# [Pw_forum] Bulk modulus under pressure?

Stefano Baroni baroni at sissa.it
Mon Feb 27 13:24:54 CET 2012

```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 [+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

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