort them first by increasing $n+\ell$ , and break ties by smaller $n$ . Reading the labels in this order gives the usual Madelung sequence, including $4s$ before $3d$ .

Step 2. Locate the outlier. In the sorted list, $4s$ sits at position 6 and $3d$ at position 7. Their Madelung sums are $4+0=4$ and $3+2=5$ : $4s$ wins because $4<5$ , despite $3d$ having the smaller $n$ . This ordering places $4s$ before $3d$ in the usual filling sequence, so the $3d$ subshell begins filling only after $4s$ has been occupied; this is an important part of the electron configurations across the transition-metal region.

Step 3. Energy minimization. In the independent-electron model the total energy is $E_{\rm tot}=\sum_{\alpha\in{\rm occ}}\epsilon_\alpha$ , a sum over the occupied spin-orbitals. Exclusion allows each spin-orbital to be occupied at most once. Choosing exactly $N$ spin-orbitals to minimize the sum is a trivial optimization: take the $N$ with the smallest $\varepsilon_a$ . That is precisely "fill lowest-first," two per spatial orbital.

Step 4. Boundary check. For a single electron ( $N=1$ ) the prescription gives the $1s$ ground state — the hydrogenic answer, where there is no screening and the $\ell$ -degeneracy is exact. For $N$ electrons, the independent-electron picture reproduces the broad shell structure and gives the usual filling sequence for most atoms. The $(n+\ell)$ rule reproduces sequence (4), including $4s$ before $3d$ , and provides a useful starting point for understanding configurations such as He, Ne, Ar, Sc, and Fe, while exceptions such as Cr and Cu require additional many-electron effects. The $(n+\ell, n)$ sort reproduces sequence (4), with $4s$ before $3d$ , and gives a useful starting point for configurations such as He, Ne, Ar, Sc, and Fe — with exceptions such as Cr and Cu requiring additional many-electron effects.

The second gear is the physics behind the outlier: why does the $(n+\ell)$ ordering come out the way it does?

Go deeper · Derivation B · 1 minWhy 4s fills before 3d: penetration and screening

Derivation BWhy 4s fills before 3d: penetration and screening

We justify the single most consequential feature of the sequence — that $4s$ sinks below $3d$ in a neutral atom being built — from the effective potential and the shapes of the orbitals. The goal is to trace the inversion to the centrifugal barrier and radial penetration, not to a numerical coincidence, and to show the same physics reverses the ordering once $3d$ is occupied. The payoff is understanding both why $4s$ fills first and why $4s$ ionizes first.

Step 1. The centrifugal barrier. In the radial equation the effective potential carries an angular term, $V_{\mathrm{eff}}^{\ell}(r)=V_{\mathrm{eff}}(r)+\hbar^2\ell(\ell+1)/2mr^2$ . For $3d$ ( $\ell=2$ ) this barrier is large and pushes the orbital outward, into the region where the nucleus is heavily screened.

Step 2. Penetration of 4s. The $4s$ orbital ( $\ell=0$ ) has no centrifugal barrier and a nonzero probability density at small $r$ . Its inner lobes penetrate the screening cloud and sample a much larger $Z_{\mathrm{eff}}$ near the nucleus. Deeper sampling means a more negative energy.

Step 3. The ordering. Greater penetration means weaker screening and therefore a lower orbital energy. In neutral atoms, this helps place $4s$ below $3d$ before the $3d$ subshell begins to fill, consistent with the usual Aufbau sequence and the configurations of potassium, $[\mathrm{Ar}]4s^1$ , and calcium, $[\mathrm{Ar}]4s^2$ .

Step 4. Role reversal on occupation. Once electrons populate $3d$ , the mutual screening among the $n=3$ electrons raises $4s$ relative to $3d$ . In the assembled neutral transition metal $3d$ lies below $4s$ , so ionization removes the $4s$ electrons first: iron $[\mathrm{Ar}]3d^6 4s^2$ ionizes to $\mathrm{Fe}^{2+}=[\mathrm{Ar}]3d^6$ , not $[\mathrm{Ar}]3d^4 4s^2$ . As expected: K $=[\mathrm{Ar}]4s^1$ and Ca $=[\mathrm{Ar}]4s^2$ (4s filled, 3d empty), and $\mathrm{Fe}^{2+}=[\mathrm{Ar}]3d^6$ from removing the two 4s electrons.

Figure

The Cr and Cu exceptions: exchange energy poaches an electron from 4s to complete a half- or full-filled d-shell

Limiting cases

The Aufbau picture lives between physical regimes; its validity is set by how well an independent-electron model with a fixed level ordering holds.

Case	Result	Meaning
Hydrogen / one electron	$E_n=-13.6\,\mathrm{eV}/n^2$	no screening: energies depend only on $n$ , the $\ell$ -degeneracy is exact, Aufbau ordering is trivial
Strong screening (many electrons)	$V_{\mathrm{eff}}(r)\approx -Z_{\mathrm{eff}}(r)e^2/4\pi\varepsilon_0 r$	inner electrons screen the nucleus; $\ell$ -degeneracy lifts; $(n+\ell)$ ordering emerges; Aufbau most useful
Near-degenerate competing subshells	$\lvert\varepsilon_{4s}-\varepsilon_{3d}\rvert\sim K_{\text{exch}}$	when level spacing $\sim$ exchange energy (transition metals, lanthanides), naive filling fails — Cr, Cu, Nb, Pd
Heavy atoms / relativistic	$v\sim Z\alpha c$	large $Z$ relativistic and spin-orbit effects reorder and split levels (gold's color); nonrelativistic ordering breaks

Common trapFilling order is not energy order — and Aufbau does not fix spins

Two traps. First, the Madelung filling order is NOT a universal statement about orbital energies after the atom is built. $4s$ is filled before $3d$ in the usual Aufbau sequence, but after the $3d$ subshell is occupied the relative orbital energies can change, and the $4s$ electrons are generally removed first on ionization. For example, Fe $^{2+}$ has the configuration $[\mathrm{Ar}]3d^6$ , not $[\mathrm{Ar}]3d^4 4s^2$ . Treating the filling sequence as a fixed energy ladder is the classic error. Second, Aufbau determines which subshells are occupied, but it does not by itself determine how electrons are distributed among degenerate orbitals or what their spins are. Hund's rules favor the maximum total spin for a partially filled subshell, so carbon has a $2p^2$ configuration with two electrons occupying different $p$ orbitals with parallel spins in its ground state.

Worked examples

Example 1 — Chromium's broken filling. Build the ground configuration of chromium ( $Z=24$ ) two ways — the naive Madelung fill and the actual state — and account for the difference using exchange energy.

Step 1. Enumerate. Pour 24 electrons into the Madelung sequence (4). Argon closes the first 18 ( $1s^2 2s^2 2p^6 3s^2 3p^6$ ). The next orbitals in order are $4s$ (holds 2) then $3d$ (holds 10). Filling naively gives $4s^2$ then $3d^4$ : the prediction is $[\mathrm{Ar}]3d^4 4s^2$ .

Step 2. Apply the correction. The measured ground state (NIST) is $[\mathrm{Ar}]3d^5 4s^1$ — one electron has moved from $4s$ to $3d$ , half-filling the $d$ -subshell. This costs a little orbital energy but the $4s$ and $3d$ levels are nearly degenerate here, so the exchange energy released by making all five $d$ -spins parallel wins. A half-filled $d^5$ has $\binom{5}{2}=10$ parallel same-spin pairs, versus $\binom{4}{2}=6$ for $d^4$ — a gain of $4$ pairs' worth of exchange stabilization, the mechanism owned by Exchange Forces and Exchange Energy.

Step 3. Boundary number. The electron count is invariant: $18+5+1=24=Z$ , so both configurations are legitimate 24-electron states — the exception is a competition between two nearly-equal energies, not a violation of exclusion. The same argument, applied to a full $d$ -shell, explains copper: $[\mathrm{Ar}]3d^{10} 4s^1$ beats $[\mathrm{Ar}]3d^9 4s^2$ because a closed $d^{10}$ is spherical and minimizes repulsion.

Example 2 — Iron, and why $\mathrm{Fe}^{2+}$ is $[\mathrm{Ar}]3d^6$ . Determine iron's ground configuration and its common ion.

Step 1. Build the neutral atom. Pour $Z=26$ into the sequence: $[\mathrm{Ar}]$ (18), then $4s^2$ (20), then $3d^6$ (26). Iron is $[\mathrm{Ar}]3d^6 4s^2$ , and $18+6+2=26$ checks.

Step 2. Ionize. Once the atom is assembled, screening among the $n=3$ electrons has pushed $3d$ below $4s$ , so the $4s$ electrons are the loosest. Removing two gives $\mathrm{Fe}^{2+}=[\mathrm{Ar}]3d^6$ (count $18+6=24$ ), and removing a third gives $\mathrm{Fe}^{3+}=[\mathrm{Ar}]3d^5$ (count $23$ ). The half-filled $3d^5$ of $\mathrm{Fe}^{3+}$ is extra-stable — which is why the $+3$ oxidation state is so prominent in iron chemistry. The filling order built $4s$ first, but the energy order at the end ionizes $4s$ first — the two orders are different, and iron is the cleanest demonstration.

Origins · 1 minaufbau principle and how this was discovered

aufbau principle

coined in the early 1920s · Copenhagen · Göttingen

The Aufbauprinzip — the "building-up" principle — grew out of Niels Bohr's program in Copenhagen in the early 1920s to explain the periodic table by adding electrons to an atom shell by shell, before quantum mechanics existed to justify it. The empirical ordering that governs the filling, the $(n+\ell)$ rule, is usually credited to Erwin Madelung, who tabulated it in 1936; the Russian chemist Vsevolod Klechkowski gave an energetic argument for it in 1962 (so it is sometimes called the Klechkowski rule), and Charles Janet had drawn a periodic table organized by $(n+\ell)$ blocks as early as 1928. None of these figures carries a portrait term in this reference, so we credit them in passing: the rule is a distilled empirical fact about penetration and screening, later grounded in the quantum theory of the multi-electron atom. Its enduring value is predictive economy — one integer, $n+\ell$ , orders the orbitals of the entire periodic table, and where it fails it fails informatively, pointing straight at the exchange energy that overrules it.

History

1921

— Bohr articulates the Aufbauprinzip, assigning electrons to shells to explain periodicity

1925

— Pauli's exclusion principle caps orbital occupancy, making shell-filling well-defined

1928

— Janet arranges a periodic table by

(n+\ell)

blocks (the "left-step" table)

1936

— Madelung tabulates the

(n+\ell)

filling rule

1962

— Klechkowski gives an energetic justification for the

(n+\ell)

ordering

Going further

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §5.2 (Atoms) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 7 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 10 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Complement $A_{XIV}$ · Bransden & Joachain, Physics of Atoms and Molecules, 2nd ed., Ch. 8 The key structural results are: the $(n+\ell,n)$ sort reproduces sequence (4) exactly including the $4s$ -before- $3d$ inversion; the subshell capacities are $2(2\ell+1)=2,6,10,14$ and shell capacities $2n^2=2,8,18,32$ ; electron counts sum to $Z$ for He, Ne, Ar, Sc, and Fe; the Cr $[\mathrm{Ar}]3d^5 4s^1$ and Cu $[\mathrm{Ar}]3d^{10} 4s^1$ exceptions follow from the $\binom{5}{2}=10$ parallel-pair exchange argument; and ionization of iron gives $\mathrm{Fe}^{2+}=[\mathrm{Ar}]3d^6$ . The exclusion cap $2(2\ell+1)$ is owned by Pauli Exclusion Principle (which also states $2n^2=2,8,18,32$ ); the exchange-energy mechanism behind the Cr/Cu exceptions is owned by Exchange Forces and Exchange Energy; the spin arrangement inside a subshell is owned by Hund's Rules; and the periodicity of the whole table is owned by The Periodic Table from Quantum Mechanics. This article owns the filling engine — the Madelung sequence, its energy-minimization justification, and its exceptions.

Builds on: Pauli Exclusion Principle · Slater Determinants · Bosons and Fermions · Spin-Statistics Theorem Leads to: Hund's Rules · The Helium Atom · Exchange Forces and Exchange Energy · Singlet and Triplet States in Two-Electron Systems

:::faq[If 4s fills before 3d, why do we write iron as [Ar]3d⁶4s² with 3d first?] Two different conventions for two different questions. The filling order (which orbital gets electrons as you build up $Z$ ) is Madelung: $4s$ before $3d$ . The written configuration conventionally groups orbitals by shell, so $3d$ is listed before $4s$ even though $4s$ filled first. Both are correct; they answer "in what order did the electrons go in?" versus "how do we group the occupied orbitals on paper?". The physically important point is the third order — the energy order in the assembled neutral atom, where $3d$ has sunk below $4s$ , which is why ionization removes $4s$ first.

Why does the (n+ℓ) rule work — is there a proof, or is it just a mnemonic?

It is an empirical rule that summarizes real physics, not a theorem derived from a single formula. Two effects compete: raising $n$ raises the energy, and raising $\ell$ also raises it (via the centrifugal barrier, which pushes high- $\ell$ orbitals out into the screened region). Their sum $n+\ell$ happens to order the effective orbital energies well across the table, with smaller $n$ breaking ties. Klechkowski gave an energetic argument for it in 1962, and self-consistent Hartree-Fock calculations reproduce the ordering, but the rule remains a distilled fact about penetration and screening rather than a first-principles identity — which is exactly why it has well-defined exceptions.

Are Cr and Cu the only exceptions?

No — there are around twenty exceptions across the periodic table, concentrated where $s$ , $d$ , and $f$ levels crowd together. Chromium and copper are the textbook cases in the $3d$ row; niobium, molybdenum, ruthenium, rhodium, palladium, and silver show similar $s\to d$ shifts in the $4d$ row, and the lanthanides and actinides have many $f$ -block irregularities. All share the same origin: two orbitals so close in energy that exchange stabilization (favoring half- or fully-filled subshells) or reduced repulsion (favoring closed shells) tips the balance. The exceptions are not random; they cluster exactly where the Madelung near-degeneracies are.

Does the Aufbau principle apply to ions and excited states?

The ground configuration of a neutral atom is what Aufbau predicts (with the noted exceptions). For ions you cannot simply run Aufbau in reverse: because the level ordering shifts once the atom is assembled, you remove the electrons that are highest in energy in the built atom, which for transition metals means the $4s$ electrons first even though they filled first. Excited configurations promote an electron to a higher orbital and are labeled by which orbital changed. So Aufbau builds the neutral ground state; ionization and excitation require the assembled atom's energy ordering, which is not the filling order.

What is the difference between the shell capacity 2n² and the subshell capacity 2(2ℓ+1)?

A subshell is a single $(n,\ell)$ : it contains $2\ell+1$ spatial orbitals (the $m_\ell$ values) and, with two spin states each, holds $2(2\ell+1)$ electrons — so $s,p,d,f$ hold $2,6,10,14$ . A shell is a full principal number $n$ , summing over all its subshells $\ell=0,\dots,n-1$ : $2\sum_\ell(2\ell+1)=2n^2$ , giving $2,8,18,32$ for $n=1,2,3,4$ . The subshell capacities are the block widths of the periodic table (s-block 2 wide, p-block 6, d-block 10, f-block 14); the shell capacities relate to period lengths but are modulated by the Madelung order (which interleaves subshells from different shells). Both caps come from exclusion, owned by Pauli Exclusion Principle.

Does Aufbau explain the shape of the periodic table by itself?

It explains the block widths and the filling order, but the full periodicity — why chemically similar elements recur, the trends in ionization energy and atomic radius, and the geometry of periods — is owned by The Periodic Table from Quantum Mechanics, which adds screening (Slater's rules) and the resulting trends. Aufbau is the engine that assigns electrons to orbitals in order; the periodic table is what you get when you tabulate the results of running that engine across all the elements and notice that closed-subshell structure repeats. The two are tightly linked but distinct: this article owns the filling, the periodicity article owns the pattern.