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Hartree atomic units
Not to be confused with Rydberg atomic units.
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics calculations. They are named after the physicist Douglas Hartree.^{[1]} In this system the numerical values of the following four fundamental physical constants are all unity by definition:
Electron mass$m_{\text{e}}$, also known as the atomic unit of mass^{[2]}^{[3]}
Inverse Coulomb constant$4\pi \epsilon _{0}=1/k_{\text{e}}$, also known as the atomic unit of permittivity^{[6]}
In Hartree atomic units, the speed of light is approximately 137 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.
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.^{[7]}
"$m=3.4~\mathrm {a.u.}$" ("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.^{[7]}
"$m=3.4$". 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 $3.4~m_{\text{e}}=3.4$.^{[8]}^{[9]}
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.
These defining constants are used to define four units that take the role of base units, in the sense that other units are generally expressed symbolically in terms of these:
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 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.
SI and Gaussian-CGS variants, and magnetism-related 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 Gaussian-CGS units.^{[37]} Although most of the units listed 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
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 are free of anthropocentric considerations. It should be kept in mind that 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 constantG 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 more slowly 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 atomic unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.