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, 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, e is the elementary charge, ε₀ is the electric constant, ℏ is the reduced Planck's constant h/(2π), and c₀ is the SI units defined speed of light in classical vacuum.

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.

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, α = fine structure constant and μ_B is the Bohr magneton.