Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, which one might name Hartree atomic units^[1] and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In atomic units, the numerical values of the following four fundamental physical constants are all unity by definition:

• electron mass Failed to parse (Missing texvc executable; please see math/README to configure.): \!m_\mathrm{e}

• elementary charge Failed to parse (Missing texvc executable; please see math/README to configure.): \!e

• reduced Planck's constant Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar = h/(2 \pi)

• Coulomb's constant Failed to parse (Missing texvc executable; please see math/README to configure.): 1/(4 \pi \epsilon_0)

.

Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in different contexts.

Use and notation[link]

Atomic units, like SI units, have a unit of mass, a unit of length, and so on. However, the use and notation is somewhat different from SI.

Suppose some particle has a mass m which is 3.4 times the mass of electron. Then, the value of m can be written in three ways:

• "Failed to parse (Missing texvc executable; please see math/README to configure.): m = 3.4~m_e

". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.^[2]

• "Failed to parse (Missing texvc executable; please see math/README to configure.): m = 3.4~\mathrm{a.u.}

" ("a.u." means "expressed in atomic units"). This notation is ambiguous: Here, it means that the mass m is 3.4 times the atomic unit of mass. But if a length L were 3.4 times the atomic unit of length, the equation would look the same, "Failed to parse (Missing texvc executable; please see math/README to configure.): L = 3.4~\mathrm{a.u.} " The dimension needs to be inferred from context.^[2]

• "Failed to parse (Missing texvc executable; please see math/README to configure.): m = 3.4

". This notation is similar to the previous one, and has the same dimensional ambiguity. It comes from formally setting the atomic units to 1, in this case Failed to parse (Missing texvc executable; please see math/README to configure.): m_e = 1 , so Failed to parse (Missing texvc executable; please see math/README to configure.): 3.4~m_e = 3.4 .^[3][4]

Fundamental atomic units[link]

These four fundamental constants form the basis of the atomic units (see above). Therefore, their numerical values in the atomic units are unity by definition.

Related physical constants[link]

Dimensionless physical constants retain their values in any system of units. Of particular importance is the fine-structure constant Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = \frac{e^2}{(4 \pi \epsilon_0)\hbar c} \approx 1/137 . This immediately gives the value of the speed of light, expressed in atomic units.

Derived atomic units[link]

Below are given a few derived units. Some of them have proper names and symbols assigned, as indicated in the table. k_B is Boltzmann constant.

SI and Gaussian-CGS variants, and magnetism-related units[link]

There are two common variants of atomic units, one where they are used in conjunction with SI units for electromagnetism, and one where they are used with Gaussian-CGS units.^[6] Although the units written above are the same either way (including the unit for electric field), the units related to magnetism are not. In the SI system, the atomic unit for magnetic field is

1 a.u. = Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\hbar}{e a_0^2}

= 2.35×105 T = 2.35×109 G,

and in the Gaussian-cgs unit system, the atomic unit for magnetic field is

1 a.u. = Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{e}{a_0^2}

= 1.72×103 T = 1.72×107 G.

(These differ by a factor of α.)

Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,^[7]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_B = \frac{e \hbar}{2 m_e} = 1/2

a.u.

and in Gaussian-based atomic units,^[8]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_B = \frac{e \hbar}{2 m_e c}=\alpha/2\approx 3.6\times 10^{-3}

a.u.

Bohr model in atomic units[link]

Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):

• Orbital velocity = 1
• Orbital radius = 1
• Angular momentum = 1
• Orbital period = 2π
• Ionization energy = ¹⁄₂
• Electric field (due to nucleus) = 1
• Electrical attractive force (due to nucleus) = 1

Non-relativistic quantum mechanics in atomic units[link]

The Schrödinger equation for an electron in SI units is

Failed to parse (Missing texvc executable; please see math/README to configure.): - \frac{\hbar^2}{2m_e} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi}{\partial t} (\mathbf{r}, t)

.

The same equation in au is

Failed to parse (Missing texvc executable; please see math/README to configure.): - \frac{1}{2} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \frac{\partial \psi}{\partial t} (\mathbf{r}, t)

.

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat H = - {{{\hbar^2} \over {2 m_e}}\nabla^2} - {1 \over {4 \pi \epsilon_0}}{{e^2} \over {r}}

,

while atomic units transform the preceding equation into

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat H = - {{{1} \over {2}}\nabla^2} - {{1} \over {r}}

.

Comparison with Planck units[link]

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. It should be kept in mind that au were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for quantum gravity and early-universe cosmology. Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and, as a result, the speed of light in atomic units is a large value, Failed to parse (Missing texvc executable; please see math/README to configure.): 1/\alpha \approx 137 . The orbital velocity of an electron around a small atom is of the order of 1 in atomic units, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms much slower than the speed of light (around 2 orders of magnitude slower).

There are much larger discrepancies in some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the Planck mass, a mass so large that if a single particle had that much mass it might collapse into a black hole. Indeed, the Planck unit of mass is 22 orders of magnitude larger than the au unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.

See also[link]

• Planck units
• Natural units
• Various extensions of the CGS system to electromagnetism.

References[link]

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature 184 (4698): 1559. Bibcode 1959Natur.184.1559S. DOI:10.1038/1841559a0. 

External links[link]

• CODATA Internationally recommended values of the Fundamental Physical Constants.

v t e SI units Base units Ampere · Candela · Kelvin · Kilogram · Metre · Mole · Second Derived units Becquerel · Coulomb · Degree Celsius · Farad · Gray · Henry · Hertz · Joule · Katal · Lumen · Lux · Newton · Ohm · Pascal · Radian · Siemens · Sievert · Steradian · Tesla · Volt · Watt · Weber Accepted for use with SI Dalton (Atomic mass unit) · Astronomical unit · Day · Decibel · Degree Celsius · Degree of arc · Electronvolt · Hectare · Hour · Litre · Minute · Minute of arc · Neper · Second of arc · Tonne Atomic units · Natural units See also SI prefixes · Systems of measurement · Conversion of units · New SI definitions · History of the metric system Book:International System of Units  · Category:SI base units