--- title: Hund's Rules and Ground-State Terms module: Many-Electron Atoms moduleNumber: 5 lessonNumber: 6 order: 506 summary: > A configuration allows several terms; Hund's three rules pick the ground one. Maximize the spin S first, then the orbital L, then set J to |L−S| for a less-than-half shell and L+S for a more-than-half shell. The first two rules come from exchange lowering the energy of apart-kept electrons; the third comes from the sign of the spin-orbit coupling, which flips as a shell passes half-filling and turns the multiplet from normal to inverted. Worked ground terms for carbon, nitrogen, oxygen, and iron show the rules in action. topics: [Many-Electron Atoms] sources: - book: Foot ref: "Ch. 5 — Hund's Rules; §5.6 The Ground Levels of Atoms" - book: Bransden & Joachain ref: "Ch. 7 — Many-Electron Atoms; §7.5 Hund's Rules and Ground Terms" - book: Demtröder ref: "Ch. 6 — Atoms with Many Electrons; Ground-State Configurations" draft: false --- The [term symbols](/atomic-physics/many-electron-atoms/ls-jj-coupling-term-symbols) of a configuration list the allowed levels but not their order. For chemistry and magnetism the one that matters is the lowest, the ground term. Hund's three rules name it directly from the configuration, without diagonalizing anything, and each rule traces to a piece of physics already assembled: exchange, the anisotropy of the Coulomb repulsion, and the sign of the spin-orbit coupling. ## The three rules For the ground term of a given configuration in LS coupling, apply in order:[^foot-56] - **Rule 1 — maximum $S$.** The ground term has the largest total spin $S$ the Pauli principle permits. - **Rule 2 — maximum $L$.** Among the terms of that largest $S$, the ground term has the largest total orbital angular momentum $L$. - **Rule 3 — the $J$ value.** For a subshell **less** than half full, the ground level has $J = |L-S|$; for a subshell **more** than half full, $J = L+S$. A half-full subshell has $L=0$, so $J=S$ and the question does not arise. The rules fix the ground level only. They are reliable for the lowest term of a configuration and are not a general ordering of the excited terms. $$ % caption: Hund's rules as an ordered cascade: fix the ground term by spin first, % then orbital, then read off J from the shell occupancy. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth, box/.style={draw, minimum width=30mm, minimum height=11mm, align=center}] \definecolor{acc}{HTML}{4A6FA5} \node[box, draw=acc, text=acc, thick] (r1) at (0,0) {1. maximize $S$}; \node[box] (r2) at (4.4,0) {2. maximize $L$}; \node[box] (r3) at (8.8,0) {3. set $J$ by occupancy}; \draw[black, ->, thick] (r1) -- (r2); \draw[black, ->, thick] (r2) -- (r3); \end{tikzpicture} $$ ## Why maximum spin and maximum L Rules 1 and 2 are electrostatic. Aligning spins forces the spatial wave function to be antisymmetric, which keeps the electrons apart and lowers their mutual Coulomb repulsion — the same [exchange](/atomic-physics/many-electron-atoms/identical-particles-hartree-fock) energy that put the helium triplet below the singlet. A larger $S$ means more parallel pairs, each contributing a favourable exchange term, so the highest-spin term lies lowest.[^foot-56] Rule 2 is subtler but the same in spirit. For fixed $S$, a larger $L$ corresponds to the electrons orbiting the nucleus in the same rotational sense, so they meet less often and their average repulsion is smaller. Both rules lower the electrostatic energy by keeping electrons out of each other's way; only the mechanism (spin correlation versus orbital correlation) differs. $$ % caption: Hund's first rule for the p³ half-shell. All three spins align (S = % 3/2), one electron per orbital, forcing an antisymmetric spatial state that % minimizes repulsion. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % three p orbital boxes \foreach \i in {0,1,2}{\draw[black, thick] (\i*1.3,0) rectangle (\i*1.3+1.0,0.75);} % one up-arrow in each \foreach \i in {0,1,2}{\draw[acc, very thick, ->] (\i*1.3+0.5,0.12) -- (\i*1.3+0.5,0.63);} \node[acc, anchor=west] at (4.2,0.38) {$p^3$: $S = \tfrac32$, all spins parallel}; \node[black, anchor=north] at (1.65,-0.1) {three p orbitals, one electron each}; \end{tikzpicture} $$ ## Why the J rule flips at half filling Rule 3 is spin-orbit. Within the ground term, the spin-orbit energy is $A\,\vec L\cdot\vec S$ with the level energies $$ E_J = \frac{A}{2}\big[J(J+1)-L(L+1)-S(S+1)\big], $$ so the sign of the constant $A$ decides which $J$ lies lowest. Summing the one-electron coefficients $\xi(r_i)\,\vec\ell_i\cdot\vec s_i$ over the electrons of a subshell gives an $A$ that is **positive** when the subshell is less than half full and **negative** when it is more than half full.[^foot-56] The reason is the particle-hole symmetry: a more-than-half subshell is better described by its positively contributing holes, whose spin-orbit coupling carries the opposite sign. - $A > 0$ (less than half): $E_J$ increases with $J$, so $J = |L-S|$ is lowest — a **normal** multiplet. - $A < 0$ (more than half): $E_J$ decreases with $J$, so $J = L+S$ is lowest — an **inverted** multiplet. $$ % caption: The multiplet inverts at half filling. Below half (A > 0) the smallest % J sits lowest; above half (A < 0) the largest J sits lowest. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % --- normal (less than half) --- \node[black] at (1.2,3.5) {less than half}; \draw[acc, thick] (0.2,0.5) -- (2.2,0.5); \node[black, anchor=east] at (0.1,0.5) {min $J$}; \draw[black, thick] (0.2,1.5) -- (2.2,1.5); \draw[black, thick] (0.2,2.7) -- (2.2,2.7); \node[black, anchor=east] at (0.1,2.7) {max $J$}; \draw[acc, ->] (2.5,0.5) -- (2.9,0.5); \node[acc, anchor=west] at (2.95,0.5) {ground}; % --- accerted (more than half) --- \begin{scope}[xshift=6.4cm] \node[black] at (1.2,3.5) {more than half}; \draw[black, thick] (0.2,0.5) -- (2.2,0.5); \node[black, anchor=east] at (0.1,0.5) {min $J$}; \draw[black, thick] (0.2,1.7) -- (2.2,1.7); \draw[acc, thick] (0.2,2.7) -- (2.2,2.7); \node[black, anchor=east] at (0.1,2.7) {max $J$}; \draw[acc, ->] (2.5,2.7) -- (2.9,2.7); \node[acc, anchor=west] at (2.95,2.7) {ground}; \end{scope} \end{tikzpicture} $$ ## Worked ground terms The recipe is mechanical. Lay out the subshell's $m_\ell$ boxes, fill them to maximize spin (one electron per box, all parallel, before pairing), then read off $S$, $L = |M_L^{\max}|$, and $J$.[^foot-56] > **Worked example.** Carbon, $2p^2$. Two electrons, both spin up (Rule 1) so > $S=1$. To maximize $M_L$ place them in $m_\ell = 1$ and $m_\ell = 0$ (they cannot > share a box), giving $M_L^{\max} = 1$, hence $L=1$. The subshell is less than > half full ($2 < 3$), so $J = |L-S| = 0$. Ground term $^3P_0$. > **Worked example.** Nitrogen, $2p^3$. Three electrons, all spin up (Rule 1), > $S=\tfrac32$, multiplicity $4$. They occupy $m_\ell = 1, 0, -1$, so > $M_L = 0$ and $L=0$. Half full, so $J = S = \tfrac32$. Ground term > $^4S_{3/2}$. > **Worked example.** Oxygen, $2p^4$. Four electrons in three boxes: three spin up > plus one spin down, so two are unpaired and $S=1$. The paired box is $m_\ell = 1$, > the singly filled boxes $m_\ell = 0, -1$, giving $M_L = 1$ and $L=1$. More than > half full ($4 > 3$), so $J = L+S = 2$. Ground term $^3P_2$ — an inverted > multiplet, the mirror of carbon's $^3P_0$. > **Worked example.** Iron, $3d^6$. Five boxes $m_\ell = 2,1,0,-1,-2$. Fill all > five spin up (five electrons), then the sixth spin down in $m_\ell = 2$. Four are > unpaired, so $S=2$, multiplicity $5$. The extra electron sits at $m_\ell = 2$, so > $M_L = 2$ and $L=2$. More than half full ($6 > 5$), so $J = L+S = 4$. Ground term > $^5D_4$. $$ % caption: Ground-term construction for oxygen (2p⁴) and iron (3d⁶). Fill boxes % for maximum spin, pair only after every box is singly occupied; the last % electron sets M_L and hence L. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % oxygen p^4: three boxes \node[black, anchor=east] at (-0.2,0.38) {$2p^4$}; \foreach \i in {0,1,2}{\draw[black, thick] (\i*1.0,0) rectangle (\i*1.0+0.8,0.75);} % box0 doubly occupied (up+down), boxes 1,2 up \draw[black, very thick, ->] (0.28,0.12) -- (0.28,0.63); \draw[acc, very thick, ->] (0.55,0.63) -- (0.55,0.12); \draw[black, very thick, ->] (1.4,0.12) -- (1.4,0.63); \draw[black, very thick, ->] (2.4,0.12) -- (2.4,0.63); \node[black, anchor=west] at (3.2,0.38) {$S=1,\ L=1:\ {}^3P_2$}; % iron d^6: five boxes \node[black, anchor=east] at (-0.2,-1.62) {$3d^6$}; \foreach \i in {0,1,2,3,4}{\draw[black, thick] (\i*1.0,-2.0) rectangle (\i*1.0+0.8,-1.25);} % box0 doubly, boxes 1-4 up \draw[black, very thick, ->] (0.28,-1.88) -- (0.28,-1.37); \draw[acc, very thick, ->] (0.55,-1.37) -- (0.55,-1.88); \foreach \i in {1,2,3,4}{\draw[black, very thick, ->] (\i*1.0+0.4,-1.88) -- (\i*1.0+0.4,-1.37);} \node[black, anchor=west] at (5.4,-1.62) {$S=2,\ L=2:\ {}^5D_4$}; \end{tikzpicture} $$ ## The two halves of a shell The pattern across a subshell is symmetric about the half-filled point. Spin rises to a maximum at half filling and falls back; $L$ is zero at both the empty and half-filled shells; and the $J$ rule switches from $J=|L-S|$ to $J=L+S$ as the shell crosses half. A subshell and its hole-complement share the same term set but sit on opposite sides of the $J$ switch, so carbon ($p^2$) and oxygen ($p^4$) both have a $^3P$ ground term, normal for carbon and inverted for oxygen. $$ % caption: Across a p-shell the ground-term J switches at half filling: normal % (J = |L−S|) for p¹–p², J = S at the p³ half shell, inverted (J = L+S) for p⁴–p⁵. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.0,0) node[right, black!70] {electrons in p-shell}; \draw[acc, dashed] (3.5,0) -- (3.5,0.85); \draw[acc, dashed] (3.5,1.55) -- (3.5,2.6); \node[acc, anchor=south] at (3.5,2.6) {half shell}; % ground terms across the shell \node[black] at (0.9,1.2) {$^2P_{\tfrac{1}{2}}$}; \node[black] at (1.9,1.2) {$^3P_0$}; \node[acc] at (3.5,1.2) {$^4S_{\tfrac{3}{2}}$}; \node[black] at (5.1,1.2) {$^3P_2$}; \node[black] at (6.1,1.2) {$^2P_{\tfrac{3}{2}}$}; \foreach \x/\lab in {0.9/1,1.9/2,3.5/3,5.1/4,6.1/5}{ \node[black, anchor=north] at (\x,-0.05) {\lab};} \node[black, anchor=east] at (2.6,0.5) {normal}; \node[black, anchor=west] at (4.4,0.5) {inverted}; \end{tikzpicture} $$ Hund's rules close the many-electron chain that began with the [central-field approximation](/atomic-physics/many-electron-atoms/central-field-self-consistent): the central field gives the configuration, the residual electrostatic interaction sorts it into terms, the spin-orbit interaction splits each term into levels, and the three rules pick the ground level. The ground terms they produce are the starting point for an atom's behaviour in an external field, taken up in the [Zeeman effect](/atomic-physics/atoms-in-external-fields/zeeman-effect). [^foot-56]: **Foot**, _Atomic Physics_, §5.6 — Hund's three rules, the exchange origin of maximum $S$ and $L$, the sign of the spin-orbit constant $A$ switching at half filling (normal versus inverted multiplet), and the worked ground terms for the $p$- and $d$-shell atoms.