[CPMD-list] Janak's theorem - under periodic boundary conditions

[Audrius Alkauskas]

Dear CPMDers

One question bothers me a little bit. A similar question was asked on
this list before, but not exactly the same.

If one has a very large cell (ultimately very very large) of some bulk
material with extended electronic states, Janak's theorem should apply.
That means, total energy differences between charged and neutral bulk
should correspond to eigenvalues of a neutral solid (provided states are
extended, changes in potentials produced by neitralizing background and 
extra or missing electrons are very small, and there are no spurious 
constant shifts in potentials - everything is kept the same).
This is what I usually get in many plane-wave codes. For example, here 
is the example of some oxide material:
Kohn-Sham eigenvalues:
e_{HOMO}=0.15 eV
e_{LUMO}=7.10 eV
Total energy differences:
e_{HOMO} from total energy differences 0.10 eV (E_{tot}^{0}-E_{tot}^{+})
e_{HOMO} from total energy differences 7.14 eV (E_{tot}^{-}-E_{tot}^{0})
Areement is very good.

Unfortunately, it does not always work in CPMD. It seems that for 
certain combination of pseudopotentials there can be shifts by up to 
several electronvolts in energies, calculated by total energy 
differences, vs. eigenvalues. That means, Janak's theorem is not 
applicable for these systems. But it is in other codes! The question: 
are eigenvalues referred to the same average local potential as the 
total energy? Or are they refered to some other potential and then this 
change is compensated somehow? This is very important for those who 
study defects in materials, because there one has often to deal with 
charged systems and in principle for bulk solids Janak's theorem should 
always work.

I do not know whether I explained the problem properly.

Audrius