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# Atomic units

Atomic units (au) form a system of units convenient for atomic physics, electromagnetism, and quantum electrodynamics, especially when the focus is on the properties of electrons. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In au, the numerical values of the following six physical constants are all unity by definition:

• Two properties of the electron, its mass and charge;
• Two properties of the hydrogen atom, its Bohr radius and the absolute value of its electric potential energy in the ground state;
• Two constants, Dirac's and that for Coulomb's Law.

## Fundamental units

Fundamental Atomic Units
Quantity Name Symbol SI value Planck unit scale
length Bohr radius a05.291 772 108(18)×10-11 m10-35 m
mass electron rest mass me 9.109 3826(16)×10-31 kg10-8 kg
charge elementary charge e 1.602 176 53(14)×10-19 C10-18 C
angular momentum Planck's constant $\hbar = h/2 \pi$ 1.054 571 68(18)×10-34 J s(same)
energy Hartree energy Eh4.359 744 17(75)×10-18 J109 J
electrostatic force constant Coulomb's constant 1/(4πε0)8.9875516×109 C-2 N m2(same)

These six quantities are not independent; to normalize all six quantities to 1, it suffices to normalize any four of them to 1. The normalizations of the Hartree energy and Coulomb's constant, for example, are only an incidental consequence of normalizing the other four quantities.

## Some derived units

Derived Atomic Units
Quantity Expression SI value Planck unit scale
time$\frac{\hbar}{E_h}$2.418 884 326 505(16)×10-17 s10-43 s
velocity$\frac{a_0 E_h}{\hbar}$2.187 691 2633(73)×106 m s-1108 m s-1
force$\frac{E_h}{a_o}$8.238 7225(14)×10-8 N1044 N
current$\frac{eE_h}{\hbar}$ 6.623 617 82(57)×10-3 A1026 A
temperature$\frac{E_h}{k_B}$ 3.157 7464(55)×105 K1032 K
pressure$\frac{E_h}{{a_o}^3}$ 2.942 1912(19)×1013 N m-210114 Pa

## Comparison with Planck units

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. To facilitate comparing the two systems of units, the above tables show the order of magnitude, in SI units, of the Planck unit corresponding to each atomic unit. Generally, when an atomic unit is "large" in SI terms, the corresponding Planck unit is "small", and vice versa. It should be kept in mind that au were designed for atomic-scale calculations in the present-day Universe, while Planck units are more suitable for quantum gravity and early-Universe cosmology.

Both au and Planck units normalize the Dirac constant and the Coulomb force constant to 1. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Letting α denote the fine structure constant, the au value of c is α-1 ≈ 137.036.

Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and a0, the Bohr radius of the hydrogen atom. Normalizing a0 to 1 amounts to normalizing the Rydberg constant, R, to 4π/α = 4πc. Given au, the Bohr magneton μB=1/2. The corresponding Planck value is e/2me. Finally, au normalize a unit of atomic energy to 1, while Planck units normalize to 1 Boltzmann's constant k, which relates energy and temperature.

## Quantum mechanics and electrodynamics simplified

The (non-relativistic) Schrödinger equation for an electron in SI units is

$- \frac{\hbar^2}{2m_e} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$.

The same equation in au is

$- \frac{1}{2} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$.

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

$\hat H = - {{{\hbar^2} \over {2 m_e}}\nabla^2} - {1 \over {4 \pi \epsilon_0}}{{e^2} \over {r}}$,

while atomic units transform the preceding equation into

$\hat H = - {{{1} \over {2}}\nabla^2} - {{1} \over {r}}$.

Finally, Maxwell's equations take the following elegant form in au:

$\nabla \cdot \mathbf{E} = 4\pi\rho$
$\nabla \cdot \mathbf{B} = 0$
$\nabla \times \mathbf{E} = -\alpha \frac{\partial \mathbf{B}} {\partial t}$
$\nabla \times \mathbf{B} = \alpha \left( \frac{\partial \mathbf{E}} {\partial t} + 4\pi \mathbf{J} \right)$

(There is actually some ambiguity in defining the atomic unit of magnetic field. The above Maxwell equations use the "Gaussian" convention, in which a plane wave has electric and magnetic fields of equal magnitude. In the "Lorentz force" convention, a factor of α is absorbed into B.)