Transition state theory: Variational formulation, dynamical corrections, and error estimates

A variant of the Kac-Zwanzig model is used to test the prediction of transition state theory (TST) and variational transition state theory (VTST). The model describes the evolution of a distinguished particle moving in a double-well external potential and coupled to N free particles through linear springs. While the Kac-Zwanzig model is deterministic, under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two-state Markov chain whose transition rate constants can be computed exactly in suitable limit. Here, these transition rate constants are compared with the predictions of TST and VTST. It is shown that the application of TST with a naive (albeit natural) choice of dividing surface leads to the wrong prediction of the transition rate constants. This is due to crossings of the dividing surface that do not correspond to actual transition events. However, optimizing over the dividing surface within VTST allows one to eliminate completely these spurious crossings, and therefore derive the correct transition rate constants for the model. The reasons why VTST is successful in this model are discussed, which allows one to speculate on the reliability of VTST in more complicated systems.

We test several approximate theories of thermal rate constants against accurate quantal equilibrium rate constants for collinear bimolecular reactions governed by given potential energy surfaces. The systems considered are F+H2 and F+D2 for the Muckerman potential energy surface no. 5 at 200–1200 K and H+F2, D+F2, and T+F2 for potential surface II of Jonathan, Okuda, and Timlin at 233–1260 K. For F+H2, conventional transition state theory overestimates the accurate rate constant by factors of 2.9–3.4, with the largest errors at the lowest and highest temperatures. Vibrationally adiabatic transmission coefficients or variational transition state theory decrease the error to factors of 1.2–1.3 at the lowest temperature and to a factor of 3.2 at the highest temperature. The unified statistical model reduces the errors to a factor of 1.1 at the lowest temperature and a factor of 2.7 at the highest temperature. For F+D2 the trends are similar but the errors in all the methods are smaller at all temperatures. For H+F2 conventional transition state theory underestimates the rate constants by a factor of 0.56 at the lowest temperature with the error decreasing to 2% at the highest temperature. Various formulations of vibrationally adiabatic transmission coefficients, in conjunction with either conventional or variational transition state theory, reduce the error to factors of 0.84–0.96 at the lowest temperature while not destroying the good agreement at the highest temperature. The unified statistical theory seems to overestimate the recrossing correction for this reaction, though only by about 10%. The results are similar for D+F2 and T+F2 except that the low-temperature underestimates of the calculations without tunneling are not as severe. The best theory, improved canonical variational theory with the Marcus–Coltrin path semiclassical adiabatic ground-state transmission coefficient, reproduces the accurate quantal results for H+F2, D+F2, and T+F2 within 5% or better in every case for the whole temperature range.

Reactions of β-hydroxypropyl radicals with O2 on the HOC3H6OO• potential energy surfaces: A theoretical study

Benzyl radical (C7H7), one of the resonantly stabilized hydrocarbon radicals, is one of the significant precursors of polycyclic aromatic hydrocarbons in interstellar media and combustion engines. The unimolecular decomposition of benzyl radical is still incompletely understood despite of its importance and relatively small molecular size. The decomposition reactions of benzyl radical were investigated in the present study by using the ab initio transition state theory (TST) and the multi‐well master equation theory. Specifically, all reaction pathways on the potential energy surface of C7H7 was calculated at the level of QCISD(T)/CBS. For the reactions with multireference characters, the CASPT2(9e,7o)/aug‐cc‐pVTZ method was used to calculate the vibrational frequencies and energies of structures along the one‐dimensional reaction coordinate of the breaking bond. The high‐pressure limits of rate constants for all the reactions were obtained by using the TST except those for C7H6 + H and C6H4 + CH3 by the variational TST. The pressure‐dependent rate constants were obtained by using the multi‐well master equation simulations. The calculated rate constants agree well with available experimental and theoretical data in the literature. Moreover, the present results identify the composition of the non‐hydrogen‐atom production observed in previous experiments, which provide new insights into the reactions of aromatic compounds.

Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. Thus, quantum effects, e.g. tunneling and zero point energy, are correctly included. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and their relation to transition state theory, describes computational schemes to obtain accurate rate constants, presents applications, and discusses approximations.KeywordsPotential Energy SurfaceDynamical FactorReaction BarrierTransition State TheoryTransition State TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The reaction path, the dynamical properties along the reaction path and CVT rate constants are computed by the ab initio MO method, the reaction path Hamiltonian theory and the variational transition state theory. The results show that the effect of the electron correlation energy on activation barrier is large, the recrossing and tunneling effects exist in the reaction.

Variational transition state theory is a powerful tool for tackling the problem of activated barrier crossing in condensed phases, enabling the exact reduction of infinite-dimensional problems to much more tractable few-dimensional problems. Variational transition state theory may be used to go beyond Kramers’ theory and consider the effects of memory friction and solute nonlinearities on the rate of barrier crossing. In addition, since variational transition state theory is a Hamiltonian based formalism, it is not tied to the limitations of generalized Langevin equation dynamics; thus it can be used to treat nonlinear solute-solvent interactions. Other recent advances include the development of new formalisms for improved optimization of the transition state dividing surface, the incorporation of intramolecular solute modes, and the development of centroid-based quantum transition state theory analogs.

State-of-the-art calculations of the C2H3O2 potential energy surface are presented. A new method is described for computing the interaction potential for R + O2 reactions. The method, which combines accurate determination of the quartet potential along the doublet minimum energy path with multireference calculations of the doublet/quartet splitting, decreases the uncertainty in the doublet potential and thence the rate constants by more than a factor of 2. The temperature- and pressure-dependent rate coefficients are computed using variable reaction coordinate transition-state theory, variational transition-state theory, and conventional transition-state theory, as implemented in a new RRKM/ME code. The main bimolecular product channels are CH2O + HCO at lower temperatures and CH2CHO + O at higher temperatures. Above 10 atm, the collisional stabilization of CH2CHOO directly competes with these two product channels. CH2CHOO decomposes primarily to CH2O + HCO. The next two most significant bimolecular products are OCHCHO + H and (3)CHCHO + OH, and not C2H2 + HO2. C2H3 + O2 will be predominantly chain branching above 1700 K. Uncertainty analysis is presented for the two most important transition states. The uncertainties in these two barrier heights result in a significant uncertainty in the temperature at which CH2CHO + O overtakes all other product channels.

Canonical variational transition state theory, microcanonical variational transition state theory, and Miller's unified statistical theory were used in an attempt to correct two major deficiencies of the conventional transition state theory. These are: (1) the necessity of extra assumptions to include quantum mechanical tunneling effects and (2) the fundamental assumption that trajectories crossing a dividing surface in phase space proceed directly to products. The accuracy of these approximate methods were tested by performing calculations for several collinear reactions of hydrogen, deuterium, chlorine, or iodine, with five isotopes of hydrogen molecules and comparison of these results with those from accurate quantitative calculations of the reaction probabilities as functions of energy and of the thermal rate constants as functions of temperature. 49 references, 28 figures, 17 tables.

The large-curvature ground-state model for the transmission coefficient of generalized transition state theory [presented in a previous paper by B. C. Garrett, D. G. Truhlar, A. F. Wagner, and T. H. Dunning, J. Chem. Phys. 78, 4400(1983)] is tested against accurate quantal calculations of the rate coefficients for collinear reactions with very large inertial effects, namely Cl+HCl→ClH+Cl, Cl+DCl→ClD+Cl, and Cl+MuCl→ClMu+Cl. The tests cover the temperature range 200–2400 K. The accurate rate calculations are based on reaction probabilities obtained by a new numerical method for solving Schrödinger’s equation in Delves’ coordinates. Improved canonical variational transition state theory predicts rate coefficients 5.0–18 times smaller than those predicted by conventional transition state theory for the H transfer; for the D transfer, the ratio is 2.0–3.4; and for Mu it is 22–2.8×107. The large-curvature model predicts transmission coefficients as large as 41, 8, and 206 for the H, D, and Mu-transfer cases at 200 K, decreasing to 1.2, 1.1, and 1.4 at 2400 K. Despite the large effect of variationally optimizing the transition state location and the large size of the tunneling effect and the wide variation of both effects with temperature, improved canonical variational transition state theory with large-curvature ground-state transmission coefficients (ICVT/LCG) is accurate within a factor of 1.7 over a temperature range of a factor of 8, 300–2400 K, for all three reactions. At 200 K, the ICVT/LCG model underestimates the rate coefficients, by factors of 2.3, 1.9, and 1.5 for H, D, and Mu, respectively.

Hydrogen abstraction from CH3NH2, (CH3)[formula omitted]NH, and (CH3)[formula omitted]N by H[formula omitted] radicals: A theoretical study

The combined use of atomistic simulations and transition state theory (TST) has enabled significant improvements in the prediction of thermally activated processes in a wide range of applications, e.g., conformation changes in molecules, chemical reactions, kinetic phase transitions, and solid-state diffusion. However, such an approach is still in its infancy with regard to mechanics of materials applications. Focusing here on dislocation nucleation in Al, we examine the utility of five TST-based approaches. Using the finite-temperature string method, we interpret the success and failure of each approach in terms of a full energetic analysis of the nucleation processes. After showing that advanced TST approaches such as variational TST can accurately predict nucleation rates, we employ variational TST to study dislocation nucleation from a free surface under ordinary laboratory conditions. The predictions (1) demonstrate that nucleation will only occur under very high stresses at room temperature, (2) provide an upper bound on the shear strength of Al in dislocation starved contexts, and (3) provide a rate sensitivity signature of the nucleation process.

The reaction of disilane with atomic hydrogen has been studied. This reaction involves both substitution and abstraction. Calculations show that the hydrogen abstraction is the strongest competing channel. The canonical variational transition state theory with a small curvature tunneling correction (SCT) has been used for the kinetic calculation. The theoretical results are in good agreement with the available experimental data. Comparing the reactions of atomic hydrogen with disilane and silane, it can be seen that the reactivity of the Si-H bond is higher in Si 2 H 6 than that in SiH 4 .

In this paper, the coupled states distorted wave (CSDW) method is used to study the quantum reactive collision dynamics of O(3P)+H2 → OH+H and its D2, HD, and DH counterparts. The potential surface used is the sum of a LEPS potential (due to Johnson and Winter) and a correction factor which raises the barrier for H+OH → HO+H exchange to a realistic value. Full basis set convergence of the CSDW transition probabilities is established at low energies where tunneling dominates the dynamics, which means that the calculated cross sections should be exact except for errors introduced by the CS approximation, and the latter are expected to be less than 30%. The results presented for all four isotopes include: reaction probabilities as a function of energy E and total angular momentum J, total and state to state integral cross sections (including an analysis of product state distributions), and thermal and state resolved rate constants. Comparison of the results with those of several previous dynamical calculations on the same or similar surfaces is made and the accuracy of the approximations made in those calculations is assessed. For example, the product rotational distributions predicted by vibrationally adiabatic distorted wave theory are found to be quite close to what we calculate, although the absolute magnitudes of the cross sections are quite different. Comparison with the results of quasiclassical trajectory calculations indicates good agreement of the reactive cross sections well above the classical threshold, but not of the rate constants (because of tunneling) or of isotope ratios. Wigner corrected conventional transition state theory is very inaccurate in predicting rate constants, but a method which uses collinear exact quantum (CEQ) transmission coefficients to correct transition state theory does quite well. Variational transition state theory estimates of the rate constants and isotope ratios are also quite good, with the CSDW results generally bracketed by results obtained using the least action ground state (LAG) and small curvature ground state (SCTSAG) tunneling approximations. Comparison with experimental rate constants and isotope ratios is studied, and we find that the CSDW results are just outside the experimental error bars in all cases. The present results on the corrected Johnson and Winter surface are found to be slightly less accurate than the best estimates of rate constants and isotope ratios obtained using the modified POLCI surface of Walch and co-workers, suggesting that the latter surface is more accurate.

Direct dynamics study on the reaction of 1, 1-difluoroethane with hydrogen trioxide radical