Free Energy Calculation of Crystalline Solids Using Normalizing Flow

Normalizing flow [1] is an invertible function, parameterized by artificial neural networks, that transforms a simple base probability distribution into a complicated targeted probability distribution function. Recently proposed Boltzmann generator [2] makes use of the normalizing flow in order to generate equilibrium atomic configuration from the canonical (NVT) ensemble. In this work, we revisit the Boltzmann generator and incorporate it into the framework of the free energy perturbation methods to determine the accurate free energy of a system of interest. Specifically, the normalizing flow is shown to transform a thermodynamic system of known free energy into another thermodynamic system of known energy that is close to the system of interest. The transformed thermodynamic system could then be used in the well-established free energy methods to find the free energy of the system of interest. We further stress the need to introduce additional potential energy to control the motion of the center of mass to mitigate the problems arising from the translation invariance of the underlying potential energy function. The training of the normalizing flow is unsupervised in that it requires no samples of the atomic configurations from the underlying equilibrium ensemble. We finally apply the normalizing flow framework to calculate the absolute Gibbs free energy of diamond cubic Si crystals and show a good agreement with the free energy values obtained from other independent free energy methods. We will conclude by discussing the challenges and approaches to applying the normalizing flow to the large atomic systems needed to model extended crystalline defects.<br/>References:<br/>[1] Ivan Kobyzev, Simon Prince, and Marcus Brubaker. Normalizing Flows: An Introduction and Review of Current Methods. IEEE Trans. Pattern Anal. Mach. Intell., pages 1–1, 2020.<br/>[2] Frank Noé, Simon Olsson, Jonas Köhler, and Hao Wu. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science, 365(6457):eaaw1147, 2019.