Free-Ion Terms and Russell-Saunders Coupling

Every UV-visible absorption spectrum of a transition-metal complex — from the pale violet of $[\text{Ti(H}_2\text{O)}_6]^{3+}$ to the intense colors of $\text{MnO}_4^-$ or $[\text{CoCl}_4]^{2-}$ — is interpreted starting from the free-ion terms generated by exactly this Russell-Saunders procedure. A ligand field does not create new electronic states from nothing; it splits the terms that L-S coupling already predicts, and the number, spacing, and spin-selection rules of the resulting bands are read directly from term symbols like $^3F$, $^3P$, $^1D$. This is the language of Tanabe-Sugano diagrams, used daily to assign d-d transitions and extract ligand-field splitting parameters ($\Delta_o$) and interelectronic repulsion parameters ($B$, the Racah parameter) from measured spectra of everything from ruby (Cr³⁺-doped Al₂O₃, whose famous red color and laser transition are $^4A_2 \to {}^4T_2$ and $^4A_2\to{}^2E$ transitions rooted in Cr³⁺'s $^4F$ ground term) to biological metal centers (high-spin versus low-spin Fe centers in hemoglobin and cytochromes, whose magnetic and spectroscopic behavior is governed by which free-ion terms are accessible and how the ligand field orders them). Beyond spectroscopy, term symbols set the stage for magnetochemistry: the spin multiplicity $2S+1$ of the ground term (selected via Hund's Rules) controls the number of unpaired electrons and hence the magnetic moment predicted by the spin-only formula, used routinely to distinguish high-spin from low-spin coordination compounds by simple susceptibility measurements. In lanthanide chemistry, where spin-orbit coupling is strong enough that $L$ and $S$ genuinely combine into good $J$ quantum numbers (unlike the 3d series, where $J$-splitting is a small perturbation), the same L-S term-generation machinery is the essential first step before adding spin-orbit coupling to get the full $^{2S+1}L_J$ ground level relevant to lanthanide luminescence (Eu³⁺, Tb³⁺ phosphors) and to single-molecule-magnet design. Racah parameters extracted by comparing observed term separations to free-ion values (the nephelauxetic effect) are a standard quantitative probe of covalency in metal-ligand bonding, used across synthetic inorganic and bioinorganic chemistry to rank ligands by their electron-donating strength. Every one of these applications inherits its vocabulary — $^3F$, $^1G$, $^2D$ — directly from the microstate-counting and peeling procedure developed here, and every one of them depends on first getting the term list right before any physical interpretation (Hund's-rule ground-state selection, ligand-field splitting into Term Symbols for d-Electron Configurations) can even be posed.

Ask the tutor: How do I apply Hund's rules to pick the ground-state term once I have the full list, and why does that same list get split further by a ligand field?

Start with a bare configuration, $np^2$ or $nd^2$, and ask: how many genuinely different ways can two indistinguishable electrons occupy these orbitals, respecting the Pauli exclusion principle? Each electron in a $d$ subshell ($\ell = 2$) has five possible $m_\ell$ values ($-2,-1,0,1,2$) and two possible $m_s$ values ($\pm\tfrac12$), giving $5 \times 2 = 10$ one-electron spin-orbitals. Choosing 2 of these 10 for two indistinguishable electrons (no two electrons can share all four quantum numbers) gives

distinct microstates. Each microstate is a legitimate, Pauli-allowed arrangement, but they are not all the same energy, because electron-electron Coulomb repulsion depends on the spatial correlation between the two electrons — two electrons crowded into orbitals with large, overlapping $|m_\ell|$ values that reinforce each other's angular distribution repel differently than two electrons in orbitals that keep them apart. Grouping microstates that belong to the same total angular momentum state is exactly what turns the unmanageable 45-way degeneracy into a short, physically meaningful list of terms. The organizing quantities are the total orbital and spin angular momenta. For light-to-medium atoms and ions (roughly through the 3d transition series), electron-electron repulsion is much stronger than any individual electron's spin-orbit coupling. In that regime, all the individual orbital angular momenta $\vec\ell_i$ couple together first, into a resultant $\vec L = \sum_i \vec\ell_i$, and independently all the individual spins $\vec s_i$ couple into a resultant $\vec S = \sum_i \vec s_i$; only afterward does the (weak) spin-orbit interaction couple $\vec L$ and $\vec S$ together into $\vec J$. This two-stage coupling scheme — orbital momenta together, spin momenta together, then combine — is the Russell-Saunders (or L-S) coupling scheme, and it is an approximation whose validity is exactly the statement that $\Delta E_{\text{repulsion}} \gg \Delta E_{\text{spin-orbit}}$. A term is a set of microstates sharing the same $L$ and same $S$ — hence the same electron-repulsion energy, to the accuracy that L-S coupling is valid. Because $L$ and $S$ are vector sums of the underlying angular momenta, they are quantized in the usual way: for two electrons with orbital quantum numbers $\ell_1,\ell_2$, the allowed total-orbital quantum numbers run

and similarly $S$ runs from $s_1+s_2$ down to $|s_1-s_2|$ in integer steps. Each allowed $L$ carries $2L+1$ orientations ($M_L = -L,\dots,+L$) and each allowed $S$ carries $2S+1$ orientations ($M_S = -S,\dots,+S$), so a term of given $L,S$ has total degeneracy $(2L+1)(2S+1)$ — this is the "orbital times spin multiplicity" count that must add up to the total microstate number. Terms are labeled with the historical atomic-spectroscopy letter code for $L$:

(note the unfortunate but standard collision: term-symbol "S" for $L=0$ versus spin quantum number $S$ — context always disambiguates). The term symbol itself is written

read as, e.g., "triplet P" for $^3P$ (here $S=1$, so $2S+1=3$; $L=1$). Physically, $2S+1$ counts spin-multiplicity — the number of ways the total spin vector can project along a chosen axis — and this is exactly the number of near-degenerate spin sublevels you would resolve by turning on a magnetic field, or (with spin-orbit coupling) the number of $J$-levels a term splits into, when $L\ge S$.

That figure shows the essential subtlety: naive vector addition of $\ell_1,\ell_2$ and $s_1,s_2$ overgenerates possible $(L,S)$ pairs; the Pauli principle for equivalent electrons (same $n,\ell$) kills some combinations. The systematic, fool-proof way to get the right answer — the one used throughout this article — is not clever vector arithmetic but brute-force enumeration of microstates followed by a mechanical extraction procedure. That is the derivation below.

The essential relationships you need are just three counting statements plus the coupling definitions already given:

where: $\ell$ is the orbital angular momentum of the subshell ($1$ for p, $2$ for d), $n$ is the number of electrons in that subshell, and $2(2\ell+1)$ is the number of one-electron spin-orbitals available. This counts all Pauli-allowed arrangements, with no reference yet to $L$ or $S$.

where: $L$ and $S$ are the term's orbital and spin quantum numbers; $g_{\text{term}}$ is the number of microstates belonging to that term (its total degeneracy, before spin-orbit splitting). This is the "size of the rectangle" you peel from the $M_L$–$M_S$ table.

where: the sum runs over every distinct term generated by the configuration $n\ell^{\,n}$. This is the master consistency check — every derivation below is only trustworthy if this sum reproduces the microstate count exactly.

Reading: the first equation tells you how big the bookkeeping problem is; the second tells you how big each labeled group within it is; the third is the audit that guarantees you have found all the terms and double-counted none.

Example 1 — Foundational: terms of p¹ (a single p electron, e.g. B atom or Al ground configuration). A single electron has $\ell=1,\ s=\tfrac12$ trivially, so $L=1,S=\tfrac12$ with no coupling ambiguity (nothing to add together). The only term is $^2P$ (doublet P), degeneracy $(2\cdot1+1)(2\cdot\tfrac12+1) = 3\times2 = 6$. Microstate count directly: $\binom{6}{1}=6$. Check:

Limiting case: for a single electron in any subshell $n\ell^1$, there is only ever one term, $^{2}\,({\rm letter}\ \ell)$, since there is nothing to couple — this is the trivial floor the whole machinery must reduce to, and it does.

Example 2 — Applied: terms of p³ (e.g. free N atom, or O²⁻ ligand valence configuration, or high-spin d⁵ formal analogy in the p-block). Three electrons among 6 spin-orbitals: $N=\binom{6}{3}=20$ microstates. Building the $M_L$–$M_S$ table (three $m_\ell\in\{-1,0,1\}$ slots, Pauli-respecting) and peeling gives three terms:

Degeneracies: $^4S$ has $(1)(4)=4$; $^2D$ has $(5)(2)=10$; $^2P$ has $(3)(2)=6$. Audit:

matching $\binom{6}{3}=20$ exactly. This is a genuinely asymmetric, non-trivial check (three unequal-degeneracy terms, none of them a round or "obvious" number), and it is the configuration behind Hund's-rule reasoning for N ($^4S$ ground term, half-filled-shell stabilization) as well as the valence hole-configuration bookkeeping used when treating O²⁻ or F⁻ ligand orbitals in ligand-field theory.

Example 3 — Exam/competition level: terms of d³ (e.g. Cr³⁺, V²⁺ free ions), using the hole/electron correspondence as a cross-check. Direct enumeration for d³ is a $\binom{10}{3}=120$-microstate problem — large enough that hand peeling is tedious, which is exactly why the hole formalism is worth knowing as an independent check: a $d^n$ configuration gives the same list of terms as $d^{10-n}$, because a "hole" (missing electron) in an otherwise full shell couples its angular momentum in the same way an electron does (the filled shell itself is a $^1S$ term, $L=S=0$, contributing nothing). So d³ must give the same terms as d⁷. The known, standard term list for d² was derived above: $^1S,{}^1D,{}^3P,{}^1G,{}^3F$ (5 terms, 45 states, for $\binom{10}{2}=45$). For d³ (120 microstates), full derivation (by the same $M_L$–$M_S$ peeling procedure, carried out over all nine $M_L$ values from $-6$ to $+6$ and four $M_S$ values from $-\tfrac32$ to $+\tfrac32$) yields 8 terms:

with degeneracies $28,\,12,\,22,\,18,\,14,\,10,\,10,\,6$. Audit:

matching $\binom{10}{3}=120$ exactly. Notice the two distinct $^2D$ terms — an example of a term label recurring more than once for the same configuration (they are different states, both with $L=2,S=\tfrac12$, but arising from different combinations of microstates; they remain degenerate at the free-ion electrostatic level only if no further term-distinguishing interaction is included, but are formally counted as separate terms in the enumeration). This is the term list relevant to Cr³⁺ (d³, the classic ruby/emerald chromophore), whose ground term ($^4F$, selected by Hund's Rules) and its ligand-field-split sublevels are the direct ancestor of the Tanabe–Sugano treatment of Cr³⁺ spectra.