Term Symbol Reference

A term symbol is a compact description of a state. For example, $^3P_2^\circ$ identifies a triplet atomic P level with $J=2$ and odd parity; $^3\Sigma_g^-$ describes a molecular electronic state with quite different projection and symmetry labels. This reference helps you read each part and recognize which labels survive when interactions mix states.

A symbol does not give the energy or a complete wavefunction. Configuration labels and LS or jj assignments often describe dominant components, whereas exact angular momentum and parity follow from symmetries of the Hamiltonian. The atomic discussion initially excludes hyperfine coupling and external fields; once nuclear spin is coupled, $F$ may replace electronic $J$ as the exact total-angular-momentum label.

Required background. Atomic term symbols explains configuration, term, and level. For detailed coupling constructions, see LS coupling and jj coupling.

Atomic and molecular symbols

System	Example	What the main letter means
atom in LS notation	$\gamma\,{}^{2S+1}L_J^\pi$	$L$ is total orbital angular momentum
atom in jj notation	$[(nl_{j_1})^{N_1}_{J_1}(n'l'_{j_2})^{N_2}_{J_2}]_J^\pi$	subshell angular momenta couple to total $J$
linear molecule	$X\,{}^{2S+1}\Lambda_\Omega$	$\Lambda$ is an axial orbital projection magnitude
nonlinear molecule	$\widetilde X\,{}^{2S+1}\Gamma$	$\Gamma$ is an irrep of a specified point group

Here $\gamma$ supplies additional labels when necessary; $X$ is an empirical ground-state name. Molecular inversion and reflection labels are added when the relevant symmetry exists.

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

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

One configuration can contain several terms. The $1s\,2p$ configuration has $12$ states: three in $^1P^\circ$ and nine in $^3P^\circ$. The triplet term contains $J=0,1,2$ levels with $1,3,5$ magnetic states. The diagram groups states; its vertical positions do not specify energy ordering.

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

Atomic jj and Intermediate Coupling

LS coupling forms total $L$ and $S$ before coupling them to $J$. In jj coupling, each electron first forms $\mathbf j_i=\mathbf l_i+\mathbf s_i$; these angular momenta, or the totals of groups of electrons, then couple to $\mathbf J$. The NIST coupling-notation guide illustrates the corresponding labels.

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 mixed level can be described by its leading components,

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

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

Brown and Carrington (2003) explain molecular coupling and electronic-state notation. 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 molecular axis as the body-fixed $z$ axis. Let $\lambda=L_z/\hbar$ and $\sigma=S_z/\hbar$ denote signed eigenvalues in the axial electronic basis. Then

$$\Lambda=|\lambda|,\qquad\Omega=|\lambda+\sigma|.$$

The term letter uses the nonnegative magnitude $\Lambda$. In a common Hund-case-(a) convention one selects the $\lambda=+\Lambda$ branch and writes the signed spin projection as $\Sigma=\sigma$, giving $\Omega=|\Lambda+\Sigma|$. The opposite branch has the corresponding opposite signed projections. This choice avoids adding an unsigned orbital magnitude to an unspecified spin projection.

These projections are distinct from the atomic magnitudes $L,S,J$. The symbol $\Sigma$ for a spin projection is also distinct from the term letter $\Sigma$ meaning zero orbital projection; context tells them apart.

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 spatial 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 term, reflection through a plane containing the molecular axis gives the electronic spatial factor 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; $\lvert \Lambda\rvert=1$; the displayed spin–orbit component has $\lvert \Omega\rvert=3/2$
$\mathrm{O}_2\;X\,{}^3\Sigma_g^-$	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 et al. 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

For the common definitions of $\mathbf R,\mathbf N,\mathbf J,\mathbf I$ and $\mathbf F$, use the molecular angular-momentum table. In particular, $J$ normally excludes nuclear spin, whereas $F$ includes it. Vibrational angular momentum can also contribute, so check the defining vector equation when moving between conventions.

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, with $A_{\mathrm{SO}}$ and $B_{\mathrm{rot}}$ in the same units. This comparison is useful when spin–orbit coupling supplies the axial splitting, as in a $^2\Pi$ state; it is not a universal criterion for $\Sigma$ states, where first-order orbital spin–orbit splitting can vanish. Compare the actual coupling matrix elements and splittings in that case. 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 $S$
spin-independent electrostatic configuration interaction	$L,S,J$, parity	configuration and additional parent labels; it does not by itself mix different $L,S$ terms
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.

Common pitfalls

One symbol, several different counts. The multiplicity $2S+1$ counts spin projections. A field-free electronic $J$ level has $2J+1$ magnetic states, while a whole LS term contains $(2L+1)(2S+1)$ states.

A symmetry sign is not an energy sign. In oxygen’s $X\,{}^3\Sigma_g^-$, $g$ means electronic inversion-even and $-$ means spatial reflection-odd. Neither says that the energy is negative or gives the parity of every rotational level.

A conventional name need not be an exact quantum number. $X,A,B$ name electronic states; a dominant LS component names an approximate coupling description. Exact labels follow from the retained Hamiltonian, and an irrep such as $A_1$ needs its point group.

Exercises

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

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

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. Assume a rotationally invariant electronic Hamiltonian without hyperfine coupling. Which labels are exact, how should the level be described, and could spin-independent electrostatic configuration interaction alone make the $^3P_2$ and $^1D_2$ components mix?

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. Electrostatic configuration interaction alone commutes with $L^2$ and $S^2$, so it cannot mix these different $L,S$ terms. Spin-dependent interactions, such as spin–orbit coupling, are needed for that mixing.

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.

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 $J$.

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.

References

• Brown, J. M., J. T. Hougen, K.-P. Huber, J. W. C. Johns, I. Kopp, H. Lefebvre-Brion, A. J. Merer, D. A. Ramsay, J. Rostas, and R. N. Zare. “The labeling of parity doublet levels in linear molecules.” Journal of Molecular Spectroscopy 55, 500–503 (1975). DOI; author-hosted article.
• Brown, J. M., and A. Carrington. Rotational Spectroscopy of Diatomic Molecules. Cambridge University Press, 2003. DOI.
• Martin, W. C., and W. L. Wiese. Atomic Spectroscopy: A Compendium of Basic Ideas, Notation, Data, and Formulas. National Institute of Standards and Technology. Online reference.
• National Institute of Standards and Technology. Atomic Spectra Database: Help and notation guide. Online documentation, accessed 14 October 2026.