Hartree atomic units

The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry 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:

• Reduced Planck constant: ${\displaystyle \hbar =1}$, also known as the atomic unit of action^[2]
• Elementary charge: ${\displaystyle e=1}$, also known as the atomic unit of charge^[3]
• Bohr radius: ${\displaystyle a_{0}=1}$, also known as the atomic unit of length^[4]
• Electron mass: ${\displaystyle m_{\text{e}}=1}$, also known as the atomic unit of mass^[5][6]

In Hartree atomic units, the speed of light is approximately 137.036 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.

Defining constants

Name	Symbol	Value in SI units
reduced Planck constant	${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s[7]
elementary charge	${\displaystyle e}$	1.602176634×10−19 C[8]
Bohr radius	${\displaystyle a_{0}}$	5.29177210903(80)×10−11 m[9]
electron rest mass	${\displaystyle m_{\mathrm {e} }}$	9.1093837015(28)×10−31 kg[10]

Note that following the 2019 redefinition of the SI base units the Planck constant ${\displaystyle h}$ is defined exactly in SI Units as 6.62607015×10⁻³⁴ J⋅Hz^−1[11], and so although the reduced Planck constant ${\displaystyle \hbar =h/2\pi }$ is exact it is also irrational in SI Units. The electron charge ${\displaystyle e}$ is also exact.

Five symbols are commonly used as units in this system, only four of them being independent:^{[12]: 94–95}

Constants used as units

Dimension	Symbol	Definition
action	${\displaystyle \hbar }$	${\displaystyle \hbar }$
electric charge	${\displaystyle e}$	${\displaystyle e}$
length	${\displaystyle a_{0}}$	${\displaystyle 4\pi \epsilon _{0}\hbar ^{2}/(m_{\text{e}}e^{2})}$
mass	${\displaystyle m_{\text{e}}}$	${\displaystyle m_{\text{e}}}$
energy	${\displaystyle E_{\text{h}}}$	${\displaystyle \hbar ^{2}/(m_{\text{e}}a_{0}^{2})}$

Units[]

Below are listed units that can be derived in the system. A few are given names, as indicated in the table.

Derived 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[13]
2nd hyperpolarizability		${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$	6.2353799905(38)×10−65 C4⋅m4⋅J−3[14]
action		${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s[15]
charge		${\displaystyle e}$	1.602176634×10−19 C[16]
charge density		${\displaystyle e/a_{0}^{3}}$	1.08120238457(49)×1012 C·m−3[17]
current		${\displaystyle eE_{\text{h}}/\hbar }$	6.623618237510(13)×10−3 A[18]
electric dipole moment		${\displaystyle ea_{0}}$	8.4783536255(13)×10−30 C·m[19]	≘ 2.541746473 D
electric field		${\displaystyle E_{\text{h}}/(ea_{0})}$	5.14220674763(78)×1011 V·m−1[20]	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[21]
electric polarizability		${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$	1.64877727436(50)×10−41 C2⋅m2⋅J−1[22]
electric potential		${\displaystyle E_{\text{h}}/e}$	27.211386245988(53) V[23]
electric quadrupole moment		${\displaystyle ea_{0}^{2}}$	4.4865515246(14)×10−40 C·m2[24]
energy	hartree	${\displaystyle E_{\text{h}}}$	4.3597447222071(85)×10−18 J[25]	${\displaystyle 2R_{\infty }hc}$, ${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$, 27.211386245988(53) eV
force		${\displaystyle E_{\text{h}}/a_{0}}$	8.2387234983(12)×10−8 N[26]	82.387 nN, 51.421 eV·Å−1
length	bohr	${\displaystyle a_{0}}$	5.29177210903(80)×10−11 m[27]	${\displaystyle \hbar /(m_{\text{e}}c\alpha )}$, 0.529177210903(80) Å
magnetic dipole moment		${\displaystyle e\hbar /m_{\text{e}}}$	1.85480201566(56)×10−23 J⋅T−1[28]	${\displaystyle 2\mu _{\text{B}}}$
magnetic flux density		${\displaystyle \hbar /(ea_{0}^{2})}$	2.35051756758(71)×105 T[29]	≘ 2.35051756758(71)×109 G
magnetizability		${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$	7.8910366008(48)×10−29 J⋅T−2[30]
mass		${\displaystyle m_{\mathrm {e} }}$	9.1093837015(28)×10−31 kg[31]
momentum		${\displaystyle \hbar /a_{0}}$	1.99285191410(30)×10−24 kg·m·s−1[32]
permittivity		${\displaystyle e^{2}/(a_{0}E_{\text{h}})}$	1.11265005545(17)×10−10 F⋅m−1[33]	${\displaystyle 4\pi \epsilon _{0}}$
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[34]
velocity		${\displaystyle a_{0}E_{\text{h}}/\hbar }$	2.18769126364(33)×106 m·s−1[35]	${\displaystyle \alpha c}$

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.

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:

• "${\displaystyle m=3.4~m_{\text{e}}}$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.^[36]
• "${\displaystyle m=3.4~{\text{a.u.}}}$" ("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.^[36]
• "${\displaystyle m=3.4}$". 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 ${\displaystyle 3.4~m_{\text{e}}=3.4}$.^[37][38]

Physical constants[]

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 ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$	${\displaystyle \alpha \,a_{0}\approx 0.007297\,a_{0}}$
Bohr radius	${\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{\text{e}}e^{2}}}}$	${\displaystyle 1\,a_{0}}$
proton mass	${\displaystyle m_{\mathrm {p} }}$	${\displaystyle m_{\mathrm {p} }\approx 1836\,m_{\text{e}}}$

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):

• 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

Non-relativistic quantum mechanics in atomic units[]

The Schrödinger equation for an electron in 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 in atomic units 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 in SI units is:

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

while atomic units transform the preceding equation into

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

Comparison with Planck units[]

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 ${\displaystyle G}$ and the speed of light in vacuum, ${\displaystyle c}$. 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, ${\displaystyle 1/\alpha \approx 137}$. 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.

See also[]

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

Notes and references[]

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0. S2CID 23692353.

External links[]

• CODATA Internationally recommended values of the Fundamental Physical Constants.