Units

Before entering into the physics in Octopus we have to address a very important issue: units. There are different unit systems that can be used at the atomic scale: the most used ones are atomic units and what we call “convenient” units. Here we present both unit systems and explain how to use them in Octopus.

Atomic Units

Atomic units are a Gaussian system of units (by “Gaussian” it means that the vacuum dielectric constant has no dimensions and is set to be $\epsilon_0 = {1 \over {4\pi}}$), in which the numerical values of the Bohr radius, the electronic charge, the electronic mass, and the reduced Planck’s constant are set to one:

$$ (1)\qquad a_0 = 1; e^2 = 1; m_e = 1; \hbar = 1. $$

This simplifies formulae (although some may feel it presents a serious hazard for dimensional analysis, interpretation and understanding of formulae, and physics in general. But this is just a personal taste). This sets directly two fundamental units, the atomic units of length and of mass:

$$ (2)\qquad {\rm au}_{\rm length} = a_0 = 5.2917721\times 10^{-11}~{\rm m};\quad {\rm au}_{\rm mass} = m_e = 9.1093819\times 10^{-31}~{\rm kg}. $$

Since the squared charge must have units of energy times length, we can thus set the atomic unit of energy

$$ (3)\qquad {\rm au}_{\rm energy} = {e^2 \over a_0} = 4.3597438\times 10^{-18}~{\rm J}, $$

which is called Hartree, Ha. And, since the energy has units of mass times length squared per time squared, this helps us get the atomic unit of time:

$$ (4)\qquad {\rm Ha} = m_e { a_0^2 \over {\rm} {\rm au}_{\rm time}^2} \to {\rm au}_{\rm time} = a_0 \sqrt{m_e \over {\rm Ha}} = {a_0 \over e} \sqrt{m_e a_0} = 2.4188843\times 10^{-17}~{\rm s}. $$

Now the catch is: what about Planck’s constant? Its dimensions are of energy times time, and thus we should be able to derive its value by now. But at the beginning we set it to one! The point is that the four physics constants used ($a_0, m_e, e^2, \hbar$) are not independent, since:

$$ (5)\qquad a_0 = { \hbar^2 \over {m_e \; {e^2 \over {4 \pi \epsilon_0} } } }. $$

In this way, we could actually have derived the atomic unit of time in an easier way, using Planck’s constant:

$$ (6)\qquad \hbar = 1\; {\rm Ha}\,{\rm au}_{\rm time} \Rightarrow {\rm au}_{\rm time} = { \hbar \over {\rm Ha}} = { {\hbar a_0} \over e^2}\,. $$

And combining (6) and (5) we retrieve (4).

Convenient Units

Much of the literature in this field is written using Ångströms and electronvolts as the units of length and of energy, respectively. So it may be “convenient” to define a system of units, derived from the atomic system of units, in which we make that substitution. And so we will call it “convenient”.

The unit mass remains the same, and thus the unit of time must change, being now $\hbar /{\rm eV},$, with $\hbar = 6.582,1220(20)\times 10^{-16}~\rm eV,s$.

Units in Octopus

Except where otherwise noted, Octopus expects all values in the input file to be in atomic units. If you prefer to use other units in the input file, the code provides some handy conversion factors. For example, to write some length value in Ångströms, you can simply multiply the value by angstrom:

A complete list of units Octopus knows about can be found in the Units variable description.

By default Octopus writes all values atomic units. You can switch to convenient units by setting the variable UnitsOutput to ev_angstrom .

Mass Units

An exception for units in Octopus is mass units. When dealing with the mass of ions, atomic mass units (amu) are always used. This unit is defined as $1/12$ of the mass of the ¹²C atom. In keeping with standard conventions in solid-state physics, effective masses of electrons are always reported in units of the electron mass (‘‘i.e.’’ the atomic unit of mass), even in the eV-Å system.

Charge Units

In both unit systems, the charge unit is the electron charge ‘‘e’’ (‘‘i.e.’’ the atomic unit of charge).

Unit Conversions

Converting units can be a very time-consuming and error-prone task when done by hand, especially when there are implicit constants set to one, as in the case of atomic units. That is why it’s better to use as specialized software like GNU Units.

In some fields, a very common unit to express the absorption spectrum is Mb. To convert a strength function from 1/eV to Mb, multiply by $\pi h c r_e,$, with $r_e=e^2/(m_e c^2),$. The numerical factor is 109.7609735.