Variational Transition State Theory in Condensed Phases

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.

The variational transition state theory (VTST) of chemical reaction rates has been further developed and two previously developed and one new version have been illustrated and tested by various applications to collinear and three-dimensional reactions of the type A + BC ..-->.. AB + C. The first two versions considered are canonical variational theory (CVT), which is based on curves of free energy of activation as functions of location of the VTST dividing surface, and microcanonical variational theory (..mu..VT), which is based on minimizing the reactive flux through the VTST dividing surface at each total energy. CVT is simpler but ..mu..VT is more accurate. The new theory, improved canonical variational theory (ICVT), is almost as simple as CVT but almost as accurate as ..mu..VT. This has been demonstrated by applications to H, O, F, Cl, and I reacting with H/sub 2/, H reacting with F/sub 2/ and Cl/sub 2/, and various isotopic analogs and model systems. It was also demonstrated that VTST leads to very good agreement with accurate quantal results for several collinear reactions. Another project used VTST to explore the systematics of kinetic isotope effects for three-dimensional reactions. The predictions sometimes differ considerably from those of the conventional theory.

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 .

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.

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

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.

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.

The elementary vapor-phase reaction between Cl atoms and HCHO has been studied by ab initio methods. Calculations at the MP2, MP3, MP4(SDTQ), CCSD, CCSD(T), and MRD-CI levels of theory show that the reaction is characterized by a low electronic barrier; excluding the effects of spin-orbit splitting in Cl, our best estimate at the MRD-CI/aug-cc-pVTZ//RHF-RCCSD(T)/aug-cc-pVTZ level of theory predicts a Born-Oppenheimer barrier height of 0.7 kJ mol-1. The energies of the lowest six electronic states as resulting from MRD-CI calculations are presented at discrete points along the reaction path, and two avoided crossings are found in the transition state region. The spin-orbit splitting in Cl is also calculated along the reaction path; it is not negligible in the transition state region and is found to increase the barrier by only 1.4 kJ mol-1 at the RCCSD(T)/aug-cc-pVTZ transition state geometry. The minimum energy path of the reaction connects an energetically weakly stabilized adduct on the flat potential surface on the reactant side and an energetically strongly stabilized postreaction adduct. The reaction rate coefficient and the kinetic isotope effects were calculated using improved canonical variational theory with small curvature tunneling (ICVT/SCT), and the results were compared to experimental data. The experimental reaction rate coefficient is reproduced within its uncertainty limits by variational transition state theory with interpolated single-point energy corrections (ISPE) at the MP4(SDTQ) level of theory and by conventional transition state theory with interpolated optimized energies (IOE) at the MRD-CI//RCCSD(T) level of theory and interpolated optimized geometries at the RCCSD(T) level of theory on an MP2/aug-cc-pVTZ potential energy surface when employing scaled vibrational frequencies.

In the present work, we use Variational Transition State Theory (VTST) to develop a practical method for transition state ensemble optimization by looking for an optimal hyperplanar dividing surface in a space of meaningful trial collective variables. These might be interatomic distances, angles, electrostatic potentials, etc. Restrained molecular dynamics simulations are used to obtain on-the-fly estimates of ensemble averages that guide the variations of the hyperplane maximizing the transmission coefficient. A central result of our work is an expression that quantitatively estimates the importance of the coordinates used for the localization of the transition state ensemble. Starting from an arbitrarily large set of trial coordinates, one can distinguish those that are indeed essential for the advance of the reaction. This facilitates the use of VTST as a practical theory to study reaction mechanisms of complex processes. The technique was applied to the reaction catalyzed by an isochorismate pyruvate lyase. This reaction involves two simultaneous chemical steps and has a shallow transition state region, making it challenging to define a good reaction coordinate. Nevertheless, the hyperplanar transition state optimized in the space of 18 geometrical coordinates provides a transmission coefficient of 0.8 and a committor histogram well-peaked about 0.5, proving the strength of the method. We have also tested the approach with the study of the NaCl dissociation in aqueous solution, a stringest test for a method based on transition state theory. We were able to find essential degrees of freedom consistent with the previous studies and to improve the transmission coefficient with respect to the value obtained using solely the NaCl distance as the reaction coordinate.

The reaction of hydroxymethyl hydroperoxide (HMHP, HOCH2OOH) with the simplest Criegee intermediate, CH2OO, has been examined using quantum chemical methods with transition state theory. Geometry optimization and IRC calculations were performed using the M06-2X, MN15-L, and B2PLYP-D3 functionals in conjunction with the aug-cc-pVTZ basis set. Single point energy calculations using QCISD(T) and BD(T) with the same basis set have been performed to determine the energy of reactants, reactive complexes, transition states, and products. Rate coefficients have been obtained using variational transition state theory. The addition of CH2OO on the three different oxygen atoms in HMHP has been considered and the ether oxide forming channel, CH2OO + HOCH2OOH → HOCH2O(O)CH2OOH (channel 2), is the most favorable. The best computed standard enthalpy of reaction (ΔH) and zero-point corrected barrier height are -20.02 and -6.33 kcal mol-1, respectively. The reaction barrier is negative and our results suggest that both the inner and outer transition states contribute to the corresponding overall reactive flux in the tropospheric temperature range (220 K to 320 K). A two-transition state model has been used to obtain reliable rate coefficients at the high-pressure limit. The pressure-dependent rate coefficient calculations using the SS-QRRK theory have shown that this channel is pressure-dependent. Moreover, our investigation has shown that the ether oxide formed may rapidly react with SO2 at 298 K to form SO3, which can, in turn, react with water to form atmospheric H2SO4. A similar calculation has been conducted for the reaction of HMHP with OH, suggesting that the titled reaction may be a significant sink of HMHP. Therefore, the reaction between CH2OO and HOCH2OOH could be an indirect source for generating atmospheric H2SO4, which is crucial to the formation of clouds, and it might relieve global warming.

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.

Theoretical kinetic estimates for the recombination of hydrogen atoms with propargyl and allyl radicals

On a recent analytical potential energy surface developed by two of the authors, an exhaustive kinetics study, using variational transition state theory with multidimensional tunneling effect, and dynamics study, using both quasi-classical trajectory and full-dimensional quantum scattering methods, was carried out to understand the reactivity of the NH(3) + H → NH(2) + H(2) gas-phase reaction. Initial state-selected time-dependent wave packet calculations using a full-dimensional model were performed, where the total reaction probabilities were calculated for the initial ground vibrational state and for four excited vibrational states of ammonia. Thermal rate constants were calculated for the temperature range 200-2000 K using the three methods and compared with available experimental data. We found that (a) the total reaction probabilities are very small, (b) the symmetric and asymmetric N-H stretch excitations enhance the reactivity, (c) the quantum-mechanical calculated thermal rate constants are about one order of magnitude smaller than the transition state theory results, which reproduce the experimental evidence, and (d) quasi-classical trajectory calculations, which were performed with the main goal of analyzing the influence of the zero-point energy problem on the final dynamics results, reproduce the quantum scattering calculations on the same surface.

We have carried out ab initio calculations for the reaction OH+CH4→H2O+CH3 using second-order Mo/ller–Plesset perturbation theory, employing a very large basis set and scaling all correlation energy for the final energy calculation, but optimizing the equilibrium and transition state structures without scaling (MP-SAC2//MP2). We found that inclusion of correlation energy has an important effect on the geometry, barrier height, and vibrational frequencies of the transition state. The final calculated values for the forward and reverse classical barrier heights are 7.4 and 20.6 kcal/mol, respectively. We have used these with interpolated canonical variational transition state theory and the centrifugal-dominant small-curvature tunneling approximation, including information at the reactants, products, transition state, and two other points along the minimum energy path, to predict the rate constants for the above reaction in the temperature range from 223 to 2400 K. The calculated rate constants agree well with experiment over a wide temperature range.

A generalization of the Kramers–Grote–Hynes theory for reaction rates in the spatial diffusion limit is derived for a general class of Hamiltonians. Previous restrictions to harmonic baths and bilinear system bath couplings are removed. The key ingredient is the systematic use of variational transition state theory (VTST) to identify the optimal dividing surface. A pair of collective modes are defined as a linear combination of all system and bath modes. A free energy surface is defined in the two degree of freedom collective mode phase space. The VTST estimate for the rate of reaction on this surface is shown to be an upper bound to the exact rate. The optimal definition of the collective modes is obtained by minimizing the rate. The resulting rate expression is formally identical to the Kramers–Grote–Hynes theory. However, the minimization procedure leads to a new definition of the time dependent friction. In consistence with transition state theory, this time dependent friction is constructed from equilibrium properties of the composite system and does not call for any dynamical computations. The friction parameters are determined from equilibrium centroid averages of partial derivatives of the full potential at the barrier of the potential of mean force. This removes previous ambiguities as to the definition of time dependent friction in condensed matter systems. A procedure is presented for finding collective modes along which the friction exerted by the bath is minimized. This result may be of substantial interest in the study of complex dynamical systems in biology, chemistry, and physics.