Oscillator Strength Reference

An oscillator strength is a dimensionless, energy-weighted measure of an electric-dipole transition. It is useful precisely because the same number can connect a structure calculation, an integrated absorption measurement, and a radiative transition probability. That portability is lost if the averaging, degeneracy, line grouping, or spectral coordinate is left implicit.

This page is a convention-explicit lookup for those translations. The derivations and physical interpretation live in Oscillator Strengths. The tables below are meant for checking a calculation, decoding a database column, or documenting a reported value without silently changing what was averaged or summed.

Canonical Scope

This page owns:

• a compact definition ledger for polarization-resolved, state-to-state, and level-averaged electric-dipole oscillator strengths;
• SI and atomic-unit relations among $f$, $gf$, $\log(gf)$, dipole matrix elements, reduced line strength $S$, and the Einstein $A$ coefficient;
• the Thomas–Reiche–Kuhn sum rule as a completeness audit;
• ordinary-frequency, angular-frequency, wavenumber, and molar-absorption area conversions;
• atomic line, multiplet, hyperfine, molecular electronic, vibronic, and rotational notation; and
• a reporting checklist for experimental and computational data.

It does not own:

• the commutator derivation of the sum rule;
• general time-dependent transition-rate theory;
• Einstein $A$–$B$ detailed balance;
• selection-rule derivations;
• Franck–Condon or Hönl–London derivations;
• forbidden-multipole transition formulas; or
• database-specific critical evaluation of an individual line.

Cross-links point to those canonical homes. Unless explicitly stated, “oscillator strength” on this page means an electric-dipole, or E1, absorption oscillator strength.

Declare the Object First

Before applying a formula, identify the object represented by the quoted number.

Question	Typical possibilities	Why it matters
What are the endpoints?	magnetic substates, fine-structure levels, LS terms, hyperfine levels, vibronic states, rotational lines, continuum channels	degeneracy factors and sums change
What was averaged?	one polarization, three Cartesian polarizations, initial magnetic substates, molecular orientations	factors of $3$, $2J+1$, or $2F+1$ can enter
What was summed?	final substates, unresolved fine structure, branches, a band, all bound states, bound plus continuum	a line value is not automatically a multiplet or band value
Which direction?	lower-to-upper absorption or upper-to-lower emission	the sign and degeneracy ratio differ
Which strength variable?	$f$, $gf$, $\log(gf)$, $S$, $A$, integrated cross section	these are related but not interchangeable
Which multipole?	E1, M1, E2, mixed M1+E2, or another process	the E1 formulas below do not transfer unchanged
Which units and axis?	SI or atomic units; $\nu$, $\omega$, $\widetilde\nu$, or $\lambda$	Jacobians and numerical constants differ

A useful one-line declaration is:

E1 absorption oscillator strength from lower fine-structure level $i$ to upper level $f$, averaged over the $g_i=2J_i+1$ lower substates and summed over upper substates and photon polarizations.

That sentence carries more information than several additional decimal places.

Convention Ledger

Throughout the main formulas:

• $i$ is the lower state or level and $f$ is the upper state or level;
• $\Delta E=E_f-E_i>0$ and $\omega_{fi}=\Delta E/\hbar>0$;
• $f_{if}>0$ is an absorption oscillator strength;
• $g_i$ and $g_f$ are the statistical weights of the lower and upper objects;
• $\widehat{\mathbf D}$ includes electric charge and has SI units $\mathrm{C\,m}$;
• $\widehat{\mathbf R}=\widehat{\mathbf D}/(-e)$ is the corresponding charge-free electronic position operator when that notation is useful;
• $S_{if}$ is an E1 reduced line strength, with its angular-momentum convention stated below; and
• line profiles are normalized to unit area in the spectral coordinate printed in their subscript.

The electron charge is $-e$, where $e>0$. A sign change in the dipole operator does not affect a strength because the matrix element is squared.

Definitions

Polarization-resolved state-to-state strength

For a specified unit polarization vector $\boldsymbol{\epsilon}$ and nondegenerate states,

$$f_{if}^{(\epsilon)} = \frac{2m_e\omega_{fi}} {\hbar e^2} \left| \boldsymbol{\epsilon}\mathbin{\cdot} \langle f|\widehat{\mathbf D}|i\rangle \right|^2.$$

This quantity describes the selected field component. It should not be averaged over three polarizations a second time.

Isotropic state-to-state strength

For an isotropic average over three orthogonal field directions,

$$\begin{aligned} f_{if} &= \frac{1}{3} \sum_{\alpha=x,y,z} f_{if}^{(\alpha)} \\ &= \frac{2m_e\omega_{fi}} {3\hbar e^2} \left| \langle f|\widehat{\mathbf D}|i\rangle \right|^2. \end{aligned}$$

Equivalently, using the charge-free position operator,

$$f_{if} = \frac{2m_e\Delta E} {3\hbar^2} \left| \langle f|\widehat{\mathbf R}|i\rangle \right|^2.$$

The factor $1/3$ is a polarization or orientation average. It is not a universal factor for every resolved experiment.

Level-averaged reduced strength

For atomic fine-structure levels, define the reduced E1 line strength

$$S_{if} = \left| \left\langle \gamma_fJ_f \left\| \widehat{\mathbf D}^{(1)} \right\| \gamma_iJ_i \right\rangle \right|^2.$$

With the standard Wigner–Eckart convention and $g_i=2J_i+1$,

$$f_{if} = \frac{2m_e\omega_{fi}} {3\hbar e^2g_i} S_{if}.$$

The lower-level average appears as $1/g_i$. The reduced matrix element has already summed the appropriate magnetic-sublevel and spherical-component content; adding another factor of three or $2J_i+1$ double-counts an average.

Atomic-unit form

When $\Delta E$ is in hartrees and $S_{if}$ is in $e^2a_0^2$,

$$f_{if} = \frac{2}{3} \frac{\Delta E}{g_i} S_{if}.$$

For nondegenerate states with $R_{if}=|\langle f|\widehat{\mathbf R}|i\rangle|$ in $a_0$,

$$f_{if} = \frac{2}{3} \Delta E\,R_{if}^2.$$

“Atomic units” must apply to both the transition energy and the matrix element. Inserting an energy in electronvolts into either formula gives a dimensionless but incorrect result.

Translation Ledger

Symbol	Definition in this reference	Units	Averaging direction
$f_{if}$	E1 absorption oscillator strength	dimensionless	lower object averaged, allowed final channels summed as declared
$g_if_{if}$ or $gf$	weighted absorption oscillator strength	dimensionless	removes the explicit lower-level average
$\log(gf)$	$\log_{10}(g_if_{if})$	dimensionless	same object as $gf$ on a base-10 logarithmic scale
$S_{if}$	squared reduced E1 matrix element	$\mathrm{C^2\,m^2}$ or $e^2a_0^2$	angular sums encoded by the reduced-matrix-element convention
$A_{fi}$	spontaneous-emission probability per unit time	$\mathrm{s^{-1}}$	upper object averaged, lower channels selected as declared
$\mathbf D_{fi}$	transition dipole including charge	$\mathrm{C\,m}$ or $ea_0$	state and polarization convention must be supplied
$\int\sigma_\nu\,d\nu$	frequency-integrated absorption cross section	$\mathrm{m^2\,Hz}$	same absorption object as $f_{if}$

Three distinctions prevent most conversion errors:

1. $f$ and $gf$ are not the same number.
2. $A_{fi}$ runs from the upper object to the lower object, whereas $f_{if}$ is conventionally an absorption strength.
3. $S$ is independent of transition energy, while $f\propto\omega$ and $A\propto\omega^3$ for a fixed E1 line strength.

Direct Conversion Formulas

Weighted strength and logarithmic strength

$$gf = g_i f_{if}, \qquad \log(gf) = \log_{10}\!\left(g_i f_{if}\right).$$

The logarithm is base ten. Recovering $f$ requires both the quoted logarithm and the lower statistical weight:

$$f_{if} = \frac{10^{\log(gf)}}{g_i}.$$

Absorption strength and spontaneous rate

For an E1 transition in vacuum SI units,

$$A_{fi} = \frac{e^2\omega_{fi}^{\,2}} {2\pi\epsilon_0m_ec^3} \frac{g_i}{g_f} f_{if}.$$

Equivalently,

$$f_{if} = \frac{2\pi\epsilon_0m_ec^3} {e^2\omega_{fi}^{\,2}} \frac{g_f}{g_i} A_{fi}.$$

In wavelength form,

$$A_{fi} = \frac{2\pi e^2} {\epsilon_0m_ec\lambda^2} \frac{g_i}{g_f} f_{if}.$$

The vacuum wavelength $\lambda=2\pi c/\omega_{fi}$ must be used in this relation. Air wavelengths can matter at the precision level of critically evaluated spectroscopy.

NIST-style numerical forms

For $\lambda$ in ångströms and $A_{fi}$ in $\mathrm{s^{-1}}$,

$$f_{if} = 1.49919\times10^{-16} \frac{g_f}{g_i} \lambda_{\mathring{\mathrm A}}^{\,2} A_{fi}.$$

The corresponding weighted form is

$$g_i f_{if} = 1.49919\times10^{-16} g_f\lambda_{\mathring{\mathrm A}}^{\,2}A_{fi}.$$

If a table prints $A$ in units of $10^8\,\mathrm{s^{-1}}$, replace the coefficient $10^{-16}$ by $10^{-8}$. Mixing those two column conventions creates an error of eight orders of magnitude.

For $S_{if}$ in atomic units,

$$A_{fi} = \frac{2.02613\times10^{18}} {g_f\lambda_{\mathring{\mathrm A}}^{\,3}} S_{if},$$

and

$$g_i f_{if} = \frac{303.756} {\lambda_{\mathring{\mathrm A}}} S_{if}.$$

These numerical forms assume an E1 reduced line strength in $e^2a_0^2$. Different multipoles use different dimensions and wavelength powers.

Reverse and signed strengths

If the same algebraic convention is extended to the reverse transition, $\omega_{if}<0$ and

$$g_i f_{if} = -g_f f_{fi}.$$

Thus an absorption strength is positive and its signed downward counterpart is negative. Some tables instead quote only positive magnitudes for both directions. Never infer the sign convention from subscripts alone.

Integrated Absorption

An oscillator strength controls the area of a weak, isolated absorption line, not its peak height. Let $L_\nu(\nu)$ be normalized by

$$\int_{-\infty}^{\infty} L_\nu(\nu)\,d\nu = 1.$$

Then

$$\sigma_\nu(\nu) = \frac{e^2} {4\epsilon_0m_ec} f_{if}L_\nu(\nu),$$

so

$$\int_{-\infty}^{\infty} \sigma_\nu(\nu)\,d\nu = \frac{e^2} {4\epsilon_0m_ec} f_{if}.$$

Numerically,

$$\frac{e^2} {4\epsilon_0m_ec} = 2.6540\times10^{-6} \ \mathrm{m^2\,Hz}.$$

Broadening changes the profile and peak while preserving this area in the linear, optically thin regime.

Angular-frequency axis

Since $\omega=2\pi\nu$ and $L_\omega=L_\nu/(2\pi)$,

$$\int \sigma_\omega(\omega)\,d\omega = \frac{\pi e^2} {2\epsilon_0m_ec} f_{if}.$$

The angular-frequency area is $2\pi$ times the ordinary-frequency area. A line profile written in $\omega$ cannot be inserted into a formula normalized in $\nu$ without the Jacobian.

Spectroscopic-wavenumber axis

For $\widetilde\nu=\nu/c$ in inverse metres,

$$\int \sigma_{\widetilde\nu}(\widetilde\nu) \,d\widetilde\nu = \frac{e^2} {4\epsilon_0m_ec^2} f_{if}.$$

If the wavenumber is reported in $\mathrm{cm^{-1}}$, the profile and integration measure must use that same unit.

Decadic molar absorption coefficient

Molecular spectroscopy often reports a decadic molar absorption coefficient $\epsilon(\widetilde\nu)$ in $\mathrm{L\,mol^{-1}\,cm^{-1}}$ and integrates it over $\widetilde\nu$ in $\mathrm{cm^{-1}}$. With Beer–Lambert absorbance

$$\mathcal A_{10} = \epsilon c_{\mathrm M}\ell_{\mathrm{cm}},$$

the corresponding band oscillator strength is

$$f = 4.319\times10^{-9} \int \epsilon(\widetilde\nu) \,d\widetilde\nu.$$

This numerical coefficient is valid only for the printed decadic, concentration, path-length, and wavenumber units. Napierian absorption, different concentration units, or integration over wavelength requires a different factor.

Sum Rules

Thomas–Reiche–Kuhn budget

For $N$ nonrelativistic electrons governed by a standard local Hamiltonian, the complete E1 strength from an initial state obeys

$$\sum_f f_{if} = N.$$

For a continuum, “sum” means

$$\sum_{f\in\mathrm{bound}} f_{if} + \int_{\mathrm{continuum}} \frac{df_i}{dE}\,dE = N.$$

The result is a spectral-weight budget. It does not say that every individual strength lies below one, and it does not imply that a short list of observed lines should nearly saturate the total.

Excited initial states

For an excited initial state, transitions to lower states contribute with the signed convention:

$$\sum_{E_f>E_i} f_{if} + \sum_{E_f<E_i} f_{if} = N, \qquad f_{if}<0 \ \text{when}\ E_f<E_i.$$

Summing positive emission magnitudes instead changes the mathematical object and does not test the same rule.

Conditions and useful failure signals

The elementary sum rule assumes:

• a complete bound-plus-continuum set;
• consistent Cartesian or isotropic normalization;
• canonical position and momentum commutators;
• a nonrelativistic Hamiltonian with the interaction assumptions used in the derivation; and
• consistent signed strengths for downward channels.

Relativistic models, nonlocal effective potentials, projected active spaces, pseudopotentials, finite basis sets, and truncated response calculations can modify the naive audit. A deficit can indicate omitted continuum strength or basis incompleteness; an excess can indicate double counting, inconsistent degeneracy factors, or use of positive downward magnitudes.

Partial-sum audit

Define a partial budget through energy cutoff $E_c$:

$$F_i(E_c) = \sum_{E_f\le E_c} f_{if} + \int_{E_{\mathrm{ion}}}^{E_c} \frac{df_i}{dE}\,dE.$$

A useful computational report gives $F_i(E_c)$ as the basis or continuum resolution is increased. Reporting only the final deviation from $N$ hides whether convergence is systematic or accidental.

Atomic Notation

Fine-structure lines

For a line

$$\gamma_iJ_i \longrightarrow \gamma_fJ_f,$$

the standard level weights are

$$g_i=2J_i+1, \qquad g_f=2J_f+1.$$

Atomic line lists commonly use lower index $i$ and upper index $k$, writing $f_{ik}$ and $A_{ki}$. This is the same direction convention as $f_{if}$ and $A_{fi}$ used here.

Database column	Read as	Do not read as
$A_{ki}$	upper-to-lower emission rate for one listed channel	total inverse lifetime unless every lower channel is included
$g_kA_{ki}$	upper-weighted transition probability	absorption $gf$
$f_{ik}$	lower-to-upper absorption oscillator strength	probability of absorption during an experiment
$\log(gf)$	$\log_{10}(g_if_{ik})$	$\log_{10}f_{ik}$
$S$ for E1	reduced dipole matrix element squared	oscillator strength without energy and degeneracy factors
relative intensity	source- and population-dependent line signal	intrinsic transition probability

The NIST Atomic Spectra Database also supplies accuracy codes and bibliographic provenance. A converted value inherits the source uncertainty; conversion does not turn a theoretical or semiempirical entry into a direct measurement.

Terms and multiplets

For an unresolved LS term, a term weight may be defined by

$$g_{\mathrm{term}} = \sum_J(2J+1) = (2S+1)(2L+1).$$

A multiplet line strength is a sum over its component line strengths:

$$S_{\mathrm M} = \sum_{\mathrm{lines}}S_{\mathrm{line}}.$$

An unweighted sum of component oscillator strengths is generally not the term-averaged multiplet oscillator strength because each component has its own lower-level weight and transition energy. Convert components to a common weighted convention before summing.

Hyperfine-resolved data

For a hyperfine level with total angular momentum $F$,

$$g_F=2F+1.$$

Use this weight only when the reported strength is genuinely resolved and averaged at the hyperfine-level scale. A fine-structure oscillator strength does not acquire a new $g_F$ merely because a later calculation partitions it among hyperfine components. In the absence of hyperfine mixing and with consistent angular factors, the resolved components reconstruct the parent fine-structure strength.

Molecular Notation

Molecular tables use “oscillator strength” for several nested objects. Always attach explicit quantum labels.

Molecular object	Example notation	Matrix element or sum represented
fixed-geometry electronic transition	$f_{if}(Q)$	electronic transition moment at nuclear geometry $Q$
vibronic line or band origin	$f_{v'v''}$	nuclear-motion integral of the coordinate-dependent electronic moment
rotational line	$f_{J'J''}$	one branch component with rotational angular factors
unresolved band	$f_{\mathrm{band}}$	declared sum over vibronic and/or rotational components
continuum strength	$df/dE$	density whose integral gives a dimensionless strength

Electronic and vibronic moments

For nuclear coordinates $Q$, an electronic transition moment is

$$\boldsymbol{\mu}_{fi}(Q) = \left\langle \psi_f(\mathbf r;Q) \left| \widehat{\mathbf D} \right| \psi_i(\mathbf r;Q) \right\rangle_{\mathbf r}.$$

The vibronic transition moment is

$$\mathbf D_{v'v''} = \int dQ\, \chi_{v'}^*(Q) \boldsymbol{\mu}_{fi}(Q) \chi_{v''}(Q).$$

In the Condon approximation, $\boldsymbol{\mu}_{fi}(Q)$ is replaced by a nearly constant value. The Franck–Condon factor then distributes squared overlap, but an oscillator strength still includes the individual transition energy:

$$f_{v'v''} \propto \Delta E_{v'v''} \left| \mathbf D_{v'v''} \right|^2.$$

Consequently, Franck–Condon fractions and oscillator-strength fractions are not exactly identical when the line energies differ. Coordinate dependence of the electronic moment introduces Herzberg–Teller intensity.

Rotational branches

Rotational line strengths are often written schematically as

$$S_{J'J''} = H_{J'J''} S_{\mathrm{vibronic}},$$

where $H_{J'J''}$ is a Hönl–London factor in a stated normalization. Some tables normalize the branch factors to one; others normalize their sum to a degeneracy. Convert the rotational factors and lower-state populations separately. A thermal absorption spectrum is not obtained by summing intrinsic $f_{J'J''}$ values without populations.

Continuum normalization

A continuum state normalized per unit energy gives a differential strength $df/dE$. One normalized per unit momentum or frequency gives a different density:

$$\frac{df}{dE}\,dE = \frac{df}{dk}\,dk = \frac{df}{d\omega}\,d\omega.$$

Only the integrated strength over the same physical interval is invariant. Quoting a continuum “oscillator strength” without its density variable and units is incomplete.

Computational Reporting

For an electronic-structure calculation, report:

1. the initial and final states, geometries, and state labels;
2. vertical, adiabatic, or vibronic transition energy as appropriate;
3. whether the value is state resolved, level averaged, orientation averaged, or band summed;
4. the dipole operator, gauge or form, and relativistic convention;
5. $f$, $gf$, $S$, or the transition moment, with units and degeneracies;
6. basis set, correlation model, active space, and continuum treatment;
7. length–velocity agreement when both are meaningful;
8. a partial or complete sum-rule audit when feasible; and
9. numerical convergence and comparison with critically evaluated data.

For exact eigenstates of a compatible Hamiltonian, length and velocity forms agree. In an approximate calculation,

$$\delta_{\mathrm{LV}} = \frac{ \left|f^{(\mathrm L)}-f^{(\mathrm V)}\right| }{ \tfrac12\left( \left|f^{(\mathrm L)}\right| + \left|f^{(\mathrm V)}\right| \right) }$$

is a useful internal diagnostic. It is not, by itself, a calibrated uncertainty. Both forms can agree while sharing the same model error, and weak lines can show large relative disagreement from small absolute matrix-element errors.

Measurement and Database Audit

Before using a tabulated strength:

1. identify the isotope, charge state, electronic configuration, term, and angular momentum of both levels;
2. distinguish observed from Ritz wavelength and air from vacuum wavelength;
3. verify whether $A$ is in $\mathrm{s^{-1}}$ or units of $10^8\,\mathrm{s^{-1}}$;
4. read $g_i$ and $g_f$ from the same level definitions used by the entry;
5. identify E1, M1, E2, mixed, or induced transition type;
6. distinguish a line from a multiplet, unresolved blend, or band;
7. retain the accuracy code, uncertainty, and transition-probability reference; and
8. compare converted quantities only after matching every convention above.

Observed emission intensity is not an intrinsic oscillator strength. Even in an optically thin source,

$$I_{fi} \propto N_fA_{fi}h\nu_{fi},$$

so the upper-level population $N_f$ and source conditions remain essential. Self-absorption, cascades, collisions, detector response, and unresolved blends can further alter the measured signal.

Common Mistakes

Treating oscillator strength as a probability

$f$ is dimensionless but is not bounded by one. It has no interaction time, photon flux, line profile, or population.

Using the wrong degeneracy

$f$ is normally averaged over the lower object, while $A$ is averaged over the upper object. The conversion therefore contains $g_i/g_f$.

Reading log(gf) as log(f)

The database quantity is usually $\log_{10}(g_if_{if})$. Omitting $g_i$ changes every line except one with $g_i=1$.

Mixing line and multiplet values

Component $f$ values cannot generally be added without weights. Use line strengths or a declared weighted convention.

Dropping the spectral Jacobian

Areas in $\nu$, $\omega$, $\widetilde\nu$, and $\lambda$ differ. Equal plotted shapes do not imply equal numerical integrals.

Applying E1 formulas to forbidden transitions

M1, E2, and mixed transitions have different operators, dimensions, and frequency powers. Use the transition type before the table column name.

Testing a sum rule with bound lines only

Continuum strength can carry a substantial part of the budget. For an excited initial state, downward terms must also be signed.

Calling gauge spread an uncertainty

Length–velocity disagreement is a diagnostic of approximation sensitivity, not a confidence interval.

Quick Reporting Template

A reproducible statement can be as compact as:

For the E1 transition $\gamma_iJ_i\rightarrow\gamma_fJ_f$, we report the absorption oscillator strength $f_{if}=x$ averaged over $g_i=2J_i+1$ lower substates and summed over final substates and photon polarizations. The transition energy is $\Delta E=y$ and the vacuum wavelength is $\lambda=z$. The corresponding values are $gf=g_if_{if}$, $S_{if}=s\,e^2a_0^2$, and $A_{fi}=a\,\mathrm{s^{-1}}$. Uncertainty, model, gauge, line grouping, and data provenance are stated separately.

For a molecule, replace the fine-structure labels by complete electronic, vibrational, rotational, parity, and branch labels, and state whether the sample is oriented or isotropic.

Exercises

Exercise 1: Dipole to oscillator strength

A nondegenerate transition has $\Delta E=0.0800\,E_{\mathrm h}$ and an isotropic charge-free transition matrix element of magnitude $R_{if}=2.00\,a_0$. Find $f_{if}$.

Solution

Use the nondegenerate atomic-unit formula:

$$\begin{aligned} f_{if} &= \frac{2}{3} \Delta E R_{if}^2 \\ &= \frac{2}{3} (0.0800)(2.00)^2 \\ &= 0.2133. \end{aligned}$$

The result is dimensionless. No extra factor of $e$, $a_0$, or $1/3$ is needed because the stated formula and matrix element already use the isotropic convention.

Exercise 2: Reduced line strength and log(gf)

An atomic transition has $J_i=1$, $\Delta E=0.0750\,E_{\mathrm h}$, and $S_{if}=6.00\,e^2a_0^2$. Find $f_{if}$, $gf$, and $\log(gf)$.

Solution

The lower weight is

$$g_i=2J_i+1=3.$$

Therefore

$$\begin{aligned} f_{if} &= \frac{2}{3} \frac{0.0750}{3} (6.00) = 0.100, \\ gf &= 3(0.100) = 0.300, \\ \log(gf) &= \log_{10}(0.300) = -0.5229. \end{aligned}$$

Taking $\log_{10}f$ instead would give $-1$, which is not the tabulated $\log(gf)$.

Exercise 3: Signed reverse strength

An absorption line has $g_i=2$, $g_f=4$, and $f_{if}=0.60$. Find the signed reverse strength $f_{fi}$.

Solution

Use detailed oscillator-strength bookkeeping:

$$g_i f_{if} = -g_f f_{fi}.$$

Thus

$$f_{fi} = -\frac{g_i}{g_f}f_{if} = -\frac{2}{4}(0.60) = -0.30.$$

A table that reports only positive emission magnitudes could print $0.30$, but that value must not be inserted as a positive term in the signed Thomas–Reiche–Kuhn sum.

Exercise 4: Convert a database A value

A listed E1 line has $\lambda=5000.0\,\mathring{\mathrm A}$, $A_{fi}=1.60\times10^7\,\mathrm{s^{-1}}$, $g_i=3$, and $g_f=5$. Find $f_{if}$ and $\log(gf)$.

Solution

Use the ångström form:

$$\begin{aligned} f_{if} &= 1.49919\times10^{-16} \frac{5}{3} (5000.0)^2 (1.60\times10^7) \\ &= 0.09995. \end{aligned}$$

Then

$$\begin{aligned} gf &= 3(0.09995) = 0.2998, \\ \log(gf) &= \log_{10}(0.2998) = -0.5231. \end{aligned}$$

Had the numerical value $1.60\times10^7$ been interpreted as a column in units of $10^8\,\mathrm{s^{-1}}$, the result would be physically and dimensionally inconsistent with the printed entry.

Exercise 5: Complete an excited-state sum-rule budget

For a two-electron system in an excited state, the known upward oscillator strengths sum to $1.35$. The signed downward contributions sum to $-0.20$. How much bound-plus-continuum strength remains unaccounted for?

Solution

The target is $N=2$. The known signed sum is

$$F_{\mathrm{known}} = 1.35-0.20 = 1.15.$$

Therefore

$$F_{\mathrm{missing}} = 2.00-1.15 = 0.85.$$

Adding the downward magnitude as $+0.20$ would incorrectly reduce the missing budget to $0.45$.

Exercise 6: Change the absorption axis

A line has $f_{if}=0.125$. Find its integrated cross section on an ordinary-frequency axis and on an angular-frequency axis.

Solution

On the ordinary-frequency axis,

$$\begin{aligned} \int\sigma_\nu\,d\nu &= (2.6540\times10^{-6})(0.125) \\ &= 3.3175\times10^{-7} \ \mathrm{m^2\,Hz}. \end{aligned}$$

The angular-frequency area is larger by $2\pi$:

$$\begin{aligned} \int\sigma_\omega\,d\omega &= 2\pi \int\sigma_\nu\,d\nu \\ &= 2.0845\times10^{-6} \ \mathrm{m^2\,rad\,s^{-1}}. \end{aligned}$$

The physical line is unchanged; only the density and integration coordinate have changed.

Exercise 7: Vibronic partition with unequal energies

In the Condon approximation, two vibronic channels have Franck–Condon factors $q_1=0.70$ and $q_2=0.30$. Their transition energies are $0.080\,E_{\mathrm h}$ and $0.100\,E_{\mathrm h}$, respectively. Neglect all other differences. What fractions of the two-channel oscillator strength do they carry?

Solution

Oscillator strength scales as $\Delta E q$ in this approximation:

$$w_1 = (0.080)(0.70) = 0.056,$$ $$w_2 = (0.100)(0.30) = 0.030.$$

After normalization,

$$\begin{aligned} \eta_1 &= \frac{0.056}{0.086} = 0.651, \\ \eta_2 &= \frac{0.030}{0.086} = 0.349. \end{aligned}$$

The Franck–Condon distribution is $70{:}30$, but the oscillator-strength distribution is approximately $65.1{:}34.9$ because the transition energies differ.

Exercise 8: Interpret gauge disagreement

A calculation gives $f^{(\mathrm L)}=0.240$ and $f^{(\mathrm V)}=0.216$. Evaluate the symmetric length–velocity diagnostic $\delta_{\mathrm{LV}}$. Can it be quoted as a one-standard-deviation uncertainty?

Solution

The diagnostic is

$$\begin{aligned} \delta_{\mathrm{LV}} &= \frac{|0.240-0.216|} {\tfrac12(0.240+0.216)} \\ &= \frac{0.024}{0.228} \\ &= 0.105 \approx 10.5\%. \end{aligned}$$

It cannot be interpreted as a calibrated one-standard-deviation uncertainty. It measures inconsistency between two approximate forms. A defensible uncertainty estimate also needs convergence tests, model comparisons, benchmarks, and experimental or critically evaluated evidence.

Cross-Links

• Oscillator Strengths
• Transition Rates
• Einstein Coefficient Reference
• Einstein Coefficients
• Absorption and Emission
• Electronic Spectroscopy
• Atomic Selection Rules
• Term Symbol Reference
• Selection Rule Tables
• Constants and Conversions
• Line Shape Reference
• Franck–Condon Factors
• Wigner–Eckart Theorem
• Sum Rules and Completeness Tricks
• Many-Body Sum Rules

References

• W. Thomas, Naturwissenschaften 13, 627 (1925), doi:10.1007/BF01558908.
• W. Kuhn, Zeitschrift für Physik 33, 408–412 (1925), doi:10.1007/BF01328322.
• F. Reiche and W. Thomas, Zeitschrift für Physik 34, 510–525 (1925), doi:10.1007/BF01328494.
• R. C. Hilborn, “Einstein coefficients, cross sections, $f$ values, dipole moments, and all that,” American Journal of Physics 50, 982–986 (1982), doi:10.1119/1.12937; revised version; erratum.
• I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer, 1992, doi:10.1007/978-3-642-76907-8.
• P. F. Bernath, Spectra of Atoms and Molecules, 5th ed., Oxford University Press, 2025, doi:10.1093/oso/9780197754498.001.0001.
• A. Kramida, “Spectral Lines: Selection Rules, Intensities, Transition Probabilities, Values, and Line Strengths”, in Atomic Spectroscopy: An Introduction, National Institute of Standards and Technology, accessed 2026-07-26.
• A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database, Standard Reference Database 78, doi:10.18434/T4W30F.
• A. Kramida and J. R. Fuhr, NIST Atomic Transition Probability Bibliographic Database, Standard Reference Database 110, version 9.0, doi:10.18434/T46C7N, accessed 2026-07-26.
• A. Kramida, “Evaluation of uncertainties in atomic data on spectral lines and transition probabilities,” European Physical Journal D 78 (2024), doi:10.1140/epjd/s10053-024-00820-y.
• International Union of Pure and Applied Chemistry, “Oscillator strength”, Compendium of Chemical Terminology, 5th ed., online version 5.0.0, 2025, doi:10.1351/goldbook.O04339.