Hartree atomic units
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. In this system the numerical values of the following four fundamental physical constants are all unity by definition:
- Reduced Planck constant,, also known as the atomic unit of action
- Elementary charge,, also known as the atomic unit of charge
- Bohr radius,, also known as the atomic unit of length
- Electron mass,, also known as the atomic unit of mass
Defining constants
Each unit in this system can be expressed as a product of powers of four physical constants without a multiplying constant. This makes it a coherent system of units, as well as making the numerical values of the defining constants in atomic units equal to unity.| Name | Symbol | Value in SI units |
| reduced Planck constant | ||
| elementary charge | ||
| Bohr radius | ||
| electron rest mass |
Five symbols are commonly used as units in this system, only four of them being independent:
| Dimension | Symbol | Definition |
| action | ||
| electric charge | ||
| length | ||
| mass | ||
| energy |
Units
Below are listed units that can be derived in the system. A few are given names, as indicated in the table.| Atomic unit of | Name | Expression | Value in SI units | Other equivalents |
| 1st hyperpolarizability | ||||
| 2nd hyperpolarizability | ||||
| action | ||||
| charge | ||||
| charge density | ||||
| current | ||||
| electric dipole moment | ||||
| electric field | , | |||
| electric field gradient | ||||
| electric polarizability | ||||
| electric potential | ||||
| electric quadrupole moment | ||||
| energy | hartree | ,, | ||
| force | , | |||
| length | bohr | , | ||
| magnetic dipole moment | ||||
| magnetic flux density | ||||
| magnetizability | ||||
| mass | ||||
| momentum | ||||
| permittivity | ||||
| pressure | ||||
| time | ||||
| velocity |
Here,
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, where the atomic unit is included explicitly as a symbol.
- "". 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 must be inferred from context.
- "". 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.
Physical constants
| Name | Symbol/Definition | Value in atomic units |
| speed of light | ||
| classical electron radius | ||
| reduced Compton wavelength of the electron | ||
| Bohr radius | ||
| proton mass |
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 :- Mass = 1 a.u. of mass
- Orbital radius = 1 a.u. of length
- Orbital velocity = 1 a.u. of velocity
- Orbital period = 2π a.u. of time
- Orbital angular velocity = 1 radian per a.u. of time
- Orbital angular momentum = 1 a.u. of momentum
- Ionization energy = a.u. of energy
- Electric field = 1 a.u. of electric field
- Electrical attractive force = 1 a.u. of force
Non-relativistic quantum mechanics in atomic units
The same equation in atomic units 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 atomic units are derived from certain fundamental properties of the physical world, and have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Atomic units 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 atomic units 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 and the speed of light in vacuum,. 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 by around 2 orders of magnitude more slowly than the speed of light.There are much larger differences for 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. The Planck unit of mass is 22 orders of magnitude larger than the atomic unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.