[DFTB-Plus-User] DFTB-Plus-User Digest, Vol 45, Issue 27

[Ben Hourahine]

Hello Fahd,

sorry for the delay in replying.

The phase DFTB+ uses for k-points is a slightly different convention
from the standard Bloch function choice.

Conventional block waves : exp( i k \dot x) where x is the position in
space.

DFTB+ : exp( i k \dot R) where R is the lattice vector to the unit cell
containing point x (this choice is fairly common in electronic structure
codes, as its slightly faster to evaluate).

The second case gives the same results for example when calculating the
density and most other properties (but isn't suitable for example when
evaluating the Berry phase of wave functions).


1) If the k-point is real, any of the valid eigenvectors (or some
subset) can be multiplied by factors of -1 and are still valid. Moving
to a complex k-point, similarly the vectors can be multiplied random
complex phases of unit magnitude. Even worse, for degenerate eigenvalues
any unitary transformation within that sub-space is also a valid set of
eigenvectors.

Adding extra constraints on the eigenvectors can fix the phases, but
there is not a standard convention for this (DFTB+ doesn't do this since
the phase choice doesn't effect standard calculated properties and costs
some computational time to set up).

2) It is random, depending on the numerical behaviour and implementation
of the eigensolver. The individual k-points are calculated separately,
so in general have ill-defined different relative phases. It is possible
to set the phases of the eigenstates between two k-points by maximally
aligning their spaces. This isn't in the standard code base, but there
is some in-development code in an old branch that can do this if
necessary along a chain of k-points (but is computationally more
expensive than standard ground state calculations).

Regards

Ben

On 31/05/18 22:34, Mohiyaddin, Fahd A. wrote:
> Hi Ben Hourahine,
>
> I also had a question regarding one of your comments (shown below).
>
>>>>> But there is one slight subtle point about the phase, which is that at each k-point there is a global phase of the >>>> wave-function
> Let's say I take a particular k-point, and I plot the wave-function in real space for the k-point.
>
> (i) Would the "global" phase be the same for all locations in the real space? I just wanted to ensure that I understand the term "global" correctly.
>
> (ii) Is there an analytical expression (in terms of the k-point) for the global phase of the wave function in DFTB ? Or is it random?
>
> Regards,
> Fahd
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Fri, 25 May 2018 08:10:10 +0100
> From: Ben Hourahine <benjamin.hourahine at strath.ac.uk>
> To: dftb-plus-user at mailman.zfn.uni-bremen.de
> Subject: Re: [DFTB-Plus-User] : Simulating Silicon Crystal : Does DFTB
>         estimate the phase of the wavefunction accurately?
> Message-ID: <233fc818-1bd9-b81c-aa76-69bd8348d673 at strath.ac.uk>
> Content-Type: text/plain; charset="utf-8"
>
> Hello Fahd,
>
> I think Bálint answered everything already. But there is one slight
> subtle point about the phase, which is that at each k-point there is a
> global phase of the wave-function. Currently this is undefined between
> different k-points (it will depend on your particular eigensolver and
> its numerical implementation). If you are calculating properties at each
> k-point separately then combining, then this is harmless, but if you
> want to calculate between k-points this can cause some complications.
>
> The global phases can be made consistent, but this is not available in
> the current public code (its related to the Berry phase of the
> wavefunction).
>
> Regards
>
> Ben
>
> On 25/05/18 07:55, Bálint Aradi wrote:
>> Dear Fahd,
>>
>>> (data from wp-1-184-5-real.cube) + 1i* (data from wp-1-184-5-imag.cube).
>>>
>>> Is the above combination a correct way for estimating the wave function,
>>> and will the relative phase between real and imaginary parts be treated
>>> correctly at all coordinates in space?
>> Yes, the real and imaginary cube files are built using the real and
>> imaginary part of the complex eigenvectors, respectively. The way you
>> combine them seems to be correct and you should get back the correct
>> complex wave function everywhere. (Unless we have a bug in waveplot.)
>>
>>> I have also tried playing with different unit cells (primitive and
>>> cubic) for silicon in DFTB. /The results from different unit cells give
>>> me the same charge densities, but the Fourier transforms of the wave
>>> functions are different. /Hence, I was wondering whether I am combining
>>> the wave functions correctly, and does DFTB take into account the phase
>>> of the wave function accurately.
>> Yes, I am pretty sure, DFTB+ takes the phase of the wave-function
>> correctly into account.
>>
>> Just to make a check, you may consider to calculate a single atomic
>> hydrogen chain (orthogonal supercell with two supercell vectors set to
>> large values). That you could compare to results obtained by other means
>> (e.g. either VASP calculation or even analytic ones) and spot, if there
>> are any bugs.
>>
>>   Best regards,
>>
>>   Bálint
>>
>>
>>
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> --
>       Dr. B. Hourahine, SUPA, Department of Physics,
>     University of Strathclyde, John Anderson Building,
>             107 Rottenrow, Glasgow G4 0NG, UK.
>     +44 141 548 2325, benjamin.hourahine at strath.ac.uk
>
> 2013/4 THE Awards Entrepreneurial University of the Year
>       2012/13 THE Awards UK University of the Year
>
>    The University of Strathclyde is a charitable body,
>         registered in Scotland, number SC015263
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-- 
      Dr. B. Hourahine, SUPA, Department of Physics,
    University of Strathclyde, John Anderson Building,
            107 Rottenrow, Glasgow G4 0NG, UK.
    +44 141 548 2325, benjamin.hourahine at strath.ac.uk

2013/4 THE Awards Entrepreneurial University of the Year
      2012/13 THE Awards UK University of the Year

   The University of Strathclyde is a charitable body,
        registered in Scotland, number SC015263