# How would one find a material's equilibrium structure at any specific temperature?

Let's take an example of a simple FCC material, a binary alloy. Above absolute zero K, the alloy will start to disorder and become an FCC solid solution. The alloy would develop defects as it disorders. What would the best strategy (workflow) to find this alloy's equilibrium structure at $$x$$ K be?

One workflow that I can think of:

1. Geometry optimization at $$0$$ K using DFT.
2. Lattice dynamics study of the alloy to get its volume (V) vs temperature (T) data, i.e. V = f(T).
3. Find V at $$x$$ K, $$V_x$$.
4. Create a DFT supercell of the alloy with $$V_x$$. Use this cell to find formation energies (thermal stability) and phonon bandstructures (dynamic stability) of various possible disordered structures.

When I think of MD calculations, I think of the time evolution of the system. What, then, does it mean to simulate temperature-based evolution of the system in MD or ab initio MD?

• +1. I suppose molecule dynamics could also be used to approximate the equilibrium structure at an arbitrary temperature. Sep 25 at 22:50
• You can use Molecular Dynamics (MD) to determine the equilibrium structure of the material at a given temperature. However, the difference between inter-atomic potentials used in MD and DFT could mean that you would either have to evaluate all required properties using MD or obtain the supercell from MD and proceed with the 4th step that you have mentioned (which probably might cause errors due to the difference in inter-atomic interactions). Use [icme.hpc.msstate.edu/mediawiki/index.php/… as an example.
– PBH
Sep 26 at 15:40
• @PBH, this makes me think of what really is the difference between DFT and MD when it comes to such calculations. Is it that DFT potentials are parameter-free, hence more universal? Also, the link in your comment doesn't open. Sep 27 at 6:13
• @HitanshuSachania, the above link is correct but I have mistakenly typed a "]" at the end of the link. I think that this post on Quora does a sufficient job in answering your question regarding the difference between DFT and MD.
– PBH
Sep 27 at 7:06
• This repeats some of the Quora link from @PBH, but just to clarify on DFT vs MD: DFT is a way to calculate energies/forces. MD is a way to use energies/forces. MD frequently uses molecular mechanics (MM), which is a different way to calculate energies/forces (much faster than DFT, but much more approximate and makes a lot of assumptions that DFT does not). People frequently incorrectly say "MD" (or "classical MD") when they mean MM. They also incorrectly refer to DFT+MD as "quantum dynamics," which is another thing entirely. (Two pet peeves!) Sep 27 at 14:20

You mentioned DFT, which is typically used to calculate the electronic energy at zero temperature. You can adjust the geometry until this electronic energy is lowest, but then you might ask why for a given pressure, if you increase the temperature, a substance's geometry changes, perhaps from an ordered lattice, to a disordered liquid, to a bunch of molecules that are effectively unbound from each other: Does the geometry of the substance stay exactly the same if you stay in the "solid" region of the above diagram, and move parallel to the horizontal axis, toward the "liquid" region? If so, does the geometry just spontaneously change drastically when you cross the line between "solid" and "liquid"? Or does the geometry (bond lengths and angles) of the solid lattice change a bit as we move away from the vertical axis along a perpendicular line?

One way you can study this is to do a molecular dynamics simulation, as I described in my answers to the following:

I'll give a simpler summary here. Instead of using PACKMOL, as I described in the above two answers, you have already suggested in step 1 of your proposed procedure, to do a DFT optimization at 0K, so this may be how you want to set the equilibrium geometry of your system of atoms. Then, you can simply use , or , or or any molecular dynamics software to propagate the system for a chosen pressure and temperature (the initial volume is already known, thanks to your input geometry, and the volume may change as the MD simulation progresses, but you expect this anyway, since you're wondering what the new geometry, including the new volume, will be). You can choose a based on the atoms you have in your system, and let the MD program find how Newton's 2nd law (F = ma) says the atoms should move. They will keep moving until hopefully an equilibrium is reached, which is now your geometry at the given temperature and pressure.

This can be made more accurate by not just choosing some predetermined forcefield, but by actually using an electronic structure method (like DFT) to calculate the forces between your atoms for their respective geometry at each step of the MD simulation. This is called or ab initio MD, and it can be very slow, so you may want to use CPMD as described in the answers by myself and Tyberius here: What are the types of ab initio Molecular Dynamics?. To explain dwhswenson's comment, I'll say that you can even take this further and do full quantum dynamics if you want even more accuracy, meaning that instead of using Newton's classical equation F = ma, you could literally solve the time-dependent Schroedinger equation $$\frac{d|\psi(t)\rangle}{dt} = -\frac{i }{\hbar}\hat{H}(t) |\psi(t)\rangle$$, but as I very recently explained in my answer to this: Deviation caused by using DFT in Non-Adiabatic Molecular Dynamics, that you would be limited to a very small system and considering that your "force" calculations from DFT (or even coupled-cluster or more more sophisticated methods) will be limited in accuracy for the types of problems for which this calculation would even be relevant.

Your best bet is therefore likely to do something quick in (for example) LAMMPS or GROMACS (using a predetermined forcefield) or to try for something a bit more accurate using CPMD (the method) with CP2K or CPMD (the software), for example.