Atomic units

Not to be confused with atomic mass units.

The atomic units are a system of natural units of measurement that is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree.^[1] Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.

Motivation

In the context of atomic physics, using the atomic units system can be a convenient shortcut, eliminating unnecessary symbols and numbers with very small orders of magnitude. For example, the Hamiltonian operator in the Schrödinger equation for the helium atom with standard quantities, such as when using SI units, is^[2]

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla _{1}^{2}-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla _{2}^{2}-{\frac {2e^{2}}{4\pi \epsilon _{0}r_{1}}}-{\frac {2e^{2}}{4\pi \epsilon _{0}r_{2}}}+{\frac {e^{2}}{4\pi \epsilon _{0}r_{12}}},}$

but adopting the convention associated with atomic units that transforms quantities into dimensionless equivalents, it becomes

${\displaystyle {\hat {H}}=-{\frac {1}{2}}\nabla _{1}^{2}-{\frac {1}{2}}\nabla _{2}^{2}-{\frac {2}{r_{1}}}-{\frac {2}{r_{2}}}+{\frac {1}{r_{12}}}.}$

In this convention, the constants ⁠${\displaystyle \hbar }$⁠, ⁠${\displaystyle m_{\text{e}}}$⁠, ⁠${\displaystyle 4\pi \epsilon _{0}}$⁠, and ⁠${\displaystyle e}$⁠ all correspond to the value ⁠${\displaystyle 1}$⁠ (see § Definition below). The distances relevant to the physics expressed in SI units are naturally on the order of ⁠${\displaystyle 10^{-10}\,\mathrm {m} }$⁠, while expressed in atomic units distances are on the order of ⁠${\displaystyle 1a_{0}}$⁠ (one Bohr radius, the atomic unit of length). An additional benefit of expressing quantities using atomic units is that their values calculated and reported in atomic units do not change when values of fundamental constants are revised. The fundamental constants are built into the conversion factors between atomic units and SI.

History

Hartree defined units based on three physical constants:^[1]: 91

Both in order to eliminate various universal constants from the equations and also to avoid high powers of 10 in numerical work, it is convenient to express quantities in terms of units, which may be called 'atomic units', defined as follows:

Unit of length, ⁠${\displaystyle a_{\text{H}}=h^{2}\,/\,4\pi ^{2}me^{2}}$⁠, on the orbital mechanics the radius of the 1-quantum circular orbit of the H-atom with fixed nucleus.

Unit of charge, ⁠${\displaystyle e}$⁠, the magnitude of the charge on the electron.

Unit of mass, ⁠${\displaystyle m}$⁠, the mass of the electron.

Consistent with these are:

Unit of action, ⁠${\displaystyle h\,/\,2\pi }$⁠.

Unit of energy, ⁠${\displaystyle e^{2}/a_{\text{H}}=2hcR=}$⁠ [...]

Unit of time, ⁠${\displaystyle 1\,/\,4\pi cR}$⁠.
 

— D.R. Hartree, The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

Here, the modern equivalent of ⁠${\displaystyle R}$⁠ is the Rydberg constant ⁠${\displaystyle R_{\infty }}$⁠, of ⁠${\displaystyle m}$⁠ is the electron mass ⁠${\displaystyle m_{\text{e}}}$⁠, of ⁠${\displaystyle a_{\text{H}}}$⁠ is the Bohr radius ⁠${\displaystyle a_{0}}$⁠, and of ⁠${\displaystyle h/2\pi }$⁠ is the reduced Planck constant ⁠${\displaystyle \hbar }$⁠. Hartree's expressions that contain ⁠${\displaystyle e}$⁠ differ from the modern form due to a change in the definition of ⁠${\displaystyle e}$⁠, as explained below.

In 1957, Bethe and Salpeter's book Quantum mechanics of one-and two-electron atoms^[3] built on Hartree's units, which they called atomic units abbreviated "a.u.". They chose to use ⁠${\displaystyle \hbar }$⁠, their unit of action and angular momentum in place of Hartree's length as the base units. They noted that the unit of length in this system is the radius of the first Bohr orbit and their velocity is the electron velocity in Bohr's model of the first orbit.

In 1959, Shull and Hall^[4] advocated atomic units based on Hartree's model but again chose to use ⁠${\displaystyle \hbar }$⁠ as the defining unit. They explicitly named the distance unit a "Bohr radius"; in addition, they wrote the unit of energy as ⁠${\displaystyle H=me^{4}/\hbar ^{2}}$⁠ and called it a Hartree. These terms came to be used widely in quantum chemistry.^[5]: 349

In 1973 McWeeny extended the system of Shull and Hall by adding permittivity in the form of ⁠${\displaystyle \kappa _{0}=4\pi \epsilon _{0}}$⁠ as a defining or base unit.^[6][7] Simultaneously he adopted the SI definition of ⁠${\displaystyle e}$⁠ so that his expression for energy in atomic units is ⁠${\displaystyle e^{2}/(4\pi \epsilon _{0}a_{0})}$⁠, matching the expression in the 8th SI brochure.^[8]

Definition

A set of base units in the atomic system as in one proposal are the electron rest mass, the magnitude of the electronic charge, the Planck constant, and the permittivity.^[6][9] In the atomic units system, each of these takes the value 1; the corresponding values in the International System of Units^[10]: 132 are given in the table.

Base atomic units^[*]

Symbol and Name	Quantity (dimensions)[†]	Atomic units[‡]	SI units
⁠${\displaystyle \hbar }$⁠, reduced Planck constant	action (ML2T−1)	1	1.054571817...×10−34 J⋅s [11]
⁠${\displaystyle e}$⁠, elementary charge	charge (Q)	1	1.602176634×10−19 C [12]
⁠${\displaystyle m_{\text{e}}}$⁠, electron rest mass	mass (M)	1	9.1093837139(28)×10−31 kg [13]
⁠${\displaystyle 4\pi \epsilon _{0}}$⁠, permittivity	permittivity (Q2W−1L−1)	1	1.11265005620(17)×10−10 F⋅m−1 [14]

Table notes

• ^ *: This arbitrary choice of base units was proposed by McWeeny.
  
• ^ †: See Dimensional analysis. W represents the dimension of energy, ML²T⁻².^[6]
  
• ^ ‡: In the 'atomic units' column, the convention that uses dimensionless equivalents has been applied.
  

Units

Three of the defining constants (reduced Planck constant, elementary charge, and electron rest mass) are atomic units themselves – of action,^[15] electric charge,^[16] and mass,^[17] respectively. Two named 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 ⁠${\displaystyle E_{\text{h}}\equiv \hbar ^{2}/m_{\text{e}}a_{0}^{2}}$⁠).

Defined atomic units

Atomic unit of	Expression	Value in SI units	Other equivalents
electric charge density	${\displaystyle e/a_{0}^{3}}$	1.08120238677(51)×1012 C⋅m−3 [18]
electric current	${\displaystyle eE_{\text{h}}/\hbar }$	6.6236182375082(72)×10−3 A [19]
electric charge	${\displaystyle e}$	1.602176634×10−19 C [20]
electric dipole moment	${\displaystyle ea_{0}}$	8.4783536198(13)×10−30 C⋅m [21]	≘ 2.541746473 D
electric quadrupole moment	${\displaystyle ea_{0}^{2}}$	4.4865515185(14)×10−40 C⋅m2 [22]
electric potential	${\displaystyle E_{\text{h}}/e}$	27.211386245981(30) V [23]
electric field	${\displaystyle E_{\text{h}}/ea_{0}}$	5.14220675112(80)×1011 V⋅m−1 [24]
electric field gradient	${\displaystyle E_{\text{h}}/ea_{0}^{2}}$	9.7173624424(30)×1021 V⋅m−2 [25]
permittivity	${\displaystyle e^{2}/a_{0}E_{\text{h}}}$	1.11265005620(17)×10−10 F⋅m−1 [14]	⁠${\displaystyle 4\pi \epsilon _{0}}$⁠
electric polarizability	${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$	1.64877727212(51)×10−41 C2⋅m2⋅J−1 [26]
1st hyperpolarizability	${\displaystyle e^{3}a_{0}^{3}/E_{\text{h}}^{2}}$	3.2063612996(15)×10−53 C3⋅m3⋅J−2 [27]
2nd hyperpolarizability	${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$	6.2353799735(39)×10−65 C4⋅m4⋅J−3 [28]
magnetic dipole moment	${\displaystyle \hbar e/m_{\text{e}}}$	1.85480201315(58)×10−23 J⋅T−1 [29]	⁠${\displaystyle 2\mu _{\text{B}}}$⁠
magnetic flux density	${\displaystyle \hbar /ea_{0}^{2}}$	2.35051757077(73)×105 T [30]	≘ 2.3505×109 G
magnetizability	${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$	7.8910365794(49)×10−29 J⋅T−2 [31]
action	${\displaystyle \hbar }$	1.054571817...×10−34 J⋅s [32]
energy	${\displaystyle E_{\text{h}}}$	4.3597447222060(48)×10−18 J [33]	⁠${\displaystyle 2hcR_{\infty }}$⁠, ⁠${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$⁠, 27.211386245988(53) eV [34]
force	${\displaystyle E_{\text{h}}/a_{0}}$	8.2387235038(13)×10−8 N [35]	82.387 nN, 51.421 eV·Å−1
length	${\displaystyle a_{0}}$	5.29177210544(82)×10−11 m [36]	⁠${\displaystyle \hbar /\alpha m_{\text{e}}c}$⁠, 0.529177 Å
mass	${\displaystyle m_{\text{e}}}$	9.1093837139(28)×10−31 kg [37]
momentum	${\displaystyle \hbar /a_{0}}$	1.99285191545(31)×10−24 kg⋅m⋅s−1 [38]
time	${\displaystyle \hbar /E_{\text{h}}}$	2.4188843265864(26)×10−17 s [39]
velocity	${\displaystyle a_{0}E_{\text{h}}/\hbar }$	2.18769126216(34)×106 m⋅s−1 [40]	⁠${\displaystyle \alpha c}$⁠
⁠${\displaystyle c}$⁠: speed of light, ⁠${\displaystyle \epsilon _{0}}$⁠: vacuum permittivity, ⁠${\displaystyle R_{\infty }}$⁠: Rydberg constant, ⁠${\displaystyle h}$⁠: Planck constant, ⁠${\displaystyle \alpha }$⁠: fine-structure constant, ⁠${\displaystyle \mu _{\text{B}}}$⁠: Bohr magneton, ≘: correspondence

Conventions

Different conventions are adopted in the use of atomic units, which vary in presentation, formality and convenience.

Explicit units

• Many texts (e.g. Jerrard & McNiell,^[7] Shull & Hall^[4]) define the atomic units as quantities, without a transformation of the equations in use. As such, they do not suggest treating either quantities as dimensionless or changing the form of any equations. This is consistent with expressing quantities in terms of dimensional quantities, where the atomic unit is included explicitly as a symbol (e.g. ⁠${\displaystyle m=3.4~m_{\text{e}}}$⁠, ⁠${\displaystyle m=3.4~{\text{a.u. of mass}}}$⁠, or more ambiguously, ⁠${\displaystyle m=3.4~{\text{a.u.}}}$⁠), and keeping equations unaltered with explicit constants.^[41]
• Provision for choosing more convenient closely related quantities that are more suited to the problem as units than universal fixed units are is also suggested, for example based on the reduced mass of an electron, albeit with careful definition thereof where used (for example, a unit ⁠${\displaystyle H_{M}=\mu e^{4}/\hbar ^{2}}$⁠, where ⁠${\displaystyle \mu =m_{\text{e}}M/(m_{\text{e}}+M)}$⁠ for a specified mass ⁠${\displaystyle M}$⁠).^[4]

A convention that eliminates units

In atomic physics, it is common to simplify mathematical expressions by a transformation of all quantities:

• Hartree suggested that expression in terms of atomic units allows us "to eliminate various universal constants from the equations", which amounts to informally suggesting a transformation of quantities and equations such that all quantities are replaced by corresponding dimensionless quantities.^[1]: 91 He does not elaborate beyond examples.
• McWeeny suggests that "... their adoption permits all the fundamental equations to be written in a dimensionless form in which constants such as ⁠${\displaystyle e}$⁠, ⁠${\displaystyle m}$⁠ and ⁠${\displaystyle h}$⁠ are absent and need not be considered at all during mathematical derivations or the processes of numerical solution; the units in which any calculated quantity must appear are implicit in its physical dimensions and may be supplied at the end." He also states that "An alternative convention is to interpret the symbols as the numerical measures of the quantities they represent, referred to some specified system of units: in this case the equations contain only pure numbers or dimensionless variables; ... the appropriate units are supplied at the end of a calculation, by reference to the physical dimensions of the quantity calculated. [This] convention has much to recommend it and is tacitly accepted in atomic and molecular physics whenever atomic units are introduced, for example for convenience in computation."
• An informal approach is often taken, in which "equations are expressed in terms of atomic units simply by setting ⁠${\displaystyle \hbar =m_{\text{e}}=e=4\pi \epsilon _{0}=1}$⁠".^[41][42][43] This is a form of shorthand for the more formal process of transformation between quantities that is suggested by others, such as McWeeny.

Physical constants

Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant ⁠${\displaystyle \alpha ={e^{2}}/{(4\pi \epsilon _{0})\hbar c}\approx 1/137}$⁠, 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, is ⁠${\displaystyle c=1/\alpha \,{\text{a.u.}}\approx 137\,{\text{a.u.}}}$⁠^[44]: 597

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	${\displaystyle r_{\text{e}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}}$	${\displaystyle \alpha ^{2}\,a_{0}\approx 0.0000532\,a_{0}}$
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	${\displaystyle m_{\text{p}}}$	${\displaystyle \approx 1836\,m_{\text{e}}}$

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
• Charge = −1 a.u. of charge
• Orbital radius = 1 a.u. of length
• Orbital velocity = 1 a.u. of velocity^[44]: 597
• Orbital period = 2π a.u. of time
• Orbital angular velocity = 1 radian per a.u. of time
• Orbital 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
• Lorentz force (due to nucleus) = 1 a.u. of force

References

^[45]

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, S2CID 122077124
2. McQuarrie, Donald A. (2008). Quantum Chemistry (2nd ed.). New York, NY: University Science Books.
3. Bethe, Hans A.; Salpeter, Edwin E. (1957). Introduction. Units. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 2–4. doi:10.1007/978-3-662-12869-5_1. ISBN 978-3-662-12871-8.
4. Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0. S2CID 23692353.
5. Levine, Ira N. (1991). Quantum chemistry. Pearson advanced chemistry series (4 ed.). Englewood Cliffs, NJ: Prentice-Hall International. ISBN 978-0-205-12770-2.
6. McWeeny, R. (May 1973). "Natural Units in Atomic and Molecular Physics". Nature. 243 (5404): 196–198. Bibcode:1973Natur.243..196M. doi:10.1038/243196a0. ISSN 0028-0836. S2CID 4164851.
7. Jerrard, H. G.; McNeill, D. B. (1992). Systems of units. Dordrecht: Springer Netherlands. pp. 3–8. doi:10.1007/978-94-011-2294-8_2. ISBN 978-0-412-46720-2.
8. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), p. 125, ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16. Note that this information is omitted in the 9th edition.
9. Paul Quincey; Peter J Mohr; William D Phillips (2019), "Angles are inherently neither length ratios nor dimensionless", Metrologia, 56 (4): 043001, arXiv:1909.08389, Bibcode:2019Metro..56d3001Q, doi:10.1088/1681-7575/ab27d7, In [the Hartree system of atomic] units, m_e, e, ħ and 1/4πε₀ are all set equal to unity. – a reference giving an equivalent set of defining constants.
10. "9th edition of the SI Brochure". BIPM. 2019. Retrieved 2019-05-20.
11. "reduced Planck constant". CODATA.
12. "elementary charge". CODATA.
13. "electron mass". CODATA.
14. "atomic unit of permittivity". CODATA.
15. "atomic unit of action". CODATA.
16. "atomic unit of charge". CODATA.
17. "atomic unit of mass". CODATA.
18. "atomic unit of charge density". CODATA.
19. "atomic unit of current". CODATA.
20. "atomic unit of charge". CODATA.
21. "atomic unit of electric dipole moment". CODATA.
22. "atomic unit of electric quadrupole moment". CODATA.
23. "atomic unit of electric potential". CODATA.
24. "atomic unit of electric field". CODATA.
25. "atomic unit of electric field gradient". CODATA.
26. "atomic unit of electric polarizability". CODATA.
27. "atomic unit of 1st hyperpolarizability". CODATA.
28. "atomic unit of 2nd hyperpolarizability". CODATA.
29. "atomic unit of magnetic dipole moment". CODATA.
30. "atomic unit of magnetic flux density". CODATA.
31. "atomic unit of magnetizability". CODATA.
32. "atomic unit of action". CODATA.
33. "atomic unit of energy". CODATA.
34. "Hartree energy in eV". CODATA.
35. "atomic unit of force". CODATA.
36. "atomic unit of length". CODATA.
37. "atomic unit of mass". CODATA.
38. "atomic unit of momentum". CODATA.
39. "atomic unit of time". CODATA.
40. "atomic unit of velocity". CODATA.
41. Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5.
42. Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4.
43. Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2.
44. Karplus, Martin; Porter, Richard Needham (1970), Atoms and Molecules: An Introduction for Students of Physical Chemistry, Netherlands: W. A. Benjamin
45. "CODATA Internationally recommended 2022 values of the Fundamental Physical Constants". NIST Reference on Constants, Units, and Uncertainty. NIST.