View source for 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 ($\rm{MDa}$).

{| id="table2.4" style="font-family: "Computer Modern";margin:auto; text-align:left;width: 100%;"
|-
! style="border-bottom: 1px solid;text-align: left;font-weight: normal;font-style: italic;width: 18%;"| Physical quantity
! style="border-bottom: 1px solid;text-align: left;font-weight: normal;font-style: italic;width: 14%;"| Physical constant
! style="border-bottom: 1px solid;text-align: left;font-weight: normal;font-style: italic;width: 8%;"| Symbol for unit
! style="border-bottom: 1px solid;text-align: left;font-weight: normal;font-style: italic;width: 5%;" | Value in SI units
! style="border-bottom: 1px solid;text-align: left;font-weight: normal;font-style: italic;width: 5%;" | Notes
|-
| <span id="mass"/>mass || electron mass || $m_{\rm{e}}$ || $\approx 9.1094 \times 10^{-31}\ \rm{kg}$ || 
|-
| <span id="charge"/>charge || elementary charge || $e$ || $= 1.602\ 176\ 634 \times 10^{-19}\ \rm{C}$ || 
|-
| <span id="action"/>action, (angular momentum) || Planck constant divided by $2\pi$ || $\hbar$ || $= 1.054\ 571\ 817 ...  \times\ 10^{-34}\ \rm{J\ s}$ || 1
|-
| <span id="length"/>length || bohr || $a_{0}$ || $\approx 5.2918 \times 10^{-11}\ \rm{m}$ || 1
|-
| <span id="energy"/>energy || hartree || $E_{\rm{h}}$ || $\approx 4.3597 \times 10^{-18}\ \rm{J}$ || 1
|-
| <span id="time"/>time || || $\hbar/E_{\rm{h}}$ || $\approx 2.1489 \times 10^{-17}\ \rm{s}$ ||
|-
| <span id="speed"/>speed || || $a_{0}E_{\rm{h}}/\hbar$ || $\approx 2.1877 \times 10^{6}\ \rm{m\ s}^{-1}$ || 2
|-
| <span id="electric field strength"/>electric field strength || || $E_{\rm{h}}/ea_{0}$ || $\approx 5.1422 \times 10^{11}\ \rm{V\ m}^{-1}$ ||
|-
| <span id="electric dipole moment"/>electric dipole moment || || $ea_{0}$ || $\approx 8.4784 \times 10^{-30}\ \rm{C\ m}$ ||
|-
| <span id="electric quadrupole moment"/>electric quadrupole moment || || $ea_{0}^2$ || $\approx 4.4866 \times 10^{-40}\ \rm{C\ m}^2$ ||
|-
| <span id="mass"/>electric polarizability || || $ea_{0}^2/E_{\rm{h}}$ || $\approx 1.6488 \times 10^{-41}\ \rm{C}^2\ \rm{m}^2\ \rm{J}^{-1}$ ||
|-
| <span id="magnetic flux densit"/>magnetic flux density || || $\hbar/ea_{0}^2$ || $\approx 2.3505 \times 10^{5}\ \rm{T}$ ||
|-
| <span id="magnetic dipole moment"/>magnetic dipole moment || || $e\hbar/m_{\rm{e}}$ || $\approx 1.8548 \times 10^{-23}\ \rm{J\ T}^{-1}$ || 3
|-
|}
<hr/>
(1) <span id="2.4_Note_1"/>$\hbar = h/2\pi; a_{0} = 4\pi \varepsilon_{0} \hbar^2/m_{e}e^2; E_{\rm{h}} = \hbar^2/m_{e}a_{0}^2$.<br/>
(2) <span id="2.4_Note_2"/>The numerical value of the speed of light, when expressed in atomic units, is equal to the reciprocal of
the fine-structure constant $\alpha$; $c/(a_{0}E_{\rm{h}}/\hbar) = c\hbar/a_{0}E_{\rm{h}} = \alpha^{-1} = 137.035\;999\;084(21)$<br/>
(3) <span id="2.4_Note_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 <em>atomic units</em> 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 <em>physical</em> 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:<br/>
$r_{e} = 2.1\ a_{0}$ &nbsp;&nbsp;&nbsp; <em>not</em> &nbsp;&nbsp;&nbsp;$r_{e} = 2.1$ a.u.<br/>
$D_{e} = 0.16\ E_{\rm{h}}$&nbsp;&nbsp; <em>not</em> &nbsp;&nbsp;$D_{e} = 0.16$ a.u.