Harmonic Superposition Approximation Research Articles

Hydrogen bond network structures of protonated 2,2,2-trifluoroethanol/ethanol mixed clusters probed by infrared spectroscopy combined with a deep-learning structure sampling approach: the origin of the linear type network preference in protonated fluoroalcohol clusters.

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.

Tetrahedral Clusters Stabilized by Alloying.

A family of nanoclusters of tetrahedral symmetry is proposed. These clusters consist of symmetrically truncated tetrahedra with additional hexagonal islands on the four facets of the starting tetrahedron. The islands are placed in stacking fault positions. The geometric magic numbers of these clusters are derived. Global optimization searches within an atomistic potential model of Pt-Pd show that the tetrahedral structures can be stabilized for intermediate compositions of these nanoalloys, even when they are not the most stable structures of the elemental clusters. These results are also confirmed by density functional theory calculations for the magic sizes 59, 100, and 180. A thermodynamic analysis by the harmonic superposition approximation shows that Pt-Pd tetrahedral nanoalloys can be stable even above room temperature.

Structural transformations in Cu, Ag, and Au metal nanoclusters.

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.

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

AgCu nanoalloys are known to exhibit a size dependent transition to the chiral stacking from the anti-Mackay stacking. In the current work, the role of composition on the structural transitions between the anti-Mackay and the chiral stacking in AgCu nanoalloys has been studied computationally using a combination of global optimization searches and heating simulations. It is observed that in addition to the size, composition also affects the surface stacking in these nanoalloys. Further, results show that the nature of the recently reported thermally induced transition to the chiral stacking is strongly dependent on the Cu content in the AgCu nanoalloys. The amount of Cu in the nanoalloy affects both the transition temperature and the “sharpness” of the transition. By analyzing the frequencies of the vibrational modes calculated using the Harmonic Superposition Approximation (HSA), the thermal stability of the chiral stacking is attributed to the vibrational entropic contribution. Finally, the findings in the current work have been consolidated in the form of a size-composition map. By referring to this size-composition map, the preference for the chiral stacking can be understood for the nanoalloys belonging to the size window between the “magic” sized nanoalloys.

Conformational diversity in deprotonated water clusters and anharmonic infrared spectra

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.

Phase Separation in AgCu and AgNi Core–Shell Icosahedral Nanoparticles: A Harmonic Thermodynamics Study

AbstractThe temperature dependence of equilibrium chemical ordering in Ag‐rich AgCu and AgNi icosahedral nanoalloys is numerically studied within the harmonic superposition approximation to the partition function. The nanoalloys are modeled by an atomistic force field derived from the second‐moment approximation to the tight‐binding model. It is found that the equilibrium chemical ordering consists of phase‐separated arrangements of the core–shell type, at least up to room temperature for AgCu and up to 600 K for AgNi. This result is rationalized in terms of the concept of preferred nucleation sites for phase separation, such as the central site in the icosahedron.

Temperature Dependence of Stability of Copper Clusters

Abstract Previously we have determined the lowest-energy structures and full vibrational spectra of Cu N with N up to 150 atoms by means of an unbiased-structure-optimization approach combined with the analytical formulas of the embedded-atom total-energy method and the harmonic superposition approximation. Here, we extend the earlier T = 0 studies to T ≠ 0 and present the equilibrium thermodynamic properties of copper clusters with N from 2 to 150. Thereby, we calculate the Helmholtz free energy quantum mechanically and examine it as a function of temperature and cluster size. It is shown, that the impact of entropic effects on the stability of the copper clusters is rather small for temperatures less than 300 K, which is about the bulk Debye temperature. For larger temperatures, the influence of the vibrational entropy on the cluster stability becomes stronger so that with increasing temperature the stability of some clusters (most often low-symmetrical) is increased and that of others (most often high-symmetrical) is reduced. For the first time, we demonstrate the existence of new magic copper clusters at elevated temperatures. These have the sizes 80, 88, 104, and 125 and will be called the temperature-dependent magic clusters.

Superposition-Enhanced Estimation of Optimal Temperature Spacings for Parallel Tempering Simulations.

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.

Proton location in (CH3)3N-H+-(CH3OH)n: A theoretical and infrared spectroscopic study

The dependence of the preferential protonated site in (CH3)3N-H+-(CH3OH)n on the cluster size was investigated using theoretical calculations and infrared spectroscopy measurements. While simple estimation from the magnitude of proton affinity suggested that the excess proton prefers the methanol site in n⩾4, density functional theory calculations of the stabilization energy indicated the clear preference as protonation of the trimethylamine site, even for n=9. Infrared spectra of the clusters were observed for n=3–7. Spectral simulations were also performed using the quantum harmonic superposition approximation. The observed (CH3)3N-H+-(CH3OH)n spectra were well interpreted by simulations of the isomers with the protonated trimethylamine ion core. It was shown that both the proton affinity and the mutual solvation energy govern the preferential location of the excess proton in binary component clusters.

Comprehensive Analysis on the Structure and Proton Switch in H+(CH3OH)m(H2O)n (m + n = 5 and 6)

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.

A Hierarchical Approach to Study the Thermal Behavior of Protonated Water Clusters H(+)(H2O)n.

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.

Structure, stability, and infrared spectroscopy of (H2O)nNH4+ clusters: A theoretical study at zero and finite temperature

The combined effects of size and temperature on the stable structures of water clusters doped with one ammonium molecule have been investigated theoretically using an empirical potential and density-functional theory (DFT) calculations. Global optimization with Monte Carlo methods has been performed using an explicit intermolecular potential based on the Kozack-Jordan polarizable model. Putative lowest-energy structures based on this empirical potential are reported. Our results indicate a high propensity for the NH(4)(+) impurity to be fully solvated by water molecules. Clathratelike patterns are formed for clusters containing more than 11 molecules. Local reoptimizations of candidate structures carried out at the DFT level with the B3LYP hybrid functional and the 6-311++G(d,p) basis set confirm the general trends obtained with the intermolecular potential. However, some reorderings between isomers often due to zero-point energy corrections are found in small clusters, leading to stable geometries in agreement with other first-principles studies. Temperature effects have been assessed using a simple harmonic superposition approximation for selected cluster sizes and using dedicated Monte Carlo simulations for (H(2)O)(20)NH(4)(+). The clusters are found to melt near 200 K, and possibly isomerize already below 50 K. The free energy barrier for core/surface isomerization of the impurity in the 21-molecule cluster is estimated to be only a few kcal/mol at 150 K. The vibrational spectroscopic signatures of the clusters obtained from the electronic structure calculations show the usual four O-H stretching bands. As the cluster size increases, the double acceptor-single donor band near 3700 cm(-1) increasingly dominates over the three other bands. While we do not find conclusive evidence for a O-H stretching spectroscopic signature of the ammonium impurity to be in the core or at the surface in the 20-molecule cluster, a possible signature via the N-H stretching bands is suggested near 2800-2900 cm(-1). In the larger (H(2)O)(49)NH(4)(+) cluster, the impurity is slightly more stable at the surface.

Solid−Solid Structural Transformations in Lennard-Jones Clusters: Accurate Simulations versus the Harmonic Superposition Approximation

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.

Low-Temperature Structural Transitions: Circumventing the Broken-Ergodicity Problem

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.

Quantum Effects on the Caloric Curves of Lithium Clusters

A tight-binding quantum Hamiltonian and an empirical embedded-atom model (EAM) potential are used to get insight into the finite-temperature behavior of small Lin clusters, n = 8, 20, and 55. Exchange Monte Carlo simulations provide an extensive sampling of configuration space, including the putative global minimum and many relevant isomers. The heat capacities obtained from the classical simulations are corrected for low-temperature quantum delocalization using the Pitzer-Gwinn approximation. Alternatively, the caloric curves are estimated from the database of local minima using the quantum harmonic superposition approximation. While the two atomistic models predict qualitatively similar features, including some premelting effects in Li20 but none in Li55, strong variations are observed in the melting temperatures, the EAM potential giving unexpectedly low values.

Free energy landscape from path-sampling: application to the structural transition in LJ38

We introduce a path-sampling scheme that allows equilibrium state-ensemble averages to be computed by means of a biased distribution of non-equilibrium paths. This non-equilibrium method is applied to the case of the 38-atom Lennard-Jones atomic cluster, which has a double-funnel energy landscape. We calculate the free energy profile along the Q4 bond orientational order parameter. At high or moderate temperature the results obtained using the non-equilibrium approach are consistent with those obtained using conventional equilibrium methods, including parallel tempering and Wang-Landau Monte Carlo simulations. At lower temperatures, the non-equilibrium approach becomes more efficient in exploring the relevant inherent structures. In particular, the free energy agrees with the predictions of the harmonic superposition approximation.

Equilibrium thermodynamics from basin-sampling

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.

Equilibrium properties of clusters in the harmonic superposition approximation

The inherent structure approach, or superposition approach is used to calculate some physical and chemical properties of clusters in thermodynamic equilibrium, for both the classical and quantum regimes. In the harmonic approximation some simple analytical estimates are obtained for the average polarizability of metal clusters, and for the photoabsorption intensity of chromophore-doped van der Waals clusters. Comparison with Monte Carlo simulations reveals good agreement, which could be further improved using systematic corrections for anharmonic perturbations.

Quantum partition functions from classical distributions: Application to rare-gas clusters

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.

Calculation of thermodynamic properties of small Lennard-Jones clusters incorporating anharmonicity

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.