Quantum Case Documentation

Definitions and quantum number conventions for ExoMol molecular datasets

Each molecular state in an ExoMol dataset is described by a set of quantum numbers in a case-by-case manner, depending on the geometry and coupling schemes of the molecule. To make these numbers unambiguous to parse and compare across different linelists, every molecule is assigned a quantum case label — a short code that specifies the set of standard quantum numbers used for that molecule. This code is listed in the definition (.def) file of each dataset.

The labelling scheme follows the Virtual Atomic and Molecular Database Consortium (VAMDC) conventions and organises molecules into five geometry families, which is then further split into a closed-shell and an open-shell variant.

It should be noted that the quantum number sets defined below are not definitive, where equivalent quantum numbers may be used instead. For example, in asymmetric top molecules, the standard (Herzberg) notation of Ka and Kc is commonly replaced by K and rotational parity (rotpar).

Molecule geometry families

Diatomic
Two atoms. One vibrational mode, one rotational axis. Described by a single vibrational quantum number v and rotational quantum number J.
Linear polyatomic
Three or more atoms in a straight line (D∞h or C∞v symmetry). Bending modes are doubly degenerate and carry vibrational angular momentum li.
Symmetric top
Non-linear molecule with at least one C3 (or higher) symmetry axis (C3v, D3h, …). Two of the three principal moments of inertia are equal.
Asymmetric top
Non-linear molecule with no symmetry axis higher than C2. All three principal moments of inertia are distinct. The most common polyatomic case.
Spherical top
Molecule with tetrahedral (Td) or higher symmetry. All three principal moments of inertia are equal.

Closed-shell vs. open-shell

A closed-shell dataset has all electrons paired, giving a singlet electronic ground state with no net electronic orbital or spin angular momentum. This label is only used when the line list contains only the singlet electronic ground state and no excited states. Datasets where the molecule has a singlet ground state but also includes excited electronic states with non-zero electronic angular momentum and/or spin are labelled as open-shell.

An open-shell dataset contains at least one electronic state where there are one or more unpaired electrons. This introduces additional quantum numbers to describe the electronic orbital angular momentum (Lambda, Omega), electron spin projection (Sigma), the electronic state label (ElecState), and spin-component indices (Fi).

In many ExoMol line lists, you may still find the electronic state and open-shell quantum numbers (Lambda, Sigma, Omega) in the states file of a closed-shell dataset due to the standardised format. However, these quantum numbers are trivial (e.g. Lambda = 0, Sigma = 0) and can be safely ignored.

Overview

Label Name Examples Quantum numbers
dcs Diatomic closed-shell H2, CaO, NaCl J, +/-, e/f, v
dos Diatomic open-shell AlH, C2, CO J, +/-, e/f, ElecState, v, Lambda, Sigma, Omega, N, Fi
lpcs Linear polyatomic closed-shell CO2, HCN, C2H2 J, +/-, e/f, Gvib, v_i, l_i, L, invpar
lpos Linear polyatomic open-shell CaOH J, +/-, e/f, ElecState, v_i, L, N, Lambda, Fi, invpar
stcs Symmetric top closed-shell NH3, CH3Cl, H3+ J, +/-, e/f, Grve, Gvib, v_i, l_i, L, Grot, K, invpar
stos Symmetric top open-shell CH3 J, +/-, e/f, Grve, ElecState, Gvib, v_i, l_i, L, Grot, K, invpar
asymcs Asymmetric top closed-shell H2O, H2CO, C2H4 J, +/-, e/f, Grve, Gvib, v_i, Grot, Ka, Kc, invpar
asymos Asymmetric top open-shell Currently not in use J, +/-, e/f, Grve, ElecState, Gvib, v_i, Grot, Ka, Kc, N, invpar
sphcs Spherical top closed-shell CH4, SiH4 J, +/-, e/f, Grve, Gvib, v_i, l_i, m_i, Grot
sphos Spherical top open-shell Currently not in use J, +/-, e/f, Grve, ElecState, Gvib, v_i, l_i, m_i, Grot, N

It should be noted that in hyperfine-resolved datasets, F becomes the rigorous quantum number, while J becomes an approximate quantum number. A hyperfine-resolved dataset is denoted with the boolean entry hyperfine_resolved_dataset in the definition file of the dataset.

Quantum Case Labels

The standard quantum number sets for each molecule geometry and shell type. Click on any quantum number chip to jump to its definition.

Diatomic cases

dcs Diatomic Closed-Shell

A two-atom molecule with all electrons paired, where the line list only includes the 1Σ ground state. No electronic angular momentum or spin-orbit coupling.

Examples: H2, CaO, NaCl

Quantum numbers:

dos Diatomic Open-Shell

A two-atom molecule with one or more unpaired electrons in any of the electronic states covered by the line list, giving non-zero electronic orbital angular momentum and/or non-zero spin. Molecules are further divided into Hund’s case (a) or (b) coupling schemes, which determines which quantum numbers are rigorous.

Examples: AlH, C2, CO

Quantum numbers:

Linear polyatomic cases

lpcs Linear Polyatomic Closed-Shell

A linear molecule consisting of three or more atoms where the line list only covers the ground state with no unpaired electrons. Symmetry group D∞h (e.g. CO2) or C∞v (e.g. HCN).

Examples: CO2, HCN, C2H2

Quantum numbers:

lpos Linear Polyatomic Open-Shell

A linear molecule with three or more atoms and one or more unpaired electrons, giving non-zero electronic angular momentum.

Examples: CaOH

Quantum numbers:

Symmetric top cases

A symmetric top molecule has one axis of at least 3-fold rotational symmetry. In a prolate symmetric top (e.g. CH3Cl) the unique axis is the longest; in an oblate top (e.g. NH3) it is the shortest. The quantum number K is the projection of J onto this axis.

stcs Symmetric Top Closed-Shell

A polyatomic symmetric top molecule where the line list only covers the ground state, which has no unpaired electrons.

Examples: NH3, CH3Cl, H3+

Quantum numbers:

stos Symmetric Top Open-Shell

A polyatomic symmetric top molecule with unpaired electrons.

Examples: CH3

Quantum numbers:

Asymmetric top cases

An asymmetric top has three distinct principal moments of inertia. Because K is not a good quantum number here, two approximate quantum numbers Ka and Kc are used: the projections of J in the prolate (Ka) and oblate (Kc) limits. They satisfy Ka + Kc = J or J + 1.

asymcs Asymmetric Top Closed-Shell

A non-linear polyatomic molecule with no 3-fold or higher symmetry axis, and the line list only covers the ground state with no unpaired electrons. The most common polyatomic case.

Examples: H2O, H2CO, C2H4

Quantum numbers:

asymos Asymmetric Top Open-Shell

An asymmetric top molecule with one or more unpaired electrons. Currently not in use in the ExoMol database.

Examples: Currently not in use

Quantum numbers:

Spherical top cases

Spherical top molecules have highly symmetrical geometries and can belong to tetrahedral (Td) or higher symmetry point groups and have three equal principal moments of inertia. Vibrational and rotational levels are labelled by irreducible representations of the relevant point group.

sphcs Spherical Top Closed-Shell

A spherical top molecule where the line list only covers the ground electronic state with no unpaired electrons.

Examples: CH4, SiH4

Quantum numbers:

sphos Spherical Top Open-Shell

A spherical top molecule with unpaired electrons. Currently not in use in the ExoMol database.

Examples: Currently not in use

Quantum numbers:

Quantum Label Types

Namespace prefixes that qualify how quantum numbers were assigned or computed

Quantum label types

Each quantum number column in an ExoMol states file may be prefixed with a quantum label type namespace, written as Type:quantum_number (e.g. hunda:Omega or TROVE:Gvib). The namespace serves three purposes:

  1. Coupling scheme disambiguation — diatomic open-shell molecules may use either Hund’s case (a) or case (b) quantum numbers; the hunda and hundb prefixes make this explicit.
  2. Notation scheme disambiguation — different spectroscopic notations (e.g. AFGL and Herzberg) sometimes use the same symbol with different meanings. The prefix identifies which convention is being used.
  3. Software provenance — the prefix records which variational nuclear-motion code computed the quantum labels or symmetry assignments, e.g. TROVE or DVR3D.

The currently defined types are listed below. New types may be added as additional line lists are incorporated into the database.

Herzberg Herzberg notation

Quantum numbers following the conventions of Herzberg (Molecular Spectra and Molecular Structure). This is also commonly referred to as normal mode quantum numbers.

AFGL AFGL notation

Quantum numbers following the Air Force Geophysics Laboratory spectroscopic conventions, as used in the HITRAN database. Commonly applied to polyatomic vibrational mode numbering.

TROVE TROVE

Local vibrational mode quantum numbers and symmetries assigned by the variational nuclear-motion program TROVE (Theoretical ROVibrational Energies; Yurchenko et al.).

DVR3D DVR3D

Symmetry assignments produced by the DVR3D program suite (Tennyson et al.). This prefix is used exclusively to namespace symmetry quantum numbers; it is not applied to other quantum number types.

Polyad Polyad

Used exclusively as a prefix for the polyad quantum number and the associated polyad counting number. Polyad quantum numbers group near-degenerate vibrational levels that interact strongly via resonance (e.g. Fermi or Darling–Dennison).

hunda Hund’s case (a)

Diatomic open-shell quantum numbers in the Hund’s case (a) coupling scheme, where both electronic orbital angular momentum (Λ) and electron spin (Σ) are well-defined projections onto the internuclear axis, giving Ω = |Λ ± Σ|. Used within the dos case label.

hundb Hund’s case (b)

Diatomic open-shell quantum numbers in the Hund’s case (b) coupling scheme, where the electron spin is not strongly coupled to the internuclear axis. The total rotational angular momentum N (excluding spin) is a good quantum number, and J = N ± S. Used within the dos case label.

Quantum Number Reference

Definitions for every quantum number that appears across the case labels

Quantum numbers

This is not an exhaustive list of all quantum labels used in the ExoMol database. It covers the standard labels defined for each quantum case. Individual datasets may contain extra molecule-specific labels or use equivalent alternative labels in place of the ones listed here. Users should always refer to the corresponding publication for the most accurate and complete description of the quantum numbers used in a given line list.
Rigorous vs approximate quantum numbers
A rigorous quantum number corresponds to an operator that commutes with the molecular Hamiltonian for the model used, so it labels exact eigenstates. An approximate quantum number is still useful for assignment and interpretation, but is not strictly conserved when interactions (for example Coriolis, spin-orbit, or resonance couplings) mix basis states.

Which labels are rigorous depends on the molecule, coupling scheme, and whether the dataset is hyperfine-resolved. For example, J, +/- and e/f, and rovibronic symmetry Grve are rigorous (F is rigorous in hyperfine-resolved datasets and J becomes approximate). Other labels, such as v, v_i, K, Ka, Kc, Lambda, Sigma, and Omega are approximate quantum numbers.
J
Total angular momentum quantum number (J)
Total angular momentum excluding nuclear spin. For closed-shell molecules J = 0, 1, 2, … For open-shell molecules J can be half-integer.
+/-
Total parity
Symmetry of the total wave function under space-fixed inversion. Electric-dipole transitions connect levels of opposite total parity.
e/f
Rotationless parity
Parity of the level with the rotational wave function removed. Distinguishes Λ-doubling or A-doubling components independently of J.
Grve
Ro-vibronic symmetry (Γrve)
Irreducible representation of the combined rotation–vibration–electronic wave function. Together with the nuclear-spin function it must satisfy the Pauli principle.
v
Vibrational quantum number (v)
Non-negative integer: v = 0, 1, 2, … where v = 0 is the vibrational ground state.
v_i
Mode vibrational quantum numbers (vi)
Vibrational quantum numbers for each normal mode of a polyatomic, indexed by mode number i (v1, v2, …). Each vi ≥ 0.
l_i
Vibrational angular momentum associated with a degenerate vibrational mode vi (li)
For a degenerate vibrational mode i, li is an integer with |li| ≤ vi. Zero for non-degenerate modes.
L
Total vibrational angular momentum (L)
Resultant of all individual vibrational angular momenta l_i. For linear molecules L = |∑ li| and couples with rotation via l-type resonance.
m_i
Vibrational multiplicity quantum number (mi)
Vibrational multiplicity quantum number associated with triply degenerate modes.
Gvib
Vibrational symmetry (Γvib)
Irreducible representation of the vibrational wave function in the molecular point group, e.g. A1, E, T2.
K
Projection of J on the symmetry axis (K)
Projection of total angular momentum J onto the molecule-fixed symmetry axis of a symmetric top. |K| ≤ J.
Ka
Prolate-limit projection quantum number (Ka)
Value of |K| in the prolate symmetric-top limit. Approximate quantum number for asymmetric tops. Ka = 0, 1, …, J. Together with Kc: Ka + Kc = J or J + 1.
Kc
Oblate-limit projection quantum number (Kc)
Value of |K| in the oblate symmetric-top limit. Approximate quantum number for asymmetric tops. Kc = 0, 1, …, J. Together with Ka: Ka + Kc = J or J + 1.
N
Rotational angular momentum before spin coupling (N)
Nuclear rotation alone before coupling with electron spin. N = 0, 1, 2, … For spin-1/2 systems J = N ± 1/2.
Fi
Spin component index (Fi)
Integer index labelling distinct levels from coupling nuclear rotation with electron spin. E.g. F1, F2, F3…
ElecState
Electronic state label
String identifying the electronic state: ground state is X, excited states A, B, … Includes spin multiplicity, symmetry symbol, and electronic inversion parity if applicable. Follows the PyValem convention, e.g. X(2Sigma+), A(2Pig).
Lambda
Electronic orbital angular momentum projection (Λ)
Absolute projection of total electronic orbital angular momentum onto the internuclear axis. Λ = 0 (Σ), 1 (Π), 2 (Δ), …
Sigma
Electron spin projection (Σ)
Projection of total electron spin S onto the internuclear axis. Ranges from −S to +S in integer steps. Not to be confused with the Σ symmetry label for Λ = 0 states.
Omega
Total electronic angular momentum projection (Ω)
Ω = |Λ ± Σ| is the total electronic angular momentum projection onto the internuclear axis. For a 2Π molecule, Ω = 1/2 or 3/2.
Grot
Rotational symmetry (Γrot)
Irreducible representation of the rotational wave function. Determines nuclear-spin statistical weights.
invpar
Inversion parity
Symmetry under inversion. For inversion-tunnelling molecules (e.g. NH3): s / a. Note that the centro-symmetric electronic inversion parity (g / u) is included in the electronic state label instead.
rotpar
Rotational state symmetry (τrot)
Symmetry label for the rotational state under the molecular symmetry group. Takes integer values (e.g. 0 or 1) that distinguish symmetry components of a given J level, in particular for molecules with permutation-inversion symmetry (e.g. A1 vs. A2, or E levels in C3v(M)).