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Elementary charge: $e=1$, also known as the atomic unit of charge^{[3]}
Bohr radius: $a_{0}=1$, also known as the atomic unit of length^{[4]}
Electron mass: $m_{\text{e}}=1$, also known as the atomic unit of mass^{[5]}^{[6]}
In Hartree atomic units, the speed of light is approximately atomic units of velocity. 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 other contexts.
Defining constants
Each unit in this system can be expressed as a product of powers of four physical constants without a multiplying constant. This makes it a coherent system of units, as well as making the numerical values of the defining constants in atomic units equal to unity.
? denotes correspondence between quantities since equality does not apply.
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.^{[35]}
"$m=3.4~{\text{a.u.}}$" ("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 must be inferred from context.^{[35]}
"$m=3.4$". 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 $3.4~m_{\text{e}}=3.4$.^{[36]}^{[37]}
Physical constants
Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant$\alpha ={\frac {e^{2}}{(4\pi \epsilon _{0})\hbar c}}\approx 1/137$, which appears in expressions as a consequence of the choice of units. For example, the numeric value of the speed of light, expressed in atomic units, has a value related to the fine structure constant.
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):
Both Planck units and atomic units are derived from certain fundamental properties of the physical world, and have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Atomic units 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 atomic units 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 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 by around 2 orders of magnitude more slowly than the speed of light.
There are much larger differences for 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. The Planck unit of mass is 22 orders of magnitude larger than the atomic unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.