Nuclear Quantum Effects in Water at the Triple Point: Using Theory as a Link Between Experiments

Quantum motion of oxygen and hydrogen in water: Atomic and total kinetic energy across melting from neutron scattering measurements

We provide a concurrent measurement of the hydrogen and oxygen nuclear kinetic energies in the water molecule across melting at 270 K in the solid phase and 276 K in the liquid phase. Experimental values are obtained by analyzing the neutron Compton profiles of each atomic species in a deep inelastic neutron scattering experiment. The concurrent measurement of the atom kinetic energy of both hydrogen and oxygen allows the estimate of the total kinetic energy per molecule due to the motion of nuclei, specifically 35.3 ± 0.8 and 34.8 ± 0.8 kJ/mol for the solid and liquid phases, respectively. Such a small difference supports results from ab initio simulations and phenomenological models from the literature on the mechanism of competing quantum effects across the phase change. Despite the experimental uncertainties, the results are consistent with the trend from state-of-the-art computer simulations, whereby the atom and molecule kinetic energies in the liquid phase would be slightly lower than in the solid phase. Moreover, the small change of nuclear kinetic energy across melting can be used to simplify the calculation of neutron-related environmental dose in complex locations, such as high altitude or polar neutron radiation research stations where liquid water and ice are both present: for neutron energies between hundreds of meV and tens of keV, the total scattering cross section per molecule in the two phases can be considered the same, with the macroscopic cross section only depending upon the density changes of water near the melting point.

A Many-Body Perspective of Nuclear Quantum Effects in Aqueous Clusters

Nuclear quantum effects play an important role in the structure and thermodynamics of aqueous systems. By performing a many-body expansion with nuclear-electronic orbital (NEO) theory, we show that proton quantization can give rise to significant energetic contributions for many-body interactions spanning several molecules in single-point energy calculations of water clusters. Although zero-point motion produces a large increase in energy at the one-body level, nuclear quantum effects serve to stabilize higher-order molecular interactions. These results are significant because they demonstrate that nuclear quantum effects play a nontrivial role in many-body interactions of aqueous systems. Our approach also provides a pathway for incorporating nuclear quantum effects into water potential energy surfaces. The NEO approach is advantageous for many-body expansion analyses because it includes nuclear quantum effects directly in the energies.

Comparing machine learning potentials for water: Kernel-based regression and Behler-Parrinello neural networks

In this paper, we investigate the performance of different machine learning potentials (MLPs) in predicting key thermodynamic properties of water using RPBE + D3. Specifically, we scrutinize kernel-based regression and high-dimensional neural networks trained on a highly accurate dataset consisting of about 1500 structures, as well as a smaller dataset, about half the size, obtained using only on-the-fly learning. This study reveals that despite minor differences between the MLPs, their agreement on observables such as the diffusion constant and pair-correlation functions is excellent, especially for the large training dataset. Variations in the predicted density isobars, albeit somewhat larger, are also acceptable, particularly given the errors inherent to approximate density functional theory. Overall, this study emphasizes the relevance of the database over the fitting method. Finally, this study underscores the limitations of root mean square errors and the need for comprehensive testing, advocating the use of multiple MLPs for enhanced certainty, particularly when simulating complex thermodynamic properties that may not be fully captured by simpler tests.

A streamlined molecular-dynamics workflow for computing solubilities of molecular and ionic crystals

Computing the solubility of crystals in a solvent using atomistic simulations is notoriously challenging due to the complexities and convergence issues associated with free-energy methods, as well as the slow equilibration in direct-coexistence simulations. This paper introduces a molecular-dynamics workflow that simplifies and robustly computes the solubility of molecular or ionic crystals. This method is considerably more straightforward than the state-of-the-art, as we have streamlined and optimised each step of the process. Specifically, we calculate the chemical potential of the crystal using the gas-phase molecule as a reference state, and employ the S0 method to determine the concentration dependence of the chemical potential of the solute. We use this workflow to predict the solubilities of sodium chloride in water, urea polymorphs in water, and paracetamol polymorphs in both water and ethanol. Our findings indicate that the predicted solubility is sensitive to the chosen potential energy surface. Furthermore, we note that the harmonic approximation often fails for both molecular crystals and gas molecules at or above room temperature, and that the assumption of an ideal solution becomes less valid for highly soluble substances.

Quadratic scaling bosonic path integral molecular dynamics

Bosonic exchange symmetry leads to fascinating quantum phenomena, from exciton condensation in quantum materials to the superfluidity of liquid 4He. Unfortunately, path integral molecular dynamics (PIMD) simulations of bosons are computationally prohibitive beyond ∼100 particles, due to a cubic scaling with the system size. We present an algorithm that reduces the complexity from cubic to quadratic, allowing the first simulations of thousands of bosons using PIMD. Our method is orders of magnitude faster, with a speedup that scales linearly with the number of particles and the number of imaginary time slices (beads). Simulations that would have otherwise taken decades can now be done in days. In practice, the new algorithm eliminates most of the added computational cost of including bosonic exchange effects, making them almost as accessible as PIMD simulations of distinguishable particles.

Coupled Cluster Molecular Dynamics of Condensed Phase Systems Enabled by Machine Learning Potentials: Liquid Water Benchmark

Coupled cluster theory is a general and systematic electronic structure method, but in particular the highly accurate "gold standard" coupled cluster singles, doubles and perturbative triples, CCSD(T), can only be applied to small systems. To overcome this limitation, we introduce a framework to transfer CCSD(T) accuracy of finite molecular clusters to extended condensed phase systems using a high-dimensional neural network potential. This approach, which is automated, allows one to perform high-quality coupled cluster molecular dynamics, CCMD, as we demonstrate for liquid water including nuclear quantum effects. The machine learning strategy is very efficient, generic, can be systematically improved, and is applicable to a variety of complex systems.

Thermodynamics of high-pressure ice phases explored with atomistic simulations

Most experimentally known high-pressure ice phases have a body-centred cubic (bcc) oxygen lattice. Our large-scale molecular-dynamics simulations with a machine-learning potential indicate that, amongst these bcc ice phases, ices VII, VII′ and X are the same thermodynamic phase under different conditions, whereas superionic ice VII″ has a first-order phase boundary with ice VII′. Moreover, at about 300 GPa, the transformation between ice X and the Pbcm phase has a sharp structural change but no apparent activation barrier, whilst at higher pressures the barrier gradually increases. Our study thus clarifies the phase behaviour of the high-pressure ices and reveals peculiar solid–solid transition mechanisms not known in other systems.

Many experimentally known high-pressure ice phase are structurally very similar. Here authors elucidate the phase behaviour of the high-pressure insulating ices and reveal solid-solid transition mechanisms not known in other systems.

Insights into lithium manganese oxide-water interfaces using machine learning potentials

Unraveling the atomistic and the electronic structure of solid-liquid interfaces is the key to the design of new materials for many important applications, from heterogeneous catalysis to battery technology. Density functional theory (DFT) calculations can, in principle, provide a reliable description of such interfaces, but the high computational costs severely restrict the accessible time and length scales. Here, we report machine learning-driven simulations of various interfaces between water and lithium manganese oxide (Li_xMn₂O₄), an important electrode material in lithium ion batteries and a catalyst for the oxygen evolution reaction. We employ a high-dimensional neural network potential to compute the energies and forces several orders of magnitude faster than DFT without loss in accuracy. In addition, a high-dimensional neural network for spin prediction is utilized to analyze the electronic structure of the manganese ions. Combining these methods, a series of interfaces is investigated by large-scale molecular dynamics. The simulations allow us to gain insights into a variety of properties, such as the dissociation of water molecules, proton transfer processes, and hydrogen bonds, as well as the geometric and electronic structure of the solid surfaces, including the manganese oxidation state distribution, Jahn-Teller distortions, and electron hopping.

Machine learning many-body potentials for colloidal systems

Simulations of colloidal suspensions consisting of mesoscopic particles and smaller species such as ions or depletants are computationally challenging as different length and time scales are involved. Here, we introduce a machine learning (ML) approach in which the degrees of freedom of the microscopic species are integrated out and the mesoscopic particles interact with effective many-body potentials, which we fit as a function of all colloid coordinates with a set of symmetry functions. We apply this approach to a colloid-polymer mixture. Remarkably, the ML potentials can be assumed to be effectively state-independent and can be used in direct-coexistence simulations. We show that our ML method reduces the computational cost by several orders of magnitude compared to a numerical evaluation and accurately describes the phase behavior and structure, even for state points where the effective potential is largely determined by many-body contributions.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Phase Diagram of a Deep Potential Water Model

Using the Deep Potential methodology, we construct a model that reproduces accurately the potential energy surface of the SCAN approximation of density functional theory for water, from low temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region. The computational efficiency of the model makes it possible to predict its phase diagram using molecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluid phases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predicted correctly, with the exception of ice III and XV that are stable in experiments, but metastable in the model. The evolution of the atomic dynamics upon heating, as ice VII transforms first into ice VII^{''} and then into an ionic fluid, reveals that molecular dissociation and breaking of the ice rules coexist with strong covalent fluctuations, explaining why only partial ionization was inferred in experiments.

Progress and challenges in ab initio simulations of quantum nuclei in weakly bonded systems

Atomistic simulations based on the first-principles of quantum mechanics are reaching unprecedented length scales. This progress is due to the growth in computational power allied with the development of new methodologies that allow the treatment of electrons and nuclei as quantum particles. In the realm of materials science, where the quest for desirable emergent properties relies increasingly on soft weakly bonded materials, such methods have become indispensable. In this Perspective, an overview of simulation methods that are applicable for large system sizes and that can capture the quantum nature of electrons and nuclei in the adiabatic approximation is given. In addition, the remaining challenges are discussed, especially regarding the inclusion of nuclear quantum effects (NQEs) beyond a harmonic or perturbative treatment, the impact of NQEs on electronic properties of weakly bonded systems, and how different first-principles potential energy surfaces can change the impact of NQEs on the atomic structure and dynamics of weakly bonded systems.

Dynamics & Spectroscopy with Neutrons-Recent Developments & Emerging Opportunities

This work provides an up-to-date overview of recent developments in neutron spectroscopic techniques and associated computational tools to interrogate the structural properties and dynamical behavior of complex and disordered materials, with a focus on those of a soft and polymeric nature. These have and continue to pave the way for new scientific opportunities simply thought unthinkable not so long ago, and have particularly benefited from advances in high-resolution, broadband techniques spanning energy transfers from the meV to the eV. Topical areas include the identification and robust assignment of low-energy modes underpinning functionality in soft solids and supramolecular frameworks, or the quantification in the laboratory of hitherto unexplored nuclear quantum effects dictating thermodynamic properties. In addition to novel classes of materials, we also discuss recent discoveries around water and its phase diagram, which continue to surprise us. All throughout, emphasis is placed on linking these ongoing and exciting experimental and computational developments to specific scientific questions in the context of the discovery of new materials for sustainable technologies.

Quantum-mechanical exploration of the phase diagram of water

The set of known stable phases of water may not be complete, and some of the phase boundaries between them are fuzzy. Starting from liquid water and a comprehensive set of 50 ice structures, we compute the phase diagram at three hybrid density-functional-theory levels of approximation, accounting for thermal and nuclear fluctuations as well as proton disorder. Such calculations are only made tractable because we combine machine-learning methods and advanced free-energy techniques. The computed phase diagram is in qualitative agreement with experiment, particularly at pressures ≲ 8000 bar, and the discrepancy in chemical potential is comparable with the subtle uncertainties introduced by proton disorder and the spread between the three hybrid functionals. None of the hypothetical ice phases considered is thermodynamically stable in our calculations, suggesting the completeness of the experimental water phase diagram in the region considered. Our work demonstrates the feasibility of predicting the phase diagram of a polymorphic system from first principles and provides a thermodynamic way of testing the limits of quantum-mechanical calculations.

Identification of synthesisable crystalline phases of water – a prototype for the challenges of computational materials design

We discuss the identification of experimentally realisable crystalline phases of water to outline and contextualise some of the diverse building blocks of a computational materials design process.

The effective isotropy of the hydrogen local potential in biphenyl and other hydrocarbons

We present an experimental investigation of the hydrogen nuclear momentum distribution in biphenyl using deep inelastic neutron scattering. Our experimental results suggest that the local potential affecting hydrogen is both harmonic and isotropic within experimental uncertainties. This feature is interpreted as a consequence of the central limit theorem, whereby the three-dimensional momentum distribution is expected to become a purely Gaussian function as the number of independent vibrational modes in a system increases. We also performed ab initio phonon calculations on biphenyl and other saturated hydrocarbons, from methane to decane. From the results of the simulations, one can observe that the nuclear momentum distribution becomes more isotropic as the number of atoms and normal modes in the molecule increases. Moreover, the predicted theoretical anisotropy in biphenyl is clearly larger than in the experiment. The reason is that the total number of normal modes necessary to reproduce the experimental results is much larger than the number of normal modes encompassed by a single unit cell due to the presence of structural disorder and intermolecular interactions in the real crystal, as well as coupling of different normal modes. Finally, experimental data were collected, over a subset of detectors on the VESUVIO spectrometer at ISIS, with a novel setup to increase the count rate and signal-to-background ratio. We envision that such an optimized experimental setup can provide faster measurements and more stringent constraints for phonon calculations.

Quantum kinetic energy and isotope fractionation in aqueous ionic solutions

At room temperature, the quantum contribution to the kinetic energy of a water molecule exceeds the classical contribution by an order of magnitude. The quantum kinetic energy (QKE) of a water molecule is modulated by its local chemical environment and leads to uneven partitioning of isotopes between different phases in thermal equilibrium, which would not occur if the nuclei behaved classically. In this work, we use ab initio path integral simulations to show that QKEs of the water molecules and the equilibrium isotope fractionation ratios of the oxygen and hydrogen isotopes are sensitive probes of the hydrogen bonding structures in aqueous ionic solutions. In particular, we demonstrate how the QKE of water molecules in path integral simulations can be decomposed into translational, rotational and vibrational degrees of freedom, and use them to determine the impact of solvation on different molecular motions. By analyzing the QKEs and isotope fractionation ratios, we show how the addition of the Na⁺, Cl^- and HPO₄^2- ions perturbs the competition between quantum effects in liquid water and impacts their local solvation structures.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH_4-xD_x/CH₄ and CH_5-xD_x ⁺/CH₅ ⁺, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

Optimization of detection strategies for epithermal neutron spectroscopy using photon-sensitive detectors

In this work, we discuss an improved detection procedure for the photon-sensitive yttrium-aluminum-perovskite detectors installed on the VESUVIO spectrometer at the ISIS pulsed neutron and muon source. By decreasing the low-level energy threshold of detected photons, we observe an increased count rate up to a factor ∼3, and a decrease of relative error bars and noise of ∼40% and 35%, respectively, for deep inelastic neutron scattering measurements. In addition, we demonstrate how the reported optimization may increase the accuracy in the line shape analysis of neutron Compton profiles, as well as in the application of the mean-force approach to detect the anisotropy and anharmonicity in the single-particle local potential. We envisage that such an upgrade of the detection procedure would have a substantial impact on the VESUVIO scientific programme based on deep inelastic neutron scattering investigations.

From Molecular Fragments to the Bulk: Development of a Neural Network Potential for MOF-5

The development of first-principles-quality reactive atomistic potentials for organic-inorganic hybrid materials is still a substantial challenge because of the very different physics of the atomic interactions-from covalent via ionic bonding to dispersion-that have to be described in an accurate and balanced way. In this work we used a prototypical metal-organic framework, MOF-5, as a benchmark case to investigate the applicability of high-dimensional neural network potentials (HDNNPs) to this class of materials. In HDNNPs, which belong to the class of machine learning potentials, the energy is constructed as a sum of environment-dependent atomic energy contributions. We demonstrate that by the use of this approach it is possible to obtain a high-quality potential for the periodic MOF-5 crystal using density functional theory (DFT) reference calculations of small molecular fragments only. The resulting HDNNP, which has a root-mean-square error (RMSE) of 1.6 meV/atom for the energies of molecular fragments not included in the training set, is able to provide the equilibrium lattice constant of the bulk MOF-5 structure with an error of about 0.1% relative to DFT, and also, the negative thermal expansion behavior is accurately predicted. The total energy RMSE of periodic structures that are completely absent in the training set is about 6.5 meV/atom, with errors on the order of 2 meV/atom for energy differences. We show that in contrast to energy differences, achieving a high accuracy for total energies requires careful variation of the stoichiometries of the training structures to avoid energy offsets, as atomic energies are not physical observables. The forces, which have RMSEs of about 94 meV/ a₀ for the molecular fragments and 130 meV/ a₀ for bulk structures not included in the training set, are insensitive to such offsets. Therefore, forces, which are the relevant properties for molecular dynamics simulations, provide a realistic estimate of the accuracy of atomistic potentials.

Ab initio thermodynamics of liquid and solid water

Thermodynamic properties of liquid water as well as hexagonal (Ih) and cubic (Ic) ice are predicted based on density functional theory at the hybrid-functional level, rigorously taking into account quantum nuclear motion, anharmonic fluctuations, and proton disorder. This is made possible by combining advanced free-energy methods and state-of-the-art machine-learning techniques. The ab initio description leads to structural properties in excellent agreement with experiments and reliable estimates of the melting points of light and heavy water. We observe that nuclear-quantum effects contribute a crucial [Formula: see text] to the stability of ice Ih, making it more stable than ice Ic. Our computational approach is general and transferable, providing a comprehensive framework for quantitative predictions of ab initio thermodynamic properties using machine-learning potentials as an intermediate step.

First Principles Neural Network Potentials for Reactive Simulations of Large Molecular and Condensed Systems

Modern simulation techniques have reached a level of maturity which allows a wide range of problems in chemistry and materials science to be addressed. Unfortunately, the application of first principles methods with predictive power is still limited to rather small systems, and despite the rapid evolution of computer hardware no fundamental change in this situation can be expected. Consequently, the development of more efficient but equally reliable atomistic potentials to reach an atomic level understanding of complex systems has received considerable attention in recent years. A promising new development has been the introduction of machine learning (ML) methods to describe the atomic interactions. Once trained with electronic structure data, ML potentials can accelerate computer simulations by several orders of magnitude, while preserving quantum mechanical accuracy. This Review considers the methodology of an important class of ML potentials that employs artificial neural networks.

Direct Measurements of Quantum Kinetic Energy Tensor in Stable and Metastable Water near the Triple Point: An Experimental Benchmark

This study presents the first direct and quantitative measurement of the nuclear momentum distribution anisotropy and the quantum kinetic energy tensor in stable and metastable (supercooled) water near its triple point, using deep inelastic neutron scattering (DINS). From the experimental spectra, accurate line shapes of the hydrogen momentum distributions are derived using an anisotropic Gaussian and a model-independent framework. The experimental results, benchmarked with those obtained for the solid phase, provide the state of the art directional values of the hydrogen mean kinetic energy in metastable water. The determinations of the direction kinetic energies in the supercooled phase, provide accurate and quantitative measurements of these dynamical observables in metastable and stable phases, that is, key insight in the physical mechanisms of the hydrogen quantum state in both disordered and polycrystalline systems. The remarkable findings of this study establish novel insight into further expand the capacity and accuracy of DINS investigations of the nuclear quantum effects in water and represent reference experimental values for theoretical investigations.

Neural network molecular dynamics simulations of solid–liquid interfaces: water at low-index copper surfaces

Molecular dynamics simulation of the water–copper interface have been carried out using high-dimensional neural network potential based on density functional theory.