Hartree atomic units

By definition, each of the following four fundamental physical constants is expressed as the numeric value 1 multiplied by a coherent unit of this system:

Defining constants

Name	Symbol	Value in SI units
reduced Planck constant	${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s[2][3]
elementary charge	${\displaystyle e}$	1.602176634×10−19 C[4][3]
electron rest mass	${\displaystyle m_{\text{e}}}$	9.1093837015(28)×10−31 kg[5]
Coulomb constant	${\displaystyle k_{\text{e}}}$	8.9875517923(14)×109 kg⋅m3⋅s−4⋅A−2[6]

Each unit in this system can be expressed as a product of powers of these four physical constants without a numerical multiplier. 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.

Three of the defining constants (reduced Planck constant, elementary charge, and electron rest mass) are atomic units themselves – of action,[7] electric charge,[8] and mass,[9] respectively. The two most important derived units are those of length (Bohr radius ${\displaystyle a_{0}\equiv \hbar ^{2}/(k_{\text{e}}m_{\text{e}}e^{2})}$) and energy (hartree ${\displaystyle E_{\text{h}}\equiv \hbar ^{2}/(m_{\text{e}}a_{0}^{2})}$). The table below lists these and many other units that can be derived in the system.

Quantities and their atomic units

Atomic unit of	Name	Expression	Value in SI units	Other equivalents
1st hyperpolarizability		${\displaystyle e^{3}a_{0}^{3}/E_{\text{h}}^{2}}$	3.2063613061(15)×10−53 C3⋅m3⋅J−2[10]
2nd hyperpolarizability		${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$	6.2353799905(38)×10−65 C4⋅m4⋅J−3[11]
action		${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s[12]
charge		${\displaystyle e}$	1.602176634×10−19 C[13]
charge density		${\displaystyle e/a_{0}^{3}}$	1.08120238457(49)×1012 C⋅m−3[14]
current		${\displaystyle eE_{\text{h}}/\hbar }$	6.623618237510(13)×10−3 A[15]
electric dipole moment		${\displaystyle ea_{0}}$	8.4783536255(13)×10−30 C⋅m[16]	≘ 2.541746473 D
electric field		${\displaystyle E_{\text{h}}/ea_{0}}$	5.14220674763(78)×1011 V⋅m−1[17]	5.14220674763(78) GV⋅cm−1, 51.4220674763(78) V⋅Å−1
electric field gradient		${\displaystyle E_{\text{h}}/ea_{0}^{2}}$	9.7173624292(29)×1021 V⋅m−2[18]
electric polarizability		${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$	1.64877727436(50)×10−41 C2⋅m2⋅J−1[19]
electric potential		${\displaystyle E_{\text{h}}/e}$	27.211386245988(53) V[20]
electric quadrupole moment		${\displaystyle ea_{0}^{2}}$	4.4865515246(14)×10−40 C⋅m2[21]
energy	hartree	${\displaystyle E_{\text{h}}}$	4.3597447222071(85)×10−18 J[22]	${\displaystyle 2R_{\infty }hc}$, ${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$, 27.211386245988(53) eV
force		${\displaystyle E_{\text{h}}/a_{0}}$	8.2387234983(12)×10−8 N[23]	82.387 nN, 51.421 eV·Å−1
length	Bohr radius	${\displaystyle a_{0}}$	5.29177210903(80)×10−11 m[24]	${\displaystyle \hbar /m_{\text{e}}c\alpha }$, 0.529177210903(80) Å
magnetic dipole moment		${\displaystyle \hbar e/m_{\text{e}}}$	1.85480201566(56)×10−23 J⋅T−1[25]	${\displaystyle 2\mu _{\text{B}}}$
magnetic flux density		${\displaystyle \hbar /ea_{0}^{2}}$	2.35051756758(71)×105 T[26]	≘ 2.35051756758(71)×109 G
magnetizability		${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$	7.8910366008(48)×10−29 J⋅T−2[27]
mass		${\displaystyle m_{\mathrm {e} }}$	9.1093837015(28)×10−31 kg[28]
momentum		${\displaystyle \hbar /a_{0}}$	1.99285191410(30)×10−24 kg·m·s−1[29]
permittivity		${\displaystyle e^{2}/a_{0}E_{\text{h}}}$	1.11265005545(17)×10−10 F⋅m−1[30]	${\displaystyle 4\pi \epsilon _{0}}$
pressure		${\displaystyle E_{\text{h}}/{a_{0}}^{3}}$	2.9421015697(13)×1013 Pa
irradiance		${\displaystyle E_{\text{h}}^{2}/\hbar {a_{0}}^{2}}$	6.4364099007(19)×1019 W⋅m−2
time		${\displaystyle \hbar /E_{\text{h}}}$	2.4188843265857(47)×10−17 s[31]
velocity		${\displaystyle a_{0}E_{\text{h}}/\hbar }$	2.18769126364(33)×106 m⋅s−1[32]	${\displaystyle \alpha c}$

Here,

• ${\displaystyle c}$ is the speed of light
• ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity
• ${\displaystyle R_{\infty }}$ is the Rydberg constant
• ${\displaystyle h}$ is the Planck constant
• ${\displaystyle \alpha }$ is the fine-structure constant
• ${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton
• ≘ denotes correspondence between quantities since equality does not apply.

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:

• "${\displaystyle m=3.4~m_{\text{e}}}$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.[33]
• "${\displaystyle m=3.4~{\text{a.u.}}}$" ("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, "${\displaystyle L=3.4~{\text{a.u.}}}$" The dimension must be inferred from context.[33]
• "${\displaystyle m=3.4}$". 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 ${\displaystyle m_{\text{e}}=1}$, so ${\displaystyle 3.4~m_{\text{e}}=3.4}$.[34][35]

Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant ${\displaystyle \alpha ={\frac {e^{2}}{(4\pi \epsilon _{0})\hbar c}}\approx 1/137}$, 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.

Some physical constants expressed in atomic units

Name	Symbol/Definition	Value in atomic units
speed of light	${\displaystyle c}$	${\displaystyle (1/\alpha )\,a_{0}E_{\text{h}}/\hbar \approx 137\,a_{0}E_{\text{h}}/\hbar }$
classical electron radius	${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$	${\displaystyle \alpha ^{2}\,a_{0}\approx 0.0000532\,a_{0}}$
reduced Compton wavelength of the electron	ƛe ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$	${\displaystyle \alpha \,a_{0}\approx 0.007297\,a_{0}}$
proton mass	${\displaystyle m_{\mathrm {p} }}$	${\displaystyle \approx 1836\,m_{\text{e}}}$

Atomic units are chosen to reflect the properties of electrons in atoms, which is particularly clear in the classical Bohr model of the hydrogen atom for the bound electron in its ground state:

• Mass = 1 a.u. of mass
• Orbital radius = 1 a.u. of length
• Orbital velocity = 1 a.u. of velocity
• Orbital period = 2π a.u. of time
• Orbital angular velocity = 1 radian per a.u. of time
• Orbital angular momentum = 1 a.u. of momentum
• Ionization energy = 1/2 a.u. of energy
• Electric field (due to nucleus) = 1 a.u. of electric field
• Electrical attractive force (due to nucleus) = 1 a.u. of force

In the context of atomic physics, nondimensionalization using the defining constants of the Hartree atomic system can be a convenient shortcut, since it can be thought of as eliminating these constants wherever they occur. Nondimesionalization involves a substitution of variables that results in equations in which these constants (${\displaystyle m_{\text{e}}}$, ${\displaystyle e}$, ${\displaystyle \hbar }$ and ${\displaystyle 4\pi \epsilon _{0}}$) "have been set to 1".[36] Though the variables are no longer the original variables, the same symbols and names are typically used.

For example, the Schrödinger equation for an electron with quantities that use SI units is

${\displaystyle -{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t).}$

The same equation with corresponding nondimensionalized quantity definitions is

${\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t).}$

For the special case of the electron around a hydrogen atom, the Hamiltonian with SI quantities is:

${\displaystyle {\hat {H}}=-{{{\hbar ^{2}} \over {2m_{\text{e}}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r}},}$

while the corresponding nondimensionalized equation is

${\displaystyle {\hat {H}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {r}}.}$

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 use the reduced Planck constant. Beyond this, Planck units use the two fundamental constants of general relativity and cosmology: the gravitational constant ${\displaystyle G}$ and the speed of light in vacuum, ${\displaystyle c}$. Atomic units, by contrast, use the mass and charge of the electron, and, as a result, the speed of light in atomic units is ${\displaystyle c=1/\alpha \,{\text{a.u.}}\approx 137\,{\text{a.u.}}}$ The orbital velocity of an electron around a small atom is of the order of 1 atomic unit, 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, which 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.

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

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0. S2CID 23692353.
• "CODATA Internationally recommended 2018 values of the Fundamental Physical Constants". NIST Reference on Constants, Units, and Uncertainty. 2019.