Electron Configurations

Electron configurations tell you exactly where every electron sits in an atom. This bridges Electron Shells and Orbitals with real chemical behavior. The arrangement of electrons decides how an atom bonds, reacts, and fits into the periodic table. Three simple rules — the Aufbau principle, the Pauli exclusion principle, and Hund's rule — are all you need. With them, you can write the correct configuration for any element in its lowest-energy (ground) state. Once you know these rules, writing a configuration becomes a filling exercise. It is like stacking books on shelves of fixed capacity. Each shelf (orbital) holds at most two electrons. Each sub-shelf group (subshell) fills in a strict order of increasing energy. Electrons spread out across equal-energy shelves before doubling up. The result is a compact notation like $1s^2 2s^2 2p^6$ . It encodes an enormous amount of chemical information in a few symbols.

Key takeaways

The Aufbau principle says electrons fill orbitals starting from the lowest available energy level.
The Pauli exclusion principle limits each orbital to two electrons, and they must have opposite spins.
Hund's rule says degenerate orbitals (same subshell) each get one electron before any is doubly occupied.
Configurations are written as $n\ell^{x}$ , e.g., $2p^4$ , read directly off an orbital-filling diagram or the periodic table.
A handful of exceptions (like Cr and Cu) arise because half-filled and fully filled subshells have extra stability.
Key idea: ground-state configurations are the unique lowest-energy way to distribute an atom's electrons among orbitals, built by filling one electron at a time according to three rules.

The core insight

Fill orbitals from lowest to highest energy (Aufbau), never put more than two electrons of opposite spin in one orbital (Pauli), and spread electrons across equal-energy orbitals before pairing them (Hund).

The physical picture

An atom's electrons do not simply pile into the lowest shell available. Each electron occupies an orbital, a region described by a specific set of quantum numbers $(n, \ell, m_\ell, m_s)$ , and no two electrons in the same atom may share all four. This is the Pauli exclusion principle. Because $m_s$ can only be $+\tfrac12$ or $-\tfrac12$ , each spatial orbital (fixed $n$ , $\ell$ , $m_\ell$ ) can hold at most two electrons, and those two must have opposite spin. Orbitals are grouped into subshells labeled $s$ , $p$ , $d$ , $f$ , corresponding to $\ell = 0, 1, 2, 3$ . Within a given shell, subshells have different energies because of electron-electron repulsion and how effectively each subshell's electron density shields the nuclear charge. This is why $4s$ fills before $3d$ despite $4s$ belonging to a higher principal shell — the $4s$ orbital's electron density has more probability near the nucleus, lowering its energy below that of $3d$ for many atoms. The Aufbau principle ("building up") says that as protons are added to build heavier atoms, electrons fill orbitals in order of increasing energy, from the bottom up. The standard filling order, obtained from the diagonal rule, is:

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d

Within a subshell containing several orbitals of identical energy (a $p$ subshell has 3, a $d$ subshell has 5, an $f$ subshell has 7), Hund's rule requires that electrons occupy separate orbitals with parallel spins before any orbital gets a second electron. This minimizes electron-electron repulsion and maximizes total spin, which lowers overall energy.

Figure

Orbital filling order across subshells, showing the diagonal (Aufbau) sequence up through 4p

The second figure shows how Hund's rule plays out concretely for a $p^4$ configuration (as in oxygen's $2p^4$ ), where four electrons fill three degenerate orbitals.

Figure

Hund's rule applied to a p⁴ subshell: three orbitals, four electrons, two unpaired

The core relations

The full electron configuration is a sequence of subshell labels with electron counts as superscripts:

1s^{a} 2s^{b} 2p^{c} 3s^{d} 3p^{e} 4s^{f} 3d^{g} \cdots

where:

$n$ is the principal quantum number (shell number),
$\ell$ is the subshell type ( $s, p, d, f$ for $\ell = 0,1,2,3$ ),
the superscript is the number of electrons in that subshell,
the sum of all superscripts equals the total number of electrons (equal to atomic number $Z$ for a neutral atom).

The maximum capacity of a subshell follows from the number of orbitals it contains and the Pauli limit of two electrons per orbital:

\text{capacity of subshell } \ell = 2(2\ell + 1)

where:

$\ell = 0$ ( $s$ ) gives capacity $2$ ,
$\ell = 1$ ( $p$ ) gives capacity $6$ ,
$\ell = 2$ ( $d$ ) gives capacity $10$ ,
$\ell = 3$ ( $f$ ) gives capacity $14$ .

The total capacity of a shell $n$ (summing over all its subshells, $\ell = 0$ to $n-1$ ) is:

\text{capacity of shell } n = 2n^2

where:

$n$ is the principal quantum number,
this counts every orbital-and-spin combination available at that shell, consistent with $\ell$ ranging from $0$ to $n-1$ and $m_\ell$ ranging over $2\ell+1$ values.

For noble-gas core notation, a configuration is abbreviated by replacing the completed inner shells with the symbol of the preceding noble gas in brackets:

\text{[core]} + \text{valence subshells}

where:

[core] stands for the full configuration of the nearest preceding noble gas,
only the electrons added beyond that noble gas are written explicitly.

Derivation ADeriving subshell and shell capacities from quantum numbers

The goal is to show why $s,p,d,f$ hold $2,6,10,14$ electrons, and why shell $n$ holds $2n^2$ electrons total.

Step 1. For a subshell with orbital angular momentum quantum number $\ell$ , the magnetic quantum number $m_\ell$ can take any integer value from $-\ell$ to $+\ell$ . The count of allowed values is:

m_\ell \in \{-\ell, -\ell+1, \dots, 0, \dots, \ell-1, \ell\}

\text{number of } m_\ell \text{ values} = 2\ell + 1

Step 2. Each distinct $(n,\ell,m_\ell)$ combination defines one spatial orbital. By the Pauli exclusion principle, each orbital holds at most two electrons, distinguished by spin quantum number $m_s = +\tfrac12$ or $-\tfrac12$ . So the subshell capacity is:

\text{subshell capacity} = 2 \times (2\ell+1) = 2(2\ell+1)

Checking each case: $\ell=0 \Rightarrow 2(1) = 2$ ( $s$ ); $\ell=1 \Rightarrow 2(3)=6$ ( $p$ ); $\ell=2 \Rightarrow 2(5)=10$ ( $d$ ); $\ell=3 \Rightarrow 2(7)=14$ ( $f$ ).

Step 3. For a given principal quantum number $n$ , the allowed values of $\ell$ run from $0$ to $n-1$ (this restriction comes from the structure of the Schrödinger equation solutions for the hydrogen-like atom). The total shell capacity is the sum of subshell capacities over all allowed $\ell$ :

\text{shell capacity} = \sum_{\ell=0}^{n-1} 2(2\ell+1)

Step 4. Expand the sum. Using $\sum_{\ell=0}^{n-1} \ell = \dfrac{(n-1)n}{2}$ :

\sum_{\ell=0}^{n-1} 2(2\ell+1) = 2\left[2\sum_{\ell=0}^{n-1}\ell + \sum_{\ell=0}^{n-1} 1\right] = 2\left[2\cdot\frac{(n-1)n}{2} + n\right]

= 2\left[(n-1)n + n\right] = 2\left[n^2 - n + n\right] = 2n^2

Step 5 (limiting check). For $n=1$ , only $\ell=0$ is allowed, giving capacity $2(1)^2 = 2$ , matching the fact that shell 1 contains only the $1s$ orbital pair. For $n=4$ , the formula gives $2(4)^2 = 32$ , which correctly matches summing $2+6+10+14 = 32$ across the $4s,4p,4d,4f$ subshells. Both checks confirm the general formula $2n^2$ .

Worked examples

Example 1 — Full and condensed configuration of arsenic (Z = 33). Arsenic has 33 electrons in its neutral ground state. Fill subshells in Aufbau order, respecting each capacity from the derivation above, until all 33 electrons are placed.

1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^{10}\,4p^3

Check the electron count by summing superscripts:

2+2+6+2+6+2+10+3 = 33

This matches $Z = 33$ exactly, confirming the configuration is complete. The noble gas immediately before arsenic is argon ( $Z=18$ , configuration $1s^2 2s^2 2p^6 3s^2 3p^6$ ), so the condensed form is:

[\text{Ar}]\,4s^2\,3d^{10}\,4p^3

The valence subshell is $4p^3$ , meaning three electrons occupy the three degenerate $4p$ orbitals singly (Hund's rule), giving arsenic three unpaired electrons in its outermost subshell — consistent with arsenic's known chemistry of forming three covalent bonds in compounds like AsH₃.

Example 2 — Determining the element and unpaired electron count from a partial configuration. Suppose an atom's ground-state configuration is:

1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^{6}

Sum the electrons:

2+2+6+2+6+2+6 = 26

An atomic number of 26 identifies this as iron (Fe). To find the number of unpaired electrons, apply Hund's rule to the $3d^6$ subshell, which has 5 orbitals. Distribute 6 electrons: first fill each of the 5 orbitals singly (5 electrons used, all unpaired, all same spin), then place the 6th electron into the first orbital, pairing it (opposite spin):

\underbrace{\uparrow\downarrow}_{\text{paired}}\;\uparrow\;\uparrow\;\uparrow\;\uparrow

This leaves $5 - 1 = 4$ orbitals with a single unpaired electron, so iron has 4 unpaired electrons in its $3d$ subshell — the basis of iron's strong paramagnetism (and, in bulk metallic form, its ferromagnetism). As a limiting check on Hund's rule itself: if the $3d$ subshell held exactly 5 electrons ( $3d^5$ , as in Mn²⁺), every orbital would hold exactly one electron with no pairing at all, giving the maximum possible 5 unpaired electrons for a $d$ subshell — the theoretical maximum consistent with $2(2\ell+1)=10$ capacity and Hund's rule.

Limiting cases & common traps

Situation	What happens	Why
$n=1$ shell only	Capacity is exactly $2n^2 = 2$ ; only $1s^2$ possible	No $\ell>0$ subshells exist when $n=1$
Subshell exactly half-filled (e.g., $3d^5$ , $4f^7$ )	Extra stability; sometimes causes exceptions to strict Aufbau order	Maximized exchange energy from parallel spins
Subshell exactly filled (e.g., $3d^{10}$ )	Extra stability; also causes exceptions (Cu, Ag, Au)	Symmetric, fully paired charge distribution lowers repulsion
Writing $3d$ before $4s$ in condensed notation	Common convention: list by increasing $n$ even though $4s$ filled first	Notation order ≠ filling order; both are chemically valid as long as electron count is right
Ion formation from transition metals	Remove $ns$ electrons before $(n-1)d$ electrons	Once occupied, $3d$ orbitals drop below $4s$ in energy for the ion

Another frequent error: when writing the configuration of a transition-metal cation, students often remove electrons from the $3d$ subshell first because it was filled last. This is wrong. For a neutral atom, $4s$ is lower in energy than $3d$ and fills first, but once electrons occupy $3d$ , that subshell drops below $4s$ in energy for the resulting ion, so $4s$ electrons are always removed first when forming cations. For example, $\text{Fe}^{2+}$ has configuration $[\text{Ar}]3d^6$ , not $[\text{Ar}]4s^2 3d^4$ .

The chemist behind it

The rules bundled together here were not the work of one person but of several physicists refining early quantum theory in the 1920s, extending the original hydrogen-atom framework toward Electron Shells and Orbitals for multi-electron atoms. Niels Bohr had proposed shell-filling ideas to explain the periodic table as early as 1922, but lacked the full quantum-mechanical justification. Wolfgang Pauli supplied the missing piece in 1925 with the exclusion principle, initially stated as a rule about quantum numbers before its deeper connection to electron spin and antisymmetric wavefunctions was understood. Friedrich Hund derived his empirical rule for electron spin arrangements in atomic ground states around 1927, based on spectroscopic evidence for atomic energy levels.

Origins · 1 minWolfgang Pauli and how this was discovered

Wolfgang Pauli

1900-1958 · Formulated the exclusion principle in 1925 while explaining anomalies in atomic spectra and the Zeeman effect; later contributed to the theory of neutrinos and quantum field theory, receiving the 1945 Nobel Prize in Physics

Ask the tutor: Why does removing an electron from a neutral iron atom change whether 4s or 3d is lower in energy?

Why does 4s fill before 3d if 3d belongs to a lower shell number?

Although $n=3$ is numerically lower than $n=4$ , the actual energy ordering of subshells in multi-electron atoms depends on both $n$ and $\ell$ , plus shielding effects. The $4s$ orbital has a higher probability density close to the nucleus (a "penetration" effect) that lowers its effective energy below $3d$ for atoms up through roughly calcium. This is why the standard Aufbau diagonal places $4s$ before $3d$ , even though ionization later removes $4s$ electrons first once $3d$ is occupied.

Are the Cr and Cu exceptions the only irregularities in the periodic table?

No. Similar half-filled/fully-filled stabilization effects appear again among the second- and third-row transition metals and especially the lanthanides and actinides, where $4f$ / $5f$ and $5d$ / $6d$ subshells lie close in energy. Common examples include molybdenum ([Kr]5s¹4d⁵) and silver ([Kr]5s¹4d¹⁰), which mirror the Cr/Cu pattern one row down. These irregularities become more numerous, and their explanation more subtle, once relativistic effects matter for heavy nuclei — a topic that belongs to more advanced treatments of Periodic Table and Electron Configurations.

Does Hund's rule apply to all atoms, or only isolated ones?

Hund's rule as presented here describes isolated gas-phase atoms or ions in their ground electronic state, which is exactly the situation used to build configurations from first principles. In molecules or crystals, orbital energies shift due to bonding and the electrostatic environment, so simple degenerate-orbital filling no longer directly applies. The atomic version remains the essential starting point before moving to bonding theories.

How do I quickly find a configuration without writing all subshells from scratch?

The periodic table is organized so that each block (s, p, d, f) corresponds to the subshell being filled across that row. Reading left to right through periods, tracking which block you pass through, reproduces the Aufbau order automatically — this shortcut is developed fully alongside Periodic Trends, but the underlying rules are exactly the three covered here.