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Atomic units

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Motivation
History
Definition

Table notes

Units
Conventions

Explicit units
A convention that elimin..

Physical constants
Bohr model in atomic uni..
References

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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. Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, , and absorbance in other contexts.

Motivation

In the context of atomic physics, using the atomic units system can be a convenient shortcut, eliminating symbols and numbers and reducing the order of magnitude of most numbers involved. For example, the Hamiltonian operator in the Schrödinger equation for the helium atom with standard quantities, such as when using SI units, is

\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

\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 , , , and all correspond to the value (see below). The distances relevant to the physics expressed in SI units are naturally on the order of , while expressed in atomic units distances are on the order of (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, since the fundamental constants are built into the conversion factors between atomic units and SI.

History

Hartree defined units based on three physical constants:

Here, the modern equivalent of is the Rydberg constant , of is the electron mass , of is the Bohr radius , and of is the reduced Planck constant . Hartree's expressions that contain differ from the modern form due to a change in the definition of , as explained below.

In 1957, Bethe and Salpeter's book Quantum mechanics of one-and two-electron atoms

(2025). 9783662128718, Springer Berlin Heidelberg. . ISBN 9783662128718

built on Hartree's units, which they called atomic units abbreviated "a.u.". They chose to use , 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 advocated atomic units based on Hartree's model but again chose to use as the defining unit. They explicitly named the distance unit a "Bohr radius"; in addition, they wrote the unit of energy as and called it a Hartree. These terms came to be used widely in quantum chemistry.

Ira N. Levine (1991). 9780205127702, Prentice-Hall International. ISBN 9780205127702

In 1973 McWeeny extended the system of Shull and Hall by adding permittivity in the form of as a defining or base unit.

(1992). 9780412467202, Springer Netherlands. . ISBN 9780412467202

Simultaneously he adopted the SI definition of so that his expression for energy in atomic units is , matching the expression in the 8th SI brochure.. Note that this information is omitted in the 9th edition.

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. In the atomic units system, each of these takes the value 1; the corresponding values in the International System of Units are given in the table.

+ Base atomic units

Table notes

Units

Three of the defining constants (reduced Planck constant, elementary charge, and electron rest mass) are atomic units themselves – of action, electric charge, and mass, respectively. Two named units are those of length (Bohr radius ) and energy (hartree ).

+ Defined atomic units
$e/a_0^3$
$e E_\text{h} / \hbar$
$e$
$e a_0$
$e a_0^2$
$E_\text{h} / e$
$E_\text{h} / e a_0$
$E_\text{h} / e a_0^2$
$e^2 / a_0 E_\text{h}$
$e^2 a_0^2 / E_\text{h}$
$e^3 a_0^3 / E_\text{h}^2$
$e^4 a_0^4 / E_\text{h}^3$
$\hbar e / m_\text{e}$
$\hbar/e a_0^2$
$e^2 a_0^2 / m_\text{e}$
$\hbar$
$E_\text{h}$		, ,
$E_\text{h} / a_0$		,
$a_0$		,
$m_\text{e}$
$\hbar/a_0$
$\hbar / E_\text{h}$
$a_0 E_\text{h} / \hbar$
speed of light, vacuum permittivity, Rydberg constant, Planck constant, fine-structure constant, 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, Shull & Hall) 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. , , or more ambiguously, ), and keeping equations unaltered with explicit constants.
Frank L. Pilar (2025). 9780486414645, Dover Publications. . ISBN 9780486414645
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 , where for a specified mass ).

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. 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 , and 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 ".
David M. Bishop (1993). 9780486673554, Dover Publications. . ISBN 9780486673554
Gordon W. F. Drake (2025). 9780387208022, Springer. . ISBN 9780387208022
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 , 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

+ Some physical constants expressed in atomic units

(1/\alpha) \,a_0 E_\text{h}/\hbar \approx 137 \,a_0 E_\text{h}/\hbar

\alpha^2 \,a_0 \approx 0.0000532 \,a_0

\alpha \,a_0 \approx 0.007297 \,a_0

\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
Orbital period = 2 π a.u. of time
Orbital angular velocity = 1 radian per a.u. of time
Orbital momentum = 1 a.u. of momentum
Ionization energy = 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

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