2.4 PHYSICAL CONSTANTS USED AS ATOMIC UNITS

Sometimes fundamental physical constants, or other well defined physical quantities, are used as though they were units in certain specialized fields of science. For example, in astronomy it may be more convenient to express the mass of a star in terms of the mass of the sun. In atomic and molecular physics it is similarly more convenient to express masses in terms of the electron mass, $m_{\rm{e}}$, or in terms of the unified atomic mass unit, $1\ \rm{u}$, and to express charges in terms of the elementary charge $e$, and energies in terms of the electronvolt, $\rm{eV}$.

The electronvolt is the kinetic energy acquired by an electron in passing through a potential difference of $1\ \rm{V}$ in vacuum, $1\ \rm{e}V = 1.602\ 176\ 634 \times 10^{−19}\ \rm{J}$. The numerical value of a quantity expressed in this unit may be converted into its value when expressed in the SI by multiplication with the value of the physical constant in the SI.

The dalton and the unified atomic mass unit are alternative names for the same unit, therefore $1\ \rm{u} = 1 \rm{Da} \approx 1.6605 \times 10^{−27}\ \rm{kg}$. The dalton may be combined with the SI prefixes to express the masses of large molecules in kilodalton ($\rm{kDa}$) or megadalton ($\rm{MDa}$).

Physical quantity	Physical constant	Symbol for unit	Value in SI units	Notes
mass	electron mass	$m_{\rm{e}}$	$\approx 9.1094 \times 10^{-31}\ \rm{kg}$
charge	elementary charge	$e$	$= 1.602\ 176\ 634 \times 10^{-19}\ \rm{C}$
action, (angular momentum)	Planck constant divided by $2\pi$	$\hbar$	$= 1.054\ 571\ 817 ... \times\ 10^{-34}\ \rm{J\ s}$	1
length	bohr	$a_{0}$	$\approx 5.2918 \times 10^{-11}\ \rm{m}$	1
energy	hartree	$E_{\rm{h}}$	$\approx 4.3597 \times 10^{-18}\ \rm{J}$	1
time		$\hbar/E_{\rm{h}}$	$\approx 2.1489 \times 10^{-17}\ \rm{s}$
speed		$a_{0}E_{\rm{h}}/\hbar$	$\approx 2.1877 \times 10^{6}\ \rm{m\ s}^{-1}$	2
electric field strength		$E_{\rm{h}}/ea_{0}$	$\approx 5.1422 \times 10^{11}\ \rm{V\ m}^{-1}$
electric dipole moment		$ea_{0}$	$\approx 8.4784 \times 10^{-30}\ \rm{C\ m}$
electric quadrupole moment		$ea_{0}^2$	$\approx 4.4866 \times 10^{-40}\ \rm{C\ m}^2$
electric polarizability		$ea_{0}^2/E_{\rm{h}}$	$\approx 1.6488 \times 10^{-41}\ \rm{C}^2\ \rm{m}^2\ \rm{J}^{-1}$
magnetic flux density		$\hbar/ea_{0}^2$	$\approx 2.3505 \times 10^{5}\ \rm{T}$
magnetic dipole moment		$e\hbar/m_{\rm{e}}$	$\approx 1.8548 \times 10^{-23}\ \rm{J\ T}^{-1}$	3

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(1) $\hbar = h/2\pi; a_{0} = 4\pi \varepsilon_{0} \hbar^2/m_{e}e^2; E_{\rm{h}} = \hbar^2/m_{e}a_{0}^2$.
(2) The numerical value of the speed of light, when expressed in atomic units, is equal to the reciprocal of the fine-structure constant $\alpha$; $c/(a_{0}E_{\rm{h}}/\hbar) = c\hbar/a_{0}E_{\rm{h}} = \alpha^{-1} = 137.035\;999\;084(21)$
(3) The atomic unit of magnetic dipole moment is twice the Bohr magneton, $\mu+{\rm{B}}$.

One particular group of physical constants that are used as though they were units deserve special mention. These are the so-called atomic units and arise in calculations of electronic wavefunctions for atoms and molecules, i.e. in quantum chemistry. Only the first five atomic units in the table above have special names and symbols.

The relation of atomic units to the corresponding SI units involves the values of the fundamental physical constants, and is therefore not exact. The numerical values in the table are rounded from the CODATA compilation [17, 18]. The numerical results of calculations in theoretical chemistry are frequently quoted in atomic units, or as numerical values in the form physical quantity divided by atomic unit, so that the reader may make the conversion using the current best estimates of the physical constants.

Many authors make no use of the symbols for the atomic units listed in the tables above, but instead use the symbol “a.u.” or “au” for all atomic units. This custom should not be followed. It leads to confusion, just as it would if we were to write “SI” as a symbol for every SI unit, or “cgs” as a symbol for every cgs unit (the ‘centimetre, gram, second’ system of units, see Section 3.2, p. 16).

Examples - For the hydrogen molecule the equilibrium bond length re, and the dissociation energy De, are given by:
$r_{e} = 2.1\ a_{0}$     not    $r_{e} = 2.1$ a.u.
$D_{e} = 0.16\ E_{\rm{h}}$   not   $D_{e} = 0.16$ a.u.