Efficient single-run implementation of generalized Einstein relation to compute transport coefficients: A binary-based time sampling.

A robust and simple implementation of the generalized Einstein formulation using single equilibrium molecular dynamics simulation is introduced to compute diffusion and shear viscosity. The unique features underlying this framework are as follows: (1) The use of a simple binary-based method to sample time-dependent transport coefficients results in a uniform distribution of data on a logarithmic time scale. Although we sample "on-the-fly," the algorithm is readily applicable for post-processing analysis. Overlapping same-length segments are not sampled as they indicate strong correlations. (2) Transport coefficients are estimated using a power law fitting function, a generalization of the standard linear relation, that accurately describes the long-time plateau. (3) The use of a generalized least squares (GLS) fitting estimator to explicitly consider correlations between fitted data points results in a reliable estimate of the statistical uncertainties in a single run. (4) The covariance matrix for the GLS method is estimated analytically using the Wiener process statistics and computed variances. (5) We provide a Python script to perform the fits and automate the procedure to determine the optimal fitting domain. The framework is applied to two fluids, binary hard sphere and a Lennard-Jones near the triple point, and the validity of the single-run estimates is verified against multiple independent runs. The approach should be applicable to other transport coefficients since the diffusive limit is universal to all of them. Given its rigor and simplicity, this methodology can be readily incorporated into standard molecular dynamics packages using on-the-fly or post-processing analysis.

Probabilistic computations of virial coefficients of polymeric structures described by rigid configurations of spherical particles: A fundamental extension of the ZENO program.

We describe an extension of the ZENO program for polymer and nanoparticle characterization that allows for precise calculation of the virial coefficients, with uncertainty estimates, of polymeric structures described by arbitrary rigid configurations of hard spheres. The probabilistic method of virial computation used for this extension employs a previously developed Mayer-sampling Monte Carlo method with overlap sampling that allows for a reduction of bias in the Monte Carlo averaging. This capability is an extension of ZENO in the sense that the existing program is also based on probabilistic sampling methods and involves the same input file formats describing polymer and nanoparticle structures. We illustrate the extension's capabilities, demonstrate its accuracy, and quantify the efficiency of this extension of ZENO by computing the second, third, and fourth virial coefficients and metrics quantifying the difficulty of their calculation, for model polymeric structures having several different shapes. We obtain good agreement with literature estimates available for some of the model structures considered.

Virial equation of state as a new frontier for computational chemistry.

The virial equation of state (VEOS) provides a rigorous bridge between molecular interactions and thermodynamic properties. The past decade has seen renewed interest in the VEOS due to advances in theory, algorithms, computing power, and quality of molecular models. Now, with the emergence of increasingly accurate first-principles computational chemistry methods, and machine-learning techniques to generate potential-energy surfaces from them, VEOS is poised to play a larger role in modeling and computing properties. Its scope of application is limited to where the density series converges, but this still admits a useful range of conditions and applications, and there is potential to expand this range further. Recent applications have shown that for simple molecules, VEOS can provide first-principles thermodynamic property data that are competitive in quality with experiment. Moreover, VEOS provides a focused and actionable test of molecular models and first-principles calculations via comparison to experiment. This Perspective presents an overview of recent advances and suggests areas of focus for further progress.

Coupled Monte Carlo and Molecular Dynamics Simulations on Interfacial Properties of Antifouling Polymer Membranes.

We use molecular simulation to study the wetting behavior of antifouling polymer-tethered membranes. We obtain the interfacial properties (e.g., contact angle) of water at various temperatures for five polymer membranes, including a base polysulfone (PSF) membrane and four other PSF membranes grafted with antifouling polymers (two poly(ethylene glycol) (PEG) tethers and two zwitterionic tethers). We implement a coupled Monte Carlo (MC)/molecular dynamics (MD) approach to determine the interface potentials of water on the membrane surfaces in an efficient manner. Within this method, short MC and MD simulations are performed in cycles to collect the surface excess free energy of a thin water film on polymer membrane surfaces. Simulation results show that the grafting of zwitterionic tethers provides a more significant enhancement in the hydrophilicity of the PSF membrane than that of the PEG tethers. Water completely wets the surface of zwitterionic polymer membranes.

Evaluation of Osmotic Virial Coefficients via Restricted Gibbs Ensemble Simulations, with Support from Gas-Phase Mixture Coefficients.

We present a method for computing osmotic virial coefficients in explicit solvent via simulation in a restricted Gibbs ensemble. Two equivalent phases are simulated at once, each in a separate box at constant volume and temperature and each in equilibrium with a solvent reservoir. For osmotic coefficient BN, a total of N solutes are individually exchanged back and forth between the boxes, and the average distribution of solute numbers between the boxes provides the key information needed to compute BN. Separately, expressions are developed for BN as a series in solvent reservoir density ρ1, with the coefficients of the series expressed in terms of the usual gas-phase mixture coefficients Bij. Normally, the Bij are defined for an infinite volume, but we suggest that the observed dependence of Bij on system size L can be used to estimate L dependence of the BN, allowing them to be computed accurately at L → ∞ while simulating much smaller system sizes than otherwise possible. The methods for N = 2 and 3 are demonstrated for two-component mixtures of size-asymmetric additive hard spheres. The proposed methods are demonstrated to have greater precision than established techniques, for a given amount of computational effort. The ρ1 series for BN when applied by itself is (for this noncondensing model) found to be the most efficient in computing accurate osmotic coefficients for the solvent densities considered here.

Cluster integrals and virial coefficients for realistic molecular models.

We present a concise, general, and efficient procedure for calculating the cluster integrals that relate thermodynamic virial coefficients to molecular interactions. The approach encompasses nonpairwise intermolecular potentials generated from quantum chemistry or other sources; a simple extension permits efficient evaluation of temperature and other derivatives of the virial coefficients. We demonstrate with a polarizable model of water. We argue that cluster-integral methods are a potent yet underutilized instrument for the development and application of first-principles molecular models and methods.

Implementation of harmonically mapped averaging in LAMMPS, and effect of potential truncation on anharmonic properties.

Implementation of the harmonically mapped averaging (HMA) framework in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is presented for on-the-fly computations of the energy, pressure, and heat capacity of crystalline systems during canonical molecular dynamics simulations. HMA has a low central processing unit and storage requirements and is straightforward to use. As a case study, the properties of the Lennard-Jones and embedded-atom model (parameterized for nickel) crystals are computed. The results demonstrate the higher efficiency of the new class compared to the inbuilt LAMMPS classes for calculating these properties. However, HMA loses its effectiveness in systems where diffusion occurs in the crystal, and an example is presented to allow this behavior to be recognized. In addition to its improved precision, HMA is less affected by small errors introduced by having a larger time step in molecular dynamics simulations. We also present an analysis of the effect of potential truncation on anharmonic properties, and show that artifacts of truncation on the HMA averages can be eliminated simply by shifting the potential energy to zero at the truncation radius. Full properties can be obtained by adding easily computed values for the lattice and harmonic properties using the untruncated potential.

A deep neural network model for packing density predictions and its application in the study of 1.5 million organic molecules.

The process of developing new compounds and materials is increasingly driven by computational modeling and simulation, which allow us to characterize candidates before pursuing them in the laboratory. One of the non-trivial properties of interest for organic materials is their packing in the bulk, which is highly dependent on their molecular structure. By controlling the latter, we can realize materials with a desired density (as well as other target properties). Molecular dynamics simulations are a popular and reasonably accurate way to compute the bulk density of molecules, however, since these calculations are computationally intensive, they are not a practically viable option for high-throughput screening studies that assess material candidates on a massive scale. In this work, we employ machine learning to develop a data-derived prediction model that is an alternative to physics-based simulations, and we utilize it for the hyperscreening of 1.5 million small organic molecules as well as to gain insights into the relationship between structural makeup and packing density. We also use this study to analyze the learning curve of the employed neural network approach and gain empirical data on the dependence of model performance and training data size, which will inform future investigations.

Combined temperature and density series for fluid-phase properties. II. Lennard-Jones spheres.

In Paper I [J. R. Elliott, A. J. Schultz, and D. A. Kofke, J. Chem. Phys. 143, 114110 (2015)] of this series, a methodology was presented for computing the coefficients of a power series of the Helmholtz energy in reciprocal temperature, ß, through density series based on cluster integral expansions. Previously, power series in ß were evaluated by thermodynamic perturbation theory (TPT) using molecular simulation of a reference fluid. The present methodology uses cluster integrals to evaluate coefficients of the density expansion at each individual order of temperature. While Paper I [J. R. Elliott, A. J. Schultz, and D. A. Kofke, J. Chem. Phys. 143, 114110 (2015)] developed this methodology for square well (SW) spheres, the present work extends the methodology to Lennard-Jones (LJ) spheres, where the reference fluid is the Weeks-Chandler-Andersen potential. Comparisons of TPT coefficients computed from cluster integrals to those from molecular simulation show good agreement through third order in ß when coefficients are expressed with effective approximants. Notably, the agreement for LJ spheres is much better than for SW spheres although fewer coefficients of the density series (B2-B5) are available than for SW spheres (B2-B6). The coefficients for Bi(ß) of the reference fluid are shown to follow a simple relationship to the virial coefficients of hard sphere fluids, corrected for the temperature dependency of the equivalent hard sphere diameter. This lays the foundation for a correlation of the second virial coefficient of LJ spheres B2(ß) that extrapolates to infinite order in temperature. This correlation of B2(ß) provides a basis for estimating the low density limit of TPT coefficients at all orders in temperature, facilitating a recursive extrapolation formula to estimate TPT coefficients of fourth order and higher over the entire density range. The applicability of the resulting equation of state is demonstrated by computing the thermodynamic properties for LJ spheres and comparing to standard simulation results.

Construction of the interface potential from a series of canonical ensemble simulations.

We introduce a method to construct the interface potential from a series of molecular dynamics simulations conducted within the canonical ensemble. The interface potential provides the surface excess free energy associated with the growth of a fluid film from a surface. We collect the force that the fluid exerts on the surface (disjoining pressure) at a series of film thicknesses. These force data are then integrated to obtain the interface potential. "Spreading" and "drying" versions of the general approach are considered. The spreading approach focuses on the growth of a thin liquid film from a solid substrate in a mother vapor. The drying approach focuses on the growth of a thin vapor film on a solid substrate in a mother liquid. The methods provide a means to compute the contact angle of a fluid droplet in contact with the surface. The general method is applied to two model systems: (1) a monatomic Lennard-Jones fluid in contact with atomistically detailed face centered cubic (FCC) substrate and (2) TIP4P/2005 water in contact with a rigid silica surface. For the Lennard-Jones model system, we generate results with both the drying and spreading methods at various temperatures and substrate strengths. These results are compared to those from previous simulation studies. For the water system, the drying method is used to obtain wetting properties over a range of temperatures. The water system also highlights challenges associated with application of the spreading method within the framework pursued here.

Application of the interface potential approach for studying wetting behavior within a molecular dynamics framework.

We introduce a means to implement the interface potential approach for computing wetting properties within a molecular dynamics framework. The general approach provides a means to determine the contact angle of a liquid droplet on a solid substrate in a mother vapor. We present a framework for implementing "spreading" and "drying" versions of the method within an isothermal-isobaric ensemble. Two free energy methods are considered: cumulative integration of average force profile and multistate Bennett acceptance ratio. An umbrella sampling strategy is used to restrain volume fluctuations and to ensure adequate sampling of a broad volume range. We explore implementation of the approach with the GROningen MAchine for Chemical Simulations and the Large-scale Atomic/Molecular Massively Parallel Simulator. We test the accuracy and efficiency of the method with models consisting of a monoatomic Lennard-Jones fluid in the vicinity of a structureless or atomistically detailed substrate. Our results show that one can successfully generate the drying potential within the framework pursued here. The efficiency of the method is strongly dependent upon how one handles the dynamics of the two confining walls. These decisions impact the rate of volume fluctuations, and therefore, the quality of the volume distributions collected. Our efforts to implement the spreading method with molecular dynamics alone proved unsuccessful. The rate at which the configuration space of the vapor phase evolves is insufficient. We show how one can overcome this challenge by implementing a coupled molecular dynamics/Monte Carlo approach. Finally, we show how one can determine the variation in interfacial properties with temperature and substrate strength.

Molecular Calculation of the Critical Parameters of Classical Helium.

We compute the vapor-liquid critical coordinates of a model of helium in which nuclear quantum effects are absent. We employ highly accurate ab initio pair and three-body potentials and calculate the critical parameters rigorously in two ways. First, we calculate the virial coefficients up to the seventh and find the point where an isotherm satisfies the critical conditions. Second, we use Gibbs Ensemble Monte Carlo (GEMC) to calculate the vapor-liquid equilibrium, and extrapolate the phase envelope to the critical point. Both methods yield results that are consistent within their uncertainties. The critical temperature of "classical helium" is 13.0 K (compared to 5.2 K for real helium), the critical pressure is 0.93 MPa, and the critical density is 28.4 mol·L-1, with expanded uncertainties (corresponding to a 95% confidence interval) on the order of 0.1 K, 0.02 MPa, and 0.5 mol·L-1, respectively. The effect of three-body interactions on the location of the critical point is small (lowering the critical temperature by roughly 0.1 K), suggesting that we are justified in ignoring four-body and higher interactions in our calculations. This work is motivated by the use of corresponding-states models for mixtures containing helium (such as some natural gases) at higher temperatures where quantum effects are expected to be negligible; in these situations, the distortion of the critical properties by quantum effects causes problems for the corresponding-states treatment.

Comprehensive high-precision high-accuracy equation of state and coexistence properties for classical Lennard-Jones crystals and low-temperature fluid phases.

We report equilibrium molecular simulation data for the classical Lennard-Jones (LJ) model, covering all thermodynamic states where the crystal is stable, as well as fluid states near coexistence with the crystal; both fcc and hcp polymorphs are considered. These data are used to compute coexistence lines and triple points for equilibrium among the fcc, hcp, and fluid phases. All results are obtained with very high accuracy and precision such that coexistence conditions are obtained with one to two significant figures more than previously reported. All properties are computed in the limit of an infinite cutoff radius of the LJ potential and in the limit of an infinite number of atoms; furthermore, the effect of vacancy defects on the free energy of the crystals is included. Data are fit to a semi-empirical equation of state to within their estimated precision, and convenient formulas for the thermodynamic and coexistence properties are provided. Of particular interest is the liquid-vapor-fcc triple point temperature, which we compute to be 0.694 55 ± 0.000 02 (in LJ units).

Best Practices for Quantification of Uncertainty and Sampling Quality in Molecular Simulations [Article v1.0].

The quantitative assessment of uncertainty and sampling quality is essential in molecular simulation. Many systems of interest are highly complex, often at the edge of current computational capabilities. Modelers must therefore analyze and communicate statistical uncertainties so that "consumers" of simulated data understand its significance and limitations. This article covers key analyses appropriate for trajectory data generated by conventional simulation methods such as molecular dynamics and (single Markov chain) Monte Carlo. It also provides guidance for analyzing some 'enhanced' sampling approaches. We do not discuss systematic errors arising, e.g., from inaccuracy in the chosen model or force field.

Effects of thermostatting in molecular dynamics on anharmonic properties of crystals: Application to fcc Al at high pressure and temperature.

The precision and accuracy of the anharmonic energy calculated in the canonical (NVT) ensemble using three different thermostats (viz., Andersen, Langevin, and Nosé-Hoover) along with no thermostat (i.e., microcanonical, NVE) are compared via application to aluminum crystals at ≈100 GPa for temperatures up to melting (4000 K) using ab initio molecular dynamics (AIMD) simulation. In addition to the role of the thermostat, the effect of using either conventional or the recently introduced harmonically mapped averaging (HMA) method is considered. The effect of AIMD time-step size Δt on the ensemble averages gauges accuracy, while for a given Δt, the stochastic uncertainty (computed using block averaging) provides the metric for precision. We identify the rate of convergence of block averages (with respect to block size) as an important issue in this context, as it imposes a minimum simulation length required to achieve reliable statistics, and it differs considerably among the methods. We observe that HMA with a Langevin thermostat in an NVT simulation shows the best performance, from the point of view of accuracy, precision, and simulation length. In addition, we introduce a novel HMA-based ensemble average for the temperature. In application to NVE simulations, the new formulation exhibits much smaller fluctuations compared to the conventional kinetic-energy approach; however, it provides only marginal improvement in uncertainty due to strong negative correlations exhibited by the conventional form (which acts to reduce its uncertainty but also slows convergence of the block averages).

No system-size anomalies in entropy of bcc iron at Earth's inner-core conditions.

New molecular modeling data show that the entropy of bcc iron exhibits no system-size anomalies, implying that it should be feasible to compute accurate free energies of this system using first-principles methods without requiring a prohibitively large number of atoms. Conclusions are based on rigorous calculations of size-dependent free energies for a Sutton-Chen model of iron previously fit to ab initio calculations, and refute statements recently appearing in the literature indicating that the size of the simulation cell is critical for stabilization of the bcc phase.

Virial Coefficients and Equations of State for Hard Polyhedron Fluids.

Hard polyhedra are a natural extension of the hard sphere model for simple fluids, but there is no general scheme for predicting the effect of shape on thermodynamic properties, even in moderate-density fluids. Only the second virial coefficient is known analytically for general convex shapes, so higher-order equations of state have been elusive. Here we investigate high-precision state functions in the fluid phase of 14 representative polyhedra with different assembly behaviors. We discuss historic efforts in analytically approximating virial coefficients up to B4 and numerically evaluating them to B8. Using virial coefficients as inputs, we show the convergence properties for four equations of state for hard convex bodies. In particular, the exponential approximant of Barlow et al. (J. Chem. Phys. 2012, 137, 204102) is found to be useful up to the first ordering transition for most polyhedra. The convergence behavior we explore can guide choices in expending additional resources for improved estimates. Fluids of arbitrary hard convex bodies are too complicated to be described in a general way at high densities, so the high-precision state data we provide can serve as a reference for future work in calculating state data or as a basis for thermodynamic integration.

Harmonically Assisted Methods for Computing the Free Energy of Classical Crystals by Molecular Simulation: A Comparative Study.

Four methods for calculation of the classical free energy of crystalline systems are compared with respect to their efficiency and accuracy. Two of the methods involve thermodynamic integration along an unphysical path (λ integration, λI), and two involve integration in temperature from the low-temperature harmonic limit (T integration, TI). Specifically, the methods considered are (1) Frenkel-Ladd integration from a noninteracting Einstein crystal reference (ECR-λI); (2) conventional integration in temperature (Conv-TI); (3) integration from an interacting quasi-harmonic reference (QHR-λI); and (4) temperature integration using harmonically mapped averaging to evaluate the integrand (HMA-TI). The latter two methods are "harmonically assisted", meaning that they exploit the harmonic nature of the crystal to greatly reduce fluctuations in the relevant averages. This feature allows them to deliver a result of much higher precision for a given computational effort, compared to ECR-λI and Conv-TI, and with no less accuracy. Regarding the harmonically assisted methods, HMA-TI has several advantages over QHR-λI with respect to the simplicity of the integration path (which promotes a more accurate result), ease of implementation, and usefulness of the data recorded along the integration path.

Calculation of high-order virial coefficients for the square-well potential.

Accurate virial coefficients B_{N}(λ,ɛ) (where ɛ is the well depth) for the three-dimensional square-well and square-step potentials are calculated for orders N=5-9 and well widths λ=1.1-2.0 using a very fast recursive method. The efficiency of the algorithm is enhanced significantly by exploiting permutation symmetry and by storing integrands for reuse during the calculation. For N=9 the storage requirements become sufficiently large that a parallel algorithm is developed. The methodology is general and is applicable to other discrete potentials. The computed coefficients are precise even near the critical temperature, and thus open up possibilities for analysis of criticality of the system, which is currently not accessible by any other means.