Four-Body Nonadditive Potential Energy Surface and the Fourth Virial Coefficient of Helium.

The four-body nonadditive contribution to the energy of four helium atoms is calculated and fitted for all geometries for which the internuclear distances exceed a small minimum value. The interpolation uses an active learning approach based on Gaussian processes. Asymptotic functions are used to calculate the nonadditive energy when the four helium atoms form distinct subclusters. The resulting four-body potential is used to compute the fourth virial coefficient D(T) for helium, at temperatures from 10 to 2000 K, with a path-integral approach that fully accounts for quantum effects. The results are in reasonable agreement with the limited and scattered experimental data for D(T), but our calculated results have much smaller uncertainties.

Parallel Implementation of Nonadditive Gaussian Process Potentials for Monte Carlo Simulations.

A strategy is presented to implement Gaussian process potentials in molecular simulations through parallel programming. Attention is focused on the three-body nonadditive energy, though all algorithms extend straightforwardly to the additive energy. The method to distribute pairs and triplets between processes is general to all potentials. Results are presented for a simulation box of argon, including full box and atom displacement calculations, which are relevant to Monte Carlo simulation. Data on speed-up are presented for up to 120 processes across four nodes. A 4-fold speed-up is observed over five processes, extending to 20-fold over 40 processes and 30-fold over 120 processes.

Machine learning for non-additive intermolecular potentials: quantum chemistry to first-principles predictions.

Prediction of thermophysical properties from molecular principles requires accurate potential energy surfaces (PES). We present a widely-applicable method to produce first-principles PES from quantum chemistry calculations. Our approach accurately interpolates three-body non-additive interaction data, using the machine learning technique, Gaussian Processes (GP). The GP approach needs no bespoke modification when the number or type of molecules is changed. Our method produces highly accurate interpolation from significantly fewer training points than typical approaches, meaning ab initio calculations can be performed at higher accuracy. As an exemplar we compute the PES for all three-body cross interactions for CO2-Ar mixtures. From these we calculate the CO2-Ar virial coefficients up to 5th order. The resulting virial equation of state (EoS) is convergent for densities up to the critical density. Where convergent, the EoS makes accurate first-principles predictions for a range of thermophysical properties for CO2-Ar mixtures, including the compressibility factor, speed of sound and Joule-Thomson coefficient. Our method has great potential to make wide-ranging first-principles predictions for mixtures of comparably sized molecules. Such predictions can replace the need for expensive, laborious and repetitive experiments and inform the continuum models required for applications.

Gaussian process models of potential energy surfaces with boundary optimization.

A strategy is outlined to reduce the number of training points required to model intermolecular potentials using Gaussian processes, without reducing accuracy. An asymptotic function is used at a long range, and the crossover distance between this model and the Gaussian process is learnt from the training data. The results are presented for different implementations of this procedure, known as boundary optimization, across the following dimer systems: CO-Ne, HF-Ne, HF-Na+, CO2-Ne, and (CO2)2. The technique reduces the number of training points, at fixed accuracy, by up to ∼49%, compared to our previous work based on a sequential learning technique. The approach is readily transferable to other statistical methods of prediction or modeling problems.

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.

Simulation of the Raman spectroscopy of multi-layered carbon nanomaterials.

Multi-layered carbon nanomaterials can have an important role in modern nanotechnology. Raman spectroscopy is a widely used analytical technique that is used to characterise the structure of these materials. In this work, an approach based upon an empirical potential for the simulation of the Raman spectroscopy of carbon nanomaterials [P. M. Tailor, R. J. Wheatley and N. A. Besley, Carbon, 2017, 113, 299-308] is extended through the addition of a term to describe the van der Waals interaction between layers of sp2 hybridised carbons. The resulting model accurately describes the properties of the shearing modes of few layer graphene and is used to characterise the low frequency modes of multi-walled carbon nanotubes and carbon nanofibres.

Active learning in Gaussian process interpolation of potential energy surfaces.

Three active learning schemes are used to generate training data for Gaussian process interpolation of intermolecular potential energy surfaces. These schemes aim to achieve the lowest predictive error using the fewest points and therefore act as an alternative to the status quo methods involving grid-based sampling or space-filling designs like Latin hypercubes (LHC). Results are presented for three molecular systems: CO2-Ne, CO2-H2, and Ar3. For each system, two of the active learning schemes proposed notably outperform LHC designs of comparable size, and in two of the systems, produce an error value an order of magnitude lower than the one produced by the LHC method. The procedures can be used to select a subset of points from a large pre-existing data set, to select points to generate data de novo, or to supplement an existing data set to improve accuracy.

Interpolation of intermolecular potentials using Gaussian processes.

A procedure is proposed to produce intermolecular potential energy surfaces from limited data. The procedure involves generation of geometrical configurations using a Latin hypercube design, with a maximin criterion, based on inverse internuclear distances. Gaussian processes are used to interpolate the data, using over-specified inverse molecular distances as covariates, greatly improving the interpolation. Symmetric covariance functions are specified so that the interpolation surface obeys all relevant symmetries, reducing prediction errors. The interpolation scheme can be applied to many important molecular interactions with trivial modifications. Results are presented for three systems involving CO2, a system with a deep energy minimum (HF-HF), and a system with 48 symmetries (CH4-N2). In each case, the procedure accurately predicts an independent test set. Training this method with high-precision ab initio evaluations of the CO2-CO interaction enables a parameter-free, first-principles prediction of the CO2-CO cross virial coefficient that agrees very well with experiments.

Reverse energy partitioning-An efficient algorithm for computing the density of states, partition functions, and free energy of solids.

A robust and model free Monte Carlo simulation method is proposed to address the challenge in computing the classical density of states and partition function of solids. Starting from the minimum configurational energy, the algorithm partitions the entire energy range in the increasing energy direction ("upward") into subdivisions whose integrated density of states is known. When combined with the density of states computed from the "downward" energy partitioning approach [H. Do, J. D. Hirst, and R. J. Wheatley, J. Chem. Phys. 135, 174105 (2011)], the equilibrium thermodynamic properties can be evaluated at any temperature and in any phase. The method is illustrated in the context of the Lennard-Jones system and can readily be extended to other molecular systems and clusters for which the structures are known.

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.

Molecular simulation of the thermophysical properties and phase behaviour of impure CO2 relevant to CCS.

Impurities from the CCS chain can greatly influence the physical properties of CO2. This has important design, safety and cost implications for the compression, transport and storage of CO2. There is an urgent need to understand and predict the properties of impure CO2 to assist with CCS implementation. However, CCS presents demanding modelling requirements. A suitable model must both accurately and robustly predict CO2 phase behaviour over a wide range of temperatures and pressures, and maintain that predictive power for CO2 mixtures with numerous, mutually interacting chemical species. A promising technique to address this task is molecular simulation. It offers a molecular approach, with foundations in firmly established physical principles, along with the potential to predict the wide range of physical properties required for CCS. The quality of predictions from molecular simulation depends on accurate force-fields to describe the interactions between CO2 and other molecules. Unfortunately, there is currently no universally applicable method to obtain force-fields suitable for molecular simulation. In this paper we present two methods of obtaining force-fields: the first being semi-empirical and the second using ab initio quantum-chemical calculations. In the first approach we optimise the impurity force-field against measurements of the phase and pressure-volume behaviour of CO2 binary mixtures with N2, O2, Ar and H2. A gradient-free optimiser allows us to use the simulation itself as the underlying model. This leads to accurate and robust predictions under conditions relevant to CCS. In the second approach we use quantum-chemical calculations to produce ab initio evaluations of the interactions between CO2 and relevant impurities, taking N2 as an exemplar. We use a modest number of these calculations to train a machine-learning algorithm, known as a Gaussian process, to describe these data. The resulting model is then able to accurately predict a much broader set of ab initio force-field calculations at comparatively low numerical cost. Although our method is not yet ready to be implemented in a molecular simulation, we outline the necessary steps here. Such simulations have the potential to deliver first-principles simulation of the thermodynamic properties of impure CO2, without fitting to experimental data.

Density of States Partitioning Method for Calculating the Free Energy of Solids.

We propose a new simulation method, which combines a cage model and a density of states partitioning technique, to compute the free energy of an arbitrary solid. The excess free energy is separated into two contributions, noninteracting and interacting. The excess free energy of the noninteracting solid is computed by partitioning its geometrical configuration space with respect to the ideal gas. This quantity depends on the lattice type and the number of molecules. The excess free energy of the interacting solid, with respect to the noninteracting solid, is calculated using density of states partitioning and a cage model. The cage model is better than the cell model in that it has a smaller configuration space and better represents the equilibrium distribution of solid configurations. Since the partition function (and hence free energy) is obtained from the density of states, which is independent of the temperature, equilibrium thermodynamic properties at any condition can be obtained by varying the density. We illustrate our method in the context of the free energy of dry ice.

Calculation of high-order virial coefficients with applications to hard and soft spheres.

A virial expansion of fluid pressure in powers of the density can be used to calculate a wealth of thermodynamic information, but the Nth virial coefficient, which multiplies the Nth power of the density in the expansion, becomes rapidly more complicated with increasing N. This Letter shows that the Nth virial coefficient can be calculated using a method that scales exponentially with N in computer time and memory. This is orders of magnitude more efficient than any existing method for large N, and the method is simple and general. New results are presented for N = 11 and 12 for hard spheres, and N = 9 and 10 for soft spheres.

Calculation of partition functions and free energies of a binary mixture using the energy partitioning method: application to carbon dioxide and methane.

It is challenging to compute the partition function (Q) for systems with enormous configurational spaces, such as fluids. Recently, we developed a Monte Carlo technique (an energy partitioning method) for computing Q [ J. Chem. Phys. 2011 , 135 , 174105 ]. In this paper, we use this approach to compute the partition function of a binary fluid mixture (carbon dioxide + methane); this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T ln Q and the Gibbs free energy (G) via G = F + pV. We then utilize G to obtain the coexisting mole fraction curves. The chemical potential of each species is also obtained. At the vapor-liquid equilibrium condition, the chemical potential of methane significantly increases, while that of carbon dioxide slightly decreases, as the pressure increases along an isotherm. Since Q is obtained from the density of states, which is independent of the temperature, equilibrium thermodynamic properties at any condition can be obtained by varying the total composition and volume of the system. Our methodology can be adapted to explore the free energies of other binary mixtures in general and of those containing CO(2) in particular. Since the method gives access to the free energy and chemical potentials, it will be useful in many other applications.

Covalent bond orders and atomic anisotropies from iterated stockholder atoms.

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.

Rapid calculation of partition functions and free energies of fluids.

The partition function (Q) is a central quantity in statistical mechanics. All the thermodynamic properties can be derived from it. Here we show how the partition function of fluids can be calculated directly from simulations; this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T ln Q. In our approach, we divide the density of states, assigning half of the configurations found in a simulation to a high-energy partition and half to a low-energy partition. By recursively dividing the low-energy partition into halves, we map out the complete density of states for a continuous system. The result allows free energy to be calculated directly as a function of temperature. We illustrate our method in the context of the free energy of water.

Molecular simulation of the binary mixture of 1-1-1-2-tetrafluoroethane and carbon dioxide.

The refrigerant 1-1-1-2-tetrafluoroethane (R134a) is being phased out in Europe from 2011. This requires the adoption of alternatives, and the mixture of R134a with carbon dioxide (CO(2)) is a promising candidate. However, limited experimental data currently stymie evaluation of its performance in industrial applications. In this paper, we employ atomistic force fields and the configurational-bias Monte Carlo technique to study the vapour-liquid equilibrium of this mixture. We also characterize the microscopic structure of the mixture, which is not readily available from experiments. At 272 K and 11.55 bar, the average coordination number of the first solvation shell of R134a is 11 and that of CO(2) is eight. CO(2) does not alter the structure of R134a, but its structure is slightly changed, due to the presence of R134a. All pair interactions are sensitive to pressure and are more structured at lower pressure. CO(2) prefers to form clusters of two in the mixture and highly extended or percolating clusters are not found.

Intermolecular potential energy surface and second virial coefficients for the water-CO2 dimer.

A five-dimensional potential energy surface is calculated for the interaction of water and CO(2), using second-order Møller-Plesset perturbation theory and coupled-cluster theory with single, double, and perturbative triple excitations. The correlation energy component of the potential energy surface is corrected for basis set incompleteness. In agreement with previous studies, the most negative interaction energy is calculated for a structure with C(2v) symmetry, where the oxygen atom of water is close to the carbon atom of CO(2). Second virial coefficients for the water-CO(2) pair are calculated for a range of temperatures, and their uncertainties are estimated. The virial coefficients are shown to be in close agreement with the available experimental data.

First principles predictions of thermophysical properties of refrigerant mixtures.

We present pair potentials for fluorinated methanes and their dimers with CO(2) based on ab initio potential energy surfaces. These potentials reproduce the experimental second virial coefficients of the pure fluorinated methanes and their mixtures with CO(2) without adjustment. Ab initio calculations on trimers are used to model the effects of nonadditive dispersion and induction. Simulations using these potentials reproduce the experimental phase-coexistence properties of CH(3)F within 10% over a wide range of temperatures. The phase coexistence curve of the mixture of CH(2)F(2) and CO(2) is reproduced with an error in the mole fractions of both phases of less than 0.1. The potentials described here are based entirely on ab initio calculations, with no empirical fits to improve the agreement with experiment.