[Pw_forum] Bulk modulus under pressure?

bahadir bahadira at buffalo.edu
Mon Feb 27 18:20:04 CET 2012

thank you for the explanation,
it is really helpfull


On 02/27/2012 07:24 AM, Stefano Baroni wrote:
> Dear Bahdir:
>> i just tried a quick calculation for my curiosity on this.  i just 
>> optimized a fcc structure under 100GPa pressure also under 0GPa
>> i calculated bulk modulus for 0GPa system by changing equilibrium 
>> celldm(1) in range of -10% and +10% and calculatin energy. and used 
>> ev.x for fitting. fiiting curve is an U-shaped plot
> +/- 10% may be too large a variation to estimate the second derivative 
> by finite differences, but assuming that you know what you are doing, 
> I would say that the procedure is correct. No surprise that you find a 
> "U-shaped" curve, given that you calculate points on that curve near 
> the minimum (the "bottom of the U")
>> then i did the same procedure for 100GPa system by calculating energy 
>> for the celldm(1) values -10% to +10%. first thing is: alatt vs 
>> energy plot is not U-shaped.
> why should it? at such a large pressure, you would probably be rather 
> far from the minimum ...
>> and if i extend the range from +-10%
>> to +-20% a get an equilibrium energy but obviously it is the same as 
>> that of 0GPa's result.
>> that is the point that i am getting confused by.
>> and still curious about, what should i do to calculate a bulk modulus 
>> for a system under 100GPa?
> I maintain that you are probably fooled by names. If you want to 
> calculate the second derivative of the E(V) curve, just do it: the 
> second derivative has well defined value even far from the minimum, 
> where the curve is not "U-shaped". Just do what any text in numerical 
> analysis tells you to do in the chapter on numerical differentiation ...
> Let's now come to names. Strictly speaking, the bulk modulus is the 
> second derivative of the energy (or free energy) *as a function of 
> volume*. It is therefore a function of volume, not of pressure. Of 
> course, once you have the bulk modulus as a function of volume, 
> B=B(V), you can always use the equation of state, V=V(P), to formally 
> write it as a function of pressure, by a simple change of variable: 
> B(P) "=" B(V(P)). Note the quotation marks that mean: "the bulk 
> modulus at the volume corresponding to the pressure P". I think that 
> some algebraic gymnastics would allow you to obtain this quantity from 
> the second derivative of the enthalpy (which is the appropriate 
> thermodynamical potential that depends on pressure as an independent 
> variable). It all depends on what you need: if you need the second 
> derivative of "A" with respect to "B", then just calculate it, 
> whatever the name of that second derivative is ...
> HTH - SB
> ---
> Stefano Baroni - SISSA&DEMOCRITOS National Simulation Center - Trieste
> http://stefano.baroni.me <http://stefano.baroni.me/> [+39] 040 3787 
> 406 (tel) -528 (fax) / stefanobaroni (skype)
> La morale est une logique de l'action comme la logique est une morale 
> de la pensée - Jean Piaget

Dr.Bahadir Altintas

* Dept. of Chemistry
  SUNY Buffalo

* Abant Izzet Baysal University
   Dept. of Computer Education

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