Thermodynamics and equilibrium structure of Ne38 cluster: Quantum mechanics versus classical

100 Years of the Lennard-Jones Potential

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Analysis and interpretation of first passage time distributions featuring rare events

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

Combining experiment and energy landscapes to explore anaerobic heme breakdown in multifunctional hemoproteins

To survive, many pathogens extract heme from their host organism and break down the porphyrin scaffold to sequester the Fe2+ ion via a heme oxygenase. Recent studies have revealed that certain pathogens can anaerobically degrade heme. Our own research has shown that one such pathway proceeds via NADH-dependent heme degradation, which has been identified in a family of hemoproteins from a range of bacteria. HemS, from Yersinia enterocolitica, is the main focus of this work, along with HmuS (Yersinia pestis), ChuS (Escherichia coli) and ShuS (Shigella dysenteriae). We combine experiments, Energy Landscape Theory, and a bioinformatic investigation to place these homologues within a wider phylogenetic context. A subset of these hemoproteins are known to bind certain DNA promoter regions, suggesting not only that they can catalytically degrade heme, but that they are also involved in transcriptional modulation responding to heme flux. Many of the bacterial species responsible for these hemoproteins (including those that produce HemS, ChuS and ShuS) are known to specifically target oxygen-depleted regions of the gastrointestinal tract. A deeper understanding of anaerobic heme breakdown processes exploited by these pathogens could therefore prove useful in the development of future strategies for disease prevention.

The Loss of Size Sensitivity in para-Hydrogen Clusters Due to the Strong Quantum Delocalization

para-Hydrogen (pH₂)_N clusters have been the focus of numerous computational studies. Originally motivated by the possibility of observing superfluidity, these studies also revealed rich and complex structural properties of (pH₂)_N. However, their structural analysis was typically limited to attempts to identify "magic number clusters" by computing their ground state energies E_N and the chemical potential μ_N = E_N-E_N-1 as a function of N. This was followed by structural analysis based on an ill-defined radial density profile. Surprisingly, however, there were remarkable discrepancies between the results reported in the literature for cluster sizes beyond approximately N = 25, and this ambiguity remained unsettled until now. In the present paper, we apply the diffusion Monte Carlo method to resolve inconsistencies in cluster sizes within the range (N = 24-28). Here, we try to avoid speculations based on the highly demanding energy calculations whose numerical accuracy harbors ambiguity. Instead, we focus on the direct and unambiguous structural analysis of the ground state wavefunctions, which supports the conclusion that the clusters are structurally the same in the size range considered. That is, there are no magic number clusters at least in the range N = 24-28, contrary to what some of the previous publications have suggested. This lack of size sensitivity of para-hydrogen clusters is a direct consequence of the strong quantum delocalization in these systems.

Rare events and first passage time statistics from the energy landscape

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Exact bound rovibrational spectra of the neon tetramer

Exact quantum dynamics calculations are performed for the bound rovibrational states of the neon tetramer (Ne₄) in its ground electronic state, using pair-wise Lennard-Jones potentials and the ScalIT suite of parallel codes. The vibrational states separate into a low-lying group mostly localized to a single potential well and a higher-energy delocalized group lying above the isomerization threshold-with such a structure serving as a testament to the "intermediate" quantum nature of the Ne₄ system. To accurately and efficiently represent both groups of states, the phase-space optimized discrete variable representation (PSO-DVR) approach was used, as implemented in the ScalIT code. The resultant 1D PSO effective potentials also shed significant light on the quantum dynamics of the system. All vibrational states were computed well up into the isomerization band and labeled up to the classical isomerization threshold-defined as the addition of the classical energy of a single bond, ϵ = 24.7 cm^-1, to the quantum ground state energy. Rovibrational energy levels for all total angular momentum values in the range J = 1-5 were also computed, treating all Coriolis coupling exactly.

Exploring Energy Landscapes

Recent advances in the potential energy landscapes approach are highlighted, including both theoretical and computational contributions. Treating the high dimensionality of molecular and condensed matter systems of contemporary interest is important for understanding how emergent properties are encoded in the landscape and for calculating these properties while faithfully representing barriers between different morphologies. The pathways characterized in full dimensionality, which are used to construct kinetic transition networks, may prove useful in guiding such calculations. The energy landscape perspective has also produced new procedures for structure prediction and analysis of thermodynamic properties. Basin-hopping global optimization, with alternative acceptance criteria and generalizations to multiple metric spaces, has been used to treat systems ranging from biomolecules to nanoalloy clusters and condensed matter. This review also illustrates how all this methodology, developed in the context of chemical physics, can be transferred to landscapes defined by cost functions associated with machine learning.

Decoding heat capacity features from the energy landscape

A general scheme is derived to connect transitions in configuration space with features in the heat capacity. A formulation in terms of occupation probabilities for local minima that define the potential energy landscape provides a quantitative description of how contributions arise from competition between different states. The theory does not rely on a structural interpretation for the local minima, so it is equally applicable to molecular energy landscapes and the landscapes defined by abstract functions. Applications are presented for low-temperature solid-solid transitions in atomic clusters, which involve just a few local minima with different morphologies, and for cluster melting, which is driven by the landscape entropy associated with the more numerous high-energy minima. Analyzing these features in terms of the balance between states with increasing and decreasing occupation probabilities provides a direct interpretation of the underlying transitions. This approach enables us to identify a qualitatively different transition that is caused by a single local minimum associated with an exceptionally large catchment volume in configuration space for a machine learning landscape.

Grand and Semigrand Canonical Basin-Hopping

We introduce grand and semigrand canonical global optimization approaches using basin-hopping with an acceptance criterion based on the local contribution of each potential energy minimum to the (semi)grand potential. The method is tested using local harmonic vibrational densities of states for atomic clusters as a function of temperature and chemical potential. The predicted global minima switch from dissociated states to clusters for larger values of the chemical potential and lower temperatures, in agreement with the predictions of a model fitted to heat capacity data for selected clusters. Semigrand canonical optimization allows us to identify particularly stable compositions in multicomponent nanoalloys as a function of increasing temperature, whereas the grand canonical potential can produce a useful survey of favorable structures as a byproduct of the global optimization search.

Mapping the phase diagram for neon to a quantum Lennard-Jones fluid using Gibbs ensemble simulations

In order to address the issue of whether neon liquid in coexistence with its gas phase can be mapped to a quantum Lennard-Jones (LJ) fluid, we perform a series of simulations using Gibbs ensemble Monte Carlo for a range of de Boer quantum parameters Λ=ℏ/(σ√(mε)). The quantum effects are incorporated by implementing the variational gaussian wavepacket method, which provides an efficient numerical framework for estimating the quantum density at thermal equilibrium. The computed data for the LJ liquid is used to produce its phase diagram as a function of the quantum parameter, 0.065 ≤ Λ ≤ 0.11. These data are then used to fit the experimental phase diagram for neon liquid. The resulting parameters, ε = 35.68 ± 0.03 K and σ = 2.7616 ± 0.0005 Å (Λ = 0.0940), of the LJ pair potential are optimized to best represent liquid neon in coexistence with its gas phase for a range of physically relevant temperatures. This multi-temperature approach towards fitting and assessing a pair-potential is much more consistent than merely fitting a single data point, such as a melting temperature or a second virial coefficient.

A rare event sampling method for diffusion Monte Carlo using smart darting

We identify a set of multidimensional potential energy surfaces sufficiently complex to cause both the classical parallel tempering and the guided or unguided diffusion Monte Carlo methods to converge too inefficiently for practical applications. The mathematical model is constructed as a linear combination of decoupled Double Wells [(DDW)(n)]. We show that the set (DDW)(n) provides a serious test for new methods aimed at addressing rare event sampling in stochastic simulations. Unlike the typical numerical tests used in these cases, the thermodynamics and the quantum dynamics for (DDW)(n) can be solved deterministically. We use the potential energy set (DDW)(n) to explore and identify methods that can enhance the diffusion Monte Carlo algorithm. We demonstrate that the smart darting method succeeds at reducing quasiergodicity for n ≫ 100 using just 1 × 10(6) moves in classical simulations (DDW)(n). Finally, we prove that smart darting, when incorporated into the regular or the guided diffusion Monte Carlo algorithm, drastically improves its convergence. The new method promises to significantly extend the range of systems computationally tractable by the diffusion Monte Carlo algorithm.

Free-energy landscapes from adaptively biased methods: application to quantum systems

Several parallel adaptive biasing methods are applied to the calculation of free-energy pathways along reaction coordinates, choosing as a difficult example the double-funnel landscape of the 38-atom Lennard-Jones cluster. In the case of classical statistics, the Wang-Landau and adaptively biased molecular-dynamics (ABMD) methods are both found efficient if multiple walkers and replication and deletion schemes are used. An extension of the ABMD technique to quantum systems, implemented through the path-integral MD framework, is presented and tested on Ne38 against the quantum superposition method.

The ground state estimation by global optimization of an effective potential. Application to binary para-H(2)/ortho-D(2) molecular clusters

It is demonstrated how the problem of ground state estimation of an n-body system can be recast as the less demanding problem of finding the global minimum of an effective potential in the 3n-dimensional coordinate space. The latter emerges when the solution of the imaginary-time Schrödinger equation is approximated by a variational Gaussian wavepacket (VGW). The VGW becomes stationary in the infinite-imaginary-time limit. Such a stationary solution is not only exact for a harmonic potential, but it also provides a good approximation for a quantum state that is still localized in one of the basins of attraction, when, for example, the harmonic approximation may fail. The landscape of the effective potential is favorable for its global optimization, and is particularly suitable for optimization by GMIN, an open source program designed for global optimization using the basin-hopping algorithm. Consequently, the methodology is applied within GMIN to estimate the ground state structures of several binary para-H(2)/ortho-D(2) molecular clusters. The results are generally consistent with the previous observations for homogeneous para-H(2) and ortho-D(2) clusters, as well as for smaller binary clusters.

Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests

Thawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wavepackets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wavepackets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

Quantum disordering versus melting in Lennard-Jones clusters

The ground states of Lennard-Jones clusters for sizes up to n=147 are estimated as a function of the de Boer quantum delocalization length Lambda , and the n-Lambda "phase diagram" is constructed. The increase in Lambda favors more disordered and diffuse structures over more symmetric and compact ones, eventually making the liquidlike motif most energetically favorable. The analogy between the quantum- and thermally-induced structural transitions is explored.

Quantum transitions in Lennard-Jones clusters

The ground states of Lennard-Jones (LJ) clusters are estimated by finding the Gaussian wave packets that minimize the energy functional. A "phase diagram" for LJ_{n} as a function of size (n=31,...,45) and de Boer quantum delocalization length (Lambdain[0;0.3]) is constructed, showing the stability ranges for the two competing structural motifs, the Mackay and anti-Mackay icosahedra. An increase of Lambda has an effect similar to heating and as such may induce structural transformations.

Thermodynamics of atomic clusters using variational quantum hydrodynamics

Small clusters of rare-gas atoms are ideal test cases for studying how quantum delocalization affects both the thermodynamics and the structure of molecular scale systems. In this paper, we use a variational quantum hydrodynamic approach to examine the structure and dynamics of (Ne)n clusters, with n up to 100 atoms, at both T = 0 K and for temperatures spanning the solid-to-liquid transition in bulk Ne. Finite temperature contributions are introduced to the grand potential in the form of an "entropy" potential. One surprising result is the prediction of a negative heat capacity for very small clusters that we attribute to the nonadditive nature of the total free-energy for very small systems.

Entropic effects in large-scale Monte Carlo simulations

The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Rényi divergence of order 1/2 . We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for nonvanishing acceptance probabilities in the limit of large dimension. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n(-1/2) for random-walk Monte Carlo and n(-1/6) for smart Monte Carlo (SMC). Numerical simulations of the Lennard-Jones 38 (LJ(38)) cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.

Rigid quantum Monte Carlo simulations of condensed molecular matter: Water clusters in the n=2→8 range

The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.

A new time evolving Gaussian series representation of the imaginary time propagator

Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] have recently demonstrated that a time evolving Gaussian approximation (TEGA) to the imaginary time propagator exp(-betaH) is useful for numerical computations of anharmonically coupled systems with many degrees of freedom. In this paper we derive a new exact series representation for the imaginary time propagator whose leading order term is the TEGA. One can thus use the TEGA not only as an approximation but also to obtain the exact imaginary time propagator. We also show how the TEGA may be generalized to provide a family of TEGA's. Finally, we find that the equations of motion governing the evolution of the center and width of the Gaussian may be thought of as introducing a quantum friction term to the classical evolution equations.

Multiple structural transformations in Lennard-Jones clusters: Generic versus size-specific behavior

The size-temperature "phase diagram" for Lennard-Jones clusters LJn with sizes up to n=147 is constructed based on the analysis of the heat capacities and orientational bond order parameter distributions computed by the exchange Monte Carlo method. Two distinct types of "phase transitions" accompanied by peaks in the heat capacities are proven to be generic. Clusters with Mackay atom packing in the overlayer undergo a lower-temperature melting (or Mackay-anti-Mackay) transition that occurs within the overlayer. All clusters undergo a higher-temperature transition, which for the three-layer clusters is proven to be the 55-atom-core-melting transition. For the two-layer clusters, the core/overlayer subdivision is ambiguous, so the higher-temperature transition is better characterized as the breaking of the local icosahedral coordination symmetry. A pronounced size-specific behavior can typically be observed at low temperatures and often occurs in clusters with highly symmetric global minima. An example of such behavior is LJ135, which undergoes a low-temperature solid-solid transition, besides the two generic transitions, i.e., the overlayer reconstruction and the core melting.