# Hartree 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 ${\displaystyle m_{\text{e}}}$, also known as the atomic unit of mass[2][3]
• Elementary charge ${\displaystyle e}$, also known as the atomic unit of charge[4]
• Reduced Planck constant ${\displaystyle \hbar =h/(2\pi )}$, also known as the atomic unit of action[5]
• Inverse Coulomb constant ${\displaystyle 4\pi \epsilon _{0}=1/k_{\text{e}}}$, 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.

## 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.[7]
• "${\displaystyle m=3.4~\mathrm {a.u.} }$" ("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, "${\displaystyle L=3.4~{\text{a.u.}}}$" The dimension needs to be inferred from context.[7]
• "${\displaystyle m=3.4}$". 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 ${\displaystyle 3.4~m_{\text{e}}=3.4}$.[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.

Defining constants
Name Symbol/Definition Value in SI units
electron rest mass ${\displaystyle m_{\mathrm {e} }}$ 9.1093837015(28)×10−31 kg[10]
elementary charge ${\displaystyle e}$ 1.602176634×10−19 C[11]
reduced Planck constant ${\displaystyle \hbar =h/(2\pi )}$ 1.054571817...×10−34 J⋅s[12]
inverse Coulomb constant ${\displaystyle 4\pi \epsilon _{0}=1/k_{\text{e}}}$ 1.11265005545(17)×10−10 F⋅m−1[13]

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:

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

A further unit is derived from these that is used for simplifying expression of all other units in terms of these five units:

Auxiliary unit symbol
Dimension Symbol Definition
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[14]
2nd hyperpolarizability ${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$ 6.2353799905(38)×10−65 C4⋅m4⋅J−3[15]
action ${\displaystyle \hbar }$ 1.054571817...×10−34 J⋅s[16]
charge ${\displaystyle e}$ 1.602176634×10−19 C[17]
charge density ${\displaystyle e/a_{0}^{3}}$ 1.08120238457(49)×1012 C·m−3[18]
current ${\displaystyle eE_{\text{h}}/\hbar }$ 6.623618237510(13)×10−3 A[19]
electric dipole moment ${\displaystyle ea_{0}}$ 8.4783536255(13)×10−30 C·m[20] 2.541746473 D
electric field ${\displaystyle E_{\text{h}}/(ea_{0})}$ 5.14220674763(78)×1011 V·m−1[21] 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[22]
electric polarizability ${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$ 1.64877727436(50)×10−41 C2⋅m2⋅J−1[23]
electric potential ${\displaystyle E_{\text{h}}/e}$ 27.211386245988(53) V[24]
electric quadrupole moment ${\displaystyle ea_{0}^{2}}$ 4.4865515246(14)×10−40 C·m2[25]
energy hartree ${\displaystyle E_{\text{h}}}$ 4.3597447222071(85)×10−18 J[26] ${\displaystyle 2R_{\infty }hc}$, ${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$, 27.211386245988(53) eV
force ${\displaystyle E_{\text{h}}/a_{0}}$ 8.2387234983(12)×10−8 N[27] 82.387 nN, 51.421 eV·Å−1
length bohr ${\displaystyle a_{0}}$ 5.29177210903(80)×10−11 m[28] ${\displaystyle \hbar /(m_{\text{e}}c\alpha )}$, 0.529177210903(80) Å
magnetic dipole moment ${\displaystyle e\hbar /m_{\text{e}}}$ 1.85480201566(56)×10−23 J⋅T−1[29] ${\displaystyle 2\mu _{\text{B}}}$
magnetic flux density ${\displaystyle \hbar /(ea_{0}^{2})}$ 2.35051756758(71)×105 T[30] 2.35051756758(71)×109 G
magnetizability ${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$ 7.8910366008(48)×10−29 J⋅T−2[31]
mass ${\displaystyle m_{\mathrm {e} }}$ 9.1093837015(28)×10−31 kg[32]
momentum ${\displaystyle \hbar /a_{0}}$ 1.99285191410(30)×10−24 kg·m·s−1[33]
permittivity ${\displaystyle e^{2}/(a_{0}E_{\text{h}})}$ 1.11265005545(17)×10−10 F⋅m−1[34] ${\displaystyle 4\pi \epsilon _{0}}$
pressure ${\displaystyle E_{\text{h}}/{a_{0}}^{3}}$ 2.9421015697(13)×1013 Pa
time ${\displaystyle \hbar /E_{\text{h}}}$ 2.4188843265857(47)×10−17 s[35]
velocity ${\displaystyle a_{0}E_{\text{h}}/\hbar }$ 2.18769126364(33)×106 m·s−1[36] ${\displaystyle \alpha c}$

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

## 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 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 ${\displaystyle c}$ ${\displaystyle (1/\alpha ){\text{ a.u.}}\approx 137{\text{ a.u.}}}$
classical electron radius ${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$ ${\displaystyle \alpha ^{2}{\text{ a.u.}}\approx 5.32\times 10^{-5}{\text{ a.u.}}}$
reduced Compton wavelength
of the electron
ƛe ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$ ${\displaystyle \alpha {\text{ a.u.}}\approx 0.007297{\text{ a.u.}}}$
proton mass ${\displaystyle m_{\mathrm {p} }}$ ${\displaystyle m_{\mathrm {p} }\approx 1836{\text{ a.u.}}}$

## 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

1 a.u. = ${\displaystyle {\frac {\hbar }{ea_{0}^{2}}}\approx }$ 2.35×105 T = 2.35×109 G,

and in the Gaussian-cgs unit system, the atomic unit for magnetic field is

1 a.u. = ${\displaystyle {\frac {e}{a_{0}^{2}c}}\approx }$ 1.72×103 T = 1.72×107 G.

(These differ by a factor of α.)

Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,[38]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}}}=1/2}$ a.u.

and in Gaussian-based atomic units,[39]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}c}}=\alpha /2\approx 3.6\times 10^{-3}}$ a.u.

## 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 au 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 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 constant G 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, ${\displaystyle 1/\alpha \approx 137}$. 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 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.

## Notes and references

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0.
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. 24 (1). Cambridge University Press. pp. 89–110. Bibcode:1928PCPS...24...89H. doi:10.1017/S0305004100011919.
2. ^ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
3. ^ Not to be confused with the unified atomic mass unit.
4. ^ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
5. ^ "2018 CODATA Value: atomic unit of action". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
6. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
7. ^ a b Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5.
8. ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4.
9. ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2.
10. ^ "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
11. ^ "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
12. ^ "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-28.
13. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
14. ^ "2018 CODATA Value: atomic unit of 1st hyperpolarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
15. ^ "2018 CODATA Value: atomic unit of 2nd hyperpolarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
16. ^ "2018 CODATA Value: atomic unit of action". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
17. ^ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
18. ^ "2018 CODATA Value: atomic unit of charge density". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
19. ^ "2018 CODATA Value: atomic unit of current". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
20. ^ "2018 CODATA Value: atomic unit of electric dipole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
21. ^ "2018 CODATA Value: atomic unit of electric field". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
22. ^ "2018 CODATA Value: atomic unit of electric field gradient". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
23. ^ "2018 CODATA Value: atomic unit of electric polarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
24. ^ "2018 CODATA Value: atomic unit of electric potential". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
25. ^ "2018 CODATA Value: atomic unit of electric quadrupole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
26. ^ "2018 CODATA Value: atomic unit of energy". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
27. ^ "2018 CODATA Value: atomic unit of force". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
28. ^ "2018 CODATA Value: atomic unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
29. ^ "2018 CODATA Value: atomic unit of magnetic dipole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
30. ^ "2018 CODATA Value: atomic unit of magnetic flux density". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
31. ^ "2018 CODATA Value: atomic unit of magnetizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
32. ^ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
33. ^ "2018 CODATA Value: atomic unit of momentum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
34. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
35. ^ "2018 CODATA Value: atomic unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
36. ^ "2018 CODATA Value: atomic unit of velocity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
37. ^ "A note on Units" (PDF). Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes.
38. ^ Chis, Vasile. "Atomic Units; Molecular Hamiltonian; Born-Oppenheimer Approximation" (PDF). Molecular Structure and Properties Calculations. Babes-Bolyai University lecture notes.
39. ^ Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. p. 380. ISBN 978-0-19-850950-9.