Assessment of Approximate Methods for Anharmonic Free Energies

Accurate nuclear quantum statistics on machine-learned classical effective potentials

The contribution of nuclear quantum effects (NQEs) to the properties of various hydrogen-bound systems, including biomolecules, is increasingly recognized. Despite the development of many acceleration techniques, the computational overhead of incorporating NQEs in complex systems is sizable, particularly at low temperatures. In this work, we leverage deep learning and multiscale coarse-graining techniques to mitigate the computational burden of path integral molecular dynamics (PIMD). In particular, we employ a machine-learned potential to accurately represent corrections to classical potentials, thereby significantly reducing the computational cost of simulating NQEs. We validate our approach using four distinct systems: Morse potential, Zundel cation, single water molecule, and bulk water. Our framework allows us to accurately compute position-dependent static properties, as demonstrated by the excellent agreement obtained between the machine-learned potential and computationally intensive PIMD calculations, even in the presence of strong NQEs. This approach opens the way to the development of transferable machine-learned potentials capable of accurately reproducing NQEs in a wide range of molecular systems.

Data-efficient fine-tuning of foundational models for first-principles quality sublimation enthalpies

Calculating sublimation enthalpies of molecular crystal polymorphs is relevant to a wide range of technological applications. However, predicting these quantities at first-principles accuracy - even with the aid of machine learning potentials - is a challenge that requires sub-kJ mol-1 accuracy in the potential energy surface and finite-temperature sampling. We present an accurate and data-efficient protocol for training machine learning interatomic potentials by fine-tuning the foundational MACE-MP-0 model and showcase its capabilities on sublimation enthalpies and physical properties of ice polymorphs. Our approach requires only a few tens of training structures to achieve sub-kJ mol-1 accuracy in the sublimation enthalpies and sub-1% error in densities at finite temperature and pressure. Exploiting this data efficiency, we perform preliminary NPT simulations of hexagonal ice at the random phase approximation level and demonstrate a good agreement with experiments. Our results show promise for finite-temperature modelling of molecular crystals with the accuracy of correlated electronic structure theory methods.

i-PI 3.0: A flexible and efficient framework for advanced atomistic simulations

Atomic-scale simulations have progressed tremendously over the past decade, largely thanks to the availability of machine-learning interatomic potentials. These potentials combine the accuracy of electronic structure calculations with the ability to reach extensive length and time scales. The i-PI package facilitates integrating the latest developments in this field with advanced modeling techniques thanks to a modular software architecture based on inter-process communication through a socket interface. The choice of Python for implementation facilitates rapid prototyping but can add computational overhead. In this new release, we carefully benchmarked and optimized i-PI for several common simulation scenarios, making such overhead negligible when i-PI is used to model systems up to tens of thousands of atoms using widely adopted machine learning interatomic potentials, such as Behler-Parinello, DeePMD, and MACE neural networks. We also present the implementation of several new features, including an efficient algorithm to model bosonic and fermionic exchange, a framework for uncertainty quantification to be used in conjunction with machine-learning potentials, a communication infrastructure that allows for deeper integration with electronic-driven simulations, and an approach to simulate coupled photon-nuclear dynamics in optical or plasmonic cavities.

How Accurate Are Simulations and Experiments for the Lattice Energies of Molecular Crystals?

Molecular crystals play a central role in a wide range of scientific fields, including pharmaceuticals and organic semiconductor devices. However, they are challenging systems to model accurately with computational approaches because of a delicate interplay of intermolecular interactions such as hydrogen bonding and Van der Waals dispersion forces. Here, by exploiting recent algorithmic developments, we report the first set of diffusion Monte Carlo lattice energies for all 23 molecular crystals in the popular and widely used X23 dataset. Comparisons with previous state-of-the-art lattice energy predictions (on a subset of the dataset) and a careful analysis of experimental sublimation enthalpies reveals that high-accuracy computational methods are now at least as reliable as (computationally derived) experiments for the lattice energies of molecular crystals. Overall, this work demonstrates the feasibility of high-level explicitly correlated electronic structure methods for broad benchmarking studies in complex condensed phase systems, and signposts a route towards closer agreement between experiment and simulation.

Nuclear Quantum Effects on the Electronic Structure of Water and Ice

The electronic properties and optical response of ice and water are intricately shaped by their molecular structure, including the quantum mechanical nature of the hydrogen atoms. Despite numerous previous studies, a comprehensive understanding of the nuclear quantum effects (NQEs) on the electronic structure of water and ice at finite temperatures remains elusive. Here, we utilize molecular simulations that harness efficient machine-learning potentials and many-body perturbation theory to assess how NQEs impact the electronic bands of water and hexagonal ice. By comparing path-integral and classical simulations, we find that NQEs lead to a larger renormalization of the fundamental gap of ice, compared to that of water, ultimately yielding similar bandgaps in the two systems, consistent with experimental estimates. Our calculations suggest that the increased quantum mechanical delocalization of protons in ice, relative to water, is a key factor leading to the enhancement of NQEs on the electronic structure of ice.

Assessment of Anharmonicities in Clusters: Developing and Validating a Minimum-Information Partition Function

Precise thermodynamic calculations are essential for understanding the dynamics of cluster systems and new particle formation. However, the widely employed harmonic statistical mechanical approach often falls short in terms of accuracy. In this study, we present an improved statistical model that incorporates vibrational anharmonicity via a novel partition function that requires only one additional system-specific input parameter. In addition to considering vibrational aspects, we also account for anharmonicity related to the configurational space. The role of anharmonicities is thoroughly examined in the case of general clusters, where the complete sets of conformers, mechanically stable spatial arrangements, are known up to clusters composed of 14 monomers. By performing consistent Monte Carlo simulations on these systems, we benchmark the statistical model's efficacy in reproducing key thermodynamic properties (formation free energy and potential energy) in the classical limit. The model exhibits exceptional alignment with simulations, accurately reproducing free energies within a precision of 2kBT and reliably capturing cluster melting temperatures. Furthermore, we demonstrate the significance and applicability of the model by reproducing thermodynamic barriers in homogeneous gas-phase nucleation of larger clusters. The transferability of our developed approach extends to more complex molecular systems and bears relevance for atmospheric multicomponent clusters, in particular.

The effect of machine learning predicted anharmonic frequencies on thermodynamic properties of fluid hydrogen fluoride

Anharmonic effects play a crucial role in determining thermochemical properties of liquids and gases. For such extended phases, the inclusion of anharmonicity in reliable electronic structure methods is computationally extremely demanding, and hence, anharmonic effects are often lacking in thermochemical calculations. In this study, we apply the quantum cluster equilibrium method to transfer density functional theory calculations at the cluster level to the macroscopic, liquid, and gaseous phase of hydrogen fluoride. This allows us to include anharmonicity, either via vibrational self-consistent field calculations for smaller clusters or using a regression model for larger clusters. We obtain the structural composition of the fluid phases in terms of the population of different clusters as well as isobaric heat capacities as an example for thermodynamic properties. We study the role of anharmonicities for these analyses and observe that, in particular, the dominating structural motifs are rather sensitive to the anharmonicity in vibrational frequencies. The regression model proves to be a promising way to get access to anharmonic features, and the extension to more sophisticated machine-learning models is promising.

Quantum chemical modeling of hydrogen binding in metal-organic frameworks: validation, insight, predictions and challenges

A detailed chemical understanding of H2 interactions with binding sites in the nanoporous crystalline structure of metal-organic frameworks (MOFs) can lay a sound basis for the design of new sorbent materials. Computational quantum chemical calculations can aid in this quest. To set the stage, we review general thermodynamic considerations that control the usable storage capacity of a sorbent. We then discuss cluster modeling of H2 ligation at MOF binding sites using state-of-the-art density functional theory (DFT) calculations, and how the binding can be understood using energy decomposition analysis (EDA). Employing these tools, we illustrate the connections between the character of the MOF binding site and the associated adsorption thermodynamics using four experimentally characterized MOFs, highlighting the role of open metal sites (OMSs) in accessing binding strengths relevant to room temperature storage. The sorbents are MOF-5, with no open metal sites, Ni2(m-dobdc), containing Lewis acidic Ni(II) sites, Cu(I)-MFU-4l, containing π basic Cu(I) sites and V2Cl2.8(btdd), also containing π-basic V(II) sites. We next explore the potential for binding multiple H2 molecules at a single metal site, with thermodynamics useful for storage at ambient temperature; a materials design goal which has not yet been experimentally demonstrated. Computations on Ca2+ or Mg2+ bound to catecholate or Ca2+ bound to porphyrin show the potential for binding up to 4 H2; there is precedent for the inclusion of both catecholate and porphyrin motifs in MOFs. Turning to transition metals, we discuss the prediction that two H2 molecules can bind at V(II)-MFU-4l, a material that has been synthesized with solvent coordinated to the V(II) site. Additional calculations demonstrate binding three equivalents of hydrogen per OMS in Sc(I) or Ti(I)-exchanged MFU-4l. Overall, the results suggest promising prospects for experimentally realizing higher capacity hydrogen storage MOFs, if nontrivial synthetic and desolvation challenges can be overcome. Coupled with the unbounded chemical diversity of MOFs, there is ample scope for additional exploration and discovery.

Realistic phase diagram of water from "first principles" data-driven quantum simulations

Since the experimental characterization of the low-pressure region of water's phase diagram in the early 1900s, scientists have been on a quest to understand the thermodynamic stability of ice polymorphs on the molecular level. In this study, we demonstrate that combining the MB-pol data-driven many-body potential for water, which was rigorously derived from "first principles" and exhibits chemical accuracy, with advanced enhanced-sampling algorithms, which correctly describe the quantum nature of molecular motion and thermodynamic equilibria, enables computer simulations of water's phase diagram with an unprecedented level of realism. Besides providing fundamental insights into how enthalpic, entropic, and nuclear quantum effects shape the free-energy landscape of water, we demonstrate that recent progress in "first principles" data-driven simulations, which rigorously encode many-body molecular interactions, has opened the door to realistic computational studies of complex molecular systems, bridging the gap between experiments and simulations.

Chemical Modifications Suppress Anharmonic Effects in the Lattice Dynamics of Organic Semiconductors

The lattice dynamics of organic semiconductors has a significant role in determining their electronic and mechanical properties. A common technique to control these macroscopic properties is to chemically modify the molecular structure. These modifications are known to change the molecular packing, but their effect on the lattice dynamics is relatively unexplored. Therefore, we investigate how chemical modifications to a core [1]benzothieno[3,2-b]benzothiophene (BTBT) semiconducting crystal affect the evolution of the crystal structural dynamics with temperature. Our study combines temperature-dependent polarization-orientation (PO) low-frequency Raman measurements with first-principles calculations and single-crystal X-ray diffraction measurements. We show that chemical modifications can indeed suppress specific expressions of vibrational anharmonicity in the lattice dynamics. Specifically, we detect in BTBT a gradual change in the PO Raman response with temperature, indicating a unique anharmonic expression. This anharmonic expression is suppressed in all examined chemically modified crystals (ditBu-BTBT and diC8-BTBT, diPh-BTBT, and DNTT). In addition, we observe solid-solid phase transitions in the alkyl-modified BTBTs. Our findings indicate that π-conjugated chemical modifications are the most effective in suppressing these anharmonic effects.

Anharmonic Coupling of Stretching Vibrations in Ice: A Periodic VSCF and VCI Description

The anharmonicity of O-H stretching vibrations of water ice is characterized by use of a periodic implementation of the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, which take phonon-phonon couplings explicitly into account through numerical evaluation of high-order terms of the nuclear potential. The low-temperature, proton-ordered phase of water ice (namely, ice XI) is investigated. The net effect of a coupled anharmonic treatment of stretching modes is not just a rigid blue-shift of the respective harmonic spectral frequencies but rather a complex change of their relative spectral positions, which cannot be captured by simple scaling strategies based on harmonic calculations. The adopted techniques allow for a hierarchical treatment of anharmonic terms of the nuclear potential, which is key to an effective identification of leading factors. We show that the anharmonic independent-mode approximation─only describing the "intrinsic anharmonicity" of the O-H stretches─is unable to capture the correct physics, and that couplings among O-H stretches must be described. Inspection of harmonic normal coordinates allows identification of specific features of the O-H stretching motions which most likely enable strong mode-mode couplings. Finally, by coupling O-H stretches to all other possible modes of ice XI (THz collective vibrations, molecular librations, bendings), we identify specific types of motion which significantly affect O-H stretching states: in particular, molecular librations are found to affect the stretching states more than molecular bendings.

Machine learning the metastable phase diagram of covalently bonded carbon

Conventional phase diagram generation involves experimentation to provide an initial estimate of the set of thermodynamically accessible phases and their boundaries, followed by use of phenomenological models to interpolate between the available experimental data points and extrapolate to experimentally inaccessible regions. Such an approach, combined with high throughput first-principles calculations and data-mining techniques, has led to exhaustive thermodynamic databases (e.g. compatible with the CALPHAD method), albeit focused on the reduced set of phases observed at distinct thermodynamic equilibria. In contrast, materials during their synthesis, operation, or processing, may not reach their thermodynamic equilibrium state but, instead, remain trapped in a local (metastable) free energy minimum, which may exhibit desirable properties. Here, we introduce an automated workflow that integrates first-principles physics and atomistic simulations with machine learning (ML), and high-performance computing to allow rapid exploration of the metastable phases to construct "metastable" phase diagrams for materials far-from-equilibrium. Using carbon as a prototypical system, we demonstrate automated metastable phase diagram construction to map hundreds of metastable states ranging from near equilibrium to far-from-equilibrium (400 meV/atom). We incorporate the free energy calculations into a neural-network-based learning of the equations of state that allows for efficient construction of metastable phase diagrams. We use the metastable phase diagram and identify domains of relative stability and synthesizability of metastable materials. High temperature high pressure experiments using a diamond anvil cell on graphite sample coupled with high-resolution transmission electron microscopy (HRTEM) confirm our metastable phase predictions. In particular, we identify the previously ambiguous structure of n-diamond as a cubic-analog of diaphite-like lonsdaelite phase.

A complete description of thermodynamic stabilities of molecular crystals

Predicting stable polymorphs of molecular crystals remains one of the grand challenges of computational science. Current methods invoke approximations to electronic structure and statistical mechanics and thus fail to consistently reproduce the delicate balance of physical effects determining thermodynamic stability. We compute the rigorous ab initio Gibbs free energies for competing polymorphs of paradigmatic compounds, using machine learning to mitigate costs. The accurate description of electronic structure and full treatment of quantum statistical mechanics allow us to predict the experimentally observed phase behavior. This constitutes a key step toward the first-principles design of functional materials for applications from photovoltaics to pharmaceuticals.

Predictions of relative stabilities of (competing) molecular crystals are of great technological relevance, most notably for the pharmaceutical industry. However, they present a long-standing challenge for modeling, as often minuscule free energy differences are sensitively affected by the description of electronic structure, the statistical mechanics of the nuclei and the cell, and thermal expansion. The importance of these effects has been individually established, but rigorous free energy calculations for general molecular compounds, which simultaneously account for all effects, have hitherto not been computationally viable. Here we present an efficient “end to end” framework that seamlessly combines state-of-the art electronic structure calculations, machine-learning potentials, and advanced free energy methods to calculate ab initio Gibbs free energies for general organic molecular materials. The facile generation of machine-learning potentials for a diverse set of polymorphic compounds—benzene, glycine, and succinic acid—and predictions of thermodynamic stabilities in qualitative and quantitative agreement with experiments highlight that predictive thermodynamic studies of industrially relevant molecular materials are no longer a daunting task.

Along the road to Crystal Structure Prediction (CSP) of pharmaceutical-like molecules

Computational methods used for predicting crystal structures of organic compounds are mature enough to be routinely used with many rigid and semi-rigid organic molecules. The usefulness of Crystal Structure Prediction...

The microscopic origin of the anomalous isotopic properties of ice relies on the strong quantum anharmonic regime of atomic vibration

Water ice is a unique material presenting intriguing physical properties, such as negative thermal expansion and anomalous volume isotope effect (VIE). They arise from the interplay between weak hydrogen bonds and nuclear quantum fluctuations, making theoretical calculations challenging. Here, we employ the stochastic self-consistent harmonic approximation to investigate how thermal and quantum fluctuations affect the physical properties of ice XI with ab initio accuracy. Regarding the anomalous VIE, our work reveals that quantum effects on hydrogen are so strong to be in a nonlinear regime: When progressively increasing the mass of hydrogen from protium to infinity (classical limit), the volume first expands and then contracts, with a maximum slightly above the mass of tritium. We observe an anharmonic renormalization of about 10% in the bending and stretching phonon frequencies probed in IR and Raman experiments. For the first time, we report an accurate comparison of the low-energy phonon dispersion with the experimental data, possible only thanks to high-level accuracy in the electronic correlation and nuclear quantum and thermal fluctuations, paving the way for the study of thermal transport in ice from first-principles and the simulation of ice under pressure.

Efficient Quantum Vibrational Spectroscopy of Water with High-Order Path Integrals: From Bulk to Interfaces

Vibrational spectroscopy is key in probing the interplay between the structure and dynamics of aqueous systems. To map different regions of experimental spectra to the microscopic structure of a system, it is important to combine them with first-principles atomistic simulations that incorporate the quantum nature of nuclei. Here we show that the large cost of calculating the quantum vibrational spectra of aqueous systems can be dramatically reduced compared with standard path integral methods by using approximate quantum dynamics based on high-order path integrals. Together with state-of-the-art machine-learned electronic properties, our approach gives an excellent description not only of the infrared and Raman spectra of bulk water but also of the 2D correlation and the more challenging sum-frequency generation spectra of the water-air interface. This paves the way for understanding complex interfaces such as water encapsulated between or in contact with hydrophobic and hydrophilic materials through robust and inexpensive surface-sensitive and multidimensional spectra with first-principles accuracy.

Importance of Nuclear Quantum Effects for NMR Crystallography

The resolving power of solid-state nuclear magnetic resonance (NMR) crystallography depends heavily on the accuracy of computational predictions of NMR chemical shieldings of candidate structures, which are usually taken to be local minima in the potential energy. To test the limits of this approximation, we systematically study the importance of finite-temperature and quantum nuclear fluctuations for ¹H, ¹³C, and ¹⁵N shieldings in polymorphs of three paradigmatic molecular crystals: benzene, glycine, and succinic acid. The effect of quantum fluctuations is comparable to the typical errors of shielding predictions for static nuclei with respect to experiments, and their inclusion improves the agreement with measurements, translating to more reliable assignment of the NMR spectra to the correct candidate structure. The use of integrated machine-learning models, trained on first-principles energies and shieldings, renders rigorous sampling of nuclear fluctuations affordable, setting a new standard for the calculations underlying NMR structure determinations.

Conformational analysis of tannic acid: Environment effects in electronic and reactivity properties

Polyphenols are natural molecules of crucial importance in many applications, of which tannic acid (TA) is one of the most abundant and established. Most high-value applications require precise control of TA interactions with the system of interest. However, the molecular structure of TA is still not comprehended at the atomic level, of which all electronic and reactivity properties depend. Here, we combine an enhanced sampling global optimization method with density functional theory (DFT)-based calculations to explore the conformational space of TA assisted by unsupervised machine learning visualization and then investigate its lowest energy conformers. We study the external environment's effect on the TA structure and properties. We find that vacuum favors compact structures by stabilizing peripheral atoms' weak interactions, while in water, the molecule adopts more open conformations. The frontier molecular orbitals of the conformers with the lowest harmonic vibrational free energy have a HOMO-LUMO energy gap of 2.21 (3.27) eV, increasing to 2.82 (3.88) eV in water, at the DFT generalized gradient approximation (and hybrid) level of theory. Structural differences also change the distribution of potential reactive sites. We establish the fundamental importance of accurate structural consideration in determining TA and related polyphenol interactions in relevant technological applications.

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.

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.

Understanding the solubilization of Ca acetylide with a new computational model for ionic pairs

The unique reactivity of the acetylenic unit in DMSO gives rise to ubiquitous synthetic methods. We theoretically consider CaC2 solubility and protolysis in DMSO and formulate a strategy for CaC2 activation in solution-phase chemical transformations. For this, we use a new strategy for the modeling of ionic compounds in strongly coordinating solvents combining Born-Oppenheimer molecular dynamics with the DFTB3-D3(BJ) Hamiltonian and static DFT computations at the PBE0-D3(BJ)/pob-TZVP-gCP level. We modeled the thermodynamics of CaC2 protolysis under ambient conditions, taking into account its known heterogeneity and considering three polymorphs of CaC2. We give a theoretical basis for the existence of the elusive intermediate HC[triple bond, length as m-dash]C-Ca-OH and show that CaC2 insolubility in DMSO is of thermodynamic nature. We confirm the unique role of water and specific properties of DMSO in CaC2 activation and explain how the activation is realized. The proposed strategy for the utilization of CaC2 in sustainable organic synthesis is outlined.

Hamiltonian-Reservoir Replica Exchange and Machine Learning Potentials for Computational Organic Chemistry

This work combines a machine learning potential energy function with a modular enhanced sampling scheme to obtain statistically converged thermodynamical properties of flexible medium-size organic molecules at high ab initio level. We offer a modular environment in the python package MORESIM that allows custom design of replica exchange simulations with any level of theory including ML-based potentials. Our specific combination of Hamiltonian and reservoir replica exchange is shown to be a powerful technique to accelerate enhanced sampling simulations and explore free energy landscapes with a quantum chemical accuracy unattainable otherwise (e.g., DLPNO-CCSD(T)/CBS quality). This engine is used to demonstrate the relevance of accessing the ab initio free energy landscapes of molecules whose stability is determined by a subtle interplay between variations in the underlying potential energy and conformational entropy (i.e., a bridged asymmetrically polarized dithiacyclophane and a widely used organocatalyst) both in the gas phase and in solution (implicit solvent).

Accurate Description of Nuclear Quantum Effects with High-Order Perturbed Path Integrals (HOPPI)

Imaginary time path-integral (PI) simulations that account for nuclear quantum effects (NQE) beyond the harmonic approximation are increasingly employed together with modern electronic-structure calculations. Existing PI methods are applicable to molecules, liquids, and solids; however, the computational cost of such simulations increases dramatically with decreasing temperature. To address this challenge, here, we propose to combine high-order PI factorization with perturbation theory (PT). Already for conventional second-order PI simulations, the PT ansatz increases the accuracy 2-fold compared to fourth-order schemes with the same settings. In turn, applying PT to high-order path integrals (HOPI) further improves the efficiency of simulations for molecular and condensed matter systems especially at low temperatures. We present results for bulk liquid water, the aspirin molecule, and the CH₅⁺ molecule. Perturbed HOPI simulations remain both efficient and accurate down to 20 K and provide a convenient method to estimate the convergence of quantum-mechanical observables.

Revisiting nuclear tunnelling in the aqueous ferrous–ferric electron transfer

We find that golden-rule quantum transition-state theory predicts nearly an order of magnitude less tunnelling than some of the previous estimates. This may indicate that the spin-boson model of electron transfer is not valid in the quantum regime.