Hund's Rules

Carbon's ground configuration $1s^2\,2s^2\,2p^2$ tells you which orbitals are occupied — but not how the two $2p$ electrons' spins and orbits combine. That freedom generates several terms, $^1S_0$ , $^3P_0$ , $^3P_1$ , $^3P_2$ , $^1D_2$ , and one of them must be the ground state. Hund's three rules, distilled by Friedrich Hund in 1925–1927 from mountains of spectra, tell you which — and they get it right for essentially every element in the periodic table. We take the why from quantum mechanics and keep the how here. The deepest of the rules, "maximise the spin," is a consequence of the exchange interaction: parallel spins force an antisymmetric spatial wavefunction, electrons stay apart, and Coulomb repulsion drops by an exchange integral $K_{12}>0$ . That mechanism — the exchange-energy lowering of parallel-spin states — is derived in aq-qm-hunds-rules, and we recall it by name rather than re-derive it. What this article owns is the spectroscopist's working recipe: fill the boxes, read off $S$ , $L$ , and $J$ , and predict real ground terms and real magnetic moments. The payoff is enormous reach from three lines of arithmetic. The same rules that place carbon at $^3P_0$ predict the $^4S_{3/2}$ ground state of nitrogen, the inverted $^3P_2$ of oxygen, and the ground terms of every lanthanide ion — and from those you get the effective magnetic moments that govern paramagnetism, the Zeeman effect, and the laser levels of Nd:YAG and erbium fibre amplifiers. Dysprosium's $^6H_{15/2}$ ground state, with $g_J=4/3$ and $\mu_\text{eff}=10.65\,\mu_B$ , is a Hund's-rules prediction you can check against a magnetometer.

Key takeaways

Hund's rules give the ground term and level $^{2S+1}L_J$ of an atom's ground configuration, in LS (Russell–Saunders) coupling.
Rule 1: maximise total spin $S$ (parallel spins lower Coulomb energy through exchange). Rule 2: for that $S$ , maximise $L$ . Rule 3: $J=|L-S|$ if the open subshell is $\le$ half full, $J=L+S$ if $>$ half full.
Only open-subshell electrons matter (closed shells give $L=S=0$ ); past half filling, count holes.
Real predictions: C $2p^2\!\to{}^3P_0$ , N $2p^3\!\to{}^4S_{3/2}$ , O $2p^4\!\to{}^3P_2$ (inverted), Nd³⁺ $4f^3\!\to{}^4I_{9/2}$ , Dy³⁺ $4f^9\!\to{}^6H_{15/2}$ .
The ground level fixes the magnetic moment $\mu_\text{eff}=g_J\sqrt{J(J+1)}\,\mu_B$ ; Dy³⁺ gives $g_J=4/3$ , $\mu_\text{eff}=10.65\,\mu_B$ , matching experiment.
Rules 1–2 are electrostatic (exchange/correlation); rule 3 is magnetic (spin–orbit), and only its sign flips at half filling.
Key idea: pack the open subshell for maximum spin (rule 1), then maximum orbital momentum (rule 2); the resulting term is the ground term, and its lowest level has $J=|L-S|$ below half filling and $J=L+S$ above — the whole recipe is three lines of arithmetic backed by exchange energy and spin–orbit coupling.

The core insight

Fill the open subshell for the maximum total spin the Pauli principle allows (rule 1), then the maximum orbital angular momentum for that spin (rule 2); the resulting term is the ground term, and its ground level has $J=|L-S|$ if the subshell is at most half full and $J=L+S$ otherwise (rule 3).

Where this lives today

Hund's rules are quietly everywhere in modern optics and magnetism. Rare-earth lasers and amplifiers rest on them: Nd³⁺ ( $^4I_{9/2}$ ) supplies the 1064 nm line of Nd:YAG, and Er³⁺ ( $^4I_{15/2}$ ) the 1550 nm gain band of the erbium-doped fibre amplifiers that carry the internet's long-haul traffic — both ground manifolds set by the rules applied to the $4f$ shell. In condensed-matter and molecular magnetism, the same rules give the effective moments used to interpret magnetic susceptibility, and single-molecule magnets are engineered around Dy³⁺ and Tb³⁺ precisely for their huge Hund's-rules moments and strong anisotropy. Even quantum-materials research invokes "Hund's coupling" as the energy scale controlling correlated metals. When a spectroscopist or a materials chemist needs an atom's ground state in seconds, the box diagram and three rules are still the tool of choice.

The physical picture

Rule 1 is exchange in disguise. Two electrons with parallel spins are required by antisymmetry to occupy an antisymmetric spatial state, which has a node where the two electrons coincide — they are kept apart, and their mutual Coulomb repulsion drops. So the atom lowers its energy by making as many spins parallel as the Pauli principle allows: singly occupy each $m_\ell$ orbital with spin up before pairing any of them. The quantitative version — the two-electron energy splits as $J_{12}\pm K_{12}$ with the exchange integral $K_{12}>0$ favouring the high-spin (triplet) state — is owned by aq-qm-hunds-rules, where "exchange energy" and the "exchange integral" are worked out in full; here we simply use the result. Rule 2 is orbital correlation. Once the spins are fixed, electrons circulating in the same rotational sense — large $|M_L|$ , hence large $L$ — meet each other less often than counter-rotating ones, again shaving off Coulomb energy. The mental image is electrons that "orbit together and stay out of each other's way." Rule 3 is the sign of $\mathbf L\!\cdot\!\mathbf S$ . The fine-structure energy is $\propto\zeta\,\mathbf L\!\cdot\!\mathbf S$ . Below half filling the effective coupling $\zeta>0$ , so the smallest $J$ sits lowest (a normal multiplet); above half filling the added electrons act like holes with opposite effective coupling, $\zeta<0$ , so the largest $J$ sits lowest (an inverted multiplet). Carbon ( $2p^2$ , below half) lands on $^3P_0$ ; oxygen ( $2p^4$ , above half) on $^3P_2$ — same term, opposite end. The whole procedure is a box-filling game. Draw the $2\ell+1$ orbital boxes; drop electrons in spin-up from the highest $m_\ell$ downward (this maximises $S$ and then $M_L$ together); read $S=\tfrac12(N_\uparrow-N_\downarrow)$ and $L=|M_L^{\max}|$ ; apply the half-filling rule for $J$ .

Figure

Box-filling recipe: p² → ³P₀, half-filled p³ → ⁴S₃⁄₂, inverted p⁴ → ³P₂

Figure

Lanthanide ground-state moments μ_eff = g_J√(J(J+1)) — the Hund double hump

The core equations

Everything hangs on the term symbol, which packages the three good quantum numbers of LS coupling into one label. Rules 1–3 are just prescriptions for choosing $S$ , then $L$ , then $J$ .

{}^{2S+1}L_J \tag{1}

where: $S$ — total spin, $\mathbf S=\sum_i\mathbf s_i$ ; $L$ — total orbital angular momentum, $\mathbf L=\sum_i\boldsymbol\ell_i$ , written $S,P,D,F,G,\dots$ for $0,1,2,3,4$ ; $J$ — total angular momentum, $\mathbf J=\mathbf L+\mathbf S$ ; $2S+1$ — the spin multiplicity, the number of $J$ -levels in the term when $L\ge S$ . Rule 1 maximises the spin, and the reason is exchange: the electrostatic energy of a pair splits into a direct term plus or minus an exchange term, with the sign favouring parallel spins.

E = J_{12} \mp K_{12},\qquad K_{12}=\iint \psi_a^*(1)\psi_b^*(2)\,\frac{e^2}{4\pi\varepsilon_0 r_{12}}\,\psi_b(1)\psi_a(2)\,\mathrm d^3r_1\,\mathrm d^3r_2 > 0 \tag{2}

where: $J_{12}$ — direct Coulomb integral; $K_{12}$ — exchange integral, positive; the $-K_{12}$ applies to the spin-triplet (higher $S$ , antisymmetric space) and $+K_{12}$ to the singlet. The high-spin state lies lower by $2K_{12}$ . This exchange machinery is owned by aq-qm-hunds-rules (grep-verified there under "exchange energy"/"exchange integral"); we quote the conclusion — maximise $S$ — and move on. Rule 2 then maximises $L$ at fixed $S$ , read straight off the box diagram as the largest allowed $M_L$ ; rule 3 selects $J$ by the half-filling test.

L_{\max}=\Big|\textstyle\sum_i m_{\ell,i}\Big|_{\max},\qquad J=\begin{cases}|L-S| & \text{subshell}\le\text{half}\\[2pt] L+S & \text{subshell}>\text{half}\end{cases} \tag{3}

where: $m_{\ell,i}$ — orbital magnetic quantum number of electron $i$ ; $M_L=\sum_i m_{\ell,i}$ is maximised after the spins are placed. The two cases in $J$ are the normal and inverted multiplets. The sign is set by the fine-structure energy, whose $J$ -dependence is the standard scalar-product identity.

E_J=\tfrac12\,\zeta\,[\,J(J+1)-L(L+1)-S(S+1)\,]\hbar^2 \tag{4}

where: $\zeta$ — effective spin–orbit constant of the term, $>0$ below half filling (normal), $<0$ above (inverted); $J,L,S$ — the term's quantum numbers. Differencing adjacent levels gives the Landé interval rule $E_J-E_{J-1}=\zeta\,J\,\hbar^2$ , a testable $2:1$ pattern for a $^3P$ term. Finally, the ground level's magnetic response is fixed by its Landé $g$ -factor, which is what a magnetometer measures.

g_J = 1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)},\qquad \mu_\text{eff}=g_J\sqrt{J(J+1)}\,\mu_B \tag{5}

where: $g_J$ — Landé factor of the ground level; $\mu_\text{eff}$ — the effective paramagnetic moment. Plugging Hund's ground term into (5) is how the rules connect to measurable magnetism — the lanthanide moment curve above is nothing but (5) evaluated across the $4f$ shell.

Derivation AGround-term census across the p-shell: C, N, O and the half-filling flip

The aim is to run the three rules end to end for the three most-quoted $p$ -shell atoms and show how the ground level $J$ flips from $|L-S|$ to $L+S$ as we cross half filling — carbon's upright $^3P_0$ versus oxygen's inverted $^3P_2$ , with nitrogen's spherically symmetric $^4S_{3/2}$ in between. The route is pure box-filling arithmetic; the one thing to watch is that oxygen is handled as two holes, which reproduces the same $S,L$ as carbon but sits on the other side of the half-filling test.

Step 1. Carbon $2p^2$ . Rule 1: two parallel spins in different $m_\ell$ boxes give $S=1$ . Rule 2: the largest allowed $M_L$ with both spins up is $(+1)+(0)=+1$ , so $L=1$ ( $^3P$ ). Rule 3: $2p^2$ is two of six, below half, so $J=|L-S|=0$ — ground level $^3P_0$ . Why $M_L$ can't reach $+2$ : Pauli forbids two spin-up electrons in the same $m_\ell=+1$ box, so the top allowed $M_L$ is $+1$ , pinning $L=1$ .

Step 2. Nitrogen $2p^3$ . Rule 1: three parallel spins fill all three boxes, $S=\tfrac32$ . Rule 2: $M_L=(+1)+(0)+(-1)=0$ , so $L=0$ — an $S$ -term, $^4S$ . Rule 3: exactly half-filled, so $L=0$ forces the single level $J=S=\tfrac32$ : $^4S_{3/2}$ , with no fine structure.

Step 3. Oxygen $2p^4$ . Treat the shell as two holes: two holes give $S=1$ , $L=1$ ( $^3P$ ), exactly as carbon. But $2p^4$ is four of six, above half filling, so $\zeta<0$ and Rule 3 gives $J=L+S=2$ : ground level $^3P_2$ — the inverted twin of carbon. Why holes work: a nearly full subshell is a nearly empty subshell of holes; the electrostatic terms are identical, but the effective spin–orbit sign flips, which is exactly what inverts the multiplet.

Step 4. Cross-check by microstate count. Two equivalent $p$ electrons span $\binom{6}{2}=15$ microstates, exhausted by $^1S_0$ (1) $+\,^3P_{0,1,2}$ (9) $+\,^1D_2$ (5). The Hund ground term $^3P$ is the lowest, and carbon's measured levels $0/16.40/43.40\ \text{cm}^{-1}$ confirm the normal ordering. Verified: C $^3P_0$ , N $^4S_{3/2}$ , O $^3P_2$ ; 15 microstates $=1+9+5$ ; carbon interval ratio $27.00/16.40=1.646$ (verify_aq-optics-hunds-rules.py; NIST ASD).

Go deeper · Derivation B · 1 minLanthanide magnetic moments: Dy³⁺ and the double hump

Derivation BLanthanide magnetic moments: Dy³⁺ and the double hump

Here we turn Hund's rules into a number a magnetometer can check. We build the ground term of a $4f^n$ ion by the three rules, feed $J,L,S$ into the Landé formula, and predict the effective moment $\mu_\text{eff}=g_J\sqrt{J(J+1)}\,\mu_B$ — showing why dysprosium's $\mu_\text{eff}=10.65\,\mu_B$ is one of the largest atomic moments in nature. The strategy exploits that the $4f$ shell is deep and well described by pure LS coupling, so the rules are unusually reliable for rare earths.

Step 1. Dy³⁺ has $4f^9$ . Rule 1: seven up-spins fill $m_\ell=+3\dots-3$ , then two down-spins pair the top two boxes, so $S=\tfrac12(7-2)=\tfrac52$ . Rule 2: $M_L=\sum m_\ell$ of the occupied set $=(+3+2)$ from the two paired-down boxes plus zero net from the seven up-spins $= 5$ , so $L=5$ ( $H$ ). Rule 3: $4f^9$ is above half (nine of fourteen), so $J=L+S=\tfrac{15}{2}$ — ground term $^6H_{15/2}$ .

Step 2. Landé factor. $g_J=1+\dfrac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$ with $J=\tfrac{15}{2},L=5,S=\tfrac52$ gives $g_J=1+\dfrac{63.75+8.75-30}{127.5}=1+\tfrac{42.5}{127.5}=\tfrac43$ . Why the clean $\tfrac43$ : the numerator $42.5$ is exactly one-third of the denominator $127.5$ , a coincidence of the $^6H_{15/2}$ quantum numbers that makes Dy³⁺ a favourite textbook check.

Step 3. Effective moment. $\mu_\text{eff}=g_J\sqrt{J(J+1)}=\tfrac43\sqrt{\tfrac{15}{2}\cdot\tfrac{17}{2}}=\tfrac43\sqrt{63.75}=\tfrac43(7.984)=10.65\,\mu_B$ — in excellent agreement with the measured $\approx10.6\,\mu_B$ .

Step 4. The double hump. Repeating across $4f^n$ gives $\mu_\text{eff}$ that rises toward each shell-end (large $J$ ), dips to zero at Eu³⁺ $4f^6$ ( $^7F_0$ , $J=0$ ), and peaks near $4f^{9,10}$ — the characteristic two-humped rare-earth curve, with Nd³⁺ ( $^4I_{9/2}$ , $3.62\,\mu_B$ ) and Er³⁺ ( $^4I_{15/2}$ , $9.59\,\mu_B$ ) on the flanks. Verified: Dy³⁺ $g_J=4/3$ , $\mu_\text{eff}=10.65\,\mu_B$ ; Nd³⁺ $3.62$ ; Er³⁺ $9.59$ ; Eu³⁺ $J=0$ (verify_aq-optics-hunds-rules.py).

Limiting cases

case	result	meaning
less than half full	$J=\lvert L-S\rvert$	normal (upright) multiplet, $\zeta>0$ , lowest $J$ is ground — e.g. C $2p^2\to{}^3P_0$
exactly half full	$L=0$ , $J=S$	$^{2S+1}S$ term, single level, no fine structure — e.g. N $2p^3\to{}^4S_{3/2}$ , Mn²⁺ $3d^5\to{}^6S_{5/2}$
more than half full	$J=L+S$	inverted multiplet, $\zeta<0$ , highest $J$ is ground — e.g. O $2p^4\to{}^3P_2$
LS coupling fails ( $\zeta_{nl}\gtrsim F$ )	rule 3 (and sometimes rule 2) unreliable	heavy atoms drift toward jj coupling — recall aq-optics-jj-coupling-intermediate

Worked examples

Example 1 — Oxygen's ground level, and why it is inverted. Predict the ground term and level of atomic oxygen ( $2p^4$ ) and contrast it with carbon.

Step 1. Rule 1 (as holes). Oxygen's $2p^4$ has two holes in the $2p$ shell; two holes act like two electrons for spin, giving $S=1$ (a triplet).

Step 2. Rule 2. The two holes maximise $M_L$ at $(+1)+(0)=+1$ , so $L=1$ ( $P$ ). The term is $^3P$ , identical to carbon's.

Step 3. Rule 3. Oxygen's $2p^4$ is above half filling, so the multiplet is inverted ( $\zeta<0$ ) and $J=L+S=1+1=2$ . Oxygen's ground level is $^3P_2$ — the same $^3P$ term as carbon, but with $J=2$ lowest instead of $J=0$ . The flip is a direct consequence of crossing half filling; measure the fine structure and you find oxygen's multiplet upside-down relative to carbon's.

Example 2 — Dysprosium's giant moment. Predict the ground level and effective magnetic moment of the Dy³⁺ ion ( $4f^9$ ).

Step 1. Ground term. Rule 1: $S=\tfrac52$ (seven up, two down). Rule 2: $L=5$ ( $H$ ). Rule 3: above half filling, $J=L+S=\tfrac{15}{2}$ . Ground term $^6H_{15/2}$ .

Step 2. Landé factor. $g_J=1+\dfrac{\tfrac{15}{2}\cdot\tfrac{17}{2}+\tfrac52\cdot\tfrac72-5\cdot6}{2\cdot\tfrac{15}{2}\cdot\tfrac{17}{2}}=1+\dfrac{63.75+8.75-30}{127.5}=\tfrac43$ .

Step 3. Effective moment. $\mu_\text{eff}=g_J\sqrt{J(J+1)}\,\mu_B=\tfrac43\sqrt{63.75}\,\mu_B$ . $\mu_\text{eff}=\mathbf{10.65\,\mu_B}$ , one of the largest ground-state moments of any ion — and it matches the measured value to within a percent. (For comparison, Nd³⁺ $^4I_{9/2}$ gives $g_J=\tfrac{8}{11}$ and $\mu_\text{eff}=3.62\,\mu_B$ .)

Physics of the portrait

Hund's rules are worth pausing over as a kind of physical law: they are empirical, exact in no rigorous sense, and yet they almost never fail for ground states. That reliability is not luck. Rule 1 encodes the exchange interaction, the same effect that underlies ferromagnetism; rule 2 encodes electron correlation; rule 3 encodes spin–orbit coupling and its particle–hole symmetry. Each rule is the shadow of a real interaction, ordered by magnitude, which is exactly why they succeed where the interactions have that hierarchy (light and moderately heavy atoms, deep shells) and degrade where they don't (heavy atoms in jj coupling). They are a masterclass in getting the physics right by ranking energies rather than solving the full problem.

Origins · 1 minhund's rules and how this was discovered

hund's rules

empirical ordering rules for atomic ground terms · atomic spectroscopy

Three prescriptions, distilled by Friedrich Hund (1896–1997) from painstaking analysis of atomic spectra in 1925–1927, that pick the ground term and level of an atom's ground configuration in LS coupling. Rooted in the exchange interaction (rule 1), orbital correlation (rule 2), and spin–orbit coupling with particle–hole symmetry (rule 3), they remain the fastest reliable route to atomic ground states and magnetic moments.

History

1925

— Hund publishes systematic analyses of multiplet spectra (Z. Phys. 33, 855; 34, 296)

1925

— Pauli's exclusion principle fixes which terms of equivalent electrons are allowed

1925

— Uhlenbeck and Goudsmit introduce electron spin, giving rule 3 its physical basis

1927

— Hund states the ground-term rules in their familiar three-part form

1929

— Slater's theory of complex spectra derives the exchange origin of the spin ordering (rule 1)

Going further

Texts. Foot, Atomic Physics (2005), §5.3 (LS coupling and Hund's rules) · Bransden & Joachain, Physics of Atoms and Molecules, Ch. 8 · Condon & Shortley, The Theory of Atomic Spectra (1935) · Demtröder, Atoms, Molecules and Photons.

Provenance (verbatim facts preserved from the base). The rules and their physical justification follow Foot §5.3, Bransden & Joachain Ch. 8, and Condon & Shortley; original papers Hund, Z. Phys. 33, 855 (1925) and 34, 296 (1925). The verifier recomputes: (i) the $2p^2$ microstate count $15=1+9+5$ yielding $^1S,^3P,^1D$ ; (ii) the carbon ground level $^3P_0$ and its inversion to $^3P_2$ for oxygen against the NIST Atomic Spectra Database; (iii) the half-filled cases N $^4S_{3/2}$ and Mn²⁺ $^6S_{5/2}$ ; (iv) the Landé interval $\Delta E_{J,J-1}=\zeta\hbar^2 J$ and $g_J$ formula; (v) the lanthanide moments Dy³⁺ $10.65\,\mu_B$ , Nd³⁺ $3.62\,\mu_B$ , Er³⁺ $9.59\,\mu_B$ . Carbon's $^3P$ cm⁻¹ values are the standard NIST values. The exchange-energy origin of rule 1 is owned by aq-qm-hunds-rules, recalled here by name.

Builds on: Quantum Numbers and Orbital Angular Momentum · Electron Spin and the Spin Magnetic Moment · Fine Structure and Spin–Orbit Coupling · LS Coupling and Term Symbols. Leads to: The Zeeman Effect · The Paschen–Back and Anomalous Zeeman Regimes · Hyperfine Structure.

Do Hund's rules tell me the order of excited states too?

No — and this is the most common misuse. The rules identify the single lowest term of the ground configuration (rules 1 and 2) and the lowest level within that term (rule 3). They say nothing reliable about the ordering of higher terms, which is set by the detailed balance of direct and exchange integrals and must be computed. For example, within carbon's $2p^2$ the ordering $^3P<{}^1D<{}^1S$ happens to hold, but that ordering is not guaranteed by the rules; only " $^3P$ is lowest" is.

Why does rule 3 flip at exactly half filling?

Because the effective spin–orbit constant $\zeta$ of the term changes sign there. Below half filling the electrons see a coupling $\zeta>0$ , so the fine-structure energy $\propto\zeta\,\mathbf L\!\cdot\!\mathbf S$ is minimised by the smallest $J=|L-S|$ . Above half filling, the nearly full shell is better described by holes, whose effective coupling has the opposite sign, $\zeta<0$ , so the largest $J=L+S$ now lies lowest. At exactly half filling $L=0$ , the term is an $S$ -term, and there is only one level, so the question of ordering doesn't even arise.

What does "count holes instead of electrons" actually mean?

Past half filling it is easier to describe the missing electrons. A subshell with capacity $4\ell+2$ holding $N$ electrons has $4\ell+2-N$ holes; the holes carry the same $S$ and $L$ possibilities as that many electrons, so oxygen's $2p^4$ (two holes) gives the same $^3P$ term as carbon's $2p^2$ (two electrons). The one thing holes change is the sign of the effective spin–orbit coupling, which is why the hole picture automatically produces the inverted multiplet. It is a bookkeeping shortcut that also encodes real physics.

Why are Hund's rules so reliable for rare-earth ions specifically?

Because the $4f$ shell is buried deep inside the filled $5s$ and $5p$ shells, so it is well shielded and its electrons interact mainly among themselves. That makes LS coupling an excellent approximation — the electrostatic term splitting comfortably exceeds the spin–orbit splitting, exactly the hierarchy the rules assume — and it makes the $4f$ levels nearly unperturbed by the chemical environment. As a result the lanthanide ground terms and effective moments predicted by the rules match experiment strikingly well, and the same buried $4f$ levels give the sharp, environment-insensitive laser lines of Nd³⁺ and Er³⁺.

How can three empirical rules be trusted if they aren't a theorem?

Because each rule is the shadow of a real, well-ordered interaction. Rule 1 tracks the exchange energy (the largest correction), rule 2 the orbital correlation energy (next largest), rule 3 the spin–orbit energy (smallest). When those interactions fall in that order of magnitude — as they do for light and moderately heavy atoms and for deep shells — ranking them one at a time gets the ground state right. The rules "fail" only when the hierarchy breaks, e.g. in heavy atoms where spin–orbit coupling rivals the electrostatic term and jj coupling takes over. So they are trustworthy exactly to the extent that the energy hierarchy they assume is real.

Does a filled or closed subshell contribute to L, S, or J?

No — a closed subshell has $L=0$ , $S=0$ , and hence $J=0$ . Every $m_\ell$ orbital is doubly occupied, so the spins cancel in pairs ( $S=0$ ) and the orbital projections sum to zero ( $M_L=0\Rightarrow L=0$ ). That is why you only ever apply Hund's rules to the open subshell: neon-core electrons, filled $d^{10}$ or $f^{14}$ shells, and the like are spectroscopically inert as far as angular momentum goes. It is also why the periodic table's chemistry and magnetism are governed almost entirely by the outermost partially filled shell.

Hund's Rules

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

Physics of the portrait

Going further

Before this

Where this leads