[Pw_forum] Problem with nbnd in clusters

[Matteo Cococcioni]

Hi Nicola,

I would like to express my opinion on this very interesting topic that 
you mentioned.
These are just my very arguable ideas or what I think I have understood 
from a little thinking and studying of this problem.

Let's start from B and H2+ or 1 e- TM atoms with only 1 d state occupied 
(e.g. Sc or Y): I think that the "real" ground
state is cylindrical. So for example in the H2+ molecule at large 
interionic distance one should get the localization of the e-
on one of the protons. The symmetry of the problem (and of the 
potential) is re-established on a statistical level because
the physical situation can be described by a statistical mixture of 
states with 1 e- on one proton or the other one that have the same 
probability.
So in the case of B it would be a misture of px, py, pz with weights of 
0.333 each (BTW this case is a bit like the Scroedinger cat because any
cartesian coordinate system you fix is arbitrary; you would probably 
need a filed to break this equivalence but in that case you would 
perturb the system....). Now problem is: would should we expect from 
DFT? I think this is at the core of the old problem that DFT "prefers" 
the solution where the charge density has the same symmetry of the 
potential (a "spherical" one for B) instead one of the possible and 
equivalent materializations of the statistical mixture for which 
symmetry is broken. Why? I think because a) you would need to break the 
symmetry "by hand"; b) there is a finite, negative contribution to the 
energy (coming from an approximate Hxc potential) even for one electron 
systems that shouldn't be there and gives lower energy for the 
symmetrical solution (e.g. the electron delocalized over the H2+ 
molecule at any distance). In fact corrections as "simple" as SIC can 
solve these problems (I guess/hope).

Situation is more involved with multi-electron systems like, for example 
H2. If you pull the two nuclei apart one gets (in real life) 1 e-
localized on each side. What is the spin of these two e- (of course they 
were in a singlet state at short distances)? Any relative orientation
is good. So again I think the infinite distance solution is a 
statistical mixture of 4 states with equal probability. How this 
solution connects to the one at finite distances? I think it 
continuously connects to a multi-reference ground state described by a 
combination of slater determinants.
What to expect from DFT? I think that to get the best out of DFT one has 
to impose a broken symmetry solution (e.g. an antiferromagnetic
one for H2); this corresponds in fact to one of the states in the 
statistical mixture of the infinite distance case and to one of the 
slater determinants
in the real ground state wavefunction. How about the charge density? If 
we sum over the spin it still has the symmetry of the problem.
If we had the perfect DFT I think that, with a broken symmetry solution 
(that has the advantage also that the two body density matrix
rho(x,x')-->0 for large interatomic distances), we would get the right 
energy and density (not wfc that is a fake one) because that is what DFT 
is meant to do. if we have a symmetrical solution with each electron 
split between both sites I think this won't come out right even with a 
perfect xc functional (which nobody has).

For multi-electron, multi-states systems of course the situation is more 
complicated than this. But I would prefer a broken symmetry solution in 
any case (and hopefully a better xc functional).

hope this helps,

Matteo








Nicola Marzari wrote:
>> I agree that symmetrization (due to the periodization) induces a 
>> degeneracy but the
>> "real" system that I want to simulate (a dimer alone in space) has even more
>> symmetries. In particular it has a rotational symmetry that will never 
>> be there
>>     
>
>
> Well, this is the part I've never understood - that's why I was 
> mentioning the boron atom. I wouldn't necessarily agree that your
> dimer charge density should have the same symmetry of the ionic
> potential - as much as a dissociated H2+ molecular ion doesn't have
> mirror symmetry on the plan bisecting the bond, as you dissociate the
> molecule, or antiferromagnetic MnO has a lower electronic-structure
> symmetry, since the Mn are inequivalent from the point of view of spin
> density.
>
> Also, the potential in atoms is spherical, but I'm sure there are
> solutions lower in energy that are cylindrical. Think at a transition
> metal with only one d electron: do you fill up 1 orbital 1.0, 2 orbitals
> 0.5, 3 orbitals 0.33333, 5 orbitals 0.2 ? Only the last is spherical,
> all others are cylindrical, although of course an ensemble of
> measurements on atoms will always converge on the spherical limit.
>
> Am I the only one worrying about this ? It seems a key problem
> that I've never seen addressed - Englisch and Englisch in 1983
> had a paper on the fact that fractional occupations are somewhat
> not compatible with v-representability, but I don't recall it very well.
>
>
> 			nicola
>
>
> ---------------------------------------------------------------------
> Prof Nicola Marzari   Department of Materials Science and Engineering
> 13-5066   MIT   77 Massachusetts Avenue   Cambridge MA 02139-4307 USA
> tel 617.4522758 fax 2586534 marzari at mit.edu http://quasiamore.mit.edu
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