Hund's Rules

Give a chemist an electron configuration — carbon is $1s^2 2s^2 2p^2$ — and they can tell you almost everything about the atom except one thing the configuration alone hides: how the open-shell electrons arrange their spins and orbital motions. A $2p^2$ configuration is not one state. It is fifteen microstates that split, under electron–electron repulsion, into three spectroscopic terms — $^1S$ , $^3P$ , $^1D$ — spread over a couple of electron-volts. Which of these is the ground state? Hund's three rules answer that question with three lines of arithmetic, and they get it right for essentially every atom in the light half of the periodic table. The rules are a hierarchy. First, maximize the total spin $S$ . Then, at that spin, maximize the total orbital angular momentum $L$ . Finally, couple them: take the smallest $J=|L-S|$ if the subshell is less than half-filled, the largest $J=L+S$ if it is more than half-filled. The first rule is the deep one — it is the Exchange Forces and Exchange Energy of the previous article wearing a periodic-table hat. Aligning spins forces the spatial wavefunction to be antisymmetric, which digs a Fermi hole between the electrons and lowers their Coulomb repulsion by the exchange integral $K$ . The rule that "electrons want parallel spins" is really "electrons want to avoid each other," and spin is the bookkeeping. This article owns the rules as a working machine. We derive carbon's ground term from scratch — enumerate the boxes, apply the three rules, read off $^3P_0$ — and then run the machine across nitrogen, oxygen, titanium, and $\mathrm{Mn}^{2+}$ , each checked against the measured ground level. The punchline is the third rule's sign flip: carbon ( $2p^2$ ) and oxygen ( $2p^4$ ) share the same term $^3P$ , yet carbon's ground level is $^3P_0$ and oxygen's is $^3P_2$ — an inverted multiplet, the fingerprint of a subshell filled past its half-way mark.

Key takeaways

Hund's three rules pick the ground-state term symbol $^{2S+1}L_J$ out of the many terms a given open-shell configuration allows: maximize $S$ ; then maximize $L$ ; then take $J=|L-S|$ below half-filling, $J=L+S$ above.
Rule 1 is exchange physics: a high-spin (symmetric spin) state forces an antisymmetric spatial wavefunction — a Fermi hole — which lowers Coulomb repulsion by the exchange integral $K$ , so the maximum-spin term sits below by $2K$ per aligned pair.
Rule 2 is the orbital echo: at fixed spin, large $|M_L|$ (electrons circulating the same way) keeps them apart better, lowering repulsion again.
Rule 3 is the small relativistic afterthought: spin–orbit coupling $\zeta\,\mathbf L\cdot\mathbf S$ has $\zeta>0$ for a less-than-half-filled subshell (minimum $J$ ) and effectively $\zeta<0$ above half-filling (maximum $J$ ).
The owned result is the machine: enumerate microstates, select the term, and reproduce carbon $^3P_0$ , nitrogen $^4S_{3/2}$ , oxygen $^3P_2$ , titanium $^3F_2$ , $\mathrm{Mn}^{2+}\,^6S_{5/2}$ — all matching NIST.
The punchline is the half-filled J-flip: carbon and oxygen are both $^3P$ , but $2p^2$ gives $^3P_0$ and $2p^4$ gives $^3P_2$ — the inverted multiplet that marks a subshell filled past half.
Key idea: the ground term is fixed by exchange energy — parallel spins buy an antisymmetric, mutually-avoiding spatial state — so maximize $S$ , then $L$ , and let spin–orbit coupling with its half-filling sign flip set $J$ .

The core insight

Apply three rules in order to a partly-filled subshell: (1) maximize the total spin $S$ ; (2) at that $S$ , maximize the total orbital angular momentum $L$ ; (3) set $J=|L-S|$ for a less-than-half-filled subshell and $J=L+S$ for a more-than-half-filled one. The first rule is exchange energy in disguise — aligning spins forces spatial antisymmetry (a Fermi hole) that lowers Coulomb repulsion — which is why carbon's ground term is $^3P_0$ and oxygen's, one J-flip later, is $^3P_2$ .

Where this lives today

Hund's rules are a standard tool in atomic physics, magnetochemistry, and materials science. They fix the ground-state magnetic moment of every transition-metal and lanthanide ion — the effective moment $\mu_{\text{eff}}=g_J\sqrt{J(J+1)}\,\mu_B$ that a chemist measures depends on the $J$ that Rule 3 selects, and the famous agreement between Hund-rule predictions and measured lanthanide moments (with the notable exceptions of Eu and Sm, where low-lying excited multiplets intrude) is a textbook triumph. In condensed matter, "Hund's coupling" — the intra-atomic exchange energy that Rule 1 harvests — is a central parameter in the physics of correlated electron systems, iron-based superconductors, and the Mott transition, where whether electrons on a site align their spins controls whether the material is a metal or an insulator. In cold-atom and quantum-information platforms, the half-filled $^6S_{5/2}$ ground state of ions like $\mathrm{Mn}^{2+}$ and $\mathrm{Fe}^{3+}$ makes them clean, orbitally-quenched spin- $\tfrac52$ qubits. The rules also underpin the routine identification of astrophysical spectral lines, where knowing the ground term and the Landé interval rule lets one predict multiplet spacings before the atom is even in the lab.

The physical picture

An electron configuration such as carbon's $2p^2$ names which orbitals are occupied but not how the electrons distribute themselves among the six spin-orbitals of the $2p$ shell. There are $\binom{6}{2}=15$ ways to place two electrons in those six boxes, and electron–electron repulsion splits those fifteen microstates into distinct energy levels. Grouped by total spin $S$ and total orbital angular momentum $L$ , they form the terms $^1S$ (one microstate), $^3P$ (nine), and $^1D$ (five) — and $1+9+5=15$ accounts for every microstate. Hund's rules are the recipe for finding the lowest of these terms without solving the many-electron Schrödinger equation. The engine underneath is exchange. The direct Coulomb interaction does not depend explicitly on spin, but antisymmetrization produces an exchange contribution that depends on the relative spin state, and it is this exchange effect that favors the high-spin ground term. The resolution comes from exchange and antisymmetrization: a spin singlet has a symmetric spatial part, allowing greater spatial overlap, while a spin triplet has an antisymmetric spatial part that suppresses the probability of finding the two electrons at the same position — the origin of the Fermi hole. The high-spin term therefore sits lower by twice the exchange integral, $2K$ , per aligned pair. Spin is free bookkeeping for a spatial correlation that costs real Coulomb energy.

Figure

Carbon versus oxygen: the same ³P term, but the third rule flips the ground level from J=0 to J=2

The figure states the thesis of the whole article. Carbon and oxygen have the same $^3P$ term because a $2p^4$ configuration is equivalent to a $2p^2$ configuration in terms of its hole structure, and the first two Hund's rules give the same $S$ and $L$ values. But the third rule is not blind to which side of half-filling the configuration lies on. For a less-than-half-filled subshell, the lowest level has the smallest allowed $J$ , whereas for a more-than-half-filled subshell, the ordering is reversed and the largest allowed $J$ lies lowest. That single sign is what separates an oxygen atom from a carbon atom at the level of angular momentum.

The three rules as a hierarchy

The rules are applied strictly in order, each acting only on the states that survive the previous one. Rule 1 fixes $S$ ; Rule 2 fixes $L$ among the maximum- $S$ arrangements; Rule 3 fixes $J$ among the $L,S$ multiplet. They are ordered by energy scale: the exchange splitting behind Rule 1 is the largest (electron-volts), the orbital correlation of Rule 2 is smaller, and the spin–orbit coupling behind Rule 3 is the smallest — often a fraction of an electron-volt for light atoms, which is why it is a fine-structure effect.

\text{Rule 1: } S=S_{\max}, \qquad \text{Rule 2: } L=L_{\max}\big|_{S=S_{\max}}, \qquad \text{Rule 3: } J=\begin{cases} |L-S| & n<2\ell+1\ (\text{below half}) \\ L+S & n>2\ell+1\ (\text{above half}) \end{cases} \qquad (1)

where: $S$ — total spin; $L$ — total orbital angular momentum; $J$ — total angular momentum; $\ell$ — the subshell's orbital quantum number; $n$ — the number of electrons in it; $2\ell+1$ — the electron count at exactly half-filling. The physical read: the two symmetry rules ( $S$ , then $L$ ) minimize electrostatic energy, and the third — a genuinely relativistic correction — orders the fine structure through the sign of the spin–orbit constant $\zeta$ . The mechanism for Rule 1 is not new physics; it is the exchange energy that Exchange Forces and Exchange Energy derives. That article establishes that two electrons in orbitals $a$ and $b$ have energies $E_\pm = J_{ab}\pm K_{ab}$ , with the spatially symmetric (spin-singlet) combination taking $+K$ and the spatially antisymmetric (spin-triplet) combination taking $-K$ , so the triplet lies below the singlet by $2K$ whenever the exchange integral $K>0$ . Hund's first rule is that same statement applied to every pair of parallel-spin electrons in an open subshell: aligning all the spins maximizes the number of antisymmetric spatial pairings and harvests the most exchange energy.

Derivation ACarbon's ground term from the three rules

Derive the ground-state term symbol of carbon ( $1s^2 2s^2 2p^2$ ) starting from nothing but the six $2p$ spin-orbitals and the Pauli principle. The goal is to show that the three rules are a concrete, countable procedure: place the electrons to maximize spin, then orbital angular momentum, then read off the total $J$ from the filling. The payoff is a number — the ground level $^3P_0$ — that we can and do check against the measured carbon spectrum, and a template that generalizes to every open-shell atom.

Step 1. List the boxes. The $2p$ subshell has $\ell=1$ , so its orbital projections are $m_\ell=+1,0,-1$ , each holding two spins $m_s=\pm\tfrac12$ : six spin-orbitals, and $\binom{6}{2}=15$ two-electron microstates.

Step 2. Rule 1 — maximize $S$ . Put both electrons spin-up. Pauli forbids two spin-up electrons in the same $m_\ell$ , but they can occupy different boxes, e.g. $m_\ell=+1$ and $m_\ell=0$ . Both spins parallel gives $M_S=+1$ , hence $S=1$ : a spin triplet, multiplicity $2S+1=3$ .

Step 3. Rule 2 — maximize $L$ at $S=1$ . With both spins parallel, the electrons must occupy different $m_\ell$ states, so the largest possible orbital projection is $M_L=(+1)+(0)=1$ (they cannot both sit in $m_\ell=+1$ ). The maximum $M_L$ available in the maximum-spin sector is $1$ , so $L=1$ — a $P$ term. Rules 1 and 2 together therefore give $^3P$ .

Step 4. Rule 3 — pick $J$ . The $2p^2$ subshell contains 2 of its 6 possible electrons, so it is less than half-filled. Hund's third rule therefore selects the minimum allowed value, $J=|L-S|=|1-1|=0$ . The ground term is consequently $^3P_0$ , matching the measured carbon ground level.

Step 5. Boundary check — the multiplets and the count. The full $^3P$ term contains $J=0,1,2$ and therefore has $1+3+5=9$ states; the $^1S$ and $^1D$ terms contribute $1$ and $5$ states, respectively. Thus the total is $9+5+1=15$ , exactly matching the $\binom{6}{2}=15$ microstates. Nothing is lost or double-counted. Microstate enumeration selects $S=1,L=1,J=0$ for $2p^2$ ; term dimensions $1+9+5=15$ ; ground level $^3P_0$ matches NIST.

Why parallel spins win: the exchange origin of Rule 1

Rule 1 deserves a derivation of its own, because "maximize spin" sounds like a magnetic statement about a spin-independent Hamiltonian, and the resolution is subtle. The Coulomb interaction $V_{ee}=e^2/4\pi\varepsilon_0 r_{12}$ contains no spin operators. What links spin to energy is the antisymmetry requirement on the total electron wavefunction, which forces a definite spatial symmetry once the spin symmetry is chosen.

Go deeper · Derivation B · 2 minThe exchange mechanism behind maximum spin

Derivation BThe exchange mechanism behind maximum spin

Show that for two electrons the maximum-spin (triplet) state lies below the minimum-spin (singlet) state by $2K$ , where $K$ is the exchange integral, and that this is why Rule 1 favours parallel spins. The goal is to trace the energy ordering back to spatial symmetry: a symmetric spin state demands an antisymmetric spatial state, whose Fermi hole cuts the Coulomb repulsion. The payoff is the sign — triplet below singlet — recognized as Exchange Forces and Exchange Energy applied to atomic shells.

Step 1. Antisymmetry couples spin to space. The total two-electron state is a product of a spatial part and a spin part, and must be antisymmetric under exchange. Thus, a symmetric spin state (the triplet, $S=1$ ) requires an antisymmetric spatial state $\psi_-$ , while an antisymmetric spin state (the singlet, $S=0$ ) requires a symmetric spatial state $\psi_+$ .

Step 2. The Coulomb expectation splits into direct and exchange parts. For orbitals $a,b$ , evaluating $\langle\psi_\pm|V_{ee}|\psi_\pm\rangle$ gives $J_{ab}\pm K_{ab}$ , with the direct integral $J_{ab}=\iint|\psi_a(1)|^2|\psi_b(2)|^2 V_{ee}\,d^3r_1 d^3r_2$ (the classical charge-cloud repulsion) and the exchange integral $K_{ab}=\iint\psi_a^*(1)\psi_b^*(2)V_{ee}\,\psi_b(1)\psi_a(2)\,d^3r_1 d^3r_2$ (a purely quantum cross term). This structure is owned and derived in Exchange Forces and Exchange Energy; here we use its result.

Step 3. Read off the energies. The singlet (symmetric space) takes $E_{\text{singlet}}=J_{ab}+K_{ab}$ ; the triplet (antisymmetric space) takes $E_{\text{triplet}}=J_{ab}-K_{ab}$ . The gap is $E_{\text{triplet}}-E_{\text{singlet}}=-2K_{ab}$ .

Step 4. The Coulomb kernel makes $K>0$ . For the repulsive Coulomb interaction, the exchange integral satisfies $K_{ab}>0$ for the usual real orbital functions. Hence $-2K_{ab}<0$ , so the triplet, the maximum-spin state, lies lower in energy. That is Hund's first rule for a pair, and summing $-2K$ over every parallel-spin pair in a subshell is why maximizing total spin maximizes the energy saving.

Step 5. The physical read. The antisymmetric spatial wavefunction $\psi_-$ vanishes when $\mathbf r_1=\mathbf r_2$ , producing an exchange or Fermi hole: parallel-spin electrons are less likely to be found close together, reducing their average Coulomb repulsion. Spin never enters the Hamiltonian; it enters the bookkeeping that decides which spatial state the electrons must occupy. A contact-interaction toy model gives $J>0$ , $K>0$ , and $E_{\text{triplet}}-E_{\text{singlet}}=-2K$ symbolically; the triplet lies below.

Figure

The p² configuration splits into three terms; Hund rule 1 puts the highest-spin ³P lowest

Rule 2 is the same idea one rung down. Among the maximum-spin arrangements, states with large $|M_L|$ correspond to electrons orbiting in the same rotational sense; like two runners on the same lane going the same way, they stay farther apart than counter-rotating electrons and pay slightly less repulsion. Rule 3 is different in kind: it is the spin–orbit interaction $\zeta\,\mathbf L\cdot\mathbf S$ , a relativistic correction whose coupling $\zeta$ is positive for a subshell below half-filling and effectively negative above it (the hole picture), which is why the fine-structure multiplet is normal (lowest $J$ on the bottom) below half-filling and inverted above.

Limiting cases

Case	Result	Meaning
Closed subshell ( $n=2(2\ell+1)$ )	$S=L=0$ , $^1S_0$	all boxes doubly filled; noble gases, alkaline-earth cores — no open-shell term
Single electron/hole ( $n=1$ or $2(2\ell+1)-1$ )	$^2L_{J}$ with $L=\ell$	one particle sets $S=\tfrac12$ , $L=\ell$ ; e.g. alkali $^2S_{1/2}$ , halogen $^2P_{3/2}$
Exactly half-filled ( $n=2\ell+1$ )	$^{2S+1}S_{S}$ , an $S$ -term	one same-spin electron per box forces $\sum m_\ell=0$ , so $L=0$ ; N is $^4S_{3/2}$ , Mn $^{2+}$ is $^6S_{5/2}$
Heavy atom, $\zeta \sim$ electrostatic gap	Rule 3 breaks (LS→jj)	spin–orbit rivals term splitting; $L,S$ cease to be good quantum numbers; use $jj$ coupling
Below vs above half-filling	normal vs inverted multiplet	$\zeta>0$ (min $J$ ) turns to $\zeta<0$ (max $J$ ); the C/O $^3P$ sign flip

Common trapTwo traps: Rule 3's half-filling sign, and applying the rules to excited terms

First, Rule 3 is the one people get wrong, because it is the only rule whose answer depends on how full the subshell is, not just on $S$ and $L$ . Below half-filling take $J=|L-S|$ ; above half-filling take $J=L+S$ ; exactly at half-filling $L=0$ so $J=S$ and the question is moot. Skipping the filling check gives carbon and oxygen the same ground level, which is wrong — carbon is $^3P_0$ and oxygen is $^3P_2$ . Second, Hund's rules order only the ground term. They do not reliably rank the excited terms: the second-lowest term is frequently not the one with the next-largest $S$ (for $p^2$ , the order is $^3P<{}^1D<{}^1S$ , and $^1D$ has lower spin than the ground $^3P$ but sits below $^1S$ ). Use the rules to find the bottom of the ladder, not to sort the whole ladder.

Worked examples

Example 1 — carbon's ground term, box by box. Find the ground-state term symbol of neutral carbon, $1s^2 2s^2 2p^2$ , and confirm the microstate bookkeeping.

Step 1. Enumerate. The open subshell is $2p^2$ : two electrons among six spin-orbitals, $\binom{6}{2}=15$ microstates.

Step 2. Apply the rules. Rule 1: both electrons spin-up in different $m_\ell$ boxes $\Rightarrow S=1$ . Rule 2: the largest orbital projection with parallel spins is $M_L=(+1)+(0)=1\Rightarrow L=1$ , a $P$ term, so $^3P$ . Rule 3: $2p^2$ is less than half-filled ( $2<3$ ), take $J=|L-S|=0$ . Ground term: $\mathbf{^3P_0}$ .

Step 3. Boundary check. The three terms of $p^2$ are $^1S,{}^3P,{}^1D$ with dimensions $1,9,5$ — total $15$ , matching the microstate count exactly. NIST lists carbon's ground level as $^3P_0$ : the rules are right. This is the boundary that closes the calculation — every microstate is accounted for, and the predicted ground level matches the measured one.

Example 2 — the nitrogen/oxygen/manganese trio and the J-flip. Predict the ground terms of nitrogen ( $2p^3$ ), oxygen ( $2p^4$ ), and $\mathrm{Mn}^{2+}$ ( $3d^5$ ), and highlight the sign flip.

Step 1. Nitrogen $2p^3$ (half-filled). Three parallel spins, one per box: $S=\tfrac32$ ( $2S+1=4$ ). All boxes singly filled forces $M_L=(-1)+0+(+1)=0\Rightarrow L=0$ , an $S$ -term. With $L=0$ , $J=S=\tfrac32$ uniquely. Ground term: $\mathbf{^4S_{3/2}}$ .

Step 2. Oxygen $2p^4$ (more than half-filled). Four electrons in six spin-orbitals correspond to two holes, which behave like a $p^2$ configuration: $S=1$ and $L=1$ , so $^3P$ . Because the shell is more than half-filled, Rule 3 gives $J=L+S=2$ . Ground term: $\mathbf{^3P_2}$ — the same term as carbon, but the opposite end of the multiplet.

Step 3. $\mathrm{Mn}^{2+}$ $3d^5$ (half-filled $d$ ). Five parallel spins fill the five $d$ -orbitals ( $m_\ell=-2,\dots,+2$ ): $S=\tfrac52$ ( $2S+1=6$ ), $\sum m_\ell=0\Rightarrow L=0$ , an $S$ -term. Ground term: $\mathbf{^6S_{5/2}}$ — the spherically symmetric, maximally-magnetic half-filled shell that makes $\mathrm{Mn}^{2+}$ and $\mathrm{Fe}^{3+}$ good $S=\tfrac52$ spin systems. All four ground levels — $^4S_{3/2}$ , $^3P_2$ , $^6S_{5/2}$ , and carbon's $^3P_0$ — match the NIST Atomic Spectra Database.

Origins · 1 minterm symbol and how this was discovered

term symbol

introduced 1920s–1930s · atomic spectroscopy

The term symbol $^{2S+1}L_J$ is the compact spectroscopic label that Hund's rules deliver: the left superscript $2S+1$ is the spin multiplicity (singlet, doublet, triplet, …), the letter $S,P,D,F,\dots$ encodes the total orbital angular momentum $L=0,1,2,3,\dots$ , and the right subscript is the total angular momentum $J$ . The notation grew out of the analysis of atomic spectra in the 1920s: the alkali doublets, the multiplet structure of transition metals, and the systematics that Russell and Saunders codified into $LS$ coupling in 1925. Friedrich Hund, working on the band and line spectra of atoms in 1925–1927, distilled the empirical regularities of which term lies lowest into the three rules that carry his name — first the spin rule, then the orbital rule, then the fine-structure rule — before their exchange-energy explanation was fully understood. The term symbol remains the standard shorthand of atomic physics: every ground state in the NIST tables is quoted as one.

History

1925

— Russell and Saunders formalize

LS

(Russell–Saunders) coupling and the term-symbol scheme

1925

— Hund begins systematizing atomic multiplet spectra; the spin and orbital rules emerge

1926

— Heisenberg identifies the exchange interaction in the helium spectrum, the mechanism behind Rule 1

1927

— Hund states the rules in essentially their modern form from spectroscopic regularities

1929

— Slater's determinant formalism makes the exchange integral

K

explicit and computable

1963

— the rules are understood as consequences of exchange and spin–orbit coupling in modern texts

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §5.2.2 (Hund's rules) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 8 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 13 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, complement on multi-electron atoms · Bransden & Joachain, Physics of Atoms and Molecules, 2nd ed., Ch. 6. The five ground terms C $^3P_0$ , N $^4S_{3/2}$ , O $^3P_2$ , Ti $^3F_2$ , $\mathrm{Mn}^{2+}$ $^6S_{5/2}$ are confirmed against the NIST Atomic Spectra Database; the $p^2$ term dimensions satisfy $1+9+5=15$ ; the exchange result $E_{\text{singlet}}-E_{\text{triplet}}=+2K$ with $K>0$ is recalled from Exchange Forces and Exchange Energy; and the Landé interval spacing is linear in $J$ . The exchange-energy structure $E_\pm=J\pm K$ and the integral $K$ are treated as pointers to Exchange Forces and Exchange Energy, applied also in The Helium Atom and Singlet and Triplet States in Two-Electron Systems; this article contributes the term-selection machine and its NIST-matched outputs.

Builds on: Pauli Exclusion Principle · Aufbau Principle and Electron Configurations · Exchange Forces and Exchange Energy Leads to: Singlet and Triplet States in Two-Electron Systems · The Helium Atom · The Periodic Table from Quantum Mechanics

Why does maximizing total spin lower the energy if the Coulomb force does not depend on spin?

Because antisymmetrization ties spin symmetry to spatial symmetry. A high-spin (symmetric spin) state must pair with an antisymmetric spatial wavefunction, which vanishes when two electrons coincide — a Fermi hole — so the electrons stay farther apart and pay less Coulomb repulsion. The energy saving is the exchange integral: the maximum-spin term sits below by $2K$ per aligned pair, where $K>0$ . Spin is bookkeeping for a spatial correlation, and the correlation carries the real energy. The mechanism is owned by Exchange Forces and Exchange Energy.

What is the difference between an electron configuration and a term symbol?

A configuration, like carbon's $1s^2 2s^2 2p^2$ , names which orbitals are occupied but says nothing about how the open-shell electrons couple their spins and orbital motions. A term symbol $^{2S+1}L_J$ names the specific coupled state: its total spin $S$ , total orbital angular momentum $L$ , and total angular momentum $J$ . One configuration generally splits into several terms — $2p^2$ gives $^1S$ , $^3P$ , $^1D$ — separated in energy by electron–electron repulsion. Hund's rules are precisely the recipe for choosing the lowest term of a given configuration.

Why is the ground term of a half-filled subshell always an S term?

Because filling every box with one same-spin electron (which maximizes $S$ ) forces the orbital projections to cancel. For $p^3$ the boxes are $m_\ell=-1,0,+1$ , and $\sum m_\ell=(-1)+0+(+1)=0$ ; no larger $M_L$ is available in the maximum-spin sector, so $L=0$ — an $S$ -term. The same holds for $d^5$ ( $\sum m_\ell=-2-1+0+1+2=0$ ) and $f^7$ . This is why nitrogen is $^4S_{3/2}$ , $\mathrm{Mn}^{2+}$ is $^6S_{5/2}$ , and $\mathrm{Gd}^{3+}$ is $^8S_{7/2}$ — spherically symmetric, maximally magnetic, orbitally quenched.

When does Hund's third rule fail?

When spin–orbit coupling grows comparable to or larger than the electrostatic term splitting — which happens for heavy atoms. Then $LS$ (Russell–Saunders) coupling breaks down toward $jj$ coupling: $L$ and $S$ stop being good quantum numbers, the $\mathbf L\cdot\mathbf S$ reasoning behind Rule 3 loses its footing, and you must couple individual electron $j$ 's first. Rules 1 and 2 also weaken, but Rule 3 is the first casualty because it is the spin–orbit interaction. For light atoms (the first few rows) $LS$ coupling and all three rules work well; for the heaviest elements they are only a starting approximation.

Are Hund's rules exact, or just a good approximation?

They are empirical rules with a physical basis, not theorems. Rule 1 (maximum spin) is the most robust — it follows directly from exchange energy and rarely fails for a ground state. Rules 2 and 3 are progressively less reliable, especially where several terms lie close together or where configuration interaction (mixing of different configurations) matters. The lanthanides are a famous partial exception: europium and samarium have low-lying excited $J$ -multiplets that spoil the naive Hund-rule magnetic moment. Treat the rules as an excellent first pass for the ground state, to be checked against measured spectra — which is exactly what the NIST database lets you do.

How does Rule 1 connect to magnetism in solids?

The intra-atomic exchange energy that Rule 1 harvests — call it "Hund's coupling" — survives when atoms assemble into a solid. On a lattice site with several electrons, Hund's coupling still favours aligning their spins, and this local moment is the seed of magnetism. In correlated-electron materials (transition-metal oxides, iron-based superconductors) the competition between Hund's coupling, the on-site Coulomb repulsion, and the kinetic energy of hopping electrons controls whether a material is a magnetic insulator or a metal. The atomic rule that "spins want to align" scales up to the collective phenomena of ferromagnetism and Mott physics.

Why do carbon and oxygen share a term but not a ground level?

A $p^4$ subshell is a $p^2$ subshell of holes, and holes obey the same first two rules as electrons, so both give $S=1$ , $L=1$ , the $^3P$ term. The difference is entirely in Rule 3, which reads the filling. Below half-filling ( $p^2$ , carbon) the spin–orbit constant is positive and the normal multiplet puts $J=|L-S|=0$ lowest: $^3P_0$ . Above half-filling ( $p^4$ , oxygen) the hole picture flips the effective sign, inverting the multiplet so $J=L+S=2$ is lowest: $^3P_2$ . Same term, opposite ends of the fine-structure ladder.