Editing Atomic units (section)

== History ==
Hartree defined units based on three physical constants:<ref name="Hartree1928"/>{{rp|91}}
{{blockquote
|text=
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'', {{tmath|1= a_\text{H} = h^2 \,/\, 4 \pi^2 m e^2 }}, on the orbital mechanics the radius of the 1-quantum circular orbit of the [[Hydrogen|H]]-atom with fixed nucleus.
: ''Unit of charge'', {{tmath|1= e }}, the magnitude of the charge on the electron.
: ''Unit of mass'', {{tmath|1= m }}, the mass of the electron.
<!--Hartree did not use other units, for example, the Boltzmann constant, nor did he use . -->
Consistent with these are:
: ''Unit of action'', {{tmath|1= h \,/\, 2 \pi }}.
: ''Unit of energy'', {{tmath|1= e^2 / a_\text{H} = 2 h c R = }} [...]
: ''Unit of time'', {{tmath|1= 1 \,/\, 4 \pi c R }}.{{br}}&nbsp;
|author=D.R. Hartree
|title=''The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods''
}}
Here, the modern equivalent of {{tmath|1= R }} is the [[Rydberg constant]] {{tmath|1= R_\infty }}, of {{tmath|1= m }} is the electron mass {{tmath|1= m_\text{e} }}, of {{tmath|1= a_\text{H} }} is the Bohr radius {{tmath|1= a_0 }}, and of {{tmath|1= h / 2 \pi }} is the reduced Planck constant {{tmath|1= \hbar }}. Hartree's expressions that contain {{tmath|1= e }} differ from the modern form due to a change in the definition of {{tmath|1= e }}, as explained below.

In 1957, Bethe and Salpeter's book ''Quantum mechanics of one-and two-electron atoms''<ref>{{cite book |last1=Bethe |first1=Hans A. |url=http://link.springer.com/10.1007/978-3-662-12869-5_1 |title=Introduction. Units |last2=Salpeter |first2=Edwin E. |date=1957 |publisher=Springer Berlin Heidelberg |isbn=978-3-662-12871-8 |location=Berlin, Heidelberg |pages=2–4 |language=en |doi=10.1007/978-3-662-12869-5_1}}</ref> built on Hartree's units, which they called '''atomic units''' abbreviated "a.u.". They chose to use {{tmath|1= \hbar }}, their unit of [[Action (physics)|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 model of the atom|Bohr orbit]] and their velocity is the electron velocity in Bohr's model of the first orbit.

In 1959, Shull and Hall<ref name="ShullHall1959">
{{cite journal
 |last1=Shull |first1=H.
 |last2=Hall |first2=G. G.
 |year=1959
 |title=Atomic Units
 |journal=[[Nature (journal)|Nature]]
 |volume=184 |issue=4698 |page=1559
 |doi=10.1038/1841559a0
 |bibcode = 1959Natur.184.1559S | s2cid=23692353
}}</ref> advocated '''atomic units''' based on Hartree's model but again chose to use {{tmath|1= \hbar }} as the defining unit. They explicitly named the distance unit a "[[Bohr radius]]"; in addition, they wrote the unit of energy as {{tmath|1= H = m e^4 / \hbar^2 }} and called it a '''Hartree'''. These terms came to be used widely in quantum chemistry.<ref>{{Cite book |last=Levine |first=Ira N. |title=Quantum chemistry |date=1991 |publisher=Prentice-Hall International |isbn=978-0-205-12770-2 |edition=4 |series=Pearson advanced chemistry series |location=Englewood Cliffs, NJ }}</ref>{{rp|349}}

In 1973 McWeeny extended the system of Shull and Hall by adding [[permittivity]] in the form of {{tmath|1= \kappa_0 = 4 \pi \epsilon_0 }} as a defining or base unit.<ref name="McWeeny1973">{{cite journal |last=McWeeny |first=R. |date=May 1973 |title=Natural Units in Atomic and Molecular Physics |url=https://www.nature.com/articles/243196a0 |journal=Nature |language=en |volume=243 |issue=5404 |pages=196–198 |doi=10.1038/243196a0 |bibcode=1973Natur.243..196M |s2cid=4164851 |issn=0028-0836}}</ref><ref name="JerrardMcNeill1992">{{cite book |last1=Jerrard |first1=H. G. |last2=McNeill |first2=D. B. |date=1992 |url=http://link.springer.com/10.1007/978-94-011-2294-8_2 |title=Systems of units |publisher=Springer Netherlands |isbn=978-0-412-46720-2 |location=Dordrecht |pages=3–8 |language=en |doi=10.1007/978-94-011-2294-8_2 }}</ref> Simultaneously he adopted the SI definition of {{tmath|1= e }} so that his expression for energy in atomic units is {{tmath|1= e^2 / (4 \pi \epsilon_0 a_0) }}, matching the expression in the 8th SI brochure.<ref>{{SIbrochure8th|page=125}}. Note that this information is omitted in the 9th edition.</ref>