[Pw_forum] FW: about Fermi Energy shift in periodic (q=0) perturbations

Stefano de Gironcoli degironc at sissa.it
Thu Dec 13 11:47:15 CET 2007

we are interested in the limit for q->0.

in this limit the total number of electrons does not change and the 
Fermi energy does not shift (since it is a q/=0 calculation) however the
delta_n(q) = \sum_G chi(q,q+G) deltaV_0(q+G)
                 = \sum_G chi0(q,q+G) [deltaV_0(q+G) + 4pi/(q+G)^2 

Now chi0(q,q) for metals is FINITE in the limit of q->0 (related to the 
DOS at the FE) and 4pi/q**2 explodes thus the only way things can still 
satisfy the above relation is that n(q) -> 0 quadratically so that
1) the limit for 4pi/q**2 delta_n(q) gives a FINITE contibution to the 
potential V_scf(q->0) that toghether with the other component of the 
response makes that
2) the system is neutral ... i.e. n(q)->0.

In a q==0 calculation the FINITE value of V_scf(q=0) coming from the 
Coulomb potential cannot be calculated calculated directly from 
condition 1) since q==0 and the above ratio is ill defined but can be 
determined by condition 2) ... whether you want to call it a potential 
shift or an (opposite) FE shift is up to you. This latter choice is made 
in the phonon code.

Hope this helps,


Ezad Shojaee wrote:
> Dear Mr. Gironcoli
> Thanx for your answer, but sorry I'm not sure i follow you, could you 
> open it more?!
> you mean in the case of periodic perturbation(like the constant one) 
> we have a change in the # of electrons? what exactly happens in this 
> case, so we have to shift the Fermi energy(in order to have neutrality 
> condition)? What happens if we do not shift it(As you see i want you 
> to answer the question you made!)?
> Thanx again
> Ezad Shojaee - Tehran University
> > At any q/=0 the perturbation is such that the number of electrons does
> > not change.
> > Even in the limit of q->0 the macroscopic term in the density response
> > is zero (because in a metal
> > the dielectic constant is infinite).
> > Now consider the case of a periodic (q=0) perturbation, the simplest
> > one, a constant, threated with or w/o the FE shift... which of the two
> > treatments corresponds to the result you would obtain in the liimit 
> q->0 ?
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