Nested sampling for materials

Abstract

We review the materials science applications of the nested sampling (NS) method, which was originally conceived for calculating the evidence in Bayesian inference. We describe how NS can be adapted to sample the potential energy surface (PES) of atomistic systems, providing a straightforward approximation for the partition function and allowing the evaluation of thermodynamic variables at arbitrary temperatures. After an overview of the basic method, we describe a number of extensions, including using variable cells for constant pressure sampling, the semi-grand-canonical approach for multicomponent systems, parallelizing the algorithm, and visualizing the results. We cover the range of materials applications of NS from the past decade, from exploring the PES of Lennard–Jones clusters to that of multicomponent condensed phase systems. We highlight examples how the information gained via NS promotes the understanding of materials properties through a novel way of visualizing the PES, identifying thermodynamically relevant basins, and calculating the entire pressure–temperature(–composition) phase diagram. Graphic abstract

Stability of the high-density Jagla liquid in 2D: sensitivity to parameterisation

“…To evaluate the expected value of an observable at a given temperature, we can calculate its weighted average over all the generated samples: 33,46 Aðb; pÞ ¼ 1 Zðb; pÞ…”

“…To evaluate the expected value of an observable at a given temperature, we can calculate its weighted average over all the generated samples: 33,46 where A i is the value of observable A at the i -th iteration, H is the enthalpy, β = 1/( k B T ) is the thermodynamic temperature, p is the pressure and Z is the partition function calculated from NS the following way:where Γ i is the nested sampling weight associated with the phase space volume at a given enthalpy level, Γ i −( i +1) = [ K /( K + 1)] i − [ K /( K + 1)] i +1 . In our study we use this to characterise structures typical at a given temperature, e.g.…”

“…NS is a Bayesian inference method developed by Skilling 43,44 that has been adapted to sample the potential energy landscape of atomistic systems. 33,[45][46][47] NS is a ''top-down'' approach, starting from randomly generated configurations representing the high enthalpy region (or the high energy in case of the canonical ensemble) of the phase-space, propagating towards the global minimum through a series of iterative steps which shrink the available phase space by a constant fraction. During the iterative step, the highest enthalpy configuration is substituted with a new lower enthalpy one, generated from a randomly chosen existing configuration through a series of uniform changes to atomic positions and simulation cell dimensions.…”

“…46,[54][55][56] For a detailed review of materials application of NS we recommend ref. 33. NS calculations were performed using the pymatnest program package 57 and configurations were sampled by utilising single-atom MC moves, and changing the volume and shape of the simulation cell.…”

“…In the current work we employ the same sampling technique as in our previous study, nested sampling, 33 to compute the phase diagram of the Jagla model in two dimensions, where the LLCP is known to be in the stable liquid region of the phase diagram. Furthermore, we also explore the parameter space by varying the three potential parameters of the model (the slope of the linear ramps and the location of the potential minimum), in order to study their effect on the phase diagram.…”