Efficient Sampling of Atomic Configurational Spaces

Abstract

We describe a method to explore the configurational phase space of chemical systems. It is based on the nested sampling algorithm recently proposed by Skilling [1, 2] and allows us to explore the entire potential energy surface (PES) efficiently in an unbiased way. The algorithm has two parameters which directly control the trade-off between the resolution with which the space is explored and the computational cost. We demonstrate the use of nested sampling on Lennard-Jones (LJ) clusters. Nested sampling provides a straightforward approximation for the partition function, thus evaluating expectation values of arbitrary smooth operators at arbitrary temperatures becomes a simple postprocessing step. Access to absolute free energies allows us to determine the temperature-density phase diagram for LJ cluster stability. Even for relatively small clusters, the efficiency gain over parallel tempering in calculating the heat capacity is an order of magnitude or more. Furthermore, by analysing the topology of the resulting samples we are able to visualise the PES in a new and illuminating way. We identify a discretely valued order parameter with basins and supra-basins of the PES allowing a straightforward and unambiguous definition of macroscopic states of an atomistic system and the evaluation of the associated free energies.

Structural and electronic properties of Σ7 grain boundaries in α-Al2O3

“…However, that Si potential is much faster to compute than ours for Al 2 O 3 , and if we were to use a simpler classical potential for Al 2 O 3 we would sacrifice the demonstrated transferability of the present potential between crystal structures. While other approaches to structure searching are being developed, such as a genetic algorithm [18,32] and direct phase-space sampling methods [19,20], which may also also prove useful for grain boundary applications, ours can be implemented easily with 620 a standard molecular dynamics code. Our present approach has enabled us to discover and characterise many candidate structures for these boundaries, which would not have been found by previous methods of energy minimization such as [7,8,9,10,11,12].…”

“…Such boundaries have many more internal degrees of freedom, which cannot be explored efficiently using this direct geometry optimization approach. Other 130 methods of structure prediction have been introduced by a number of authors to overcome this problem, such as a genetic algorithm used to determine non-stoichiometric grain boundary structures in SrTiO 3 (001) [18], random structure searches [19], or nested sampling [20]. For the present cases we have chosen to adapt a simulated annealing method, similar to one used for silicon grain boundaries [21,…”

“…We are nevertheless motivated to study so-called 'special' boundaries, which have rather small interfacial unit cells, for two reasons, unrelated to the inevitable limitations of high-performance computers. One is the possibility of studying and interpreting their structure 20 and electronic properties experimentally with high resolution transmission electron microscopy (HRTEM) and electron energy loss spectroscopy (EELS), which can help validate theories and experiments. The second reason is that more complex boundaries can be built from the structural units and defects displayed by simpler boundaries.…”

Conformational ensembles and sampled energy landscapes: Analysis and comparison

“…Studying merges between energy basins then requires scanning the landscape upward with a horizontal plane, so as to discover when connected components of sublevel sets appear and merge. [55] In fact, all three methods, namely ours and those from [54] and [55] exploit the same construction from Morse theory, [51] which consists of studying the evolution of sub (or super-)level sets of a height function, discretized using a NNG. But our approach goes beyond the previous two in two ways.…”

Perturbation method to calculate the density of states

“…This, or a mathematically equivalent, route to the CPF has been exploited in numerical methods such as the reference system equilibration (RSE) method [2], the histogramming [3,4] and multihistogram [5,6] methods, the histogram reweighting method [7,8], the Wang-Landau (WL) sampling [9,10], multicanonical methods [11][12][13], transition matrix methods [14,15] or the nested sampling (NS) algorithm [16][17][18][19]. The CPF is directly related to the free energy by…”