Reference and Data

Reference material is useful when it shortens a lookup without hiding the conditions that make the answer true. This chapter is the AMO checking layer: use it to translate a scale, identify a model, qualify a selection rule, locate a Hamiltonian, interpret a spectroscopic quantity, or find an evaluated data source. Then follow the linked canonical page whenever a derivation or validity argument affects the result.

A table entry is not a free-standing law of nature. Numerical constants have release dates, line strengths have convention choices, term labels depend on coupling regimes, and “linewidth” can mean several different widths. Reliable lookup means recording those choices rather than relying on a familiar symbol.

What This Chapter Owns

This chapter owns:

compact AMO-specific tables, conversion workflows, nomenclature, and routing indexes;
a common protocol for versioning numerical data and conventions;
links from reference entries to the one canonical derivation for each subject;
quick distinctions among quantities that are often conflated; and
reproducibility requirements for database and literature lookups.

It does not own the full derivation of atomic structure, molecular motion, transition rates, line shapes, laser dynamics, or quantum-optical models. Those remain in their subject chapters. The site-wide Reference Library owns cross-volume formulas, operators, model cards, theorems, symbols, and conventions.

Choose by Task

If you need to…	Start here	Follow for depth
convert among joules, eV, hertz, inverse centimetres, kelvin, or atomic units	Constants and Conversions	Units and Constants
identify a Hartree-unit quantity, convert it to SI, or diagnose a Hartree–Rydberg mismatch	Atomic Units	Atomic Units and Scales
decide whether an E1, M1, E2, rotational, vibrational, or Raman transition is symmetry-allowed	Selection Rule Tables	Selection Rules in Spectroscopy
choose among two-level, optical Bloch, Rabi, Jaynes–Cummings, Dicke, Hubbard, or Gross–Pitaevskii models	AMO Model Index	the canonical page linked in its model row
identify the terms, units, symmetries, and validity regime of a standard atomic Hamiltonian	Common Atomic Hamiltonians	the hydrogenic, multi-electron, fine-, hyperfine-, Stark, Zeeman, or Rydberg page linked from its operator row
identify the electronic, nuclear, rotational, rovibrational, spin–rotation, and hyperfine pieces of a molecular Hamiltonian	Common Molecular Hamiltonians	the Coulomb, Born–Oppenheimer, vibration, rotation, and rovibrational pages linked from its operator rows
decode an atomic or molecular term symbol, parity marker, or coupling label	Term Symbol Reference	Atomic Term Symbols, LS Coupling, jj Coupling, and Molecular Symmetry
distinguish Lorentzian, Gaussian, and Voigt conventions or natural, Doppler, collisional, transit-time, and power broadening	Line Shape Reference	Line Shapes and Broadening and the mechanism-specific sources cited there
translate $f$ , $gf$ , $\log(gf)$ , line strength, dipole matrix elements, radiative rates, or integrated absorption	Oscillator Strength Reference	Oscillator Strengths and Transition Rates
connect Einstein coefficients, lifetimes, branching, stimulated processes, oscillator strengths, and radiation density	Einstein Coefficient Reference	Einstein Coefficients and Spontaneous Emission
check wavelength, wavenumber, absorbance, optical depth, branch, or linewidth language	Spectroscopy Nomenclature	Spectroscopy and Constants and Conversions
check gain, threshold, finesse, $Q$ , mode volume, coherence, detuning, Rabi-frequency, or saturation language	Laser Nomenclature	Lasers, Optical Cavities, and Optical Bloch Equations
compare the record, observable, inference, and limitation of an AMO landmark	AMO Experiment Index	the canonical experiment page, the site-wide Experiment and Historical Index, and the primary sources
choose an AMO textbook, review, primary paper, database, or computational source	AMO Bibliography and Reading Guide	the site-wide Bibliography and Reading Guides and the subject page’s references
verify a numerical constant, level, wavelength, or transition probability	the versioned sources below	the source record and primary references attached to the evaluated value

Lookup Workflow

The shortest trustworthy route is

\begin{gathered} \text{name the physical object} \\ \downarrow \\ \text{fix units, symbols, and conventions} \\ \downarrow \\ \text{state the model and approximation regime} \\ \downarrow \\ \text{record source and version} \\ \downarrow \\ \text{check a limit, identity, or benchmark} \\ \downarrow \\ \text{follow the canonical derivation if needed}. \end{gathered}

1. Name the object

Decide whether the requested number is an energy, cyclic frequency, angular frequency, vacuum wavelength, medium wavelength, spectroscopic wavenumber, decay rate, half-width, full width, oscillator strength, cross section, integrated intensity, or fitted parameter. Several of these can share the same dimension while answering different questions.

2. Fix the convention

Write a quantity as a numerical value times a unit,

Q = \{Q\}_{[Q]}\,[Q].

Changing the unit changes $\{Q\}_{[Q]}$ , not $Q$ . A symbol alone is not a convention. For example, $\nu$ , $f$ , and $\omega$ are used inconsistently across subfields, so the defining equation is safer:

E = h\nu = \hbar\omega, \qquad \omega=2\pi\nu.

3. State the model boundary

A transition wavelength calculated from a nonrelativistic clamped-nucleus Hamiltonian and one measured in a field-shifted, hyperfine-resolved sample do not refer to the same model quantity. Record the isotope, charge state, electronic level, angular-momentum labels, external fields, and retained corrections.

4. Record provenance

For a database value, save:

database or compilation name;
version, release, or last-update identifier;
query date;
species, isotope, state labels, and search filters;
displayed value and uncertainty;
units and air/vacuum convention where applicable;
source reference attached to the entry.

A screenshot without this metadata is not a reproducible lookup.

5. Run a structural check

Use dimensional analysis, normalization, a sum rule, detailed balance, a known limit, or an independent unit conversion. A value can be copied perfectly from the wrong row.

Reference Layers

Constants, units, and scales

Constants and Conversions owns AMO translations among energy, frequency, wavenumber, wavelength, temperature-equivalent energy, magnetic moments, and dipole units. It distinguishes exact SI defining constants from measured CODATA quantities.

Atomic Units owns the AMO lookup contract: the Hartree convention, versioned SI multipliers, dimensional restoration, and Hartree–Rydberg diagnostics. Atomic Units and Scales derives the system and explains why atomic energies, lengths, times, fields, and polarizabilities acquire natural scales. Use the site-wide Atomic Units translator when moving a formula across volumes.

The universal energy identity is

E = h\nu = \hbar\omega = hc\tilde\nu = k_{\mathrm B}T_E.

$T_E$ is an energy-equivalent temperature. It is not automatically the thermodynamic temperature of an ensemble.

Hamiltonians and models

The Common Atomic Hamiltonians entry compares hydrogenic, many-electron, central-field, relativistic, hyperfine, external-field, and Rydberg operators. It records the Hilbert space, unit, exact-label, perturbative-regime, and double-counting checks that must accompany a compact formula.

The Common Molecular Hamiltonians entry follows the corresponding molecular reduction from the all-particle Coulomb operator through clamped-nuclei surfaces, nuclear vibration, rotation–vibration coupling, and spin-resolved effective Hamiltonians. It also separates electronic spin–rotation from nuclear spin–rotation and declares the angular-momentum and unit conventions needed to compare fitted constants.

The AMO Model Index begins from retained degrees of freedom rather than model names. It routes internal-structure, molecular, light–matter, cavity, laser, cooling, trapping, lattice, and condensate models to their canonical homes.

A Hamiltonian lookup should identify at least

\left( \mathcal H,\, H,\, \boldsymbol\theta,\, \mathcal S,\, \mathcal A \right),

where $\mathcal H$ is the Hilbert space, $\boldsymbol\theta$ is the parameter set, $\mathcal S$ lists symmetries, and $\mathcal A$ records approximations. Two equations with the same operator pattern can represent different physical models if these entries differ.

Transition permission and strength

Selection Rule Tables answer whether symmetry forces a matrix element to vanish under stated assumptions. They do not predict the magnitude of every allowed line.

For an operator $\hat O$ ,

\mathcal M_{fi} = \langle f|\hat O|i\rangle.

A selection rule may show $\mathcal M_{fi}=0$ . If it does not, radial integrals, reduced matrix elements, state mixing, population, geometry, and detector response still determine the observed strength.

Oscillator strengths, Einstein coefficients, line strengths, transition dipoles, cross sections, and integrated absorbances are related quantities, not interchangeable names. Translate them only after matching degeneracy, polarization, SI/cgs, angular-frequency, and spectral-density conventions.

Line positions, shapes, and instruments

A measured spectrum is not merely a list of energy differences. A useful forward model separates

S_{\mathrm{obs}}(x) = \left[ S_{\mathrm{stick}} * \phi_{\mathrm{physical}} * R_{\mathrm{instrument}} \right](x) + B(x),

where $x$ is a declared spectral coordinate, $\phi_{\mathrm{physical}}$ collects physical broadening, $R_{\mathrm{instrument}}$ is the instrument response, and $B$ is background. The convolution symbol does not imply every mechanism is stationary or independent; that assumption must be checked.

Use Line Shapes and Broadening for Lorentzian, Gaussian, Voigt, natural, Doppler, collisional, power, transit- time, and instrumental effects. Use Precision Spectroscopy when line-center estimation and uncertainty budgets matter.

Convention Gates

Frequency and wavelength

Record all of the following that apply:

cyclic frequency $\nu$ or angular frequency $\omega$ ;
vacuum or in-medium wavelength;
air or vacuum wavelength for tabulated optical lines;
spectroscopic wavenumber $\tilde\nu=1/\lambda_{\mathrm{vac}}$ or wavevector magnitude $k=2\pi/\lambda$ ;
ordinary hertz or angular-frequency linewidth.

Width and lifetime

For a simple exponentially decaying excited-state population,

P_e(t) = P_e(0)e^{-\Gamma t}, \qquad \tau=\Gamma^{-1}.

The corresponding natural line has convention-dependent width statements. Before using “ $\Gamma$ is the linewidth,” specify the spectral coordinate and whether the width is HWHM or FWHM. Coherence decay can also contain pure dephasing and need not equal the population-decay rate.

Angular momentum and parity

Term symbols are meaningful only in a stated coupling regime. Distinguish exact quantum numbers from dominant-component labels, and record whether parity, inversion parity, reflection labels, hyperfine $F$ , or field-dressed projections remain good quantum numbers.

Intensity and logarithms

Absorbance, optical depth, transmittance, absorption coefficient, cross section, and molar absorptivity use different normalizations. In particular,

T = \frac{I}{I_0}, \qquad \mathrm{OD} = -\ln T, \qquad A_{10} = -\log_{10}T.

Thus $\mathrm{OD}=(\ln10)A_{10}$ . Reporting “absorbance” without the logarithm base can create a factor of $\ln10$ .

Evaluated Data and Source Hierarchy

Defining constants and recommended values

Use the current BIPM SI Brochure for SI definitions. Use a named NIST/CODATA release for recommended measured constants. Do not mix values from different adjustments merely because they have more displayed digits.

Atomic levels and transitions

The NIST Atomic Spectra Database is an evaluated starting point for atomic levels, lines, and transition probabilities. Its records can combine measurements, calculations, and critical evaluations. Follow the record’s references when method provenance or uncertainty matters.

Molecular and chemical data

The NIST Chemistry WebBook provides evaluated and compiled thermochemical, spectroscopic, and related molecular data. The IUPAC Gold Book is an authority for chemical terminology, not a substitute for a measured spectrum or a derivation.

Literature roles

Source type	Best use	Main caution
standards body	definitions, units, metrological conventions	edition and effective date matter
evaluated database	recommended or critically assessed numerical data	inspect flags, uncertainties, and source records
primary paper	original method, measurement, or theoretical result	later corrections or re-evaluations may supersede a number
review article	field map and synthesis	not every tabulated value is independently re-evaluated
textbook or monograph	durable derivation and notation	conventions may differ from current databases or SI practice
software documentation	implementation contract and version behavior	documentation does not validate the physical model

Scientific Status and Versioning

Status	Meaning in a reference entry
exact definition	fixed by a declared mathematical or metrological convention
defining constant	numerical value fixed by the SI
recommended value	evaluated estimate with uncertainty and release identifier
measured datum	result tied to an experiment, calibration, and uncertainty model
calculated datum	result tied to a Hamiltonian, method, basis, and convergence record
fitted parameter	value conditional on a model, dataset, and fitting protocol
scaling estimate	order-of-magnitude guide, not a recommended datum
active interpretation	evidence is developing or competing models remain viable

Precision without status is misleading. A fitted line center with twelve digits is not an exact constant; an exact conversion factor does not make its measured input exact.

Reproducible Lookup Record

For a calculation or publication, retain a compact record.

Field	Example of what to state
quantity	vacuum transition frequency between fully specified levels
value and uncertainty	central value, standard uncertainty, and coverage convention
unit and coordinate	Hz, rad s⁻¹, cm⁻¹, vacuum nm, or another declared coordinate
system identity	isotope, charge state, electronic configuration, term, hyperfine level
environment	field, pressure, temperature, trap, polarization, and reference frame
source	database or paper with version, table/record, DOI, and access date
transformation	constants and equations used to convert the source value
model status	measured, evaluated, calculated, fitted, or estimated
validation	independent conversion, sum rule, residual, or benchmark

If uncertainty is transformed through a nonlinear function $y=f(x)$ , use the appropriate covariance propagation or a documented numerical method. For a single small uncertainty,

u_y \simeq \left| \frac{df}{dx} \right| u_x.

Do not attach the source’s original uncertainty unchanged after inverting a wavelength or combining correlated constants.

Common Lookup Failures

Copying digits without the release

The value cannot be reproduced or updated systematically. Record the adjustment, database version, and access date.

Mixing vacuum and air wavelengths

The difference can exceed a precision experiment’s uncertainty by many orders of magnitude. Record the refractive-index convention and environmental conditions.

Treating an allowed transition as a strong transition

Selection permission is only one factor. Evaluate the matrix element, population, polarization, branching, broadening, and detection chain.

Reading a fitted parameter as an observable

A Lorentzian width, quantum defect, rotational constant, or effective temperature can depend on the fitting model and interval. Report the model and residuals with the number.

Combining incompatible degeneracy conventions

Oscillator strengths, Einstein coefficients, and line strengths may be averaged over initial substates or summed over final substates. Match those conventions before applying a conversion formula.

Using a compact entry as a derivation

When signs, domains, approximations, or uncertainty depend on the reasoning, follow the canonical page. A reference table is a map back to that reasoning.

Lookup Exercises

Exercise 1: One energy, four numbers

A transition is reported as a cyclic frequency $\nu$ . State the equations needed to express it as angular frequency, photon energy, and spectroscopic wavenumber.

Solution

Use

\omega=2\pi\nu, \qquad E=h\nu, \qquad \tilde\nu=\frac{\nu}{c}.

The last relation gives inverse metres; divide by $100$ to express the result in $\mathrm{cm^{-1}}$ . The symbol $\tilde\nu$ is spectroscopic wavenumber, not the wavevector magnitude $k=2\pi\tilde\nu$ .

Exercise 2: Allowed but absent

An E1 transition passes the angular-momentum and parity rules but is absent from a measured spectrum. List four checks before calling the selection rule wrong.

Solution

Check the reduced and radial matrix elements or configuration mixing; initial population; polarization and geometry; branching and competing decay; line overlap or broadening; detector sensitivity; and whether the assigned quantum numbers remain valid in the applied fields. Selection rules are normally necessary conditions, not guaranteed line-strength predictions.

Exercise 3: Two linewidths called gamma

One source defines $\Gamma$ through population decay, while another calls $\gamma$ the optical-coherence HWHM in angular frequency. Can the symbols be equated directly?

Solution

No. Derive the coherence equation in each convention. For an isolated radiative two-level system without pure dephasing, the coherence decays at $\Gamma/2$ , but extra dephasing changes that relation. Also verify whether the reported width is HWHM or FWHM and whether it uses $\omega$ or $\nu$ .

Exercise 4: Database provenance

You record a NIST wavelength in a notebook. What minimum metadata should accompany it?

Solution

Record the database name and version or update identifier, access date, species and isotope, charge state, both level labels, vacuum or air convention, displayed value and uncertainty, units, query filters, and the source reference attached to the record. Also save any conversion performed after the lookup.

Exercise 5: Choosing the first reference

A calculation needs the steady fluorescence of a driven transition with spontaneous emission. Should the selection-rule table or model index be the first stop?

Solution

Use both for different questions. The selection-rule table checks whether the chosen coupling is symmetry-permitted. The AMO Model Index then routes the dissipative driven problem to the optical Bloch equations. Neither the permission rule nor the model name alone supplies the dipole matrix element, decay branching, collection efficiency, and detector response needed for absolute counts.

Cross-Links

References

Bureau International des Poids et Mesures, The International System of Units (SI), 9th ed., version 4.01, 2026, doi:10.59161/AUEZ1291.
E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA Recommended Values of the Fundamental Physical Constants: 2022”, Journal of Physical and Chemical Reference Data 54, 033105, 2025.
A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASD Team, NIST Atomic Spectra Database, version 5.12, National Institute of Standards and Technology, 2024, doi:10.18434/T4W30F.
P. J. Linstrom and W. G. Mallard, eds., NIST Chemistry WebBook, NIST Standard Reference Database 69, National Institute of Standards and Technology.
International Union of Pure and Applied Chemistry, Compendium of Chemical Terminology, the Gold Book, 5th ed., online version 5.0.0, 2025.
P. F. Bernath, Spectra of Atoms and Molecules, 5th ed., Oxford University Press, 2025, doi:10.1093/oso/9780197754498.001.0001.
C. J. Foot, Atomic Physics, Oxford University Press, 2005.
W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed., Springer, 2014.