[Pw_forum] Re: Free energy w.r.t. T and QHA

[Eduardo Ariel Menendez P]

> >
> >My question is: Is this calculation (F as a function
> >of T)of any real significance? Since under QHA, free
> >energy F is a funciton of temperature T and volume V.
I don't know if this is the goal of fqha or if it is useful,
but thermodynamics books state that -dF/dT at constant volume is the
entropy.


> >In a real situation, the system volume would change
> >with temperature T, so does the phonon spectrum. Thus,
> >in calculation of F(T) curve, we need phonon spectrum
> >at each specific T which is sure different under QHA.
After you read the foundations of QHA in Laundau&Lihshits or in
Ashcroft&Mermin, consider the following method

1) For a certain volume you calculate the phonon DOS.
2) With phonon DOS you (fqha.f90) calculate F(V,T), i.e. the free energy
at this volume and at the temperature T that you want.
3) Repeat 1 and 2 for other volumes. You will have a different phonon DOS
   for each volume  (just do it and look). Note that here you have gone beyond the QHA,
  because you are calculating with DFT. The same would happen if you use classical
  interatomic potentials like Lennard-Jones or Buckingham. However, if you
  use only spring-like interatomic potentials, then you obtain the same
  phonon  DOS for any volume. In other words, if you use realistic
  potentials or DFT,
  the elastic constants (the second derivative of the energy funtion) depend on the volume.

4) If you did that, now you have a table of values of the function
   F(V,T). If you want the thermal expansion, consider that the pressure
   is P=-dF/dV at constant T.
   Let f(V)=F(V,T) for any particular T. Solving (numerically) P=-df/dV
   you get V for
   this temperature. Repeating this for different temperatures you get a
   table of the function V(T) for this particular pressure.

5) If you want the equation of state, just calculate P=-dF/dV for all the
   T, then  you have P(V,T).

It is laborious, maybe you can automate it.

Regards
Eduardo