Atomic Units

Hartree atomic units are the default working units of much of atomic physics, electronic-structure theory, and quantum chemistry. They make electron-scale Coulomb Hamiltonians compact, but the printed number is interpretable only after the unit convention, quantity, and conversion release have been stated.

This page is the AMO quick-lookup entry. It gives convention checks, versioned conversion factors, and dimensional restoration rules. The Atomic Units and Scales page owns the derivation from the Coulomb problem and the hierarchy of physical scales. The site-wide Atomic Units translator gives the short cross-volume convention entry, while Constants and Conversions owns general conversions among eV, Hz, $\mathrm{cm}^{-1}$, K, magnetic units, and laboratory dipole units.

Canonical Scope

Use this page when a calculation, table, or code output says “a.u.” and you need to answer one of four questions:

1. Does “a.u.” mean Hartree atomic units or a Rydberg-based convention?
2. Which atomic unit belongs to the reported quantity?
3. What multiplier converts the number to SI or a laboratory unit?
4. Which metadata are needed so another researcher can reproduce the conversion?

The central rule is

$$X=X_{\mathrm{au}}X_0,$$

where $X$ is the physical quantity, $X_{\mathrm{au}}$ is its numerical value in Hartree atomic units, and $X_0$ is the atomic unit with the same physical dimensions. A bare statement such as “$X=0.03$ a.u.” is incomplete unless $X$ is named: the unit for an energy is $E_{\mathrm h}$, the unit for a time is $\hbar/E_{\mathrm h}$, and the unit for an electric field is $E_{\mathrm h}/(ea_0)$.

Hartree Convention in One Line

Hartree atomic units set the numerical values

$$\hbar=m_e=e=4\pi\varepsilon_0=1.$$

This compact statement means:

• action is measured in units of $\hbar$;
• mass is measured in units of the electron mass $m_e$;
• charge is measured in units of the positive elementary-charge magnitude $e$; and
• the rationalized Coulomb constant $1/(4\pi\varepsilon_0)$ has numerical value one.

It does not mean that the physical constants cease to exist. They define the conversion between the dimensionless numerical representation and SI. Nor does it imply

$$\varepsilon_0=1.$$

Because the convention sets $4\pi\varepsilon_0=1$, the numerical value of $\varepsilon_0$ itself is $1/(4\pi)$ when it appears separately.

Signs and constants that remain visible

The symbol $e$ is a positive magnitude. The electron charge remains

$$q_e=-e=-1$$

in Hartree atomic units. Nuclear charges are $q_A=Z_A$, and nuclear masses are large dimensionless ratios $M_A/m_e$, not unity.

The speed of light is also not set to one:

$$c=\alpha^{-1}\approx137.036.$$

This makes the fine-structure constant $\alpha$ explicit as the parameter that separates nonrelativistic electron velocities from $c$. An expression that simultaneously assumes $c=1$ and Hartree atomic units is using an additional natural-unit convention and must say so.

Unit convention versus numerical conversion

There are two different kinds of equality on this page:

$$\hbar=m_e=e=4\pi\varepsilon_0=1$$

is exact by unit convention, whereas

$$a_0 =5.291\,772\,105\,44(82)\times10^{-11}\ \mathrm m$$

is a measured conversion factor from the 2022 CODATA adjustment. The parentheses give the standard uncertainty in the final quoted digits. Choosing atomic units exactly does not make their SI realizations exact.

Source Version and Status

The conversion factors below use the 2022 CODATA recommended values, NIST Web Version 9.0. NIST identifies this as the latest available CODATA adjustment as of 26 July 2026; the next regularly scheduled adjustment is the 2026 adjustment. SI status and defining constants follow version 4.01 of the ninth edition of the BIPM SI Brochure, published in June 2026.

This page uses three status labels:

Status	Meaning in this entry	Example
convention exact	fixed to one by the Hartree unit convention	$\hbar$, $m_e$, $e$, $4\pi\varepsilon_0$ as atomic-unit numerical values
SI exact or derived exact	fixed by the SI, perhaps through a nonterminating expression	$e$, $h$, $\hbar=h/(2\pi)$
CODATA measured	has a recommended value and standard uncertainty	$a_0$, $E_{\mathrm h}$, $t_0$

The first and third labels can apply to the same physical constant in different statements. For example, $m_e=1$ is exact in the atomic-unit representation, while $m_e=9.109\,383\,7139(28)\times10^{-31}\ \mathrm{kg}$ is a measured SI conversion.

Do not splice final digits from different adjustments. A reproducible result should name the release, such as “2022 CODATA,” or cite the source table.

Defining Scales

Bohr radius

The atomic unit of length is the Bohr radius,

$$\begin{aligned} a_0 &=\frac{4\pi\varepsilon_0\hbar^2}{m_e e^2},\\ &=\frac{\hbar}{\alpha m_e c}. \end{aligned}$$

Its 2022 CODATA value is

$$\begin{aligned} a_0 &=5.291\,772\,105\,44(82)\times10^{-11}\ \mathrm m,\\ &=0.529\,177\,210\,544(82)\ \text{\AA},\\ &=52.917\,721\,0544(82)\ \mathrm{pm}. \end{aligned}$$

The bohr is an electron-scale length, not a generic atomic radius. Many electron densities and bond lengths are of order a few $a_0$, but their values depend on state, charge, bonding, and the radius definition.

Useful reciprocal conversions are

$$1\ \text{\AA} \approx1.889\,726\,126\,a_0, \qquad 1\ \mathrm{nm} \approx18.897\,261\,26\,a_0.$$

Hartree energy

The atomic unit of energy is the Hartree energy,

$$\begin{aligned} E_{\mathrm h} &=\frac{\hbar^2}{m_ea_0^2},\\ &=\frac{e^2}{4\pi\varepsilon_0a_0},\\ &=\alpha^2m_ec^2,\\ &=2hcR_\infty. \end{aligned}$$

The identities expose several interpretations: kinetic energy on the scale $a_0$, Coulomb energy at separation $a_0$, a small fraction $\alpha^2$ of the electron rest energy, and twice the infinite-mass Rydberg energy.

The 2022 CODATA conversion is

$$\begin{aligned} E_{\mathrm h} &=4.359\,744\,722\,2060(48)\times10^{-18}\ \mathrm J,\\ &=27.211\,386\,245\,981(30)\ \mathrm{eV}. \end{aligned}$$

Rydberg energy and Rydberg constant

Define the infinite-mass Rydberg energy by

$$E_{\mathrm{Ry}} =hcR_\infty =\frac{E_{\mathrm h}}{2}.$$

Therefore

$$1E_{\mathrm h} =2E_{\mathrm{Ry}}, \qquad E_{\mathrm{Ry}} =13.605\,693\,122\,990(15)\ \mathrm{eV}.$$

Three objects must not be collapsed into the same symbol:

Object	Quantity type	Relation
Hartree energy $E_{\mathrm h}$	energy	$2hcR_\infty$
Rydberg energy $E_{\mathrm{Ry}}$	energy	$hcR_\infty=E_{\mathrm h}/2$
Rydberg constant $R_\infty$	inverse length	$E_{\mathrm{Ry}}/(hc)$

$E_{\mathrm{Ry}}$ is an infinite-nuclear-mass scale. It is not exactly the measured ionization energy of ordinary hydrogen, which includes reduced-mass, recoil, relativistic, radiative, finite-size, and hyperfine effects.

Atomic unit of time

The atomic unit of time is

$$t_0=\frac{\hbar}{E_{\mathrm h}} =2.418\,884\,326\,5864(26)\times10^{-17}\ \mathrm s.$$

Equivalently,

$$t_0 =24.188\,843\,265\,864(26)\ \mathrm{as} =0.024\,188\,843\,265\,864(26)\ \mathrm{fs}.$$

It is the inverse angular-frequency scale:

$$\frac{1}{t_0} =\frac{E_{\mathrm h}}{\hbar}.$$

The ordinary frequency corresponding to one Hartree is instead $E_{\mathrm h}/h$ and is smaller by $2\pi$.

Core Conversion Tables

The tables give multipliers for converting a numerical Hartree-atomic-unit value to SI:

$$X_{\mathrm{SI}} =X_{\mathrm{au}}\times \left(\text{SI value of }X_0\right).$$

To convert SI to atomic units, divide by the same multiplier. Parenthetical digits are standard uncertainties; displayed trailing digits should not be treated as the accuracy of a physical model.

Mechanical and kinematic quantities

Quantity	Atomic unit $X_0$	2022 CODATA SI value
charge	$e$	$1.602\,176\,634\times10^{-19}\ \mathrm C$ exact
mass	$m_e$	$9.109\,383\,7139(28)\times10^{-31}\ \mathrm{kg}$
action	$\hbar$	$1.054\,571\,817\ldots\times10^{-34}\ \mathrm{J\,s}$ derived exact
length	$a_0$	$5.291\,772\,105\,44(82)\times10^{-11}\ \mathrm m$
energy	$E_{\mathrm h}$	$4.359\,744\,722\,2060(48)\times10^{-18}\ \mathrm J$
time	$t_0=\hbar/E_{\mathrm h}$	$2.418\,884\,326\,5864(26)\times10^{-17}\ \mathrm s$
velocity	$a_0/t_0=\alpha c$	$2.187\,691\,262\,16(34)\times10^6\ \mathrm{m\,s^{-1}}$
momentum	$\hbar/a_0$	$1.992\,851\,915\,45(31)\times10^{-24}\ \mathrm{kg\,m\,s^{-1}}$
force	$E_{\mathrm h}/a_0$	$8.238\,723\,5038(13)\times10^{-8}\ \mathrm N$

Electromagnetic quantities

Quantity	Atomic unit $X_0$	2022 CODATA SI value
electric potential	$E_{\mathrm h}/e$	$27.211\,386\,245\,981(30)\ \mathrm V$
electric field	$E_{\mathrm h}/(ea_0)$	$5.142\,206\,751\,12(80)\times10^{11}\ \mathrm{V\,m^{-1}}$
electric-field gradient	$E_{\mathrm h}/(ea_0^2)$	$9.717\,362\,4424(30)\times10^{21}\ \mathrm{V\,m^{-2}}$
electric dipole moment	$ea_0$	$8.478\,353\,6198(13)\times10^{-30}\ \mathrm{C\,m}$
electric quadrupole moment	$ea_0^2$	$4.486\,551\,5185(14)\times10^{-40}\ \mathrm{C\,m^2}$
electric polarizability	$e^2a_0^2/E_{\mathrm h}$	$1.648\,777\,272\,12(51)\times10^{-41}\ \mathrm{C^2\,m^2\,J^{-1}}$
magnetic flux density	$\hbar/(ea_0^2)$	$2.350\,517\,570\,77(73)\times10^5\ \mathrm T$
magnetic dipole moment	$\hbar e/m_e=2\mu_{\mathrm B}$	$1.854\,802\,013\,15(58)\times10^{-23}\ \mathrm{J\,T^{-1}}$

The atomic unit of magnetic dipole moment is $2\mu_{\mathrm B}$, not $\mu_{\mathrm B}$. Many atomic-structure tables deliberately use the Bohr magneton instead. A value labeled only “a.u.” is therefore not enough to identify a magnetic-moment convention.

One-Hartree equivalents

Representation of $E_{\mathrm h}$	2022 CODATA value	Conversion used
joules	$4.359\,744\,722\,2060(48)\times10^{-18}\ \mathrm J$	direct
electronvolts	$27.211\,386\,245\,981(30)\ \mathrm{eV}$	$E_{\mathrm h}/e$
ordinary frequency	$6.579\,683\,920\,5000(72)\times10^{15}\ \mathrm{Hz}$	$E_{\mathrm h}/h$
angular frequency	$4.134\,137\,333\,517(5)\times10^{16}\ \mathrm{rad\,s^{-1}}$	$E_{\mathrm h}/\hbar$
vacuum wavenumber	$219\,474.631\,363\,14(24)\ \mathrm{cm^{-1}}$	$E_{\mathrm h}/(hc)$
energy-equivalent temperature	$315\,775.024\,804(35)\ \mathrm K$	$E_{\mathrm h}/k_{\mathrm B}$

The last displayed digits in values derived here from rounded table entries are for lookup, not a replacement for machine-readable CODATA data. For precision metrology, retrieve the value and covariance information directly from NIST.

Common compact conversions

Starting quantity	Equivalent
$1\ \text{\AA}$	$1.889\,726\,126\,a_0$
$1a_0$	$0.529\,177\,210\,544\ \text{\AA}$
$1E_{\mathrm h}$	$27.211\,386\,246\ \mathrm{eV}$
$1E_{\mathrm h}$	$2E_{\mathrm{Ry}}$
$1E_{\mathrm h}$	$219\,474.6314\ \mathrm{cm^{-1}}$
$1t_0$	$24.188\,8433\ \mathrm{as}$
$1ea_0$	$2.541\,746\,47\ \mathrm D$
$1\ \mathrm D$	$0.393\,430\,27\,ea_0$
$1\ \mathrm{kV\,cm^{-1}}$	$1.944\,690\,38\times10^{-7}$ a.u. of electric field
$1a_0^2$	$2.800\,285\,20\times10^{-21}\ \mathrm{m^2}$

Here the debye conversion uses

$$1\ \mathrm D =3.335\,640\,951\ldots\times10^{-30}\ \mathrm{C\,m}.$$

Hamiltonian Diagnostics

The coefficients in a Hamiltonian are often the fastest way to identify the unit convention.

One electron in a Coulomb field

For an infinitely heavy nucleus of charge $Z$, the Hartree-unit Hamiltonian is

$$H =-\frac12\nabla^2-\frac{Z}{r}.$$

The hydrogenic bound energies are

$$E_n=-\frac{Z^2}{2n^2}.$$

Thus hydrogen has $E_{1s}=-1/2$ in Hartree. The full derivation and wavefunction normalization live in Hydrogen Atom and Atomic Units and Scales.

Fixed-nucleus many-electron system

For nuclei at fixed positions $\mathbf R_A$, the electronic Hamiltonian in Hartree units is

$$\begin{aligned} H_{\mathrm{el}} ={}&-\frac12\sum_i\nabla_i^2 -\sum_{i,A}\frac{Z_A}{r_{iA}}\\ &+\sum_{i<j}\frac{1}{r_{ij}} +\sum_{A<B}\frac{Z_AZ_B}{R_{AB}}. \end{aligned}$$

Some electronic-structure programs omit the final nuclear-repulsion term from the electronic eigenvalue and add it when forming the total Born–Oppenheimer energy. The unit convention does not determine this bookkeeping choice; the program documentation must.

Coupling to an electric field

In the electric-dipole approximation,

$$H_{\mathrm{int}} =-\mathbf d\cdot\boldsymbol{\mathcal E}.$$

With $\mathbf d$ in $ea_0$ and $\boldsymbol{\mathcal E}$ in $E_{\mathrm h}/(ea_0)$, their product is automatically in Hartree:

$$(ea_0) \left(\frac{E_{\mathrm h}}{ea_0}\right) =E_{\mathrm h}.$$

The field may be a static value, a peak optical amplitude, a root-mean-square amplitude, or a complex positive-frequency amplitude. Atomic units do not remove that convention choice.

Time and Frequency

Time-dependent Schrödinger evolution illustrates why inverse atomic time is an angular-frequency unit. In SI,

$$\lvert\psi(t)\rangle =e^{-iHt/\hbar}\lvert\psi(0)\rangle.$$

Writing

$$H=E_{\mathrm h}H_{\mathrm{au}}, \qquad t=t_0t_{\mathrm{au}},$$

and using $E_{\mathrm h}t_0/\hbar=1$ gives

$$\lvert\psi(t_{\mathrm{au}})\rangle =e^{-iH_{\mathrm{au}}t_{\mathrm{au}}} \lvert\psi(0)\rangle.$$

For an energy gap $\Delta E$,

$$\begin{aligned} \omega &=\frac{\Delta E}{\hbar},\\ \nu &=\frac{\Delta E}{h} =\frac{\omega}{2\pi}. \end{aligned}$$

Consequently,

$$\begin{aligned} 1\ \text{a.u. of inverse time} &=\frac{E_{\mathrm h}}{\hbar} \approx4.1341\times10^{16}\ \mathrm{rad\,s^{-1}},\\ \frac{1E_{\mathrm h}}{h} &\approx6.5797\times10^{15}\ \mathrm{Hz}. \end{aligned}$$

Calling $E_{\mathrm h}/\hbar$ “hertz” creates a factor-of-$2\pi$ error. For a reported Rabi frequency, linewidth, or decay parameter, retain the defining equation and state whether the displayed number is divided by $2\pi$.

Electric Fields, Dipoles, and Polarizabilities

Electric-field scale

The atomic field

$$\mathcal E_0 =\frac{E_{\mathrm h}}{ea_0} \approx5.1422\times10^{11}\ \mathrm{V\,m^{-1}}$$

changes the potential energy of one elementary charge by one Hartree over one Bohr radius. Laboratory fields are therefore often small as atomic-unit numbers:

$$10\ \mathrm{kV\,cm^{-1}} \approx1.9447\times10^{-6}\ \text{a.u.}$$

Small in atomic units does not automatically mean perturbatively weak. The relevant comparison is between the coupling matrix element and the energy separation of the states being mixed.

Dipole scale

The atomic dipole unit is

$$d_0=ea_0.$$

A calculated transition dipole $d_{\mathrm{au}}=2.0$ therefore means

$$\begin{aligned} d &=2.0ea_0,\\ &\approx5.0835\ \mathrm D,\\ &\approx1.6957\times10^{-29}\ \mathrm{C\,m}. \end{aligned}$$

The sign or phase of an off-diagonal dipole matrix element depends on state phase conventions. Observable line strengths use phase-invariant combinations, together with angular and degeneracy factors.

Polarizability convention

For a static scalar polarizability defined by

$$\Delta E =-\frac12\alpha_E\mathcal E^2,$$

the Hartree atomic unit is

$$\alpha_{E,0} =\frac{e^2a_0^2}{E_{\mathrm h}} =4\pi\varepsilon_0a_0^3.$$

The final identity explains a persistent notation trap. A tabulated polarizability “in $a_0^3$” usually reports the same numerical value as the Hartree atomic-unit polarizability, but the SI polarizability is not obtained by multiplying by a bare volume. It is obtained from

$$\alpha_E^{(\mathrm{SI})} =\alpha_{E,\mathrm{au}} \left(4\pi\varepsilon_0a_0^3\right).$$

If a source instead tabulates the polarizability volume $\alpha_E/(4\pi\varepsilon_0)$, the conversion multiplier is $a_0^3$. Record which quantity the table means.

Optical intensity is amplitude-dependent

For a plane wave in vacuum whose electric-field peak amplitude is $\mathcal E_{\mathrm{pk}}$,

$$I =\frac12c\varepsilon_0 \mathcal E_{\mathrm{pk}}^2.$$

If the quoted field is root-mean-square,

$$I =c\varepsilon_0 \mathcal E_{\mathrm{rms}}^2.$$

An “atomic unit of intensity” can therefore differ by a factor of two when the amplitude convention is omitted. Convert the explicitly defined field amplitude first, then apply the corresponding intensity formula.

Dimensional Restoration

Memorized tables are useful, but dimensional restoration handles quantities that are not listed.

Suppose a quantity $X$ has dimensions

$$[X] =M^aL^bT^cQ^d,$$

where $Q$ denotes electric-charge dimension. Its Hartree atomic unit is

$$X_0 =m_e^a a_0^b t_0^c e^d.$$

Then

$$X=X_{\mathrm{au}}X_0.$$

Examples follow directly:

Quantity	Dimensions	Atomic unit
area or cross section	$L^2$	$a_0^2$
volume	$L^3$	$a_0^3$
number density	$L^{-3}$	$a_0^{-3}$
rate	$T^{-1}$	$t_0^{-1}$
current	$QT^{-1}$	$e/t_0$
momentum	$MLT^{-1}$	$m_ea_0/t_0=\hbar/a_0$
force	$MLT^{-2}$	$m_ea_0/t_0^2=E_{\mathrm h}/a_0$
dipole moment	$QL$	$ea_0$

Wavefunctions and densities

A normalized three-dimensional wavefunction satisfies

$$\int \lvert\psi(\mathbf r)\rvert^2\,d^3r =1.$$

If $\mathbf r=a_0\mathbf r_{\mathrm{au}}$, then

$$\psi(\mathbf r) =a_0^{-3/2} \psi_{\mathrm{au}}(\mathbf r_{\mathrm{au}}).$$

Therefore a probability density reported in atomic units carries the scale $a_0^{-3}$. A radial function may use a different normalization and power of $a_0$; inspect its defining integral before restoring units.

Rates and cross sections

A decay rate $\Gamma_{\mathrm{au}}$ converts as

$$\Gamma_{\mathrm{SI}} =\frac{\Gamma_{\mathrm{au}}}{t_0}.$$

A cross section converts as

$$\sigma_{\mathrm{SI}} =\sigma_{\mathrm{au}}a_0^2.$$

Because

$$a_0^2 \approx2.8003\times10^{-17}\ \mathrm{cm^2},$$

a cross section of $10^{-4}$ a.u. is approximately $2.80\times10^{-21}\ \mathrm{cm^2}$. This conversion says nothing about whether the underlying scattering approximation is accurate.

Hartree and Rydberg Conventions

The abbreviation “a.u.” is sometimes used for two related but numerically different systems.

In Hartree units,

$$H_{\mathrm H} =-\frac12\nabla^2-\frac{Z}{r}, \qquad E_{1s}=-\frac12.$$

Dividing the same physical Hamiltonian by $E_{\mathrm{Ry}}=E_{\mathrm h}/2$ gives Rydberg units:

$$H_{\mathrm{Ry}} =-\nabla^2-\frac{2Z}{r}, \qquad E_{1s}=-1.$$

For a many-electron Coulomb Hamiltonian, every energy coefficient doubles under the same change of energy unit. A number expressed in Rydberg converts to Hartree through

$$E[\mathrm{Ha}] =\frac12E[\mathrm{Ry}].$$

Do not infer the convention from a file extension or a field name alone. Check:

1. the kinetic-energy coefficient;
2. the documented energy unit;
3. a known benchmark such as hydrogen $1s$;
4. whether a cutoff, eigenvalue, total energy, and force use the same unit;
5. whether the output labels Hartree as Ha, Eh, or hartree.

For example, official Quantum ESPRESSO documentation states that much of PWscf uses Rydberg units while CP uses Hartree atomic units. This is a program-specific contract, not a universal rule for plane-wave codes.

Nuclear Mass and Isotope Conventions

The atomic unit of mass is the electron mass:

$$m_0=m_e.$$

It is not the unified atomic mass unit $\mathrm u$. A nuclear mass supplied in unified atomic mass units must be converted to electron-mass units before entering a Hartree-unit Hamiltonian:

$$M_{\mathrm{au}} =\frac{M}{m_e}.$$

For a two-body Coulomb problem, the reduced mass is

$$\mu =\frac{m_eM}{m_e+M},$$

or numerically in electron-mass units,

$$\mu_{\mathrm{au}} =\frac{M_{\mathrm{au}}}{1+M_{\mathrm{au}}}.$$

One may introduce problem-specific reduced-mass-scaled units, but those are not the CODATA bohr and Hartree unless explicitly declared. In standard Hartree units, isotope and recoil effects belong in the Hamiltonian rather than in a silent redefinition of $a_0$.

The same care applies to molecular coordinates. Electronic coordinates are often in bohr, nuclear masses in electron masses, vibrational frequencies in $E_{\mathrm h}/\hbar$, and final spectra in $\mathrm{cm}^{-1}$. Every handoff must preserve the quantity and unit, not only the number.

Worked Conversions

Length from ångström to bohr

For a bond length $R=1.40\ \text{\AA}$,

$$\begin{aligned} R_{\mathrm{au}} &=\frac{1.40\ \text{\AA}} {0.529\,177\,210\,544\ \text{\AA}},\\ &\approx2.646. \end{aligned}$$

The result should not be printed to twelve digits when the input was given to three significant figures.

Energy in spectroscopic forms

For $\Delta E=0.125E_{\mathrm h}$,

$$\begin{aligned} \Delta E &\approx3.4014\ \mathrm{eV},\\ \frac{\Delta E}{hc} &\approx27\,434.3\ \mathrm{cm^{-1}},\\ \frac{\Delta E}{h} &\approx822.460\ \mathrm{THz},\\ \frac{\Delta E}{k_{\mathrm B}} &\approx39\,471.9\ \mathrm K. \end{aligned}$$

The kelvin value is an energy-equivalent temperature, not a claim that the system is in thermal equilibrium at that temperature.

Propagation time

A real-time calculation propagated for $2500t_0$ covers

$$\begin{aligned} t &=2500 \left(2.418\,884\,3266\times10^{-17}\ \mathrm s\right),\\ &\approx60.4721\ \mathrm{fs}. \end{aligned}$$

The timestep must also be converted. Reporting only the number of steps does not specify the simulated time interval.

Field–dipole interaction

Take a dipole matrix element $d=2.3ea_0$ and a peak field $\mathcal E=50\ \mathrm{kV\,cm^{-1}}=5.0\times10^6\ \mathrm{V\,m^{-1}}$. The field in atomic units is

$$\mathcal E_{\mathrm{au}} =\frac{5.0\times10^6} {5.1422\times10^{11}} \approx9.72\times10^{-6}.$$

The coupling scale is

$$\begin{aligned} d\mathcal E &\approx2.24\times10^{-5}E_{\mathrm h},\\ &\approx6.09\times10^{-4}\ \mathrm{eV},\\ \frac{d\mathcal E}{h} &\approx147\ \mathrm{GHz}. \end{aligned}$$

This is the magnitude before angular factors, detuning, polarization projection, and any peak-versus-complex-amplitude factor are applied.

Reporting a Reproducible Result

At minimum, a computational or experimental handoff should record:

Field	Example
quantity	total electronic energy
numerical value	$-75.983\,948$
unit	Hartree, not merely a.u.
convention	Hartree atomic units
physical model	clamped nuclei, nonrelativistic electronic Hamiltonian
reference zero	separated nuclei and electrons, or code-specific zero
conversion release	2022 CODATA
output provenance	program, version, input, method, basis, and convergence settings

For a field or time-dependent interaction, also record:

• peak, root-mean-square, or complex field amplitude;
• cyclic or angular frequency;
• gauge and interaction convention when relevant;
• pulse-envelope definition and full-width convention; and
• whether quoted rates are population or amplitude decay rates.

The converted number should not imply more knowledge than the original result. If

$$E=0.125E_{\mathrm h}$$

is specified to three significant figures, conversion with a twelve-significant-figure CODATA multiplier does not make the physical energy twelve-digit accurate.

Common Mistakes

Treating “a.u.” as a complete unit label

Atomic-unit dimensions depend on the quantity. Write Hartree for energy, bohr for length, $ea_0$ for dipole moment, or explicitly define the unit.

Confusing Hartree and Rydberg

$1E_{\mathrm h}=2E_{\mathrm{Ry}}$. Inspect the kinetic coefficient and a known eigenvalue before comparing outputs.

Confusing atomic mass and unified atomic mass

The Hartree atomic mass unit is $m_e$. The unified atomic mass unit $\mathrm u$ is roughly a nucleon mass and is about $1823m_e$.

Forgetting the electron charge sign

$e=1$ denotes a positive magnitude. The electron charge is $-1$.

Calling inverse atomic time hertz

$t_0^{-1}=E_{\mathrm h}/\hbar$ is an angular-frequency scale. Divide by $2\pi$ to obtain cycles per second.

Multiplying mixed-unit quantities

A dipole in $ea_0$ times a field in V/m is not yet an energy. Convert both quantities to one coherent system before multiplying.

Converting polarizability as a bare volume

The SI multiplier for Hartree atomic polarizability is $4\pi\varepsilon_0a_0^3$, not just $a_0^3$.

Hiding an optical-amplitude convention

Peak and root-mean-square fields differ by $\sqrt{2}$ for a sinusoid, so intensities inferred from them differ by a factor of two.

Redefining the bohr for each isotope

Reduced-mass-scaled coordinates can be useful, but they are a separate declared convention. Standard $a_0$ uses $m_e$.

Combining CODATA releases

Values are correlated products of one adjustment. Name one release and keep it throughout a precision calculation.

Reporting converted digits as model accuracy

Unit conversion preserves uncertainty and model error; it does not improve either.

Exercises

Exercise 1: Identify the energy convention

Two programs give the hydrogen ground-state energy as $-0.5$ and $-1.0$, respectively. Program A uses

$$H_A=-\frac12\nabla^2-\frac1r,$$

while program B uses

$$H_B=-\nabla^2-\frac2r.$$

Are the physical predictions inconsistent?

Solution

No. Program A divides physical energy by $E_{\mathrm h}$, while program B divides by $E_{\mathrm{Ry}}=E_{\mathrm h}/2$. Therefore

$$-0.5E_{\mathrm h} =-1.0E_{\mathrm{Ry}}.$$

The kinetic and Coulomb coefficients change together under the factor-of-two rescaling. A valid comparison first converts both outputs to the same unit and checks that their reference zeros and physical approximations also agree.

Exercise 2: Convert an electronic energy gap

Convert $\Delta E=0.0200E_{\mathrm h}$ to eV, ordinary frequency in THz, and vacuum wavenumber in $\mathrm{cm}^{-1}$. State an appropriate number of significant figures.

Solution

Use the one-Hartree table:

$$\begin{aligned} \Delta E &=0.0200 \left(27.211\,386\,246\ \mathrm{eV}\right),\\ &=0.544\ \mathrm{eV},\\ \frac{\Delta E}{h} &=0.0200 \left(6579.6839\ \mathrm{THz}\right),\\ &=132\ \mathrm{THz},\\ \frac{\Delta E}{hc} &=0.0200 \left(219\,474.6314\ \mathrm{cm^{-1}}\right),\\ &=4.39\times10^3\ \mathrm{cm^{-1}}. \end{aligned}$$

The input has three significant figures, so defensible reported values are $0.544\ \mathrm{eV}$, $132\ \mathrm{THz}$, and $4.39\times10^3\ \mathrm{cm^{-1}}$. Extra CODATA digits belong in the intermediate calculation, not the final claim.

Exercise 3: Time and angular frequency

A two-state energy gap is $0.010E_{\mathrm h}$. Find its angular frequency, ordinary frequency, and period. Explain the role of $2\pi$.

Solution

The angular frequency is

$$\begin{aligned} \omega &=0.010\frac{E_{\mathrm h}}{\hbar},\\ &\approx4.13\times10^{14}\ \mathrm{rad\,s^{-1}}. \end{aligned}$$

The ordinary frequency is

$$\begin{aligned} \nu &=\frac{\omega}{2\pi},\\ &=0.010\frac{E_{\mathrm h}}{h},\\ &\approx6.58\times10^{13}\ \mathrm{Hz}. \end{aligned}$$

Hence

$$T=\frac1\nu\approx1.52\times10^{-14}\ \mathrm s=15.2\ \mathrm{fs}.$$

$E_{\mathrm h}/\hbar$ counts phase in radians per second, while $E_{\mathrm h}/h$ counts cycles per second. Omitting $2\pi$ would confuse these two quantities.

Exercise 4: Dipole and laboratory field

A transition dipole is $3.00\ \mathrm D$ and a field is $1.00\ \mathrm{kV\,cm^{-1}}$. Convert each to atomic units and estimate $d\mathcal E$ in Hartree.

Solution

The dipole is

$$d_{\mathrm{au}} =3.00 \left(0.393\,430\,27\right) \approx1.18.$$

The field is

$$\mathcal E_{\mathrm{au}} =1.944\,69\times10^{-7}.$$

Their product is

$$\begin{aligned} d\mathcal E &\approx (1.18)(1.94469\times10^{-7})E_{\mathrm h},\\ &\approx2.30\times10^{-7}E_{\mathrm h}. \end{aligned}$$

This is a coupling scale. A transition Hamiltonian may include polarization, angular-momentum, rotating-wave, or field-amplitude factors, so one should not identify it with an observed splitting without specifying the model.

Exercise 5: Restore units by dimensions

A scattering calculation reports

$$\sigma=6.0\times10^{-3}$$

in Hartree atomic units. Convert the cross section to $\mathrm{cm^2}$.

Solution

A cross section has dimensions of area, so its atomic unit is $a_0^2$:

$$\sigma =6.0\times10^{-3}a_0^2.$$

Using

$$a_0^2 \approx2.8003\times10^{-17}\ \mathrm{cm^2},$$

gives

$$\sigma \approx1.7\times10^{-19}\ \mathrm{cm^2}.$$

The two-significant-figure result reflects the precision of the input. No special cross-section conversion rule was required; dimensional restoration was enough.

Exercise 6: Polarizability convention

A database lists a static polarizability as $120$ a.u. Write the conversion to SI polarizability and to the polarizability volume in $a_0^3$. Why are these not the same dimensional statement?

Solution

For the Stark-shift definition

$$\Delta E=-\frac12\alpha_E\mathcal E^2,$$

the SI polarizability is

$$\begin{aligned} \alpha_E^{(\mathrm{SI})} &=120 \left( 1.648\,777\,272\,12\times10^{-41} \right)\\ &\quad\times \mathrm{C^2\,m^2\,J^{-1}},\\ &\approx1.98\times10^{-39} \ \mathrm{C^2\,m^2\,J^{-1}}. \end{aligned}$$

The corresponding polarizability volume is

$$\frac{\alpha_E}{4\pi\varepsilon_0} =120a_0^3.$$

The numerical value $120$ is shared because

$$\alpha_{E,0}=4\pi\varepsilon_0a_0^3,$$

but SI polarizability and volume have different dimensions. A source that prints only “$120a_0^3$” should state that it is reporting $\alpha_E/(4\pi\varepsilon_0)$.

Exercise 7: Isotope-scaled units

A paper replaces $m_e$ by the electron–nucleus reduced mass $\mu$ when defining its length scale and then calls the result “one bohr.” Is that standard Hartree atomic-unit usage?

Solution

No. The standard CODATA bohr is

$$a_0 =\frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}$$

and uses $m_e$. Replacing $m_e$ by $\mu$ defines a useful problem-specific scaled length,

$$a_\mu =\frac{4\pi\varepsilon_0\hbar^2}{\mu e^2},$$

but $a_\mu$ is isotope-dependent and is not numerically identical to $a_0$. The paper can use that convention if it declares it; a data handoff must preserve the distinction. In standard Hartree units, one instead keeps $a_0$ and inserts the reduced-mass factor in the Hamiltonian.

Exercise 8: Audit a reproducibility record

A methods section states: “The pulse amplitude was $0.001$ a.u., the frequency was $0.05$ a.u., and the calculation ran for 5000 steps.” List the missing information needed to reproduce the physical field and duration.

Solution

At least the following are missing:

1. whether the amplitude is an electric field, vector potential, or another quantity;
2. whether the field amplitude is peak, root-mean-square, or a complex positive-frequency amplitude;
3. whether “frequency” means angular frequency in $t_0^{-1}$ or cyclic frequency;
4. the timestep in $t_0$ or seconds;
5. the pulse-envelope definition, carrier phase, and duration convention;
6. whether the code uses Hartree or Rydberg units;
7. the code and version implementing the convention; and
8. the CODATA release used for any SI conversion.

If the first number is a peak electric field in Hartree units, it would mean $0.001E_{\mathrm h}/(ea_0)$. If the second is an angular frequency, it means $0.05E_{\mathrm h}/\hbar$. The total physical duration cannot be recovered from 5000 steps without the timestep.

Cross-Links

• Reference and Data is the task-oriented gateway for AMO lookups.
• Common Atomic Hamiltonians shows where each atomic-unit quantity enters standard operator models.
• Atomic Units and Scales derives the unit system and develops physical scale intuition.
• Constants and Conversions gives the general eV, Hz, $\mathrm{cm}^{-1}$, K, magnetic, and dipole conversion spine.
• Site-Wide Atomic Units Translator gives the short convention entry used across volumes.
• Units and Constants states the project-wide reporting policy.
• Hydrogen as Atomic Prototype applies the Coulomb scales to real atomic structure.
• Hydrogen Atom is the canonical solution of the nonrelativistic Coulomb bound-state problem.
• Computational AMO and Quantum Chemistry connects unit declarations to electronic-structure and time-dependent workflows.
• Reproducibility Benchmarks gives artifact-backed checks for computational handoffs.

References

1. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Journal of Physical and Chemical Reference Data 54, 033105 (2025), DOI: 10.1063/5.0279860. Table XXXV is the primary source for the atomic-unit conversions used here.
2. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Reviews of Modern Physics 97, 025002 (2025), DOI: 10.1103/RevModPhys.97.025002.
3. E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, The 2022 CODATA Recommended Values of the Fundamental Physical Constants, Web Version 9.0, National Institute of Standards and Technology (2024).
4. Bureau International des Poids et Mesures, The International System of Units (SI), 9th ed., version 4.01 (2026), DOI: 10.59161/AUEZ1291.
5. C. M. A. Brett, J. G. Frey, R. Hinde, Y. Kuroda, R. Marquardt, F. Pavese, M. Quack, J. Stohner, and A. J. Thor, eds., Quantities, Units and Symbols in Physical Chemistry, abridged 4th ed., IUPAC and Royal Society of Chemistry, 2023.
6. E. Tiesinga, “Units and constants”, in Springer Handbook of Atomic, Molecular, and Optical Physics (2023).
7. T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory, Wiley, 2000.
8. P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 5th ed., Oxford University Press, 2010.
9. Quantum ESPRESSO Foundation, “What are the units for quantity XYZ?”, official user documentation, accessed 26 July 2026.