The atomic units are a system of natural units of measurement which is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree.^[1] Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.

Motivation

In the context of atomic physics, using the atomic units system can be a convenient shortcut, eliminating unnecessary symbols and numbers with very small orders of magnitude. For example, the Hamiltonian operator in the Schrödinger equation for the helium atom with standard quantities, such as when using SI units, is^[2]

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla _{1}^{2}-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla _{2}^{2}-{\frac {2e^{2}}{4\pi \epsilon _{0}r_{1}}}-{\frac {2e^{2}}{4\pi \epsilon _{0}r_{2}}}+{\frac {e^{2}}{4\pi \epsilon _{0}r_{12}}},}$

but adopting the convention associated with atomic units that transforms quantities into dimensionless equivalents, it becomes

${\displaystyle {\hat {H}}=-{\frac {1}{2}}\nabla _{1}^{2}-{\frac {1}{2}}\nabla _{2}^{2}-{\frac {2}{r_{1}}}-{\frac {2}{r_{2}}}+{\frac {1}{r_{12}}}.}$

In this convention, the constants ${\displaystyle \hbar }$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle 4\pi \epsilon _{0}}$, and ${\displaystyle e}$ all correspond to the value ${\displaystyle 1}$ (see § Definition below). The distances relevant to the physics expressed in SI units are naturally on the order of ${\displaystyle 10^{-10}\,\mathrm {m} }$, while expressed in atomic units distances are on the order of ${\displaystyle 1a_{0}}$ (one Bohr radius, the atomic unit of length). An additional benefit of expressing quantities using atomic units is that their values calculated and reported in atomic units do not change when values of fundamental constants are revised. The fundamental constants are built into the conversion factors between atomic units and SI.

History

Hartree defined units based on three physical constants:^[1]: 91

Both in order to eliminate various universal constants from the equations and also to avoid high powers of 10 in numerical work, it is convenient to express quantities in terms of units, which may be called 'atomic units', defined as follows:

Unit of length, ${\displaystyle a_{\text{H}}=h^{2}\,/\,4\pi ^{2}me^{2}}$, on the orbital mechanics the radius of the 1-quantum circular orbit of the H-atom with fixed nucleus.

Unit of charge, ${\displaystyle e}$, the magnitude of the charge on the electron.

Unit of mass, ${\displaystyle m}$, the mass of the electron.

Consistent with these are:

Unit of action, ${\displaystyle h\,/\,2\pi }$.

Unit of energy, ${\displaystyle e^{2}/a_{\text{H}}=2hcR=}$ [...]

Unit of time, ${\displaystyle 1\,/\,4\pi cR}$.
 

— D.R. Hartree, The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

Here, the modern equivalent of ${\displaystyle R}$ is the Rydberg constant ${\displaystyle R_{\infty }}$, of ${\displaystyle m}$ is the electron mass ${\displaystyle m_{\text{e}}}$, of ${\displaystyle a_{\text{H}}}$ is the Bohr radius ${\displaystyle a_{0}}$, and of ${\displaystyle h/2\pi }$ is the reduced Planck constant ${\displaystyle \hbar }$. Hartree's expressions that contain ${\displaystyle e}$ differ from the modern form due to a change in the definition of ${\displaystyle e}$, as explained below.

In 1957, Bethe and Salpeter's book Quantum mechanics of one-and two-electron atoms^[3] built on Hartree's units, which they called atomic units abbreviated "a.u.". They chose to use ${\displaystyle \hbar }$, their unit of action and angular momentum in place of Hartree's length as the base units. They noted that the unit of length in this system is the radius of the first Bohr orbit and their velocity is the electron velocity in Bohr's model of the first orbit.

In 1959, Shull and Hall^[4] advocated atomic units based on Hartree's model but again chose to use ${\displaystyle \hbar }$ as the defining unit. They explicitly named the distance unit a "Bohr radius"; in addition, they wrote the unit of energy as ${\displaystyle H=me^{4}/\hbar ^{2}}$ and called it a Hartree. These terms came to be used widely in quantum chemistry.^[5]: 349

In 1973 McWeeny extended the system of Shull and Hall by adding permittivity in the form of ${\displaystyle \kappa _{0}=4\pi \epsilon _{0}}$ as a defining or base unit.^[6][7] Simultaneously he adopted the SI definition of ${\displaystyle e}$ so that his expression for energy in atomic units is ${\displaystyle e^{2}/(4\pi \epsilon _{0}a_{0})}$, matching the expression in the 8th SI brochure.^[8]

Definition

A set of base units in the atomic system as in one proposal are the electron rest mass, the magnitude of the electronic charge, the Planck constant, and the permittivity.^[6] In the atomic units system, each of these takes the value 1; the corresponding values in the International System of Units^[9]: 132 are given in the table.

Base atomic units^†

Symbol and Name	Quantity (dimensions)‡	Atomic units⹋	SI units
${\displaystyle \hbar }$, reduced Planck constant	action (MLT−1)	1	1.054571817...×10−34 J⋅s[10]
${\displaystyle e}$, elementary charge	charge (Q)	1	1.602176634×10−19 C[11]
${\displaystyle m_{\text{e}}}$, electron rest mass	mass (M)	1	9.1093837015(28)×10−31 kg[12]
${\displaystyle 4\pi \epsilon _{0}}$, permittivity	permittivity (Q2W−1L−1)	1	1.11265005545(17)×10−10 F⋅m−1[13]
† McWeeny's proposed base units; the choice is essentially arbitrary. ‡ W represents the dimensions of energy, ML2T−2.[6] ⹋ In this column, the convention on atomic units that uses dimensionless equivalents has been applied.

Units

Three of the defining constants (reduced Planck constant, elementary charge, and electron rest mass) are atomic units themselves – of action,^[14] electric charge,^[15] and mass,^[16] respectively. Two named units are those of length (Bohr radius ${\displaystyle a_{0}\equiv 4\pi \epsilon _{0}\hbar ^{2}/m_{\text{e}}e^{2}}$) and energy (hartree ${\displaystyle E_{\text{h}}\equiv \hbar ^{2}/m_{\text{e}}a_{0}^{2}}$).

Defined atomic units

Atomic unit of	Expression	Value in SI units	Other equivalents
1st hyperpolarizability	${\displaystyle e^{3}a_{0}^{3}/E_{\text{h}}^{2}}$	3.2063613061(15)×10−53 C3⋅m3⋅J−2[17]
2nd hyperpolarizability	${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$	6.2353799905(38)×10−65 C4⋅m4⋅J−3[18]
action	${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s[19]
charge	${\displaystyle e}$	1.602176634×10−19 C[20]
charge density	${\displaystyle e/a_{0}^{3}}$	1.08120238457(49)×1012 C⋅m−3[21]
current	${\displaystyle eE_{\text{h}}/\hbar }$	6.623618237510(13)×10−3 A[22]
electric dipole moment	${\displaystyle ea_{0}}$	8.4783536255(13)×10−30 C⋅m[23]	≘ 2.541746473 D
electric field	${\displaystyle E_{\text{h}}/ea_{0}}$	5.14220674763(78)×1011 V⋅m−1[24]	5.14220674763(78) GV⋅cm−1, 51.4220674763(78) V⋅Å−1
electric field gradient	${\displaystyle E_{\text{h}}/ea_{0}^{2}}$	9.7173624292(29)×1021 V⋅m−2[25]
electric polarizability	${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$	1.64877727436(50)×10−41 C2⋅m2⋅J−1[26]
electric potential	${\displaystyle E_{\text{h}}/e}$	27.211386245988(53) V[27]
electric quadrupole moment	${\displaystyle ea_{0}^{2}}$	4.4865515246(14)×10−40 C⋅m2[28]
energy	${\displaystyle E_{\text{h}}}$	4.3597447222071(85)×10−18 J[29]	${\displaystyle 2hcR_{\infty }}$, ${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$, 27.211386245988(53) eV[30]
force	${\displaystyle E_{\text{h}}/a_{0}}$	8.2387234983(12)×10−8 N[31]	82.387 nN, 51.421 eV·Å−1
length	${\displaystyle a_{0}}$	5.29177210903(80)×10−11 m[32]	${\displaystyle \hbar /\alpha m_{\text{e}}c}$, 0.529177210903(80) Å
magnetic dipole moment	${\displaystyle \hbar e/m_{\text{e}}}$	1.85480201566(56)×10−23 J⋅T−1[33]	${\displaystyle 2\mu _{\text{B}}}$
magnetic flux density	${\displaystyle \hbar /ea_{0}^{2}}$	2.35051756758(71)×105 T[34]	≘ 2.35051756758(71)×109 G
magnetizability	${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$	7.8910366008(48)×10−29 J⋅T−2[35]
mass	${\displaystyle m_{\text{e}}}$	9.1093837015(28)×10−31 kg[36]
momentum	${\displaystyle \hbar /a_{0}}$	1.99285191410(30)×10−24 kg·m·s−1[37]
permittivity	${\displaystyle e^{2}/a_{0}E_{\text{h}}}$	1.11265005545(17)×10−10 F⋅m−1[38]	${\displaystyle 4\pi \epsilon _{0}}$
time	${\displaystyle \hbar /E_{\text{h}}}$	2.4188843265857(47)×10−17 s[39]
velocity	${\displaystyle a_{0}E_{\text{h}}/\hbar }$	2.18769126364(33)×106 m⋅s−1[40]	${\displaystyle \alpha c}$
${\displaystyle c}$: speed of light, ${\displaystyle \epsilon _{0}}$: vacuum permittivity, ${\displaystyle R_{\infty }}$: Rydberg constant, ${\displaystyle h}$: Planck constant, ${\displaystyle \alpha }$: fine-structure constant, ${\displaystyle \mu _{\text{B}}}$: Bohr magneton, ≘: correspondence

Conventions

Different conventions are adopted in the use of atomic units, which vary in presentation, formality and convenience.

Physical constants

Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant ${\displaystyle \alpha ={e^{2}}/{(4\pi \epsilon _{0})\hbar c}\approx 1/137}$, which appears in expressions as a consequence of the choice of units. For example, the numeric value of the speed of light, expressed in atomic units, is ${\displaystyle c=1/\alpha \,{\text{a.u.}}\approx 137\,{\text{a.u.}}}$^[44]: 597

Some physical constants expressed in atomic units

Name	Symbol/Definition	Value in atomic units
speed of light	${\displaystyle c}$	${\displaystyle (1/\alpha )\,a_{0}E_{\text{h}}/\hbar \approx 137\,a_{0}E_{\text{h}}/\hbar }$
classical electron radius	${\displaystyle r_{\text{e}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}}$	${\displaystyle \alpha ^{2}\,a_{0}\approx 0.0000532\,a_{0}}$
reduced Compton wavelength of the electron	ƛe ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$	${\displaystyle \alpha \,a_{0}\approx 0.007297\,a_{0}}$
proton mass	${\displaystyle m_{\text{p}}}$	${\displaystyle \approx 1836\,m_{\text{e}}}$

Bohr model in atomic units

Atomic units are chosen to reflect the properties of electrons in atoms, which is particularly clear in the classical Bohr model of the hydrogen atom for the bound electron in its ground state:

• Mass = 1 a.u. of mass
• Charge = −1 a.u. of charge
• Orbital radius = 1 a.u. of length
• Orbital velocity = 1 a.u. of velocity^[44]: 597
• Orbital period = 2π a.u. of time
• Orbital angular velocity = 1 radian per a.u. of time
• Orbital momentum = 1 a.u. of momentum
• Ionization energy = 1/2 a.u. of energy
• Electric field (due to nucleus) = 1 a.u. of electric field
• Lorentz force (due to nucleus) = 1 a.u. of force

References

^[45]