# Hartree atomic units

The Hartree atomic units are a system of natural units of measurement which is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They are named after the physicist Douglas Hartree.[1] 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

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 ${\displaystyle m_{\mathrm {e} }}$ 9.1093837015(28)×10−31 kg[5]
Coulomb constant ${\displaystyle k_{e}}$ 8.9875517923(14)×109 kg⋅m3⋅s−4⋅A−2[6]

## Units

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 contants (reduced Planck constant, elementary charge, and electron rest mass) are basic 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 ${\displaystyle a_{0}\equiv 4\pi \epsilon _{0}\hbar ^{2}/(m_{\text{e}}e^{2})}$) 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.

Basic and 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[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.

## 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.[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]

## 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 ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$ ${\displaystyle \alpha \,a_{0}\approx 0.007297\,a_{0}}$
proton mass ${\displaystyle m_{\mathrm {p} }}$ ${\displaystyle \approx 1836\,m_{\text{e}}}$

## Bohr model in atomic units

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

## Non-relativistic quantum mechanics in atomic units

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

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

## 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 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.

## Notes and references

1. ^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 24, no. 1. Cambridge University Press. pp. 89–110. Bibcode:1928PCPS...24...89H. doi:10.1017/S0305004100011919.
2. ^ "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-28.
3. ^ a b Note that as of the 2019 redefinition of the SI base units, the elementary charge ${\displaystyle e}$ and the Planck constant ${\displaystyle h}$ (and consequently also the reduced Planck constant ${\displaystyle \hbar }$) are defined as having exact numerical values in SI units.
4. ^ "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
5. ^ "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
6. ^ Derived from ${\displaystyle k_{e}\equiv (4\pi \epsilon _{0})^{-1}}$ "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
7. ^ "Atomic unit of action". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-15.
8. ^ "Atomic unit of charge". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-15.
9. ^ "Atomic unit of mass". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-15.
10. ^ "Atomic unit of 1st hyperpolarizability". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
11. ^ "Atomic unit of 2nd hyperpolarizability". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
12. ^ "Atomic unit of action". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
13. ^ "Atomic unit of charge". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
14. ^ "Atomic unit of charge density". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
15. ^ "Atomic unit of current". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
16. ^ "Atomic unit of electric dipole moment". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
17. ^ "Atomic unit of electric field". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
18. ^ "Atomic unit of electric field gradient". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
19. ^ "Atomic unit of electric polarizability". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
20. ^ "Atomic unit of electric potential". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
21. ^ "Atomic unit of electric quadrupole moment". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
22. ^ "Atomic unit of energy". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
23. ^ "Atomic unit of force". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
24. ^ "Atomic unit of length". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
25. ^ "Atomic unit of magnetic dipole moment". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
26. ^ "Atomic unit of magnetic flux density". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
27. ^ "Atomic unit of magnetizability". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
28. ^ "Atomic unit of mass". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
29. ^ "Atomic unit of momentum". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
30. ^ "Atomic unit of permittivity". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
31. ^ "Atomic unit of time". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
32. ^ "Atomic unit of velocity". NIST CODATA 2018. 20 May 2019. Retrieved 2019-08-31.
33. ^ a b Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5.
34. ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4.
35. ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2.
36. ^ McQuarrie, Donald A. (2008). Quantum Chemistry (2nd ed.). New York, NY: University Science Books. p. 489.