[Pw_forum] a questoin about pressure and bulk modulus of system

[Stefano Baroni]

Dear Meysam,

your post seems the result of an intricate intertwining of thermodynamics, quantum mechanics, and condensed matter misunderstandings ... Let me try to sort them out.

1) In the Born-Oppenheimer approximation, which you are correctly referring too, you can just forget about electronic degrees of freedom, when dealing with the thermodynamics of the system. The chemical potential that may (or may not, see below) appear in your thermodynamics equation has nothing to do with the Fermi energy, in this case: it is the energy required to add or subtract one "particle" (atom, molecule) to/from your system. In fact, you should have one different chemical potential per atomic/molecular species.

2) When quoting any thermodynamic relation, you should always keep in mind which are the "natural" variables you are adopting. In standard treatments, the "natural" variables of the energy as an extensive quantity are the volume, the entropy, and the number of of particles: E=E(S,V,N) [see e.g. D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, New York (1987), Chapt. 1]. This being the case, the pressure is defined as p=-∂E/∂V - no other derivatives are involved in this definition.

2') The thermodynamic relation you are quoting looks very obscure to me. Where have you taken it from? Which "natural" variables would the energy depend on?

3) All the above is a complication of real life. In real-life DFT calculations (which, I presume, are what you are interested in) the number of atoms is fixed, and the temperature (and, hence, the entropy) is zero (unless you do molecular dynamics / Monte Carlo, which I understand is not the case). Hence the energy is a function of the volume alone (as well as of any other applied field, but this would open another thread), and the pressure is just the negative of the derivative of the energy with respect to the only extensive quantity it can depend on, i.e. the volume ... as simple as that!

4) The Fermi energy of any metal does depend on volume (think of the simple jellium model of simple metals), both at zero and at finite temperature, but, as said, this has nothing to do with the calculation of the pressure (but for the fact that when calculating the total energy of a metal as a function of volume, you should recalculate the Fermi energy for every volume you are considering).

Hope this clarifies/helps a bit

Stefano B.

On Oct 14, 2010, at 1:27 PM, meysam pazoki wrote:

> Dear PWSCF users
> 
> I have a queston about bulk modulus and pressure of a system.
> We know from thermodynamics that the energy of our electronic
> system is E=TS-PV+μ N.In the zero temrature(T=0) we have E=-PV+μN and  p=-∂E/∂v.In the limit of born openhimer approximation with freezed ions, we can calculate the pressure of our system(electrons+ions) by partial derrivative of energy with respect to volume of system.But in literature I see that calculate pressure from this term: p=-dE/dV and neglect the ∂E/∂μ term.Is it correct?
> In the zero temrature chemical potential is equal to fermi energy and We should expect that the fermi energy have  no changes by variation in volume of system.Is it correct for finite temratures,too?
> 
> Best Regards
> Meysam Pazoki
> SUT
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---
Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center - Trieste
http://stefano.baroni.me [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)

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