Hartree atomic units

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The Hartree atomic units are a system of natural units which is especially convenient for atomic physics calculations. They are named after the physicist Douglas Hartree.[1] In this system the numerical values of the following four fundamental physical constants are all unity by definition:

• Electron mass ${\displaystyle m_{\text{e}}}$;
• Elementary charge ${\displaystyle e}$;
• Reduced Planck's constant ${\displaystyle \hbar =h/(2\pi )}$;
• Coulomb's constant ${\displaystyle k_{\text{e}}=1/(4\pi \epsilon _{0})}$.

In Hartree units, the speed of light is approximately ${\displaystyle 137}$. Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in different contexts.

Use and notation

Atomic units, like SI units, have a unit of mass, a unit of length, and so on. However, the use and notation is somewhat different from SI.

Suppose a particle with a mass of m has 3.4 times the mass of electron. The value of m can be written in three ways:

• "${\displaystyle m=3.4~m_{\text{e}}}$ ". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.[2]
• "${\displaystyle m=3.4~\mathrm {a.u.} }$ " ("a.u." means "expressed in atomic units"). This notation is ambiguous: Here, it means that the mass m is 3.4 times the atomic unit of mass. But if a length L were 3.4 times the atomic unit of length, the equation would look the same, "${\displaystyle L=3.4~{\text{a.u.}}}$ " The dimension needs to be inferred from context.[2]
• "${\displaystyle m=3.4}$ ". This notation is similar to the previous one, and has the same dimensional ambiguity. It comes from formally setting the atomic units to 1, in this case ${\displaystyle m_{\text{e}}=1}$ , so ${\displaystyle 3.4~m_{\text{e}}=3.4}$ .[3][4]

Fundamental atomic units

These four fundamental constants form the basis of the atomic units (see above). Therefore, their numerical values in the atomic units are unity by definition.

Fundamental atomic units
Dimension Name Symbol/Definition Value in SI units[5]
mass electron rest mass ${\displaystyle m_{\mathrm {e} }}$  9.1093837015(28)×10−31 kg
charge elementary charge ${\displaystyle e}$  1.602176634(0)×10−19 C
action reduced Planck's constant ${\displaystyle \hbar =h/(2\pi )}$  1.054571817(0)×10−34 J·s
electric constant−1 Coulomb force constant ${\displaystyle k_{\text{e}}=1/(4\pi \epsilon _{0})}$  8.987551792(1)×109 kg·m3·s−2·C−2

Related physical constants

Dimensionless physical constants retain their values in any system of units. Of particular importance is the fine-structure constant ${\displaystyle \alpha ={\frac {e^{2}}{(4\pi \epsilon _{0})\hbar c}}\approx 1/137}$ . This immediately gives the value of the speed of light, expressed in atomic units.

Some physical constants expressed in atomic units
Name Symbol/Definition Value in atomic units
speed of light ${\displaystyle c}$  ${\displaystyle 1/\alpha \approx 137}$
classical electron radius ${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$  ${\displaystyle \alpha ^{2}\approx 5.32\times 10^{-5}}$
proton mass ${\displaystyle m_{\mathrm {p} }}$  ${\displaystyle m_{\mathrm {p} }/m_{\mathrm {e} }\approx 1836}$

Derived atomic units

Below are given a few derived units. Some of them have proper names and symbols assigned, as indicated in the table. kB is the Boltzmann constant.

Derived atomic units
Dimension Name Symbol Expression Value in SI units Value in other units
length bohr ${\displaystyle a_{0}}$  ${\displaystyle 4\pi \epsilon _{0}\hbar ^{2}/(m_{\mathrm {e} }e^{2})=\hbar /(m_{\mathrm {e} }c\alpha )}$  5.29177210903(80)×10−11 m[6] 0.0529177210903(80) nm = 0.529177210903(80) Å
energy hartree ${\displaystyle E_{\mathrm {h} }}$  ${\displaystyle m_{\mathrm {e} }e^{4}/(4\pi \epsilon _{0}\hbar )^{2}=\alpha ^{2}m_{\mathrm {e} }c^{2}}$  4.3597447222071(85)×10−18 J 27.211386245988 eV = 627.509 kcal·mol−1
time ${\displaystyle \hbar /E_{\mathrm {h} }}$  2.4188843265857(47)×10−17 s
momentum ${\displaystyle \hbar /a_{0}}$  1.99285191410(30)×10−24 kg·m·s−1[7]
velocity ${\displaystyle a_{0}E_{\mathrm {h} }/\hbar =\alpha c}$  2.18769126364(33)×106 m·s−1
force ${\displaystyle E_{\mathrm {h} }/a_{0}}$  8.2387234983(12)×10−8 N 82.387 nN = 51.421 eV·Å−1
temperature ${\displaystyle E_{\mathrm {h} }/k_{\mathrm {B} }}$  3.1577464(55)×105 K
pressure ${\displaystyle E_{\mathrm {h} }/{a_{0}}^{3}}$  2.9421912(19)×1013 Pa
electric field ${\displaystyle E_{\mathrm {h} }/(ea_{0})}$  5.14220674763(78)×1011 V·m−1 5.14220674763(78) GV·cm−1 = 51.4220674763(78) V·Å−1
electric potential ${\displaystyle E_{\mathrm {h} }/e}$  2.7211386245988(53)×101 V
electric dipole moment ${\displaystyle ea_{0}}$  8.4783536255(13)×10−30 C·m 2.541746473 D
Magnetic field (SI) ${\displaystyle {\frac {\hbar }{ea_{0}^{2}}}}$  2.35051756758(71)×105 T 2.35051756758(71)×109 G
Magnetic dipole moment (SI) ${\displaystyle {\frac {e\hbar }{m_{e}}}}$  2 (Bohr magneton)
Magnetic field (cgs) ${\displaystyle {\frac {e}{a_{0}^{2}c}}}$  1.715255528(11)×103 T 1.715255528(11)×107 G
Magnetic dipole moment (cgs) ${\displaystyle {\frac {e\hbar }{m_{e}c}}}$  2 (Bohr magneton)

SI and Gaussian-CGS variants, and magnetism-related units

There are two common variants of atomic units, one where they are used in conjunction with SI units for electromagnetism, and one where they are used with Gaussian-CGS units.[8] Although most of the units listed above are the same either way (including the unit for electric field), the units related to magnetism are not. In the SI system, the atomic unit for magnetic field is

1 a.u. = ${\displaystyle {\frac {\hbar }{ea_{0}^{2}}}\approx }$  2.35×105 T = 2.35×109 G,

and in the Gaussian-cgs unit system, the atomic unit for magnetic field is

1 a.u. = ${\displaystyle {\frac {e}{a_{0}^{2}c}}\approx }$  1.72×103 T = 1.72×107 G.

(These differ by a factor of α.)

Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,[9]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}}}=1/2}$  a.u.

and in Gaussian-based atomic units,[10]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}c}}=\alpha /2\approx 3.6\times 10^{-3}}$  a.u.

Bohr model in atomic units

Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):

• Orbital velocity = 1
• Angular momentum = 1
• Orbital period = 2π
• Ionization energy = ​12
• Electric field (due to nucleus) = 1
• Electrical attractive force (due to nucleus) = 1

Non-relativistic quantum mechanics in atomic units

The Schrödinger equation for an electron in SI units is

${\displaystyle -{\frac {\hbar ^{2}}{2m_{e}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}$ .

The same equation in au is

${\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}$ .

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

${\displaystyle {\hat {H}}=-{{{\hbar ^{2}} \over {2m_{e}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r}}}$ ,

while atomic units transform the preceding equation into

${\displaystyle {\hat {H}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {r}}}$ .

Comparison with Planck units

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. It should be kept in mind that au were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for quantum gravity and early-universe cosmology. Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and, as a result, the speed of light in atomic units is a large value, ${\displaystyle 1/\alpha \approx 137}$ . The orbital velocity of an electron around a small atom is of the order of 1 in atomic units, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms much slower than the speed of light (around 2 orders of magnitude slower).

There are much larger discrepancies in some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the Planck mass, a mass so large that if a single particle had that much mass it might collapse into a black hole. Indeed, the Planck unit of mass is 22 orders of magnitude larger than the au unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.

Notes and references

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0.
1. ^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society. 24 (1). Cambridge University Press. pp. 89–110. Bibcode:1928PCPS...24...89H. doi:10.1017/S0305004100011919.
2. ^ a b Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5.
3. ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4.
4. ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2.
5. ^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 31 May 2019.
6. ^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 31 May 2019.
7. ^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 31 May 2019.
8. ^ "A note on Units" (PDF). Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes.
9. ^ Chis, Vasile. "Atomic Units; Molecular Hamiltonian; Born-Oppenheimer Approximation" (PDF). Molecular Structure and Properties Calculations. Babes-Bolyai University lecture notes.
10. ^ Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. p. 380. ISBN 978-0-19-850950-9.