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Atomic units
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:
 Electron mass <math>m_\text{e}</math>;
 Elementary charge <math>e</math>;
 Reduced Planck's constant <math>\hbar = h/(2 \pi)</math>;
 Coulomb's constant <math>k_\text{e} = 1/(4 \pi \epsilon_0)</math>.
In Hartree units, the speed of light is approximately <math>137</math>. 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
 1 Use and notation
 2 Fundamental atomic units
 3 Related physical constants
 4 Derived atomic units
 5 SI and GaussianCGS variants, and magnetismrelated units
 6 Bohr model in atomic units
 7 Nonrelativistic quantum mechanics in atomic units
 8 Comparison with Planck units
 9 See also
 10 Notes and references
 11 External links
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:
 "<math>m = 3.4~m_\text{e}</math>". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.^{[2]}
 "<math>m = 3.4~\mathrm{a.u.}</math>" ("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, "<math>L = 3.4~\text{a.u.}</math>" The dimension needs to be inferred from context.^{[2]}
 "<math>m = 3.4</math>". 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 <math>m_\text{e} = 1</math>, so <math>3.4~m_\text{e} = 3.4</math>.^{[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.
Dimension  Name  Symbol/Definition  Value in SI units^{[5]} 

mass  electron rest mass  <math>\!m_\mathrm{e}</math>  9.10938291(40)×10^{−31}Lua error: Unmatched closebracket at pattern character 67. 
charge  elementary charge  <math>\!e</math>  1.602176565(35)×10^{−19}Lua error: Unmatched closebracket at pattern character 67. 
action  reduced Planck's constant  <math>\hbar = h/(2 \pi)</math>  1.054571726(47)×10^{−34}Lua error: Unmatched closebracket at pattern character 67. 
electric constant^{−1}  Coulomb force constant  <math>k_\text{e} = 1/(4 \pi \epsilon_0)</math>  8.9875517873681×10^{9}Lua error: Unmatched closebracket at pattern character 67. 
Related physical constants
Dimensionless physical constants retain their values in any system of units. Of particular importance is the finestructure constant <math>\alpha = \frac{e^2}{(4 \pi \epsilon_0)\hbar c} \approx 1/137</math>. This immediately gives the value of the speed of light, expressed in atomic units.
Name  Symbol/Definition  Value in atomic units 

speed of light  <math>\!c</math>  <math>\!1/\alpha \approx 137</math> 
classical electron radius  <math>r_\mathrm{e}=\frac{1}{4\pi\epsilon_0}\frac{e^2}{m_\mathrm{e} c^2}</math>  <math>\!\alpha^2 \approx 5.32\times10^{5}</math> 
proton mass  <math>m_\mathrm{p}</math>  <math>m_\mathrm{p}/m_\mathrm{e} \approx 1836</math> 
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.
Dimension  Name  Symbol  Expression  Value in SI units  Value in more common units 

length  bohr  <math>\!a_0</math>  <math>4\pi \epsilon_0 \hbar^2 / (m_\mathrm{e} e^2) = \hbar / (m_\mathrm{e} c \alpha) </math>  5.2917721092(17)×10^{−11}Lua error: Unmatched closebracket at pattern character 67.^{[5]}  0.052917721092(17)Lua error: Unmatched closebracket at pattern character 67. = 0.52917721092(17)Lua error: Unmatched closebracket at pattern character 67. 
energy  hartree  <math>\!E_\mathrm{h}</math>  <math>m_\mathrm{e} e^4/(4\pi\epsilon_0\hbar)^2 = \alpha^2 m_\mathrm{e} c^2 </math>  4.35974417(75)×10^{−18}Lua error: Unmatched closebracket at pattern character 67.  27.211Lua error: Unmatched closebracket at pattern character 67. = 627.509Lua error: Unmatched closebracket at pattern character 67. 
time  <math>\hbar / E_\mathrm{h}</math>  2.418884326505(16)×10^{−17}Lua error: Unmatched closebracket at pattern character 67.  
velocity  <math> a_0 E_\mathrm{h} / \hbar = \alpha c</math>  2.1876912633(73)×10^{6}Lua error: Unmatched closebracket at pattern character 67.  
force  <math>\! E_\mathrm{h} / a_0 </math>  8.2387225(14)×10^{−8}Lua error: Unmatched closebracket at pattern character 67.  82.387Lua error: Unmatched closebracket at pattern character 67. = 51.421Lua error: Unmatched closebracket at pattern character 67.  
temperature  <math>\! E_\mathrm{h} / k_\mathrm{B} </math>  3.1577464(55)×10^{5}Lua error: Unmatched closebracket at pattern character 67.  
pressure  <math> E_\mathrm{h} / {a_0}^3 </math>  2.9421912(19)×10^{13}Lua error: Unmatched closebracket at pattern character 67.  
electric field  <math>\!E_\mathrm{h} / (ea_0) </math>  5.14220652(11)×10^{11}Lua error: Unmatched closebracket at pattern character 67.  5.14220652(11)Lua error: Unmatched closebracket at pattern character 67. = 51.4220652(11)Lua error: Unmatched closebracket at pattern character 67.  
electric potential  <math>\!E_\mathrm{h} / e </math>  2.721138505(60)×10^{1}Lua error: Unmatched closebracket at pattern character 67.  
electric dipole moment  <math> e a_0 </math>  8.47835326(19)×10^{−30}Lua error: Unmatched closebracket at pattern character 67.  2.541746Lua error: Unmatched closebracket at pattern character 67. 
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.^{[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
and in the Gaussiancgs unit system, the atomic unit for magnetic field is
(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,^{[7]}
 <math>\mu_\text{B} = \frac{e \hbar}{2 m_\text{e}} = 1/2</math> a.u.
and in Gaussianbased atomic units,^{[8]}
 <math>\mu_\text{B} = \frac{e \hbar}{2 m_\text{e} c}=\alpha/2\approx 3.6\times 10^{3}</math> 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
 <math> \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)</math>.
The same equation in au is
 <math> \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)</math>.
For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:
 <math>\hat H =  {{{\hbar^2} \over {2 m_e}}\nabla^2}  {1 \over {4 \pi \epsilon_0}}{{e^2} \over {r}}</math>,
while atomic units transform the preceding equation into
 <math>\hat H =  {{{1} \over {2}}\nabla^2}  {{1} \over {r}}</math>.
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, <math>1/\alpha \approx 137</math>. 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". Nature 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0.
 ^ 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. doi:10.1017/S0305004100011919.
 ^ ^{a} ^{b} Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 9780486414645.
 ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 9780486673554.
 ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 9780387208022.
 ^ ^{a} ^{b} "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 1 April 2012.
 ^ "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. ISBN 9780198509509.
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