Hartree

The hartree (symbol: E_h 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 2R_∞hc, where R_∞ is the Rydberg constant, h is the Planck constant and c is the speed of light. The 2014 CODATA recommended value is E_h = 4.359 744 650(54)×10⁻¹⁸ J = 27.211 386 02(17) eV.^[1]

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,

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 $a_{0}$ 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 $e^{2}/\epsilon$ where $\epsilon$ 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 mili-electron volts (meV) ^[2]

References

↑
↑ Tsuneya Ando, Alan B. Fowler, and Frank Stern Rev. Mod. Phys. 54, 437 (1982)

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[1] ↑

[2] Tsuneya Ando, Alan B. Fowler, and Frank Stern Rev. Mod. Phys. 54, 437 (1982)