Manual:Units

From OctopusWiki

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 are atomic units and what we call "convenient" units. Here we present both unit systems and explain how to use them in Octopus.

1 Atomic Units
2 Convenient Units
3 Units in Octopus
4 Mass Units
5 Unit Conversions

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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, in my opinion, it seriously hazards dimensionality analysis, formulae interpretation and understanding, 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 from 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).

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Convenient Units

A lot of the literature in this field is written using Ångströms and electron-volts 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 $\hbar = 6.582\,1220(20)\times 10^{-16}~\rm eV\,s$ .

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Units in Octopus

By default Octopus reads and writes atomic units; you can switch to convenient units by setting the variable Units to "eVA". If you prefer different units for input and output, there are the variables UnitsInput and UnitsOutput.

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Mass Units

The only exception for units in Octopus is mass units. When dealing with the mass of ions, always atomic mass units (amu) are used. This unit is defined as $1 / 12$ of the mass of the Carbon atom.

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Unit Conversions

Converting units can be a very time-consuming and error-prone task when doing it by hand, especially when there are implicit constants set to one like in this case. 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 $r_e=e^2/(m_e c^e)\,$ . The numerical factor is 109.7609735.