# Calculation of linear coefficient of thermal expansion using VASP-Phonopy for Hexagonal material

We use quasi-harmonic approximation using VASP-Phonopy. In whole calculation we only assume Free energy as function of volume but in case of HCP material which have different coefficient of linear expansion along $$a$$ and $$c$$ direction creates a scenario that free energy is function of both volume and c/a ratio. As 'phonopy' doen't have such features to include c/a ratio effect, hence any discussion will be highly appreciated?

• I have not used phonopy before, but you should be able to do this "by hand" by separately varying the a and c lattice parameters and doing a phonon calculation at each (a,c) pair. – ProfM Mar 27 at 20:37
• @ProfM Does that mean he must find the equilibrium c/a ratio for each expanded volume and then do the usual phonopy calculation for those phases? That would make the calculations for large disordered systems very exhaustive. Could there be a workaround? – Hitanshu Sachania Apr 3 at 0:29
• @HitanshuSachania in general the quasiharmonic approximation becomes increasingly difficult very rapidly with the number of degrees of freedom in the system, which can be up to six (three lengths and three angles). To do this properly, there is no way around doing a full minimization over all degrees of freedom. However, whenever there is more than one degree of freedom many people simplify the problem by simply treating the "volume" as the only degree of freedom to optimize. (1/2) – ProfM Apr 3 at 7:22
• This means fixing the volume of the calculation, and allowing all cell lengths and angles to vary. This assumes that DFT can correctly capture the various lengths and angles for a given volume, which is not generally true. I have not thoroughly tested how reliable this "fixed volume approximation" is, so cannot fully comment on its merits. I have seen plenty of people using it -- but I am not sure they have properly tested it either. It looks more like a "we cannot do better, so we do this" type of approach. (2/2) – ProfM Apr 3 at 7:26

## 1 Answer

You can minimize the free energy F(a,c) as a function of (a,c), which is simply high school math. An example is provided in the section III (A) of the Supplementary Material of Phys. Rev. B 100, 161101(R) (2019):

https://journals.aps.org/prb/supplemental/10.1103/PhysRevB.100.161101/supplemental.pdf