# [Pw_forum] Periodicity

Mike Marchywka marchywka at hotmail.com
Thu Jun 27 12:45:40 CEST 2013

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> Date: Thu, 27 Jun 2013 06:06:37 +0200
> From: akohlmey at gmail.com
> To: pw_forum at pwscf.org
> Subject: Re: [Pw_forum] Periodicity
>
> On Thu, Jun 27, 2013 at 5:53 AM, Yantao Wu <ywu at g.hmc.edu> wrote:
>> Dear Axel,
>>
>> Thank you very much for your reply. I understand that the energy wouldn't
>> exactly be doubled if you just put two identical systems together. But if QE
>> does treat the system as an infinite lattice, what is exactly the meaning of
>> the total energy in the output? How does QE obtain this output energy from
>> the calculated band energies in a manner that is proportional to the size of
>> the supercell. That is, I don't quite understand why you say
>>
>> "the reason that you *do* get exactly twice the energy if you double
>> the system (and reduce the kpoint grid corresspondingly) is exactly
>> proof of that."
>
> the total energy of an infinite system is ... infinity. so that is a
> pointless quantity.
> what you do get is the energy of _one unit cell_ *in* the infinite system.
> this one unit cell has interactions with its periodic neighbors (is
> the cell itself).
> perhaps you should look up how Ewald summation works to see how you
> can compute the energy of a unit cell in an intinitely large lattice.
>
> if you have an isolated unit cell, you have no interactions with
> periodic neighbors. so doubling the system, *significantly* changes
> the energy from double the single system. however, doubling a unit
> cell in a periodic system should give you exactly double the energy.
> the one thing that may make this not exactly double is k-point
> sampling you have to make sure that your sampling of k-space is the
> same, so you have to reduce the k-point grid, if you double the unit
> cell.
>
> if this still confuses you, you should look things up in a text book.
> there are several really good ones around that cover electronic
> structure calculations in condensed matter physics.

Many universities have good lecture notes online and of course full text
theses on google scholar can provide a lot of introductory. citeseer has lots of math and computer
related academic work. I usually end up going to wikipedia
for many quick references- in fact I think I did just check them for Ewald sum and looking up how
lattice vectors are defined. You can not always assume it to be accurate but often it is fine.

>
> axel.
>
> p.s.: the arguments also hold for classical interaction models of
> point charges and empirical potentials
>
>
>>
>> Thanks,
>> Yantao
>>
>>
>> On Wed, Jun 26, 2013 at 8:28 PM, Axel Kohlmeyer <akohlmey at gmail.com> wrote:
>>>
>>>
>>> QE *does* treat your system as if it was periodic and thus a _unit
>>> cell of an infinite crystal_.
>>> the reason that you *do* get exactly twice the energy if you double
>>> the system (and reduce the kpoint grid corresspondingly) is exactly
>>> proof of that. if it would treat just the input system, the energy of
>>> the double size system would be the energy of the two halves *plus*
>>> the interaction energy between them.
>>>
>>> does that make sense?
>>> axel.
>>>
>>> On Thu, Jun 27, 2013 at 5:10 AM, Yantao Wu <ywu at g.hmc.edu> wrote:
>>>> Dear QE users,
>>>>
>>>> I have a general question on how QE treats periodicity of the system,
>>>> e.g. a
>>>> bcc lattice. I originally thought QE only calculates the wave function
>>>> for
>>>> the system you input into the program and enforce the Bloch periodic
>>>> boundary condition of Psi(r+R) = exp(ikR)Psi(r) on the wave-functions,
>>>> where
>>>> R, the lattice constant, QE knows from the input of ibrav. In this case,
>>>> QE
>>>> doesn't "know" anything about the infinite lattice that may be inferred
>>>> from
>>>> the input system.
>>>>
>>>> But then I was told the other day that QE actually infers the infinite
>>>> lattice from the input system and calculates the wave-function for the
>>>> infinite system. Even though this sounds appealing to me, I'm still
>>>> confused
>>>> by one seeming paradox. That is, if I double the size of the input
>>>> system
>>>> but making sure that the infinite lattice inferred from this
>>>> double-sized
>>>> system be the same as the original system, then the energy calculated
>>>> for
>>>> the double-sized system will be almost precisely double of the energy of
>>>> the
>>>> original system. This makes me feel like that QE treats the system of
>>>> interest to be just what the input system is.
>>>>
>>>> Can anyone clarify how exactly is periodicity treated in QE?
>>>>
>>>> Thank you much,
>>>> Yantao Wu,
>>>> Harvey Mudd College 15'
>>>>
>>>> _______________________________________________
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>>>
>>>
>>>
>>> --
>>> Dr. Axel Kohlmeyer akohlmey at gmail.com http://goo.gl/1wk0
>>> International Centre for Theoretical Physics, Trieste. Italy.
>>> _______________________________________________
>>> Pw_forum mailing list
>>> Pw_forum at pwscf.org
>>> http://pwscf.org/mailman/listinfo/pw_forum
>>
>>
>>
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>
>
>
> --
> Dr. Axel Kohlmeyer akohlmey at gmail.com http://goo.gl/1wk0
> International Centre for Theoretical Physics, Trieste. Italy.
> _______________________________________________
> Pw_forum mailing list
> Pw_forum at pwscf.org
> http://pwscf.org/mailman/listinfo/pw_forum

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