Chemical Reaction Rate Coefficients from Ring Polymer Molecular Dynamics: Theory and Practical Applications

Rate coefficients for the O + H2 and O + D2 reactions: how well ring polymer molecular dynamics accounts for tunelling

We present here extensive calculations of the O(3P) + H2 and O(3P) + D2 reaction dynamics spanning the temperature range from 200 K to 2500 K. The calculations have been carried out using fully converged time-independent quantum mechanics (TI QM), quasiclassical trajectories (QCT) and ring polymer molecular dynamics (RPMD) on the two lowest lying adiabatic potential energy surfaces (PESs), 13A' and 13A'', calculated by Zanchet et al. [J. Chem. Phys., 2019, 151, 094307]. TI QM rate coefficients were determined using the cumulative reaction probability formalism on each PES including all of the total angular momenta and the Coriolis coupling and can be considered to be essentially exact within the Born-Oppenheimer approximation. The agreement between the rate coefficients calculated by using QM and RPMD is excellent for the reaction with D2 in almost the whole temperature range. For the reaction with H2, although the agreement is very good above 500 K, the deviations are significant at lower temperatures. In contrast, the QCT calculations largely underestimate the rate coefficients for the two isotopic variants due to their inability to account for tunelling. The differences found in the disagreements between RPMD and QM rate coefficients for the reactions for both the isotopologues are indicative of the ability of the RPMD method to accurately describe systems where tunelling plays a relevant role. Considering that both reactions are dominated by tunelling below 500 K, the present results show that RPMD is a very powerful tool for determining rate coefficients. The present QM rate coefficients calculated on adiabatic PESs slightly underestimate the best global fits of the experimental measurements, which we attribute to the intersystem crossing with the singlet 11A' PES.

Quantum rates in dissipative systems with spatially varying friction

We investigate whether making the friction spatially dependent on the reaction coordinate introduces quantum effects into the thermal reaction rates for dissipative reactions. Quantum rates are calculated using the numerically exact multi-configuration time-dependent Hartree method, as well as the approximate ring-polymer molecular dynamics (RPMD), ring-polymer instanton methods, and classical molecular dynamics. By conducting simulations across a wide range of temperatures and friction strengths, we can identify the various regimes that govern the reactive dynamics. At high temperatures, in addition to the spatial-diffusion and energy-diffusion regimes predicted by Kramer's rate theory, a (coherent) tunneling-dominated regime is identified at low friction. At low temperatures, incoherent tunneling dominates most of Kramer's curve, except at very low friction, when coherent tunneling becomes dominant. Unlike in classical mechanics, the bath's influence changes the equilibrium time-independent properties of the system, leading to a complex interplay between spatially dependent friction and nuclear quantum effects even at high temperatures. More specifically, a realistic friction profile can lead to an increase (or decrease) of the quantum (classical) rates with friction within the spatial-diffusion regime, showing that classical and quantum rates display qualitatively different behaviors. Except at very low frictions, we find that RPMD captures most of the quantum effects in the thermal reaction rates.

Reactions dynamics for X + H2 insertion reactions (X = C(1D), N(2D), O(1D), S(1D)) with Cayley propagator ring-polymer molecular dynamics

In this work, rate coefficients of four prototypical insertion reactions, X + H2 → H + XH (X = C(1D), N(2D), O(1D), S(1D)), and associated isotope reactions are calculated based on ring polymer molecular dynamics (RPMD) with Cayley propagator (Cayley-RPMD). The associated kinetic isotope effects are systematically studied too. The Cayley propagator used in this work increases the stability of numerical integration in RPMD calculations and also supports a larger evolution time interval, allowing us to reach both high accuracy and efficiency. So, our results do not only provide chemical kinetic data for the title reactions in an extended temperature range but also consist of experimental results, standard RPMD, and other theoretical methods. The results in this work also reflect that Cayley-RPMD has strong consistency and high reliability in its investigations of chemical dynamics for insertion reactions.

A personal perspective of the present status and future challenges facing thermal reaction rate theory

Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

Accurate Reaction Probabilities for Translational Energies on Both Sides of the Barrier of Dissociative Chemisorption on Metal Surfaces

Molecular dynamics simulations are essential for a better understanding of dissociative chemisorption on metal surfaces, which is often the rate-controlling step in heterogeneous and plasma catalysis. The workhorse quasi-classical trajectory approach ubiquitous in molecular dynamics is able to accurately predict reactivity only for high translational and low vibrational energies. In contrast, catalytically relevant conditions generally involve low translational and elevated vibrational energies. Existing quantum dynamics approaches are intractable or approximate as a result of the large number of degrees of freedom present in molecule-metal surface reactions. Here, we extend a ring polymer molecular dynamics approach to fully include, for the first time, the degrees of freedom of a moving metal surface. With this approach, experimental sticking probabilities for the dissociative chemisorption of methane on Pt(111) are reproduced for a large range of translational and vibrational energies by including nuclear quantum effects and employing full-dimensional simulations.

Use of QM/MM Surface Hopping Simulations to Understand Thermally Activated Rare-Event Nonadiabatic Transitions in the Condensed Phase

We implement a rare-event sampling scheme for quantifying the rate of thermally activated nonadiabatic transitions in the condensed phase. Our Quantum mechanics/molecular mechanics (QM/MM) methodology uses the recently developed Interface for NonAdiabatic QM/MM in Solvent (INAQS) package to interface an elementary electronic structure package and a popular open-source molecular dynamics software (GROMACS) to simulate an electron transfer event between two stationary ions in a solution of acetonitrile solvent molecules. Nonadiabatic effects are implemented through a surface hopping scheme, and our simulations allow further quantitative insight into the participation ratio of a solvent and the effect of ion separation distance as far as facilitating electron transfer. We also demonstrate that the standard gas-phase approaches for treating frustrated hops and velocity reversal must be refined when working in the condensed phase with many degrees of freedom. The code and methodology developed here can be easily expanded upon and modified to incorporate other systems and should provide a great deal of new insight into a wide variety of condensed phase nonadiabatic phenomena.

Ring Polymer Molecular Dynamics Approach to Quantum Dissociative Chemisorption Rates

A ring polymer molecular dynamics (RPMD) method is proposed for the calculation of the dissociative chemisorption rate coefficient on surfaces. The RPMD rate theory is capable of handling quantum effects such as the zero-point energy and tunneling in dissociative chemisorption, while it relies on classical trajectories for the simulation. Applications to H2 dissociative chemisorption are demonstrated. For the highly activated process on Ag(111), strong deviations from Arrhenius behavior are found at low temperatures and attributed to tunneling. On Pt(111), where the dissociation has a barrierless pathway, the RPMD rate coefficient is found to agree with the experimentally derived thermal sticking coefficient within a factor of 2 over a large temperature range. Significant quantum effects are also found.

First-principles surface reaction rates by ring polymer molecular dynamics and neural network potential: role of anharmonicity and lattice motion

Elementary gas-surface processes are essential steps in heterogeneous catalysis. A predictive understanding of catalytic mechanisms remains challenging due largely to difficulties in accurately characterizing the kinetics of such steps. Experimentally, thermal rates for elementary surface reactions can now be measured using a novel velocity imaging technique, providing a stringent testing ground for ab initio rate theories. Here, we propose to combine ring polymer molecular dynamics (RPMD) rate theory with state-of-the-art first-principles-determined neural network potential to calculate surface reaction rates. Taking NO desorption from Pd(111) as an example, we show that the harmonic approximation and the neglect of lattice motion in the commonly-used transition state theory overestimates and underestimates the entropy change during the desorption process, respectively, leading to opposite errors in rate coefficient predictions and artificial error cancellations. Including anharmonicity and lattice motion, our results reveal a generally neglected surface entropy change due to significant local structural change during desorption and obtain the right answer for the right reasons. Although quantum effects are found to be less important in this system, the proposed approach establishes a more reliable theoretical benchmark for accurately predicting the kinetics of elementary gas-surface processes.

Dynamics study of the post-transition-state-bifurcation process of the (HCOOH)H+ → CO + H3O+/HCO+ + H2O dissociation: application of machine-learning techniques

The process of protonated formic acid dissociating from the transition state was studied using ring-polymer molecular dynamics (RPMD), classical MD, and quasi-classical trajectory (QCT) simulations. Temperature had a strong influence on the branching fractions for the HCO⁺ + H₂O and CO + H₃O⁺ dissociation channels. The RPMD and classical MD simulations showed similar behavior, but the QCT dynamics were significantly different owing to the excess energies in the quasi-classical trajectories. Machine-learning analysis identified several key features in the phase information of the vibrational motions at the transition state. We found that the initial configuration and momentum of a hydrogen atom connected to a carbon atom and the shrinking coordinate of the CO bond at the transition state play a role in the dynamics of HCO⁺ + H₂O production.

Nuclear quantum effects on zeolite proton hopping kinetics explored with machine learning potentials and path integral molecular dynamics

Proton hopping is a key reactive process within zeolite catalysis. However, the accurate determination of its kinetics poses major challenges both for theoreticians and experimentalists. Nuclear quantum effects (NQEs) are known to influence the structure and dynamics of protons, but their rigorous inclusion through the path integral molecular dynamics (PIMD) formalism was so far beyond reach for zeolite catalyzed processes due to the excessive computational cost of evaluating all forces and energies at the Density Functional Theory (DFT) level. Herein, we overcome this limitation by training first a reactive machine learning potential (MLP) that can reproduce with high fidelity the DFT potential energy surface of proton hopping around the first Al coordination sphere in the H-CHA zeolite. The MLP offers an immense computational speedup, enabling us to derive accurate reaction kinetics beyond standard transition state theory for the proton hopping reaction. Overall, more than 0.6 μs of simulation time was needed, which is far beyond reach of any standard DFT approach. NQEs are found to significantly impact the proton hopping kinetics up to ~473 K. Moreover, PIMD simulations with deuterium can be performed without any additional training to compute kinetic isotope effects over a broad range of temperatures.

An efficient algorithm for capturing quantum effects in classical reactive scattering: application to D + H+3 → H2D+ + H

Motivated by a recent semiclassical analysis of chemical reaction thresholds [Bonnet et al., J. Chem. Phys., 2022, 157, 094114], we present an efficient algorithm for including zero-point energy (ZPE) effects in classical reactive scattering. The algorithm is an extension of the quasi-classical trajectory (QCT) Gaussian binning method. We apply it to the astrophysically important D + H+3 reaction, where there are significant quantum effects and where application of other methods is problematic [Braunstein et al., Phys. Chem. Chem. Phys., 2022, 24, 5489]. The rate constants computed with the new, general algorithm closely match recent Ring Polymer Molecular Dynamics (RPMD) [Bulut et al., J. Phys. Chem. A, 2019, 123, 8766] and experimentally derived [Bowen et al., J. Chem. Phys., 2021, 154, 084307] ones spanning ∼4 orders of magnitude from 70 to 1500 K.

The quantum dynamics of H2 on Cu(111) at a surface temperature of 925 K: Comparing state-of-the-art theory to state-of-the-art experiments 2

State-of-the-art 6D quantum dynamics simulations for the dissociative chemisorption of H₂ on a thermally distorted Cu(111) surface, using the static corrugation model, were analyzed to produce several (experimentally available) observables. The expected error, especially important for lower reaction probabilities, was quantified using wavepackets on several different grids as well as two different analysis approaches to obtain more accurate results in the region where a slow reaction channel was experimentally shown to be dominant. The lowest reaction barrier sites for different thermally distorted surface slabs are shown to not just be energetically, but also geometrically, different between surface configurations, which can be used to explain several dynamical effects found when including surface temperature effects. Direct comparison of simulated time-of-flight spectra to those obtained from state-of-the-art desorption experiments showed much improved agreement compared to the perfect lattice BOSS approach. Agreement with experimental rotational and vibrational efficacies also somewhat improved when thermally excited surfaces were included in the theoretical model. Finally, we present clear quantum effects in the rotational quadrupole alignment parameters found for the lower rotationally excited states, which underlines the importance of careful quantum dynamical analyses of this system.

Quantum versus classical unimolecular fragmentation rate constants and activation energies at finite temperature from direct dynamics simulations

In the present work, we investigate how nuclear quantum effects modify the temperature dependent rate constants and, consequently, the activation energies in unimolecular reactions. In the reactions under study, nuclear quantum effects mainly stem from the presence of a large zero point energy. Thus, we investigate the behavior of methods compatible with direct dynamics simulations, the quantum thermal bath (QTB) and ring polymer molecular dynamics (RPMD). To this end, we first compare them with quantum reaction theory for a model Morse potential before extending this comparison to molecular models. Our results show that, in particular in the temperature range comparable with or lower than the zero point energy of the system, the RPMD method is able to correctly capture nuclear quantum effects on rate constants and activation energies. On the other hand, although the QTB provides a good description of equilibrium properties including zero-point energy effects, it largely overestimates the rate constants. The origin of the different behaviours is in the different distance distributions provided by the two methods and in particular how they differently describe the tails of such distributions. The comparison with transition state theory shows that RPMD can be used to study fragmentation of complex systems for which it may be difficult to determine the multiple reaction pathways and associated transition states.

New Stable and Fast Ring-Polymer Molecular Dynamics for Calculating Bimolecular Rate Coefficients with an Example of OH + CH4

The accurate and efficient calculation of the rate coefficients of chemical reactions is a key issue in the research of chemical dynamics. In this work, by applying the dimension-free ultrastable Cayley propagator, the thermal rate coefficients of a prototypic high dimensional chemical reaction OH + CH₄ → H₂O + CH₃ in the temperature range of 200 to 1500 K are investigated with ring polymer molecular dynamics (RPMD) on a highly accurate full-dimensional potential energy surface. Kinetic isotope effects (KIEs) for three isotopologues of the title reaction are also studied. The results demonstrate excellent agreement with experimental data, even in the deep tunneling region. Especially, the Cayley propagator shows a high calculation efficiency with little loss of accuracy. The present results confirmed the applicability of the RPMD method, particularly the speed-up using a Cayley propagator, in theoretical calculations of bimolecular reaction rates.

Quantum effects in thermal reaction rates at metal surfaces

There is wide interest in developing accurate theories for predicting rates of chemical reactions that occur at metal surfaces, especially for applications in industrial catalysis. Conventional methods contain many approximations that lack experimental validation. In practice, there are few reactions where sufficiently accurate experimental data exist to even allow meaningful comparisons to theory. Here, we present experimentally derived thermal rate constants for hydrogen atom recombination on platinum single-crystal surfaces, which are accurate enough to test established theoretical approximations. A quantum rate model is also presented, making possible a direct evaluation of the accuracy of commonly used approximations to adsorbate entropy. We find that neglecting the wave nature of adsorbed hydrogen atoms and their electronic spin degeneracy leads to a 10× to 1000× overestimation of the rate constant for temperatures relevant to heterogeneous catalysis. These quantum effects are also found to be important for nanoparticle catalysts.

Equilibrium and Dynamical Characteristics of Hydrogen Bond Bifurcations in Water-Water and Water-Ammonia Dimers: A Path Integral Molecular Dynamics Study

We present path integral molecular dynamics results that describe the effects of nuclear quantum fluctuations on equilibrium and dynamical characteristics pertaining to bifurcation pathways in hydrogen bonded dimers combining water and ammonia, at cryogenic temperatures of the order of 20 K. Along these isomerizations, the hydrogen atoms in the molecules acting as hydrogen-bond donors interchange their original dangling/connective characters. Our results reveal that the resulting quantum transition paths comprise three stages: the initial and final ones involve overall rotations during which the two protons retain their classical-like characteristics. Effects from quantum fluctuation are clearly manifested in the changes operated at the intermediate passages over transition states, as the spatial extents of the protons stretch over typical lengths comparable to the distances between connective and dangling basins of attractions. Consequently, the classical over-the-hill path is replaced by a tunneling controlled mechanism which, within the path integral perspective, can be cast in terms of concerted inter-basin migrations of polymer beads from dangling-to-connective and from connective-to-dangling, at practically no energy costs. We also estimated the characteristic timescales describing such interconversions within the approximate ring polymer rate theory. Effects derived from full and partial deuteration are also discussed.

Molecular Modeling in Anion Exchange Membrane Research: A Brief Review of Recent Applications

Anion Exchange Membrane (AEM) fuel cells have attracted growing interest, due to their encouraging advantages, including high power density and relatively low cost. AEM is a polymer matrix, which conducts hydroxide (OH-) ions, prevents physical contact of electrodes, and has positively charged head groups (mainly quaternary ammonium (QA) groups), covalently bound to the polymer backbone. The chemical instability of the quaternary ammonium (QA)-based head groups, at alkaline pH and elevated temperature, is a significant threshold in AEMFC technology. This review work aims to introduce recent studies on the chemical stability of various QA-based head groups and transportation of OH- ions in AEMFC, via modeling and simulation techniques, at different scales. It starts by introducing the fundamental theories behind AEM-based fuel-cell technology. In the main body of this review, we present selected computational studies that deal with the effects of various parameters on AEMs, via a variety of multi-length and multi-time-scale modeling and simulation methods. Such methods include electronic structure calculations via the quantum Density Functional Theory (DFT), ab initio, classical all-atom Molecular Dynamics (MD) simulations, and coarse-grained MD simulations. The explored processing and structural parameters include temperature, hydration levels, several QA-based head groups, various types of QA-based head groups and backbones, etc. Nowadays, many methods and software packages for molecular and materials modeling are available. Applications of such methods may help to understand the transportation mechanisms of OH- ions, the chemical stability of functional head groups, and many other relevant properties, leading to a performance-based molecular and structure design as well as, ultimately, improved AEM-based fuel cell performances. This contribution aims to introduce those molecular modeling methods and their recent applications to the AEM-based fuel cells research community.

Equilibrium-nonequilibrium ring-polymer molecular dynamics for nonlinear spectroscopy

Two-dimensional Raman and hybrid terahertz-Raman spectroscopic techniques provide invaluable insight into molecular structures and dynamics of condensed-phase systems. However, corroborating experimental results with theory is difficult due to the high computational cost of incorporating quantum-mechanical effects in the simulations. Here, we present the equilibrium-nonequilibrium ring-polymer molecular dynamics (RPMD), a practical computational method that can account for nuclear quantum effects on the two-time response function of nonlinear optical spectroscopy. Unlike a recently developed approach based on the double Kubo transformed (DKT) correlation function, our method is exact in the classical limit, where it reduces to the established equilibrium-nonequilibrium classical molecular dynamics method. Using benchmark model calculations, we demonstrate the advantages of the equilibrium-nonequilibrium RPMD over classical and DKT-based approaches. Importantly, its derivation, which is based on the nonequilibrium RPMD, obviates the need for identifying an appropriate Kubo transformed correlation function and paves the way for applying real-time path-integral techniques to multidimensional spectroscopy.

Capturing quantum effects with quasi-classical trajectories in the D + H+3 → H2D+ + H reaction

We present quasi-classical trajectory (QCT) cross sections, rate constants, and product state distributions for the D + H+3 → H₂D⁺ + H reaction. Using the same H+4 potential surface, the rate constants obtained from several QCT-based methods correcting for zero-point effects by Gaussian binning the product H₂D⁺ are compared to ring polymer molecular dynamics (RPMD) rate constants [Bulut et al., J. Phys. Chem. A, 2019, 123, 8766] which include quantum effects and to recent experimentally derived rate constants [Bowen et al., J. Chem. Phys., 2021, 154, 084307]. QCT with standard binning predicts rate constants that increase slowly as the temperature decreases from 1500 to 100 K. In contrast, the RPMD rate constants decrease rapidly with decreasing temperature. By 100 K, the QCT standard binning rate constant is more than 3 orders of magnitude larger than the RPMD rate constant. We show that QCT with Gaussian binning and proper normalization captures the zero-point effects and reproduces the RPMD rate constants over a large temperature range. Furthermore, the simple technique of counting only reactive trajectories with vibrational energy above the product zero-point energy matches the RPMD results well down to ∼300 K. The present Gaussian binned rate constants are in fair agreement with new experimentally derived rate constants from 100 to 1500 K. However, because the Gaussian binned rate constants do not include tunneling, important at lower temperatures, and the RPMD and experimentally derived rate constants have significant differences, the roles of the competing effects of zero-point energy, internal excitation of the H+3, and quantum tunneling are not simple and require further study for a consistent picture of the dynamics. Since rate constants for complex forming reactions, such as the title reaction, are difficult to converge with RPMD, alternative QCT-based methods, which include quantum effects and in addition provide product state distributions as described here, are highly desirable.

Chemical reactivity prediction: current methods and different application areas

The ability to predict chemical reactivity of a molecule is highly desirable in drug discovery, both ex vivo (synthetic route planning, formulation, stability) and in vivo: metabolic reactions determine pharmacodynamics, pharmacokinetics and potential toxic effects, and early assessment of liabilities is vital to reduce attrition rates in later stages of development. Quantum mechanics offer a precise description of the interactions between electrons and orbitals in the breaking and forming of new bonds. Modern algorithms and faster computers have allowed the study of more complex systems in a punctual and accurate fashion, and answers for chemical questions around stability and reactivity can now be provided. Through machine learning, predictive models can be built out of descriptors derived from quantum mechanics and cheminformatics, even in the absence of experimental data to train on. In this article, current progress on computational reactivity prediction is reviewed: applications to problems in drug design, such as modelling of metabolism and covalent inhibition, are highlighted and unmet challenges are posed.

Experimental and theoretical studies of the gas-phase reactions of O(1D) with H2O and D2O at low temperature

Here we report the results of an experimental and theoretical study of the gas-phase reactions between O(¹D) and H₂O and O(¹D) and D₂O at room temperature and below. On the experimental side, the kinetics of these reactions have been investigated over the 50-127 K range using a continuous flow Laval nozzle apparatus, coupled with pulsed laser photolysis and pulsed laser induced fluorescence for the production and detection of O(¹D) atoms respectively. Experiments were also performed at 296 K in the absence of a Laval nozzle. On the theoretical side, the existing full-dimensional ground X ¹A potential energy surface for the H₂O₂ system involved in this process has been reinvestigated and enhanced to provide a better description of the barrierless H-atom abstraction pathway. Based on this enhanced potential energy surface, quasiclassical trajectory calculations and ring polymer molecular dynamics simulations have been performed to obtain low temperature rate constants. The measured and calculated rate constants display similar behaviour above 100 K, showing little or no variation as a function of temperature. Below 100 K, the experimental rate constants increase dramatically, in contrast to the essentially temperature independent theoretical values. The possible origins of the divergence between experiment and theory at low temperatures are discussed.

Testing the quasicentroid molecular dynamics method on gas-phase ammonia

Quasicentroid molecular dynamics (QCMD) is a path-integral method for approximating nuclear quantum effects in dynamics simulations, which has given promising results for gas- and condensed-phase water. In this work, by simulating the infrared spectrum of gas-phase ammonia, we test the feasibility of extending QCMD beyond water. Overall, QCMD works as well for ammonia as for water, reducing or eliminating blue shifts from the classical spectrum without introducing the artificial red shifts or broadening associated with other imaginary-time path-integral methods. However, QCMD gives only a modest improvement over the classical spectrum for the position of the symmetric bend mode, which is highly anharmonic (since it correlates with the inversion pathway). We expect QCMD to have similar problems with large-amplitude degrees of freedom in other molecules but otherwise to work as well as for water.

Isotopic separation of helium through graphyne membranes: a ring polymer molecular dynamics study

Microscopic-level understanding of the separation mechanism for two-dimensional (2D) membranes is an active area of research due to potential implications of this class of membranes for various technological processes. Helium (He) purification from the natural resources is of particular interest due to the shortfall in its production. In this work, we applied the ring polymer molecular dynamics (RPMD) method to graphdiyne (Gr2) and graphtriyne (Gr3) 2D membranes having variable pore sizes for the separation of He isotopes, and compare for the first time with rigorous quantum calculations. We found that the transmission rate through Gr3 is many orders of magnitude greater than Gr2. The selectivity of either isotope at low temperatures is a consequence of a delicate balance between the zero-point energy effect and tunneling of ⁴He and ³He. In particular, a remarkable tunneling effect is reported on the Gr2 membrane at 10 K, leading to a much larger permeation of the lighter species as compared to the heavier isotope. RPMD provides an efficient approach for studying the separation of He isotopes, taking into account quantum effects of light nuclei motions at low temperatures, which classical methods fail to capture.

Computational approaches to dissociative chemisorption on metals: towards chemical accuracy

We review the state-of-the-art in the theory of dissociative chemisorption (DC) of small gas phase molecules on metal surfaces, which is important to modeling heterogeneous catalysis for practical reasons, and for achieving an understanding of the wealth of experimental information that exists for this topic, for fundamental reasons. We first give a quick overview of the experimental state of the field. Turning to the theory, we address the challenge that barrier heights (E_b, which are not observables) for DC on metals cannot yet be calculated with chemical accuracy, although embedded correlated wave function theory and diffusion Monte-Carlo are moving in this direction. For benchmarking, at present chemically accurate E_b can only be derived from dynamics calculations based on a semi-empirically derived density functional (DF), by computing a sticking curve and demonstrating that it is shifted from the curve measured in a supersonic beam experiment by no more than 1 kcal mol^-1. The approach capable of delivering this accuracy is called the specific reaction parameter (SRP) approach to density functional theory (DFT). SRP-DFT relies on DFT and on dynamics calculations, which are most efficiently performed if a potential energy surface (PES) is available. We therefore present a brief review of the DFs that now exist, also considering their performance on databases for E_b for gas phase reactions and DC on metals, and for adsorption to metals. We also consider expressions for SRP-DFs and briefly discuss other electronic structure methods that have addressed the interaction of molecules with metal surfaces. An overview is presented of dynamical models, which make a distinction as to whether or not, and which dissipative channels are modeled, the dissipative channels being surface phonons and electronically non-adiabatic channels such as electron-hole pair excitation. We also discuss the dynamical methods that have been used, such as the quasi-classical trajectory method and quantum dynamical methods like the time-dependent wave packet method and the reaction path Hamiltonian method. Limits on the accuracy of these methods are discussed for DC of diatomic and polyatomic molecules on metal surfaces, paying particular attention to reduced dimensionality approximations that still have to be invoked in wave packet calculations on polyatomic molecules like CH₄. We also address the accuracy of fitting methods, such as recent machine learning methods (like neural network methods) and the corrugation reducing procedure. In discussing the calculation of observables we emphasize the importance of modeling the properties of the supersonic beams in simulating the sticking probability curves measured in the associated experiments. We show that chemically accurate barrier heights have now been extracted for DC in 11 molecule-metal surface systems, some of which form the most accurate core of the only existing database of E_b for DC reactions on metal surfaces (SBH10). The SRP-DFs (or candidate SRP-DFs) that have been derived show transferability in many cases, i.e., they have been shown also to yield chemically accurate E_b for chemically related systems. This can in principle be exploited in simulating rates of catalyzed reactions on nano-particles containing facets and edges, as SRP-DFs may be transferable among systems in which a molecule dissociates on low index and stepped surfaces of the same metal. In many instances SRP-DFs have allowed important conclusions regarding the mechanisms underlying observed experimental trends. An important recent observation is that SRP-DFT based on semi-local exchange DFs has so far only been successful for systems for which the difference of the metal work function and the molecule's electron affinity exceeds 7 eV. A main challenge to SRP-DFT is to extend its applicability to the other systems, which involve a range of important DC reactions of e.g. O₂, H₂O, NH₃, CO₂, and CH₃OH. Recent calculations employing a PES based on a screened hybrid exchange functional suggest that the road to success may be based on using exchange functionals of this category.

Neural network potential energy surface for the low temperature ring polymer molecular dynamics of the H2CO + OH reaction

A new potential energy surface (PES) and dynamical study of the reactive process of H₂CO + OH toward the formation of HCO + H₂O and HCOOH + H are presented. In this work, a source of spurious long range interactions in symmetry adapted neural network (NN) schemes is identified, which prevents their direct application for low temperature dynamical studies. For this reason, a partition of the PES into a diabatic matrix plus a NN many-body term has been used, fitted with a novel artificial neural network scheme that prevents spurious asymptotic interactions. Quasi-classical trajectory (QCT) and ring polymer molecular dynamics (RPMD) studies have been carried on this PES to evaluate the rate constant temperature dependence for the different reactive processes, showing good agreement with the available experimental data. Of special interest is the analysis of the previously identified trapping mechanism in the RPMD study, which can be attributed to spurious resonances associated with excitations of the normal modes of the ring polymer.

Insights into the Formation of Hydroxyl Radicals with Nonthermal Vibrational Excitation in the Meinel Airglow

To understand night time airglow in the Meinel bands and heat conversion from the highly excited OH radicals in the upper atmosphere via the important atmospheric reaction H + O₃ → OH + O₂, we report here a quasi-classical trajectory study of the reaction dynamics on a recently developed full-dimensional potential energy surface (PES). Our results indicate that the reaction energy of this highly exoergic reaction is almost exclusively channeled into the vibration of the OH product, underscoring an extreme departure from the statistical limit. The calculated OH vibrational distribution is highly inverted and peaks near the highest accessible vibrational state, in excellent agreement with experimental observations, validating the accuracy of the PES. More importantly, the dynamical origin of the nonthermal excitation of the OH vibrational mode is identified by its large projection onto the reaction coordinate at a small potential barrier in the entrance channel, which controls the energy flow into various degrees of freedom in the products.

Small Nuclear Quantum Effects in Scattering of H and D from Graphene

We study nuclear quantum effects in H/D sticking to graphene, comparing scattering experiments at near-zero coverage with classical, quantized, and transition-state calculations. The experiment shows H/D sticking probabilities that are indistinguishable from one another and markedly smaller than those expected from a consideration of zero-point energy shifts of the chemisorption transition state. Inclusion of dynamical effects and vibrational anharmonicity via ring-polymer molecular dynamics (RPMD) yields results that are in good agreement with the experimental results. RPMD also reveals that nuclear quantum effects, while modest, arise primarily from carbon and not from H/D motion, confirming the importance of a C atom rehybridization mechanism associated with H/D sticking on graphene.

Theoretical H + O3 rate coefficients from ring polymer molecular dynamics on an accurate global potential energy surface: assessing experimental uncertainties

Thermal rate coefficients and kinetic isotope effects have been calculated for an important atmospheric reaction H/D + O3 → OH/OD + O2 based on an accurate permutation invariant polynomial-neural network potential energy surface, using ring polymer molecular dynamics (RPMD), quasi-classical trajectory (QCT) and variational transition-state theory (VTST) with multidimensional tunneling. The RPMD approach yielded results that are generally in better agreement with experimental rate coefficients than the VTST and QCT ones, especially at low temperatures, attributable to its capacity to capture quantum effects such as tunneling and zero-point energy. The theoretical results support one group of existing experiments over the other. In addition, rate coefficients for the D + O3 → OD + O2 reaction are also reported using the same methods, which will allow a stringent assessment of future experimental measurements, thus helping to reduce the uncertainty in the recommended rate coefficients of this reaction.

Experimental study of the proton-transfer reaction C + H2+ → CH+ + H and its isotopic variant (D2+)

We report absolute integral cross section (ICS) measurements using a dual-source merged-fast-beams apparatus to study the titular reactions over the relative translational energy range of Er ∼ 0.01-10 eV. We used photodetachment of C- to produce a pure beam of atomic C in the ground electronic 3P term, with statistically populated fine-structure levels. The H2+ and D2+ were formed in an electron impact ionization source, with well known vibrational and rotational distributions. The experimental work is complemented by a theoretical study of the CH2+ electronic system in the reactant and product channels, which helps to clarify the possible reaction mechanisms underlying the ICS measurements. Our measurements provide evidence that the reactions are barrierless and exoergic. They also indicate the apparent absence of an intermolecular isotope effect, to within the total experimental uncertainties. Capture models, taking into account either the charge-induced dipole interaction potential or the combined charge-quadrupole and charge-induced dipole interaction potentials, produce reaction cross sections that lie a factor of ∼4 above the experimental results. Based on our theoretical study, we hypothesize that the reaction is most likely to proceed adiabatically through the 14A' and 14A'' states of CH2+via the reaction C(3P) + H2+(2Σ+g) → CH+(3Π) + H(2S). We also hypothesize that at low collision energies only H2+(v ≤ 2) and D2+(v ≤ 3) contribute to the titular reactions, due to the onset of dissociative charge transfer for higher vibrational v levels. Incorporating these assumptions into the capture models brings them into better agreement with the experimental results. Still, for energies ⪅0.1 eV where capture models are most relevant, the modified charge-induced dipole model yields reaction cross sections with an incorrect energy dependence and lying ∼10% below the experimental results. The capture cross section obtained from the combined charge-quadrupole and charge-induced dipole model better matches the measured energy dependence but lies ∼30-50% above the experimental results. These findings provide important guidance for future quasiclassical trajectory and quantum mechanical treatments of this reaction.

Kinetic study of the OH + HO2 → H2O + O2 reaction using ring polymer molecular dynamics and quantum dynamics

The reaction OH + HO2 → H2O + O2 is a prototype of radical-radical reactions. It plays an important role in interstellar/atmospheric chemistry and combustion, and considerable attention has thus been dedicated to its kinetics. In our previous work, we reported an accurate full-dimensional potential energy surface for the title reaction on the ground triplet electronic state. The quasi-classical trajectory (QCT) approach was employed to investigate its kinetics. Although the QCT rate coefficients were in good agreement with some experimental and theoretical results, QCT cannot account for the quantum mechanical effects, such as zero-point vibrational energy, recrossing, and tunneling, which may significantly affect the rate coefficients, particularly at low temperatures. In this work, the reduced-dimensional quantum dynamics and ring polymer molecular dynamics calculations were carried out to examine these effects and their impact on rate coefficients over the temperature range of 300-1300 K.

Advances and New Challenges to Bimolecular Reaction Dynamics Theory

Dynamics of bimolecular reactions in the gas phase are of foundational importance in combustion, atmospheric chemistry, interstellar chemistry, and plasma chemistry. These collision-induced chemical transformations are a sensitive probe of the underlying potential energy surface(s). Despite tremendous progress in past decades, our understanding is still not complete. In this Perspective, we survey the recent advances in theoretical characterization of bimolecular reaction dynamics, stimulated by new experimental observations, and identify key new challenges.

Calculations of quantum tunnelling rates for muonium reactions with methane, ethane and propane

Thermal rate constants for Mu + CH₄, Mu + C₂H₆ and Mu + C₃H₈ and their equivalent reactions with H were evaluated with ab initio instanton rate theory. The potential-energy surfaces are fitted using Gaussian process regression to high-level electronic-structure calculations evaluated around the tunnelling pathway. This method was able to successfully reproduce various experimental measurements for the rate constant of these reactions. However, it was not able to reproduce the faster-than-expected rate of Mu + C₃H₈ at 300 K reported by Fleming et al. [Phys. Chem. Chem. Phys., 2015, 17, 19901 and Phys. Chem. Chem. Phys., 2020, 22, 6326]. Analysis of our results indicates that the kinetic isotope effect at this temperature is not significantly influenced by quantum tunnelling. We consider many possible factors for the discrepancy between theory and experiment but conclude that in each case, the instanton approximation is unlikely to be the cause of the error. This is in part based on the good agreement we find between the instanton predictions and new multiconfigurational time-dependent Hartree (MCTDH) calculations for Mu + CH₄ using the same potential-energy surface. Further experiments will therefore be needed to resolve this issue.

VTST and RPMD kinetics study of the nine-body X + C2H6 (X ≡ H, Cl, F) reactions based on analytical potential energy surfaces

Thermal rate constants of nine-atom hydrogen abstraction reactions, X + C2H6 → HX + C2H5 (X ≡ H, Cl, F) with qualitatively different reaction paths, have been investigated using two kinetics approaches - variational transition state theory with multidimensional tunnelling (VTST/MT) and ring polymer molecular dynamics (RPMD) - and full dimensional analytical potential energy surfaces. For the H + C2H6 reaction, which proceeds through a noticeable barrier height of 11.62 kcal mol-1, kinetics approaches showed excellent agreement between them (with differences less than 30%) and with the experiment (with differences less than 60%) in the wide temperature range of 200-2000 K. For X = Cl and F, however, the situation is very different. The barrier height is either low or very low, 2.44 and 0.23 kcal mol-1, respectively, and the presence of van der Waals complexes in the entrance channel leads to a very flat topography and, consequently, imposes theoretical challenges. For the Cl(2P) reaction, VTST/MT underestimates the experimental rate constants (with differences less than 86%), and RPMD demonstrates better agreement (with differences less than 47%), although the temperature dependence is opposite to the experiment at low temperatures. Finally, for the F(2P) reaction, available experimental information shows discrepancies, both in the absolute values of the rate constants and also in the temperature dependence. Unfortunately, kinetics theories did not resolve this discrepancy. Different possible causes of these theory/experiment discrepancies were analyzed.

Capturing Deuteration Effects in a Molecular Mechanics Force Field: Deuterated THF and the THF-Water Miscibility Gap

Deuteration is a common chemical modification used in conjunction with experiments such as neutron scattering, NMR, and Fourier-transform infrared for the study of molecular systems. Under the Born-Oppenheimer (BO) approximation, while the underlying potential energy surface remains unchanged by isotopic substitutions, isotopic substitution still alters intramolecular vibrations, which in turn may alter intermolecular interactions. Molecular mechanics (MM) force fields used in classical molecular dynamics simulations are assumed to represent local approximations of the BO potential energy surfaces, and hence, MD simulations using simple isotopic mass substitutions should capture BO-compatible isotope effects. However, standard MM force-field parameterizations do not directly fit to the local harmonic quantum mechanical (QM) Hessian that describes the BO surface, but rather to QM normal-modes and/or mass-dependent internal-coordinate derived distortion energies. Here, using tetrahydrofuran (THF)-water mixtures as our model system, we show that not only does a simple mass-substitution approach fail to capture an experimentally characterized deuteration effect (the loss of the closed-loop miscibility gap associated with the complete deuteration of THF) but also it is necessary to generate new MM force-field parameters that correctly describe isotopic dependent vibrations to capture the experimental deuteration effect. We show that the origin of this failure is a result of using mass-dependent features to fit the THF MM force field, which unintentionally biases the bonded terms of the force field to represent only the isotopologue used during the original force-field parameterization. In addition, we make use of our isotopologue-corrected force field for D₈THF to examine the molecular origins of the isotope-dependent loss of the THF-water miscibility gap.

An accurate potential energy surface and ring polymer molecular dynamics study of the Cl + CH4→ HCl + CH3reaction

Thermal rate coefficients for the Cl + CH₄/CD₄reactions were studied on a new full-dimensional accurate potential energy surface with the spin–orbit corrections considered in the entrance channel.

Ring Polymer Molecular Dynamics in Gas-Surface Reactions: Inclusion of Quantum Effects Made Simple

Accurately modeling gas-surface collision dynamics presents a great challenge for theory, especially in the low-energy (or temperature) regime where quantum effects are important. Here, a path integral-based nonequilibrium ring polymer molecular dynamics (NE-RPMD) approach is adapted to calculate dissociative initial sticking probabilities (S₀) of H₂ on Cu(111) and D₂O on Ni(111), revealing the distinct quantum nature in the two benchmark surface reactions. NE-RPMD successfully captures quantum tunneling in H₂ dissociation at very low energies, where the quasi-classical trajectory (QCT) method suddenly fails. Additionally, QCT substantially overestimates S₀ of D₂O because of severe zero point energy (ZPE) leakage, even at collision energies greater than the ZPE-corrected barrier. Instead, NE-RPMD predicts S₀ values of D₂O in much improved agreement with reference results obtained by the quantum wavepacket method with reasonable corrections of the thermal contribution. Our results suggest NE-RPMD as a promising approach to model quantum effects in gas-surface reactions.

Dynamics of H + HeH+(v = 0, j = 0) → H2+ + He: Insight on the Possible Complex-Forming Behavior of the Reaction

The H + HeH⁺→ He + H₂⁺ reaction has been studied by means of a combination of theoretical approaches: a statistical quantum method (SQM), ring polymer molecular dynamics (RPMD), and the quasiclassical trajectory (QCT) method. Cross sections and rate constants have been calculated in an attempt to investigate the dynamics of the process. The comparison with previous calculations and experimental results reveals that despite the fact that statistical predictions seem to reproduce some of the overall observed features, the analysis at a more detailed state-to-state level shows noticeable deviations from a complex-forming dynamics. We find some differences in cross sections and rate constants obtained in the QCT calculation with a Gaussian binning procedure with respect to previous works in which the standard histogram binning was employed.

Quantum Effects on the D + H3+ → H2D+ + H Deuteration Reaction and Isotopic Variants

The title reaction and its isotopic variants are studied using quasi-classical trajectory (QCT) (without taking into account corrections to account for the possible zero point energy breakdown) and ring polymer molecular dynamics (RPMD) methods with a full dimensional and accurate potential energy surface which presents an exchange barrier of approximately 0.144 eV. The QCT rate constant increases when the temperature decreases from 1500 to 10 K. On the contrary, the RPMD rate constant decreases with decreasing temperature, in semiquantitative agreement with recent experimental results. The present RPMD results are in between the thermal and translational experimental rate constants, extracted from the measured data to eliminate the initial vibrational excitation of H₃⁺, obtained in an arc discharge. The difference between the present RPMD results and experimental values is attributed to the possible existence of non thermal vibrational excitation of H₃⁺, not completely removed by the semiempirical model used for the analysis of the experimental results. Also, it is found that, below 200 K, the RPMD trajectories are trapped, forming long-lived collision complexes, with lifetimes longer than 1 ns. These collision complexes can fragment by either redissociating back to reactants or react to products, in the two cases tunneling through the centrifugal and reaction barriers, respectively. The contribution of the formation of the complex to the total deuteration rate should be calculated with more accurate quantum methods, as has been found recently for reactions of larger systems, and the present four atoms system is a good candidate to benchmark the adequacy of RPMD method at temperatures below 100 K.