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:

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

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.").