Atomic Units

Atomic units are tuned to electron-scale Coulomb problems. They are common in atomic physics, quantum chemistry, and electronic-structure calculations.

For versioned conversion factors, dimensional restoration, and Hartree–Rydberg code diagnostics, use the AMO Atomic Units reference. The present entry is the short cross-volume convention translator.

Hartree Atomic Units

The usual Hartree atomic-unit convention sets

$$\hbar=1, \qquad m_e=1, \qquad e=1, \qquad 4\pi\epsilon_0=1.$$

Then the Coulomb potential between charges $q_1$ and $q_2$ is written without the SI prefactor:

$$V(r)=\frac{q_1q_2}{r}.$$

Basic Atomic Units

Quantity	Atomic Unit	SI Meaning
Length	Bohr radius $a_0$	see Constants
Energy	Hartree $E_h$	$E_h\approx27.211386246\ \mathrm{eV}$ using 2022 CODATA
Charge	elementary charge magnitude $e$	positive charge unit
Mass	electron mass $m_e$	electron mass unit
Angular momentum	$\hbar$	action unit

Hydrogenic Reminder

In atomic units, the nonrelativistic hydrogen Hamiltonian for an infinitely heavy nucleus is often written

$$H=-\frac12\nabla^2-\frac{1}{r}.$$

The simplicity is convention-dependent. Restoring SI units brings back $\hbar$, $m_e$, $e$, and $4\pi\epsilon_0$.

Common Mistakes

• Treating $e=1$ as meaning the electron charge is $+1$; the electron charge is still negative.
• Mixing atomic-unit Coulomb expressions with SI electromagnetic fields.
• Forgetting that energies in Hartree are twice energies in Rydberg units.
• Comparing code outputs without checking whether the code uses Hartree, Rydberg, or electron volt units.

Canonical Links

• Hydrogen Atom
• AMO Atomic Units Reference
• Atomic Units and Scales
• Hydrogen Spectrum
• Constants
• Units

References

• P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 5th ed., Oxford University Press, 2010.
• I. N. Levine, Quantum Chemistry, 7th ed., Pearson, 2014.
• NIST CODATA Fundamental Physical Constants