Hartree

For other articles and the surname, see Hartree (disambiguation).

The hartree (symbol: E_h), also known as the Hartree energy, is the unit of energy in the atomic units system, named after the British physicist Douglas Hartree. Its CODATA recommended value is E_h = 4.3597447222060(48)×10⁻¹⁸ J‍^[1] = 27.211386245981(30) eV.^[2] The name "hartree" was suggested for this unit of energy.^[3][4]

The hartree is approximately the negative electric potential energy of the electron in a 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

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

= 2 Ry = 2 R_∞hc

= 27.211386245981(30) eV‍^[2]

= 4.3597447222060(48)×10⁻¹⁸ J‍^[1]

= 4.3597447222060(48)×10⁻¹¹ erg

≘ 2625.4996394799(50) kJ/mol

≘ 627.5094740631(12) kcal/mol

≘ 219474.63136320(43) cm⁻¹

≘ 6579.683920502(13) THz

where:

• ħ is the reduced Planck constant,
• m_e is the electron 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.

Effective hartree units are used in semiconductor physics where ${\displaystyle e^{2}}$  is replaced by ${\displaystyle e^{2}/\varepsilon }$  and ${\displaystyle \varepsilon }$  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).^[5]

References

1. "2022 CODATA Value: Hartree energy". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
2. "2022 CODATA Value: Hartree energy in eV". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
3. Shull, H.; Hall, G.G. (1959). "Atomic Units". Nature. 184 (4698). Nature Publishing Group: 1559–1560. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0.
4. McWeeny, R. (May 1973). "Natural Units in Atomic and Molecular Physics". Nature. 243 (5404): 196–198. Bibcode:1973Natur.243..196M. doi:10.1038/243196a0. ISSN 0028-0836. S2CID 4164851.
5. Tsuneya Ando, Alan B. Fowler, and Frank Stern Rev. Mod. Phys. 54, 437 (1982)