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. In particular, 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. Because length is a basic unit, the speed of light is a measured quantity in a.u., c=1/α a.u. of velocity where the (dimensionless) fine structure constant 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).

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 vacuum. In SI units the a.u. unit of length, the Bohr radius a₀, 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.

Units

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. 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 ≡ αc = 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/c2	permittivity	1.112 650 056... × 10−10 F m−1 (exact)

Here, c = measured speed of light, α = fine structure constant and μ_B is the Bohr magneton.