Atomic units

Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, Hartree atomic units[1] and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units, where the numerical values of the following four fundamental physical constants are all unity by definition:

In Hartree units, the speed of light is approximately . 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:

  • "". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.[2]
  • "" ("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, "" The dimension needs to be inferred from context.[2]
  • "". 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 , so .[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 9.10938291(40)×10−31 kg
charge elementary charge 1.602176565(35)×10−19 C
action reduced Planck's constant 1.054571726(47)×10−34 J·s
electric constant−1 Coulomb force constant 8.9875517873681×109 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 . 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
classical electron radius
proton mass

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 5.2917721092(17)×10−11 m[6] 0.052917721092(17) nm = 0.52917721092(17) Å
energy hartree 4.35974417(75)×10−18 J 27.211385 eV = 627.509 kcal·mol−1
time 2.418884326505(16)×10−17 s
momentum 1.992851882(24)×10−24 kg·m·s−1[7]
velocity 2.1876912633(73)×106 m·s−1
force 8.2387225(14)×10−8 N 82.387 nN = 51.421 eV·Å−1
temperature 3.1577464(55)×105 K
pressure 2.9421912(19)×1013 Pa
electric field 5.14220652(11)×1011 V·m−1 5.14220652(11) GV·cm−1 = 51.4220652(11) V·Å−1
electric potential 2.721138505(60)×101 V
electric dipole moment 8.47835326(19)×10−30 C·m 2.541746 D
Magnetic field (SI) 2.350517550(14)×105 T 2.350517550(14)×109 G
Magnetic dipole moment (SI) 2 (Bohr magneton)
Magnetic field (cgs) 1.715255528(11)×103 T 1.715255528(11)×107 G
Magnetic dipole moment (cgs) 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. = 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. = 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]


and in Gaussian-based atomic units,[10]


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
  • Orbital radius = 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


The same equation in au is


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


while atomic units transform the preceding equation into


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, . 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.

See also

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 1 April 2012.
  6. ^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 21 January 2014.
  7. ^ "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 29 October 2017.
  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.

External links

Atomic mass unit

The unified atomic mass unit or dalton (SI symbols: u, or Da; Deprecated/colloquial symbol: amu) is a standard unit of mass that quantifies mass on an atomic or molecular scale (atomic mass). One unified atomic mass unit is approximately the mass of one nucleon (either a single proton or neutron) and is effectively numerically equivalent to 1 g/mol. It is defined as one twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest, and has a value approaching 1.66053906660(50)×10−27 kg, or approximately 1.66 yoctograms. The CIPM has categorised it as a non-SI unit accepted for use with the SI, and whose value in SI units must be obtained experimentally.The atomic mass unit (amu) without the "unified" prefix is technically an obsolete unit based on oxygen, which was replaced in 1961. However, some nontechnical and preparatory sources continue to occasionally use the term amu but now define it in the same way as u (i.e., based on carbon-12). In this sense, most uses of the terms atomic mass units and amu, today, actually refer to unified atomic mass unit. For standardization, a specific atomic nucleus (carbon-12 vs. oxygen-16) had to be chosen because the average mass of a nucleon depends on the count of the nucleons in the atomic nucleus due to mass defect. This is also why the mass of a proton or neutron by itself is more than (and not equal to) 1 u.

The atomic mass unit is not the unit of mass in the atomic units system, which is rather the electron rest mass (me).

Prior to the 2019 redefinition of SI base units, the number of daltons in a gram was exactly the Avogadro number by definition, or equivalently, a dalton was exactly equivalent to 1 gram/mol. Currently, these relationships are no longer exact, but are nonetheless extremely accurate approximations.

Bohr magneton

In atomic physics, the Bohr magneton (symbol μB) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum.

The Bohr magneton is defined in SI units by

and in Gaussian CGS units by


e is the elementary charge,
ħ is the reduced Planck constant,
me is the electron rest mass and
c is the speed of light.

The electron magnetic moment, which is the electron's intrinsic spin magnetic moment, is approximately one Bohr magneton.

Bohr radius

The Bohr radius (a0 or rBohr) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. Its value is 5.2917721067(12)×10−11 m.

Bond length

In molecular geometry, bond length or bond distance is the average distance between nuclei of two bonded atoms in a molecule. It is a transferable property of a bond between atoms of fixed types, relatively independent of the rest of the molecule.

Completely Fair Scheduler

The Completely Fair Scheduler (CFS) is a process scheduler which was merged into the 2.6.23 (October 2007) release of the Linux kernel and is the default scheduler. It handles CPU resource allocation for executing processes, and aims to maximize overall CPU utilization while also maximizing interactive performance.

Con Kolivas's work with scheduling, most significantly his implementation of "fair scheduling" named Rotating Staircase Deadline, inspired Ingo Molnár to develop his CFS, as a replacement for the earlier O(1) scheduler, crediting Kolivas in his announcement.In contrast to the previous O(1) scheduler used in older Linux 2.6 kernels, the CFS scheduler implementation is not based on run queues. Instead, a red–black tree implements a "timeline" of future task execution. Additionally, the scheduler uses nanosecond granularity accounting, the atomic units by which an individual process' share of the CPU was allocated (thus making redundant the previous notion of timeslices). This precise knowledge also means that no specific heuristics are required to determine the interactivity of a process, for example.Like the old O(1) scheduler, CFS uses a concept called "sleeper fairness", which considers sleeping or waiting tasks equivalent to those on the runqueue. This means that interactive tasks which spend most of their time waiting for user input or other events get a comparable share of CPU time when they need it.


Core-and-pod design is a network design that uses individual pods that hang off the core layer as atomic units. Within the pod, there may be only a single access layer or a “leaf and spine” network in the pod. The routed core layer serves as a fast and simple way to connect many generations of pods to each other. When the “leaf and spine” network is used within the pod, the core layer can be referred to as the “spine of spines,” since it is the thing that connects the “spines” of the pods. This design then resembles some kind of large and wide tree, with many “branches,” or pods, off the main “trunk,” or core.

This new design differs from the original “three-tier” architecture through the fact that pods can be bundled as a unit. The units can be manipulated as however the staff of the network pleases. The new design also requires the staff of the network to understand the many designs and tools that can be used to manage the different pod iterations.


The coulomb (symbol: C) is the International System of Units (SI) unit of electric charge. It is the charge (symbol: Q or q) transported by a constant current of one ampere in one second:

Thus, it is also the amount of excess charge on a capacitor of one farad charged to a potential difference of one volt:

The coulomb is equivalent to the charge of approximately 6.242×1018 (1.036×10−5 mol) protons, and −1 C is equivalent to the charge of approximately 6.242×1018 electrons.

A new definition, in terms of the elementary charge, took effect on 20 May 2019. The new definition defines the elementary charge (the charge of the proton) as exactly 1.602176634×10−19 coulombs.

Elementary charge

The elementary charge, usually denoted by e or sometimes qe, is the electric charge carried by a single proton or, equivalently, the magnitude of the electric charge carried by a single electron, which has charge −1 e. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the elementary positive charge.

This charge has a measured value of approximately 1.602176634×10−19 C (coulombs). When the 2019 redefinition of SI base units takes effect on 20 May 2019, its value will be exactly 1.602176634×10−19 C by definition of the coulomb. In the centimetre–gram–second system of units (CGS), it is 4.80320425(10)×10−10 statcoulombs.Robert A. Millikan's oil drop experiment first measured the magnitude of the elementary charge in 1909.

Hartree equation

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the self-consistent field method.

In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radial function and a spherical harmonic with an angular quantum number , namely . The equation for the radial function was

In mathematics, the Hartree equation, named after Douglas Hartree, is

in where


The non-linear Schrödinger equation is in some sense a limiting case.

Holstein–Herring method

The Holstein–Herring method, also called the surface Integral method, also called Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei (Pauli exclusion principle).

Natural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly 1. These constants are then typically omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for dimensional analysis. It precludes the interpretation of an expression in terms of fundamental physical constants, such as e and c, unless it is known which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of e, c, etc., can be uniquely determined.

Phase-space formulation

The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.

The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner.The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit). Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory (see Kontsevich quantization formula) and noncommutative geometry.

Planck constant

The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum. He assumed that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. He was able to calculate the proportionality constant, h, from the experimental measurements, and that constant is named in his honor. In 1905, the value E was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to the project of replacing the International Prototype of the Kilogram, a metal cylinder that had defined the kilogram since 1889. The new definition was unanimously approved at the General Conference on Weights and Measures (CGPM) on 16 November 2018 as part of the 2019 redefinition of SI base units. For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.62607015×10−34 J⋅s exactly. The kilogram was the last SI base unit to be re-defined by a fundamental physical property to replace a physical artifact.

Ponderomotive energy

In strong-field laser physics, ponderomotive energy is the cycle-averaged quiver energy of a free electron in an electromagnetic field.

Quantities, Units and Symbols in Physical Chemistry

Quantities, Units and Symbols in Physical Chemistry, also known as the Green Book, is a compilation of terms and symbols widely used in the field of physical chemistry. It also includes a table of physical constants, tables listing the properties of elementary particles, chemical elements, and nuclides, and information about conversion factors that are commonly used in physical chemistry. The Green Book is published by the International Union of Pure and Applied Chemistry (IUPAC) and is based on published, citeable sources. Information in the Green Book is synthesized from recommendations made by IUPAC, the International Union of Pure and Applied Physics (IUPAP) and the International Organization for Standardization (ISO), including recommendations listed in the IUPAP Red Book Symbols, Units, Nomenclature and Fundamental Constants in Physics and in the ISO 31 standards.


Ryd may refer to:

Ryd, Tingsryd Municipality, a village in Tingsryd Municipality in the south of Sweden

Ryd, Linköping, a residential area in Linköping, Sweden

Ryd, a suburb of the city of Skövde, Sweden

Ryd, an abbreviation used when stating physical quantities in Rydberg atomic units, a type of atomic units

System of measurement

A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement in use include the International System of Units (SI), the modern form of the metric system, the imperial system, and United States customary units.

Thomas–Fermi screening

Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid. It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit. It is named after Llewellyn Thomas and Enrico Fermi.

The Thomas-Fermi wavevector (in Gaussian-cgs units) is


where μ is the chemical potential (fermi level), n is the electron concentration and e is the elementary charge.

Under many circumstances, including semiconductors that are not too heavily doped, neμ/kBT, where kB is Boltzmann constant and T is temperature. In this case,


i.e. 1/k0 is given by the familiar formula for Debye length. In the opposite extreme, in the low-temperature limit T=0, electrons behave as quantum particles (Fermions). Such an approximation is valid for metals at room temperature, and the Thomas-Fermi screening wavevector kTF given in atomic units is


If we restore the electron mass and the Planck constant , the screening wavevector in Gaussian units is .

For more details and discussion, including the one-dimensional and two-dimensional cases, see the article: Lindhard theory.

Base units
Derived units
with special names
Other accepted units
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