Atomic units

# Atomic units

Discussion

Encyclopedia
Atomic units (au or a.u.) form a system
Systems of measurement
A system of measurement is a set of units which can be used to specify anything which can be measured and were historically important, regulated and defined because of trade and internal commerce...

of natural units
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, or the speed of light c is a natural unit of speed...

which is especially convenient for atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg
Rydberg constant
The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

atomic units
, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In atomic units, the numerical values of the following four fundamental physical constants are all unity by definition:NEWLINE
NEWLINE
• electron mass $\!m_\mathrm\left\{e\right\}$;
• NEWLINE
• elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$;
• NEWLINE
• reduced Planck's constant $\hbar = h/\left(2 \pi\right)$;
• NEWLINE
• Coulomb's constant $1/\left(4 \pi \epsilon_0\right)$.
NEWLINE Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

s, arbitrary unit
Arbitrary unit
In science and technology, an arbitrary unit or procedure defined unit is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined reference measurement...

s, 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 some particle has a mass m which is 3.4 times the mass of electron. Then, the value of m can be written in three ways:NEWLINE
NEWLINE
• "$m = 3.4~m_e$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.
• NEWLINE
• "$m = 3.4~\mathrm\left\{a.u.\right\}$" ("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, "$L = 3.4~\mathrm\left\{a.u.\right\}$" The dimension needs to be inferred from context.
• NEWLINE
• "$m = 3.4$". 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 $m_e = 1$, so $3.4~m_e = 3.4$.
NEWLINE

## 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. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Fundamental atomic units
DimensionNameSymbol/DefinitionValue in SI units
mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

electron rest mass
Electron rest mass
The electron rest mass is the mass of a stationary electron. It is one of the fundamental constants of physics, and is also very important in chemistry because of its relation to the Avogadro constant...

$\!m_\mathrm\left\{e\right\}$ {{val|9.1093826|(16)|e=-31|u=kg}}
charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$ {{val|1.60217653|(14)|e=-19|u=C}}
angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

reduced Planck's constant $\hbar = h/\left(2 \pi\right)$ {{val|1.05457168|(18)|e=-34|u=J·s}}
electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

Coulomb force constant $1/\left(4 \pi \epsilon_0\right)$ {{val|8.9875517873681|e=9|u=kg·m3·s-2·C-2}}
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## Related physical constants

Evidently, dimensionless physical constants
Dimensionless physical constant
In physics, a dimensionless physical constant is a universal physical constant that is dimensionless - having no unit attached, so its numerical value is the same under all possible systems of units...

retain their values in any system of units. Of particular importance is the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

$\alpha = \frac\left\{e^2\right\}\left\{\left(4 \pi \epsilon_0\right)\hbar c\right\} \approx 1/137$. This immediately gives the value of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, expressed in atomic units. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Some physical constants expressed in atomic units
NameSymbol/DefinitionValue in atomic units
speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

$\!c$ $\!1/\alpha \approx 137$
The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...

$r_\mathrm\left\{e\right\}=\frac\left\{1\right\}\left\{4\pi\epsilon_0\right\}\frac\left\{e^2\right\}\left\{m_\mathrm\left\{e\right\} c^2\right\}$ $\!\alpha^2 \approx 5.32\times10^\left\{-5\right\}$
proton mass  $m_\mathrm\left\{p\right\}$ $m_\mathrm\left\{p\right\}/m_\mathrm\left\{e\right\} \approx 1836$
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## 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 Boltzmann constant. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Derived atomic units
DimensionNameSymbolExpressionValue in SI unitsValue in more common units
length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and 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...

$\!a_0$ $4\pi \epsilon_0 \hbar^2 / \left(m_\mathrm\left\{e\right\} e^2\right) = \hbar / \left(m_\mathrm\left\{e\right\} c \alpha\right)$ {{val|5.2917720859|(36)|e=-11|u=m}} {{val|0.052918|u=nm}}={{val|0.52918|u=Å}}
energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

Hartree energy
Hartree energy
The hartree , also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. It is defined as...

$\!E_\mathrm\left\{h\right\}$ $m_\mathrm\left\{e\right\} e^4/\left(4\pi\epsilon_0\hbar\right)^2 = \alpha^2 m_\mathrm\left\{e\right\} c^2$ {{val|4.35974417|(75)|e=-18|u=J}} {{val|27.211|u=eV}}={{val|627.509|u=kcal·mol−1}}
time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

$\hbar / E_\mathrm\left\{h\right\}$ {{val|2.418884326505|(16)|e=-17|u=s}}
velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

$a_0 E_\mathrm\left\{h\right\} / \hbar = \alpha c$ {{val|2.1876912633|(73)|e=6|u=m·s−1}}
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

$\! E_\mathrm\left\{h\right\} / a_0$ {{val|8.2387225|(14)|e=-8|u=N}}{{val|82.387|u=nN}}={{val|51.421|u=eV·Å−1}}
temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

$\! E_\mathrm\left\{h\right\} / k_\mathrm\left\{B\right\}$ {{val|3.1577464|(55)|e=5|u=K}}
pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

$E_\mathrm\left\{h\right\} / \left\{a_0\right\}^3$ {{val|2.9421912|(19)|e=13|u=Pa}}
electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

$\!E_\mathrm\left\{h\right\} / \left(ea_0\right)$ {{val|5.1421|e=11|u=V·m−1}} {{val|5.1421|u=GV·cm−1}}={{val|51.421|u=V·Å−1}}
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## 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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, and one where they are used with Gaussian-CGS units
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...

. Although the units written 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 isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{\hbar\right\}\left\{e a_0^2\right\}$ = {{val|2.35|e=5}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|2.35|e=9}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

,
NEWLINE and in the Gaussian-cgs unit system, the atomic unit for magnetic field isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{e\right\}\left\{a_0^2\right\}$ = {{val|1.72|e=3}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|1.72|e=7}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

.
NEWLINE (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,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e\right\} = 1/2$ a.u.
NEWLINE and in Gaussian-based atomic units,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e c\right\}=\alpha/2\approx 3.6\times 10^\left\{-3\right\}$ a.u.
NEWLINE

## 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
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

in its ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):NEWLINE
NEWLINE
• Orbital velocity = 1
• NEWLINE
• NEWLINE
• Angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

= 1
• NEWLINE
• Orbital period = 2π
• NEWLINE
• Ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...

= {{frac|1|2}}
• NEWLINE
• Electric field (due to nucleus) = 1
• NEWLINE
• Electrical attractive force (due to nucleus) = 1
NEWLINE

## Non-relativistic quantum mechanics in atomic units

The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for an electron in SI units isNEWLINE
NEWLINE
$- \frac\left\{\hbar^2\right\}\left\{2m_e\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \hbar \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE The same equation in au isNEWLINE
NEWLINE
$- \frac\left\{1\right\}\left\{2\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE For the special case of the electron around a hydrogen atom, the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

in SI units is:NEWLINE
NEWLINE
$\hat H = -$
NEWLINE Atomic units (au or a.u.) form a system
Systems of measurement
A system of measurement is a set of units which can be used to specify anything which can be measured and were historically important, regulated and defined because of trade and internal commerce...

of natural units
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, or the speed of light c is a natural unit of speed...

which is especially convenient for atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg
Rydberg constant
The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

atomic units
, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In atomic units, the numerical values of the following four fundamental physical constants are all unity by definition:NEWLINE
NEWLINE
• electron mass $\!m_\mathrm\left\{e\right\}$;
• NEWLINE
• elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$;
• NEWLINE
• reduced Planck's constant $\hbar = h/\left(2 \pi\right)$;
• NEWLINE
• Coulomb's constant $1/\left(4 \pi \epsilon_0\right)$.
NEWLINE Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

s, arbitrary unit
Arbitrary unit
In science and technology, an arbitrary unit or procedure defined unit is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined reference measurement...

s, 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 some particle has a mass m which is 3.4 times the mass of electron. Then, the value of m can be written in three ways:NEWLINE
NEWLINE
• "$m = 3.4~m_e$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.
• NEWLINE
• "$m = 3.4~\mathrm\left\{a.u.\right\}$" ("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, "$L = 3.4~\mathrm\left\{a.u.\right\}$" The dimension needs to be inferred from context.
• NEWLINE
• "$m = 3.4$". 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 $m_e = 1$, so $3.4~m_e = 3.4$.
NEWLINE

## 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. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Fundamental atomic units
DimensionNameSymbol/DefinitionValue in SI units
mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

electron rest mass
Electron rest mass
The electron rest mass is the mass of a stationary electron. It is one of the fundamental constants of physics, and is also very important in chemistry because of its relation to the Avogadro constant...

$\!m_\mathrm\left\{e\right\}$ {{val|9.1093826|(16)|e=-31|u=kg}}
charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$ {{val|1.60217653|(14)|e=-19|u=C}}
angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

reduced Planck's constant $\hbar = h/\left(2 \pi\right)$ {{val|1.05457168|(18)|e=-34|u=J·s}}
electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

Coulomb force constant $1/\left(4 \pi \epsilon_0\right)$ {{val|8.9875517873681|e=9|u=kg·m3·s-2·C-2}}
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## Related physical constants

Evidently, dimensionless physical constants
Dimensionless physical constant
In physics, a dimensionless physical constant is a universal physical constant that is dimensionless - having no unit attached, so its numerical value is the same under all possible systems of units...

retain their values in any system of units. Of particular importance is the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

$\alpha = \frac\left\{e^2\right\}\left\{\left(4 \pi \epsilon_0\right)\hbar c\right\} \approx 1/137$. This immediately gives the value of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, expressed in atomic units. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Some physical constants expressed in atomic units
NameSymbol/DefinitionValue in atomic units
speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

$\!c$ $\!1/\alpha \approx 137$
The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...

$r_\mathrm\left\{e\right\}=\frac\left\{1\right\}\left\{4\pi\epsilon_0\right\}\frac\left\{e^2\right\}\left\{m_\mathrm\left\{e\right\} c^2\right\}$ $\!\alpha^2 \approx 5.32\times10^\left\{-5\right\}$
proton mass  $m_\mathrm\left\{p\right\}$ $m_\mathrm\left\{p\right\}/m_\mathrm\left\{e\right\} \approx 1836$
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## 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 Boltzmann constant. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Derived atomic units
DimensionNameSymbolExpressionValue in SI unitsValue in more common units
length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and 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...

$\!a_0$ $4\pi \epsilon_0 \hbar^2 / \left(m_\mathrm\left\{e\right\} e^2\right) = \hbar / \left(m_\mathrm\left\{e\right\} c \alpha\right)$ {{val|5.2917720859|(36)|e=-11|u=m}} {{val|0.052918|u=nm}}={{val|0.52918|u=Å}}
energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

Hartree energy
Hartree energy
The hartree , also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. It is defined as...

$\!E_\mathrm\left\{h\right\}$ $m_\mathrm\left\{e\right\} e^4/\left(4\pi\epsilon_0\hbar\right)^2 = \alpha^2 m_\mathrm\left\{e\right\} c^2$ {{val|4.35974417|(75)|e=-18|u=J}} {{val|27.211|u=eV}}={{val|627.509|u=kcal·mol−1}}
time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

$\hbar / E_\mathrm\left\{h\right\}$ {{val|2.418884326505|(16)|e=-17|u=s}}
velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

$a_0 E_\mathrm\left\{h\right\} / \hbar = \alpha c$ {{val|2.1876912633|(73)|e=6|u=m·s−1}}
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

$\! E_\mathrm\left\{h\right\} / a_0$ {{val|8.2387225|(14)|e=-8|u=N}}{{val|82.387|u=nN}}={{val|51.421|u=eV·Å−1}}
temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

$\! E_\mathrm\left\{h\right\} / k_\mathrm\left\{B\right\}$ {{val|3.1577464|(55)|e=5|u=K}}
pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

$E_\mathrm\left\{h\right\} / \left\{a_0\right\}^3$ {{val|2.9421912|(19)|e=13|u=Pa}}
electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

$\!E_\mathrm\left\{h\right\} / \left(ea_0\right)$ {{val|5.1421|e=11|u=V·m−1}} {{val|5.1421|u=GV·cm−1}}={{val|51.421|u=V·Å−1}}
NEWLINENEWLINE

## 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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, and one where they are used with Gaussian-CGS units
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...

. Although the units written 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 isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{\hbar\right\}\left\{e a_0^2\right\}$ = {{val|2.35|e=5}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|2.35|e=9}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

,
NEWLINE and in the Gaussian-cgs unit system, the atomic unit for magnetic field isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{e\right\}\left\{a_0^2\right\}$ = {{val|1.72|e=3}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|1.72|e=7}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

.
NEWLINE (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,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e\right\} = 1/2$ a.u.
NEWLINE and in Gaussian-based atomic units,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e c\right\}=\alpha/2\approx 3.6\times 10^\left\{-3\right\}$ a.u.
NEWLINE

## 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
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

in its ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):NEWLINE
NEWLINE
• Orbital velocity = 1
• NEWLINE
• NEWLINE
• Angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

= 1
• NEWLINE
• Orbital period = 2π
• NEWLINE
• Ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...

= {{frac|1|2}}
• NEWLINE
• Electric field (due to nucleus) = 1
• NEWLINE
• Electrical attractive force (due to nucleus) = 1
NEWLINE

## Non-relativistic quantum mechanics in atomic units

The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for an electron in SI units isNEWLINE
NEWLINE
$- \frac\left\{\hbar^2\right\}\left\{2m_e\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \hbar \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE The same equation in au isNEWLINE
NEWLINE
$- \frac\left\{1\right\}\left\{2\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE For the special case of the electron around a hydrogen atom, the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

in SI units is:NEWLINE
NEWLINE
$\hat H = -$
NEWLINE Atomic units (au or a.u.) form a system
Systems of measurement
A system of measurement is a set of units which can be used to specify anything which can be measured and were historically important, regulated and defined because of trade and internal commerce...

of natural units
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, or the speed of light c is a natural unit of speed...

which is especially convenient for atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg
Rydberg constant
The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

atomic units
, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In atomic units, the numerical values of the following four fundamental physical constants are all unity by definition:NEWLINE
NEWLINE
• electron mass $\!m_\mathrm\left\{e\right\}$;
• NEWLINE
• elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$;
• NEWLINE
• reduced Planck's constant $\hbar = h/\left(2 \pi\right)$;
• NEWLINE
• Coulomb's constant $1/\left(4 \pi \epsilon_0\right)$.
NEWLINE Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

s, arbitrary unit
Arbitrary unit
In science and technology, an arbitrary unit or procedure defined unit is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined reference measurement...

s, 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 some particle has a mass m which is 3.4 times the mass of electron. Then, the value of m can be written in three ways:NEWLINE
NEWLINE
• "$m = 3.4~m_e$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.
• NEWLINE
• "$m = 3.4~\mathrm\left\{a.u.\right\}$" ("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, "$L = 3.4~\mathrm\left\{a.u.\right\}$" The dimension needs to be inferred from context.
• NEWLINE
• "$m = 3.4$". 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 $m_e = 1$, so $3.4~m_e = 3.4$.
NEWLINE

## 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. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Fundamental atomic units
DimensionNameSymbol/DefinitionValue in SI units
mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

electron rest mass
Electron rest mass
The electron rest mass is the mass of a stationary electron. It is one of the fundamental constants of physics, and is also very important in chemistry because of its relation to the Avogadro constant...

$\!m_\mathrm\left\{e\right\}$ {{val|9.1093826|(16)|e=-31|u=kg}}
charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

$\!e$ {{val|1.60217653|(14)|e=-19|u=C}}
angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

reduced Planck's constant $\hbar = h/\left(2 \pi\right)$ {{val|1.05457168|(18)|e=-34|u=J·s}}
electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

Coulomb force constant $1/\left(4 \pi \epsilon_0\right)$ {{val|8.9875517873681|e=9|u=kg·m3·s-2·C-2}}
NEWLINENEWLINE

## Related physical constants

Evidently, dimensionless physical constants
Dimensionless physical constant
In physics, a dimensionless physical constant is a universal physical constant that is dimensionless - having no unit attached, so its numerical value is the same under all possible systems of units...

retain their values in any system of units. Of particular importance is the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

$\alpha = \frac\left\{e^2\right\}\left\{\left(4 \pi \epsilon_0\right)\hbar c\right\} \approx 1/137$. This immediately gives the value of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, expressed in atomic units. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Some physical constants expressed in atomic units
NameSymbol/DefinitionValue in atomic units
speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

$\!c$ $\!1/\alpha \approx 137$
The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...

$r_\mathrm\left\{e\right\}=\frac\left\{1\right\}\left\{4\pi\epsilon_0\right\}\frac\left\{e^2\right\}\left\{m_\mathrm\left\{e\right\} c^2\right\}$ $\!\alpha^2 \approx 5.32\times10^\left\{-5\right\}$
proton mass  $m_\mathrm\left\{p\right\}$ $m_\mathrm\left\{p\right\}/m_\mathrm\left\{e\right\} \approx 1836$
NEWLINENEWLINE

## 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 Boltzmann constant. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
Derived atomic units
DimensionNameSymbolExpressionValue in SI unitsValue in more common units
length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and 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...

$\!a_0$ $4\pi \epsilon_0 \hbar^2 / \left(m_\mathrm\left\{e\right\} e^2\right) = \hbar / \left(m_\mathrm\left\{e\right\} c \alpha\right)$ {{val|5.2917720859|(36)|e=-11|u=m}} {{val|0.052918|u=nm}}={{val|0.52918|u=Å}}
energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

Hartree energy
Hartree energy
The hartree , also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. It is defined as...

$\!E_\mathrm\left\{h\right\}$ $m_\mathrm\left\{e\right\} e^4/\left(4\pi\epsilon_0\hbar\right)^2 = \alpha^2 m_\mathrm\left\{e\right\} c^2$ {{val|4.35974417|(75)|e=-18|u=J}} {{val|27.211|u=eV}}={{val|627.509|u=kcal·mol−1}}
time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

$\hbar / E_\mathrm\left\{h\right\}$ {{val|2.418884326505|(16)|e=-17|u=s}}
velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

$a_0 E_\mathrm\left\{h\right\} / \hbar = \alpha c$ {{val|2.1876912633|(73)|e=6|u=m·s−1}}
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

$\! E_\mathrm\left\{h\right\} / a_0$ {{val|8.2387225|(14)|e=-8|u=N}}{{val|82.387|u=nN}}={{val|51.421|u=eV·Å−1}}
temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

$\! E_\mathrm\left\{h\right\} / k_\mathrm\left\{B\right\}$ {{val|3.1577464|(55)|e=5|u=K}}
pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

$E_\mathrm\left\{h\right\} / \left\{a_0\right\}^3$ {{val|2.9421912|(19)|e=13|u=Pa}}
electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

$\!E_\mathrm\left\{h\right\} / \left(ea_0\right)$ {{val|5.1421|e=11|u=V·m−1}} {{val|5.1421|u=GV·cm−1}}={{val|51.421|u=V·Å−1}}
NEWLINENEWLINE

## 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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, and one where they are used with Gaussian-CGS units
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...

. Although the units written 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 isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{\hbar\right\}\left\{e a_0^2\right\}$ = {{val|2.35|e=5}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|2.35|e=9}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

,
NEWLINE and in the Gaussian-cgs unit system, the atomic unit for magnetic field isNEWLINE
NEWLINE
1 a.u. = $\frac\left\{e\right\}\left\{a_0^2\right\}$ = {{val|1.72|e=3}} T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= {{val|1.72|e=7}} G
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

.
NEWLINE (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,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e\right\} = 1/2$ a.u.
NEWLINE and in Gaussian-based atomic units,NEWLINE
NEWLINE
$\mu_B = \frac\left\{e \hbar\right\}\left\{2 m_e c\right\}=\alpha/2\approx 3.6\times 10^\left\{-3\right\}$ a.u.
NEWLINE

## 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
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

in its ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):NEWLINE
NEWLINE
• Orbital velocity = 1
• NEWLINE
• NEWLINE
• Angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

= 1
• NEWLINE
• Orbital period = 2π
• NEWLINE
• Ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...

= {{frac|1|2}}
• NEWLINE
• Electric field (due to nucleus) = 1
• NEWLINE
• Electrical attractive force (due to nucleus) = 1
NEWLINE

## Non-relativistic quantum mechanics in atomic units

The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for an electron in SI units isNEWLINE
NEWLINE
$- \frac\left\{\hbar^2\right\}\left\{2m_e\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \hbar \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE The same equation in au isNEWLINE
NEWLINE
$- \frac\left\{1\right\}\left\{2\right\} \nabla^2 \psi\left(\mathbf\left\{r\right\}, t\right) + V\left(\mathbf\left\{r\right\}\right) \psi\left(\mathbf\left\{r\right\}, t\right) = i \frac\left\{\partial \psi\right\}\left\{\partial t\right\} \left(\mathbf\left\{r\right\}, t\right)$.
NEWLINE For the special case of the electron around a hydrogen atom, the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

in SI units is:NEWLINE
NEWLINE
$\hat H = - \left\{\left\{\left\{\hbar^2\right\} \over \left\{2 m_e\right\}\right\}\nabla^2\right\} - \left\{1 \over \left\{4 \pi \epsilon_0\right\}\right\}\left\{\left\{e^2\right\} \over \left\{r\right\}\right\}$,
NEWLINE while atomic units transform the preceding equation intoNEWLINE
NEWLINE
$\hat H = - \left\{\left\{\left\{1\right\} \over \left\{2\right\}\right\}\nabla^2\right\} - \left\{\left\{1\right\} \over \left\{r\right\}\right\}$.
NEWLINE

## Comparison with Planck units

Both Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...

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
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

and early-universe cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

. Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

and cosmology: the gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

G and the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

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, $1/\alpha \approx 137$. 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
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

. 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.

NEWLINE
NEWLINE
• Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...

• NEWLINE
• Natural units
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, or the speed of light c is a natural unit of speed...

• NEWLINE
• Various extensions of the CGS system to electromagnetism.
NEWLINE