[Wien] missing electrons

[Peter Blaha]

Hi,
I could verify your "problem" and got the same results, i.e. the integrated
DOS (case.dos1) does not yield the proper number of electrons.

First I was a little shocked, but now it appears that everything is in
principle correct, only your k-mesh is too crude to give a good DOS.

Let me explain what the problem is:

The DOS is an integral over k; done numerically by the tetrahedron
method. Thus the BZ is divided into tetrahedra and the DOS is a sum of
all those tetrahedra integrals (which can be done analytically).
For small k-meshes it may now occur that a single tetrahedron has 4
identical vertices !, i.e. that all 4 k-points for this tetrahedron are
the same. If this happens, it means that the DOS contribution due to this
tetrahedron is a delta-function at e=e_k (with a normalization equal to
the volume of this tetrahedron). Of course with any finite E-mesh
(the mesh where you calculate the DOS (eg. -1, 0.001, 1.)), even a very fine
one you will "never" catch these spikes of the delta-functions in your
DOS nor will you be able to integrate the DOS numerically over energy, when
it has such singularities.

It will always occur when in case.kgen you find tetrahedra with identical
vertices like:

         4        19  0.185185185185E-01       101         1
         <6         1         1         1         1>         <4
         1         1         1         2>         ...

The first line means that there are 4 k-points with 19 tetrahedra.
In the 2nd and 3rd line I've added < > to indicate their meaning:
The first tetrahedron has "weight" 6 and four identical vertices (1,1,1,1)
(first k-point), while the second tetrahedron has weight 4, but vertices
1,1,1,2 (thus one k-point is different),....

Thus the first tetrahedron is such an example. All eigenvalues of band i
will be identical leading to a delta-function at this eigenvalue and give
zero contribution to DOS(E) on your energy-mesh E.

Hope this explains the problem.

Without shift the probablility of tetrahedra with identical vertices
decreases. With a shift it seems that even for very dense meshes such
critical tetrahedra occur (always the first one ?), but their weight get
drastically reduced so that you don't see the problem anymore.
However, from some tests on tio2 this does not necessarely mean that
the DOS of an unshifted mesh is "better". The DOS with a crude unshifted
mesh "integrates" properly, but has nothing in common with the final DOS
with a converged (shifted or unshifted) k-mesh.

It also explains, why you do not "see" any problems with bulk Rh or other
(smaller) cells, because there you probably have very dense k-meshes
(or no shift).

> However, when we integrate  the Density of States (case.dos1ev)  by
> "xmgrace" over the energy interval : -0.1 Ry to the Fermi Level, we get
> 150 electrons (i.e., we lose 50 electrons). However, when we do the same
> for the DOS calculated by another DFT code (with pseudopotential
> approximation), we get almost the right number of electrons. Now, if  we
> plot these DOS's together on the same graph, it becomes obvious that the
> DOS calculated by WIEN2k has a lot of electrons or states missing
> (because the DOS-curves corresponding to the pseudopotential code lie
> far above that of the WIEN2k code).
>
> When I add up the partial DOS's and that of the interstitial, I get the
> total DOS. This part is alright. Therefore, I would guess that tetra
> commits some error in projecting the partial DOS's, which leads to
> partial loss of electrons per muffin-tin sphere and these losses add up
> to 50 electrons for the whole supercell.
>
> However, I do not encounter this problem when I integrate the DOS over
> the same energy interval for Ru bulk, RuO2 bulk and the Ru(0001) surface
> by the same software, "xmgrace". These 3 systems are perfectly ok.


                                      P.Blaha
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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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