LS Coupling and Term Symbols

Send the light of a discharge lamp through a good spectrograph and the lines refuse to come singly. They arrive in groups with a fixed internal rhythm: sodium's yellow line is a close doublet, the alkaline earths show triplets, and iron throws up multiplets of five and seven lines whose spacings follow a simple arithmetic. That regularity is the term structure of the atom, and the term symbol $^{2S+1}L_J$ is the grammar that names it. This article builds that grammar and reads the measured multiplets it labels. From quantum mechanics we take the machinery and do not rebuild it: that two angular momenta add by the triangle rule, each total appearing once, is owned by aq-qm-addition-of-angular-momenta; the Clebsch-Gordan coefficients that convert between the uncoupled and coupled bases are owned by aq-qm-clebsch-gordan-coefficients; and the two-electron exchange energetics that split a singlet from a triplet are worked out for helium in aq-qm-helium-atom. What belongs here is the spectroscopy. A configuration such as $2p^2$ is highly degenerate; the residual electron repulsion left over from the central field, with the weaker spin-orbit coupling, fans it into terms and then levels. We show why $p^2$ yields exactly $^1S$ , $^3P$ and $^1D$ , what the Landé interval rule measures in silicon versus carbon, and how helium's two ladders — para and ortho — display the exchange splitting in real electron-volts. LS coupling is not a law but a scheme: good for light atoms, fraying as the nuclear charge grows, yet its term symbols survive as the universal labels of atomic spectra.

Key takeaways

In LS (Russell-Saunders) coupling the valence electrons' orbital momenta first add to a total $\mathbf L=\sum_i\mathbf l_i$ and their spins to $\mathbf S=\sum_i\mathbf s_i$ ; only then do $\mathbf L$ and $\mathbf S$ couple to $\mathbf J=\mathbf L+\mathbf S$ .
A term symbol $^{2S+1}L_J$ encodes three numbers: the spin multiplicity $2S+1$ , the orbital letter $L=S,P,D,F$ for $L=0,1,2,3$ , and the total angular momentum $J$ .
The exclusion principle deletes many naive combinations for equivalent electrons: $p^2$ gives only $^1S_0,\,^3P_{0,1,2},\,^1D_2$ — fifteen microstates, $1+9+5$ .
Within a term, spin-orbit coupling splits the levels by the Landé interval rule $\Delta E_{J,J-1}=\beta J$ , so a $^3P$ multiplet should space $2{:}1$ — silicon measures $1.894$ , carbon $1.644$ .
Helium's excited configurations split by the exchange integral into a lower triplet (ortho) and higher singlet (para): $1s2s$ by $0.796$ eV, $1s2p$ by $0.254$ eV.
The scheme is good when the residual electrostatic interaction beats spin-orbit coupling — light atoms — and gives way to $jj$ coupling as $Z$ grows.
Key idea: when residual Coulomb repulsion dominates spin-orbit coupling, $\mathbf L$ and $\mathbf S$ are separately good, the term $^{2S+1}L_J$ names them in order of the splitting they govern, and the Pauli principle plus the measured multiplet spacings decide which terms exist and how they are arranged.

The core insight

When the residual Coulomb repulsion dominates the spin-orbit interaction, $\mathbf L=\sum\mathbf l_i$ and $\mathbf S=\sum\mathbf s_i$ are separately good angular momenta; the term $^{2S+1}L_J$ names them in order of the energy splitting they govern, and the Pauli principle plus the measured multiplet structure fix which terms exist and how they are spaced.

Where this lives today

Term symbols are the universal address system of atomic physics. Every level in the NIST Atomic Spectra Database is tagged with its $^{2S+1}L_J$ , and every laser and lamp transition is quoted as a jump between two of them — the sodium D lines as $^2P_{3/2,1/2}\to{}^2S_{1/2}$ . The scheme reaches its sharpest modern use in optical atomic clocks: the strontium clock runs on the $^1S_0\to{}^3P_0$ transition, an intercombination line that changes $S$ and is therefore almost perfectly forbidden — which is exactly why it is so narrow and so good as a frequency reference. That a $\Delta S=0$ selection rule, born from helium's two ladders, enables the world's best clocks is a direct dividend of term structure. In astrophysics, multiplet analysis still turns stellar and nebular spectra into temperatures, densities and abundances, and the Landé $g$ -factors read off term symbols (developed in The Zeeman Effect) let magnetic fields be measured light-years away — while where the interval rule visibly breaks, in heavy atoms drifting toward $jj$ coupling, the deviation itself becomes the measurement.

The physical picture

Read the term symbol as a two-stage story. First the strong, spin-independent electron repulsion cares about how the electrons are arranged in space and — through exchange — how their spins line up, so it locks in a collective orbital momentum $\mathbf L$ and a collective spin $\mathbf S$ . Only afterwards, and far more weakly, does the magnetic spin-orbit interaction couple those two totals into $\mathbf J$ . The symbol $^{2S+1}L_J$ simply reports that hierarchy, listing the good quantum numbers in the order of the splitting each controls. The first figure is that pipeline, with the term-symbol anatomy and the $p^2$ result attached.

Figure

The LS pipeline: orbital momenta add to L and spins to S, then L and S couple to J, producing the term symbol ²ˢ⁺¹L_J — read as multiplicity, orbital letter, total angular momentum. Two equivalent p-electrons give exactly ¹S₀, ³P₀,₁,₂ and ¹D₂, fifteen microstates in all

Exchange is the hidden hand behind the ordering of the terms. Electrons with parallel spins are forced by antisymmetry into a spatially antisymmetric state, which keeps them apart and lowers their Coulomb repulsion — so the higher-spin term of a configuration tends to lie lowest, not because spin carries energy but because it labels the spatial correlation. The cleanest place to watch this is helium, where an excited electron and the $1s$ core split into a lower triplet (orthohelium) and a higher singlet (parahelium). The gap is the exchange energy, and it is large — a fraction of an electron-volt — because it is electrostatic in origin, not magnetic. The second figure shows helium's two ladders at their measured energies.

Figure

Helium's two ladders: each excited configuration splits by the exchange integral into a lower triplet (ortho) and a higher singlet (para). Measured NIST energies above the ground state — 1s2s: ³S₁ 19.8196 eV, ¹S₀ 20.6158 eV (gap 0.796 eV = 2K); 1s2p: ³P 20.9641 eV, ¹P₁ 21.2180 eV (gap 0.254 eV). Ortho lies below para in both

The core equations

LS coupling is defined by the order in which the momenta are added: the individual orbital and spin momenta form collective totals first, and only those totals are coupled. This ordering is a physical statement about which interaction is stronger, not a mathematical convention.

\mathbf L=\sum_i\mathbf l_i,\qquad \mathbf S=\sum_i\mathbf s_i,\qquad \mathbf J=\mathbf L+\mathbf S \qquad (1)

where: $\mathbf l_i,\mathbf s_i$ — orbital and spin momenta of electron $i$ ; $\mathbf L,\mathbf S$ — total orbital and spin momenta; $\mathbf J$ — grand total. Each sum runs only over the open subshells, since closed shells contribute nothing. The totals are quantized, and the allowed values follow the triangle rule for adding two angular momenta — a counting theorem we recall rather than prove, owned by aq-qm-addition-of-angular-momenta.

|l_1-l_2|\le L\le l_1+l_2,\qquad |L-S|\le J\le L+S \qquad (2)

where: each combination steps in integer units and appears once for non-equivalent electrons. For equivalent electrons the exclusion principle then removes many of these, which is why explicit microstate counting is needed. The Clebsch-Gordan coefficients that transform the uncoupled $|m_l\,m_s\rangle$ basis into the coupled $|L\,S\,J\rangle$ basis are owned by aq-qm-clebsch-gordan-coefficients. The result is packed into a term symbol, three numbers in a compact glyph.

^{2S+1}L_J \qquad (3)

where: $2S+1$ — spin multiplicity (singlet, doublet, triplet, $\dots$ ); $L$ — the letter $S,P,D,F,G,\dots$ for $L=0,1,2,3,4,\dots$ ; $J$ — the level's total angular momentum, a subscript. A term is a fixed $(L,S)$ ; each of its $J$ values is a level. Inside a term the spin-orbit interaction lifts the $J$ -degeneracy. Replacing $\mathbf L\cdot\mathbf S$ by its eigenvalue — the identity $\mathbf L\cdot\mathbf S=\tfrac12(\mathbf J^2-\mathbf L^2-\mathbf S^2)$ is diagonal in the coupled basis and is recalled from aq-qm-addition-of-angular-momenta — gives each level's shift.

E_{\mathrm{SO}}(J)=\tfrac12\beta\big[J(J+1)-L(L+1)-S(S+1)\big] \qquad (4)

where: $\beta$ — the spin-orbit constant, constant within a term. Differencing adjacent levels leaves only the $J$ -dependent piece, the Landé interval rule.

\Delta E_{J,J-1}=E(J)-E(J-1)=\beta J \qquad (5)

where: $\Delta E_{J,J-1}$ — spacing between adjacent levels; the larger $J$ sets the interval, so a $^3P$ multiplet ( $J=0,1,2$ ) should space $2{:}1$ . Measured deviations from this ratio diagnose departure from pure LS coupling. Finally, the terms of a configuration are themselves split — before spin-orbit coupling — by the electrostatic exchange interaction, seen most cleanly in two-electron helium.

E_\pm=E_0+J_{\text{d}}\pm K,\qquad \Delta_{\text{singlet-triplet}}=2K \qquad (6)

where: $J_{\text{d}}$ — direct (classical) repulsion integral; $K$ — exchange integral, $K>0$ ; the triplet ( $E_0+J_{\text{d}}-K$ ) lies below the singlet ( $E_0+J_{\text{d}}+K$ ). That the triplet lies lower for excited configurations is owned by aq-qm-helium-atom.

Derivation ATerms of the p² configuration by microstate counting

The goal is to derive, by honest microstate counting, that two equivalent p-electrons give exactly the three terms $^1S$ , $^3P$ and $^1D$ — not the larger unrestricted list — because the exclusion principle deletes every combination in which the two electrons would share a spin-orbital. We enumerate the antisymmetric states, tabulate their $(M_L,M_S)$ , and peel complete $(L,S)$ multiplets off the top; the addition machinery behind each peel is recalled from aq-qm-addition-of-angular-momenta.

Step 1. The $p$ subshell ( $l=1$ ) has six spin-orbitals $(m_l,m_s)$ with $m_l\in\{-1,0,1\}$ and $m_s=\pm\tfrac12$ . Placing two indistinguishable electrons in distinct spin-orbitals gives $\binom{6}{2}=15$ antisymmetric microstates. Why: for equivalent electrons the count is a choice of two seats from six, and it is this restricted count — not the unrestricted $(2L+1)(2S+1)$ product — that the terms must resum to.

Step 2. The largest $M_L=2$ occurs only for both electrons at $m_l=+1$ , which forces opposite spins, so $M_S=0$ . A state with $M_L=2,M_S=0$ and no partner at higher $M_S$ signals an $L=2,S=0$ term: $^1D$ , with $2L+1=5$ states and $J=2$ , i.e. $^1D_2$ . Remove these five; ten remain. Why: the highest weight of a multiplet is unique, so reading it off the top identifies one whole term unambiguously.

Step 3. The highest remaining state has $M_L=1,M_S=1$ , signalling $L=1,S=1$ : the term $^3P$ , with $(2L+1)(2S+1)=9$ states and $J=0,1,2$ . Remove these nine; one remains. Why: an $S=1$ state cannot come from the already-removed singlet $^1D$ , so the leftover $M_S=1$ box forces a triplet.

Step 4. The single leftover has $M_L=0,M_S=0$ : an $L=0,S=0$ term, $^1S_0$ . The census closes as $5+9+1=15$ . Why: the exclusion principle has deleted $^3D$ , $^3S$ and $^1P$ — which the unrestricted rule would have allowed — and the surviving three sum exactly to the microstate count, the built-in check.

Step 5. Of these, the term of highest multiplicity, $^3P$ , is observed to lie lowest, with $^3P_0$ the measured ground level of carbon ( $2p^2$ ) and silicon ( $3p^2$ ). The systematic rules that order a configuration's terms are the subject of Hund's Rules; here the ordering is stated as a spectroscopic fact. Why: the term list is fixed by counting alone, but which term is the ground term is an energetic question answered by measurement and by the rules of the companion article. *Verified in verify_aq-optics-ls-coupling-term-symbols.py: $\binom{6}{2}=15$ and $(2L+1)(2S+1)$ over $^1S,^3P,^1D$ sums to $1+9+5=15$ .*

Go deeper · Derivation B · 1 minThe Landé interval rule and what silicon and carbon measure

Derivation BThe Landé interval rule and what silicon and carbon measure

The goal is to turn the spin-orbit operator inside a single term into a testable prediction about the spacing of its fine-structure levels, and then to confront it with two measured $^3P$ multiplets — silicon's and carbon's — seeing both the rule working and its light-atom deviation quantified. We replace $\mathbf L\cdot\mathbf S$ by its coupled-basis eigenvalue (the identity owned by aq-qm-addition-of-angular-momenta), difference adjacent levels, and compare the $2{:}1$ ratio to the NIST intervals.

Step 1. From equation (4), $E(J)=\tfrac12\beta[J(J+1)-L(L+1)-S(S+1)]$ ; the $L,S$ constants shift the whole term but do not affect spacings. Why: only the $J(J+1)$ piece varies within a term, so the multiplet's internal structure is controlled by $\beta$ alone.

Step 2. Difference adjacent levels: $E(J)-E(J-1)=\tfrac12\beta[J(J+1)-(J-1)J]=\beta J$ . Hence the ratio of successive intervals equals the ratio of the larger $J$ values, independent of $\beta$ . Why: because $\beta$ cancels, the interval ratio is a pure prediction of LS coupling — a parameter-free test.

Step 3. For a $^3P$ term ( $L=1,S=1$ , levels $J=0,1,2$ ) the intervals are $E_1-E_0=\beta$ and $E_2-E_1=2\beta$ , so the ratio is exactly $2$ . Why: this $2{:}1$ signature is what a spectroscopist looks for to confirm a $^3P$ assignment and pure LS coupling.

Step 4. Silicon's $^3P$ has NIST levels at $0,\,77.11,\,223.16\ \mathrm{cm^{-1}}$ , giving intervals $77.11$ and $146.05\ \mathrm{cm^{-1}}$ and a ratio $146.05/77.11=1.894$ — close to $2$ . Carbon's $^3P$ at $0,\,16.417,\,43.413\ \mathrm{cm^{-1}}$ gives $1.644$ , a larger deviation. Why: the rule works well for silicon but visibly less so for lighter carbon, where the small spin-orbit energy is more contaminated by higher-order electrostatic and configuration-mixing effects — the deviation is a measurable diagnostic, not a failure. *Verified in verify_aq-optics-ls-coupling-term-symbols.py: symbolic $\Delta E=\beta J$ and $^3P$ ratio $=2$ ; Si $146.05/77.11=1.894$ ; C $26.996/16.417=1.644$ .*

Go deeper · Derivation C · 1 minExchange splitting: why helium is almost two atoms

Derivation CExchange splitting: why helium is almost two atoms

The goal is to show why the terms of a configuration are already split before spin-orbit coupling enters at all — that the electrostatic repulsion alone, through the exchange integral, separates states of different total spin — and to anchor it in helium's measured singlet-triplet gaps, the cleanest two-electron example there is. The sign of the splitting and the para/ortho structure are owned by aq-qm-helium-atom; the electron-volt numbers are the payload we read here.

Step 1. For an excited configuration $1s\,nl$ the two electrons occupy different orbitals, so both a spatially symmetric (singlet spin) and a spatially antisymmetric (triplet spin) combination exist, with diagonal energies $E_\pm=E_0+J_{\text{d}}\pm K$ . Why: total antisymmetry ties spatial symmetry to spin, so the two spin choices carry genuinely different spatial wavefunctions and hence different repulsion energies.

Step 2. Because the exchange integral $K$ is positive, the triplet ( $E_0+J_{\text{d}}-K$ , orthohelium) lies below the singlet ( $E_0+J_{\text{d}}+K$ , parahelium), separated by $2K$ . Why: the spatially antisymmetric triplet vanishes when the electrons coincide, keeping them apart and lowering the Coulomb energy — the same Fermi-hole effect that underlies the ordering of atomic terms.

Step 3. Read the measured gaps. For $1s2s$ , orthohelium $^3S_1$ sits at $19.8196$ eV and parahelium $^1S_0$ at $20.6158$ eV above the ground state — a splitting of $0.796$ eV. For $1s2p$ , $^3P$ at $20.9641$ eV and $^1P_1$ at $21.2180$ eV give $0.254$ eV. Why: these are electrostatic-scale energies, thousands of times larger than any magnetic spin-orbit splitting, which is why the singlet-triplet gap dominates the term structure of light atoms.

Step 4. The two ladders barely communicate: a singlet-to-triplet transition changes $S$ and is optically forbidden, so parahelium and orthohelium behave almost like two separate gases. Why: this near-decoupling is the extreme expression of the $\Delta S=0$ selection rule that spin conservation imposes. *Verified in verify_aq-optics-ls-coupling-term-symbols.py: $20.6158-19.8196=0.796$ eV and $21.2180-20.9641=0.254$ eV.*

Limiting cases

Case	Result	Meaning
Light atoms (LS limit)	$H_{\text{res}}\gg H_{\text{SO}}$	$L,S$ are good quantum numbers; $^{2S+1}L_J$ labels near-exact eigenstates and the interval rule holds.
Heavy atoms	$H_{\text{SO}}\gg H_{\text{res}}$	Each electron forms $\mathbf j_i=\mathbf l_i+\mathbf s_i$ first; $L,S$ cease to be good — the $jj$ -coupling regime (jj Coupling in Heavy Atoms).
Half-filled subshell	$L=0\Rightarrow$ no fine structure	Nitrogen $2p^3$ has ground term $^4S_{3/2}$ ; with $L=0$ the spin-orbit term vanishes and the level is single.
Closed subshell	$L=S=J=0\Rightarrow{}^1S_0$	A filled subshell contributes nothing to $L,S,J$ ; only open subshells make terms.

Common trapThe term symbol describes a single electron

It does not. The quantum numbers $S$ , $L$ and $J$ in $^{2S+1}L_J$ are totals for the whole atom's open-shell electrons, not properties of any one of them: the $^3P_0$ of carbon is a statement about how its two $2p$ electrons combine, and reducing it to a single electron is meaningless. A second, deeper caution: LS coupling is a coupling scheme, not a law of nature. It is the correct organizing picture only while the residual electrostatic interaction outweighs spin-orbit coupling, which fails as the nuclear charge climbs — the spin-orbit energy grows roughly as $Z^4$ for inner electrons while the electrostatic splitting grows only mildly with $Z$ . In heavy atoms $L$ and $S$ stop being good quantum numbers and the levels are better described by $jj$ coupling, though the term symbols survive as approximate labels.

Worked examples

1 · The ground term of nitrogen

Nitrogen has the open configuration $2p^3$ . Find its allowed terms by counting, and state its measured ground term.

Step 1. Three electrons in the six $p$ spin-orbitals give $\binom{6}{3}=20$ antisymmetric microstates — the number the terms must resum to. Nitrogen's $2p^3$ has 20 microstates.

Step 2. Peeling multiplets from the top (as for $p^2$ ) yields three Pauli-allowed terms: $^4S$ , $^2D$ and $^2P$ , with dimensions $(2\cdot 0+1)(2\cdot\tfrac32+1)=4$ , $(2\cdot 2+1)(2)=10$ , and $(2\cdot 1+1)(2)=6$ . The census checks: $4+10+6=20$ . The allowed terms are $^4S$ , $^2D$ , $^2P$ .

Step 3. The measured ground term (NIST) is $^4S_{3/2}$ — the maximum-multiplicity term, with $L=0$ . Because $L=0$ makes $\mathbf L\cdot\mathbf S=0$ , this ground level has no first-order fine structure: a half-filled $p$ shell shows a single line where its neighbours show multiplets. The ordering that puts $^4S$ lowest is the province of Hund's Rules; here it is a spectroscopic fact. Nitrogen's ground term is $^4S_{3/2}$ , with no fine-structure splitting.

2 · Terms of two equivalent d-electrons

How many microstates does a $d^2$ configuration have, and into which terms do they split?

Step 1. A $d$ subshell ( $l=2$ ) has $2(2\cdot2+1)=10$ spin-orbitals, so two equivalent $d$ -electrons give $\binom{10}{2}=45$ microstates. $d^2$ has 45 microstates.

Step 2. The Pauli-allowed terms are $^1S$ , $^3P$ , $^1D$ , $^3F$ and $^1G$ . Their dimensions $(2L+1)(2S+1)$ are $1$ , $9$ , $5$ , $21$ and $9$ . The terms are $^1S$ , $^3P$ , $^1D$ , $^3F$ , $^1G$ .

Step 3. Sum the dimensions: $1+9+5+21+9=45$ , matching the microstate count exactly — the same self-consistency check as $p^2$ , now with five terms. The many closely spaced terms of open $d$ shells are what give transition-metal spectra their richness. The five terms sum to $45$ , confirming the decomposition.

Physics of the portrait

The term symbol has astronomical roots — its coupling scheme was found by people reading the spectra of stars and the alkaline earths.

Origins · 1 minrussell-saunders coupling and how this was discovered

russell-saunders coupling

the term-symbol scheme born from stellar and alkaline-earth spectra · Catalán → Landé → Russell & Saunders

Russell-Saunders coupling carries the names of an astronomer and a physicist. Henry Norris Russell was Princeton's leading astrophysicist — the Russell of the Hertzsprung-Russell diagram — and his motivation for understanding atomic spectra was to read the composition of stars. Frederick Saunders brought meticulous laboratory data on the alkaline earths. Their groundwork was laid by others: in 1922 Miguel Catalán, examining the manganese arc spectrum, recognized that its lines fell into regular groups he called multiplets; and in 1923 Alfred Landé formulated the interval rule showing that the spacings inside a multiplet follow a simple arithmetic in $J$ . In 1925 Russell and Saunders, in a paper on "new regularities in the spectra of the alkaline earths," synthesized these clues into the coupling scheme now named for them: add the orbital momenta into a total $L$ and the spins into a total $S$ , and the multiplet structure falls out. Friedrich Hund soon supplied the rules ordering the terms, and by 1929 Russell, Shenstone and Turner had standardized the $^{2S+1}L_J$ notation still used. It is a scheme with stellar roots — fitting, since its first job was to make the light of distant suns legible.

History

1868

— Janssen and Lockyer see the yellow D₃ line of an unknown element (helium) in the solar spectrum

1922

— Catalán recognizes multiplets in the manganese arc spectrum

1923

— Landé formulates the fine-structure interval rule

1925

— Russell and Saunders publish the LS coupling scheme for the alkaline earths

1925

— Hund begins his systematic analysis of the terms of complex spectra

1929

— Russell, Shenstone and Turner standardize the term-symbol notation

Going further

Texts. Foot, Atomic Physics, ch. 5 (two-electron atoms and LS coupling) · Demtröder, Atoms, Molecules and Photons, term symbols and multiplets · Bransden & Joachain, Physics of Atoms and Molecules, ch. 8 · Condon & Shortley, The Theory of Atomic Spectra · Woodgate, Elementary Atomic Structure, ch. 5-6.

Provenance. The $p^2$ term list ( $^1S_0,^3P_{0,1,2},^1D_2$ ), the $p^3$ list ( $^4S,^2D,^2P$ ) and the $d^2$ list ( $^1S,^3P,^1D,^3F,^1G$ ) follow from microstate counting (Foot ch. 5; Condon & Shortley) and are reproduced in the verifier. The addition/triangle-rule and Clebsch-Gordan machinery are owned by aq-qm-addition-of-angular-momenta and aq-qm-clebsch-gordan-coefficients; the singlet-triplet ordering by exchange is owned by aq-qm-helium-atom. Measured data: silicon $^3P$ levels $0,77.11,223.16\ \mathrm{cm^{-1}}$ (ratio $1.894$ ) and carbon $^3P$ $0,16.417,43.413\ \mathrm{cm^{-1}}$ (ratio $1.644$ ); helium $1s2s$ $^3S_1$ $19.8196$ / $^1S_0$ $20.6158$ eV (gap $0.796$ ) and $1s2p$ $^3P$ $20.9641$ / $^1P_1$ $21.2180$ eV (gap $0.254$ ), all NIST ASD values. Ground terms C $^3P_0$ , Si $^3P_0$ , N $^4S_{3/2}$ are standard.

Prerequisites. Quantum Numbers and Orbital Angular Momentum · Electron Spin and the Spin Magnetic Moment · Fine Structure and Spin-Orbit Coupling · Electron Configurations and the Aufbau Principle. Leads to. Hund's Rules · jj Coupling in Heavy Atoms · The Zeeman Effect · The Paschen-Back and Anomalous Zeeman Regimes.

Why can't I just add every L and S by the triangle rule for equivalent electrons?

Because the Pauli principle deletes many of the naive combinations. The triangle rule tells you which totals two angular momenta can form, and for non-equivalent electrons (different $n$ or $l$ ) every one is realized. But equivalent electrons — same $n$ and $l$ — cannot share a spin-orbital, so the two-electron state must be antisymmetric, and that forbids the combinations with the wrong spatial-spin symmetry pairing. For $p^2$ the unrestricted rule would allow $^1S,^1P,^1D,^3S,^3P,^3D$ ; the exclusion principle keeps only $^1S,^3P,^1D$ . The reliable way to find the survivors is microstate counting: enumerate the antisymmetric states, tabulate $(M_L,M_S)$ , and peel off complete multiplets until the boxes are used up.

What does the superscript 2S+1 actually count?

It is the spin multiplicity — the number of orientations the total spin $\mathbf S$ can take, $2S+1$ : singlet for $S=0$ , doublet for $S=\tfrac12$ , triplet for $S=1$ , and so on. There is a subtlety worth knowing: $2S+1$ equals the number of fine-structure levels $J$ in the term only when $S\le L$ . If $S>L$ the number of $J$ values is $2L+1$ , smaller — yet the term keeps its name from $2S+1$ . So a $^4S$ term ( $S=\tfrac32$ , $L=0$ ) is called a quartet but has just one level, since $L=0$ allows only $J=\tfrac32$ . The multiplicity names the spin; it counts the levels only when $S\le L$ .

Why does the triplet lie below the singlet if parallel spins seem like they should be higher energy?

This is the most counterintuitive point in the subject, and the answer is that spin itself carries no significant energy here — it is a label for spatial correlation. Total antisymmetry ties the spin state to the spatial state: a triplet (parallel spins, symmetric spin part) must pair with a spatially antisymmetric wavefunction, which vanishes when the two electrons coincide. That spatial "Fermi hole" keeps the electrons apart and lowers their repulsion, so the triplet ends up lower; the singlet, symmetric in space, lets the electrons crowd and pays a higher price. Helium makes this concrete: its excited triplets lie below the singlets by fractions of an electron-volt — an electrostatic energy, not a magnetic one. Parallel spins are lower not from any spin force but because antisymmetry hands them a roomier spatial state.

How do I know which term of a configuration is the ground term?

Counting gives you the list of terms; it does not by itself order them. The ordering is what Hund's Rules address, and the definitive answer for any real atom is its measured spectrum, tabulated in databases like NIST ASD. Empirically the term of highest spin multiplicity lies lowest, and among equal-multiplicity terms the one of highest $L$ — which is why carbon's ground term is $^3P$ and nitrogen's is $^4S$ . Rather than restate those rules here — that is the companion article's job — treat the ground term as a spectroscopic fact you can look up, and use the counting of this article to know what the candidates are.

Why does nitrogen's ground term have no fine structure?

Because its ground term is $^4S_{3/2}$ , and the $S$ means $L=0$ . Fine structure comes from the spin-orbit interaction $\beta\,\mathbf L\cdot\mathbf S$ , and when $L=0$ that operator is identically zero — there is no orbital angular momentum for the spin to couple to. So the term has a single level, $J=\tfrac32$ , with no first-order splitting, and nitrogen shows a clean single line where carbon and oxygen (with $L=1$ ground terms) show $^3P$ multiplets. A half-filled $p^3$ shell gives $L=0$ because its three electrons, one in each of $m_l=-1,0,+1$ with parallel spins, sum to $M_L=0$ : half-filled shells are the quiet ones in a fine-structure spectrum.

When does LS coupling break down, and what replaces it?

LS coupling assumes the residual electrostatic interaction (which fixes $L$ and $S$ ) is larger than the spin-orbit interaction (which fixes $J$ ). That ordering holds for light atoms but reverses as the nuclear charge grows, because the spin-orbit energy scales roughly as $Z^4$ for inner electrons while the electrostatic splitting grows only slowly. When spin-orbit wins, each electron's own $\mathbf l_i$ and $\mathbf s_i$ couple into $\mathbf j_i$ first, and the $\mathbf j_i$ then combine into $J$ — the $jj$ -coupling scheme of heavy atoms. Most real atoms lie in between, where neither picture is exact but the LS term symbols persist as approximate labels because the true eigenstates connect continuously to the pure-LS basis. You can spot the crossover in data: the Landé interval ratio drifts from its LS prediction, exactly the silicon-versus-carbon deviation seen here, only far more pronounced.

What is the difference between a term and a level?

A term is a set of states sharing the same $L$ and $S$ — one value of the electrostatic energy — written $^{2S+1}L$ . A level is one specific $J$ within that term, written $^{2S+1}L_J$ , and it is what the spin-orbit interaction resolves. So $^3P$ is a term; $^3P_0$ , $^3P_1$ and $^3P_2$ are its three levels, and each splits further in a magnetic field into $2J+1$ sublevels $M_J$ . The hierarchy mirrors the energy scales: configuration, then term (residual electrostatic), then level (spin-orbit), then Zeeman sublevel. The term symbol $^{2S+1}L_J$ names a level; drop the subscript and you name the whole term.