# [Pw_forum] Bulk modulus under pressure?

Mon Feb 27 18:20:04 CET 2012

```thank you for the explanation,

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
>

--

* Dept. of Chemistry
SUNY Buffalo
NY,USA

* Abant Izzet Baysal University
Dept. of Computer Education
Bolu,Turkey

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