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["Allan, Douglas C Dr" <AllanDC@Corning.com>]

For a supercell to be effective as a good approximation to your nonperiodic 
defect, it should be large enough that k=0 is a good approximation.  This 
would not be the case if your actual defect were really periodic, in which 
case multiple k-points would make sense, but usually we use a supercell to 
mimic a nonperiodic point defect like a vacancy.  In this case, there are 
artificial periodic images of the vacancy in nearby cells, and the supercell 
needs to be large enough to avoid too much interaction between the periodic 
images.

Having said that, the physics you are trying to calculate or approximate is 
that of a nonperiodic defect and its surrounding atoms.  The use of multiple 
k-points is only appropriate if the underlying system IS periodic.  Even 
though it is true that your computational system is artificially periodic, 
the physical system (the vacancy) is not.  In principle if the supercell were 
large enough, there would be no dispersion at all with k and it would make no 
difference whether you used one or many k-points.  For example, you could use 
the zone center k=0 or a zone boundary point and you would get the same 
answer.  In practice of course the supercell is never that large, and there 
might be a case made for calculating with the (1/4,1/4,1/4) point and its 
related points because in some sense that small set "averages" between the 
slighly different results you would get for zone center and zone boundary.  
If the supercell results depend strongly on these choices then the supercell 
is definitely too small.

Zone folding of the Brillouin zone is a topic you should read up on in a 
textbook, but I will give a brief description/definition.  Consider the 
primitive cell in real space and its reciprocal (the original Brillouin 
zone).  Then consider the larger supercell and its smaller reciprocal cell 
(the new BZ).  The new BZ is smaller because it has smaller G-vectors that 
are associated with the larger real space translations of the supercell.  If 
you start on any k-point of the original BZ (the larger one), you can map or 
"fold" any points outside the new BZ (the smaller one) to the inside of the 
new BZ by using the G-vectors (reciprocal space translations) that are 
reciprocal to the supercell.

Unfolding the smaller BZ out into the larger one, you can start at any 
k-point in the smaller BZ (e.g. k=0) and add all possible small G-vectors 
associated with the small BZ until you find all (k+G) points that can lie 
inside the larger BZ.  This is the zone folding I recommended.  These 
k-points in the larger BZ would constitute the best grid for the primitive 
cell calculation to be compared with the supercell calculation.

My experience is from semiconductors and insulators.  If you are working with 
metals where you need enough k-points to handle a fermi surface integration, 
then you might need multiple k-points even for a supercell that is adequate 
for your calculation.  Others should comment on these issues for metals.

If you have computer cycles to burn, you can save yourself a lot of trouble 
just by calculating the nondefective structure in the same supercell as the 
defective structure.  Then the differences are very straightforward.  Also, 
as an exercise, you can try calculating something easy (like Si) and practice 
zone-folding between a 2-atom primitive cell and the related 8-atom supercell 
(the latter is a simple cube).  If you do it right the resulting 
energies/atom should be identical within the tolerance on convergence.  In 
the past I tested abinit by pushing the tolerances essentially to machine 
precision (or as close as I could get) and proved that this idea works as 
expected.

-----Original Message-----
From: Steven Homolya [mailto:Steven.Homolya@spme.monash.edu.au] 
Sent: Thursday, October 09, 2003 9:02 AM
To: 'forum@abinit.org'
Subject: RE: [abinit-forum] Vacancy-formation energy with abinit


On Thu, 9 Oct 2003, Allan, Douglas C Dr wrote:

> Another and perhaps better way to select compatible k-points is to 
> start with a single k-point in the supercell (since it is a supercell 
> a grid of multiple k-points does not really make sense) and then do 
> zone-folding to make a set of k-points in the primitive cell.  In the 
> absence of the vacancy in the supercell this should produce the same 
> energy/atom in both cases to very good accuracy; they results should 
> agree within the tolerance you supply for convergence.  Then when you 
> put the vacancy in the supercell you can better trust the energy 
> comparisons between the two cells.
> 

I'm still sorting k-mesh compatibility out for the systems I'm looking at, 
so I'm happy to read your suggestions.

My questions are:

* Why shouldn't one use multiple k-points for a supercell?
* How does one "unfold" the B.Z.?

I found that one k-point is not enough for 16, 27, 32, 64, or even 108 site 
fcc supercells (when I want at least 0.0005 Ha accuracy in total energy).

E.g. fcc-Cu:

- happy with 1-atom prim. cell using ngkpt 16 16 16
- get the same result using 8 atom fcc supercell with ngkpt 8 8 8
- get the same using 64 atom fcc supercell with ngkpt 4 4 4

If I use only one k-point for 64 atom supercell, that would equate to 4x4x4 
== 64 k-points in the B.Z. of the one atom fcc unit cell, which is 
insufficient for anything but a rough initial estimate of optimal geometry.

Steve

-- 
Steven Homolya
School of Physics and Materials Engineering
Monash University, VIC 3800
Australia
Tel: +61 3 9905 3694
Fax: +61 3 9905 3637