Atomic units
A natural unit system for atomic physics calculations.
Atomic units are a system of natural units of measurement designed for convenience in atomic physics, computational chemistry, and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree to improve accuracy and stability in reporting values, as earlier reliance on fundamental constants with fluctuating accepted values caused confusion. Hartree defined his original units using three physical constants, though his expressions differ from modern forms due to a later change in the definition of one constant. In 1957, Bethe and Salpeter built on Hartree’s units in their book, calling them atomic units and abbreviating them "a.u."; they chose to use a unit of action and angular momentum as a base unit, noting that the unit of length was the radius of the first Bohr orbit and the unit of velocity was the electron velocity in Bohr’s first orbit. In 1959, Shull and Hall advocated for atomic units based on Hartree’s model but again used that same unit of action as defining; they explicitly named the distance unit the Bohr radius and the energy unit the Hartree, terms that became widely used in quantum chemistry. In 1973, McWeeny extended the system by adding permittivity as a base unit, simultaneously adopting the SI definition of the relevant constant. A set of base units in one proposal includes the electron rest mass, the magnitude of the electronic charge, the Planck constant, and the permittivity; in the convention that treats quantities as dimensionless, each takes the value 1. Different conventions exist: some texts define atomic units as explicit dimensional quantities without transforming equations, while others adopt a transformation that eliminates universal constants from the equations, allowing all quantities to be expressed as dimensionless numbers. This latter approach permits fundamental equations to be written without constants like Planck’s constant or the electron charge, with units supplied at the end based on physical dimensions.
- field
- Atomic physics, computational chemistry, atomic spectroscopy
- known_for
- Proposing and naming the atomic units system
- named_by
- Douglas Hartree
Lore & Background
Atomic units were motivated by the need for stable and accurate reporting of quantum-mechanical calculations. In early atomic physics, the accepted values of fundamental constants such as ℏ, mₑ, e, and c were not sufficiently stable or accurate, making it difficult to directly compare results from different years. This led to suggestions that results should be reported using units based directly on these constants, eliminating confusion. Douglas Hartree defined atomic units based on three physical constants: the unit of length (a_H = h² / 4π² m e²), the unit of charge (e, the magnitude of the electron's charge), and the unit of mass (m, the mass of the electron). Consistent with these were the unit of action (h / 2π), the unit of energy (e² / a_H = 2 h c R), and the unit of time (1 / 4π c R). Using atomic units simplifies equations by setting ℏ, mₑ, 4πϵ₀, and e all to 1. For example, the Hamiltonian for the helium atom in SI units contains many constants, but in atomic units it becomes a simpler expression with only numerical coefficients and distances. Additionally, values reported in atomic units do not change when fundamental constants are revised, because those constants are built into the conversion factors.
Reader's Guide
Atomic units are significant because they provide a stable and convenient framework for atomic physics calculations. By setting fundamental constants to unity, they eliminate symbols and numbers, reducing the order of magnitude of most numbers involved. This makes calculations easier to perform and compare across different eras, as the values do not shift when the accepted values of constants like ℏ or mₑ are updated. The system was originally suggested by Douglas Hartree and remains widely used in fields such as computational chemistry and atomic spectroscopy. Its abbreviation "a.u." or "au" should not be confused with astronomical units, arbitrary units, or absorbance units. The legacy of atomic units lies in their ability to streamline complex quantum-mechanical equations—such as the Schrödinger equation for multi-electron atoms—while ensuring that reported results remain consistent over time, independent of revisions to fundamental constants.
Did You Know?
- Atomic units were originally suggested and named by physicist Douglas Hartree.
- The abbreviation 'a.u.' or 'au' is also used for astronomical units, arbitrary units, and absorbance units in other contexts.
- In atomic units, the constants ℏ, mₑ, 4πϵ₀, and e all correspond to the value 1.
- Hartree defined the atomic unit of length as a_H = h² / 4π² m e², based on the radius of the 1-quantum circular orbit of the hydrogen atom.
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