Accelerated Entropic Path Sampling with a Bidirectional Generative Adversarial Network

The role of entropy in mediating the dynamic outcomes of chemical reactions remains largely unknown. To evaluate the change of entropy along post-transition state paths, we have previously developed entropic path sampling that computes configurational entropy from an ensemble of reaction trajectories. However, one major caveat of this approach lies in its high computational demand: about 2000 trajectories are needed to converge the computation of an entropic profile. Here, by leveraging a deep generative model, we developed an accelerated entropic path sampling approach that evaluates entropic profiles using merely a few hundred reaction dynamic trajectories. The new method, called bidirectional generative adversarial network-entropic path sampling, can enhance the estimation of probability density functions of molecular configurations by generating pseudo-molecular configurations that are statistically indistinguishable from the true data. The method was established using cyclopentadiene dimerization, in which we reproduced the reference entropic profiles (derived from 2480 trajectories) using merely 124 trajectories. The method was further benchmarked using three reactions with symmetric post-transition-state bifurcation, including endo-butadiene dimerization, 5-fluoro-1,3-cyclopentadiene dimerization, and 5-methyl-1,3-cyclopentadiene dimerization. The results indicate the existence of a "hidden entropic intermediate", which is a dynamic species that binds to a local entropic maximum where no free energy minimum is formed.

Symmetric Post-Transition State Bifurcation Reactions with Berry Pseudomagnetic Fields

We investigate how the Berry force (i.e., the pseudomagnetic force operating on nuclei as induced by electronic degeneracy and spin-orbit coupling (SOC)) might modify a post-transition state bifurcation (PTSB) reaction path and affect product selectivity for situations when multiple products share the same transition state. To estimate the magnitude of this effect, Langevin dynamics are performed on a model system with a valley-ridge inflection (VRI) point in the presence of a magnetic field (that mimics the Berry curvature). We also develop an analytic model for such selectivity that depends on key parameters such as the surface topology, the magnitude of the Berry force, and the nuclear friction. Within this dynamical model, static electronic structure calculations (at the level of generalized Hartree-Fock with spin-orbit coupling (GHF+SOC) theory) suggest that electronic spin induced Berry force effects may indeed lead to noticeable changes in methoxy radical isomerization.

Prediction of Kinetic Product Ratios: Investigation of a Dynamically Controlled Case

Of the various factors influencing kinetically controlled product ratios, the role of nonstatistical dynamics is arguably the least well understood. In this paper, reactions were chosen in which dynamics played a dominant role in product selection, by design. Specifically, the reactions studied were the ring openings of cyclopropylidene to allene and tetramethylcyclopropylidene to tetramethylallene (2,4-dimethylpenta-2,3-diene). Both reactions have intrinsic reaction coordinates that bifurcate symmetrically, leading to products that are enantiomeric once the atoms are uniquely labeled. The question addressed in the study was whether the outcomes─that is, which product well on the potential energy surface was selected─could be predicted from their initial conditions for individual trajectories in quasiclassical dynamics simulations. Hybrid potentials were developed based on cooperative interaction between molecular mechanics and artificial neural networks, trained against data from electronic structure calculations. These potentials allowed simulations of both gas-phase and condensed-phase reactions. The outcome was that, for both reactions, prediction of initial selection of product wells could be made with >95% success from initial conditions of the trajectories in the gas phase. However, when trajectories were run for longer, looking for "final" products for each trajectory, the predictability dropped off dramatically. In the gas-phase simulations, this drop off was caused by trajectories hopping between product wells on the potential energy surface. That behavior could be suppressed in condensed phases, but then new uncertainty was introduced because the intermolecular interactions between solute and bath, necessary to permit intermolecular energy transfer and cooling of the hot initial products, often led to perturbations of the initial directions of trajectories on the potential energy surface. It would consequently appear that a general ability to predict outcomes for reactions in which nonstatistical dynamics dominate remains a challenge even in the age of sophisticated machine-learning capabilities.

Efficient and Accurate Description of Diels-Alder Reactions Using Density Functional Theory

Modeling chemical reactions using Quantum Chemistry is a widely used predictive strategy capable to complement experiments in order to understand the intrinsic mechanisms guiding the chemicals towards the most favorable reaction products. However, at this purpose, it is mandatory to use reliable and computationally tractable theoretical methods. In this work, we focus on six Diels-Alder reactions of increasing complexity and perform an extensive benchmark of middle- to low-cost computational approaches to predict the characteristic reactions energy barriers. We found that Density Functional Theory, using the ωB97XD, LC-ωPBE, CAM-B3LYP, M11 and MN12SX functionals, with empirical dispersion corrections coupled to an affordable 6-31G basis set, provides quality results for this class of reactions, at a small computational effort. Such efficient and reliable simulation protocol opens perspectives for hybrid QM/MM molecular dynamics simulations of Diels-Alder reactions including explicit solvation.

Atomistic Simulations for Reactions and Vibrational Spectroscopy in the Era of Machine Learning─Quo Vadis?

Atomistic simulations using accurate energy functions can provide molecular-level insight into functional motions of molecules in the gas and in the condensed phase. This Perspective delineates the present status of the field from the efforts of others and some of our own work and discusses open questions and future prospects. The combination of physics-based long-range representations using multipolar charge distributions and kernel representations for the bonded interactions is shown to provide realistic models for the exploration of the infrared spectroscopy of molecules in solution. For reactions, empirical models connecting dedicated energy functions for the reactant and product states allow statistically meaningful sampling of conformational space whereas machine-learned energy functions are superior in accuracy. The future combination of physics-based models with machine-learning techniques and integration into all-purpose molecular simulation software provides a unique opportunity to bring such dynamics simulations closer to reality.

Effect of solvent motions on the dynamics of the Diels-Alder reaction

How solvent motions affect the dynamics of chemical reactions in which the solute undergoes a substantial shape change is a fundamental but elusive issue. This work utilizes reactive simulation and Grote-Hynes theory to explore the effect of solvent motions on the dynamics of the Diels-Alder reaction (in the reverse direction, this reaction involves very substantial solute expansion) in aprotic solvents. The results reveal that the solvent environment is not sufficiently constraining to influence transition state passage dynamics, with the calculated transmission coefficients being close to unity. Even when solvent motions are suppressed or artificially slowed down, the solvent only affects the reaction dynamics in the transition state region to a very small extent. The only notable effect of solvent occurs far from the transition state region and corresponds to caging of the reactants within the reactant well.

Influence of Water and Enzyme on the Post-Transition State Bifurcation of NgnD-Catalyzed Ambimodal [6+4]/[4+2] Cycloaddition

The enzyme NgnD catalyzes an ambimodal cycloaddition that bifurcates to [6+4]- and [4+2]-adducts. Both products have been isolated in experiments, but it remains unknown how enzyme and water influence the bifurcation selectivity at the femtosecond time scale. Here, we study the impact of water and enzyme on the post-transition state bifurcation of NgnD-catalyzed [6+4]/[4+2] cycloaddition by integrating quantum mechanics/molecular mechanics quasiclassical dynamics simulations and biochemical assays. The ratio of [6+4]/[4+2] products significantly differs in the gas phase, water, and enzyme. Biochemical assays were employed to validate computational predictions. The study informs how water and enzyme affect the bifurcation selectivity through perturbation of the reaction dynamics in the femtosecond time scale, revealing the fundamental roles of condensed media in dynamically controlling the chemical selectivity for biosynthetic reactions.

Entropic Path Sampling: Computational Protocol to Evaluate Entropic Profile along a Reaction Path

Fleeting intermediates constitute dynamically stepwise mechanisms. They have been characterized in molecular dynamics trajectories, but whether these intermediates form a free energy minimum to become entropic intermediates remains elusively defined. We developed a computational protocol known as entropic path sampling to evaluate the entropic variation of reacting species along a reaction path based on an ensemble of trajectories. Using cyclopentadiene dimerization as a model reaction, we observed an entropy maximum along the reaction path which originates from an enhanced conformational flexibility as the reacting species enter into a flat energy region. As the reacting species further approach product formation, unfavorable entropic restriction fails to offset the potential energy drop, resulting in no free energy minimum along the post-TS pathway. Our results show that cyclopentadiene dimerization involves an entropy maximum that leads to dynamic intermediates with elongated lifetimes, but the reaction does not involve entropic intermediates.

Vibrational Energy Relaxation of Deuterium Fluoride in d-Dichloromethane: Insights from Different Potentials

Vibrationally excited deuterium fluoride (DF) formed by fluorine atom reaction with a solvent was found (Science, 2015, 347, 530) to relax rapidly (less than 10 ps) in acetonitrile-d₃ (CD₃CN) and dichloromethane-d₂ (CD₂Cl₂). However, insights into how CD₂Cl₂ facilitates this energy relaxation have so far been lacking, given the weak interaction between DF and a single CD₂Cl₂. In this work, we report the results of reactive simulations with a two-state reactive empirical valence bond (EVB) potential to study the energy deposited into nascent DF after transition-state passage and of nonequilibrium molecular dynamics simulations using multiple different potential energy functions to model the relaxation dynamics. For these second simulations, we used the standard Merck molecular force field (MMFF) potential, an MMFF-based covalent-ionic empirical valence bond (EVB) potential (EVB^CI), a newly developed potential [referred to as MMFF(rDF)] which extends upon the MMFF potential by making the DF/CD₂Cl₂ interaction depend on the value of the D-F bond stretching coordinate and by taking the anisotropic charge distribution of the solvent molecules into account, the polarizable atomic multipole optimized energetics for biomolecular applications (AMOEBA) potential, and the quantum mechanics/molecular mechanics (QM/MM) potential. The relaxation is revealed to be highly sensitive to the potential used. Neither standard MMFF nor EVB^CI reproduces the experimentally observed rapid relaxation dynamics, and they also fail to provide a good description of the interaction potential between DF and CD₂Cl₂ as calculated using CCSD(T)-F12. This is attributed to the use of a point-charge model for the solute and to failing to model the anisotropic electrostatic properties of CD₂Cl₂. The MMFF(rDF), AMOEBA, and QM/MM potentials all reproduce the CCSD(T)-F12 two-body DF---CD₂Cl₂ interaction potential rather well but only with the QM/MM approach is fast vibrational relaxation obtained (lifetimes of ∼288, ∼186, and ∼8 ps, respectively), which we attribute to differences in the solute-solvent local structure. With QM/MM, a unique "many-body" interaction pattern in which DF is in close contact with two solvent Cl atoms and more than three solvent D atoms is found, but this structure is not seen with other potentials. The QM/MM dynamics also display enhanced solute-solvent interactions with vibrationally excited DF that induce a DF band redshift and hence a resonant overlap with solvent C-D modes, which facilitate the intermolecular energy transfer. Our work also suggests that potentials used to model energy relaxation need to capture the fine structure of solute-solvent interactions and not just the two-body part.

Reactive Molecular Dynamics at Constant Pressure via Nonreactive Force Fields: Extending the Empirical Valence Bond Method to the Isothermal-Isobaric Ensemble

The Empirical Valence Bond (EVB) method offers a suitable framework to obtain reactive potentials through the coupling of nonreactive force fields. In this formalism, most of the implemented coupling terms are built using functional forms that depend on spatial coordinates, while parameters are fitted against reference data to model the change of chemistry between the participating nonreactive states. In this work, we demonstrate that the use of such coupling terms precludes the computation of the stress tensor for condensed phase systems and prevents the possibility to carry out EVB molecular dynamics in the isothermal-isobaric (NPT) ensemble. Alternatively, we make use of coupling terms that depend on the energy gaps, defined as the energy differences between the participating nonreactive force fields, and derive a general expression for the EVB stress tensor suitable for computation. Implementation of this new methodology is tested for a model of a single reactive malonaldehyde solvated in nonreactive water. Mass densities and probability distributions for the values of the energy gaps computed in the NPT ensemble reveal a negligible role of the reactive potential in the limit of low concentrated solutions, thus corroborating for the first time the validity of approximations based on the canonical NVT ensemble, customarily adopted for EVB simulations. The presented formalism also aims to contribute to future implementations and extensions of the EVB method to research the limit of highly concentrated solutions.

Bond Memory in Dynamically Determined Stereoselectivity

The carboborative ring contraction of cyclohexenes exhibits an abnormal selectivity pattern in which a formally concerted double migration gives rise to predominant but not exclusive inversion products. In dynamic trajectories, the inversion and retention products are formed from the same transition state, and the trajectories accurately account for the experimental product ratios. The unusual origin of the selectivity is the dynamically retained non-equivalence of newly formed versus pre-existing bonds after the first bond migration.

Low dimensional representations along intrinsic reaction coordinates and molecular dynamics trajectories using interatomic distance matrices

Most chemical transformations (reactions or conformational changes) that are of interest to researchers have many degrees of freedom, usually too many to visualize without reducing the dimensionality of the system to include only the most important atomic motions. In this article, we describe a method of using Principal Component Analysis (PCA) for analyzing a series of molecular geometries (e.g., a reaction pathway or molecular dynamics trajectory) and determining the reduced dimensional space that captures the most structural variance in the fewest dimensions. The software written to carry out this method is called PathReducer, which permits (1) visualizing the geometries in a reduced dimensional space, (2) determining the axes that make up the reduced dimensional space, and (3) projecting the series of geometries into the low-dimensional space for visualization. We investigated two options to represent molecular structures within PathReducer: aligned Cartesian coordinates and matrices of interatomic distances. We found that interatomic distance matrices better captured non-linear motions in a smaller number of dimensions. To demonstrate the utility of PathReducer, we have carried out a number of applications where we have projected molecular dynamics trajectories into a reduced dimensional space defined by an intrinsic reaction coordinate. The visualizations provided by this analysis show that dynamic paths can differ greatly from the minimum energy pathway on a potential energy surface. Viewing intrinsic reaction coordinates and trajectories in this way provides a quick way to gather qualitative information about the pathways trajectories take relative to a minimum energy path. Given that the outputs from PCA are linear combinations of the input molecular structure coordinates (i.e., Cartesian coordinates or interatomic distances), they can be easily transferred to other types of calculations that require the definition of a reduced dimensional space (e.g., biased molecular dynamics simulations).

Anharmonic Molecular Mechanics: Ab Initio Based Morse Parametrizations for the Popular MM3 Force Field

Methodologies for creating reactive potential energy surfaces from molecular mechanics force-fields are becoming increasingly popular. To date, molecular mechanics force-fields in biochemistry and small molecule organic chemistry tend to use harmonic expressions to treat bonding stretches, which is a poor approximation in reactive and nonequilibirum molecular dynamics simulations since bonds are often displaced significantly from their equilibrium positions. For such applications there is need for a better treatment of anharmonicity. In this contribution, Morse bonding potentials have been extensively parametrized for the atom types in the MM3 force field of Allinger and co-workers using high level CCSD(T)(F12*) energies. To our knowledge this is among the first instances of a comprehensive parametrization of Morse potentials in a popular organic chemistry force field. In the context of molecular dynamics simulations, these data will: (1) facilitate the fitting of reactive potential energy surfaces using empirical valence bond approaches and (2) enable more accurate treatments of energy transfer.

Mechanisms and Beyond: Elucidation of Fluxional Dynamics by Exchange NMR Spectroscopy

Detailed mechanistic information is crucial to our understanding of reaction pathways and selectivity. Dynamic exchange NMR techniques, in particular 2D exchange spectroscopy (EXSY) and its modifications, provide indispensable intricate information on the mechanisms of organic and inorganic reactions and other phenomena, for example, the dynamics of interfacial processes. In this Review, key results from exchange NMR studies of small molecules over the last few decades are systemised and discussed. After a brief introduction to the theory, the key types of dynamic processes are identified and fundamental examples given of intra- and intermolecular reactions, which, in turn, could involve, or not, bond-making and bond-breaking events. Following that logic, internal molecular rotation, intramolecular stereomutation and molecular recognition will first be considered because they do not typically involve bond breaking. Then, rearrangements, substitution-type reactions, cyclisations, additions and other processes affecting chemical bonds will be discussed. Finally, interfacial molecular dynamics and unexpected combinations of different types of fluxional processes will also be highlighted. How exchange NMR spectroscopy helps to identify conformational changes, coordination and molecular recognition processes as well as quantify reaction energy barriers and extract detailed mechanistic information by using reaction rate theory in conjunction with computational techniques will be shown.

EVB and polarizable MM study of energy relaxation in fluorine–acetonitrile reactions

Many-body effects can impact on rates of energy transfer from a ‘hot’ DF solute to acetonitrile solvent.

Post-transition state bifurcations induce dynamical detours in Pummerer-like reactions

A post-transition state bifurcation (PTSB) involved in a Pummerer-type rearrangement is characterized using density functional theory (DFT) calculations on potential energy stationary points and direct dynamics simulations. A sensitivity of the ratio of products produced via this PTSB to solvent dielectric constant is revealed and implications of such a dependence for selectivity control of organic reactions are discussed.

Alkyl hydrogen atom abstraction reactions of the CN radical with ethanol

We present a study of the abstraction of alkyl hydrogen atoms from the β and α positions of ethanol by the CN radical in solution using the Empirical Valence Bond (EVB) method. We have built separate 2 × 2 EVB models for the H_β and H_α reactions, where the atom transfer is parameterized using ab initio calculations. The intra- and intermolecular potentials of the reactant and product molecules were modelled with the General AMBER Force Field, with some modifications. We have carried out the dynamics in water and chloroform, which are solvents of contrasting polarity. We have computed the potential of mean force for both abstractions in each of the solvents. They are found to have a small and early barrier along the reaction coordinate with a large energy release. Analyzing the solvent structure around the reaction system, we have found two solvents to have little effect on either reaction. Simulating the dynamics from the transition state, we also study the fate of the energies in the HCN vibrational modes. The HCN molecule is born vibrationally hot in the CH stretch in both reactions and additionally in the HCN bends for the H_α abstraction reaction. In the early stage of the dynamics, we find that the CN stretch mode gains energy at the expense of the energy in CH stretch mode.

Failure and Redemption of Statistical and Nonstatistical Rate Theories in the Hydroboration of Alkenes

Our previous work found that canonical forms of transition state theory incorrectly predict the regioselectivity of the hydroboration of propene with BH3 in solution. In response, it has been suggested that alternative statistical and nonstatistical rate theories can adequately account for the selectivity. This paper uses a combination of experimental and theoretical studies to critically evaluate the ability of these rate theories, as well as dynamic trajectories and newly developed localized statistical models, to predict quantitative selectivities and qualitative trends in hydroborations on a broader scale. The hydroboration of a series of terminally substituted alkenes with BH3 was examined experimentally, and a classically unexpected trend is that the selectivity increases as the alkyl chain is lengthened far from the reactive centers. Conventional and variational transition state theories can predict neither the selectivities nor the trends. The canonical competitive nonstatistical model makes somewhat better predictions for some alkenes but fails to predict trends, and it performs poorly with an alkene chosen to test a specific prediction of the model. Added nonstatistical corrections to this model make the predictions worse. Parametrized Rice-Ramsperger-Kassel-Marcus (RRKM)-master equation calculations correctly predict the direction of the trend in selectivity versus alkene size but overpredict its magnitude, and the selectivity with large alkenes remains unpredictable with any parametrization. Trajectory studies in explicit solvent can predict selectivities without parametrization but are impractical for predicting small changes in selectivity. From a lifetime and energy analysis of the trajectories, "localized RRKM-ME" and "competitive localized noncanonical" rate models are suggested as steps toward a general model. These provide the best predictions of the experimental observations and insight into the selectivities.

Post-transition state bifurcations gain momentum – current state of the field

The existence of post-transition state bifurcations on potential energy surfaces for organic and biological reaction mechanisms has been known for decades, but recently, new reports of bifurcations have been occurring at a much higher rate. Beyond simply discovering bifurcations, computational chemists are developing techniques to understand what aspects of molecular structure and vibrations control the product selectivity in systems containing bifurcations. For example, the distribution of products seen in simulations has been found to be extremely sensitive to the local environment of the reacting system (i.e. the presence of a catalyst, enzyme, or explicit solvent molecules). The outlook for the future of this field is discussed, with an eye towards the application of the principles discussed here by experimental chemists to design a reaction setup to efficiently generate desired products.

Navigating Past a Fork in the Road: Carbocation-π Interactions Can Manipulate Dynamic Behavior of Reactions Facing Post-Transition-State Bifurcations

Dynamics calculations are described for carbocation rearrangements involving product-forming pathways with post-transition-state bifurcations. We show that noncovalent interactions with associated benzene rings (a simple model of aromatic amino acid side chains) can switch inherent dynamical tendencies for competing modes of disrotation, establishing that meaningful changes in dynamically controlled product selectivity can be achieved with few weak noncovalent interactions.

Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces

In this Introduction, we show the basic problems of non-statistical and non-equilibrium phenomena related to the papers collected in this themed issue. Over the past few years, significant advances in both computing power and development of theories have allowed the study of larger systems, increasing the time length of simulations and improving the quality of potential energy surfaces. In particular, the possibility of using quantum chemistry to calculate energies and forces 'on the fly' has paved the way to directly study chemical reactions. This has provided a valuable tool to explore molecular mechanisms at given temperatures and energies and to see whether these reactive trajectories follow statistical laws and/or minimum energy pathways. This themed issue collects different aspects of the problem and gives an overview of recent works and developments in different contexts, from the gas phase to the condensed phase to excited states.This article is part of the themed issue 'Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces'.

Analysis of the Acting Forces in a Theory of Catalysis and Mechanochemistry

The theoretical description of a chemical process resulting from the application of mechanical or catalytical stress to a molecule is performed by the generation of an effective potential energy surface (PES). Changes for minima and saddle points by the stress are described by Newton trajectories (NTs) on the original PES. From the analysis of the acting forces we postulate the existence of pulling corridors built by families of NTs that connect the same stationary points. For different exit saddles of different height we discuss the corresponding pulling corridors; mainly by simple two-dimensional surface models. If there are different exit saddles then there can exist saddles of index two, at least, between. Then the case that a full pulling corridor crosses a saddle of index two is the normal case. It leads to an intrinsic hysteresis of such pullings for the forward or the backward reaction. Assuming such relations we can explain some results in the literature. A new finding is the existence of roundabout corridors that can switch between different saddle points by a reversion of the direction. The findings concern the mechanochemistry of molecular systems under a mechanical load as well as the electrostatic force and can be extended to catalytic and enzymatic accelerated reactions. The basic and ground ansatz includes both kinds of forces in a natural way without an extra modification.

Migration of small ligands in globins: Xe diffusion in truncated hemoglobin N

In heme proteins, the efficient transport of ligands such as NO or O₂ to the binding site is achieved via ligand migration networks. A quantitative assessment of ligand diffusion in these networks is thus essential for a better understanding of the function of these proteins. For this, Xe migration in truncated hemoglobin N (trHbN) of Mycobacterium Tuberculosis was studied using molecular dynamics simulations. Transitions between pockets of the migration network and intra-pocket relaxation occur on similar time scales (10 ps and 20 ps), consistent with low free energy barriers (1-2 kcal/mol). Depending on the pocket from where Xe enters a particular transition, the conformation of the side chains lining the transition region differs which highlights the coupling between ligand and protein degrees of freedom. Furthermore, comparison of transition probabilities shows that Xe migration in trHbN is a non-Markovian process. Memory effects arise due to protein rearrangements and coupled dynamics as Xe moves through it.

Binding and transport of ligands in proteins is essential, in particular in globular proteins which often exhibit internal cavities. In truncated Hemoglobin N (trHbN) these cavities are arranged as a network with particular connectivities. The present work supports the notion that ligand diffusion in trHbN is an active process and coupled to the protein dynamics. Furthermore, transition probabilities between neighboring pockets depend on the location from where the ligand entered the transition, which is typical for non-Markovian processes. Hence, ligand migration in trHbN exhibits memory effects due to dynamical coupling between the protein and ligand motion.

Reactive force fields made simple

Combining QMDFF with EVB allows to generate reactive force fields of useful quality for widely varying reactions with minimal effort.