Term Symbols and Atomic States

Term symbols are not a historical curiosity confined to hydrogen-like textbook problems — they remain the operational language of atomic and near-atomic spectroscopy across chemistry and physics. Emission and absorption line assignments in astrophysical spectra (stellar atmospheres, nebular emission lines, solar coronal lines) are routinely labeled by term symbol, and the famous "forbidden lines" of ionized oxygen and nitrogen that reveal the composition of planetary nebulae are transitions between terms (such as $^1D_2 \to {}^3P_1$ in [O III]) that are electric-dipole forbidden by the selection rules built on term quantum numbers — a topic that leads directly into Atomic Spectra and Selection Rules. In lanthanide and actinide chemistry, ground-state term symbols (via the Hund's-rules-plus-$J$ machinery of this article) directly predict the magnetic moments used to characterize $f$-block coordination complexes and materials, since the total angular momentum $J$ (not simply spin, as for most $3d$ transition metals) governs the effective magnetic moment through the Landé $g$-factor. This underlies the design of single-molecule magnets and lanthanide-based MRI contrast agents (e.g., Gd$^3+$, an $f^7$ half-filled-shell ion with an exceptionally large $^8S_{7/2}$ ground term and correspondingly large, isotropic magnetic moment, which is precisely why gadolinium complexes dominate clinical contrast agent chemistry). In laser physics and atomic clock technology, term symbol assignments of specific fine-structure levels (identifying exact $J$ values) are essential to selecting clock transitions with minimal sensitivity to external field perturbations — the strontium and ytterbium optical lattice clocks that currently define the most precise timekeeping on Earth rely on term-symbol-labeled "clock transitions" between states of carefully chosen $J$. Photoelectron spectroscopy of transition-metal complexes and gas-phase atomic ions likewise reports ionic states by term symbol, connecting directly to configuration-interaction and multireference quantum chemistry calculations that must reproduce the correct term ordering — a stringent test of any electronic structure method's treatment of electron correlation, since Hartree–Fock alone (a single Slater Determinants configuration) cannot distinguish terms of the same configuration at all without going beyond a single determinant.

Ask the tutor: Can you show me, step by step, how to derive all the terms for a $3d^3$ configuration and identify which one is the ground term?

Consider two independent electrons, each with orbital angular momentum quantum number $\ell$ and spin $s = 1/2$. If the electrons occupied different subshells (a "non-equivalent" configuration such as $2p^13p^1$), every combination of individual quantum numbers would be allowed, and the total orbital angular momentum $L$ and total spin $S$ would range freely: $L = |\ell_1 - \ell_2|, \dots, \ell_1+\ell_2$ and $S = |s_1-s_2|, \dots, s_1+s_2$, i.e., $S = 0$ or $1$. Every $(L,S)$ pair would be a valid term, and each would appear exactly once. The complication — and the actual content of this topic — arises for equivalent electrons: two or more electrons in the same subshell, such as $2p^2$ or $3d^3$. Here the Pauli exclusion principle, encoded automatically in a Slater Determinants antisymmetrized wavefunction, forbids certain combinations of $(m_\ell, m_s)$ that would otherwise be allowed. Not every $(L,S)$ pair that naive addition of angular momenta would suggest is actually realized; some are excluded, and the count of surviving microstates must match a definite combinatorial number. A microstate is a single, fully-specified assignment of $m_\ell$ and $m_s$ quantum numbers to every electron in the open subshell, consistent with the exclusion principle (no two electrons with identical $m_\ell$ and identical $m_s$). For $n$ electrons distributed among the $4l+2$ available spin-orbitals of a subshell with orbital quantum number $l$, the total number of microstates is

For $2p^2$ ($l=1$, so $4l+2 = 6$ spin-orbitals, $n=2$ electrons): $N = \binom{6}{2} = 15$. These 15 microstates are not 15 different physical states of the atom in the sense of 15 different energies — they group into a small number of terms, each term being a set of $(2L+1)(2S+1)$ microstates degenerate under the orbital part of the Hamiltonian (though split further by spin-orbit coupling into $J$ levels). The quantity $M_L = \sum_i m_{\ell,i}$ and $M_S = \sum_i m_{s,i}$ are good quantum numbers for any microstate — they are simply sums of individual (conserved, non-interacting-electron) magnetic quantum numbers — and a term with definite $L$ contains microstates with $M_L = -L, \dots, +L$ while a term with definite $S$ contains microstates with $M_S = -S,\dots,+S$. This is the operational key to extracting terms from a raw list of microstates: build a table of microstates sorted by $(M_L, M_S)$, then repeatedly identify the highest $M_L, M_S$ box, deduce the $L, S$ of the term it must belong to, and subtract off an entire $(2L+1)\times(2S+1)$ block from the table.

Once the terms for a configuration are known, Hund's rules rank them in energy for the ground configuration (they say nothing rigorous about excited configurations, though they are often qualitatively extended). Hund's first rule maximizes $S$: parallel spins keep electrons apart via the Pauli principle (Fermi hole), reducing Coulomb repulsion. Hund's second rule, among states of the same maximal $S$, maximizes $L$: larger $L$ corresponds to electrons orbiting "in the same sense," which on average also increases their spatial separation. Hund's third rule fixes $J$: spin-orbit coupling $H_{SO} = \zeta\, \mathbf{L}\cdot\mathbf{S}$ favors $J = |L-S|$ if the subshell is less than half-filled and $J = L+S$ if more than half-filled (this comes from the sign of the effective spin-orbit constant $\zeta$, which flips with the sign of the "hole vs. particle" character of the shell). The full term symbol assembles all these quantum numbers along with the parity-free letter code for $L$ ($L = 0,1,2,3,4,\dots \to S,P,D,F,G,\dots$, unrelated to the orbital-shell letter despite the notation clash):

The left superscript $2S+1$ is called the multiplicity (1 = singlet, 2 = doublet, 3 = triplet, etc. — the number of $M_S$ values, hence the number of $J$-sublevels the term splits into under spin-orbit coupling, for $L \ge S$). The right subscript $J$ ranges over $J = L+S, L+S-1, \dots, |L-S|$ by the ordinary vector-addition rule for combining $\mathbf{L}$ and $\mathbf{S}$ into total angular momentum $\mathbf{J} = \mathbf{L}+\mathbf{S}$.

Each $J$-level within a term has degeneracy $2J+1$ (the number of $M_J$ values), and summing $2J+1$ over all $J$ in a term recovers $(2L+1)(2S+1)$, the total microstate count for that term — an essential internal consistency check.

where: $l$ is the orbital angular momentum quantum number of the subshell ($l=0,1,2,3$ for $s,p,d,f$), $4l+2$ is the number of spin-orbitals in that subshell, and $n$ is the number of equivalent electrons occupying it. Reading: this is simply "choose $n$ occupied spin-orbitals out of $4l+2$ available slots," since each spin-orbital can hold at most one electron (Pauli).

where: $m_{\ell,i} \in \{-l,\dots,+l\}$ and $m_{s,i} = \pm \tfrac12$ are the individual quantum numbers of electron $i$ in a given microstate. Reading: these totals are additive because $\hat L_z = \sum_i \hat\ell_{z,i}$ and $\hat S_z = \sum_i \hat s_{z,i}$ are one-electron-operator sums, so their eigenvalues simply add across independent electrons in a product (or antisymmetrized product) state.

where: $(2L+1)(2S+1)$ is the total microstate count belonging to that term. Reading: this is the extraction rule — the highest available $(M_L, M_S)$ combination in a microstate table must be the "corner" of exactly one term, whose full $(2L+1)\times(2S+1)$ rectangle of microstates can then be subtracted from the table.

where: $J$ is the total angular momentum quantum number, obtained from ordinary quantum addition of two angular momenta $\mathbf L$ and $\mathbf S$. Reading: each value of $J$ labels a distinct energy sublevel once spin-orbit coupling is included, with degeneracy $2J+1$.

where: $\zeta$ is the spin-orbit coupling constant for the term (positive for less-than-half-filled shells, negative for more-than-half-filled shells), and $J(J+1)$, $L(L+1)$, $S(S+1)$ are eigenvalues of $\mathbf J^2, \mathbf L^2, \mathbf S^2$ in units of $\hbar^2$. Reading: this is the Landé interval rule — it says spin-orbit splitting between adjacent $J$ and $J-1$ levels within a term scales linearly with $J$, $\Delta E = \zeta J$, a directly testable spectroscopic signature of $\mathbf L\cdot\mathbf S$ coupling.

Example 1 — Foundational: terms of $np^1$ (a single $p$ electron), verified against hydrogen's $2p^1$ excited state. A single electron in a $p$ subshell ($l=1$) has $m_\ell \in \{-1,0,1\}$ and $m_s = \pm\tfrac12$: 6 microstates, matching $N = \binom{6}{1}=6$. Since there is only one electron, $L = l = 1$ and $S = s = \tfrac12$ trivially — no combinatorics needed, no Pauli restriction bites (there's nothing for a lone electron to conflict with). The single term is $^2P$, with multiplicity $2S+1=2$ and $(2L+1)(2S+1) = 3\times 2 = 6$ microstates, exactly matching the count. This is the term symbol for the excited $2p^1$ configuration of hydrogen (ground configuration $1s^1$ has term $^2S_{1/2}$; the first excited configuration $2p^1$ has term $^2P$). Applying $J = L+S,\dots,|L-S|$: $J = 3/2, 1/2$, giving $^2P_{3/2}$ and $^2P_{1/2}$ — the famous sodium D-line doublet arises from exactly this term structure in $3p^1$, split by spin-orbit coupling into two closely spaced lines (589.0 and 589.6 nm).

Example 2 — Applied: terms of $3d^2$ (relevant to early transition-metal ions such as Ti$^{2+}$ or V$^{3+}$, both $d^2$). For $l=2$ ($d$ subshell), there are $4l+2 = 10$ spin-orbitals, so $N = \binom{10}{2} = 45$ microstates for two equivalent $d$ electrons. Tabulating $(M_L, M_S)$ (with $m_\ell \in \{-2,-1,0,1,2\}$) and peeling from the top: The maximum $M_L$ is $4$ (both electrons at $m_\ell=+2$, forced to opposite spins, $M_S=0$), signaling $L=4$ with only $M_S=0$ present at that corner — a $^1G$ term ($(2\times4+1)(1) = 9$ microstates). Removing $^1G$'s 9 states, the next maximum $M_L$ is $3$, appearing at all three $M_S = -1,0,1$ — a $^3F$ term ($(2\times3+1)(3) = 21$ microstates). Removing $^3F$'s 21 states (total removed so far: $9+21=30$, leaving $15$), the next maximum $M_L$ is $2$ at $M_S=0$ only — a $^1D$ term ($5\times1=5$ microstates, leaving $10$). Continuing, the next maximum $M_L$ is $1$ at all three $M_S$ — a $^3P$ term ($3\times3=9$ microstates, leaving $1$). The final leftover state sits at $M_L=0, M_S=0$: a $^1S$ term (1 microstate, leaving $0$). Checking the total: $9+21+5+9+1 = 45$ ✓, matching $\binom{10}{2}$ exactly. The five terms of $d^2$ are $^1G, {}^3F, {}^1D, {}^3P, {}^1S$. By Hund's rules, maximum multiplicity picks out $^3F$ (multiplicity 3) over the singlets, and since $^3F$ is the only triplet, it is immediately the ground term — no need to invoke the second rule here. With $L=3, S=1$: $J = 4,3,2$. The $d^2$ shell is less than half-filled (2 of 10 electrons), so Hund's third rule gives $J=|L-S|=2$ as lowest: the ground term of a free $d^2$ ion is $^3F_2$. (In real transition-metal complexes, crystal-field splitting typically overwhelms this free-ion term structure, but the free-ion $^3F$ ground term is exactly why Ti$^{2+}$ and V$^{3+}$ have characteristic magnetic moments close to the spin-only value for two unpaired electrons before ligand-field effects are added.)

Example 3 — Competition-level: terms of $2p^13p^1$ (non-equivalent electrons) contrasted with $2p^2$, and the trap of forgetting equivalence. Suppose one electron sits in a $2p$ orbital and another in a $3p$ orbital — these are non-equivalent electrons, since they occupy different principal shells despite the same $l=1$. Here the Pauli exclusion principle places no orbital restriction at all between the two electrons (their spatial wavefunctions differ by the principal quantum number, so no two-electron quantum state is actually forbidden by antisymmetry beyond the ordinary combination rules). Both electrons independently range over $l_1=1$ and $l_2=1$, giving $L = |1-1|,\dots,(1+1) = 0,1,2$, and $S = |{\tfrac12-\tfrac12}|,\dots,(\tfrac12+\tfrac12) = 0,1$. Every combination is realized once: terms $^1S, {}^3S, {}^1P, {}^3P, {}^1D, {}^3D$ — six terms in all. Counting microstates: $(2\times0+1)(1) + (2\times0+1)(3) + (2\times1+1)(1) + (2\times1+1)(3) + (2\times2+1)(1) + (2\times2+1)(3)$

This matches the unrestricted count for two distinguishable electrons each with 6 spin-orbital choices: $6\times6=36$ (no combinatorial "choose," since the electrons are not equivalent and swapping which one occupies $2p$ vs. $3p$ is a distinct assignment before antisymmetrization, but the antisymmetrized states are still in one-to-one correspondence with these 36 term-slots because the spatial orbitals already distinguish the electrons). Contrast sharply with $2p^2$ (equivalent electrons): there, the Pauli principle prunes this same angular-momentum-addition list down from what would naively be $^1S,{}^3S,{}^1P,{}^3P,{}^1D,{}^3D$ (36 states) to only $^1S, {}^1D, {}^3P$ (15 states) — precisely because equivalent electrons cannot occupy identical $(n,l,m_\ell,m_s)$, eliminating $^3S$, $^1P$, and $^3D$ entirely, along with parts of the surviving terms. A common mistake is to apply the non-equivalent-electron term list (from simple vector-coupling of $L$'s and $S$'s) to an equivalent-electron configuration — this manufactures 21 phantom microstates that violate the exclusion principle and predicts spurious terms like $^3S$ and $^1P$ for carbon's ground configuration, which do not exist.

The term symbol formalism is the joint product of atomic spectroscopy's long empirical tradition and the theoretical framework built by Henry Norris Russell and Frederick Saunders in 1925, refined through the angular-momentum coupling theory of quantum mechanics developed in the following years. But the rule that gives the formalism its predictive teeth for ground states — the rule for ranking terms by energy from a configuration — is due to Friedrich Hund, working from careful analysis of atomic spectral series in the 1920s.