Structure-resolved free energy estimation of the 38-atom Lennard Jones cluster via population annealing

Abstract

We systematically investigate the thermodynamic landscape of the 38-atom Lennard--Jones cluster LJ$_{38}$ using Population Annealing (PA), a method suited for systems with challenging double-funnel energy landscapes. By employing an adaptive temperature schedule, we demonstrate that thermodynamic observables, such as internal energy and heat capacity, converge robustly when the population size is sufficiently large. To gain deeper insights into the competing basins, we introduce an integrated framework that combines PA reweighting factors with structure-resolved analysis. Using quenched configurations characterized by potential energy and Steinhardt's bond-orientational order parameters, we identify three structural basins, FCC-like, icosahedral, and liquid-like, via dimensionality reduction and clustering. This framework enables the direct computation of structure-resolved free energy differences from population fractions, providing a quantitative mapping of the thermodynamic competition between the funnels. The resulting structural crossovers are consistent with the heat-capacity peak, demonstrating PA as a promising and scalable framework for structure-resolved thermodynamics in complex molecular systems.

Figures (8)

• Figure 1: Representative low-energy structures of LJ$_{38}$: the FCC truncated-octahedral global minimum (left) and a competing icosahedral local minimum (right). Despite having similar energies, the funnels are separated by a substantial free energy barrier, making LJ$_{38}$ a demanding test case for sampling algorithms.
• Figure 2: Schematic overview of the population annealing procedure (top to bottom). Walkers equilibrated at inverse temperature $\beta_i$ are reweighted to the next inverse temperature $\beta_{i+1}$, followed by a systematic resampling process to maintain a constant population size. Subsequently, short MCMC updates are applied to decorrelate the walkers and enhance configurational diversity at $\beta_{i+1}$. The bottom of the schematic presents the overlap between the energy distributions at $\beta_{i+1}$ and $\beta_{i+2}$, which determines the importance weights for the next step. Circle size indicates the relative weights, and the bell-shaped curves schematically represent the energy distributions at each temperature. The procedure is repeated along the annealing schedule.
• Figure 3: Reduced internal energy $U^*(T^*)$ as a function of temperature $T^*$ for various population sizes $R$. Data points represent averages over $10$ independent PA runs, with error bars indicating the standard error. The consistent overlap of the curves, despite the adaptive temperature schedule being determined independently for each $R$, demonstrates the robustness of the sampling procedure. The symbol at $T^*=0$ denotes the global minimum energy reported in literature $(\simeq -173.92)$wales1997global. Our finite-temperature estimates extrapolate linearly toward this ground-state value as $T^*\to 0$. The pronounced change in slope at $T^*\simeq 0.17$ reflects the structural transition of the LJ$_{38}$ cluster.
• Figure 4: Reduced heat capacity $C_v^*(T^*)(=C_v/k_{\mathrm B})$ estimated via PA for population sizes $R=1000$--$16000$. Error bars denote the standard error estimated from $10$ independent runs. The primary peak at $T^*\simeq 0.17$ signals the structural crossover of the LJ$_{38}$ cluster. The unstable feature observed at $T^*\simeq 0.08$ (most markedly for $R=4000$) is a numerical artifact arising from substantial inter-run fluctuations in the low-temperature regime.
• Figure 5: Heat map of squared loadings for the first three principal components (PC1--PC3). The PCA was performed on quenched configurations ($R=16000$), using a feature vector composed of the inherent-structure energy $U^*$ and the bond-orientational order parameters $Q_2$--$Q_{12}$ evaluated at the corresponding local minima. PC1, PC2, and PC3 explain 46.8%, 14.7%, and 10.9% of the total variance, respectively.