Atomic units

The atomic units (English: atomic units, au) form a natural system of units , which is mainly used in atomic and molecular physics and theoretical chemistry . The atomic units are based on the properties of the electron in the hydrogen atom .

The atomic units are:

• Length : the Bohr radius  ${\displaystyle a_{0}}$
• Mass : the atomic mass unit  and the electron mass ${\displaystyle \mathrm {amu} }$  ${\displaystyle m_{\mathrm {e} }}$
• Charge : the elementary charge  ${\displaystyle e}$
• Energy : the Hartree energy  ${\displaystyle E_{\text{h}}.}$

Also common:

• Angular momentum : Planck's constant  ${\displaystyle \hbar }$

• Time : the atomic time unit  ${\displaystyle \mathrm {a.t.u.} ={\frac {\hbar }{E_{\text{h}}}}.}$

The use of atomic units simplifies the Schrödinger equation . For example, the Hamilton operator for an electron in the hydrogen atom results in:

• in SI units :

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m_{\mathrm {e} }}}\nabla ^{2}-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{r}}}$

• in atomic units :

${\displaystyle {\hat {H}}=-{\frac {\nabla ^{2}}{2}}-{\frac {1}{r}}}$

Is in atomic units . The vacuum speed of light has the value where is the fine structure constant . With CODATA 2014, the SI units for precisely these basic units and for a further 18 derived atomic units were defined. ${\displaystyle e=m_{\mathrm {e} }=\hbar ={\frac {1}{4\pi \varepsilon _{0}}}=1\,\mathrm {a.u.} }$ ${\displaystyle 1/\alpha \approx 137\,\mathrm {a.u.} }$ ${\displaystyle \alpha }$

Values ​​in atomic units are formally dimensionless, quantities that are not dimensionless in SI units are usually identified by the formal "unit symbol" au (the dots are part of the unit symbol). For example, a mass of 2 au is twice the mass of electrons, while an electric field strength of is the field strength that prevails at a distance of a Bohr radius from an elementary charge. ${\displaystyle 1\,\mathrm {a.u.} ={\frac {\hbar ^{2}}{m\cdot e\cdot a_{0}^{3}}}}$

Atomic units for magnetic quantities

Atomic units for quantities of the magnetic field such as the magnetic flux density are not clearly defined . Either applies to an electromagnetic wave in a vacuum as in SI units or as in Gaussian units . It denotes the electric field strength and the vacuum speed of light . These different definitions have an impact on all quantities that are derived from the magnetic flux density. For example, Bohr's magneton corresponds only in SI-based atomic units . Various constant pre-factors result when calculating the intensity of an electromagnetic wave from the electric field strength. ${\displaystyle B}$ ${\displaystyle B=E/c_{0}}$ ${\displaystyle B=E}$ ${\displaystyle E}$ ${\displaystyle c_{0}}$ ${\displaystyle 1/2\,\mathrm {a.u.} }$

With CODATA 2014, the SI units for the atomic units of the magnetic flux density , the electric field strength , the speed and, based on this, the Bohr magneton etc. were specified. ${\displaystyle B_{\mathrm {au} }=\hbar /(a_{0}^{2}e)}$ ${\displaystyle {\vec {E}}_{\mathrm {au} }=E_{\mathrm {h} }/(a_{0}e)}$ ${\displaystyle v_{\mathrm {au} }=c_{0}\alpha }$

See also

• Planck units
• natural units

Individual evidence

1. ^ Douglas Rayner Hartree : The calculation of atomic structures . Wiley, New York, NY 1957 (IX, 181 p., The chapter ATOMIC UNITS can be found on p. 5 ff.).
2. ↑ Entry on atomic units . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.A00504 Version: 2.1.6.