Hartree energy

Physical constant
Surname	Hartree energy
Formula symbol	${\displaystyle E_{\mathrm {h} }}$
value
SI	4th.359 744 722 2071 (85)e-18 y
Uncertainty  (rel.)	1.9e-12
Relation to other constants
${\displaystyle E_{\mathrm {h} }={\frac {m_{\mathrm {e} }e^{4}}{4\varepsilon _{0}^{2}h^{2}}}}$ ${\displaystyle \varepsilon _{0}}$- Permittivity of the vacuum - Electron mass - Elementary charge - Planck's quantum of action ${\displaystyle m_{\mathrm {e} }}$ ${\displaystyle e}$ ${\displaystyle h}$
Sources and Notes
Source SI value: CODATA  2018 ( direct link )

The Hartree energy (after the English physicist Douglas Rayner Hartree ) is a physical constant that is used in atomic units as a unit of energy : ${\displaystyle E_{\mathrm {h} }}$

${\displaystyle {\begin{aligned}E_{\mathrm {h} }&={\frac {\hbar ^{2}}{m_{\mathrm {e} }{a_{0}}^{2}}}\\&={\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {e^{2}}{a_{0}}}\\&={\frac {m_{\mathrm {e} }\cdot e^{4}}{4\varepsilon _{0}^{2}\cdot h^{2}}}\\&=m_{\mathrm {e} }(c\alpha )^{2}\\&={\frac {\hbar c\alpha }{a_{0}}}\end{aligned}}}$

With

• ${\displaystyle 2\pi \cdot \hbar =h}$the Planck constant
• ${\displaystyle m_{\mathrm {e} }}$the mass of the electron
• ${\displaystyle a_{0}}$the Bohr radius
• ${\displaystyle e}$the elementary charge
• ${\displaystyle \varepsilon _{0}}$the electric field constant
• ${\displaystyle c}$the speed of light
• ${\displaystyle \alpha }$the fine structure constant .

The Hartree energy has twice the value of the unit Ry , which corresponds to the binding energy of the electron in the ground state of the hydrogen atom :

${\displaystyle E_{\mathrm {h} }=2\,\mathrm {Ry} =27{,}211\,386\,245\,988(53)\,\mathrm {eV} =4{,}359\,744\,722\,2071(85)\cdot 10^{-18}\,\mathrm {J,} }$

The numbers in brackets denote the uncertainty in the last digits of the value, this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.

Related Hartree energies:

• on the amount of substance : ${\displaystyle n}$

${\displaystyle E_{\mathrm {h} }/n=2{,}625\,499\,639\,479(5)\,\mathrm {MJ} /\mathrm {mol} =627{,}509\,474\,\mathrm {kcal} /\mathrm {mol} \ }$

• on (useful for wave numbers in spectroscopy): ${\displaystyle hc}$

${\displaystyle E_{\mathrm {h} }/(hc)=219\,474{,}631\,363\,\mathrm {cm} ^{-1}}$.

Hartree defined the energy unit, which was later named after him, in his book The calculation of atomic structures as "mutual potential energy of two charge units that are at a unit distance from each other". As a load unit it has previously the amount of the charge of the electron, and as a unit of distance of the radius of the "first electron orbital of the hydrogen atom in the normal state", the Bohr radius is defined.

See also

• Hartree-Fock method (many-body systems)

Individual evidence

1. ↑ CODATA Recommended Values. Hartree energy in eV E _h . National Institute of Standards and Technology, accessed July 20, 2019 . 
2. ↑ CODATA Recommended Values. Hartree energy E _h . National Institute of Standards and Technology, accessed July 20, 2019 . 
3. ↑ The thermochemical calorie 1 cal _th = 4.184 J was used to convert to kcal .
4. ^ Douglas Rayner Hartree: The calculation of atomic structures . Wiley, New York, NY 1957 (IX, 181 S., Chapter ATOMIC UNITS on p. 5: "Unit of energy (...) the mutual potential energy of two unit charges at unit distance appart.").