[Pw_forum] More on F-D smearing

[Stefano de Gironcoli]

Dear Paul,

    I just wanted to confirm all what Nicola was writing.

    The quantity called total energy in PWSCF is the variational 
quantity in presence of finite smearing, whose functional derivatives 
are the K-S equations, as solved by the code, and whose derivatives 
(forces and stress) can be computed exploiting Hellmann-Feynman theorem .

    Demet is the quantity that needs to be added to the band energy  
\sum f_i \epsilon_i in order to make it equal to the integral computed 
from the smeared density of state  \int \epsilon n(\epsilon) d\epsilon.  
It is essential to recover the variational character of the functional.
In this sense it is a correction to the usual definition of eband needed 
in the metallic case.  Since it generates confusion "correction for 
metals" is probably an unlucky name. 

     If the smearing function is chosen to be Fermi-Dirac and degauss is 
interpreted as a temperature it happens that the variational quantity at 
finite temperature is the free energy and this is the quantity computed 
and printed as total energy by the code.
I agree total free energy would be more appropriate in this case. Demet 
in this case equals  -TS.
 
     However, if the smearing is Gaussian, cold-smearing or 
Methfessel-Paxton there is no real temperature around and no need to 
talk about free energy and my feeling is that calling it total free 
energy would also generate some confusion. The termodynamic analogue is 
sometimes useful to understand, for instance, why the "total (free) 
energy" decreases quadratically with increasing "temperature" with 
gaussian and F-D smearing while is much flatted with M-V or M-P smearing.

      Note that in the case of Gaussian or F-D smearing, although the 
quantity  (F+E)/2 = E - TS/2 is closer to the T=0 limit, it SHOULD NOT 
be used as a the estimate for the energy since it IS NOT variational and 
therefore not consistent with computed  forces . One should use instead 
M-V or M-P smearing that are both more insensitive to the value of 
smearing AND variational.

     Finally, in order to perform accurate finite temperature 
calcualtion with reasonable number of k-points one should combine M-V or 
M-P smearing (in order to speed up k-point convergence) with F-D (in 
order to have finite temperature). This is probably what is done in the 
paper by Gonze and coworkers mentioned by Nicola (I did not read it yet 
so I may be wrong).  This is presently not implemented in PWSCF but it 
should be rather straightforward to use as smearing width the 
convolution of a M-V/M-P function (with a given degauss) and a F-D 
function (with the desired physical temperature).
     In this case the "total energy" printed by PWSCF would be the 
approximate "total free energy" but the "correction for metals" will not 
be immediately related to  -TS  (one should deconvolute it somehow) .

     Since this issues keep generating confusion among users, I agree we 
should make an effort to clarify them.

     stefano

Paul Tangney wrote:

>Thanks Nicola,
>
>I think that answered my question.
>If you're right and what PWSCF calls "total energy"
>is E-TS, then "demet" is -TS and is incorrectly
>called the "correction for metals" in the code.
>As you say, the "correction" should be -TS/2.
>
>If F-D smearing is not a good computational device
>then definitely, I agree, the code should call this
>"Total Energy" "Free Energy" instead. Otherwise, if F-D
>is to be used purely as a smearing technique, the
>correct correction should be used...i.e. "demet"
>should be changed in the code from -TS to -TS/2.
>
>I'm interested in using F-D in a *physical* way. I want F = E - TS.
>So I guess I don't need to change the code at all.
>
>By the way, on the system that I'm studying (crystalline
>Tellurium), the application of a finite temperature actually
>*lowers* the free energy. Obviously this is unphysical....although
>perhaps not totally surprising, given that I'm using a finite
>(but dense : 10x10x10) k-point grid. Its still a little worrying though.
>
>Regards,
>
>Paul
>
>
>  
>