Short-range Pair Potentials Research Articles

Microscopic derivation of Vlasov–Dirac–Benney equation with short-range pair potentials

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with pair potentials of the form N3β−1ϕ(Nβx) for β∈[0,17] and ϕ∈ L∞(R3)∩L1(R3) . In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov–Dirac–Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov–Dirac–Benney equation from the microscopic N-particle dynamics with a strong short range force.

Improved cutoff functions for short-range potentials and the Wolf summation

ABSTRACT A class of radial, polynomial cutoff functions for short-ranged pair potentials or related expressions is proposed. Their derivatives up to order n and n + 1 vanish at the outer cutoff and an inner radius , respectively. Moreover, and . It is shown that the used order n can qualitatively affect results: stress and bulk moduli of ideal crystals are unavoidably discontinuous with density for n = 0 and n = 1, respectively. Systematic errors on energies and computing times decrease by 20–50% for Lennard-Jones with n = 2 or n = 3 compared to standard cutting procedures. Another cutoff function turns out beneficial to compute Coulomb interactions using the Wolf summation, which is shown to not properly converge when local charge neutrality is obeyed only in a stochastic sense. However, for all investigated homogeneous systems with thermal noise (ionic crystals and liquids), the modified Wolf summation, despite being infinitely differentiable at , converges similarly quickly as the original summation. Finally, it is discussed how to reduce the computational burden of numerically exact Monte Carlo simulations using the Wolf summation even when it does not properly converge.

Extended law of corresponding states: square-well oblates

The vapour–liquid coexistence collapse in the reduced temperature, T r = T/T c , reduced density, ρ r = ρ/ρ c , plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ = 1/3) with varying short-range square-well interactions collapse into a single master curve in the –ρ r plane, where , B 2 is the second virial coefficient, and v 0 is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Baseline Comparisons of Complementary Sampling Methods for Assembly Driven by Short-Ranged Pair Potentials toward Fast and Flexible Hybridization.

This work measures baseline sampling characteristics that highlight fundamental differences between sampling methods for assembly driven by short-ranged pair potentials. Such granular comparison is essential for fast, flexible, and accurate hybridization of complementary methods. Besides sampling speed, efficiency, and accuracy of uniform grid coverage, other sampling characteristics measured are (i) accuracy of covering narrow low energy regions that have low effective dimension (ii) ability to localize sampling to specific basins, and (iii) flexibility in sampling distributions. As a proof of concept, we compare a recently developed geometric methodology EASAL (Efficient Atlasing and Search of Assembly Landscapes) and the traditional Monte Carlo (MC) method for sampling the energy landscape of two assembling trans-membrane helices, driven by short-range pair potentials. By measuring the above-mentioned sampling characteristics, we demonstrate that EASAL provides localized and accurate coverage of crucial regions of the energy landscape of low effective dimension, under flexible sampling distributions, with much fewer samples and computational resources than MC sampling. EASAL's empirically validated theoretical guarantees permit credible extrapolation of these measurements and comparisons to arbitrary number and size of assembling units. Promising avenues for hybridizing the complementary advantages of the two methods are discussed.

Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity.

This Article describes a novel geometric methodology for analyzing free energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent and provides a proof-of-concept illustration of its unique capabilities. An atlas is a labeled partition of the assembly landscape into a roadmap of maximal, contiguous, nearly-equipotential-energy conformational regions or macrostates, together with their neighborhood relationships. The new methodology decouples the roadmap generation from sampling and produces: (1) a queryable atlas of local potential energy minima, their basin structure, energy barriers, and neighboring basins; (2) paths between a specified pair of basins, each path being a sequence of conformational regions or macrostates below a desired energy threshold; and (3) approximations of relative path lengths, basin volumes (configurational entropy), and path probabilities. Results demonstrating the core algorithm's capabilities and high computational efficiency have been generated by a resource-light, curated open source software implementation EASAL (Efficient Atlasing and Search of Assembly Landscapes, ACM Trans. Math. Softw. 2018 44, 1-48. 10.1145/3204472; see software, Efficient Atlasing and Search of Assembly Landscapes, 2016. https://bitbucket.org/geoplexity/easal; video, Video Illustrating the opensource software EASAL, 2016. https://cise.ufl.edu/~sitharam/EASALvideo.mpeg; and user guide, EASAL software user guide, 2016. https://bitbucket.org/geoplexity/easal/src/master/CompleteUserGuide.pdf). Running on a laptop with Intel(R) Core(TM) i7-7700@3.60 GHz CPU with 16GB of RAM, EASAL atlases several hundred thousand conformational regions or macrostates in minutes using a single compute core. Subsequent path and basin computations each take seconds. A parallelized EASAL version running on the same laptop with 4 cores gives a 3× speedup for atlas generation. The core algorithm's correctness, time complexity, and efficiency-accuracy trade-offs are formally guaranteed using modern distance geometry, geometric constraint systems and combinatorial rigidity. The methodology further links the shape of the input assembling units to a type of intuitive and queryable bar-code of the output atlas, which in turn determine stable assembled structures and kinetics. This succinct input-output relationship facilitates reverse analysis and control toward design. A novel feature that is crucial to both the high sampling efficiency and decoupling of roadmap generation from sampling is a recently developed theory of convex Cayley (distance-based) custom parametrizations specific to assembly, as opposed to folding. Representing microstates with macrostate-specific Cayley parameters, to generate microstate samples, avoids gradient-descent search used by all prevailing methods. Further, these parametrizations convexify conformational regions or macrostates. This ratchets up sampling efficiency, significantly reducing number of repeated and discarded samples. These features of the new stand-alone methodology can also be used to complement the strengths of prevailing methodologies including Molecular Dynamics, Monte Carlo, and Fast Fourier Transform based methods.

Generalized Moment Correction for Long-Ranged Electrostatics.

Describing long-ranged electrostatics using short-ranged pair potentials is appealing because the computational complexity scales linearly with the number of particles. The foundation of the approach presented here is to mimic the long-ranged medium response by cancelling electric multipoles within a small cutoff sphere. We propose a rigorous and formally exact new method that cancels up to infinitely many multipole moments and is free of operational damping parameters often required in existing theories. Using molecular dynamics simulations of water with and without added salt, we discuss radial distribution functions, Kirkwood–Buff integrals, dielectrics, diffusion coefficients, and angular correlations in relation to existing electrostatic models. We find that the proposed method is an efficient and accurate alternative for handling long-ranged electrostatics as compared to Ewald summation schemes. The methodology and proposed parameterization are applicable also for dipole–dipole interactions.

On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion

The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.

Existence of bound states of N-body problem in an optical lattice

We provide sufficient conditions to have at least one N-particle bound state below the essential spectrum of a large class of N-particle discrete Schrödinger operators associated with the Hamiltonian of (not necessarily identical) N particles, moving on a lattice and interacting via short-range pair potentials. We also describe the essential spectrum of

Structure Formation in 2D Assemblies Comprising Functional Tripod Molecules with Reduced Symmetry

Surface-confined self-assembly of organic building blocks is a versatile method of creation of low dimensional structures with potential applications in nanotechnology and material engineering. A key factor which often determines morphology of such 2D supramolecular assemblies is the geometry and functionality of a molecular tecton at play. In this contribution, we use theoretical modeling to predict the structure of adsorbed systems in which tripod molecules with reduced symmetry interact via a short-range anisotropic pair potential. To that end the self-assembly of a tripod molecule with one shortened/elongated arm, adsorbed on a triangular lattice, is modeled using the Monte Carlo simulation method. The probe asymmetric molecule consists of a few interconnected segments, and it is equipped with terminal interaction centers (active segments) with differently assigned interaction directions. Our study focuses on the effect of directionality of intermolecular interactions on the morphology of the resulting supramolecular 2D assemblies. It is demonstrated that a suitable encoding of the interaction directions allows for the creation of largely diversified adsorbed structures including aperiodic porous networks, molecular ladders, strings, ribbons and dispersed aggregates. Main differences and similarities between the findings reported for the asymmetric molecules and their C3-symmetric counterparts are also discussed. The obtained results can be helpful in designing new molecular tectons for controlled on-surface self-assembly and coupling reactions, as they provide useful information on the molecule–superstructure relation. This preliminary information can, for example, facilitate screening of molecular libraries to select an optimal molecule able to create low-dimensional structures with predefined properties.

Theory of model electrolyte solutions: Assessing the short- and long-ranged contributions by molecular simulations

In this work, we conduct a comprehensive simulation study on Helmholtz energies for non-primitive model electrolyte solutions in order to assess fluid theories. Simulation data is obtained with two different methods. Using a new thermodynamic integration path, we calculate Helmholtz energy contributions arising from damped, short-ranged electrostatic pair potentials. The division of the potential into a damped, short-ranged and a long-ranged part using a complementary error function corresponds to the division of a standard Ewald sum. Additionally, we applied a standard thermodynamic integration method to obtain the full electrostatic contribution to the Helmholtz energy. We evaluate the analytical long-range contribution developed by Rodgers and Weeks [J. M. Rodgers and J. D. Weeks, J. Chem. Phys. 131, 244108 (2009)] and find excellent agreement to our simulation data for any damping parameter α*≥1.5 for liquid-like densities. Further, the simulation data is compared to a third order perturbation theory we recently presented. By analyzing the theory against results of molecular simulations we were able to introduce refinements to our theory, such as a rearranged Padé approximation. We show, however, that third order perturbation theory, despite these improvements, is only reliable for strongly damped potentials (α*<1) for ion-dipole as well as for purely dipolar fluids. This study shows that a perturbation theory of third order is insufficient for such systems.

Colloidal clusters from a global optimization perspective

Modern microscopy techniques opened new perspectives to look at colloidal clusters. It is now possible to visualize the colloidal particles forming the cluster, which provides relevant experimental data that can be combined with detailed information resulting from structural optimization to understand phenomena at the mesoscale. From the theoretical view point, short-ranged Morse pair potentials have been successfully applied to model interactions between colloidal particles, while the development of efficient global optimization methods has allowed one to face the difficult task of searching for the geometry of the cluster that minimizes energy. In this work, we have carried out a thorough review concerning the global optimization methods proposed in the literature. The optimization of the title systems relies on the methods especially designed to deal with atomic clusters. Studies involving the global optimization of colloidal clusters were also surveyed.We employ the evolutionary algorithm developed in our group to search low-energy structures of colloidal clusters that are modeled with short-ranged Morse potentials. To the best of our knowledge, this is the first time that such kind of global optimization algorithms are applied to colloidal clusters. The method shows to be efficient for discovering the putative global minima in the cluster size-range N≤38. The results appear to indicate that, for colloidal clusters, significant structural changes due to the variation in the range of the interaction are unlikely to occur. Also the quasi-degeneracy of both global and local minima, that is usually present in these aggregates, is not raised up by modifying the composition of the cluster or decreasing the range of the interaction between the colloidal particles. In contrast, charged particles have shown a great influence on the stable structure of the clusters.

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrodinger operators depending on the quasimomentum of a two-particle system k ∈ \(\mathbb{T}^d \), where \(\mathbb{T}^d \) is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ℤd, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrodinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ \(\mathbb{T}^d \) if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrodinger operator H+(k), k ∈ \(\mathbb{T}^d \), corresponding to a two-particle system with repulsive interaction.

Bridging interactions of proteins with silica nanoparticles: the influence of pH, ionic strength and protein concentration.

Charge-driven bridging of nanoparticles by macromolecules represents a promising route for engineering functional structures, but the strong electrostatic interactions involved when using conventional polyelectrolytes impart irreversible complexation and ill-defined structures. Recently it was found that the electrostatic interaction of silica nanoparticles with small globular proteins leads to aggregate structures that can be controlled by pH. Here we study the combined influence of pH and electrolyte concentration on the bridging aggregation of silica nanoparticles with lysozyme in dilute aqueous dispersions. We find that protein binding to the silica particles is determined by pH irrespective of the ionic strength. The hetero-aggregate structures formed by the silica particles with the protein were studied by small-angle X-ray scattering (SAXS) and the structure factor data were analyzed on the basis of a short-range square-well attractive pair potential (close to the sticky-hard-sphere limit). It is found that the electrolyte concentration has a strong influence on the stickiness near pH 5, where the weakly charged silica particles are bridged by the strongly charged protein. An even stronger influence of the electrolyte is found in the vicinity of the isoelectric point of the protein (pI = 10.7) and is attributed to shielding of the repulsion between the highly charged silica particles and hydrophobic interactions between the bridging protein molecules.

Multiscale modelling of material failure using particles

In this paper, we explore a prototypical model of material fragmentation – the penetration of a plate by a fast moving ball in two dimensions using both smooth particle applied mechanics (SPAM) and molecular dynamics (MD). Using a family of short-ranged pair potentials, we investigate the fracture as a function of impact velocity and plate strength. The MD provides data from which we make a direct comparison with SPAM simulations of the same system. The parameters for the SPAM simulations: equation of state, bulk and shear moduli, yield strength and tensile strength are obtained from the pair potentials and via non-equilibrium MD (tension test). By careful parameterisation of the SPAM model, we are able to observe qualitatively similar behaviour to that shown in the MD simulations of a single crystal (brittle) specimen; bending at low penetrator speed, cracking at higher speeds and eventually, complete penetration. The behaviour depends on the plate strength, which is directly related to details of the force law. For plates characterised as high strength, the agreement between MD and SPAM is evidently poor. We discuss the possible causes for this discrepancy. Nevertheless, SPAM, when used in conjunction with accurate atomistic simulation, is shown to be a highly promising tool for studying failure in materials.

Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Numerical construction of the continuous spectrum Eigenfunctions of the three body Schrödinger operator: Three particles on the axis with short-range pair potentials

Based on a new method of the numerical construction of the three-body Schrodinger operator continuous spectrum eigenfunctions an analysis of the solutions of the problem of three identical particles on the axis with quickly decreasing repulsive pair potentials is offered. The initial problem is reduced to solving an inhomogeneous boundary problem for an elliptical partial differential equation in a twodimensional domain as a circle with radiation boundary conditions, with a ray approximation of the solution with diffraction corrections, contributing to a smoothness of a solution sought, being used. The approach offered allows a natural generalization for a case of slowly decreasing potentials of the Coulomb type and higher configuration space dimensions.

Coarse-Grained Model DNA with Explicit H-Bonding and Implicit Solvent

We present a coarse-grained model for DNA that is intended to function realistically at the level of individual bases. The model is composed of residues with up to eight coarse-grained beads each, which is sufficient for DNA-like base stacking and base-base recognition by hydrogen bonding. The beads interact by means of short-ranged pair potentials and a simple implicit solvent model. Movement is simulated by Brownian dynamics without hydrodynamic coupling. The main stabilizing forces are base stacking and hydrogen bonding, as modified by the effects of solvation. Complementary double-stranded chains of such residues form stable double helices over long runs (∼10 μs) at or near room temperature, with structural parameters close to those of B-form DNA. Most mismatched chains or mismatched regions within a complementary molecule melt and become disordered. Long-range fluctuations and elastic properties, as measured by bending and twisting persistence lengths, are close to experimental values. Single-stranded chains are flexible, with transient stretches of free bases in equilibrium with globules stabilized by intrastrand stacking and hydrogen bonding. Model DNAs in covalently closed loops form right or left-handed supercoils, depending on the sign of overtwist or undertwist. Short stem-loop structures melt at elevated temperatures and reanneal when the temperature is carefully lowered. Overall, most qualitative properties of real DNA arise naturally in the model from local interactions at the base pair level.

Coarse-Grained Model DNA: Structure, Sequences, Stems, Circles, Hairpins

A coarse-grained model for DNA that is intended to function realistically at the level of individual bases is reported. The model is composed of residues with up to eight coarse-grained beads each, which is sufficient for DNA-like base stacking and base-base recognition by hydrogen bonding. The beads interact by means of short-ranged pair potentials and a simple implicit solvent model. Movement is simulated by Brownian dynamics without hydrodynamic coupling. The main stabilizing forces are base stacking and hydrogen bonding, as modified by the effects of solvation. Complementary double-stranded chains of such residues form stable double helices over long runs (~10 μs) at or near room temperature, with structural parameters close to those of B-form DNA. Most mismatched chains or mismatched regions within a complementary molecule melt and become disordered. Long-range fluctuations and elastic properties, as measured by bending and twisting persistence lengths, are close to experimental values. Single-stranded chains are flexible, with transient stretches of free bases in equilibrium with globules stabilized by intrastrand stacking and hydrogen bonding. Model DNAs in covalently closed loops form right- or left-handed supercoils, depending on the sign of overtwist or undertwist. Short stem-loop structures melt at elevated temperatures and reanneal when the temperature is carefully lowered. Overall, most qualitative properties of real DNA arise naturally in the model from local interactions at the base-pair level.

Influence of polarizability on metal oxide properties studied by molecular dynamics simulations

We have studied the dependence of metal oxide properties in molecular dynamics (MD) simulations on the polarizability of oxygen ions. We present studies of both liquid and crystalline structures of silica (SiO2), magnesia (MgO) and alumina (Al2O3). For each of the three oxides, two separately optimized sets of force fields were used: (i) long-range Coulomb interactions between oxide and metal ions combined with a short-range pair potential; (ii) extension of force field (i) by adding polarizability to the oxygen ions. We show that while an effective potential of type (i) without polarizable oxygen ions can describe radial distributions and lattice constants reasonably well, potentials of type (ii) are required to obtain correct values for bond angles and the equation of state. The importance of polarizability for metal oxide properties decreases with increasing temperature.

Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape

We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic cluster of finite surface area and constant density, the latter being given by the density of atoms per unit volume in the triangular lattice. In the special case of the Heitmann–Radin sticky disc potential and exact ground states, we show that the macroscopic cluster has a (unique) Wulff shape. This is done by showing that the atomistic energy of crystalline configurations, after subtracting off a bulk part and re-scaling, Gamma-converges to a macroscopic anisotropic surface energy.