Chapter 4 - Global optimisation of gold-based nanoalloys: AuCo, AuCu, and AuRh

ABSTRACTInfrared spectroscopy has become a powerful tool for structural assignment of ionic compounds in the gas phase. Using the reactive OSS2 potential, the stable conformers of deprotonated water clusters (HO)OH− have been explored for n = 6–9 and n = 20 using different computational approaches, namely conventional molecular dynamics, replica-exchange molecular dynamics (REMD), and the harmonic superposition approximation (HSA). The stable structures generally exhibit a rather open character, the missing proton giving rise to either a well-localised hydroxide ion or, occasionally, to a bihydroxide anion with one proton shared between two oxygens. At finite temperature, the predictions of REMD and HSA agree reasonably well with one another. The anharmonic infrared absorption spectra obtained from the equilibrium samples of conformers were also determined from the three methods. At 100 K, the spectra resulting from conventional MD sampling are found to be possibly unreliable owing to isomerisation over times scales that exceed fluxional motion which we evaluate to be on the order of a few tens of picoseconds.

We consider systems undergoing very-low-temperature solid-solid transitions associated with minima of similar energy but different symmetry, and separated by a high potential barrier. In such cases the well-known "broken-ergodicity" problem is often difficult to overcome, even using the most advanced Monte Carlo (MC) techniques, including the replica exchange method (REM). The methodology that we develop in this paper is suitable for the above specified cases and is numerically accurate and efficient. It is based on a new MC move implemented within the REM framework, in which trial points are generated analytically using an auxiliary harmonic superposition system that mimics well the true system at low temperatures. Due to the new move, the low-temperature random walks are able to frequently switch the relevant potential energy funnels leading to an efficient sampling. Numerically accurate results are obtained for a number of Lennard-Jones clusters, including those that have so far been treated only by the harmonic superposition approximation (HSA). The latter is believed to provide good estimates for low-temperature equilibrium properties but is manifestly uncontrollable and is difficult to validate. The present results provide a good test for the HSA and demonstrate its reliability, particularly for estimation of the solid-solid transition temperatures in most cases considered.

We consider systems undergoing very-low-temperature solid-solid transitions, exhibiting the well-known "broken-ergodicity" problem that is often so severe that even the replica exchange method converges too slowly. We propose an improvement of the latter, which consists of coupling the lower-temperature random walks to analytically generated random walks corresponding to an auxiliary harmonic superposition system. Numerically accurate results are obtained for several Lennard-Jones clusters, which have so far been treated only by the harmonic superposition approximation.

On the nature of the structural transitions between anti-Mackay stacking, chiral stacking and their thermal stability in AgCu nanoalloys

Equilibrium properties of clusters in the harmonic superposition approximation

Effective parallel tempering simulations rely crucially on a properly chosen sequence of temperatures. While it is desirable to achieve a uniform exchange acceptance rate across neighboring replicas, finding a set of temperatures that achieves this end is often a difficult task, in particular for systems undergoing phase transitions. Here we present a method for determination of optimal replica spacings, which is based upon knowledge of local minima in the potential energy landscape. Working within the harmonic superposition approximation, we derive an analytic expression for the parallel tempering acceptance rate as a function of the replica temperatures. For a particular system and a given database of minima, we show how this expression can be used to determine optimal temperatures that achieve a desired uniform acceptance rate. We test our strategy for two atomic clusters that exhibit broken ergodicity, demonstrating that our method achieves uniform acceptance as well as significant efficiency gains.

The energy landscape of protonated water clusters H(+)(H2O)n is thoroughly explored at the first-principle level using a hierarchical search methodology. In particular, the distinct configurational isomers of OSS2 empirical potential for n = 5-9 are uncovered and archived systematically using an asynchronous genetic algorithm and are subsequently refined with first-principle calculations. Using the OSS2 model, quantitative agreements in the thermal properties between Monte Carlo and harmonic superposition approximation (HSA) highlighted the reliability of the latter approach for the study of small- to medium-sized protonated water clusters. From the large sets of collected isomers, finite temperature behavior of the clusters can be efficiently examined at first-principle accuracy with the use of HSA. From the results obtained, evidence of structural changes from single-ring to treelike (n = 5-7) and multi-ring to single-ring structures (n = 7-9) is observed, as expected for the empirical model. Finally, the relevance of these findings to recent experimental data is discussed.

While preferential hydrogen bond network structures of cold protonated alcohol clusters H+(ROH)n are generally switched from a linear type to a cyclic one at n = 4-5, those of protonated 2,2,2-trifluoroethanol (TFE) clusters maintain linear type structures at least in the size range of n = 3-7. To explore the origin of the strong linear type network preference of H+(TFE)n, infrared spectra of protonated mixed clusters H+(TFE)m(ethanol)n (m + n = 5) were measured. An efficient structure sampling technique using parallelized basin-hopping algorithms and deep-learning neural network potentials is developed to search for essential isomers of the mixed clusters. Vibrational simulations based on the harmonic superposition approximation were compared with the observed spectra to identify the major isomer component at each mixing ratio. It was found that the formation of the cyclic structure occurs only in n ≥ 3 of the mixed clusters, in which the proton solvating sites and the double acceptor site are occupied by ethanol. The crucial role of the stability of the double acceptor site in the cyclic structure formation is discussed.

We investigate the thermodynamic behavior of quantum many-body systems using several methods based on classical calculations. These approaches are compared for the melting of Lennard-Jones (LJ) clusters, where path-integral Monte Carlo (PIMC) results are also available. First, we examine two quasiclassical approaches where the classical potential is replaced by effective potentials accounting for quantum corrections of low order in ℏ. Of the Wigner–Kirkwood and Feynman–Hibbs effective potentials, only the latter is found to be in quantitative agreement with quantum simulations. However, both potentials fail to describe even qualitatively the low-temperature regime, where quantum effects are strong. Our second approach is based on the harmonic superposition approximation, but with explicit quantum oscillators. In its basic form, this approach is in good qualitative agreement with PIMC results, and becomes more accurate at low temperatures. By including anharmonic corrections in the form of temperature-dependent frequency shifts, the agreement between the quantum superposition and the PIMC results becomes quantitative for the caloric curve of neon clusters. The superposition method is then applied to larger clusters to study the influence of quantum delocalization on the melting and premelting of LJ19, LJ31, LJ38, and LJ55. The quantum character strongly affects the thermodynamics via changes in the ground state structure due to increasing zero-point energies. Finally, we focus on the lowest temperature range, and we estimate the Debye temperatures of argon clusters and their size variation. A strong sensitivity to the cluster structure is found, especially when many surface atoms reorganize as in the anti-Mackay/Mackay transition. In the large size regime, the Debye temperature smoothly rises to its bulk limit, but still depends slightly on the growth sequence considered.

We present a "basin-sampling" approach for calculation of the potential energy density of states for classical statistical models. It combines a Wang-Landau-type uniform sampling of local minima and a novel approach for approximating the relative contributions from local minima in terms of the volumes of basins of attraction. We have employed basin-sampling to study phase changes in atomic clusters modeled by the Lennard-Jones potential and for ionic clusters. The approach proves to be efficient for systems involving broken ergodicity and has allowed us to calculate converged heat capacity curves for systems that could previously only be treated using the harmonic superposition approximation. Benchmarks are also provided by comparison with parallel tempering and Wang-Landau simulations, where these proved feasible.

A method for calculating thermodynamic properties of clusters from knowledge of a sample of minima on the potential energy surface using a harmonic superposition approximation is extended to incorporate anharmonicity using Morse correction terms to the density of states. Anharmonicity parameters are found for different regions of the potential energy surface by fitting to simulation results using the short-time averaged temperature as an order parameter. The resulting analytical expression for the density of states can be used to calculate many thermodynamic properties in a variety of ensembles, which accurately reproduce simulation results. This method is illustrated for 13-atom and 55-atom Lennard-Jones clusters.

Theoretical and experimental methods were integrated to investigate the structures of H(+)(CH(3)OH)(m)(H(2)O)(n) clusters for m + n = 5 and 6. An effective theoretical approach is presented to search for extensive sets of structural isomers using an empirical model and substitution schemes. Stable isomers were then reoptimized by the B3LYP level of computations with the 6-31+G* basis set. Canonical averages of these structural isomers were analyzed by harmonic superposition approximation (HSA) to study their finite temperature behavior and enable quantitative comparisons with experimental results. Thermal energy is found to have a significant effect on the structure of these clusters. Our calculations show that cyclic isomers are preferred at low temperature, while linear and tree forms become more favorable at high temperature (>200 K). Furthermore, we found that proton can reside on both water and methanol ion cores and the proton switch is associated with morphology change. Experimental IR spectra in the free OH stretching region were also obtained and compared with calculated spectra.

This invited review attempts to draw together recent advances in the structural characterisation of clusters and our theoretical understanding of dynamics, especially coexistence phenomena. It is now possible to characterise the potential energy surface of a small cluster in great detail, both in terms of local minima and transition states. A selection of results is collected includingab initio calculations on main group ligated clusters and a wide variety of systems bound by model analytic potentials. Useful comparisons may be made between the rearrangement mechanisms supported by the various potential energy surfaces. Furthermore, knowledge of transition states enables us to explain the results of dynamical simulations in great detail, and make comparisons with thermodynamic models. For larger systems, however, the number of stationary points is daunting, yet progress is still possible in terms of the underlying potential energy surface using the harmonic superposition approximation.

Finite-temperature structures of Cu, Ag, and Au metal nanoclusters are calculated in the entire temperature range from 0K to melting using a computational methodology that we proposed recently [M. Settem et al., Nanoscale 14, 939 (2022)]. In this method, Harmonic Superposition Approximation (HSA) and Parallel Tempering Molecular Dynamics (PTMD) are combined in a complementary manner. HSA is accurate at low temperatures and fails at higher temperatures. PTMD, on the other hand, effectively samples the high temperature region and melts. This method is used to study the size- and system-dependent competition between various structural motifs of Cu, Ag, and Au nanoclusters in the size range 1-2nm. Results show that there are mainly three types of structural changes in metal nanoclusters, depending on whether a solid-solid transformation occurs. In the first type, the global minimum is the dominant motif in the entire temperature range. In contrast, when a solid-solid transformation occurs, the global minimum transforms either completely to a different motif or partially, resulting in the co-existence of multiple motifs. Finally, nanocluster structures are analyzed to highlight the system-specific differences across the three metals.

The multicanonical basin hopping (MUBH) method, which uses a multicanonical weight in the basin hopping (BH) Monte Carlo method, was found to be very efficient for global optimization of large-scale systems such as Lennard-Jones clusters containing more than 150 atoms. We have implemented an asynchronous parallel version of the MUBH method using the message passing interface (MPI) to take advantage of the full usage of multiprocessors in either a homogeneous or heterogeneous computational environment. Based on the intrinsic properties of the Monte Carlo method, this MPI implementation used the task parallelism to minimize interthread data communication. For a Co nanocluster consisting of N atoms, we have applied the asynchronous multicanonical basin hopping (AMUBH) method (for 181 < N < or = 200), together with BH (for 2 < or = N < 150) and MUBH (for 150 < or = N < or = 180), to search for the molecular configuration of the global energy minimum. AMUBH becomes the only practical computational scheme for locating the energy minimum within realistic computational time for a relatively large cluster.