Electron Configurations and the Aufbau Principle

Electron configurations are not a museum piece confined to introductory tables — they are load-bearing infrastructure across chemistry and materials science. In coordination chemistry and bioinorganic chemistry, the $d^n$ configuration of a transition-metal ion (obtained by the same "remove $ns$ electrons first" rule used above for Mn²⁺) directly sets crystal-field stabilization energy, spin state (high- vs. low-spin), and magnetic moment — essential for understanding oxygen binding in hemoglobin (Fe²⁺, $d^6$), electron transport in cytochromes, and the design of single-molecule magnets built on lanthanide $4f^n$ ions. In materials science, the valence electron count derived from configurations underlies band-structure predictions for semiconductors and the 18-electron rule used to predict stability of organometallic complexes in catalysis (hydroformylation, cross-coupling catalysts). In X-ray photoelectron spectroscopy (XPS), core-electron binding energies — a direct experimental probe of the core/valence partition — are used routinely to fingerprint oxidation states at surfaces, from battery electrode characterization to catalyst degradation studies. In nuclear and superheavy-element chemistry, predicted configurations (with relativistic corrections) for elements like oganesson or flerovium are essentially the only accessible information about their expected chemical behavior, since these elements exist for microseconds and in single-atom quantities — Aufbau-type reasoning, suitably relativistically corrected, is the entire predictive apparatus available. And pedagogically, Aufbau remains the gateway concept connecting Orbital Energy Ordering in Multielectron Atoms and The Four Quantum Numbers to the large-scale architecture of the Structure of the Periodic Table, while its exceptions are the natural on-ramp to the deeper story of Configuration Anomalies.

:::quiz[When forming the Fe²⁺ ion from neutral iron ([Ar]3d⁶4s²), which electrons are removed first, and why?]

• Two 3d electrons, because 3d is always higher in energy than 4s
• Two 4s electrons, because once the ion forms, the compact 3d orbitals lie below the diffuse 4s orbital in energy ✓
• One 4s and one 3d electron, split evenly
• It is ambiguous and depends only on temperature

:::quiz[Chromium's ground-state configuration is [Ar]3d⁵4s¹ rather than the naively predicted [Ar]3d⁴4s². What is the principal driving force for this exception?]

• Relativistic contraction of the 4s orbital
• Extra exchange stabilization gained by achieving a half-filled 3d⁵ subshell with maximal unpaired spins ✓
• A typographical convention with no physical basis
• The Pauli exclusion principle forbids 3d⁴4s² configurations entirely

:::faq[If 3d is lower in energy than 4s for transition-metal ions, why do textbooks still write configurations as [Ar]3d⁵4s² instead of [Ar]4s²3d⁵ or [Ar]3d⁷?] Two separate things are being conflated. First, the listing convention groups electrons by principal quantum number for readability, writing $3d$ before $4s$ simply because $3d$ belongs to the third shell — this has nothing to do with energy ordering, only with alphabetizing by shell number. Second, and more substantively, for a neutral atom the $4s$ orbital genuinely is filled (and is the lower-energy choice) during the building-up process; it is only after ionization, once electrons are removed and the remaining ones rearrange, that $3d$ drops below $4s$ in the resulting ion's own orbital energy ordering. Both configurations ($3d^54s^2$ for atom, $3d^5$ for the 2+ ion) are correct for their respective species.

Ask the tutor: Why doesn't tungsten show the same $5d$/$6s$ promotion anomaly that gold does, given they're both period-6 transition metals?

Every neutral atom is defined by two numbers: the nuclear charge $Z$ (which fixes the number of protons and, for neutral atoms, the number of electrons) and the arrangement of those electrons among the available orbitals. The Aufbau principle states that in the ground state, electrons occupy the lowest-energy orbitals available, one at a time, subject to two constraints already established elsewhere: the Pauli exclusion principle (no two electrons in an atom share all four quantum numbers) and, for degenerate orbitals within a subshell, Hund's rule (maximize unpaired spins before pairing). Aufbau itself answers a narrower question than either of those: given a stack of orbitals ranked by energy, in what order do subshells get populated as $Z$ increases? For a one-electron atom (hydrogen-like), orbital energy depends only on the principal quantum number $n$:

so all orbitals with the same $n$ (the $2s$, $2p$; or the $3s$, $3p$, $3d$) are degenerate. But real atoms have more than one electron, and electron–electron repulsion breaks that degeneracy. An electron in an $s$ orbital has a nonzero probability density right at the nucleus (finite $\psi(0)$), so it penetrates the electron cloud of inner shells more effectively than a $p$ electron of the same $n$, which in turn penetrates more than a $d$ electron. Better penetration means less shielding from the other electrons, means a lower (more negative, more stable) orbital energy. This is the physical root of the split

for a given $n$, once more than one electron is present — this ordering was itself established as a consequence of shielding and penetration. The empirical bookkeeping tool that reproduces the correct filling order for the vast majority of the periodic table is the Madelung rule, also called the $(n+\ell)$ rule:

• orbitals fill in order of increasing $n + \ell$;
• for equal $n + \ell$, the orbital with smaller $n$ fills first.

This single rule, applied mechanically, generates the entire filling sequence:

Notice immediately that $4s$ (with $n+\ell = 4+0=4$) fills before $3d$ (with $n+\ell = 3+2=5$), even though $3d$ has a smaller principal quantum number. This is the origin of the transition-metal filling pattern and the reason the fourth period starts $[\text{Ar}]4s^1$ (K), $[\text{Ar}]4s^2$ (Ca), then moves into $3d$ from scandium onward.

Once the filling order is fixed, writing a configuration is direct bookkeeping: distribute $Z$ electrons into that ordered list of subshells, respecting subshell capacities ($s$: 2, $p$: 6, $d$: 10, $f$: 14, from $2(2\ell+1)$). For compactness, chemists write condensed (noble-gas core) configurations: replace the electrons corresponding to the most recent preceding noble gas with its bracketed symbol, then list only the remaining ("valence-region") subshells in the order they were filled, but conventionally regrouped by principal quantum number for clarity. For example bromine ($Z=35$):

Argon ($Z=18$) accounts for the first 18 electrons; the remaining 17 are written explicitly, with $3d$ conventionally placed before $4s,4p$ in the final written form even though it filled after $4s$ — this is a matter of listing convention (grouping by shell $n$) and does not change what fills when. This convention split — filling order vs. listed order — is a frequent point of confusion, so it is worth stating plainly as its own rule.

Core electrons are those belonging to filled inner shells that reproduce a noble-gas closed-shell configuration; they are strongly bound, spherically symmetric in aggregate, largely chemically inert, and act mainly to shield the valence electrons from the full nuclear charge. Valence electrons are those in the outermost occupied principal shell, together with any incompletely filled $d$ or $f$ subshells of lower $n$ that lie close in energy and participate in bonding — this is why for transition metals both the $ns$ and $(n-1)d$ electrons count as valence, and for lanthanides/actinides the $(n-2)f$ electrons often do too. Valence electrons set oxidation states, coordination chemistry, color, and magnetism; core electrons do not (to first approximation) participate in bond formation, though core-level energies matter for X-ray photoelectron spectroscopy (XPS) and core-hole spectroscopy.

where: $n$ is the principal quantum number, $\ell$ is the azimuthal (orbital angular momentum) quantum number ($\ell = 0,1,2,3$ for $s,p,d,f$). Reading: this single integer sum reproduces, almost without exception, the correct order in which subshells acquire their first electron across the entire periodic table; it is an empirical distillation of the shielding/penetration physics, not a separate law.

where: $2\ell+1$ is the number of degenerate orbitals ($m_\ell$ values) in the subshell, and the factor of 2 comes from the two spin states ($m_s = \pm\tfrac12$) allowed by the Pauli principle per orbital. Reading: this fixes $s,p,d,f$ capacities at 2, 6, 10, 14 electrons, which is what makes period lengths 2, 8, 18, 32 — the block structure of the periodic table is a direct printout of this formula.

where: $Z$ is the true nuclear charge, $S$ is the shielding (screening) constant from all other electrons (estimated, e.g., by Slater's rules), and $Z_{\text{eff}}$ is the effective nuclear charge felt by a given electron. Reading: orbital energy tracks $-Z_{\text{eff}}^2/n^2$ to a first approximation; because $S$ depends on which other orbitals are occupied, $Z_{\text{eff}}$ — and hence the true energy ordering of subshells — shifts as electrons are added or removed, which is exactly why filling order and ionization order can differ.

where: for a neutral atom being built up, $4s$ is lower until a small nuclear charge threshold, but once several $3d$ electrons are present in a cation, the compact, poorly-shielded $3d$ orbital drops below $4s$. Reading: this inequality flipping is precisely why $4s$ fills before $3d$ in neutral atoms, yet $4s$ electrons are lost before $3d$ electrons on ionization — both facts follow from the same physics evaluated at different electron counts.

Example 1 (foundational): Full and condensed configuration of vanadium, Z = 23. Apply the Madelung filling order and subshell capacities directly. Filling order up to the needed subshells:

with capacities $2,2,6,2,6,2,10,\dots$. Running electron count:

The count reaches exactly $23$ after placing 3 electrons in $3d$, since $20 + 3 = 23$. Full configuration:

Condensed (using argon, $Z=18$, and conventional shell-grouped listing order):

Core electrons: the 18 electrons of the $[\text{Ar}]$ core ($1s^22s^22p^63s^23p^6$). Valence electrons: the remaining 5 ($3d^3 4s^2$), consistent with vanadium's common oxidation states up to $+5$ (as in $\text{VO}_2^+$ or $\text{VO}_4^{3-}$, vanadium(V)), since all five valence electrons can in principle be removed or engaged in bonding. Limiting-case check: setting $Z=18$ (argon itself) should return exactly the noble-gas configuration with no partially filled outer subshell — running the same procedure gives $1s^22s^22p^63s^23p^6 = [\text{Ne}]3s^23p^6$, a closed-shell, chemically inert result, consistent with argon's known lack of a stable higher- or lower-charge ion and its position as a full period-3 noble gas.

Example 2 (applied): Configuration and valence count for the Mn²⁺ ion, relevant to bioinorganic and EPR spectroscopy. Start from neutral manganese, $Z = 25$:

(check: $18 + 5 + 2 = 25$ ✓; note manganese is not one of the Cr/Cu-type exceptions — the naive $(n+\ell)$ prediction and the true ground state agree here, because $3d^5 4s^2$ already has the half-filled $3d^5$ advantage without needing to promote an electron). To form $\text{Mn}^{2+}$, remove two electrons from the $4s$ orbital first, per the ionization rule established above (once ions form, $3d$ lies below $4s$ in energy, so $4s$ empties preferentially):

This is a genuinely important result: $\text{Mn}^{2+}$ is $3d^5$, a half-filled subshell with five unpaired electrons (high-spin, $S = 5/2$), which is why $\text{Mn}^{2+}$ complexes typically show very weak, spin-forbidden d–d transitions (pale color) and give rise to characteristic six-line EPR hyperfine spectra (from coupling to the $I=5/2$ $^{55}$Mn nucleus) — both signatures trace directly back to this configuration. Limiting-case check: if instead one (incorrectly) removed electrons from $3d$ first, the result would be $[\text{Ar}]3d^34s^2$ for "$\text{Mn}^{2+}$," predicting only three unpaired electrons ($S=3/2$) — directly contradicted by measured magnetic moments (Mn²⁺ salts show $\mu_{\text{eff}} \approx 5.9\ \mu_B$, matching $S=5/2$ via the spin-only formula $\mu = \sqrt{n(n+2)}\,\mu_B$ with $n=5$, not $n=3$). This confirms, independently of any bookkeeping rule, that $4s$ must empty before $3d$.

Example 3 (exam/competition level): Ground-state configuration of lanthanum vs. cerium, and why the "regular" $(n+\ell)$ rule needs a spectroscopic correction at the start of the lanthanides. Naive Madelung filling for lanthanum, $Z = 57$: after $[\text{Xe}]$ ($Z=54$) and $6s^2$ ($Z=56$), the $(n+\ell)$ rule says the next subshell is $4f$ ($n+\ell = 4+3=7$) before $5d$ ($n+\ell=5+2=7$, but higher $n$, so $4f$ should fill first at equal $n+\ell$). This predicts

The experimentally observed ground state, however, is

with the $4f$ orbital left empty. The reason is that $4f$ and $5d$ are extremely close in energy at the start of the lanthanide series — closer than the Madelung rule's coarse bookkeeping can resolve — and for lanthanum specifically, the absence of any other $4f$ or $5d$ electrons means there is no exchange stabilization yet to favor populating the compact $4f$ orbital over the more accessible $5d$. This makes La (along with Ce, in a milder form) a genuine, spectroscopically confirmed exception at the very start of the $f$-block. For cerium, $Z = 58$, the naive rule predicts $[\text{Xe}]4f^2 6s^2$; the measured ground state is

another one-electron rearrangement between $4f$ and $5d$, again reflecting their extreme near-degeneracy at low $4f$-occupation. By the time the series reaches gadolinium, $Z = 64$, exchange stabilization of a half-filled $4f^7$ subshell reasserts a Cr-like pattern:

confirming that the same exchange-stabilization logic used for Cr/Cu governs part of the lanthanide series too, layered on top of the intrinsically small $4f$–$5d$ gap. Limiting-case / consistency check: by mid-to-late lanthanides where $4f$ has dropped well below $5d$ in energy (once several $4f$ electrons are present and have contracted the $4f$ shell via reduced self-shielding), the naive $(n+\ell)$ rule reliably resumes — e.g., ytterbium ($Z=70$) is correctly predicted and observed as $[\text{Xe}]4f^{14}6s^2$, with no anomaly, because at that point $4f^{14}$ is both complete (maximal stabilization, no competing exchange calculus needed) and far enough below $5d$ that no near-degeneracy ambiguity remains. This shows the La/Ce/Gd anomalies are confined to the region of the $4f$/$5d$ near-crossing, exactly as the shielding-based argument predicts, rather than being scattered unpredictably across the whole series.