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At least 289 records · Page 16

Dynamic Interconversion of Metal Active Site Ensembles in Zeolite Catalysis

Catalysis science is founded on understanding the structure, number, and reactivity of active sites. Kinetic models that consider active sites to be static and noninteracting entities are routinely successful in describing the behavior of heterogeneous catalysts. Yet, active site ensembles often restructure in response to their external environment and even during steady-state catalytic turnover, sometimes requiring non-mean-field kinetic treatments to describe distance-dependent interactions among sites. Such behavior is being recognized more frequently in modern catalysis research, with the advent of experimental methods to quantify turnover rates with increasing precision, an expanding arsenal of operando characterization tools, and computational descriptions of atomic structure and motion at chemical potentials and timescales increasingly relevant to reaction conditions. This review focuses on dynamic changes to metal active site ensembles on zeolite supports, which are silica-based crystalline materials substituted with Al that generate binding sites for isolated and low-nuclearity metal site ensembles. Metal sites can become solvated and mobilized during reaction, facilitating interactions among sites that change their nuclearity and function. Such intersite communication can be regulated by the zeolite support, resulting in non-single-site and potentially non-mean-field kinetic behavior arising from mechanisms of catalytic action that combine elements of those canonically associated with homogeneous and heterogeneous catalysis. Here, we discuss recent literature examples that document dynamic active site behavior in metal-zeolites and outline methodologies to identify and interpret such behavior. We conclude with our outlook on future research directions to develop this evolving branch of catalysis science and harness it for practical applications.

operando characterization↗

Nuclear quantum effects on the quasiparticle properties of the chloride anion aqueous solution within the GW approximation

Photoelectron spectroscopy experiments in ionic solutions reveal important electronic structure information, in which the interaction between hydrated ions and water solvent can be inferred. Based on many-body perturbation theory with GW approximation, we theoretically compute the quasiparticle electronic structure of chloride anion solution, which is modeled by path-integral abinitio molecular dynamics simulation by taking account the nuclear quantum effects (NQEs). The electronic levels of hydrated anion as well as water are determined and compared with the recent experimental photoelectron spectra. It is found that NQEs improve the agreement between theoretical prediction and experiment because NQEs effectively weaken the hybridization of the between the Cl – anion and water. Furthermore, our study indicates that NQEs plays a small but non-negligible role in predicting the electronic structure of the aqueous solvation of ions of the Hofmeister series.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Using Density-Corrected DFT to Understand Density-Driven and Functional-Dependent Errors in Ab Initio Simulations of the Hydrated Electron

The hydrated electron, an excess electron in liquid water, plays a crucial role in a plethora of chemical processes, motivating extensive research efforts to characterize its structure, dynamics, and reactivity in solution. Recent theoretical approaches to understanding this intriguing object have involved ab initio simulations based on density functional theory (DFT). Although DFT allows for the study of hydrated electron reactivity and quantum mechanical behavior, it is well-known that anionic systems can suffer from significant density-driven errors (DDEs). Density-corrected DFT (DC-DFT) provides a framework to mitigate such errors; the method reduces DDEs by replacing the self-consistent (SC) density associated with a given density functional with the Hartree–Fock (HF) density. Since HF densities tend to be more localized than DFT SC densities, the DC-DFT scheme significantly improves errors in calculations where the SC density is spuriously delocalized. Here, we investigate how the use of density correction affects the calculated properties of the DFT-simulated (PBEh) hydrated electron, a particularly challenging diffuse anionic system to simulate. First, we analyze charge delocalization in a system consisting of a model octahedral hydrated electron water cluster (the so-called Kevan structure) along with a spatially separated sulfur atom. We show that the use of density correction indeed reduces DDEs in comparison to a standard DFT global hybrid functional. We then propagate molecular dynamics trajectories of the hydrated electron using DC-DFT, where we find that DC further localizes electron density in the cavity region, a signature of reduced charge delocalization. Unfortunately, the decreased radius of gyration of the spin density and corresponding tightening of the local solvation structure from density correction causes predicted observables to deviate further from experimental measurements than when density correction is not employed. Here, we argue that DC’s worse agreement with experiment results from the removal of a fortuitous cancellation of errors that is intrinsic to the PBEh functional. This indicates that the difficulties with DFT to simulate hydrated electrons are primarily due to the inherent approximations in DFT rather than to density-driven errors.

Density functional theory↗

Molecular-based analysis of nanoparticle solvation: Classical density functional approach

Proper statistical mechanics understanding of nanoparticle solvation processes requires an accurate description of the molecular structure of the solvent. Achieving this goal with standard molecular dynamics (MD) simulation methods is challenging due to large length scales. An alternative approach to this problem can be formulated using classical density functional theory (cDFT), where a full configurational description of the positions of all the atoms is replaced by collective atomic site densities in the molecule. Using an example of the negatively charged silica-like system in an aqueous polar environment represented by a two-site water model, we demonstrate here that cDFT can reproduce MD data at a fraction of the computational cost. An important implication of this result is the ability to understand how the solvent molecular features may affect the system’s properties at the macroscopic scale. A concrete example highlighted in this work is the analysis of nanoparticle interactions with sizes of up to 100 nm in diameter.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Liquid ToF-SIMS revealing the oil, water, and surfactant interface evolution

Bilge water from ships is regarded as a major pollutant in the marine environment. Bilge water exists in a stable oil-in-water (O/W) emulsion form. However, little is known about the O/W liquid–liquid (l–l) interface. Traditional bulk characterization approaches are not capable of capturing the chemical changes at the O/W l–l interface. Although surfactants are deemed essential in droplet formation, their roles in bilge water stabilization have not been fully revealed. We have utilized novel in situ chemical imaging tools including in situ scanning electron microscopy (SEM) and in situ time-of-flight secondary ion mass spectrometry (ToF-SIMS) to study the evolving O/W interface using a NAVY bilge model for the first time. The droplet size distribution (DSD) does not change significantly without the addition of X-100 surfactants under static or rocking conditions. Both the oil components and the water clusters are shown to evolve over time at the O/W droplet interface by in situ liquid SIMS imaging. Of particular interest to droplet stabilization, the contribution of surfactants to the aged bilge droplets becomes more significant as the droplet size increases. The higher mass surfactant component does not appear on the droplet surface immediately while many lower mass surfactants are solvated inside the droplet. We have provided the first three-dimensional images of the evolving O/W interface and demonstrated that in situ surface chemical mapping is powerful enough to reveal the complex and dynamic l–l interface in the liquid state. Our observational insights suggest that surfactants are important in mediating droplet growth and facilitating effective separation of bilge water emulsion.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Dielectric relaxation of neodymium chloride in water and in methanol

In this work, an open-ended coaxial probe was used to study the dielectric relaxation of neodymium chloride (NdCl 3 ) in water and in methanol at room temperature over the frequency range 0.2–40.0 GHz. An equivalent circuit model based on the antenna model was used to obtain the dielectric spectra with the effective removal of spurious effects. This extends the use of the open-ended coaxial probe technique for highly conductive substances. The structural and thermodynamic parameters obtained from the dielectric spectra of aqueous NdCl 3 solutions suggest that solvent-separated ion pairs and solvent-shared ion pairs dominate the low-frequency behavior at low concentrations. In methanol, solvent-shared ion pair relaxation modes were detected. No contact pair relaxation was observed in either solvent. Under all conditions, Nd 3+ was strongly solvated, as reflected by its high effective hydration number. The estimated association constant of NdCl 3 in water at infinite dilution was within the literature values and was lower than that observed in methanol.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Thermodynamically Optimized Machine-Learned Reaction Coordinates for Hydrophobic Ligand Dissociation

Ligand unbinding is mediated by its free energy change, which has intertwined contributions from both energy and entropy. It is important, but not easy, to quantify their individual contributions to the free energy profile. We model hydrophobic ligand unbinding for two systems, a methane particle and a C 60 fullerene, both unbinding from hydrophobic pockets in all-atom water. Using a modified deep learning framework, we learn a thermodynamically optimized reaction coordinate to describe the hydrophobic ligand dissociation for both systems. Interpretation of these reaction coordinates reveals the roles of entropic and enthalpic forces as the ligand and pocket sizes change. In both cases, we observe that the free-energy barrier to unbinding is dominated by entropy considerations. Furthermore, the process of methane unbinding is driven by methane solvation, while fullerene unbinding is driven first by pocket wetting and then fullerene wetting. For both solutes, the direct importance of the distance from the binding pocket to the learned reaction coordinate is present, but low. Furthermore, our framework and subsequent feature important analysis thus give useful thermodynamic insight into hydrophobic ligand dissociation problems that are otherwise difficult to glean.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Self‐Assembly Methods Induce Different Individual‐Polymer‐Block Solvation Responses

An understanding of block-specific responses to stimuli in self-assembled copolymers is essential for the effective design of responsive materials. Here, materials formed from different post-reaction processing methods of the same ROMP block copolymer exhibit different block-specific solvent responses, as characterized in situ by fluorescence lifetime imaging microscopy (FLIM). FLIM data were acquired by tagging the polar or nonpolar block separately with a covalently incorporated viscosity-sensitive fluorescent molecular rotor to measure the changing tightness or looseness of assembly upon changes in solvent composition. Rapid precipitation from an insoluble solvent to form a film results in tighter assembly and less difference in tightness/looseness between the two blocks. Further, it interrupts individual block-solvent response behavior. Both results indicate an apparent disruption of the core–shell assembly. This model from FLIM is further supported by differential scanning calorimetry (DSC) and small-angle X-ray scattering (SAXS) data from isolated solids, which show tighter long-range interactions and more long-range order in these kinetically trapped states. Here, the experiments develop FLIM as a method for pinpointing stimuli-responsive behaviors from different processing methods to individual blocks. Together these outcomes provide a future handle for tailoring solvent-triggered assembly/disassembly behavior.

López, Pía A. [University of California, Irvine, C↗

Elucidating the Structure of the Eu‐EDTA Complex in Solution at Various Protonation States

Abstract Ethylenediaminetetraacetic acid (EDTA), which has two amine and four carboxylate protonation sites, forms stable complexes with lanthanide ions. This work analyzes the coordination structure, in atomic resolution, of the Eu 3+ ion complexed with EDTA in all its protonation states in aqueous solution. Eu‐EDTA complexes were modeled using classical molecular dynamics (MD) simulations using force field parameters optimized with ab initio molecular dynamics (AIMD) simulations. Structures from the MD simulations were used to predict extended X‐ray absorption fine structure (EXAFS) spectra and compared with EXAFS measurements of the Eu 3+ aqua ion and Eu‐EDTA complexes at pH 3 and 11. This work details how Eu‐EDTA complex coordination structures change with increasing protonation of the EDTA ligand in the complex, from the tightly bound unprotonated complex to the unbinding of the fully protonated EDTA ligand from the Eu 3+ ion as both become solvated by water. Agreement between predicted and measured EXAFS spectra supports the findings from simulation.

Chemistry↗

Modeling prebiotic chemistries with quantum accuracy at classical costs

Molecular Dynamics (MD) simulations using classical force-fields are commonly employed in numerous scientific investigations. However, many natural processes involve bond breaking and quantum forces. This complexity is compounded by the presence of multiple competing length and timescales. For example, accurately modeling the thermodynamics and dynamics of a chemical reaction requires accounting for the concerted movements of numerous solvent molecules and ions with their own fast or slow timescales. While widely used static Density Functional Theory (DFT) calculations at 0 temperature can be beneficial for such investigations, they do not account for dynamics, and lack precision in describing the molecular environments. They particularly fail at correct, rigorous treatments of finite-temperature fluctuations, and thus generalization to experimentally relevant conditions. In PNAS Benayad et al develop a scalable, generalizable approach for designing Neural Network Potentials (NNPs) that can handle chemical reactivity in solvated systems with quantum accuracy at classical costs. Specifically, they study phosphoester bond formation and rupture, which is fundamentally relevant to the Phosphorus-Oxygen bond formation central to life, and especially for the RNA world hypothesis. The framework developed here has the potential to generalize to different chemical reactions of energy and biological relevance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Sand Migration Simulation during Gas Production from Gas Hydrate Reservoir at Kuparuk 7–11–12 site in the Prodhoe Bay Unit, Alaska

Uncontrolled sand production impedes continuous gas production from a hydrate reservoir as observed in the past field-scale gas production tests. Sand mobilization is strongly linked with sediment deformation and high pressure gradient. Sand production occurs when the mobilized sands reach the well. Throughout gas production from a hydrate reservoir, because of hydrate dissociation, deformation and pressure gradient change both in time and space. Therefore, it is necessary to consider the entire process to identify where sand is likely mobilized and how much mobilized sand could reach the well. Here, this study utilizes a coupled thermal, hydrological, chemical, and mechanical (thermo-hydro-chemo-mechanical) sand migration model to simulate a year long gas production from the hydrate reservoir at the Kuparuk 7–11–12 site in the Prodhoe Bay Unit, Alaska. It is found that sand would mainly come from the lower portion of the production zone where faster hydrate dissociation occurs. The relatively faster hydrate dissociation coupled with the fact that hydrate-bearing sediments and hydrate-free sediments have similar stiffness but different strengths causes complex stress transfer, which results in excessive shear deformation when the upper portion of the production zone starts to dissociate. This is evident not only near the well but also away from the well, leading to a large amount of sand mobilization. Another focus of the modeling study is evaluation of the effect of sand migration on gas production. The comparison of two extreme cases–with sandscreen and without sandscreen–suggests that migrated and settled solids around a sand control device can prevent pressure drop across the near-wellbore zone and lead to reduction of the gas production rate.

59 BASIC BIOLOGICAL SCIENCES↗

Tracking structural solvent reorganization and recombination dynamics following e – photoabstraction from aqueous I – with femtosecond x-ray spectroscopy and scattering

Here we present a sub-picosecond resolved investigation of the structural solvent reorganization and geminate recombination dynamics following 400 nm two-photon excitation and photodetachment of a valence p electron from the aqueous atomic solute, I–(aq). The measurements utilized time-resolved X-ray Absorption Near Edge Structure (TR-XANES) spectroscopy and X-ray Solution Scattering (TR-XSS) at the Linac Coherent Light Source x-ray free electron laser in a laser pump/x-ray probe experiment. The XANES measurements around the L 1 -edge of the generated nascent iodine atoms (I 0 ) yield an average electron ejection distance from the iodine parent of 7.4 ± 1.5 Å with an excitation yield of about 1/3 of the 0.1M NaI aqueous solution. The kinetic traces of the XANES measurement are in agreement with a purely diffusion-driven geminate iodine–electron recombination model without the need for a long-lived (I 0 :e–) contact pair. Nonequilibrium classical molecular dynamics simulations indicate a delayed response of the caging H 2 O solvent shell and this is supported by the structural analysis of the XSS data: We identify a two-step process exhibiting a 0.1 ps delayed solvent shell reorganization time within the tight H-bond network and a 0.3 ps time constant for the mean iodine–oxygen distance changes. The results indicate that most of the reorganization can be explained classically by a transition from a hydrophilic cavity with a well-ordered first solvation shell (hydrogens pointing toward I–) to an expanded cavity around I 0 with a more random orientation of the H 2 O molecules in a broadened first solvation shell.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

The Energetic Origins of Pi–Pi Contacts in Proteins

Accurate potential energy models of proteins must describe the many different types of noncovalent interactions that contribute to a protein’s stability and structure. Pi–pi contacts are ubiquitous structural motifs in all proteins, occurring between aromatic and nonaromatic residues and play a nontrivial role in protein folding and in the formation of biomolecular condensates. Guided by a geometric criterion for isolating pi–pi contacts from classical molecular dynamics simulations of proteins, we use quantum mechanical energy decomposition analysis to determine the molecular interactions that stabilize different pi–pi contact motifs. We find that neutral pi–pi interactions in proteins are dominated by Pauli repulsion and London dispersion rather than repulsive quadrupole electrostatics, which is central to the textbook Hunter–Sanders model. This results in a notable lack of variability in the interaction profiles of neutral pi–pi contacts even with extreme changes in the dielectric medium, explaining the prevalence of pi-stacked arrangements in and between proteins. We also find interactions involving pi-containing anions and cations to be extremely malleable, interacting like neutral pi–pi contacts in polar media and like typical ion–pi interactions in nonpolar environments. Like-charged pairs such as arginine–arginine contacts are particularly sensitive to the polarity of their immediate surroundings and exhibit canonical pi–pi stacking behavior only if the interaction is mediated by environmental effects, such as aqueous solvation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Unexpected Kinetic Solvent Effects Enhance Activity and Selectivity in Biphasic Systems

Biphasic dehydration of fructose to 5-hydroxymethylfurfural (HMF) has shown unprecedented increases in productivity, but a mechanistic understanding is lacking. Herein, we couple fast experimental reaction kinetics, multiscale modeling (phase behavior, classical molecular dynamics(MD), and quantum mechanics/molecular mechanics MD), in situ sampling, and IR and 13 C-NMR spectroscopy to elucidate the complex effects of nonpolar extracting organic solvents on the kinetics of fructose dehydration. We show that these organic solvents can reach significant mutual solubility with water at reaction temperatures, enabling the partition of the sugar and catalyst into the extracting phase. In the organic-rich environment, the dehydration of fructose proceeds faster and more selectively than in water due to increased relative abundance of the reactive furanose isomer, enhanced water–catalyst–substrate interactions driven by nanophase separation, and higher product stability stemming from preferential solvation. Furthermore, we demonstrate that these solvent effects impact other critical biphasic reactions in biomass upgrading and provide qualitative principles for solvent selection.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Solvation Structure of 237 Np 4+ in a Noncomplexing Environment

Here, the solvation structure of an Np 4+ ion in an aqueous, noncomplexing and nonoxidizing environment of trifluoromethanesulfonic (triflic) acid was investigated with X-ray absorption spectroscopy (XAS) combined with ab initio molecular dynamics (AIMD) and time-dependent density functional theory (TDDFT) calculations. Np L III -edge X-ray absorption near-edge structure (XANES) and extended X-ray absorption fine structure (EXAFS) data were collected for Np 4+ in 1, 3, and 7 M triflic acid using a laboratory-scale spectrometer and separately at a synchrotron facility, producing data sets in excellent agreement. TDDFT calculations revealed a weak pre-edge feature not previously reported for Np L III -edge XANES. AIMD modeling results showed differences in the hydration shell of the Np 4+ ion at different concentrations of triflic acid; these results are supported by the experiment. EXAFS fit models to the experiment resulted in similar coordination of Np 4+ in noncomplexing aqueous media as reported in the literature for 1 M perchloric acid but, together with calculations, revealed more than one distance between Np and O atoms in 7 M triflic acid. These results imply monodentate coordination with sulfonate groups in 7 M triflic acid and suggest the possibility of proto-neptunyl species in relatively low-concentration Np 4+ acid solutions.

Boglaienko, Daria V. [Pacific Northwest National L↗

The Influence of Charge Correlation and Ion Solvation on the Phase Behavior of Single-Ion Conducting Polymer Blend Electrolytes Using SAXS/SANS

Single-ion conducting polymer blends (SICPBs) have demonstrated exceptional electrochemical performance as solid-state battery electrolytes; however, their nanoscale morphology and thermodynamic behavior remain unexplored. In this work, we investigate blends composed of deuterated poly(ethylene oxide) and poly[lithium sulfonyl(trifluoromethane sulfonyl)imide methacrylate], dPEO/P(LiMTFSI), and report the first experimental study of the nanostructures of charge-neutral polymer blends using small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS). Despite the macroscopic miscibility indicated by a single glass-transition temperature, SANS and SAXS results reveal disordered, charge-correlated nanostructures that are strongly influenced by blend composition and temperature. At low concentrations of charge polymer, the scattering is dominated by concentration fluctuations, and the random phase approximation is applied to extract values of the Flory–Huggins interaction parameter, χ SC . At higher charged polymer content, concentration fluctuations are suppressed, and a correlation model is used to characterize the nanostructures of the charge correlations. We find that the structures of the charge correlations are highly dependent on blend composition─consistent with predictions from Sing’s self-consistent field theory-liquid state models. Understanding these features is essential for uncovering the ion transport mechanism that leads to improved electrochemical performance previously reported in SICPB systems.

25 ENERGY STORAGE↗

Reaction Energetics and 13 C Fractionation of Alanine Transamination in the Aqueous and Gas Phases

The alanine transaminase enzyme catalyzes the transfer of an amino group from alanine to α-ketoglutarate to produce pyruvate and glutamate. Isotope fractionation factors (IFFs) for the reaction +H 3 NCH(CH 3 )COO – + – OOCCH 2 CH 2 C(O)COO – ↔ CH 3 C(O)COO – + + H 3 NCH(CH 2 CH 2 COO – )COO – (zwitterionic neutral alanine + doubly deprotonated α-ketoglutarate ↔ pyruvate + zwitterionic glutamate anion) were calculated from the partition functions of explicitly and implicitly solvated molecules at 298 K. Calculations were done for alanine (noncharge separated, zwitterion, deprotonated), pyruvic acid (neutral, deprotonated), glutamic acid (noncharge separated, zwitterion, deprotonated, doubly deprotonated), and α-ketoglutaric acid (neutral, deprotonated, doubly deprotonated). The computational results, calculated from gas phase- and aqueous-optimized clusters with explicit H 2 O molecules at the MP2/aug-cc-pVDZ and MP2/aug-cc-pVDZ/COSMO levels, respectively, predict that substitution of 13 C at the C2 position of alanine and pyruvic acid and their various forms leads to the C2 position of pyruvic acid/pyruvate being enriched in 13 C/ 12 C ratio by 9‰. Simpler approaches that estimate the IFFs based solely on changes in the zero-point energies (ZPEs) are consistent with the higher-level model. ZPE-based IFFs calculated for simple analogues formaldehyde and methylamine (analogous to the C2 positions of pyruvate and alanine, respectively) predict a 13 C enrichment in formaldehyde of 7–8‰ at the MP2/aug-cc-pVDZ and aug-cc-pVTZ levels. A simple predictive model using canonical functional group frequencies and reduced masses for 13 C exchange between R 2 C=O and R 2 CH–NH 2 predicted enrichment in R 2 C=O that is too large by a factor of two but is qualitatively accurate compared with the more sophisticated models. Our models are all in agreement with the expectation that pyruvate and formaldehyde will be preferentially enriched in 13 C because of the strength of their >C=O bond relative to that of ≡C–NH 2 in alanine and methylamine. 13 C/ 12 C substitution is also modeled at the methyl and carboxylic acid sites of alanine and pyruvic acid, respectively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Effect of Solvent on the Local Structure, Dynamics, and Vibrational Density of States in Sn-BEA Zeolite

Lewis acid zeolites are attractive catalysts for epoxidation and biomass valorization, as they are highly active and selective in the liquid phase and can operate at or near ambient conditions. While a rich experimental literature exists on liquid-phase Lewis acid zeolite catalysis, our understanding of the molecular organization and solvent dynamics in the vicinity of Lewis acid sites with differing metal site speciation remains limited. In this work, we investigate the molecular coordination and diffusion of two common solvents (methanol and water) around the closed and open Sn-BEA zeolite active sites using molecular dynamics simulations with a machine-learned interatomic potential trained on ab initio molecular dynamics trajectories. Molecular dynamics simulations reveal that introducing active sites significantly enhances local order in the first and second solvation shells compared to the pure silica case. For methanol, both closed and open active sites are singly coordinated, while more than two water molecules coordinate the open site. In contrast to methanol, we observed that water molecules dissociate, leading to the formation of additional Sn-OH and silanol groups away from the active site. The diffusion coefficients of water and methanol are functions of the solvent population in the pore. Here, our work provides insights into how active site speciation in Lewis acid zeolites affects solvent coordination, diffusion, and vibrational signature. This information is foundational for catalyst design and optimization of liquid-phase catalytic processes in zeolites. It also demonstrates the suitability of machine-learned interatomic potentials for modeling reactive systems, enabling sufficiently long trajectories for appropriate statistical averaging.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗