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.

Example units

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

Length

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

$a_{0}=\alpha /4{\rm {\pi }}R_{\infty }=4{\rm {\pi }}\epsilon _{0}\hbar ^{2}/m_{\rm {e}}e^{2}=\mathrm {0.52917720859(36)\ \times \ 10^{-10}\ m} \ ,$

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:

α =7.297 352 5376(50) × 10^-3 = 1/137.035 999 679(94).

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 (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 E_h = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{a_0} \ =\ \mathrm{ 4.359\ 744\ 17(75)\ \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.

ℏ/E_h = 2.418 884 326 505(16) × 10⁻¹⁷ 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.

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 53(14) × 10−19 C
Bohr radius (bohr)	a0	length	0.529 177 2108(18) × 10−10 m
electron mass	me	mass	9.109 3826(16) × 10−31 kg
reduced Planck constant	ℏ	action	1.054 571 68(18) × 10−34 Js
Hartree energy (hartree)	Eh	energy	4.359 744 17(75) × 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 2633(73) × 106 m/s
a.u. time	ℏ/Eh	time	2.418 884 326 505(16) × 10−17 s
a.u. current	eEh/ℏ	current	6.623 617 63(17) × 10−3 A
a.u. electric potential	Eh/e	electric potential	27.211 383 86(68) V
a.u. magnetic flux density	ℏ/ea02	magnetic flux density	2.350 517 382(59) × 105 T
a.u. magnetic dipole moment	ℏe/me = 2μB	magnetic dipole moment	1.854 801 830(46) × 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.