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
