Atomic Term Symbols

An atomic term symbol is a compact label for angular momentum, spin multiplicity, and often parity. In its familiar Russell–Saunders form,

$${}^{2S+1}L_J,$$

it can identify an entire pattern of magnetic behavior and transition constraints in a few characters. It is not a wavefunction, an electron configuration, or a guarantee that $L$ and $S$ are exact quantum numbers.

The notation is useful precisely because it records a controlled approximation. A good reader should be able to decode a symbol, test whether its angular momenta are mutually consistent, recover parity from the configuration, and recognize when configuration or coupling admixture makes the label approximate.

Required background. Atomic orbitals define the one-electron functions, angular-momentum coupling supplies the triangle rules, and Pauli exclusion restricts equivalent-electron states.

Configuration, Term, Level, and State

Atomic spectroscopy uses a hierarchy of labels. The distinctions matter because different interactions resolve different layers.

Object	Labels in an $LS$ description	What is grouped together
configuration	$\mathcal C=(n_1\ell_1)^{N_1}(n_2\ell_2)^{N_2}\cdots$	states with the same orbital occupations
term	$\gamma\mathcal C\;{}^{2S+1}L^{\pi}$	states with common $L$, $S$, and parity
level	$\gamma\mathcal C\;{}^{2S+1}L^{\pi}_J$	a fixed total electronic $J$
magnetic state	$\lvert\gamma\mathcal C LSJM_J\rangle$	a fixed projection $M_J$

The auxiliary label $\gamma$ stands for whatever else is needed to distinguish repeated terms: parent terms, seniority, configuration labels, or an enumeration index. In informal usage, “term symbol” often means the full level label including $J$. Strictly, ${}^{2S+1}L$ denotes a term and the subscript $J$ selects one fine-structure level of that term.

The $1s2p$ configuration has $12$ antisymmetric two-electron states. In $LS$ coupling they separate into a three-state ${}^1P^\circ$ term and a nine-state ${}^3P^\circ$ term. The latter contains $J=0,1,2$ levels with $1$, $3$, and $5$ magnetic states, respectively.

The hierarchy also clarifies terminology for spectral features. A transition array connects configurations, a multiplet connects terms, a line connects levels, and a resolved Zeeman or hyperfine component connects individual states. Real spectra do not always resolve every layer.

The Grammar of an LS Label

A sufficiently explicit label may be written schematically as

$$\gamma\,\mathcal C\; {}^{2S+1}L^{\pi}_J.$$

Each piece answers a different question:

Symbol	Meaning	Typical check
$\mathcal C$	dominant electron configuration	occupations obey Pauli exclusion
$2S+1$	spin multiplicity	$S=(\text{multiplicity}-1)/2$
$L$	total electronic orbital angular momentum	capital letter maps to a nonnegative integer
$J$	total electronic angular momentum	$\lvert L-S\rvert\leq J\leq L+S$
$\pi$	total electronic parity	$\pi=(-1)^{\sum_i\ell_i}$ for a configuration
$\gamma$	additional ancestry or state identifier	needed when the other labels recur

Nuclear spin is not included. Coupling electronic $J$ to nuclear $I$ produces hyperfine $F$, treated on Hyperfine Structure. Likewise, $M_J$ labels a magnetic substate but is normally omitted from the zero-field level name.

Total L, S, and J

In $LS$ coupling, individual orbital and spin angular momenta are added separately:

$$\begin{aligned} \mathbf L&=\sum_i\mathbf \ell_i,\\ \mathbf S&=\sum_i\mathbf s_i,\\ \mathbf J&=\mathbf L+\mathbf S. \end{aligned}$$

The corresponding eigenvalues are

$$\begin{aligned} \mathbf L^2&=\hbar^2L(L+1),\\ \mathbf S^2&=\hbar^2S(S+1),\\ \mathbf J^2&=\hbar^2J(J+1). \end{aligned}$$

For fixed $L$ and $S$, angular-momentum addition permits

$$J=|L-S|, |L-S|+1,\ldots,L+S.$$

Each $J$ level contains

$$2J+1$$

magnetic states before hyperfine interactions or external fields are resolved. Across the complete $LS$ term,

$$\sum_{J=|L-S|}^{L+S}(2J+1) =(2L+1)(2S+1).$$

This identity is an excellent consistency check: changing from the uncoupled $|LM_L;SM_S\rangle$ basis to the coupled $|LS;JM_J\rangle$ basis cannot change the dimension of the space.

Spectroscopic Letters

The total orbital quantum number is written as a capital letter:

$L$	0	1	2	3	4	5	6	7
term letter	$S$	$P$	$D$	$F$	$G$	$H$	$I$	$K$

The letter $J$ is skipped in this sequence to avoid confusion with total angular momentum. Lowercase $s,p,d,f,\ldots$ labels a one-electron subshell with angular momentum $\ell$; uppercase $S,P,D,F,\ldots$ labels the total $L$ of all electrons. Thus a $p^2$ configuration can contain an $S$, $P$, or $D$ term.

Multiplicity

The left superscript is

$$2S+1,$$

called the spin multiplicity. Common names are

$S$	$2S+1$	Name
$0$	$1$	singlet
$1/2$	$2$	doublet
$1$	$3$	triplet
$3/2$	$4$	quartet
$2$	$5$	quintet

Multiplicity counts the possible $M_S$ values of spin $S$. It does not generally equal the number of magnetic states of a level, which is $2J+1$. It also need not equal the number of $J$ levels in a term: the latter is

$$2\min(L,S)+1.$$

Only when $L\geq S$ does that number equal $2S+1$. For instance, ${}^5S$ has multiplicity five but only one level, ${}^5S_2$, because $L=0$ forces $J=S=2$.

Parity Notation

Each one-electron orbital has parity $(-1)^\ell$. A configuration therefore has total electronic parity

$$\pi =\prod_i(-1)^{\ell_i} =(-1)^{\sum_i\ell_i}.$$

Closed subshells contribute even parity, so in practice only open subshells need to be counted. Standard atomic notation usually leaves even parity unmarked and appends a degree sign to an odd-parity term:

$$\begin{aligned} {}^3P_2 &\quad &&\text{even parity},\\ {}^3P^\circ_2 &\quad &&\text{odd parity}. \end{aligned}$$

Databases may use an asterisk in plain-text output because a superscript degree sign is unavailable. Some tables instead print explicit $e$ and $o$ parity labels, especially when no reliable $LS$ assignment exists.

The term letter does not determine parity. Two $p$ electrons give

$$\pi(p^2)=(-1)^{1+1}=+1,$$

so a $p^2\;{}^3P_J$ level is even. By contrast, an $sp\;{}^3P^\circ_J$ level is odd because $0+1$ is odd. Confusing the letter $P$ with parity is one of the most common notation errors.

For the usual electromagnetic atomic Hamiltonian, parity and $J$ often remain exact even when configuration mixing makes $L$ and $S$ approximate. External electric fields can mix opposite parities; Stark Effect in Atoms develops that failure mode.

Helium Examples

Helium shows how configuration, antisymmetry, terms, and levels fit together.

Ground configuration

The $1s^2$ ground configuration has $L=0$ and even parity. Because both electrons occupy the same spatial orbital, Pauli antisymmetry permits only the spin singlet. The level is

$$1s^2\;{}^1S_0.$$

A hypothetical $1s^2\;{}^3S_1$ would have a symmetric spin state and the same symmetric spatial occupation, so it is excluded.

Excited 1s2s configuration

The electrons occupy nonequivalent orbitals, and both spin couplings occur:

$$1s2s\;{}^1S_0, \qquad 1s2s\;{}^3S_1.$$

Both are even because $\ell_{1s}+\ell_{2s}=0$. Exchange and correlation give the singlet and triplet different energies; that energy ordering is not encoded in the symbols themselves.

Excited 1s2p configuration

Here $L=1$ and parity is odd. The allowed terms and levels are

$$\begin{aligned} 1s2p\;{}^1P^\circ_1, \qquad 1s2p\;{}^3P^\circ_{0,1,2}. \end{aligned}$$

The singlet contributes $2J+1=3$ magnetic states. The triplet contributes $1+3+5=9$, accounting for all $12$ states of the configuration. For a spin-independent electric-dipole operator between pure $LS$ states, $\Delta S=0$ exactly within that approximation, which is why singlet and triplet systems have sharply different line strengths. The precise hierarchy of exact and approximate rules is given in Atomic Selection Rules.

Alkali Examples

An alkali atom has one valence electron outside closed subshells. The core has $L=S=J=0$ and even parity, so the term labels reduce to those of the valence electron:

$$\begin{array}{ccl} ns &\longrightarrow& ns\;{}^2S_{1/2},\\ np &\longrightarrow& np\;{}^2P^{\circ}_{1/2,3/2},\\ nd &\longrightarrow& nd\;{}^2D_{3/2,5/2}. \end{array}$$

The sodium ground level, for example, is $3s\;{}^2S_{1/2}$. Its resonance-line upper configuration $3p$ produces the two fine-structure levels $3p\;{}^2P^{\circ}_{1/2}$ and $3p\;{}^2P^{\circ}_{3/2}$. Their separation is the familiar alkali doublet. Alkali Atoms explains the effective one-electron picture, while Fine Structure develops the splitting mechanism.

This simplicity is special. For multiple open subshells, the same configuration can support several $L$ and $S$ values, and equivalent-electron antisymmetry removes some combinations that naive vector addition would allow.

Equivalent Electrons: A Counting Check

Pauli Principle in Atoms derives the exchange-parity restriction that selects the allowed terms. Here the result is used as a notation and state-counting check.

For a $p^2$ configuration, one $p$ subshell contains six spin-orbitals. Choosing two gives

$$\binom{6}{2}=15$$

antisymmetric determinants. The allowed $LS$ levels are

$${}^1S_0, \qquad {}^1D_2, \qquad {}^3P_{0,1,2}.$$

Their magnetic-state counts are

$$1+5+(1+3+5)=15.$$

The count closes, and every term has even parity because the two $p$ electrons contribute $(-1)^2$. This example is useful as a diagnostic, but it does not replace the antisymmetrized construction that determines which terms survive.

LS Coupling, jj Coupling, and Intermediate Coupling

The superscript-multiplicity notation assumes that $L$ and $S$ are useful labels. That is most natural when residual electrostatic interactions organize the spectrum more strongly than individual-electron spin–orbit interactions.

Regime	Coupling order	Natural label
$LS$ coupling	$\sum_i\mathbf\ell_i\to\mathbf L$, $\sum_i\mathbf s_i\to\mathbf S$, then $\mathbf L+\mathbf S\to\mathbf J$	$\gamma\mathcal C\;{}^{2S+1}L^\pi_J$
$jj$ coupling	each $\mathbf\ell_i+\mathbf s_i\to\mathbf j_i$, then $\sum_i\mathbf j_i\to\mathbf J$	configuration with $j_i$ subscripts and coupled $J$
intermediate coupling	diagonalize across several basis labels with the same exact symmetries	dominant-component label plus composition data

A schematic two-electron $jj$ basis is

$$\left| (\ell_1s_1)j_1, (\ell_2s_2)j_2; JM_J \right\rangle .$$

For example, a closed pair in a relativistic $p_{1/2}$ subshell may be labeled $(p_{1/2}^{\,2})_0$. There is no generally useful spin multiplicity in that label because $L$ and $S$ are not coupled first. The full construction and its Pauli restrictions are developed in jj Coupling.

Real atoms often lie between the limits. A level of exact $J$ and parity may have an expansion

$$|\Psi_{J\pi}\rangle = \sum_r c_r |\gamma_r\mathcal C_rL_rS_r;J\pi\rangle .$$

If one weight $|c_r|^2$ dominates, its $LS$ term is a useful short name. If several weights are comparable, a single symbol hides important physics: nominally spin-forbidden lines can gain strength, Landé factors shift away from pure-$LS$ values, and labels may exchange character through an avoided crossing. The Hamiltonian has not changed when the basis notation changes; only the most informative description has.

In the NIST Atomic Spectra Database, configuration, term, $J$, and energy occupy separate fields. First check parity and the angular-momentum triangle rule; then apply the equivalent-electron restrictions. A valid symbol still need not specify an energy ordering: Hund’s rules add an approximate energetic argument.

Exercises

Exercise 1: Decode and check a quartet

Decode ${}^4F^{\circ}_{9/2}$. Give $S$, $L$, $J$, parity, the allowed $J$ values of the term, and the number of magnetic states in the displayed level.

Solution

The multiplicity gives

$$2S+1=4 \quad\Longrightarrow\quad S=\frac{3}{2}.$$

The letter $F$ means $L=3$, and the subscript gives $J=9/2$. The degree sign marks odd parity. Angular-momentum addition permits

$$J=\left|3-\frac32\right|,\ldots, 3+\frac32 = \frac32,\frac52,\frac72,\frac92.$$

The displayed $J=9/2$ level has

$$2J+1=10$$

magnetic states.

Exercise 2: Multiplicity is not level degeneracy

Compare ${}^5S_2$ and ${}^3P_2$. State the multiplicity, number of fine-structure $J$ levels in the complete term, and number of $M_J$ states in the displayed level.

Solution

For ${}^5S_2$, $S=2$ and $L=0$. The multiplicity is five, but only $J=2$ is possible, so the term has one fine-structure level. That level has $2J+1=5$ magnetic states.

For ${}^3P_2$, $S=1$ and $L=1$. Its multiplicity is three and the complete term has $J=0,1,2$, hence three fine-structure levels. The displayed $J=2$ level has five magnetic states. The same number five appears for two different reasons in these examples.

Exercise 3: Electron number and multiplicity

Can a free atom with an even number of electrons have an electronic ${}^4F$ term? Explain without constructing its configuration, and distinguish the electronic result from hyperfine $F$.

Solution

Adding an even number of spin-$1/2$ electrons gives integer $S$, so $2S+1$ must be odd. A quartet has $S=3/2$ and requires an odd number of electrons. Orbital angular momenta are integers, so electronic $J$ has the same integer/half-integer character as $S$. Coupling to nuclear spin can change the integer character of total hyperfine $F$; nuclear spin is not part of the electronic term symbol.

Exercise 4: A mixed level

Suppose a normalized odd-parity $J=1$ eigenstate is approximated by

$$|\Psi\rangle = \sqrt{0.82}\;|{}^1P^\circ_1\rangle + \sqrt{0.18}\;|{}^3P^\circ_1\rangle.$$

Which labels are exact in a rotationally and parity-invariant Hamiltonian, and how should the level be named?

Solution

Both basis components have $J=1$ and odd parity, so those labels can remain exact. The state is not an eigenstate of $S^2$: singlet and triplet are approximate component labels. If a short $LS$ name is needed, the dominant-component convention gives ${}^1P^\circ_1$, accompanied by the composition $82\%$ singlet and $18\%$ triplet. Omitting the composition would hide the mechanism that can open an intercombination transition.

References

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