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Derivation

LS Coupling, Term Symbols, and Hund's Rules

Statement

For a many-electron atom in which the residual electrostatic repulsion left over from the central field dominates the spin–orbit interaction, the good quantum numbers are the total orbital angular momentum \(L\) and total spin \(S\), and then the total angular momentum \(J\). A configuration therefore splits first into terms \(^{2S+1}L\), each of which splits into levels \(^{2S+1}L_J\); the residual electrostatic energy ordering of the ground configuration is fixed by Hund's first two rules (maximise \(S\), then \(L\)) and the spin–orbit ordering within the ground term by Hund's third rule (via the Landé interval rule), giving \(J=|L-S|\) lowest for a less-than-half-filled shell and \(J=L+S\) lowest for a more-than-half-filled shell.

Why it matters

Term symbols are the working language of atomic spectroscopy: they label the stationary states that optical transitions connect, and they encode the angular-momentum selection rules (\(\Delta L=0,\pm1\); \(\Delta S=0\); \(\Delta J=0,\pm1\)) that decide which spectral lines appear. Without the LS classification the tangle of levels in a transition-metal or lanthanide spectrum is unreadable.

Hund's rules turn a first-principles energy hierarchy into a one-line recipe for the ground state of every neutral atom in the periodic table. That ground term controls magnetism (the effective moment \(g_J\sqrt{J(J+1)}\,\mu_B\)), chemical valence, and the low-temperature behaviour of paramagnetic salts, so the same reasoning that orders spectral terms also underlies condensed-matter and materials physics.

Assumptions
LS (Russell–Saunders) regime.The residual electrostatic term dominates spin–orbit, \(\langle\hat H_1\rangle\gg\langle\hat H_{\mathrm{SO}}\rangle\); dropped, one is in the \(jj\)-coupling regime of heavy atoms where \(L\) and \(S\) are no longer good quantum numbers and Hund's first two rules lose their meaning.
Central-field starting point.A self-consistent (Hartree–Fock) central potential \(V_c(r)\) already defines a single dominant electron configuration; dropped, strong configuration interaction mixes configurations and the term picture built on one configuration is no longer even approximately valid.
First-order, non-degenerate perturbation within a term.Spin–orbit is diagonalised inside a single term, ignoring its off-diagonal matrix elements to other terms of the same \(J\); dropped (intermediate coupling), terms of equal \(J\) repel and mix, the Landé interval rule is violated, and \(S\) is only approximately conserved.
Equivalent-electron antisymmetry.The total wavefunction is a fully antisymmetric determinant, so Pauli forbids many \((L,S)\) combinations for electrons in the same \(nl\) shell; dropped, one would predict spurious terms (e.g. \(^3\!P\) for \(2p^2\) with both \(m_l=1\)) that violate the exclusion principle.
Free, spherically symmetric atom.No external field and no crystal environment, so total angular momentum \(J\) is conserved and each level is \((2J+1)\)-fold degenerate; dropped, a magnetic field (Zeeman) or ligand field lifts the \(M_J\) degeneracy and \(J\) may cease to be good.
Derivation
1
\[ \hat H=\sum_{i}\left(-\frac{\hbar^2}{2m}\nabla_i^2-\frac{Ze^2}{4\pi\varepsilon_0 r_i}\right)+\sum_{i<j}\frac{e^2}{4\pi\varepsilon_0 r_{ij}}+\sum_i \xi(r_i)\,\hat{\vec l}_i\cdot\hat{\vec s}_i \]
The non-relativistic atomic Hamiltonian: kinetic + nuclear attraction, pairwise electron repulsion, and the one-body spin–orbit term inherited from the Dirac reduction. A
2
\[ \hat H=\underbrace{\sum_i\!\left(-\frac{\hbar^2}{2m}\nabla_i^2+V_c(r_i)\right)}_{\hat H_0}+\underbrace{\sum_{i<j}\frac{e^2}{4\pi\varepsilon_0 r_{ij}}-\sum_i\!\Big(V_c(r_i)+\tfrac{Ze^2}{4\pi\varepsilon_0 r_i}\Big)}_{\hat H_1}+\underbrace{\sum_i\xi(r_i)\,\hat{\vec l}_i\cdot\hat{\vec s}_i}_{\hat H_{\mathrm{SO}}} \]
Add and subtract the central field \(V_c\) fixed by the Hartree–Fock SCF. \(\hat H_0\) is exactly solvable and defines the configuration; \(\hat H_1\) is the residual (non-central) electrostatic repulsion; \(\hat H_{\mathrm{SO}}\) is spin–orbit. B
3
\[ \langle \hat H_1\rangle \gg \langle \hat H_{\mathrm{SO}}\rangle \qquad\text{(light atoms, low }Z\text{)} \]
The LS hierarchy: \(\hat H_1\) scales roughly as the Slater integrals \(F^k,G^k\sim\) few eV and grows slowly with \(Z\), while \(\zeta_{nl}\sim Z^4/n^3\) is small for light atoms. We therefore diagonalise \(\hat H_1\) first and treat \(\hat H_{\mathrm{SO}}\) as a later perturbation. B
4
\[ \bigl[\hat H_0+\hat H_1,\ \hat{\vec L}^2\bigr]=\bigl[\hat H_0+\hat H_1,\ \hat{\vec S}^2\bigr]=\bigl[\hat H_0+\hat H_1,\ \hat L_z\bigr]=\bigl[\hat H_0+\hat H_1,\ \hat S_z\bigr]=0 \]
\(\hat H_1\) depends only on the interelectron distances \(r_{ij}\); it is a scalar under simultaneous rotation of all spatial coordinates, so it commutes with \(\hat{\vec L}=\sum_i\hat{\vec l}_i\). Being spin-independent it commutes with \(\hat{\vec S}=\sum_i\hat{\vec s}_i\). Hence \(L,S,M_L,M_S\) are simultaneously good. C
5
\[ \hat H_0+\hat H_1 \;\longrightarrow\; E_{\text{term}}(L,S)\ \text{ on each }\ ^{2S+1}L,\qquad \dim=(2L+1)(2S+1) \]
Within a configuration, degenerate perturbation theory in \(\hat H_1\) block-diagonalises by \((L,S)\); each block is a term \(^{2S+1}L\) with energy independent of \(M_L,M_S\) by the Wigner–Eckart theorem (rotational invariance). Antisymmetry of the determinant restricts which \((L,S)\) occur for equivalent electrons. C
6
\[ E_{\text{term}}=E_0+\sum_{k}\bigl(a_k\,F^{k}+b_k(L,S)\,G^{k}\bigr),\qquad E(\text{singlet})-E(\text{triplet})=+2K_{ab}>0 \]
Expanding \(1/r_{ij}\) in multipoles gives the Slater–Condon form; the spin-dependence enters only through the exchange integrals \(G^k\) (\(K_{ab}\) for two orbitals). A symmetric spatial function (spin singlet) forces electrons together and costs \(+K\); an antisymmetric spatial function (spin triplet) keeps them apart and gains \(-K\). This is the exchange origin of the ordering. C
7
\[ \boxed{\text{Hund 1: maximise } S}\qquad\text{then}\qquad\boxed{\text{Hund 2: for that }S,\ \text{maximise }L} \]
Maximising \(S\) maximises the number of antisymmetric spatial pairs, each lowering the Coulomb energy by \(2K>0\) (Step 6). For fixed \(S\), the largest \(L\) corresponds to electrons circulating the same way, meeting less often, so a smaller mean \(1/r_{ij}\) and lower repulsion. Both follow from the sign of the exchange term. B
8
\[ \hat H_{\mathrm{SO}}=\sum_i\xi(r_i)\,\hat{\vec l}_i\cdot\hat{\vec s}_i \;\xrightarrow{\ \text{projection within a term}\ }\; \zeta_{LS}\,\hat{\vec L}\cdot\hat{\vec S} \]
Inside a fixed term the vectors \(\hat{\vec l}_i,\hat{\vec s}_i\) act as irreducible tensors of rank 1; the projection (Landé/Wigner–Eckart) theorem replaces \(\sum_i\xi(r_i)\hat{\vec l}_i\cdot\hat{\vec s}_i\) by a single scalar \(\zeta_{LS}\,\hat{\vec L}\cdot\hat{\vec S}\), with \(\zeta_{LS}\) a term-dependent constant. C
9
\[ \hat{\vec J}=\hat{\vec L}+\hat{\vec S}\ \Rightarrow\ \hat{\vec L}\cdot\hat{\vec S}=\tfrac12\!\left(\hat{\vec J}^2-\hat{\vec L}^2-\hat{\vec S}^2\right),\quad |L-S|\le J\le L+S \]
Coupling \(\vec L\) and \(\vec S\) makes \(J\) (and \(M_J\)) good instead of \(M_L,M_S\); squaring \(\hat{\vec J}=\hat{\vec L}+\hat{\vec S}\) isolates the operator \(\hat{\vec L}\cdot\hat{\vec S}\) in terms of the Casimir operators, whose eigenvalues are known. B
10
\[ E_{\mathrm{SO}}(J)=\frac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr],\qquad A\equiv\zeta_{LS}\hbar^2 \]
Substituting the eigenvalues into Step 8's operator: the term of degeneracy \((2L+1)(2S+1)\) fans out into levels \(^{2S+1}L_J\), one for each allowed \(J\). \(A\) carries units of energy. B
11
\[ E_{\mathrm{SO}}(J)-E_{\mathrm{SO}}(J-1)=A\,J \qquad(\text{Landé interval rule}) \]
Differencing Step 10 over adjacent \(J\): the fine-structure spacing is proportional to the larger \(J\). A regular multiplet (\(A>0\)) has \(J=|L-S|\) lowest; an inverted one (\(A<0\)) has \(J=L+S\) lowest. A
12
\[ A=\zeta_{LS}\hbar^2 \propto \frac{\zeta_{nl}}{2S}\times\begin{cases}+1 & \text{shell }<\text{ half full}\\[2pt] -1 & \text{shell }>\text{ half full}\end{cases} \]
Summing \(\xi(r_i)\hat{\vec l}_i\cdot\hat{\vec s}_i\) over the maximally-aligned Hund ground term: for a less-than-half shell the spins add to \(\vec L\) with a positive coefficient; a hole picture flips the sign for a more-than-half shell (a half-filled shell has \(L=0\), so \(A\) vanishes). This is Hund's third rule. C
Result
\[ E\bigl(^{2S+1}L_J\bigr)=\underbrace{E_0+E_{\text{term}}(L,S)}_{\text{electrostatic}}+\underbrace{\frac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]}_{\text{spin–orbit}} \]
\[ \text{Ground term: }S_{\max},\ \text{then }L_{\max};\qquad J_{\text{gs}}=\begin{cases}|L-S| & n_e\le \tfrac12(2l+1)\!\cdot\!2\\[2pt] L+S & n_e> \tfrac12(2l+1)\!\cdot\!2\end{cases} \]

Reading. A configuration splits into terms \(^{2S+1}L\) by the residual electrostatic repulsion (spacing set by exchange integrals), and each term splits into levels \(^{2S+1}L_J\) by spin–orbit coupling (spacing set by the Landé rule \(\Delta E=AJ\)). The ground level is found by maximising \(S\), then \(L\), then choosing \(J\) minimal below half-filling and maximal above; a half-filled shell gives \(L=0\) and a single \(J=S\) level.

Units check. Every energy term is in joules: the Coulomb integrals \(F^k,G^k\sim e^2/4\pi\varepsilon_0\langle r\rangle\) have units \(\mathrm{J}\), and \(A=\zeta_{LS}\hbar^2\) also has units of energy since \(\hat{\vec L}\cdot\hat{\vec S}\) is measured in \(\hbar^2\) and the bracket \([J(J+1)-\dots]\) is a pure number. The term symbol \(^{2S+1}L_J\) is dimensionless. \(\checkmark\)

Limiting cases
  • \(A\to 0\) (half-filled shell, \(L=0\)): the term is an \(S\) state, a single level \(^{2S+1}S_S\) with no fine structure — e.g. \(3d^5\Rightarrow{}^6S_{5/2}\).
  • Single electron outside closed shells (\(S=\tfrac12\)): only two levels \(J=l\pm\tfrac12\); the Landé rule reduces to the alkali doublet, e.g. Na \(^2P_{1/2},{}^2P_{3/2}\).
  • \(\zeta_{LS}\to\infty\) relative to \(F^k\): LS coupling breaks down and the problem crosses over to \(jj\) coupling — the \(J\) values are preserved but their groupings change.
  • Closed shell/subshell: \(L=S=J=0\), a single \(^1S_0\) term — noble gases and alkaline-earth ground states.
  • \(\hat H_{\mathrm{SO}}\to 0\): the \((2J+1)\) levels collapse back to a single \((2L+1)(2S+1)\)-fold degenerate term (pure Russell–Saunders limit).
Breaks when
  • Heavy atoms (large \(Z\)). When \(\zeta_{nl}\gtrsim F^k,G^k\), spin–orbit is no longer a small perturbation on the terms; \(L\) and \(S\) stop being good quantum numbers and the levels regroup into \(jj\) coupling (e.g. Pb \(6p^2\)). Only \(J\) and parity survive.
  • Strong configuration interaction. If two configurations lie close in energy (near-degeneracy, e.g. \(3d^n4s^2\) vs \(3d^{n+1}4s\)), the single-configuration term picture fails, Hund's rules can predict the wrong ground term, and Slater-integral fits break down.
  • Intermediate coupling for excited terms. Even in light atoms, terms of the same \(J\) mix through the off-diagonal part of \(\hat H_{\mathrm{SO}}\); the Landé interval rule is then only approximate (carbon \(^3P\) already shows a \(\sim\)30% deviation).
  • External perturbations comparable to fine structure. A strong magnetic field (Paschen–Back regime) or a ligand/crystal field decouples \(\vec L\) and \(\vec S\) or quenches \(\vec L\), so \(^{2S+1}L_J\) is no longer the right basis.
Failure modes
  • Building terms with the Pauli-forbidden micro-state. Assigning both \(2p^2\) electrons \(m_l=+1\) with parallel spins to "maximise \(L\) and \(S\) at once" — this determinant vanishes; the true ground term is \(^3P\), not \(^3D\).
  • Applying Hund's third rule to excited terms. The \(J_{\min}/J_{\max}\) rule is derived only for the ground term of a single shell; using it for \(^1D\) or an excited term is meaningless (those terms have their own \(J\) from a single value).
  • Forgetting the sign flip past half-filling. Treating oxygen \(2p^4\) like carbon \(2p^2\) and taking \(J=0\) lowest; because the shell is more than half full the multiplet is inverted and \(^3P_2\) is the ground level.
  • Reading multiplicity as "number of levels." \(2S+1\) is the spin multiplicity, not the level count; when \(L<S\) the number of \(J\) levels is \(2L+1\), not \(2S+1\) (e.g. \(^4S\) is a single level).
  • Coupling \(j\) first in a light atom. Applying \(jj\) coupling (\(\vec j_i=\vec l_i+\vec s_i\) then \(\vec J=\sum\vec j_i\)) to carbon; the correct scheme for low \(Z\) is LS, and the two give different level orderings.
  • Mislabelling \(L\). Writing the numerical \(L\) instead of the spectroscopic letter, or vice versa (\(L=0,1,2,3,4\to S,P,D,F,G\)).
Discussion

The whole construction is a statement about a hierarchy of energy scales. The central-field Hartree–Fock problem sets the coarsest scale (tens of eV, the configuration); the residual electrostatic repulsion is the next scale down (\(\sim\)1 eV, splitting a configuration into terms); spin–orbit is smaller again (from meV in light atoms to eV in heavy ones, splitting terms into levels); and Zeeman or hyperfine effects are smaller still. LS coupling is simply the assertion that the second scale beats the third. That single inequality is what makes \(L\) and \(S\) individually good quantum numbers, because \(\hat H_1\) — depending only on the scalars \(r_{ij}\) — is invariant under separate rotations of the spatial and spin spaces.

Physically, Hund's first two rules are the exchange interaction wearing spectroscopic clothing. Parallel spins demand an antisymmetric spatial wavefunction, which pushes electrons apart and lowers their mutual Coulomb energy; this is the same exchange energy \(K\) that drives ferromagnetism in solids. There is no new force — magnetism does not order the spins; the electrostatic repulsion does, and the spins merely follow through the antisymmetry requirement. Maximising \(L\) is the residual, weaker version of the same avoidance: electrons that orbit in the same sense meet less often.

Hund's third rule, by contrast, is purely spin–orbit and depends on the sign of \(A=\zeta_{LS}\hbar^2\). The projection theorem is what makes a sum of one-electron operators collapse to a single \(\hat{\vec L}\cdot\hat{\vec S}\); its coefficient changes sign at half-filling because a more-than-half-filled shell is best described by holes, whose spin–orbit coupling is opposite in sign to that of electrons. This is why the iron-group and lanthanide series show regular multiplets in their first halves and inverted ones in their second halves — a striking, directly measurable fingerprint of the electron/hole duality.

At the deepest level the good quantum numbers are dictated by which subgroup of the full rotation–spin symmetry each interaction preserves. \(\hat H_0+\hat H_1\) is invariant under the independent groups \(SO(3)_L\times SU(2)_S\), giving the Casimirs \(\hat{\vec L}^2,\hat{\vec S}^2\); \(\hat H_{\mathrm{SO}}\) breaks this to the diagonal \(SU(2)_J\), leaving only \(\hat{\vec J}^2\). LS versus \(jj\) coupling are two different chains of subgroups of the same symmetry group, and "intermediate coupling" is the generic case in which no chain is exact — the atom lives somewhere on the continuous path between the two idealised limits, which is why real fine-structure intervals deviate from the exact \(1:2:3\dots\) Landé ratios.

Common misconceptions. Hund's rules do not follow from any spin–spin magnetic force — the energy that orders spins is electrostatic, mediated by the antisymmetry of the wavefunction. And the rules give only the ground term's ordering; they say nothing reliable about the relative energies of excited terms, which require the full Slater-integral calculation or experiment.

Worked examples

Example 1 — Ground level of carbon, \([\mathrm{He}]\,2s^2\,2p^2\).

1
\[ \text{Two equivalent }p\text{ electrons}\ (l=1):\quad \text{allowed terms }= {}^1S,\ {}^1D,\ {}^3P \]
Antisymmetry of the two-electron determinant permits only these three terms (of the naive \(3\times3\times2\times2=36\) micro-states, \(\binom{6}{2}=15\) survive Pauli, distributed as \(1+5+9\)). B
2
\[ S_{\max}:\ \text{the }{}^3P\ (S=1)\ \text{lies lowest}\quad\Rightarrow\quad L=1,\ S=1 \]
Hund 1: highest multiplicity wins. Only \(^3P\) has \(S=1\); \(^1D\) and \(^1S\) have \(S=0\). A
3
\[ J\in\{|L-S|,\dots,L+S\}=\{0,1,2\};\qquad n_e=2<\tfrac12(2\cdot1+1)\cdot2=3 \]
The \(2p\) shell holds 6 electrons; half-full is 3. With only 2 electrons the shell is less than half full, so the multiplet is regular (\(A>0\)) and \(J_{\min}\) is lowest. A
4
\[ J_{\text{gs}}=|L-S|=|1-1|=0 \]
Landé rule with \(A>0\): the lowest interval sits at the smallest \(J\). A
\[ \boxed{\ ^3P_0\ }\quad(\text{carbon ground level}) \]

Reading. Carbon's ground state is the \(J=0\) level of the \(^3P\) term; the levels rise as \(^3P_0<{}^3P_1<{}^3P_2\), and the singlets \(^1D_2,{}^1S_0\) sit above. Units check. Term symbol dimensionless; measured intervals \(^3P_1{-}^3P_0=16.4\ \mathrm{cm^{-1}}\), \(^3P_2{-}^3P_1=43.4\ \mathrm{cm^{-1}}\) (energies), ratio \(2.6\) vs the ideal \(2{:}1\) — the expected intermediate-coupling deviation. \(\checkmark\)

Example 2 — Spin–orbit constant from the sodium D-doublet, \([\mathrm{Ne}]\,3p^1\).

1
\[ l=1,\ s=\tfrac12\ \Rightarrow\ L=1,\ S=\tfrac12,\quad \text{term }{}^2P,\quad J=\tfrac12,\tfrac32 \]
A single valence electron: \(L=l\), \(S=s\); the two allowed \(J\) give the \(^2P_{1/2}\) and \(^2P_{3/2}\) levels. A
2
\[ E\!\left(^2P_{3/2}\right)-E\!\left(^2P_{1/2}\right)=A\,J_{\text{upper}}=A\cdot\tfrac32 \]
Landé interval rule \(E_J-E_{J-1}=AJ\) with \(J_{\text{upper}}=\tfrac32\). Symbols first. A
3
\[ \Delta\tilde\nu=\tilde\nu(589.0\,\mathrm{nm})-\tilde\nu(589.6\,\mathrm{nm})=\frac{1}{589.0\times10^{-7}\,\mathrm{cm}}-\frac{1}{589.6\times10^{-7}\,\mathrm{cm}}\approx 17.2\ \mathrm{cm^{-1}} \]
The two D lines share the same lower level (\(3s\ ^2S_{1/2}\)), so their wavenumber difference is exactly the \(^2P\) fine-structure splitting. B
4
\[ A=\frac{2}{3}\,\Delta\tilde\nu=\frac{2}{3}(17.2\ \mathrm{cm^{-1}})\approx 11.5\ \mathrm{cm^{-1}}\;\equiv\;1.42\times10^{-3}\ \mathrm{eV} \]
Solve Step 2 for \(A\) and insert the measurement; convert with \(1\ \mathrm{cm^{-1}}=1.240\times10^{-4}\ \mathrm{eV}\). A
\[ \boxed{\ A=\zeta_{3p}\hbar^2\approx 11.5\ \mathrm{cm^{-1}}\approx 1.4\ \mathrm{meV}\ } \]

Reading. The sodium \(3p\) spin–orbit constant is about \(1.4\ \mathrm{meV}\); since \(A>0\) the multiplet is regular and \(^2P_{1/2}\) lies below \(^2P_{3/2}\), consistent with a less-than-half-filled shell. Units check. \(\Delta\tilde\nu\) in \(\mathrm{cm^{-1}}\) (energy, via \(hc\)); the factor \(\tfrac23\) is dimensionless, so \(A\) inherits energy units. \(\checkmark\)

Problems
  1. (Easy) Give the ground-state term symbol of chlorine, \([\mathrm{Ne}]\,3s^2\,3p^5\).
    Solution

    The \(3p^5\) shell is one hole in a \(p\) shell, so its terms match \(p^1\): a single \(^2P\) term with \(L=1,S=\tfrac12\), giving \(J=\tfrac12,\tfrac32\). The shell is more than half full (5 of 6), so the multiplet is inverted (\(A<0\)) and \(J_{\max}\) lies lowest. Hence the ground level is \(\boxed{^2P_{3/2}}\) (compare Na, where the regular multiplet puts \(^2P_{1/2}\) lowest).

  2. (Easy–medium) Determine the ground term of titanium, \([\mathrm{Ar}]\,3d^2\,4s^2\).
    Solution

    Closed \(4s^2\) contributes nothing; work with \(3d^2\) (\(l=2\)). Hund 1: two parallel spins give \(S=1\). Hund 2: place them in the two highest distinct \(m_l\) values \(2\) and \(1\), so \(M_L^{\max}=3\Rightarrow L=3\) (an \(F\) term). Allowed \(J=|3-1|\dots(3+1)=2,3,4\). The shell (2 of 10) is less than half full, so \(J_{\min}=2\). Ground term \(\boxed{^3F_2}\).

  3. (Medium) A \(^3P\) term (\(S=1,L=1\)) has levels \(J=0,1,2\). Predict the ratio of the two fine-structure intervals and compare with carbon's measured values \(16.4\) and \(43.4\ \mathrm{cm^{-1}}\).
    Solution

    Landé rule \(E_J-E_{J-1}=AJ\): interval \((1\!\leftarrow\!0)=A\cdot1\), interval \((2\!\leftarrow\!1)=A\cdot2\). Ideal ratio \((2\!\leftarrow\!1):(1\!\leftarrow\!0)=2:1\). Carbon: \(43.4/16.4=2.65\), about 30% above the ideal — a signature of intermediate coupling (mixing of \(^3P_2\) with \(^1D_2\) and \(^3P_0\) with \(^1S_0\) shifts the levels). Since \(A>0\) the ordering is \(^3P_0<{}^3P_1<{}^3P_2\).

  4. (Medium) Find the ground term of the \(\mathrm{Mn^{2+}}\) ion, \([\mathrm{Ar}]\,3d^5\).
    Solution

    Five electrons exactly half-fill the \(d\) shell. Hund 1: all five spins parallel, one per \(m_l\in\{2,1,0,-1,-2\}\), so \(S=\tfrac52\) and multiplicity \(2S+1=6\). Hund 2: the \(m_l\) sum is \(2+1+0-1-2=0\), so \(L=0\) — an \(S\) term. With \(L=0\), \(J=S=\tfrac52\) is the only value and there is no fine structure (\(A\to0\)). Ground term \(\boxed{^6S_{5/2}}\), an isotropic, half-filled-shell state (hence \(\mathrm{Mn^{2+}}\)'s well-known magnetic simplicity).

  5. (Harder) (a) Give the ground term of nickel's \([\mathrm{Ar}]\,3d^8\) shell. (b) How many distinct \((L,S,M_L,M_S)\) micro-states does a \(d^2\) configuration have, and check it against the sum of its term degeneracies?
    Solution

    (a) \(d^8\) is two holes in a \(d\) shell, so its terms match \(d^2\): the maximum \(S=1\) and (as in Problem 2) maximum \(L=3\), an \(F\) term. \(J=2,3,4\). The shell is more than half full (8 of 10), so the multiplet is inverted and \(J_{\max}\) is lowest: \(\boxed{^3F_4}\).

    (b) Two equivalent electrons in a 10-fold-degenerate \(d\) shell give \(\binom{10}{2}=45\) micro-states. The allowed \(d^2\) terms are \(^1S,{}^3P,{}^1D,{}^3F,{}^1G\), with degeneracies \((2L+1)(2S+1)\): \(1+9+5+21+9=45\). \(\checkmark\) The same 45 states apply to \(d^8\) by the particle–hole correspondence.