Term Symbol Reference

A term symbol is compressed state metadata. It can report spin multiplicity, angular momentum, spatial symmetry, inversion behavior, reflection behavior, or an empirical ordering label. It does not by itself report an electron configuration, an energy, a complete wavefunction, or the Hamiltonian under which all of its labels are exact.

The safest way to read a symbol is therefore not to memorize its typography. Ask which operator each token refers to and whether that operator commutes with the Hamiltonian being used. The degree sign on an atomic term, the $g/u$ subscript of a centrosymmetric molecule, the $+/-$ superscript of a linear molecule, and an $e/f$ rovibronic label all encode different symmetry statements.

This page is a lookup and translation layer. The derivation of allowed atomic terms belongs to Atomic Term Symbols, the coupling limits belong to LS Coupling and jj Coupling, and the group-theoretic construction of molecular labels belongs to Molecular Symmetry.

Canonical Scope

This page owns:

• a compact grammar for atomic LS, atomic jj, linear-molecule, and nonlinear-molecule labels;
• a dictionary of multiplicity, parity, inversion, reflection, and projection symbols;
• examples that translate a printed symbol into physical statements;
• tests for deciding which labels are exact, approximate, or merely conventional; and
• cautions for moving between databases, papers, and effective Hamiltonians.

It does not own:

• microstate counting or antisymmetrization for equivalent electrons;
• derivations of LS-to-jj recoupling coefficients;
• molecular character tables or projection operators;
• Hund-case Hamiltonians and their matrix elements;
• line-strength formulas or complete selection-rule derivations; or
• assignments of a particular measured spectrum.

Follow the linked canonical page whenever one of those tasks matters.

Four-Pass Decoder

Read any unfamiliar label in four passes.

Pass	Question	Typical answer
physical object	Is this an orbital, configuration, term, fine-structure level, rovibronic level, or hyperfine level?	atomic $^{3}P_2$ is a level; molecular $X\,{}^2\Pi$ names an electronic term
symmetry domain	Is the system an atom, a linear molecule, or a nonlinear molecule in a specified point group?	$P$ means $L=1$ for an atom, while $\Pi$ means $
token grammar	Which angular momenta, projections, and symmetry eigenvalues are printed?	$2S+1$, $J$, $\Omega$, $g/u$, or a point-group irrep
validity	Which printed labels commute with the stated Hamiltonian, and which only identify a dominant basis component?	$J$ and parity may remain exact after $L$ and $S$ cease to be exact

A compact comparison is:

System and regime	Common skeleton	Central symmetry content
atom, LS coupling	$\gamma\mathcal C\,{}^{2S+1}L_J^\pi$	total $L$, total $S$, total $J$, spatial parity
atom, jj coupling	$\gamma[(n_1l_1)_{j_1}^{N_1}(n_2l_2)_{j_2}^{N_2}\cdots]_J^\pi$	subshell or group $j$ values coupled to total $J$, spatial parity
linear molecule	$T\,{}^{2S+1}\Lambda_\Omega$, with applicable $g/u$ and $+/-$ labels	axial projections and linear-molecule symmetry
nonlinear molecule	$\widetilde T\,{}^{2S+1}\Gamma$	spin multiplicity and a point-group irrep

Here $\gamma$ stands for additional labels needed to distinguish states, $\mathcal C$ is a configuration label, $T$ is an empirical electronic-state label such as $X$ or $A$, and $\Gamma$ is an irreducible representation. Not every slot is present in every source.

Atomic Structural Hierarchy

Atomic spectroscopy distinguishes four nested objects. Keeping them separate prevents most degeneracy and selection-rule mistakes.

Object	Schematic label	What is fixed
configuration	$\mathcal C=(n_1l_1)^{N_1}(n_2l_2)^{N_2}\cdots$	orbital occupations
term	$\gamma\mathcal C\,{}^{2S+1}L^\pi$	configuration, total $L$, total $S$, parity, and any extra labels
level	$\gamma\mathcal C\,{}^{2S+1}L_J^\pi$	a particular total $J$ within the term
magnetic state	$\gamma\mathcal C\,{}^{2S+1}L_J^\pi M_J$	a particular projection $M_J$

For an isolated field-free atom with rotational invariance,

$$\hat{\mathbf J}^2|\alpha JM_J\rangle = \hbar^2J(J+1)|\alpha JM_J\rangle,$$ $$\hat J_z|\alpha JM_J\rangle = \hbar M_J|\alpha JM_J\rangle.$$

The level contains $2J+1$ magnetic states before an external field or another anisotropy resolves them:

$$M_J=-J,-J+1,\ldots,J.$$

The symbol $\alpha$ or $\gamma$ is deliberately unspecific. Two distinct levels can have the same $J$ and parity, so those labels are not always a complete state identifier.

Term, level, and state in data tables

Database columns often separate configuration, term, and $J$. A row displaying

$$3p\,{}^2P^\circ, \qquad J=\frac32$$

refers to the level conventionally written

$$3p\,{}^2P^\circ_{3/2}.$$

In formatted NIST Atomic Spectra Database output, a degree sign marks odd parity. In ASCII output, the same information is represented by an asterisk. The typography changes; the parity does not.

Atomic LS Grammar

In the Russell–Saunders or LS limit, individual orbital angular momenta first form a total orbital angular momentum and individual spins form a total spin:

$$\mathbf L=\sum_i\mathbf l_i, \qquad \mathbf S=\sum_i\mathbf s_i.$$

Those totals then couple to

$$\mathbf J=\mathbf L+\mathbf S.$$

A full level label can be organized as

$$\gamma\mathcal C\,{}^{2S+1}L_J^\pi.$$

Token	Meaning	Diagnostic
$\gamma$	parent terms, seniority, or other distinguishing information	needed when the visible labels are not unique
$\mathcal C$	electron configuration	occupations, not a term
$2S+1$	spin multiplicity	singlet, doublet, triplet, quartet, and so on
$L$	total electronic orbital angular momentum	encoded by an uppercase spectroscopic letter
$J$	total electronic angular momentum	labels a fine-structure level
$\pi$	spatial inversion parity	even or odd

Spectroscopic letters

The atomic letter code is:

$L$	0	1	2	3	4	5	6	7	8
term letter	S	P	D	F	G	H	I	K	L

The letter J is skipped to avoid confusion with total angular momentum $J$. Lowercase $s,p,d,f,\ldots$ label one-electron orbital angular momentum $l=0,1,2,3,\ldots$; uppercase S, P, D, F, $\ldots$ label the many-electron total $L$.

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

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

Thus a $^3P$ term has $S=1$ and $L=1$, and contains

$${}^3P_0,\qquad {}^3P_1,\qquad {}^3P_2.$$

This statement identifies possible $J$ values. It does not determine their energy ordering.

Multiplicity is not total degeneracy

The left superscript is

$$2S+1,$$

the number of spin projections $M_S$ for a pure spin-$S$ object. It is not the number of fine-structure levels and not the degeneracy of one $J$ level.

For a pure LS term, the uncoupled dimension is

$$(2L+1)(2S+1).$$

The same dimension appears after coupling:

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

For $^3P$,

$$(2L+1)(2S+1)=3\times3=9,$$

while the three fine-structure levels have degeneracies

$$1,\quad 3,\quad 5.$$

Atomic parity

The spatial parity of an electron configuration is

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

Closed subshells contribute even powers and therefore do not change the result. Common notations are:

Parity	Formatted term	Other common forms
even	$^3P_2$	$^3P_2^{\mathrm e}$ or $^3P_2^+$ when the convention is explicit
odd	$^3P_2^\circ$	$^3P_2^{\mathrm o}$; an ASCII database may use 3P*2 or separate the $J$ column

The absence of a degree sign conventionally means even parity in many atomic tables. That convention should not be transferred blindly to nuclear, molecular, or condensed-matter notation.

Atomic examples

Label	Reading
He $1s^2\,{}^1S_0$	$S=0$, $L=0$, $J=0$, even parity
He $1s2p\,{}^3P^\circ_2$	$S=1$, $L=1$, $J=2$, odd parity
Na $3p\,{}^2P^\circ_{3/2}$	one valence $p$ electron, $S=1/2$, $L=1$, $J=3/2$, odd parity
C $2p^2\,{}^3P_0$	triplet P level, $J=0$, even parity because $1+1$ is even
an alkali $ns\,{}^2S_{1/2}$ level	doublet S, $J=1/2$, even parity

For the Na example,

$$2S+1=2 \quad\Longrightarrow\quad S=\frac12,$$ $$P\quad\Longrightarrow\quad L=1, \qquad J=\frac32.$$

The degree sign is independently required by

$$\pi=(-1)^1=-1.$$

The configuration is what makes the parity check possible; the bare $^2P_{3/2}$ label does not identify which radial orbital is occupied.

Atomic jj and Intermediate Coupling

The LS symbol records the order

$$(\mathbf l_1+\mathbf l_2+\cdots) + (\mathbf s_1+\mathbf s_2+\cdots) \longrightarrow \mathbf J.$$

In the jj limit, each electron or relativistic subshell instead forms

$$\mathbf j_i=\mathbf l_i+\mathbf s_i,$$

and the resulting $j_i$ values are coupled:

$$\mathbf J=\sum_i\mathbf j_i.$$

The notation must therefore expose different intermediate angular momenta.

One-electron and subshell labels

For an electron with spin $1/2$,

$$j=l\pm\frac12,$$

except that $l=0$ has only $j=1/2$. A relativistic subshell is written

$$nl_j,$$

so a $p$ shell splits into $p_{1/2}$ and $p_{3/2}$ subshells. An equivalent electron group can be written

$$(nl_j)^N.$$

The subscript on $p_{3/2}$ is a one-electron $j$, whereas the final subscript outside a coupled configuration is the total $J$. Their positions are part of the grammar.

Coupled-group labels

For two groups with angular momenta $J_1$ and $J_2$,

$$|J_1-J_2| \le J\le J_1+J_2.$$

A schematic level label is

$$\bigl[(nl_{j_1})^{N_1}_{J_1} (n'l'_{j_2})^{N_2}_{J_2}\bigr]_J^\pi.$$

Authors omit brackets or redundant intermediate labels when the coupling order is clear. For example,

$$(p_{1/2}^2)_0$$

states that two equivalent electrons occupy the $j=1/2$ subshell and couple to total $J=0$. The Pauli principle still restricts which total values are allowed; the triangle rule alone is not sufficient for equivalent electrons.

LS and jj labels are basis statements

The same exact field-free level can be expanded in either basis:

$$|\Psi_{\alpha JM_J}\rangle = \sum_{\gamma LS} c_{\gamma LS} |\gamma LSJM_J\rangle,$$

or

$$|\Psi_{\alpha JM_J}\rangle = \sum_{\eta\{j_i\}} d_{\eta\{j_i\}} |\eta\{j_i\}JM_J\rangle.$$

The basis changes; the exact $J$, $M_J$, and parity labels do not. A named LS or jj label is exact only in an ideal coupling limit. In intermediate coupling, it normally identifies the largest component or a historically assigned parent.

Label in a rotationally invariant, parity-conserving atomic model	Typical status
$J$ and $M_J$	exact in zero field, with $M_J$ basis-dependent inside a degenerate level
parity $\pi$	exact if the Hamiltonian preserves spatial inversion
$L$ and $S$	exact in the nonrelativistic electrostatic LS limit; approximate with spin-dependent relativistic terms
individual $j_i$ or subshell $J_i$	exact only in the corresponding jj or group-coupling limit
configuration	approximate when configuration interaction is retained

A responsible assignment can report a leading component,

$$|\Psi\rangle = \sqrt{0.62}\,|\gamma\,{}^3P_2\rangle + \sqrt{0.31}\,|\gamma'\,{}^1D_2\rangle + \cdots,$$

while calling the level “predominantly $^3P_2$.” The percentages are basis-dependent and should be accompanied by the basis and calculation.

Molecular Electronic-State Grammar

For a molecule, the spatial part of an electronic state is classified by an irreducible representation of the molecular symmetry group appropriate to the geometry and model. The atomic letter $L$ is generally not a molecular quantum number because the nuclear framework is not spherically symmetric.

The two principal grammars are:

$$\text{linear molecule:} \qquad T\,{}^{2S+1}\Lambda_\Omega,$$

with applicable inversion and reflection labels, and

$$\text{nonlinear molecule:} \qquad \widetilde T\,{}^{2S+1}\Gamma.$$

The empirical state label $T$, the symmetry label, and the spin multiplicity answer different questions. A state may require all three.

Linear Molecules

Choose the internuclear or molecular axis as the body-fixed $z$ axis. The electronic orbital projection is

$$\Lambda = \frac{|\mathbf L\cdot\hat{\mathbf z}|}{\hbar},$$

and the electronic spin projection is

$$\Sigma = \frac{\mathbf S\cdot\hat{\mathbf z}}{\hbar}.$$

In a Hund-case-(a) description, the projection of total electronic angular momentum is

$$\Omega = |\Lambda+\Sigma|.$$

These are body-fixed projections, not the atomic magnitudes $L$, $S$, and $J$.

Projection letters

$\Lambda$	0	1	2	3	4
electronic term	$\Sigma$	$\Pi$	$\Delta$	$\Phi$	$\Gamma$
one-electron orbital	$\sigma$	$\pi$	$\delta$	$\phi$	$\gamma$

The uppercase letter labels the many-electron electronic state. A lowercase letter labels a molecular orbital. Thus $(1\pi)^3$ is part of a configuration, whereas $^2\Pi$ is a term symmetry.

Multiplicity and spin–orbit components

The left superscript remains

$$2S+1.$$

For a $^2\Pi$ term,

$$S=\frac12, \qquad \Lambda=1,$$

so the case-(a) components are

$$\Omega=\frac12,\frac32.$$

They may be written

$${}^2\Pi_{1/2}, \qquad {}^2\Pi_{3/2}.$$

An $\Omega$ subscript is most informative when the molecular-axis projection is nearly good. If rotation or other couplings strongly mix $\Omega$, the subscript becomes an approximate component label.

Empirical state letters

The conventional electronic-state prefix is separate from the term symbol:

Prefix	Conventional use
$X$	ground electronic state
$A,B,C,\ldots$	excited states of the same multiplicity as $X$, in increasing energy order
$a,b,c,\ldots$	excited states of a different multiplicity

For polyatomic molecules, a tilde is customarily added, $\widetilde X,\widetilde A,\ldots$, to distinguish empirical state names from point-group irreps. Historical literature contains exceptions and reassignments, so the prefix is an empirical identifier, not a theorem about the Hamiltonian.

Inversion labels g and u

If the fixed-nuclei molecular Hamiltonian has a center of inversion, an electronic eigenfunction may satisfy

$$\hat i\,\psi_{\mathrm{el}} = +\psi_{\mathrm{el}} \qquad \text{or} \qquad \hat i\,\psi_{\mathrm{el}} = -\psi_{\mathrm{el}}.$$

The corresponding labels are

$$g\quad\text{(gerade)}, \qquad u\quad\text{(ungerade)}.$$

For a homonuclear diatomic, the electronic term may therefore be written

$${}^{2S+1}\Lambda_g \qquad\text{or}\qquad {}^{2S+1}\Lambda_u.$$

A heteronuclear diatomic such as CO has no inversion center that leaves the Hamiltonian invariant, so $g/u$ is not an allowed exact label. “Bonding” and “antibonding” are not replacements for $g$ and $u$.

Reflection labels plus and minus

For a $\Sigma$ electronic state, reflection through any plane containing the molecular axis can return the wavefunction with eigenvalue $+1$ or $-1$:

$$\hat\sigma_v\psi_{\mathrm{el}} = \pm\psi_{\mathrm{el}}.$$

This is denoted by a right superscript:

$${}^{2S+1}\Sigma^+, \qquad {}^{2S+1}\Sigma^-.$$

For $\Lambda>0$, the degenerate $+\Lambda$ and $-\Lambda$ partners are interchanged by such a reflection, so a single $+/-$ electronic-term superscript is generally not used in the same way. Rotation and other interactions can later produce parity doublets, which are instead labeled $e/f$.

Representative linear-molecule labels

Label	Decode
$\mathrm{H}_2^+\;X\,{}^2\Sigma_g^+$	ground electronic state; $S=1/2$; $\Lambda=0$; gerade under inversion; even under a plane containing the axis
$\mathrm{CO}\;X\,{}^1\Sigma^+$	ground state; singlet; $\Lambda=0$; reflection-even; no $g/u$ label
$\mathrm{OH}\;X\,{}^2\Pi_{3/2}$	ground state; doublet; $
$\mathrm{O}_2\;X\,{}^3\Sigma_g^-$	ground state; triplet; $\Lambda=0$; gerade electronic inversion symmetry; reflection-odd electronic $\Sigma$ symmetry

The minus sign in $X\,{}^3\Sigma_g^-$ for oxygen is not a negative energy, not $u$ symmetry, and not by itself the total parity of every rotational level.

Nonlinear Molecules

For a nonlinear equilibrium geometry, the electronic spatial label is an irrep $\Gamma$ of a point group:

$$\widetilde T\,{}^{2S+1}\Gamma.$$

Examples include

$$\mathrm{H_2O}\; \widetilde X\,{}^1A_1 \quad(C_{2v}),$$

and a schematic octahedral triplet

$${}^3T_{1g} \quad(O_h).$$

Token	Meaning
$\widetilde X$	empirical ground-state name
$2S+1$	spin multiplicity
$A_1$ or $T_{1g}$	irrep in the declared point group
$g/u$, prime/double-prime, or numeric subscripts	parts of the point-group irrep, when that group has the relevant operations

The same letters can carry very different meanings in different point groups. $A_1$ in $C_{2v}$ and $A_{1g}$ in $D_{4h}$ are not universal state types; each is defined by a particular character table and axis convention.

Geometry and model are part of the label

A point-group label is attached to a geometry and a symmetry model. At a distorted geometry, a degenerate irrep can split into irreps of a subgroup. Isotopic substitution can also reduce the symmetry of the full nuclear Hamiltonian even when the clamped-nuclei electronic potential retains a visually symmetric geometry.

For large-amplitude motion, tunneling, or rovibronic coupling, the exact labels may belong to a molecular symmetry or permutation–inversion group rather than the equilibrium point group. A bare $A_1$ assignment is then insufficient unless its group is stated.

Orbital, configuration, and state labels

For a nonlinear molecule, a one-electron orbital may transform as $a_1$, $b_2$, or $e_g$, while a many-electron electronic state transforms as $A_1$, $B_2$, or $E_g$. A configuration such as

$$(1a_1)^2(2a_1)^2(1b_2)^2$$

does not by itself determine the total many-electron state symmetry when open shells are present. The occupied orbitals must be coupled, antisymmetrized, and decomposed into total spin and spatial irreps.

Parity and Symmetry Dictionary

The word “parity” is used at several layers. The operator, not the placement of the sign, determines the meaning.

Printed label	Operator or transformation	Eigenvalue statement	Typical domain
atomic $\pi=\pm1$ or odd degree sign	inversion of all electronic spatial coordinates through the nucleus	$\hat P\psi=\pi\psi$	atom
molecular $g/u$	inversion through a molecular center	$\hat i\psi_{\mathrm{el}}=\pm\psi_{\mathrm{el}}$	centrosymmetric molecule
linear-molecule $\Sigma^+/\Sigma^-$	reflection in a plane containing the molecular axis	$\hat\sigma_v\psi_{\mathrm{el}}=\pm\psi_{\mathrm{el}}$	electronic $\Sigma$ state
$e/f$	total field-free parity correlated with $J$	convention defined below	linear-molecule rovibronic level
prime/double prime	reflection in the horizontal plane in groups such as $D_{3h}$	even/odd under $\sigma_h$	point-group irrep
explicit $+/-$ level parity	full spatial inversion of the specified total state	convention must identify included degrees of freedom	atomic, molecular, nuclear, or particle context

These labels can coexist because they refer to different transformations. A centrosymmetric linear molecule can have both a $g/u$ electronic label and a total rovibronic parity.

The e/f convention

Field-free rotational levels of a linear molecule often occur in opposite-parity pairs. The Brown–Hougen convention labels the two parity classes independently of the Hund coupling case.

For integral $J$,

$$\begin{aligned} p(e)&=+(-1)^J,\\ p(f)&=-(-1)^J. \end{aligned}$$

For half-integral $J$,

$$\begin{aligned} p(e)&=+(-1)^{J-1/2},\\ p(f)&=-(-1)^{J-1/2}. \end{aligned}$$

Here $J$ excludes nuclear spin. Consequently, $e$ does not universally mean even parity and $f$ does not universally mean odd parity. Their parity alternates with $J$.

For example, at $J=3/2$,

$$p(e)=-1, \qquad p(f)=+1.$$

The labels also do not mean “lower” and “upper” member of a doublet. Level ordering can change with $J$, perturbations, and effective-Hamiltonian parameters while the $e/f$ assignment remains tied to parity.

Electronic symmetry versus total parity

The electronic label $g/u$ acts on the clamped-nuclei electronic wavefunction. The parity of a rovibronic level also includes rotational and vibrational transformation properties. Therefore,

$$g\not\equiv p=+1, \qquad u\not\equiv p=-1$$

for an arbitrary total molecular level. Nuclear-spin permutation symmetry can add yet another independent classification.

Molecular Angular-Momentum Dictionary

Molecular papers often use several angular momenta in one label or Hamiltonian. Record the author’s definitions before translating.

Symbol	Common meaning
$\mathbf L$	total electronic orbital angular momentum
$\Lambda$	magnitude of the body-fixed projection of $\mathbf L$
$\mathbf S$	total electronic spin
$\Sigma$	signed body-fixed projection of $\mathbf S$
$\Omega$	magnitude of the body-fixed projection $\Lambda+\Sigma$ in the case-(a) picture
$\mathbf R$	mechanical rotation of the nuclear framework
$\mathbf N$	angular momentum excluding electron spin; often $\mathbf N=\mathbf J-\mathbf S$
$\mathbf J$	total angular momentum excluding nuclear spin
$\mathbf I$	total nuclear spin, or a specified nuclear spin
$\mathbf F$	total angular momentum including nuclear spin, commonly $\mathbf F=\mathbf J+\mathbf I$

Conventions for $\mathbf R$ and $\mathbf N$ vary in rovibronic problems, especially when vibrational angular momentum is present. The defining vector equation is more reliable than the letter alone.

Hund cases are regimes, not species

In a simplified Hund-case-(a) hierarchy, electronic orbital and spin projections are strongly tied to the molecular axis:

$$|A_{\mathrm{SO}}| \gg B_{\mathrm{rot}}J,$$

schematically, so $\Lambda$, $\Sigma$, and $\Omega$ are useful labels. In a case-(b) hierarchy, $\Lambda$ remains tied to the axis but spin is more weakly coupled; $\mathbf N$ first describes rotation excluding spin and then couples to $\mathbf S$:

$$\mathbf J=\mathbf N+\mathbf S.$$

Cases (c), (d), and (e) describe other limiting orders. Real states can move between cases with $J$, vibrational excitation, electronic perturbations, or external fields. A Hund-case label says which basis organizes the spectrum; it does not create an exact symmetry.

Exactness Test

For a proposed label $q$, the mathematical question is whether an operator $\hat Q$ exists such that

$$\hat Q|\psi\rangle=q|\psi\rangle$$

and

$$[\hat H,\hat Q]=0.$$

Three outcomes should be distinguished:

Status	Meaning	Reporting language
exact	symmetry enforces the label for the declared Hamiltonian	“the level has $J=2$ and odd parity”
approximate	the label is nearly conserved because off-diagonal couplings are small	“predominantly $^3P_2$ in the stated LS basis”
empirical	a conventional name tracks a state through observations or calculations	“the $A$ state”

Degeneracy is not required for exactness. Conversely, near-degeneracy does not prove that two levels share a symmetry.

What common interactions preserve

Interaction or model change	Labels often preserved	Labels that can fail
atomic spin–orbit coupling in zero field	total $J$, $M_J$ as a basis label, parity	separate $L$ and $S$
atomic configuration interaction	$J$, parity	configuration and pure LS parentage
static electric field	projection about the field axis when axial symmetry remains	field-free parity; often total $J$
static magnetic field	projection about the field axis when axial symmetry remains	time-reversal degeneracy; often total $J$
molecular rotation	total $J$ and full parity in zero field	exact body-fixed $\Omega$ in intermediate cases
vibronic coupling	exact total symmetry of the coupled problem	separate electronic and vibrational irreps
isotopic substitution	angular momentum and symmetries retained by the isotopologue	exchange or point-group labels lost by mass asymmetry

The Common Atomic Hamiltonians and Common Molecular Hamiltonians pages identify the operators behind these statements.

Worked Translations

Atomic LS level

Consider

$$2p^2\,{}^3P_2.$$

Read it left to right:

1. $2p^2$ is the configuration.
2. $2S+1=3$, so $S=1$.
3. $P$ means total $L=1$.
4. The displayed fine-structure level has $J=2$.
5. No odd-parity mark appears, and independently $\pi=(-1)^{1+1}=+1$.
6. The level contains $2J+1=5$ magnetic substates before field splitting.

The whole $^3P$ term contains $J=0,1,2$, but this symbol names only the $J=2$ level.

Atomic jj level

Consider the schematic label

$$(p_{1/2}^2)_0.$$

It says:

1. each occupied one-electron state belongs to a relativistic $p_{1/2}$ subshell;
2. two equivalent electrons occupy that subshell;
3. the allowed pair shown here has total $J=0$; and
4. the configuration has even parity because two $p$ electrons contribute $(-1)^{1+1}=+1$.

It does not claim $L=0$ or $S=0$. Those are LS-basis questions, and the recoupled state can contain more than one LS component when the relevant $J^\pi$ sector permits it.

Hydrogen molecular ion

Consider

$$\mathrm{H}_2^+\; X\,{}^2\Sigma_g^+.$$

The prefix $X$ identifies the ground electronic state. The multiplicity gives $S=1/2$. The $\Sigma$ letter gives $\Lambda=0$. The $g$ subscript says that the electronic wavefunction is even under inversion through the molecular center. The $+$ superscript says that it is even under reflection in a plane containing the internuclear axis.

The canonical construction of its bonding and antibonding states is in the H₂⁺ Ion article.

Hydroxyl spin–orbit component

Consider

$$\mathrm{OH}\; X\,{}^2\Pi_{3/2}.$$

The state is a doublet with $S=1/2$ and $|\Lambda|=1$. The subscript selects the $|\Omega|=3/2$ spin–orbit component. OH is heteronuclear, so no exact $g/u$ label applies. Each rotational $J$ level can be further resolved into opposite-parity components and labeled $e/f$; those labels are not already contained in the bare electronic term symbol.

Molecular oxygen

Consider

$$\mathrm{O}_2\; X\,{}^3\Sigma_g^-.$$

This combines three distinct statements:

$$S=1, \qquad \Lambda=0, \qquad \hat i\psi_{\mathrm{el}}=+\psi_{\mathrm{el}}.$$

It also states

$$\hat\sigma_v\psi_{\mathrm{el}} = -\psi_{\mathrm{el}}.$$

The subscript $g$ and superscript $-$ therefore cannot be collapsed into one notion of “even” or “odd.”

Water

Consider

$$\mathrm{H_2O}\; \widetilde X\,{}^1A_1 \quad(C_{2v}).$$

The tilde-bearing $X$ is the empirical ground-state name. The state is a singlet. Its electronic spatial wavefunction transforms as the $A_1$ irrep of $C_{2v}$ for the declared equilibrium geometry and axis convention. No $\Lambda$, $\Omega$, or $g/u$ slot belongs in this nonlinear-molecule label.

Assignment Workflow

1. Copy the label exactly

Preserve superscripts, subscripts, parentheses, primes, degree signs, tildes, and capitalization. Plain-text export can erase the very distinctions being interpreted.

2. Identify the object

Ask whether the source labels an orbital, configuration, electronic term, fine-structure level, rotational level, rovibronic level, or hyperfine level. Do not infer this from one symbol in isolation; read the table headings and Hamiltonian definitions.

3. State the symmetry domain

Record atom versus molecule, linear versus nonlinear geometry, point group, isotopologue, field configuration, and whether nuclear spin is included in the reported total angular momentum.

4. Expand every token

Write a ledger:

Token	Operator or convention	Value	Exactness
example: $^3$	$\hat{\mathbf S}^2$	$S=1$	approximate if spin is strongly mixed
example: $J=2$	$\hat{\mathbf J}^2$	$2$	exact in a rotationally invariant field-free model
example: $g$	inversion $\hat i$	$+1$	exact only when inversion is a symmetry
example: $A$	empirical ordering	first same-multiplicity excited state by convention	conventional

5. Check allowed ranges

Use angular-momentum triangle rules, parity from the configuration, allowed point-group labels, and equivalent-particle restrictions. A syntactically valid symbol can still describe an impossible state.

6. Check the Hamiltonian

List retained interactions and test which labels commute with them. If the calculation reports mixing coefficients, state the basis and leading percentage rather than presenting a mixed label as exact.

7. Cross-check observables

Energy intervals, Zeeman factors, hyperfine patterns, polarization, branch structure, isotope shifts, and relative line strengths can challenge an assignment. Agreement with one line position is not sufficient.

8. Preserve source conventions

When importing a database row, record the database version or query date, formatted versus ASCII notation, configuration, term, $J$, parity, and any uncertainty or leading-percentage field. Translate into house notation only after retaining the original.

Common Mistakes

Treating the multiplicity as a level degeneracy

$2S+1$ counts spin projections in a spin-$S$ representation. A particular atomic $J$ level has magnetic degeneracy $2J+1$ in zero field.

Reading atomic P and molecular Pi as the same quantum number

Atomic $P$ means the magnitude $L=1$. Molecular $\Pi$ means the axial projection $|\Lambda|=1$. A molecule need not have a good total $L$.

Reading an orbital label as a term label

Lowercase $p$ or $\pi$ labels a one-electron orbital class. Uppercase $P$ or $\Pi$ labels a many-electron term.

Calling every sign “parity”

Atomic inversion, molecular $g/u$, $\Sigma^+/\Sigma^-$ reflection, point-group prime/double-prime labels, and rovibronic $e/f$ classes refer to distinct operations.

Assigning g or u to a heteronuclear diatomic

The inversion operation exchanges unlike nuclei and is not a symmetry of the fixed-nuclei Hamiltonian. A heteronuclear diatomic therefore has no exact electronic $g/u$ classification.

Interpreting e and f as even and odd

The $e/f$ parity alternates with $J$. Use the defining formulas, and remember that $J$ excludes nuclear spin in this convention.

Interpreting e and f as lower and upper

They are parity classes, not energy-order labels. A perturbation can reverse the ordering without changing the labels.

Treating LS or jj parentage as exact

In intermediate coupling, $J$ and parity may be exact while LS and jj components are basis-dependent. Report leading percentages and the chosen basis.

Omitting the point group

$A_1$, $E$, and $T_{2g}$ acquire meaning only within a declared group and axis convention.

Assuming X, A, and B are symmetry labels

They are empirical electronic-state names. The symmetry information follows in the term symbol.

Exercises

Exercise 1: Decode an atomic level

Decode

$$4s4p\,{}^3P^\circ_1$$

and state the multiplicity, $L$, $J$, parity, number of magnetic substates, and other fine-structure levels in the same ideal LS term.

Solution

The multiplicity is three, so

$$2S+1=3 \quad\Longrightarrow\quad S=1.$$

The letter P gives $L=1$, and the right subscript gives $J=1$. The degree sign states odd parity, consistent with

$$\pi=(-1)^{l_s+l_p} = (-1)^{0+1} = -1.$$

The displayed level has

$$2J+1=3$$

magnetic substates. Coupling $L=1$ and $S=1$ also permits $J=0$ and $J=2$, so the ideal term contains $^3P^\circ_0$, $^3P^\circ_1$, and $^3P^\circ_2$.

Exercise 2: Multiplicity and dimension

How many magnetic basis states belong to a pure $^4D$ term? Into which fine-structure $J$ levels do they organize?

Solution

For a quartet,

$$2S+1=4 \quad\Longrightarrow\quad S=\frac32.$$

For a D term, $L=2$. The uncoupled dimension is

$$(2L+1)(2S+1) = 5\times4 = 20.$$

The allowed total angular momenta are

$$J= \left|2-\frac32\right|, \ldots, 2+\frac32 = \frac12,\frac32,\frac52,\frac72.$$

Their dimensions sum to

$$2+4+6+8=20.$$

Exercise 3: Parity audit

Find the parity of the configurations $p^2$, $pd$, $d^3$, and $sf$. Which ones would conventionally carry an odd degree sign on an atomic term?

Solution

Use

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

Therefore,

$$\begin{array}{c|c|c} \text{configuration} & \sum_i l_i & \pi\\ \hline p^2 & 1+1=2 & +1\\ pd & 1+2=3 & -1\\ d^3 & 2+2+2=6 & +1\\ sf & 0+3=3 & -1 \end{array}$$

Terms of $pd$ and $sf$ have odd parity and conventionally carry the degree sign. Terms of $p^2$ and $d^3$ are even.

Exercise 4: LS label for a mixed level

A calculated $J=2$, even-parity atomic level has LS-basis weights $0.54$ in $^3P_2$, $0.41$ in $^1D_2$, and $0.05$ in other components. Which labels are exact in a zero-field parity-conserving calculation, and how should the level be described?

Solution

Total $J=2$ and even parity are exact under the stated symmetries. Neither pure $L=1,S=1$ nor pure $L=2,S=0$ describes the eigenvector. A defensible description is:

an even-parity $J=2$ level, predominantly $^3P_2$ with 54% weight and a substantial 41% $^1D_2$ component in the stated LS basis.

Calling it simply an exact $^3P_2$ level would hide the intermediate coupling. The basis and calculation should accompany the percentages.

Exercise 5: Decode a linear-molecule symbol

Decode

$$X\,{}^2\Sigma_u^+.$$

Which molecular class can carry every printed label, and which information is not supplied?

Solution

$X$ names the ground electronic state. The multiplicity gives $S=1/2$. $\Sigma$ gives $\Lambda=0$. The $u$ subscript means odd electronic behavior under inversion through the molecular center, and the $+$ superscript means even behavior under reflection in a plane containing the molecular axis.

All of these labels can apply to a centrosymmetric linear molecule, notably a homonuclear diatomic. The symbol does not supply a vibrational quantum number, rotational $J$, total parity, $e/f$ component, nuclear spin, hyperfine $F$, energy, or electronic configuration.

Exercise 6: Why oxygen’s minus sign is not u

For

$$\mathrm{O}_2\; X\,{}^3\Sigma_g^-,$$

identify the two symmetry operations associated with $g$ and $-$. Could the same electronic state have $g$ and $+$ instead without changing the meaning of $g$?

Solution

The $g$ label is the $+1$ eigenvalue of inversion through the molecular center:

$$\hat i\psi_{\mathrm{el}}=+\psi_{\mathrm{el}}.$$

The $-$ label is the $-1$ eigenvalue of reflection in a plane containing the molecular axis:

$$\hat\sigma_v\psi_{\mathrm{el}}=-\psi_{\mathrm{el}}.$$

They refer to independent symmetry operations. A different electronic $\Sigma$ state could indeed be $^3\Sigma_g^+$: changing the reflection eigenvalue would not alter the meaning of the inversion label $g$.

Exercise 7: Compute e/f parity

For a linear-molecule level with $J=3/2$, find the parity of its $e$ and $f$ components. Repeat for integral $J=2$.

Solution

For half-integral $J$,

$$p(e)=+(-1)^{J-1/2}, \qquad p(f)=-(-1)^{J-1/2}.$$

At $J=3/2$,

$$J-\frac12=1,$$

so

$$p(e)=-1, \qquad p(f)=+1.$$

For integral $J$,

$$p(e)=+(-1)^J, \qquad p(f)=-(-1)^J.$$

At $J=2$,

$$p(e)=+1, \qquad p(f)=-1.$$

Thus neither letter has one fixed parity for all $J$.

Exercise 8: Classify molecular labels

For

$$\mathrm{H_2O}\; \widetilde X\,{}^1A_1 \quad(C_{2v}),$$

classify $\widetilde X$, the left superscript, and $A_1$ as empirical, angular-momentum, or symmetry information. Explain why replacing $A_1$ by $\Sigma^+$ would be inappropriate.

Solution

$\widetilde X$ is an empirical electronic-state name. The left superscript gives angular-momentum information:

$$2S+1=1 \quad\Longrightarrow\quad S=0.$$

$A_1$ is the spatial irrep of the electronic wavefunction in $C_{2v}$. Water is nonlinear, so it has no unique molecular axis supporting a linear molecule’s $\Lambda$ classification. $\Sigma^+$ would therefore use the wrong symmetry group and the wrong body-fixed projection grammar.

Key Takeaways

• Identify the physical object before decoding the typography.
• Atomic $L$ and molecular $\Lambda$ are different: a magnitude versus an axial projection.
• The multiplicity $2S+1$, a level degeneracy $2J+1$, and the number of fine-structure levels are different counts.
• In intermediate atomic coupling, $J$ and parity can remain exact while LS and jj names become dominant-component labels.
• Molecular $g/u$, $\Sigma^+/\Sigma^-$, point-group primes, and rovibronic $e/f$ labels refer to different transformations.
• $e/f$ labels are coupling-case independent parity classes, not universal synonyms for even/odd or lower/upper.
• A nonlinear molecular term symbol is incomplete unless its point group and geometry convention are known.
• The Hamiltonian determines which labels are exact.

Cross-Links

• Atomic Term Symbols derives the atomic configuration–term–level hierarchy and works through microstate checks.
• LS Coupling develops the electrostatic coupling limit, fine structure, magnetic diagnostics, and intermediate-coupling tests.
• jj Coupling constructs relativistic subshell labels and LS-to-jj recoupling.
• Molecular Symmetry develops point groups, irreps, direct products, and selection rules.
• Common Molecular Hamiltonians defines $\mathbf N$, $\mathbf J$, $\mathbf I$, $\mathbf F$, spin–rotation, hyperfine, and field terms.
• Electronic Spectroscopy connects state labels to vibronic bands, intensities, and assignments.
• Selection Rule Tables gives a qualified lookup for atomic, molecular, rotational, vibrational, and Raman transitions.

References

1. International Union of Pure and Applied Chemistry, “Term symbols”, Compendium of Chemical Terminology, 5th ed. (2025).
2. E. R. Cohen et al., Quantities, Units and Symbols in Physical Chemistry, 3rd ed., IUPAC and RSC Publishing (2007), Sec. 2.6.3; searchable IUPAC edition.
3. W. C. Martin and W. L. Wiese, Atomic Spectroscopy: An Introduction, National Institute of Standards and Technology, especially “Hierarchy of Atomic Structure” and “Notations for Different Coupling Schemes.”
4. A. Kramida et al., NIST Atomic Spectra Database help and notation guide, National Institute of Standards and Technology.
5. C. J. H. Schutte et al., “Notations and conventions in molecular spectroscopy: Part 2. Symmetry notation (IUPAC Recommendations 1997),” Pure and Applied Chemistry 69, 1641–1650 (1997), doi:10.1351/pac199769081641.
6. J. M. Brown et al., “The labeling of parity doublet levels in linear molecules,” Journal of Molecular Spectroscopy 55, 500–503 (1975), doi:10.1016/0022-2852(75)90291-X.
7. F. J. Lovas, NIST Molecular Spectral Databases: list of symbols and quantum-number conventions, National Institute of Standards and Technology.
8. J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press (2003).
9. H. Lefebvre-Brion and R. W. Field, The Spectra and Dynamics of Diatomic Molecules, Elsevier (2004).
10. P. F. Bernath, Spectra of Atoms and Molecules, 4th ed., Oxford University Press (2020).