[Pw_forum] something about Pt.pw91-n-van.UPF

[Agostino Migliore]

Hello prof!

Quoting Nicola Marzari <marzari at MIT.EDU>:

>
>
> Dear Agostino,
>
>
> a couple of points - we are dealing with an atom, and for
> smearings small enough that occupations do not change as
> a function of temperature - fractional occupations that are
> there just because of degeneracies. This is a bit different
> from the ideal case of the free-electron metal - so for an atom
> as a function of temperature the total energy E and the entropy
> S do not change (provided the temperature is smaller than the
> distance to the next  set of empty orbitals), E-TS changes
> only because of T changing, and S is not zero just because
> of degeneracy. So my previous post re the atom energies should
> still hold.
>
> Regarding the issue of taking (E+F)/2 (a good idea for
> a metal, not an atom), that was first introduced by Mike
> Gillan in 1989 (there is a JPcondmatt from then, I believe,
> and a later one in 1991 with Alessandro de Vita). That suggestion
> works only for the energy, but not for antyhing else (forces, stresses,
> etc...). The Methfessel-Paxton or Marzari-Vanderbilt smearings
> achieve the same goal of taking (E+F)/2 , but do that variationally
> (i.e. consistently for forces, stresses, etc...). A long discussion
> is in chap 4 of http://quasiamore.mit.edu/phd/ .
>
> 			nicola

Thanks for your remarks, which are interesting and allow me to specify
something about what I said before.

Indeed, in the paper I cited before, the limit for n in the relation
[F(T)+E(T)]/2=E(0)+O(T^n), with n>2,
(there demonstrated under rather general assumptions) has been pushed up
to n=4 (i.e., n even and n>2, so n=4), while it was thought to be n=3
before (e.g., both in M. J. Gillan, J. Phys.: Condens. Matter 1, 689
(1989) and F. Wagner, Th. Laloyaux, and M. Scheffler, Phys. Rev. B 57,
2102 (1998)). So I cited it as the last useful example. And clearly it
works only for energy.

I agree with you that in the specific case where some levels are
degenerate, in a range of smearing values where the occupations don't
change, the only change in the free energy associated to fractional
occupations is a linear one, depending on T through -TS. In fact, it is a
particular case where dS/dT=0 (for a suitable range of T) and dE/dT = T(dS
dT)=0 too. Obviously, things can change even with a small fractional
occupation of the higher lying levels or some departure from degeneracy of
the involved multiplet. So, I spoke in general, having in mind that even
in an atomic system you often have a multiplet and there is no exact
degeneracy. This was not in contrast with the specific case you were
referring to.

Best,
Agostino Migliore
CMM, Chemistry Department, UPenn
Philadelphia, PA