[Pw_forum] implementation of variation of occupations and Fermi level with perturbations

[David Strubbe]

Dear developers,

I am trying to understand how the variation of occupations and Fermi level
is implemented in density-functional perturbation theory for metals. In the
Baroni, de Gironcoli, and Dal Corso Rev Mod Phys paper on DFPT, this issue
appears in equations 68 and 75-79. I see that routine ef_shift can implement
the result of a shift in the Fermi level as in either eq. 68 or eq. 75
depending on the value of "flag". But for the calculation of the shift
itself, I do not see the correspondence between what is done in the code and
the equations in the paper. Eq. 79 refers to a quantity \Delta n_{ext} and
an integral of the LDOS with \Delta V_{SCF} to calculate the shift in Fermi
level. However in ef_shift it appears that the density response drhoscf is
used instead of these quantities in the numerator, which doesn't seem like
the same thing. Can you explain what the relation is between the calculation
in the code and the equations in the RMP paper?

Thank you,
David Strubbe
UC Berkeley
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