# Temperature effect on elastic constant using VASP

I want to incorporate the temperature effect on elastic constant (EC) of a system using VASP.
From literature, there are two ways which can be used to get the temperature effect on EC i) ab initio molecular dynamics ii) standard DFT with quasi harmonic approximation (as far as I know). Based on the second approach, standard DFT calculation with IBRION = 5 or 6 with ISIF = 3 tags can generate EC at 0 Kelvin for a particular volume. Again, volume vs. temperature data is one of the outputs of the phonon codes (QHA). But somehow I am not able to correlate this two quantities.

Can somebody please explain a detailed way for getting the temperature effect by considering both theories and code (VASP)?

• I assume from the keywords that you are interested in doing this using VASP, but it would be important to clarify it in the question. Dec 7, 2020 at 16:14
• You are very correct @ProfM. Dec 7, 2020 at 17:06

It is always better to write your own code to get elastic constants. For example , In case of cubic system, we need three types of distorsion to unit cell. Now elastic constant is simply slope of second order fit of energies at different value of distorsion (see different publications) normalized with Volume minima (volume corresponds to minimum energy) . To calculate elastic constants at higher temperature, we need two things

1. Free energy of crystal at different temperature( use phonopy)
2. Volume minima at higher temp( we can get from v-t)

Second order fit of free energy (at higher temp) normalized with volume minima of that temp will give elastic constants at higher temperature.

• "Second order fit of free energy (at higher temp) normalized with volume minima of that temp" -- Can you please elaborate a bit about this part? Thanks Dec 18, 2020 at 19:42
• Elastic constant is basically second order derivative of free energy, but it need to be normalized with equilibrium volume. Volume is function of temperature so you need to find equilibrium volume at that temp. Basically you need volume temp profile. Sorry for late reply Jan 14, 2021 at 17:34

The following paper is quite nice, in my opinion:

Shang et al., Temperature-dependent elastic stiffness constants of α- and θ-Al2O3 from first-principles calculations, J. Phys. Condens. Matter 22 (2010) 375403 doi:10.1088/0953-8984/22/37/375403

The key concept is outlined in Section 2.2, which is the same approach suggested by Pranav. To summarize:

1. Construct a set of volumes to build an equation of state for a quasiharmonic approximation.
2. For each volume, compute the elastic constants at 0K, creating some relationship for your elastic constants, $$c_{ij}(V)$$.
3. Use the quasiharmonic approximation ($$\min_V F(V,T)$$) to find the equilibrium volume as a function of temperature, $$V_{eq}(T)$$.
4. At each temperature of interest, you know the equilibrium volume at that temperature from (2) and you can obtain the elastic constants at that temperature using the calculations from (1), i.e. $$c_{ij}(T) = c_{ij}(V=V_{eq}(T))$$.