README.md

Stochastic sampling with TDEP: sTDEP

Briefly put, the idea is to generate force constants self-consistently by using them to approximate the atomic displacement distribution in the (harmonic) canonical ensemble, and iteratively improve the approximation by true forces in the system. Check out the documentation of canonical_configuration for some background. The scheme was first introduced in [Shulumba2017] and we refer to it as stochastic TDEP (sTDEP).

In the classical case and very simplified mathematical terms, we solve this equation self-consistently for the force constants $\Phi$:

$$ \begin{align} \langle V_2 \rangle &= \int {\rm d} R ~ {\rm e}^{- \beta V({\bf R})} V_2 ({\bf R}) \\ &{\color{red} \approx} \int {\rm d} R ~ {\rm e}^{- \beta {\color{red} V_2} ({\bf R}) } V_2 ({\bf R})~, \end{align} $$

where

$$ \begin{align} V_2({\bf R}) = \frac{1}{2} \sum_{ij} \Phi_{i \alpha, j \beta}({\bf R^0})U^{i \alpha} U^{j \beta}~, \end{align} $$

i.e., instead of sampling the true nuclear distribution as it would be obtained with MD simulations (at much higher cost!), we sample the approximate (effective) harmonic distribution and update the force constants self-consistently after each iteration until convergence. The solution will correspond to the (effective) harmonic model that best mimics the true thermodynamic behavior of the system at the given temperature (defined by $1/\beta = k_{\rm B} T$), and gives the best trial free energy similar to self-consistent phonon schemes.

Outline

• General sTDEP sampling tutorial
• Some practical examples:
  • Magnesium oxide at 0K using a neural network potential
  • Zirconium at 1300K: dynamical stabilization
• Most importantly: Convergence tutorials!

General comment on Convergence

You should always check the convergence of the property you are interested in! We cannot stress this enough!

On purely harmonic level, this can be the density of states (DOS). Compare the bandstructure and DOS at each step in the self-consistent loop. There is an explicit example in the MgO tutorial

Suggested reading

• N. Shulumba, O. Hellman, and A. J. Minnich, Phys. Rev. B 95, 014302 (2017)
• Appendix of N. Benshalom, G. Reuveni, R. Korobko, O. Yaffe, and O. Hellman, Phys Rev Mater 6, 033607 (2022)

Prerequisites

• TDEP is installed
• TDEP tools are installed
• gnuplot is installed