Atomic units

History

Hartree defined units based on three physical constants:

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, ⁠${\displaystyle a_{\text{H}}=h^2/4{\pi }^{2}m{e}^2}$⁠, 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 ⁠${\displaystyle R_{\mathrm{\infty }}}$⁠, 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 ⁠${\displaystyle \hslash }$⁠. 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 built on Hartree's units, which they called atomic units abbreviated "a.u.". They chose to use ⁠${\displaystyle \hslash }$⁠, 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 ⁠${\displaystyle \hslash }$⁠ as the defining unit. They explicitly named the distance unit a "Bohr radius"; in addition, they wrote the unit of energy as ⁠${\displaystyle H=m{e}^4/{\hslash }^2}$⁠ and called it a Hartree. These terms came to be used widely in quantum chemistry.

In 1973 McWeeny extended the system of Shull and Hall by adding permittivity in the form of ⁠${\displaystyle {\kappa }_0=4\pi {ϵ}_0}$⁠ as a defining or base unit. Simultaneously he adopted the SI definition of ⁠${\displaystyle e}$⁠ so that his expression for energy in atomic units is ⁠${\displaystyle e^2/(4\pi {ϵ}_0{a}_0)}$⁠, matching the expression in the 8th SI brochure.

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 convention of the atomic units system that treats quantities as dimensionless, each of these takes the value 1; the corresponding values in the International System of Units are given in the table.

Table notes

• ^ *: This choice of base units was proposed by McWeeny; the choice is essentially arbitrary.
  
• ^ †: The base dimensions for dimensional analysis may chosen as M, L, T, Q. For convenience, the derived dimensions A (action, ML²T⁻¹) and W (energy, ML²T⁻²) are also used.
  
• ^ ‡: This column use the convention that uses the dimensionless equivalents of quantities.
  

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 ⁠${\displaystyle a_0\equiv 4\pi {ϵ}_0{\hslash }^2/m_{\text{e}}{e}^2}$⁠) and energy (hartree ⁠${\displaystyle E_{\text{h}}\equiv {\hslash }^2/m_{\text{e}}{a}_0^2}$⁠).

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. ⁠${\displaystyle m=3.4m_{\text{e}}}$⁠, ⁠${\displaystyle m=3.4\text{a.u. of mass}}$⁠, or more ambiguously, ⁠${\displaystyle m=3.4\text{a.u.}}$⁠), and keeping equations unaltered with explicit constants.
• 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 ⁠${\displaystyle H_M=\mu e^4/{\hslash }^2}$⁠, where ⁠${\displaystyle \mu =m_{\text{e}}M/(m_{\text{e}}+M)}$⁠ for a specified mass ⁠${\displaystyle M}$⁠).

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 ⁠${\displaystyle e}$⁠, ⁠${\displaystyle m}$⁠ 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 ⁠${\displaystyle \hslash =m_{\text{e}}=e=4\pi {ϵ}_0=1}$⁠". 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 ⁠${\displaystyle \alpha =e^2/(4\pi {ϵ}_0\hslash c)\approx 1/137}$⁠, 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 ⁠${\displaystyle c=1/\alpha \text{a.u.}\approx 137\text{a.u.}}$⁠

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 = 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