Atomic units are a system used in atomic physics and quantum chemistry where the electron mass, elementary charge, reduced Planck constant and Coulomb constant equal one, making the Bohr radius and the Hartree energy the natural scales of atomic structure.Atomic units are a measurement system designed for the world of atoms and molecules, where the natural scales are set by the electron itself. In this system, the electron mass (mₑ), the elementary charge (e), the reduced Planck constant (ħ), and the Coulomb constant (1/4πε₀) are all set to 1. This choice makes the Bohr radius — the typical distance of an electron from a hydrogen nucleus — equal to 1 atomic unit of length, and the Hartree energy — the binding energy of the hydrogen atom — equal to 1 atomic unit of energy. Everything in atomic physics then becomes a simple number: the hydrogen atom's ground state energy is -0.5, the electron's orbital radius is 1, and so on.
Atomic units were introduced by the physicist Douglas Hartree in the 1920s, who developed them to simplify his numerical calculations of atomic structure, and they were later formalized and widely adopted in quantum chemistry.
To set up atomic units, you define the unit of mass as the electron mass, the unit of charge as the electron charge, and the unit of angular momentum as ħ. From these, the unit of length emerges as the Bohr radius (a₀ ≈ 5.29×10⁻¹¹ m), and the unit of energy as the Hartree (Eₕ ≈ 4.36×10⁻¹⁸ J). The Coulomb constant is then automatically 1, because the force between two unit charges at unit distance is 1. In practice, you write the Schrödinger equation with mₑ = 1, e = 1, ħ = 1, and 1/4πε₀ = 1, and all quantities are expressed in these units.
Atomic units are the standard in quantum chemistry software (like Gaussian and GAMESS) and in atomic physics research, used for calculating molecular structures, spectra, and chemical reactions.
In atomic units, the speed of light is about 137 — the inverse of the fine-structure constant — which is why relativistic effects in atoms are small but not negligible.
