# Hartree

Jump to navigation Jump to search

The hartree (symbol: Eh or Ha), also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. It is defined as 2Rhc, where R is the Rydberg constant, h is the Planck constant and c is the speed of light. The 2014 CODATA recommended value is Eh = 4.359 744 650(54)×10−18 J = 27.211 386 02(17) eV.

The hartree energy is approximately the electric potential energy of the hydrogen atom in its ground state and, by the virial theorem, approximately twice its ionization energy; the relationships are not exact because of the finite mass of the nucleus of the hydrogen atom and relativistic corrections.

The hartree is usually used as a unit of energy in atomic physics and computational chemistry: for experimental measurements at the atomic scale, the electronvolt (eV) or the reciprocal centimetre (cm−1) are much more widely used.

## Other relationships

$E_{\mathrm {h} }={\hbar ^{2} \over {m_{\mathrm {e} }a_{0}^{2}}}=m_{\mathrm {e} }\left({\frac {e^{2}}{4\pi \epsilon _{0}\hbar }}\right)^{2}=m_{\mathrm {e} }c^{2}\alpha ^{2}={\hbar c\alpha \over {a_{0}}}$ = 2 Ry
27.21138602(17) eV
4.359744650(54)×10−18 J
4.359744650(54)×10−11 erg
2625.499638(65) kJ/mol
627.509474(15) kcal/mol
219474.6313702(13) cm−1
6579.683920711(39) THz
315775.13(18) K

where:

ħ is the reduced Planck constant,
me is the electron rest mass,
e is the elementary charge,
a0 is the Bohr radius,
ε0 is the electric constant,
c is the speed of light in vacuum, and
α is the fine structure constant.

Note that since the Bohr radius $a_{0}$ is defined as $a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{m_{\mathrm {e} }e^{2}}}={\frac {\hbar }{m_{\mathrm {e} }c\alpha }}$ , one may write the Hartree energy as $E_{\mathrm {h} }=e^{2}/a_{0}$ in Gaussian units where $4\pi \varepsilon _{0}=1$ . Effective Hartree units are used in semiconductor physics where $e^{2}$ is replaced by $e^{2}/\epsilon$ and $\epsilon$ is the static dielectric constant. Also, the electron mass is replaced by the effective band mass $m^{*}$ . The effective Hartree in semiconductors becomes small enough to be measured in millielectronvolts (meV).