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

Predicting Selectivity with a Bifurcating Surface: Inaccurate Model or Inaccurate Statistics of Dynamics?

Reactions on post-transition-state bifurcation (PTSB) energy surfaces are an important class of reaction in which classical rate theories, such as the transition state theory, fail to account for the selectivity. Quasiclassical trajectory molecular dynamic (QCT-MD) simulation is an important computational approach to understanding reactions mechanisms, especially for reactions that cannot be predicted from conventional rate theories. However, the applicability of direct dynamic simulations is hampered by huge computational costs for collecting a statistically meaningful set of trajectories, making it difficult to compare simulation results with theoretical or physical insights-based predictions (non-MD predictions). In this work, we examine the PTSB of Schmidt-Aubé reactions studied by Tantillo and co-workers. With machine-learning using kernel-ridge regression (KRR) to predict atomic forces, statistical reliability was enhanced by significantly increasing the number of trajectories. With KRR, the bottleneck of simulating dynamics (atomic forces in QCT-MD with density functional theory) was accelerated more than 100-fold. We found that this KRR-aided QCT-MD approach is successful in predicting branching ratios with a much larger number of trajectories. With our approach, statistical errors are greatly reduced, and hypothetical non-MD models for predicting selectivity are tested with much higher confidence. By comparison with non-MD models, dynamical properties that affect branching ratios become more clearly described.

Trajectory-Based Time-Resolved Mechanism for Benzene Reductive Elimination from Cyclopentadienyl Mo/W Phenyl Hydride Complexes

Calculated potential energy structures and landscapes are very often used to define the sequence of reaction steps in an organometallic reaction mechanism and interpret kinetic isotope effect (KIE) measurements. Underlying most of this structure-to-mechanism translation is the use of statistical rate theories without consideration of atomic/molecular motion. Here we report direct dynamics simulations for an organometallic benzene reductive elimination reaction, where nonstatistical intermediates and dynamic-controlled pathways were identified. Specifically, we report single spin state as well as mixed spin state quasiclassical direct dynamics trajectories in the gas phase and explicit solvent for benzene reductive elimination from Mo and W bridged cyclopentadienyl phenyl hydride complexes ([Me2Si(C5Me4)2]M(H)(Ph), M = Mo and W). Different from the energy landscape mechanistic sequence, the dynamics trajectories revealed that after the benzene C-H bond forming transition state (often called reductive coupling), σ-coordination and π-coordination intermediates are either skipped or circumvented and that there is a direct pathway to forming a spin flipped solvent caged intermediate, which occurs in just a few hundred femtoseconds. Classical molecular dynamics simulations were then used to estimate the lifetime of the caged intermediate, which is between 200 and 400 picoseconds. This indicates that when the η2-π-coordination intermediate is formed, it occurs only after the first formation of the solvent-caged intermediate. This dynamic mechanism intriguingly suggests the possibility that the solvent-caged intermediate rather than a coordination intermediate is responsible (or partially responsible) for the inverse KIE value experimentally measured for W. Additionally, this dynamic mechanism prompted us to calculate the kH/kD KIE value for the C-H bonding forming transition states of Mo and W. Surprisingly, Mo gave a normal value, while W gave an inverse value, albeit small, due to a much later transition state position.

Bispericyclic Ambimodal Dimerization of Pentafulvene: The Origin of Asynchronicity and Kinetic Selectivity of the Endo Transition State

The propensity of fulvenes to undergo dimerization has long been known, although the in-depth mechanism and electronic behavior during dimerization are still elusive. Herein, we made an attempt to gain insights into the reactivity of pentafulvene for Diels-Alder (DA) and [6 + 4]-cycloadditions via conventional and ambimodal routes. The result emphasizes that pentafulvene dimerization preferentially proceeds through a unique bifurcation mechanism where two DA pathways merge together to produce two degenerate [4 + 2]-cycloadducts from a single TS. Despite the [6 + 4]-cycloadduct being thermodynamically preferred, [4 + 2]-cycloaddition reactions are kinetically driven. Singlet biradicaloid is involved in through-space 6e- delocalization as a secondary orbital interaction that originates asynchronicity and stabilizes the bispericyclic transition state (TS). The transformation of various actively participating intrinsic bonding orbitals (IBOs) unambiguously forecasts the formation of multiple products from a single TS and rationalizes the mechanism of ambimodal reactions that are rather difficult to probe with other analyses. The changes in active IBOs clearly distinguish the conventional reactions from bifurcation reactions and can be employed to characterize and confirm the ambimodal mechanism. This report gains a crucial theoretical insight into the mechanism of bifurcation, the origin of asynchronicity, and electronic behavior in ambimodal TS, which will certainly be of enormous value for future studies.

Vibrational synchronization and its reaction pathway influence from an entropic intermediate in a dirhodium catalyzed allylic C-H activation/Cope rearrangement reaction

In reactions with consecutive transition states without an intermediate, and an energy surface bifurcation, atomic motion generally determines product selectivity. Understanding this dynamic-based selectivity can be straightforward if there is extremely fast descent from the first transition state to a product. However, in cases where a nonstatistical roaming/entropic intermediate occurs prior to product formation the motion that influences selectivity can be difficult to identify. Here we report quasiclassical direct dynamics trajectories for the dirhodium catalyzed reaction between styryldiazoacetate and 1,4-cyclohexadiene and prior experiments by Davies showed competitive allylic C-H insertion and Cope products. Trajectories confirmed the proposed energy surface bifurcation and revealed that dirhodium vinylcarbenoid when reacting with 1,4-cyclohexadiene can induce either a dynamically concerted pathway or a dynamically stepwise pathway with a nonstatistical entropic tight ion-pair intermediate. In the dynamically stepwise reaction pathway C-H insertion versus Cope selectivity is highly influenced by whether or not vibrational synchronization occurs in the nonstatistical entropic intermediate. This vibrational synchronization highlights the possible need for an entropic intermediate to have organized transition state-like motion to proceed to a product.

Substituent effects on the selectivity of ambimodal [6+4]/[4+2] cycloaddition

In this work, we report a density functional theory (DFT) study and a dynamical trajectory study of substituent effects on the ambimodal [6+4]/[4+2] cycloaddition proposed for 1,3,5,10,12-cycloheptadecapentaene, referred to as cycloheptadecapentaene. The proposed cycloaddition proceeds through an ambimodal transition state, which results in both a [6+4] adduct a [4+2] adduct directly. The [6+4] adduct can be readily converted to the [4+2] adduct via a Cope rearrangement. We study the selectivity of the reaction with regard to the position of substituents, steric effects of substituents, and electronic effects of substituents. In the dynamical trajectory study, we find that nitro-substituted reactants lead to a new product from the ambimodal transition state via the hetero Diels-Alder reaction, and this new product can then be converted to a [4+2] adduct by a hetero [3, 3]-sigmatropic rearrangement. These results may provide insights for designing more bridged heterocyclic compounds.

Binomial Uncertainty in Molecular Dynamics-Based Reactions Analysis

Molecular Dynamics-based reaction analysis is an indispensable tool for studying processes defying the transition-state theory (TST), where the product ratios do not follow energies of transition states. The main class of such processes is ambimodal reactions, which have a post-transition-state bifurcation, so that several products form via a single transition state. Multiple runs of molecular dynamics allow one to sample the space of possibilities and ultimately predict the product ratio without relying on TST; however, no techniques for estimating the reliability of the prediction were proposed so far. Here we show that dynamics runs follow the same rules as die rolls, which paves a simple way for estimating their uncertainty and, accordingly, the number of runs necessary to achieve the required accuracy. Remarkably, we find that the majority of such studies carried out in the last 5 years use far too few runs, so that the product ratios predicted in them can be off by >50% in more than 50% of cases.

Origins of Periselectivity and Regioselectivity in Ambimodal Tripericyclic [8+6]/[6+4]/[4+2] Intramolecular Cycloadditions of a Heptafulvenyl-Fulvene

Quantum mechanical calculations and molecular dynamics simulations have elucidated the reaction mechanism for intramolecular cycloadditions of a heptafulvenyl-fulvene tethered by a trimethylene chain. Prior experiments by Liu and Houk reported the formation of only an endo-[8+6] cycloadduct at 185 °C. Liu et al. later reported an exo-[4+2] Diels-Alder cycloadduct as the major product at 140 °C (Tetrahedron, 1999, 55, 9171). Cycloadditions involve Diels-Alder and an ambimodal intramolecular tripericyclic [8+6]/[6+4]/[4+2] cycloaddition. The mechanistic details explain the experimental reports of temperature dependence on the periselectivity of intramolecular cycloadditions. Additional calculations with multireference-based methods CASSCF and NEVPT2 highlight the artifacts of DFT methods and single-reference wavefunction-based CCSD(T) in the description of complete potential energy surface involving various cycloadditions of the heptafulvenyl-fulvene.

Dynamic-dependent selectivity in a bisphosphine iron spin crossover C-H insertion/π-coordination reaction

Reaction pathway selectivity is generally controlled by competitive transition states. Organometallic reactions are complicated by the possibility that electronic spin state changes rather than transition states can control the relative rates of pathways, which can be modeled as minimum energy crossing points (MECPs). Here we show that in the reaction between bisphosphine Fe and ethylene involving spin state crossover (singlet and triplet spin states) that neither transition states nor MECPs model pathway selectivity consistent with experiment. Instead, single spin state and mixed spin state quasiclassical trajectories demonstrate nonstatistical intermediates and that C-H insertion versus π-coordination pathway selectivity is determined by the dynamic motion during reactive collisions. This example of dynamic-dependent product outcome provides a new selectivity model for organometallic reactions with spin crossover.

Metal-Free Arylation of Benzothiophenes at C4 by Activation as their Benzothiophene S-Oxides

Benzothiophenes, activated by oxidation to the corresponding S-oxides, undergo C-H/C-H-type coupling with phenols to give C4 arylation products. While an electron-withdrawing group at C3 of the benzothiophene is important, the process operates without a directing group and a metal catalyst, thus rendering it compatible with sensitive functionalities-e.g. halides and formyl groups. Quantum chemical calculations suggest a formal stepwise mechanism involving heterolytic cleavage of an aryloxysulfur species to give a π-complex of the corresponding benzothiophene and a phenoxonium cation. Subsequent addition of the phenoxonium cation to the C4 position of the benzothiophene is favored over the addition to C3; Fukui functions predict that the major regioisomer is formed at the more electron-rich position between C3 and C4. Varied selective manipulation of the benzothiophene products showcase the synthetic utility of the metal-free arylation process.

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.

C-H Insertion in Dirhodium Tetracarboxylate-Catalyzed Reactions despite Dynamical Tendencies toward Fragmentation: Implications for Reaction Efficiency and Catalyst Design

Rh-catalyzed C-H insertion reactions to form β-lactones suffer from post-transition state bifurcations, with the same transition states leading to ketones and ketenes via fragmentation in addition to β-lactones. In such a circumstance, traditional transition state theory cannot predict product selectivity, so we employed ab initio molecular dynamics simulations to do so and provide a framework for rationalizing the origins of said selectivity. Weak interactions between the catalyst and substrate were studied using energy decomposition and noncovalent interaction analyses, which unmasked an important role of the 2-bromophenyl substituent that has been used in multiple β-lactone-forming C-H insertion reactions. Small and large catalysts were shown to behave differently, with the latter providing a means of overcoming dynamically preferred fragmentation by lowering the barrier for the recombination of the product fragments in the grip of the large catalyst active site cavity.

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.

Reaction Space Projector (ReSPer) for Visualizing Dynamic Reaction Routes Based on Reduced-Dimension Space

To analyze chemical reaction dynamics based on a reaction path network, we have developed the "Reaction Space Projector" (ReSPer) method with the aid of the dimensionality reduction method. This program has two functions: the construction of a reduced-dimensionality reaction space from a molecular structure dataset, and the projection of dynamic trajectories into the low-dimensional reaction space. In this paper, we apply ReSPer to isomerization and bifurcation reactions of the Au₅ cluster and succeed in analyzing dynamic reaction routes involved in multiple elementary reaction processes, constructing complicated networks (called "closed islands") of nuclear permutation-inversion (NPI) isomerization reactions, and elucidating dynamic behaviors in bifurcation reactions with reference to bundles of trajectories. Interestingly, in the second application, we find a correspondence between the contribution ratios in the ability to visualize and the symmetry of the morphology of closed islands. In addition, the third application suggests the existence of boundaries that determine the selectivity in bifurcation reactions, which was discussed in the phase space. The ReSPer program is a versatile and robust tool to clarify dynamic reaction mechanisms based on the reduced-dimensionality reaction space without prior knowledge of target reactions.

Kinetic Analysis of a Reaction Path Network Including Ambimodal Transition States: A Case Study of an Intramolecular Diels-Alder Reaction

This study proposes a methodology for the kinetic analysis of a reaction path network including ambimodal transition states (TSs), through which an ensemble of trajectories bifurcates to multiple minima in a phenomenon called dynamical bifurcation. The proposed methodology consists of three techniques: an automated reaction path search to construct a reaction path network including ambimodal TSs, an ab initio molecular dynamics simulation to evaluate the branching ratio, and the definition of rate constants incorporating this ratio. Applying the procedure to a Diels-Alder reaction, it was found that the inclusion of dynamical bifurcations is necessary to explain the experimental reaction yield of a byproduct. In addition, it was verified that the products take 10¹³ s to reach thermal equilibrium and that the experimental selectivity is determined by the dynamical bifurcations.

Stereoelectronic and dynamical effects dictate nitrogen inversion during valence isomerism in benzene imine

Non-classical processes such as heavy-atom tunneling and post transition-state dynamics govern stereoselectivity for benzene imine ⇌ 1H-azepine.

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.

Quasiclassical Direct Dynamics Trajectory Simulations of Organometallic Reactions

Homogeneous metal-mediated organometallic reactions represent a very large and diverse reaction class. Density functional theory calculations are now routinely carried out and reported for analyzing organometallic mechanisms and reaction pathways. While density functional theory calculations are extremely powerful to understand the energy and structure of organometallic reactions, there are several assumptions in their use and interpretation to define reaction mechanisms and to analyze reaction selectivity. Almost always it is assumed that potential energy structures calculated with density functional theory adequately describe mechanisms and selectivity within the framework of statistical theories, for example, transition state theory and RRKM theory. However, these static structures and corresponding energy landscapes do not provide atomic motion information during reactions that could reveal nonstatistical intermediates without complete intramolecular vibrational redistribution and nonintrinsic reaction coordinate (non-IRC) pathways. While nonstatistical intermediates and non-IRC reaction pathways are now relatively well established for organic reactions, these dynamic effects have heretofore been highly underexplored in organometallic reactions. Through a series of quasiclassical density functional theory direct dynamics trajectory studies, my group has recently demonstrated that dynamic effects occur in a variety of fundamental organometallic reactions, especially bond activation reactions. For example, in the C-H activation reaction between methane and [Cp*(PMe₃)Ir^III(CH₃)]⁺, while the density functional theory energy landscape showed a two-step oxidative cleavage and reductive coupling mechanism, trajectories revealed a mixture of this two-step mechanism and a dynamic one-step mechanism that skipped the [Cp*(PMe₃)Ir^V(H)(CH₃)₂]⁺ intermediate. This study also showed that despite a methane σ-complex being located on the density functional theory surface before oxidative cleavage and after reductive coupling, this intermediate is always skipped and should not be considered an intermediate during reactive trajectories. For non-IRC reaction pathways, quasiclassical direct dynamics trajectories showed that for the isomerization of [Tp(NO)(PMe₃)W(η²-benzene)] to [Tp(NO)(PMe₃)W(H)(Ph)], there are many dynamic reaction pathway connections due to a relatively flat energy landscape and π coordination is not necessary for C-H bond activation through oxidative cleavage. Trajectories also showed that dynamic effects are important in selectivity for ethylene C-H activation versus π coordination in reaction with Cp(PMe₃)₂Re, and trajectories provide a more quantitative model of selectivity than transition state theory. Quasiclassical trajectories examining Au-catalyzed monoallylic diol cyclizations showed dynamic coupling of several reaction steps that include alkoxylation π bond addition, proton shuttling, and water elimination reaction steps. Overall, these studies highlight the need to use direct dynamics trajectory simulations to consider atomic motion during reactions to understand organometallic reaction mechanisms and selectivity.

Visualization of reaction route map and dynamical trajectory in reduced dimension

In the quantum chemical approach, chemical reaction mechanisms are investigated based on a potential energy surface (PES). Automated reaction path search methods enable us to construct a global reaction route map containing multiple reaction paths corresponding to a series of elementary reaction processes. The on-the-fly molecular dynamics (MD) method provides a classical trajectory exploring the full-dimensional PES based on electronic structure calculations. We have developed two reaction analysis methods, the on-the-fly trajectory mapping method and the reaction space projector (ReSPer) method, by introducing a structural similarity to a pair of geometric structures and revealed dynamic aspects affecting chemical reaction mechanisms. In this review, we will present the details of these analysis methods and discuss the dynamics effects of reaction path curvature and reaction path bifurcation with applications to the CH₃OH + OH^- collision reaction and the Au₅ cluster branching and isomerization reactions.

Bifurcating reactions: distribution of products from energy distribution in a shared reactive mode

Bifurcating reactions yield two different products emerging from one single transition state and are therefore archetypal examples of reactions that cannot be described within the framework of the traditional Eyring's transition state theory (TST). With the growing number and importance of these reactions in organic and biosynthetic chemistry, there is also an increasing demand for a theoretical tool that would allow for the accurate quantification of reaction outcome at low cost. Here, we introduce such an approach that fulfils these criteria, by evaluating bifurcation selectivity through the energy distribution within the reactive mode of the key transition state. The presented method yields an excellent agreement with experimentally reported product ratios and predicts the correct selectivity for 89% of nearly 50 various cases, covering pericyclic reactions, rearrangements, fragmentations and metal-catalyzed processes as well as a series of trifurcating reactions. With 71% of product ratios determined within the error of less than 20%, we also found that the methodology outperforms three other tested protocols introduced recently in the literature. Given its predictive power, the procedure makes reaction design feasible even in the presence of complex non-TST chemical steps.

Reactive Mode Composition Factor (RMCF) analysis is a powerful tool to forecast the product distribution of bifurcating reactions through analysis of the kinetic energy distribution within the first transition state traversed by the reacting system.

Mechanistic Investigation of a Synthetic Route to Biaryls by the Sigmatropic Rearrangement of Arylsulfonium Species

A comprehensive mechanistic investigation was conducted on the coupling reaction of aryl sulfoxides with phenols by using trifluoroacetic anhydride to yield biaryls. NMR experiments revealed that our previously proposed mechanism, which consists of a cascade of an interrupted Pummerer reaction and a rate-determining [3,3] sigmatropic rearrangement, is reasonable. The electronic effects of the substrates were also evaluated to elucidate the nature of the rearrangement step. Based on experimental observations and theoretical calculations, we conclude that the rearrangement is highly asynchronous and stepwise rather than concerted when electron-rich phenols are employed for the reaction.

Allenylboronic Acid Pinacol Ester: A Selective Partner for [4 + 2] Cycloadditions

We have studied the reaction of allenylboronic acid pinacol ester with cyclopentadiene with experimental and computational methods. The reaction occurred efficiently with complete Diels-Alder periselectivity and regioselectivity at the proximal double bond. The concerted mechanism for the observed transformation was computed to be favored over competitive addition to the distal double bond, [3,3]-sigmatropic rearrangements, and stepwise radical mechanism. This unprecedented Diels-Alder reaction enables the construction of synthetically versatile boron-substituted cycloadducts.

Pathway Bifurcations in the Activation of Allylic Halides by Palladium and Their Influence on the Dynamics of η1 and η3 Allyl Intermediates

Transition-metal-catalyzed allylic substitution often exhibits complex product selectivity patterns, which have been primarily attributed to π ↔ σ ↔ π isomerization of the η¹ and η³ allyl intermediates. Product selectivity may be even further complicated if η¹- and η³-allyls share a single transition state (TS), leading to their formation resulting in a post-transition-state bifurcation (PTSB). In this work, density functional theory calculations using ab initio molecular dynamics (AIMD) have been carried out that support the presence of a PTSB in Pd-catalyzed allylic halide activation directly influencing product selectivity. The AIMD results initiated from the TS predict the η¹-allyl to be favored in the gas phase and a low dielectric (ε < 2.5) for trialkylphosphines, while the selectivity shifts toward the η³-allyl in higher dielectrics. The minimum energy path is also predicted to shift in product preference, consistent with the dynamics predictions. The bifurcation in allylic chloride activation is predicted to largely favor the η³-allyl at any solvent polarity. A PTSB was also discovered to be present in Ni and Pt allylic activation but with less bifurcation. These results offer a unique view into the mechanism of metal-catalyzed allylic substitution.

Direct Dynamics Trajectories Reveal Nonstatistical Coordination Intermediates and Demonstrate that σ and π-Coordination Are Not Required for Rhenium(I)-Mediated Ethylene C-H Activation

The C-H activation reaction between Cp(PMe₃)₂Re and ethylene results in kinetic selectivity for the Re-vinyl hydride I over the thermodynamically more stable Cp(PMe₃)₂Re(η²-ethylene) π-complex II. While transition-state and variational transition-state structures were located for individual pathways leading to I and II, DFT and CCSD(T) energies predict a large kinetic selectivity of 10²-10⁴, which is incompatible with the experimental 10:1 ratio. DFT direct quasiclassical trajectories revealed that the transition states do not provide a qualitatively correct reaction mechanism or a quantitatively correct selectivity due to a nonstatistical σ-CH coordination intermediate that precedes the transition states for C-H activation and π coordination. Using metadynamics and quasiclassical direct dynamics, we show that trajectories for the reaction between Cp(PMe₃)₂Re and ethylene result in direct formation of either the Re-vinyl hydride I or the π-complex II. Trajectories leading to the Re-vinyl hydride skip σ-coordination and do not require π-coordination. Consistent with experiments, trajectory selectivity provides a relatively small kinetic selectivity for the Re-vinyl hydride.

Machine learning classification of disrotatory IRC and conrotatory non-IRC trajectory motion for cyclopropyl radical ring opening

Quasiclassical trajectory analysis is now a standard tool to analyze non-minimum energy pathway motion of organic reactions. However, due to the large amount of information associated with trajectories, quantitative analysis of the dynamic origin of reaction selectivity is complex. For the electrocyclic ring opening of cyclopropyl radical, more than 4000 trajectories were run showing that allyl radicals are formed through a mixture of disrotatory intrinsic reaction coordinate (IRC) motion as well as conrotatory non-IRC motion. Geometric, vibrational mode, and atomic velocity transition-state features from these trajectories were used for supervised machine learning analysis with classification algorithms. Accuracy >80% with a random forest model enabled quantitative and qualitative assessment of transition-state trajectory features controlling disrotatory IRC versus conrotatory non-IRC motion. This analysis revealed that there are two key vibrational modes where their directional combination provides prediction of IRC versus non-IRC motion.

Dynamic Effects in Intramolecular Schmidt Reactions: Entropy, Electrostatic Drag, and Selectivity Prediction

Electrostatic drag in the intramolecular Schmidt reactions of azidopropylcyclohexanones is characterized using density functional theory (DFT) calculations and direct dynamics simulations. Despite resulting from enthalpically favorable interactions, electrostatic drag slows down N₂ loss during formation of bridged lactam products, an effect with implications for controlling product selectivity.

Dynamical Trajectory Study of the Transannular [6+4] and Ambimodal Cycloaddition in the Biosynthesis of Heronamides

We report the dynamical trajectory study of the transannular [6+4] and ambimodal cycloaddition proposed in the biosynthesis of heronamide natural products. The originally proposed bifurcation of the potential energy surface is found to strongly favor the formation of the [6+4] product, both in the gas phase and in an explicit water environment, as evidenced by our trajectory simulations. The detailed information on how the bonds are formed and how water influences the bonding dynamics was analyzed at the femtosecond time scale.

Machine Learning Analysis of Direct Dynamics Trajectory Outcomes for Thermal Deazetization of 2,3-Diazabicyclo[2.2.1]hept-2-ene

Experimentally, the thermal gas-phase deazetization of 2,3-diazabicyclo[2.2.1]hept-2-ene (1) results in the loss of N₂ and the formation of bicyclo products 3 (exo) and 4 (endo) in a nonstatistical ratio, with preference for the exo product. Here, we report unrestricted M06-2X quasiclassical trajectories initialized from the concerted N₂ ejection transition state that were able to replicate the experimental preference to form 3. We found that the 3:4 ratio results from the relative amounts of very fast (ballistic) exotype trajectories versus trajectories that lead to the 1,3-diradical intermediate 2. These quasiclassical trajectories provided a set of transition-state vibrational, velocity, momenta, and geometric features for the machine learning analysis. A selection of popular supervised classification algorithms (e.g., random forest) provided poor prediction of trajectory outcomes based on only transition-state vibrational quanta and energy features. However, these machine learning models provided more accurate predictions using atomic velocities and atomic positions, attaining ∼70% accuracy using initial conditions and between 85 and 95% accuracy at later reaction time steps. This increased accuracy allowed the feature importance analysis to reveal that, at the later-time analysis, the methylene bridge out-of-plane bending is correlated with trajectory outcomes for the formation of either the exo product or toward the diradical intermediate. Possible reasons for the struggle of machine learning algorithms to classify trajectories based on transition-state features is the heavily overlapping feature values, the finite but very large possible vibrational mode combinations, and the possibility of chaos as trajectories propagate. We examined this chaos by comparing a set of nearly identical trajectories that differed by only a very small scaling of the kinetic energies resulting from the transition-state reaction coordinate.

Mechanistic molecular motion of transition-metal mediated β-hydrogen transfer: quasiclassical trajectories reveal dynamically ballistic, dynamically unrelaxed, two step, and concerted mechanisms

Quasiclassical direct dynamics reveal new dynamical mechanisms for metal-alkyl to ethylene β-hydrogen transfer.

Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective

The onset of facile intramolecular vibrational energy flow can be related to features in the connected network of anharmonic resonances in the classical phase space.

AFIR explorations of transition states of extended unsaturated systems: automatic location of ambimodal transition states

Based on the artificial force induced reaction (AFIR) method, we proposed a procedure to systematically explore ambimodal transition states (TSs) that cause the dynamical bifurcation.

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).