Hartree

The Hartree E_h, also known as the Hartree energy, is a physical constant, which is used in the Hartree atomic units system and named after the British physicist Douglas Hartree. It is defined as 2R_∞hc, where R_∞ is the Rydberg constant, h is the Planck constant and c is the speed of light. Its CODATA recommended value is E_h = 4.3597447222071(85)×10⁻¹⁸ J[1] = 27.211386245988(53) eV.[2]

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 like 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

${\displaystyle E_{\mathrm {h} }={\hbar ^{2} \over {m_{\mathrm {e} }a_{0}^{2}}}=m_{\mathrm {e} }\left({\frac {e^{2}}{4\pi \epsilon _{0}\hbar }}\right)^{2}=m_{\mathrm {e} }c^{2}\alpha ^{2}={\hbar c\alpha \over {a_{0}}}}$

= 2 Ry

≜ 27.211386245988(53) eV

≜ 4.3597447222071(85)×10⁻¹⁸ J

≜ 4.3597447222071(85)×10⁻¹¹ erg

≜ 2625.4996394799(50) kJ/mol

≜ 627.5094740631(12) kcal/mol

≜ 219474.63136320(43) cm⁻¹

≜ 6579.683920502(13) THz

≜ 315775.02480407(61) K

where:

ħ is the reduced Planck constant,

m_e 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 ${\displaystyle a_{0}}$ is defined as ${\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{m_{\mathrm {e} }e^{2}}}={\frac {\hbar }{m_{\mathrm {e} }c\alpha }}}$, one may write the Hartree energy as ${\displaystyle E_{\mathrm {h} }=e^{2}/a_{0}}$ in Gaussian units where ${\displaystyle 4\pi \varepsilon _{0}=1}$. Effective Hartree units are used in semiconductor physics where ${\displaystyle e^{2}}$ is replaced by ${\displaystyle e^{2}/\epsilon }$ and ${\displaystyle \epsilon }$ is the static dielectric constant. Also, the electron mass is replaced by the effective band mass ${\displaystyle m^{*}}$. The effective Hartree in semiconductors becomes small enough to be measured in millielectronvolts (meV).[3]

References

1. "2018 CODATA Value: Hartree energy". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. "2018 CODATA Value: Hartree energy in eV". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-09-01.
3. Tsuneya Ando, Alan B. Fowler, and Frank Stern Rev. Mod. Phys. 54, 437 (1982)