Atomic units

Use of atomic units has been motivated on the grounds of accuracy and stability of reported values: since the values of the accepted values of the fundamental constants in atomic physics such as , {{tmath| m_\text{e} }}, and were not sufficiently stable or accurate, the values of calculations and measurements performed in different years could not be directly compared, which resulted in confusion. This led to suggestions that the results of quantum-mechanical calculations should be reported using units based directly on such constants. In the context of atomic physics, using the atomic units system can be a convenient shortcut, eliminating symbols and numbers and reducing the order of magnitude of most numbers involved. For example, the Hamiltonian operator in the Schrödinger equation for the helium atom with standard quantities, such as when using SI units, is : \hat{H} = - \frac{\hbar^2}{2m_\text{e}} \nabla_1^2 - \frac{\hbar^2}{2m_\text{e}} \nabla_2^2 - \frac{2e^2}{4\pi\epsilon_0 r_1} - \frac{2e^2}{4\pi\epsilon_0 r_2} + \frac{e^2}{4\pi\epsilon_0 r_{12}} , but adopting the convention associated with atomic units that transforms quantities into dimensionless equivalents, it becomes : \hat{H} = - \frac{1}{2} \nabla_1^2 - \frac{1}{2} \nabla_2^2 - \frac{2}{r_1} - \frac{2}{r_2} + \frac{1}{r_{12}} . In this convention, the constants , {{tmath|1= m_\text{e} }}, , and all correspond to the value (see '''' below). The distances relevant to the physics expressed in SI units are naturally on the order of , while expressed in atomic units distances are on the order of (one Bohr radius, the atomic unit of length). An additional benefit of expressing quantities using atomic units is that their values calculated and reported in atomic units do not change when values of fundamental constants are revised, since the fundamental constants are built into the conversion factors between atomic units and SI. == History ==

Hartree defined units based on three physical constants: built on Hartree's units, which they called atomic units abbreviated "a.u.". They chose to use , their unit of action and angular momentum in place of Hartree's length as the base units. They noted that the unit of length in this system is the radius of the first Bohr orbit and their velocity is the electron velocity in Bohr's model of the first orbit. In 1959, Shull and Hall advocated atomic units based on Hartree's model but again chose to use as the defining unit. They explicitly named the distance unit a "Bohr radius"; in addition, they wrote the unit of energy as and called it a Hartree. These terms came to be used widely in quantum chemistry. In 1973 McWeeny extended the system of Shull and Hall by adding permittivity in the form of as a defining or base unit. Simultaneously he adopted the SI definition of so that his expression for energy in atomic units is , matching the expression in the 8th SI brochure. == Definition ==

A set of base units in the atomic system as in one proposal are the electron rest mass, the magnitude of the electronic charge, the Planck constant, and the permittivity. are given in the table. Table notes • • • == Units ==

Three of the defining constants (reduced Planck constant, elementary charge, and electron rest mass) are atomic units themselves – of action, electric charge, and mass, respectively. Two named units are those of length (Bohr radius {{tmath|1= a_0 \equiv 4 \pi \epsilon_0 \hbar^2 / m_\text{e} e^2 }}) and energy (hartree {{tmath|1= E_\text{h} \equiv \hbar^2 / m_\text{e} a_0^2 }}). == Conventions ==

Different conventions are adopted in the use of atomic units, which vary in presentation, formality and convenience. Explicit units • Many texts (e.g. Jerrard & McNiell, • Provision for choosing more convenient closely related quantities that are more suited to the problem as units than universal fixed units are is also suggested, for example based on the reduced mass of an electron, albeit with careful definition thereof where used (for example, a unit , where {{tmath|1= \mu = m_\text{e} M / (m_\text{e} + M) }} for a specified mass ). This is a form of shorthand for the more formal process of transformation between quantities that is suggested by others, such as McWeeny. == Physical constants ==

Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant {{tmath|1= \alpha = {e^2} / {(4 \pi \epsilon_0\,\hbar c)} \approx 1/137 }}, which appears in expressions as a consequence of the choice of units. For example, the numeric value of the speed of light, expressed in atomic units, is {{tmath|1= c = 1/\alpha\,\text{a.u.} \approx 137\,\text{a.u.} }} == Bohr model in atomic units ==

Atomic units are chosen to reflect the properties of electrons in atoms, which is particularly clear in the classical Bohr model of the hydrogen atom for the bound electron in its ground state: • Mass = 1 a.u. of mass • Charge = −1 a.u. of charge • 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 momentum = 1 a.u. of momentum • Ionization energy = a.u. of energy • Electric field (due to nucleus) = 1 a.u. of electric field • Lorentz force (due to nucleus) = 1 a.u. of force == References ==