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, 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, where the numerical values of the following four fundamental physical constants are all unity by definition:
In Hartree units, the speed of light is approximately 137. 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.
Contents

Use and notation 1

Fundamental atomic units 2

Related physical constants 3

Derived atomic units 4

SI and GaussianCGS variants, and magnetismrelated units 5

Bohr model in atomic units 6

Nonrelativistic quantum mechanics in atomic units 7

Comparison with Planck units 8

See also 9

Notes and references 10

External links 11
Use and notation
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 a particle with a mass of m has 3.4 times the mass of electron. The value of m can be written in three ways:

"m = 3.4~m_\text{e}". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.^{[2]}

"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, "L = 3.4~\text{a.u.}" The dimension needs to be inferred from context.^{[2]}

"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 m_\text{e} = 1, so 3.4~m_\text{e} = 3.4.^{[3]}^{[4]}
Fundamental atomic units
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
Dimensionless physical constants retain their values in any system of units. Of particular importance is the finestructure constant \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.
Some physical constants expressed in atomic units
Name

Symbol/Definition

Value in atomic units

speed of light

\!c

\!1/\alpha \approx 137

classical electron radius

r_\mathrm{e}=\frac{1}{4\pi\epsilon_0}\frac{e^2}{m_\mathrm{e} c^2}

\!\alpha^2 \approx 5.32\times10^{5}

proton mass

m_\mathrm{p}

m_\mathrm{p}/m_\mathrm{e} \approx 1836

Derived atomic units
Below are given a few derived units. Some of them have proper names and symbols assigned, as indicated in the table. k_{B} is the Boltzmann constant.
Derived atomic units
Dimension

Name

Symbol

Expression

Value in SI units

Value in more common units

length

bohr

\!a_0

4\pi \epsilon_0 \hbar^2 / (m_\mathrm{e} e^2) = \hbar / (m_\mathrm{e} c \alpha)

6989529177210920000♠5.2917721092(17)×10^{−11} m^{[6]}

6989529177210920000♠0.052917721092(17) nm = 6989529177210920000♠0.52917721092(17) Å

energy

hartree

\!E_\mathrm{h}

m_\mathrm{e} e^4/(4\pi\epsilon_0\hbar)^2 = \alpha^2 m_\mathrm{e} c^2

6982435974417000000♠4.35974417(75)×10^{−18} J

6982435968243877570♠27.211 eV = 7002627509000000000♠627.509 kcal·mol^{−1}

time



\hbar / E_\mathrm{h}

6983241888432650500♠2.418884326505(16)×10^{−17} s


velocity



a_0 E_\mathrm{h} / \hbar = \alpha c

7006218769126330000♠2.1876912633(73)×10^{6} m·s^{−1}


force



\! E_\mathrm{h} / a_0

6992823872250000000♠8.2387225(14)×10^{−8} N

6992823870000000000♠82.387 nN = 7001514210000000000♠51.421 eV·Å^{−1}

temperature



\! E_\mathrm{h} / k_\mathrm{B}

7005315774640000000♠3.1577464(55)×10^{5} K


pressure



E_\mathrm{h} / {a_0}^3

7013294219120000000♠2.9421912(19)×10^{13} Pa


electric field



\!E_\mathrm{h} / (ea_0)

7011514220652000000♠5.14220652(11)×10^{11} V·m^{−1}

7000514220652000000♠5.14220652(11) GV·cm^{−1} = 7001514220652000000♠51.4220652(11) V·Å^{−1}

electric potential



\!E_\mathrm{h} / e

7001272113850499999♠2.721138505(60)×10^{1} V


electric dipole moment



e a_0

6970847835326000000♠8.47835326(19)×10^{−30} C·m

7000254174599999999♠2.541746 D

SI and GaussianCGS variants, and magnetismrelated units
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 GaussianCGS units.^{[7]} 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. = \frac{\hbar}{e a_0^2} = 7005235000000000000♠2.35×10^{5} T = 7009235000000000000♠2.35×10^{9} G,
and in the Gaussiancgs unit system, the atomic unit for magnetic field is

1 a.u. = \frac{e}{a_0^2} = 7003172000000000000♠1.72×10^{3} T = 7007172000000000000♠1.72×10^{7} G.
(These differ by a factor of α.)
Other magnetismrelated quantities are also different in the two systems. An important example is the Bohr magneton: In SIbased atomic units,^{[8]}

\mu_\text{B} = \frac{e \hbar}{2 m_\text{e}} = 1/2 a.u.
and in Gaussianbased atomic units,^{[9]}

\mu_\text{B} = \frac{e \hbar}{2 m_\text{e} c}=\alpha/2\approx 3.6\times 10^{3} a.u.
Bohr model in atomic units
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
Nonrelativistic quantum mechanics in atomic units
The Schrödinger equation for an electron in SI units is

 \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

 \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:

\hat H =  \nabla^2}  {1 \over {4 \pi \epsilon_0}},
while atomic units transform the preceding equation into

\hat H =  \nabla^2}  .
Comparison with Planck units
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 atomicscale calculations in the presentday universe, while Planck units are more suitable for quantum gravity and earlyuniverse 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, 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
Notes and references

Shull, H.; Hall, G. G. (1959). "Atomic Units".

^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a NonCoulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society 24 (1) (Cambridge University Press). pp. 89–110.

^ ^{a} ^{b} Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155.

^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217.

^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5.

^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 1 April 2012.

^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 21 January 2014.

^ "A note on Units" (PDF). Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes.

^ Chis, Vasile. "Atomic Units; Molecular Hamiltonian; BornOppenheimer Approximation" (PDF). Molecular Structure and Properties Calculations. BabesBolyai University lecture notes. )

^ Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. p. 380.
External links

CODATA Internationally recommended values of the Fundamental Physical Constants.


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