Temperature effect on elastic constant using VASP

Question

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)?

Answer 1

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.

Answer 2

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:

0. Construct a set of volumes to build an equation of state for a quasiharmonic approximation.
1. For each volume, compute the elastic constants at 0K, creating some relationship for your elastic constants, $c_{ij}(V)$.
2. Use the quasiharmonic approximation ($\min_V F(V,T)$) to find the equilibrium volume as a function of temperature, $V_{eq}(T)$.
3. 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. $c_{ij}(T) = c_{ij}(V=V_{eq}(T))$.