Atomic units

The atomic units, abbreviated a.u. is a set of units used in atomic calculations. In the a.u. system any four of the five quantities charge e, mass m_e, action ℏ, length a₀, and energy E_h may be taken as base quantities, and other quantities are derived. A listing of numerical values in terms of SI units can be found on the NIST website.

Example units

The expressions for a few example atomic units in terms of formulas in SI units are discussed next.

Mass

The atomic unit of mass is the electron mass, m_e. However, instead, the unified atomic mass unit often is used, the Dalton, symbol u. The Dalton is defined as 1/12 the mass of a free carbon 12 atom, with a value of 1.660538921(73) × 10^-27 kg or 931.494 061(21) MeV. A proton has a mass of approximately 1.007 276 466 812 u. The electron mass is 5.485 799 0946 × 10^-4 u.

Length

In SI units the a.u. unit of length, the Bohr radius a₀ or bohr, is:

$a_{0}=\alpha /4{\rm {\pi }}R_{\infty }=4{\rm {\pi }}\epsilon _{0}\hbar ^{2}/m_{\rm {e}}e^{2}=\mathrm {0.52917721092(17)\ \times \ 10^{-10}\ m} \ ,$

where R_∞ is the Rydberg constant, e is the elementary charge, ε₀ is the electric constant, ℏ is the reduced Planck's constant h/(2π), m_e is the electron mass, and where the (dimensionless) fine structure constant α is given by (in SI units):

$\alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c_{0}}}\ ,$

and has the value:

$\alpha =\mathrm {7.2973525698(24)\times 10^{-3}} =\mathrm {1/137.035999074(44)} \ .$

The Bohr radius was the distance of an electron from the nucleus of a hydrogen atom predicted by the Bohr theory of the atom, which required an integer number of wavelengths around the electron orbit. In modern quantum mechanics the Bohr radius is the distance of maximum likelihood for finding the electron in the hydrogen atom in its ground state.

Energy

The unit of energy, the hartree, is the energy of two a.u. charges separated by one bohr in a medium of permittivity given by 1 a.u. of permittivity, 4πε₀:

Failed to parse (syntax error): {\displaystyle E_h = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{a_0} \ =\ \mathrm{ 4.359 744 34(19)\ \times \ 10^{−18}\ J}\ . }

Time

Somewhat unusually, time is a derived quantity, ℏ/E_h, with the interpretation as the period of an electron circling in the first Bohr orbit divided by 2π.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar/E_h = \mathrm{2.418 884 326 502(12)} ×\times 10^{-17}\mathrm{ s} }

Velocity

Using the unit of time, and the expression for the hartree, the a.u. unit of velocity is one bohr per a.u. unit of time:

$v_{B}=a_{0}/(\hbar /E_{h})={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar }}=\alpha c_{0}\ .$

Here c₀ is the SI units defined speed of light in classical vacuum and α is the fine structure constant. Its value is:

$a_{0}/(\hbar /E_{h})=\mathrm {2.18769126379(71)} \times 10^{6}\mathrm {m/s} \ .$

The a.u. unit of velocity changes size with refinement in measurement of the fine structure constant. However, the defined value of c₀ = v_B/α is unaffected by such refinements. See the articles speed of light and metre for more detail about the defined value c₀ = 299 792 458 m/s (exactly).

Tabulation

Basic atomic units
Name	Symbol	Quantity	Value in SI units
elementary charge	e	charge	1.602 176 565(35) × 10−19 C
Bohr radius (bohr)	a0	length	0.529 177 210 92(17) × 10−10 m
electron mass	me	mass	9.109 382 91(40) × 10−31 kg
reduced Planck constant	ℏ	action	1.054 571 726(47) × 10−34 Js
Hartree energy (hartree)	Eh	energy	4.359 744 34(19) × 10−18 J

Evidently in atomic units, if we choose e, ℏ, m_e, and a₀ as the four basic units, then these entities all become 1 in algebraic expressions, and because a₀ = (4πε₀)(ℏ/e)²/m_e, it follows that 4πε₀ = 1 as well, defining the a.u. unit of permittivity. The hartree, E_h = (4πε₀)⁻¹ e²/a₀ also is automatically 1.

Derived atomic units
Name	Formula	Quantity	Value in SI units
a.u. velocity	vB = αc0 = a0Eh/ℏ	velocity	2.187 691 263 79(71) × 106 m/s
a.u. time	ℏ/Eh	time	2.418 884 326 502(12) × 10−17 s
a.u. current	eEh/ℏ	current	6.623 617 95(15) × 10−3 A
a.u. electric potential	Eh/e	electric potential	27.211 385 05(60) V
a.u. magnetic flux density	ℏ/ea02	magnetic flux density	2.350 517 464(52) × 105 T
a.u. magnetic dipole moment	ℏe/me = 2μB	magnetic dipole moment	1.854 801 936(41) × 10−23 J T−1
a.u. permittivity	e2/a0Eh = 107/c02=4πε0	permittivity	1.112 650 056... × 10−10 F m−1 (exact)

Here, c₀ = SI units defined value for the speed of light in classical vacuum, ε₀ is the electric constant, α = fine structure constant and μ_B is the Bohr magneton.