[Wien] Charge Convergence is not achieved
Laurence Marks
L-marks at northwestern.edu
Thu Jul 7 16:24:27 CEST 2011
While what you say is probably right (it normally is), for this to
prevent convergence one has to hypothesize that these occupancy
changes correspond to excitation of different eigenvectors of the
charge density (not just the wavefunctions) with respect to the
appropriate dielectric response matrix, so near the solution the
mixing Jacobian only obtains information about the off-diagonal terms,
minimal information about the diagonal terms. (In effect the Jacobian
then become very ill-conditioned.) In effect this implies that in hard
cases this response matrix varies rapidly with k so needs to be finely
sampled to include all the different eigenvectors and avoid
ill-conditioning. While this is a very reasonable hypothesis, proving
it is not so simple....
N.B., with respect to the comment by Shamik Chakrabarti, yes, if the
mixing has converged then this is a sufficient condition that you have
satisifed the KS equations for the functional used, i.e. found a
variational minimum and the problem is feasible. If the problem does
not converge it may be unfeasible, and all that is being found is a
trap not a fixed-point variational minimum. Or it is too hard for the
mixer.
On Thu, Jul 7, 2011 at 8:24 AM, Peter Blaha
<pblaha at theochem.tuwien.ac.at> wrote:
> Basically, for a metal the convergence depends on the details of the
> bandstructure around EF and on the method to determine EF and the occupation
> of all eigenvalues.
>
> Suppose you have two bands crossing EF, one has A character, the other one
> B.
> Now you start with a coarse k-mesh and represent the band with only a few
> k-points,
> such that the weight (number of electrons) for each eigenvalue E_n_k is
> large (e.g 0.1 e)
>
> At some iteration it can happen that E_n1_k1 is just a tiny little bit lower
> than E_n2_k2
> (k1 and k2 come from different bands) and both are close to EF. Than E_n1_k1
> is
> "fully" occupied", while E_n2_k2" is completely empty when using the TETRA
> method (because this
> interpolates only within the same band n!) and thus you get more charge
> at atom A.
> Even when the mixer now adds only very little of this new density, it may
> lead to a potential where
> E_n_k1 is now HIGHER than E_n_k2, and thus in the next iteration we get a
> density
> which has 0.1 e more at site B (and not A). Thus the newly generated charge
> densities
> differ by a huge (0.1 e) amount from the previous one.
>
> If you now increase the k-mesh, the weight of an individual k-point will go
> down
> (eg. be only 0.01 e) and thus such oszillations will be an order of
> magnitude smaller.
> In addition, an integration (TETRAHEDRON method) becomes better with more
> sample
> points and convergence will be better.....
>
> On the other hand when using TEMP(S) instead of TETRA, you may be able to
> damp these
> oszillations, since the occupation depends only on the energy, but not on
> the
> "topology" of the bands (i.e. which eigenvalues are connected to each other
> via band n
> and k-index k). This is a clear advantage of TEMP, however, you run into the
> problem
> that a final solution eventually has ALWAYS some occupation of "unoccupied"
> states,
> which should be zero for an "exact method" (and you may even loose or
> greatly reduce
> your magnetic moment).
>
> Basically, there is no absolute rule and convergence has to be checked for
> each individual
> case because you do not know the band-details.
>
> Of coarse there are general considerations like:
>
> bad - good convergence
> metal - nonmetal
> flat bands - steep bands at EF, or equivalently
> elements with f,d-states at EF - no d,f states at EF
> many non-equivalent atoms of the same type - onyl ONE equivalent atom on
> nuclear charge Z
>
> Some examples derived from those rules:
>
> fcc Cu converges very quick (only ONE very STEEP S-like band at EF), bcc V
> is more difficult
> (MANY D-BANDS cross EF).
> fcc Ni is even worse, because of spin polarization you DOUBLE the number of
> bands at EF
> and one can easily shuffle electrons from spin-up to dn,...
>
> A supercell or surface of Ni becomes even worse, because you may have
> several different
> Ni atoms (surface, sub-surface,.... bulk) and thus have with X-layers
> X-TIMES as many bands
> around EF, all of them VERY SIMILAR (because they are all Ni), but still
> clearly distinct
> (surface,....).....
>
>
>
> Am 07.07.2011 14:42, schrieb Laurence Marks:
>>
>> 2011/7/7 Shamik Chakrabarti<shamikiitkgp at gmail.com>:> Dear Peter Blaha
>> Sir,> Indeed by increasing number of K points we
>> got the> convergence. Sir I have now some basic queries on this topic. You
>> have said> that> "sometimes you cannot reach (easily)
>> arbitrary> convergence"> why in some cases we can not reach convergence up
>> to our desired limit?...is> it the limitation of DFT?....or it means that
>> the feasibility of the> solution is only up to the achieved convergence?
>> This is in fact a deep, and very good question, at least in my opinion.
>> Unfortunately that does not mean that there is a good answer to it!
>> With the perfect functional convergence should (I believe, others
>> maydisagree) always be good. With a very imperfect functional it is
>> quitepossible that a DFT calculation will not converge, i.e. it
>> isunfeasible. Empirically many (but not all) metals do not converge wellwith
>> small numbers of k-points, but some others do. Why....I do notunderstand as
>> I cannot write down a mathematical analysis to explainthis and do not
>> believe that there is a formal analysis in theliterature, it is just
>> empirical knowledge (folklore).
>>
>> -- Laurence MarksDepartment of Materials Science and EngineeringMSE Rm
>> 2036 Cook Hall2220 N Campus DriveNorthwestern UniversityEvanston, IL 60208,
>> USATel: (847) 491-3996 Fax: (847) 491-7820email: L-marks at northwestern dot
>> eduWeb: www.numis.northwestern.eduChair, Commission on Electron
>> Crystallography of IUCRwww.numis.northwestern.edu/Research is to see what
>> everybody else has seen, and to think whatnobody else has thoughtAlbert
>> Szent-Gyorgi_______________________________________________Wien mailing
>> listWien at zeus.theochem.tuwien.ac.athttp://zeus.theochem.tuwien.ac.at/mailman/listinfo/wien
>
> --
>
> P.Blaha
> --------------------------------------------------------------------------
> Peter BLAHA, Inst.f. Materials Chemistry, TU Vienna, A-1060 Vienna
> Phone: +43-1-58801-15671 FAX: +43-1-58801-15698
> Email: blaha at theochem.tuwien.ac.at WWW:
> http://info.tuwien.ac.at/theochem/
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--
Laurence Marks
Department of Materials Science and Engineering
MSE Rm 2036 Cook Hall
2220 N Campus Drive
Northwestern University
Evanston, IL 60208, USA
Tel: (847) 491-3996 Fax: (847) 491-7820
email: L-marks at northwestern dot edu
Web: www.numis.northwestern.edu
Chair, Commission on Electron Crystallography of IUCR
www.numis.northwestern.edu/
Research is to see what everybody else has seen, and to think what
nobody else has thought
Albert Szent-Gyorgi
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