Nested Sampling for Exploring Lennard-Jones Clusters

TL;DR

The paper tackles the challenge of computing the partition function $Z(\beta)$ for Lennard-Jones clusters by applying nested sampling via the $nested\_fit$ program, enabling efficient exploration of the potential-energy surface and density of states. It derives a partition-function decomposition $Z(\beta)=Z_k Z_c$, with $Z_k$ and $Z_c$ computed through nested-sampling contributions $c_i$ and weights $w_i$, and investigates two slice-sampling implementations (transformed vs real space) alongside parallelization and covariance-cadence strategies. The authors demonstrate that 7-atom clusters recover evaporation and melting transitions, while 36-atom clusters reveal a low-temperature solid–solid transition when sufficiently many live points are used, highlighting the method's ability to resolve competing basins in finite systems. They show that real-space slice sampling and reduced-frequency covariance updates significantly reduce computational cost, and that parallelization on many cores yields substantial speedups, making it feasible to study larger and more complex clusters and paving the way for quantum extensions with greater computational demands.

Abstract

Lennard-Jones clusters, while an easy system, have a significant number of non equivalent configurations that increases rapidly with the number of atoms in the cluster. Here, we aim at determining the cluster partition function; we use the nested sampling algorithm, which transforms the multidimensional integral into a one-dimensional one, to perform this task. In particular, we use the nested_fit program, which implements slice sampling as search algorithm. We study here the 7-atom and 36-atom clusters to benchmark nested_fit for the exploration of potential energy surfaces. We find that nested_fit is able to recover phase transitions and find different stable configurations of the cluster. Furthermore, the implementation of the slice sampling algorithm has a clear impact on the computational cost.

Figures (5)

• Figure S1: (a) Heat capacity for the Lennard-Jones cluster with $7$ atoms. Comparison with the results in Ref. partay_efficient_2010. The configurations shown were obtained with XCrySDenkokalj_xcrysdennew_1999. (b) Evolution of slice sampling's acceptance ratio during one of the eight runs.
• Figure S2: Heat capacity for the Lennard-Jones cluster with $36$ atoms. The configurations shown were obtained with XCrySDenkokalj_xcrysdennew_1999. The full line is the mean obtained from the eight runs and the dashed curves correspond to the first standard deviation.
• Figure S3: Profiling for slice sampling transformed with the calculation of the covariance matrix at each iteration
• Figure S4: Profiling for slice sampling transformed with the calculation of the covariance matrix every $0.05K$ iterations
• Figure S5: Profiling for slice sampling real with the calculation of the covariance matrix every $0.05K$ iterations