Russell-Saunders (LS) Coupling

LS coupling is not a historical curiosity — it is the working vocabulary of modern atomic, molecular, and materials spectroscopy. Every atomic emission or absorption line list (used, for instance, in analyzing stellar spectra, plasma diagnostics in fusion research, or interstellar medium composition) is indexed by Russell-Saunders term symbols, because selection rules for electric-dipole transitions are stated most simply in terms of $\Delta L, \Delta S, \Delta J$ changes between LS-coupled levels. In magnetochemistry, the effective magnetic moment of a transition-metal or lanthanide ion in the free-ion limit is computed directly from its Russell-Saunders ground term via the Landé $g$-factor,

and this underlies the interpretation of magnetic susceptibility measurements used to characterize spin states in coordination complexes, molecular magnets, and single-molecule magnets built from lanthanide ions (where LS coupling, with subsequent weak crystal-field perturbation, remains an excellent starting point because the $4f$ orbitals are shielded from the ligand environment). In laser physics and atomic clock technology, hyperfine and fine-structure levels labeled by $^{2S+1}L_J$ term symbols are precisely the addressed transitions — for example, the cesium clock transition and optical lattice clock transitions in strontium and ytterbium are described starting from their Russell-Saunders ground terms before hyperfine and external-field corrections are added. In inorganic and organometallic chemistry, Tanabe-Sugano and Orgel diagrams — the standard tools for interpreting d-d transition energies and colors of transition-metal complexes — are built by taking the free-ion Russell-Saunders terms (exactly as derived above for $d^2$, $d^3$, etc.) and tracking how each splits under an octahedral or tetrahedral ligand field, making LS coupling the essential zeroth-order starting point for all crystal-field and ligand-field theory used to rationalize the colors and magnetism of transition-metal compounds.

Ask the tutor: How would the term diagram and ground-level assignment change for a $d^7$ ion (like Co²⁺), where the shell is more than half-filled?

Consider an atom with several electrons outside closed shells. Closed subshells contribute zero net orbital and spin angular momentum (all $m_l$ and $m_s$ values are filled and cancel), so only the open-shell electrons matter. Each electron $i$ carries an orbital angular momentum operator $\vec{l}_i$ (quantum number $l_i$) and a spin angular momentum operator $\vec{s}_i$ (quantum number $s_i = 1/2$). These angular momenta interact with each other in two physically distinct ways, and the relative strength of those interactions decides how we should combine them. The dominant interaction in light and medium atoms is electrostatic (Coulomb) repulsion between electrons. This interaction commutes with the total orbital angular momentum $\vec{L} = \sum_i \vec{l}_i$ and the total spin $\vec{S} = \sum_i \vec{s}_i$ separately — because the Coulomb Hamiltonian is spin-independent and rotationally invariant in orbital space. So $L$ and $S$ are (approximately) good quantum numbers, and the electrons' orbital motions couple among themselves, while spins couple among themselves, independently of each other. This is Russell-Saunders coupling. The weaker interaction is spin-orbit coupling: each electron's own spin interacts with the magnetic field generated by its own orbital motion, $H_{SO} = \sum_i \xi(r_i)\, \vec{l}_i \cdot \vec{s}_i$. This does not commute with $L_z$ or $S_z$ individually but does commute with the total $\vec{J} = \vec{L} + \vec{S}$. Because $H_{SO}$ is weak compared to electron repulsion (true for atomic number roughly $Z \lesssim 40$), we treat it as a small perturbation after $L$ and $S$ have already been formed — hence the two-stage coupling: first $l_i \to L$ and $s_i \to S$, then $L, S \to J$. The result is a hierarchy of energy scales. Differences in configuration (which orbitals are occupied) are largest — tens of eV. Differences in term ($L, S$ for a fixed configuration) are next — driven by electron repulsion, typically 1–5 eV. Differences in level ($J$ within a term) are smallest — driven by spin-orbit coupling, typically $10^{-3}$–$10^{-1}$ eV for light atoms, growing roughly as $Z^4$ so becoming substantial for heavy elements. Each level is further $(2J+1)$-fold degenerate in the space-fixed magnetic quantum number $M_J$, a degeneracy lifted only by an external field (Zeeman effect).

The vector-model picture makes the coupling concrete: draw $L$ as a vector of length $\sqrt{L(L+1)}\hbar$ precessing rapidly, $S$ likewise, both precessing around their resultant $J = |L+S|, |L+S-1|, \ldots, |L-S|$. Only integer steps between $|L-S|$ and $L+S$ are allowed, exactly as in ordinary angular momentum addition — because that is precisely what this is: successive applications of the Clebsch-Gordan coupling rule, first to combine all $l_i$ into $L$ and all $s_i$ into $S$, then to combine $L$ and $S$ into $J$.

The order of coupling matters conceptually even though the final atom, of course, has one true Hamiltonian. LS coupling is an approximation valid when spin-orbit splitting is much smaller than the term splitting from electron repulsion; the opposite extreme, where spin-orbit coupling for each electron dominates over inter-electron repulsion (heavy atoms, high $Z$), is called $jj$-coupling, in which each electron's own $j_i = l_i + s_i$ forms first, and the $j_i$ only afterward combine into total $J$. Real heavy atoms are intermediate, but LS coupling remains an excellent zeroth-order language for essentially all of first- and second-row chemistry and for describing atomic spectra and selection rules.

For $N$ open-shell electrons with individual quantum numbers $l_i$, $s_i = 1/2$, the total orbital and spin angular momentum quantum numbers arise by vector-model (Clebsch-Gordan) addition, applied pairwise:

where: $l_1, l_2$ are the orbital angular momentum quantum numbers of the coupling electrons; each allowed $L$ differs from its neighbor by an integer, spanning $|l_1-l_2|$ to $l_1+l_2$.

where: for two electrons ($s_i = 1/2$ each) this gives only $S = 0$ (singlet) or $S=1$ (triplet); for $N$ electrons, couple sequentially. The term symbol collects $L$ and $S$ with the multiplicity as a left superscript:

where: $2S+1$ is the spin multiplicity (number of $M_S$ values, $=$ number of $J$-levels when $L \ge S$); $L$ is written as a letter code inherited from the old spectroscopic $s,p,d,f$ notation but capitalized for the total momentum. Total angular momentum then couples $L$ and $S$:

where: each allowed $J$ labels a distinct level, written as a right subscript, $^{2S+1}L_J$; there are $\min(2L+1, 2S+1)$ such levels per term. Each level's degeneracy — the number of magnetic sublevels $M_J = -J, \ldots, +J$ — is

where: $g_J$ counts the states that remain degenerate in the absence of an external field, split only by the Zeeman effect when a field is applied. The spin-orbit energy splitting within a term follows the Landé interval rule, derived by treating $H_{SO} = A\, \vec{L}\cdot\vec{S}$ as a perturbation with $\vec{L}\cdot\vec{S} = \tfrac12[J(J+1)-L(L+1)-S(S+1)]\hbar^2$:

where: $A$ is the spin-orbit coupling constant for that term (positive for less-than-half-filled shells, negative for more-than-half-filled shells); $J$ here is the larger of the two adjacent $J$ values. This interval rule is a direct, checkable consequence of the $\vec L\cdot\vec S$ operator and is what makes fine-structure spacings in real spectra (e.g., the Na D-line doublet) ratios predictable from a single constant $A$. Finally, the number of microstates in a full term (before spin-orbit splitting) must match the number obtained by direct counting of $m_l, m_s$ combinations for the configuration — this is the master consistency check used to build term tables:

where: the right side is the number of ways to place $N$ indistinguishable electrons in the $2(2l+1)$ spin-orbitals of a subshell with orbital quantum number $l$ (Pauli-allowed microstates only); the left side sums $(2L+1)(2S+1)$ over every term that the configuration produces.

Example 1 — Foundational: term symbols and level ordering for carbon ($2p^2$). This restates the derivation above with explicit numbers, to fix the method as a template. Carbon's ground configuration is $1s^2 2s^2 2p^2$; the closed $1s^2, 2s^2$ shells contribute nothing, so only the two $2p$ electrons matter. Step 1: Microstate count is $\binom{6}{2}=15$. Step 2: Building the $(M_L, M_S)$ table (as derived above) peels off $^1D$ (5 states), $^3P$ (9 states), $^1S$ (1 state), totaling 15. Step 3: Apply Hund's first rule — maximum $S$ is $S=1$ (the $^3P$ term) — so $^3P$ is the ground term. Step 4: Hund's second rule is not needed (only one $S=1$ term here). Step 5: Hund's third rule — $2p^2$ is less than half of the 6-electron capacity of a $p$ shell, so the term is "regular" ($A>0$) and minimum $J$ lies lowest:

giving levels $^3P_0 < {}^3P_1 < {}^3P_2$. Degeneracies: $g_{J=0}=1$, $g_{J=1}=3$, $g_{J=2}=5$, summing to $1+3+5=9=(2L+1)(2S+1)=3\times3$, confirming the full $^3P$ term's microstate count. Final ordering with the excited terms:

Example 2 — Applied: full term diagram of atomic nitrogen ($2p^3$), an asymmetric half-filled case. Nitrogen's open subshell is $2p^3$ — three electrons among six $2p$ spin-orbitals, so

microstates. Building the $(M_L,M_S)$ table (three unequal, non-special quantum numbers to track: three electrons each with $l=1, s=1/2$) proceeds as follows. Maximum $M_L$: place electrons at $m_l = +1, 0, -1$ (cannot put two in $m_l=+1$ with the same spin, and using all three distinct $m_l$ values with parallel spins is the unique way to get large $M_S$); the maximum $M_S = 3/2$ configuration is $m_l=+1,0,-1$ all with $m_s=+1/2$, giving $M_L=0, M_S=3/2$. Since the highest $M_S$ seen is $3/2$ and it occurs only at $M_L=0$, this signals $S=3/2, L=0$: a $^4S$ term, contributing $(2\times0+1)(2\times\tfrac32+1) = 1\times4 = 4$ states. Removing those 4, the next extremal state has $M_L=2$ (electrons at $m_l=1,1,0$ is Pauli-forbidden for identical $m_l$ with same spin, but $m_l = 1,0$ both occupied twice with opposite spins plus a third electron — working through the table carefully) at $M_S=1/2$, signaling $L=2, S=1/2$: a $^2D$ term, contributing $(2\times2+1)(2\times\tfrac12+1)=5\times2=10$ states. Removing those 10, the remaining 6 states resolve into a $^2P$ term ($L=1,S=1/2$), contributing $3\times2=6$ states. Check: $4+10+6=20$, matching Step 1 exactly. Hund's first rule: maximum $S=3/2$ picks $^4S$ as ground term — nitrogen's famous half-filled-shell ground term, with all three electrons' spins parallel (maximally exchange-stabilized), consistent with nitrogen's known extra stability (half-filled $2p^3$ resists ionization more than would be predicted by simple orbital energy trends). Hund's third rule for the ground term: $^4S$ has $L=0$, so $J=S=3/2$ only — a single level, $^4S_{3/2}$, with degeneracy $g_J = 2(3/2)+1=4$. There is no fine structure to order because there is only one $J$ value — exactly the "half-filled shell has $L=0$, rule is moot" caveat from Step 3 of the second derivation. The excited terms $^2D$ (further split into $J=3/2,5/2$) and $^2P$ (further split into $J=1/2,3/2$) lie above $^4S$ by several eV (experimentally, $^2D$ is about $2.38$ eV above $^4S$, and $^2P$ about $3.58$ eV above), with fine-structure spacings again three to four orders of magnitude smaller than the term gaps — the same energy hierarchy as carbon, now demonstrated for a different, asymmetric ($N=3$, half-filled) configuration, which cross-checks that the method does not depend on the particular electron count chosen.

Example 3 — Competition level: term and level ordering for $d^2$, then compare to the experimentally observed ground term of Ti²⁺. A $d^2$ configuration (two electrons, $l_1=l_2=2$, in a $d$ subshell with $2l+1=5$ orbitals, 10 spin-orbitals) is the analogue of $p^2$ but richer, directly relevant to transition-metal ions such as Ti$^{2+}$ ($[\text{Ar}]3d^2$). Step 1 — total microstates: $\binom{10}{2} = 45$. Step 2 — Coupling two equivalent $d$ electrons ($l=2$ each) using the non-equivalent-electron rule as a first pass would allow $L=0,1,2,3,4$ combined with $S=0,1$, i.e. terms $^1S,^3S,^1P,^3P,^1D,^3D,^1F,^3F,^1G,^3G$ — ten terms. The Pauli principle for equivalent electrons removes exactly those terms whose spatial symmetry is incompatible with the required overall antisymmetry; the surviving terms for $d^2$ (a standard, checkable result) are

i.e. singlets with even $L$ and triplets with odd $L$ (this parity rule follows because for two equivalent electrons in the same $l$-shell, the spatial part's exchange symmetry under swapping the two electrons is $(-1)^L$, and antisymmetric spin — singlet — must pair with symmetric spatial part, $(-1)^L=+1 \Rightarrow L$ even; symmetric spin — triplet — must pair with antisymmetric spatial part, $(-1)^L=-1\Rightarrow L$ odd). Step 3 — check the count: $(2L+1)(2S+1)$ for each term:

Sum: $1+5+9+9+21=45$, matching Step 1 exactly. Step 4 — Hund's first rule: the maximum-$S$ terms are the triplets, $S=1$: candidates are $^3P$ and $^3F$. Step 5 — Hund's second rule: among the $S=1$ terms, pick maximum $L$: $^3F$ ($L=3$) beats $^3P$ ($L=1$). So the ground term is $^3F$. Step 6 — Hund's third rule: $d^2$ has 2 of 10 possible electrons — less than half-filled, so $A>0$ and minimum $J$ lies lowest. For $^3F$, $L=3,S=1$, so

giving level ordering $^3F_2 < {}^3F_3 < {}^3F_4$, with ground level $^3F_2$. This is exactly the observed ground level of free Ti$^{2+}$ ($3d^2$): $^3F_2$, confirming both the term-selection logic (Hund 1 and 2) and the level-ordering logic (Hund 3) against real spectroscopic data. (In an actual crystal or complex, the five-fold orbital degeneracy of the free-ion $D$/$F$/$G$ terms is further split by the ligand field — a separate, subsequent effect layered on top of the free-ion LS terms computed here.)

Henry Norris Russell and Frederick Albert Saunders, American and Canadian-American astronomers/physicists, published their coupling scheme in 1925 based on classifying the term structure of complex atomic spectra recorded at Princeton and elsewhere — motivated by astrophysical spectroscopy of stellar atmospheres, where correctly assigning atomic energy levels was essential to interpreting absorption lines. Their scheme, though empirical in origin like Hund's rules, turned out to be the correct weak-spin-orbit limit of the full quantum-mechanical multi-electron Hamiltonian, a fact only appreciated after the Pauli principle (1925) and Dirac/Heisenberg's full quantum mechanics (1925–1926) matured.