Microcanonical Rate Constants for Unimolecular Reactions in the Low-Pressure Limit

Isomer-resolved unimolecular dynamics of the hydroperoxyalkyl intermediate (•QOOH) in cyclohexane oxidation

The oxidation of cycloalkanes is important in the combustion of transportation fuels and in atmospheric secondary organic aerosol formation. A transient carbon-centered radical intermediate (•QOOH) in the oxidation of cyclohexane is identified through its infrared fingerprint and time- and energy-resolved unimolecular dissociation dynamics to hydroxyl (OH) radical and bicyclic ether products. Although the cyclohexyl ring structure leads to three nearly degenerate •QOOH isomers (β-, γ-, and δ-QOOH), their transition state (TS) barriers to OH products are predicted to differ considerably. Selective characterization of the β-QOOH isomer is achieved at excitation energies associated with the lowest TS barrier, resulting in rapid unimolecular decay to OH products that are detected. A benchmarking approach is employed for the calculation of high-accuracy stationary point energies, in particular TS barriers, for cyclohexane oxidation (C6H11O2), building on higher-level reference calculations for the smaller ethane oxidation (C2H5O2) system. The isomer-specific characterization of β-QOOH is validated by comparison of experimental OH product appearance rates with computed statistical microcanonical rates, including significant heavy-atom tunneling, at energies in the vicinity of the TS barrier. Master-equation modeling is utilized to extend the results to thermal unimolecular decay rate constants at temperatures and pressures relevant to cyclohexane combustion.

ABC+D: A time-independent coupled-channel quantum dynamics program for elastic and ro-vibrational inelastic scattering between atoms and triatomic molecules in full dimensionality

We discuss the details of a time-independent quantum mechanical method and its implementation for full-dimensional non-reactive scattering between a closed-shell triatomic molecule and a closed-shell atom. By solving the time-independent Schrödinger equation within the coupled-channel framework using a log-derivative method, the state-to-state scattering matrix (S-matrix) can be determined for inelastic scattering involving both the rotational and vibrational modes of the molecule. Various approximations are also implemented. The ABC+D code provides an important platform for understanding an array of physical phenomena involving collisions between atoms and molecules.

Recent advances in quantum theory on ro-vibrationally inelastic scattering

Molecular collisions are of fundamental importance in understanding intermolecular interaction and dynamics. Its importance is accentuated in cold and ultra-cold collisions because of the dominant quantum mechanical nature of the scattering. We review recent advances in the time-independent approach to quantum mechanical characterization of non-reactive scattering in tetratomic systems, which is ideally suited for large collisional de Broglie wavelengths characteristic in cold and ultracold conditions. We discuss quantum scattering algorithms between two diatoms and between a triatom and an atom and their implementation, as well as various approximate schemes. They not only enable the characterization of collision dynamics in realistic systems but also serve as benchmarks for developing more approximate methods.

Full-Dimensional Quantum Dynamics Studies of Ro-vibrationally Inelastic Scattering of H2O with Ar: A Benchmark Test of the Rigid-Rotor Approximation

While the rigid-rotor (RR) approximation is usually considered to be accurate for describing pure rotationally inelastic scattering involving diatoms in their ground or low-lying vibrational states, its validity in scattering involving polyatomic molecules has not been fully examined. The existence of soft/anharmonic vibrational modes in polyatomic molecules could make rotational-vibrational energy transfer rather efficient, thus undermining the premise of the RR approximation. In this work, we conduct a benchmark test of the RR approximation in the rotationally inelastic scattering of the H₂O(v₂ = 0, 1) + Ar system by comparing with full-dimensional quantum scattering calculations. We demonstrate that the error in the RR rate coefficient for v₂ = 0 is less than 5%, while it can reach up to 20% for some initial states within the v₂ = 1 manifold. These results indicate that the RR approximation gradually deteriorates with increasing quantum number v₂. Vibrational relaxation dynamics of this system was also studied, and it is found that transitions from initial states with a large rotational quantum number of projection on the a principal axis are more efficient. These results shed valuable light on ro-vibrationally inelastic scattering involving polyatomic molecules.

Improved microcanonical instanton theory

Canonical (thermal) instanton theory is now routinely applicable to complex gas-phase reactions and allows for the accurate description of tunnelling in highly non-separable systems. Microcanonical instanton theory is by contrast far less well established. Here, we demonstrate that the best established microcanonical theory [S. Chapman, B. C. Garrett and W. H. Miller, J. Chem. Phys., 1975, 63, 2710-2716], fails to accurately describe the deep-tunnelling regime for systems where the frequencies of the orthogonal modes change rapidly along the instanton path. By taking a first principles approach to the derivation of microcanonical instanton theory, we obtain an improved method, which accurately recovers the thermal instanton rate when integrated over energy. The resulting theory also correctly recovers the separable limit and can be thought of as an instanton generalisation of Rice-Ramsperger-Kassel-Marcus (RRKM) theory. When combined with the density-of-states approach [W. Fang, P. Winter and J. O. Richardson, J. Chem. Theory Comput., 2021, 17, 40-55], this new method can be straightforwardly applied to real molecular systems.

Full-dimensional quantum studies of vibrational energy transfer dynamics between H2O and Ar: theory assessing experiment

We report the first full-dimensional quantum mechanical calculations of the ro-vibrational inelastic scattering dynamics between water molecules and argon atoms on an accurate potential energy surface, using a recently developed time-independent quantum method based on the close-coupling approach. The state-to-state integral cross-sections and rate coefficients show strong observance of gap laws. The calculated thermal rate coefficients for the relaxation of the stretching fundamental states of H₂O are in good agreement with experimental values, while those for the bending overtone state are approximately five times smaller than the values extracted through a previous kinetic modeling of fluorescence decay data. Our state-specific quantum scattering results suggest the need to reassess the kinetic modeling of the experimental data. This work advanced our understanding of the quantum dynamics of vibrationally inelastic energy transfer processes involving polyatomic molecules.

Predicting third-body collision efficiencies for water and other polyatomic baths

Low-pressure-limit microcanonical (collisional activation) and thermal rate constants are predicted using a combination of automated ab initio potential energy surface construction, classical trajectories, transition state theory, and a detailed kinetic...

Spiers Memorial Lecture: theory of unimolecular reactions

One hundred years ago, at an earlier Faraday Discussion meeting, Lindemann presented a mechanism that provides the foundation for contemplating the pressure dependence of unimolecular reactions. Since that time, our...

Global Master Equation Analysis of Rate Data for the Reaction C2H4 + H ⇄ C2H5: ΔfH0⊖C2H5

While forward and reverse rate constants are frequently used to determine enthalpies of reaction and formation, this process is more difficult for pressure-dependent association/dissociation reactions, especially since the forward and reverse reactions are usually studied at very different temperatures. The problems can be overcome by using a data-fitting procedure based on a master equation model. This approach has been applied to existing experimental pressure-dependent forward and reverse rate coefficients for the reaction C₂H₄ + H ⇄ C₂H₅ (k₁, k_-1) using the MESMER code to determine Δ_fH₀^⊖C₂H₅ from the enthalpy of the reaction. New measurements of k₁, k_-1 were included in analysis. They are based on laser flash photolysis with direct observation of H atom time profiles by vacuum ultraviolet laser-induced fluorescence under conditions where the approach to equilibrium could be observed. Measurements were made over the temperature range 798-828 K and with [He] from 2.33 to 7.21 × 10¹⁸ molecule cm^-3. These data were then combined with a wide range of existing experimental data with helium as the bath gas (112 measurements of k₁ and k_-1, covering the temperature range 285-1094 K, and [He] = 7.1 × 10¹⁵-1.9 × 10¹⁹ molecule cm^-3) and fitted using the master equation solver MESMER. The required vibrational frequencies and rotational constants of the system were obtained from ab initio calculations, and the activation threshold for association (ΔE_thresh), enthalpy of reaction (Δ_rH₀^⊖), imaginary frequency (υ_imag), and helium energy-transfer parameters (⟨ΔE⟩_d,298(T/298)ⁿ) were optimized. The resulting parameters (errors are 2σ) are ΔE_thresh = 11.43 ± 0.34 kJ mol^-1, Δ_rH₀^⊖ = -145.34 ± 0.60 kJ mol^-1, υ_imag = 730 ± 130 cm^-1, ⟨ΔE⟩_d,298 = 54.2 ± 7.6 cm^-1, and n = 1.17 ± 0.12. A value of Δ_fH_298.15^⊖(C₂H₅) = 120.49 ± 0.57 kJ mol^-1 is obtained by combining Δ_rH₀^⊖ with standard enthalpies of formation for H and C₂H₄ and making the appropriate temperature corrections. The dependence of these parameters on how the internal rotor and CH₂ inversion modes are treated has been explored. The experimental data for other bath gases have been analyzed, and data sets compatible with the potential energy surface parameters determined above have been identified. The parameters are virtually identical but with slightly smaller error limits. Parameterization of k₁, k_-1 using the Troe formalization has been used to investigate competition between ethyl decomposition and reaction with oxygen under combustion conditions.

Vibrational energy pooling via collisions between asymmetric stretching excited CO2: a quasi-classical trajectory study on an accurate full-dimensional potential energy surface

In low temperature plasmas, energy transfer between asymmetric stretching excited CO₂ molecules can be highly efficient, which leads to further excitation (and de-excitation) of the CO₂ molecules: CO₂(v_as) + CO₂(v_as) → CO₂(v_as + 1) + CO₂(v_as - 1). Through such a vibrational ladder climbing mechanism, CO₂ can be activated and eventually dissociates. To gain mechanistic insight of such processes, a full-dimensional accurate potential energy surface (PES) for the CO₂ + CO₂ system is developed using the permutational invariant polynomial-neural network method based on CCSD(T)-F12a/AVTZ energies at about 39 000 geometries. This PES is used in quasi-classical trajectory (QCT) studies of the vibrational energy transfer between CO₂ molecules excited in the asymmetric stretching mode. A machine learning algorithm is used to determine state-specific rate coefficients for the vibrational transfer processes from a limited data set. In addition to the CO₂(v_as + 1) + CO₂(v_as - 1) channel, the QCT simulations revealed significant contributions from the CO₂(v_as + 2,3) + CO₂(v_as - 2,3) channels, particularly at low collision energies/temperatures. These multi-vibrational-quantum processes are attributed to enhanced energy flow in the collisional complex formed by enhanced dipole-dipole interaction between asymmetric stretching excited CO₂ molecules.

Permutationally Invariant Polynomial Expansions with Unrestricted Complexity

A general strategy is presented for constructing and validating permutationally invariant polynomial (PIP) expansions for chemical systems of any stoichiometry. Demonstrations are made for three categories of gas-phase dynamics and kinetics: collisional energy-transfer trajectories for predicting pressure-dependent kinetics, three-body collisions for describing transient van der Waals adducts relevant to atmospheric chemistry, and nonthermal reactivity via quasiclassical trajectories. In total, 30 systems are considered with up to 15 atoms and 39 degrees of freedom. Permutational invariance is enforced in PIP expansions with as many as 13 million terms and 13 permutationally distinct atom types by taking advantage of petascale computational resources. The quality of the PIP expansions is demonstrated through the systematic convergence of in-sample and out-of-sample errors with respect to both the number of training data and the order of the expansion, and these errors are shown to predict errors in the dynamics for both reactive and nonreactive applications. The parallelized code distributed as part of this work enables the automation of PIP generation for complex systems with multiple channels and flexible user-defined symmetry constraints and for automatically removing unphysical unconnected terms from the basis set expansions, all of which are required for simulating complex reactive systems.

Watching a hydroperoxyalkyl radical (•QOOH) dissociate

A prototypical hydroperoxyalkyl radical (•QOOH) intermediate, transiently formed in the oxidation of volatile organic compounds, was directly observed through its infrared fingerprint and energy-dependent unimolecular decay to hydroxyl radical and cyclic ether products. Direct time-domain measurements of •QOOH unimolecular dissociation rates over a wide range of energies were found to be in accord with those predicted theoretically using state-of-the-art electronic structure characterizations of the transition state barrier region. Unimolecular decay was enhanced by substantial heavy-atom tunneling involving O-O elongation and C-C-O angle contraction along the reaction pathway. Master equation modeling yielded a fully a priori prediction of the pressure-dependent thermal unimolecular dissociation rates for the •QOOH intermediate-again increased by heavy-atom tunneling-which are required for global models of atmospheric and combustion chemistry.

Energy Dependence of Ensemble-Averaged Energy Transfer Moments and Its Effect on Competing Decomposition Reactions

We carried out a direct dynamics study on the internal-energy dependence of the ensemble-averaged energy transfer moments of the isobutyl radical in collisions with N₂ bath gas. We find a linear dependence of the downward moment ⟨ΔE_d⟩ and the root-mean-square moment ⟨ΔE2⟩ on the initial internal energy, but the upward moment ⟨ΔE_u⟩ is found to be independent of the molecule's internal energy. We improved the exponential-down relaxation model by including a linear dependence of ⟨ΔE_d⟩ on the initial energy, and we used the improved treatment in the 1D master equation for isobutyl radical decomposition reactions and for a model of competitive reactions with a larger difference in barrier heights. We calculated phenomenological rate constants and branching ratios from chemically significant eigenmodes of the master equation and showed that the energy dependence of ⟨ΔE_d⟩ has a greater influence on channels with higher barriers in competitive reactions. Rate constants and branching ratios from master equation calculations indicate that for a given temperature and pressure, there is a constant ⟨ΔE_d⟩ that can reproduce results obtained with an E-dependent ⟨ΔE_d⟩. But a constant ⟨ΔE_d⟩ cannot do this for all temperatures and pressures, with larger differences when the barriers for the competing channels differ more. We conclude that when the branching ratio of competitive reactions is sensitive to pressure, including the energy dependence of ⟨ΔE_d⟩ in master equation simulations can make a significant difference in the results.

Two-Dimensional Master Equation Modeling of Some Multichannel Unimolecular Reactions

Multichannel thermal decomposition reactions of n-propyl radicals, 1-pentyl radicals, and toluene are investigated by solving a two-dimensional master equation formulated as a function of total energy (E) and angular momentum (J). The primary aim of this study is to elucidate the role of angular momentum in the kinetics of multichannel unimolecular reactions. The collisional transition processes of the reactants colliding with argon are characterized based on the classical trajectory calculations and implemented in the master equation. The rate constants calculated by using the two-dimensional master equation are compared with those of one-dimensional master equations. The consequence of the explicit treatment of angular momentum depends on the J dependence of the microscopic rate constants and is particularly emphasized in the thermal decomposition of toluene, for which the C-H and C-C bond fission channels are considered. The centrifugal effect is insignificant in the energetically favored C-H bond fission but is substantial in the energetically higher C-C bond fission, which causes rotational channel switching of the microscopic rate constants. The proper treatment of the J-dependent channel coupling effect, weak collisional transfer of J, and initial-J-dependent collisional energy transfer are found to be essential for predicting the branching fractions at low pressures.

On the Rate Constant for NH2+HO2 and Third-Body Collision Efficiencies for NH2+H(+M) and NH2+NH2(+M)

In low-temperature flash photolysis of NH₃/O₂/N₂ mixtures, the NH₂ consumption rate and the product distribution is controlled by the reactions NH₂ + HO₂ → products (R1), NH₂ + H (+M) → NH₃ (+M) (R2), and NH₂ + NH₂ (+M) → N₂H₄ (+M) (R3). In the present work, published flash photolysis experiments by, among others, Cheskis and co-workers, are re-interpreted using recent direct measurements of NH₂ + H (+N₂) and NH₂ + NH₂ (+N₂) from Altinay and Macdonald. To facilitate analysis of the FP data, relative third-body collision efficiencies compared to N₂ for R2 and R3 were calculated for O₂ and NH₃ as well as for other selected molecules. Results were in good agreement with the limited experimental data. Based on reported NH₂ decay rates in flash photolysis of NH₃/O₂/N₂, a rate constant for NH₂ + HO₂ → NH₃ + O₂ (R1a) of k_1a = 1.5(±0.5) × 10¹⁴ cm³ mol^-1 s^-1 at 295 K was derived. This value is higher than earlier determinations based on the FP results but in good agreement with recent theoretical work. Kinetic modeling of reported N₂O yields indicates that NH₂ + HO₂ → H₂NO + O (R1c) is competing with R1a, but perturbation experiments with addition of CH₄ indicate that it is not a dominating channel. Measured HNO profiles indicate that this component is formed directly by NH₂ + HO₂ → HNO + H₂O (R1b), but theoretical work indicates that R1b is only a minor channel. Based on this analysis, we estimate k_1c = 2.5 × 10¹³ cm³ mol^-1 s^-1 and k_1b = 2.5 × 10¹² cm³ mol^-1 s^-1 at 295 K, with significant uncertainty margins.

Microcanonical Tunneling Rates from Density-of-States Instanton Theory

Semiclassical instanton theory is a form of quantum transition-state theory which can be applied to the computation of thermal reaction rates in complex molecular systems including quantum tunneling effects. There have been a number of attempts to extend the theory to treat microcanonical rates. However, the previous formulations are either computationally unfeasible for large systems due to an explicit sum over states or they involve extra approximations, which make them less reliable. We propose a robust and practical microcanonical formulation called density-of-states instanton theory, which avoids the sum over states altogether. In line with the semiclassical approximations inherent to the instanton approach, we employ the stationary-phase approximation to the inverse Laplace transform to obtain the densities of states. This can be evaluated using only post-processing of the data available from a small set of instanton calculations, such that our approach remains computationally efficient. We show that the new formulation predicts results that agree well with quantum scattering theory for an atom-diatom reaction and with experiments for a photoexcited unimolecular hydrogen transfer in a Criegee intermediate. When the thermal rate is evaluated from a Boltzmann average over our new microcanonical formalism, it can overcome some problems of conventional instanton theory. In particular, it predicts a smooth transition at the crossover temperature and is able to describe bimolecular reactions with pre-reactive complexes such as CH₃OH + OH.

Modeling Collisional Transitions in Thermal Unimolecular Reactions: Successive Trajectories and Two-Dimensional Master Equation for Trifluoromethane Decomposition in an Argon Bath

Collisional transition processes in thermal unimolecular reactions are modeled by collision frequency, Z, and probability distribution function, P(E, J; E', J'), which describes the probabilities of collisional transitions from the initial state specified by the total energy and angular momentum, (E', J'), to the final states, (E, J). The validity of the collisional transition model, consisting of Z and P(E, J; E', J'), is assessed here for the title reaction. The present model and its parameters are derived from the moments of transition probabilities calculated by classical trajectory simulations. The model explicitly accounts for coupling between the energy and angular momentum transfer and the dependence of transition probability on the initial state. The performance of the model is evaluated by comparing the rate constants calculated by solving the two-dimensional master equation with those obtained from the classical trajectory calculations of the sequence of successive collisions. The rate constants are also compared with available experimental data. The present collisional transition model is found to perform fairly well for predicting the pressure-dependent rate constants. The uncertainty in the prediction and sensitivities of the rate constants to the model parameters are discussed. A simplified version of the model is proposed, which performs as well as the full model. The simplifications and robust procedures for calculating the model parameters are described.

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.