Application of the large-curvature tunneling approximation to polyatomic molecules: Abstraction of H or D by methyl radical

Rate coefficients and product branching ratios for (E)-2-butenal + H reactions

Thermal rate constants for the hydrogen abstraction reactions of (E)-2-butenal by hydrogen atoms were calculated, for the first time, using the multipath canonical variational theory with small-curvature tunneling (MP-CVT/SCT). After a torsional potential energy surface exploration, ten conformations of the transition states (including the mirror images) were found and separated into four conformational reaction channels (CRCs). Individual energy paths of each CRC were built, recrossing and quantum tunneling effects estimated, and the thermal rate constants obtained. Due to the hindered rotors, the torsional anharmonicity was incorporated in the rate coefficient through the calculations of the rovibrational partition functions using the extended two-dimensional torsional method (E2DT). For comparison, the one-well (1W-CVT/SCT) and harmonic multipath (MP-CVT/SCT) thermal rate constants were also estimated. In addition, kinetic Monte Carlo (KMC) simulations were performed to predict the product branching ratios. For all kinetic approaches, the formation of products of (R1) is predominant. Compared to the harmonic multipath estimation, the percentage of reaction (R4) increases by approximately 9% when the torsional anharmonicity is taken into account. For the reactions (R2) and (R3), the product branching ratio is slightly decreased when compared with the harmonic simulation.

Quantum Suppression of Intramolecular Deuterium Kinetic Isotope Effects in a Pericyclic Hydrogen Transfer Reaction

It is generally accepted that hydrogen tunneling enhances both primary and secondary H/D kinetic isotope effects (KIEs) over what would be expected under the assumptions of classical barrier transition. Previous studies have exclusively shown that the effects of tunneling upon primary H/D KIEs have been much larger than those observed for secondary H/D KIEs. Here we present a series of experimental H/D KIE results associated with the Chugaev elimination of methyl xanthate derived from β-phenylethanol over the temperature range of 180 to 290 °C. Intramolecular H/D KIEs computed according to classical transition state theory (TST) are markedly overestimated relative to experimentally measured values. Experimental intermolecular H/D KIEs and direct dynamic calculations based on canonical variational transition state theory (CVT) with small-curvature tunneling correction (SCT) reveal that this result is largely the consequence of extraordinary tunneling enhancement of the secondary H/D KIE. This unexpected behavior is examined in the context of other similar hydrogen transfer reactions.

Variational transition state theory: theoretical framework and recent developments

This article reviews the fundamentals of variational transition state theory (VTST), its recent theoretical development, and some modern applications.

Stress Test for Quantum Dynamics Approximations: Deep Tunneling in the Muonium Exchange Reaction D + HMu → DMu + H

Quantum effects play a crucial role in chemical reactions involving light atoms at low temperatures, especially when a light particle is exchanged between two heavier partners. Different theoretical methodologies have been developed in the last decades attempting to describe zero-point energy and tunneling effects without abandoning a classical or semiclassical framework. In this work, we have chosen the D + HMu → DMu + H reaction as a stress test system for three well-established methods: two representative versions of transition state theory (TST), canonical variational theory and semiclassical instanton, and ring polymer molecular dynamics (RPMD). These calculations will be compared with accurate quantum mechanical results. Despite its apparent simplicity, the exchange of the extremely light muonium atom (0.114 u) becomes a most challenging reaction for conventional methods. The main result of this work is that RPMD provides an overall better performance than TST-based methods for such a demanding reaction. RPMD might well turn out to be a useful tool beyond TST applicability.

Substrate-dependent H/D kinetic isotope effects and the role of the di(μ-oxo)diiron(IV) core in soluble methane monooxygenase: a theoretical study

Soluble methane monooxygenase (sMMO) is an enzyme that converts alkanes to alcohols using a di(μ-oxo)diiron(IV) intermediate Q at the active site. Very large kinetic isotope effects (KIEs) indicative of significant tunneling are observed for the hydrogen transfer (H-transfer) of CH4 and CH3 CN; however, a relatively small KIE is observed for CH3NO2. The detailed mechanism of the enzymatic H-transfer responsible for the diverse range of KIEs is not yet fully understood. In this study, variational transition-state theory including the multidimensional tunneling approximation is used to calculate rate constants to predict KIEs based on the quantum-mechanically generated intrinsic reaction coordinates of the H-transfer by the di(μ-oxo)diiron(IV) complex. The results of our study reveal that the role of the di(μ-oxo)diiron(IV) core and the H-transfer mechanism are dependent on the substrate. For CH4 , substrate binding induces an electron transfer from the oxygen to one Fe(IV) center, which in turn makes the μ-O ligand more electrophilic and assists the H-transfer by abstracting an electron from the C-H σ orbital. For CH3CN, the reduction of Fe(IV) to Fe(III) occurs gradually with substrate binding and H-transfer. The charge density and electrophilicity of the μ-O ligand hardly change upon substrate binding; however, for CH3NO2, there seems to be no electron movement from μ-O to Fe(IV) during the H-transfer. Thus, the μ-O ligand appears to abstract a proton without an electron from the C-H σ orbital. The calculated KIEs for CH4, CH3CN, and CH3NO2 are 24.4, 49.0, and 8.27, respectively, at 293 K, in remarkably good agreement with the experimental values. This study reveals that diverse KIE values originate mainly from tunneling to the same di(μ-oxo)diiron(IV) core for all substrates, and demonstrate that the reaction dynamics are essential for reproducing experimental results and understanding the role of the diiron core for methane oxidation in sMMO.

Biofuel combustion. Energetics and kinetics of hydrogen abstraction from carbon-1 in n-butanol by the hydroperoxyl radical calculated by coupled cluster and density functional theories and multistructural variational transition-state theory with multidimensional tunneling

Multistructural canonical variational transition-state theory with small-curvature multidimensional tunneling (MS-CVT/SCT) is employed to calculate thermal rate constants for hydrogen-atom abstraction from carbon-1 of n-butanol by the hydroperoxyl radical over the temperature range 250-2000 K. The M08-SO hybrid meta-GGA density functional was validated against CCSD(T)-F12a explicitly correlated wave function calculations with the jul-cc-pVTZ basis set. It was then used to compute the properties of all stationary points and the energies and Hessians of a few nonstationary points along the reaction path, which were then used to generate a potential energy surface by the multiconfiguration Shepard interpolation (MCSI) method. The internal rotations in the transition state for this reaction (like those in the reactant alcohol) are strongly coupled to each other and generate multiple stable conformations, which make important contributions to the partition functions. It is shown that neglecting to account for the multiple-structure effects and torsional potential anharmonicity effects that arise from the torsional modes would lead to order-of-magnitude errors in the calculated rate constants at temperatures of interest in combustion.

Direct Dynamics Implementation of the Least-Action Tunneling Transmission Coefficient. Application to the CH4/CD3H/CD4 + CF3 Abstraction Reactions

We present two new direct dynamics algorithms for calculating transmission coefficients of polyatomic chemical reactions by the multidimensional least-action tunneling approximation. The new algorithms are called the interpolated least-action tunneling method based on one-dimensional interpolation (ILAT1D) and the double interpolated least-action tunneling (DILAT) method. The DILAT algorithm, which uses a one-dimensional spline under tension to interpolate both of the effective potentials along the nonadiabatic portions of tunneling paths and the imaginary action integrals as functions of tunneling energies, was designed for the calculation of multidimensional LAT transmission coefficients for very large polyatomic systems. The performance of this algorithm has been tested for the CH4/CD3H/CD4 + CF3 hydrogen abstraction reactions with encouraging results, i.e., when the fitting is performed using 13 points, the algorithm is about 30 times faster than the full calculation with deviations that are smaller than 5%. This makes direct dynamics least-action tunneling calculations practical for larger systems, higher levels of electron correlation, and/or larger basis sets.

The calculation of multidimensional semiclassical wave functions in the forbidden region using real valued coordinates

A method that uses only real valued coordinates is presented for integrating the many dimensional semiclassical wave function into the forbidden region. The procedure first determines a surface of caustic points by running the set of trajectories that define the wave function in the allowed region. In the forbidden region, the momentum and the action integral are both complex functions of position, and their imaginary parts vanish on the caustic surface. The direction of the imaginary part of the momentum p(I) can be chosen to the perpendicular to the caustic surface at all points on that surface. Equations are derived for integrating the values of the real and imaginary parts of the momentum along the curves that follow the direction of p(I). The equations for the change in the action integral and the prefactor for the semiclassical wave function along these curves are also obtained, allowing for the determination of the semiclassical wave function in the forbidden region. Calculations are performed for a two dimensional problem, and the semiclassical wave function is found to be is excellent agreement with the results of exact quantum calculations.

Combining the nuclear-electronic orbital approach with vibronic coupling theory: calculation of the tunneling splitting for malonaldehyde

The nuclear-electronic orbital (NEO) method is combined with vibronic coupling theory to calculate hydrogen tunneling splittings in polyatomic molecules. In this NEO-vibronic coupling approach, the transferring proton and all electrons are treated quantum mechanically at the NEO level, and the other nuclei are treated quantum mechanically using vibronic coupling theory. The dynamics of the molecule are described by a vibronic Hamiltonian in a diabatic basis of two localized nuclear-electronic states for the electrons and transferring proton. This ab initio approach is computationally practical and efficient for relatively large molecules, and the accuracy can be improved systematically. The NEO-vibronic coupling approach is used to calculate the hydrogen tunneling splitting for malonaldehyde. The calculated tunneling splitting of 24.5 cm(-1) is in excellent agreement with the experimental value of 21.6 cm(-1). This approach also enables the identification of the dominant modes coupled to the transferring hydrogen motion and provides insight into their roles in the hydrogen tunneling process.

Direct-Dynamics VTST Study of the [1,7] Hydrogen Shift in 7-Methylocta-1,3(Z),5(Z)-triene. A Model System for the Hydrogen Transfer Reaction in Previtamin D3

Direct-dynamics canonical variational transition-state theory calculations with microcanonically optimized multidimensional transmission coefficient (CVT/muOMT) for tunneling were carried out at the MPWB1K/6-31+G(d,p) level to study the [1,7] sigmatropic hydrogen rearrangement in 7-methylocta-1,3(Z),5(Z)-triene. This compound has seven conformers, of which only one leads to products, although all of them have to be included in the theoretical treatment. The calculated CVT/muOMT rate constants are in good agreement with the available experimental data. To try to understand the role of tunneling in the hydrogen shift reaction, we have also calculated the thermal rate constants for the monodeuterated compound in the interval T = 333.2-388.2 K. This allowed us to evaluate primary kinetic isotope effects (KIEs) and make a direct comparison with the experiment. Our calculations show that both the large measured KIE and the large measured difference in the activation energies between the deuterated and root compounds are due to the quantum tunneling. The tunneling contribution to the KIE becomes noticeable only when the coupling between the reaction coordinate and the transverse modes is taken into account. Our results confirm previous experimental and theoretical works, which guessed that the obtained kinetic parameters pointed to a reaction with an important contribution due to tunneling. The above conclusion would be essentially valid for the case of the [1,7] hydrogen shift in previtamin D3 because of the similarity to the studied model system.

Double hydrogen tunneling revisited: the breakdown of experimental tunneling criteria

Formic acid dimer was chosen as a model system to investigate synchronous double proton transfer by means of variational transition state theory (VTST) for various isotopically modified hydrogen species. The electronic barrier for the double proton transfer was evaluated to be 7.9 kcal/mol, thus being significantly lower than it was determined in previous studies. The tunneling probabilities were evaluated at temperatures from 100 up to 400 K and typical Arrhenius behavior with enhancement by tunneling is observed. When comparing the transmission factors kappa in dependence of the mass of the tunneling hydrogen, it was found that there are two maxima, one at very low masses (e.g., 0.114 amu, corresponding to the muonium entity) and one maximum at around 2 amu (corresponding to deuterium). With the knowledge of the VTST-hydrogen transfer rates and the corresponding tunneling corrections, various tunneling criteria were tested (e.g., Swain-Schaad exponents) and were shown to fail in this reaction in predicting the extent of tunneling. This finding adds another aspect in the ongoing "Tunneling-Enhancement by Enzymes" discussion, as the used tunneling criteria based on experimental reaction rates may fail to predict tunneling behavior correctly.

Comparative study of cluster- and supercell-approaches for investigating heterogeneous catalysis by electronic structure methods: Tunneling in the reaction N + H → NH on Ru(0001)

Different ruthenium clusters of various sizes are constructed with the aim to model the Ru(0001) surface with a sufficient accuracy for predicting catalysis by hybrid density functional methods (B3LYP). As an example reaction the hydrogenation step N(ads) + H(ads) --> NH(ads) from the catalytic production cycle of ammonia is chosen. A cluster of 12 ruthenium atoms is found to reproduce experimental geometries and frequencies of the various reactants on the surface satisfyingly. To get the geometries of adsorbed hydrogen qualitatively correct it is shown that second layer atoms have to be included in the model cluster. Boundary effects are believed to have minor effects on optimized geometries, whereas the effects on reaction barriers are significant. A comparison of model cluster calculations to a periodic supercell approach employing plane waves and density functional methods (RPBE) reveals similar barriers for reaction. The influence of tunneling in this reaction is determined by the small curvature tunneling approach on the electronic surfaces.

The importance of tunneling in the first hydrogenation step in ammonia synthesis over a Ru(0001) surface

The hydrogenation of nitrogen (N(ads)+H(ads)-->NH(ads)) on metal surfaces is an important step in ammonia catalysis. We investigate the reaction dynamics of this hydrogenation step by time independent scattering theory and variational transition state theory (VTST) including tunneling corrections. The potential energy surface is derived by hybrid density functional theory on a model cluster composed of 12 ruthenium atoms resembling a Ru(0001) surface. The scattering calculations are performed on a reduced dimensionality potential energy hypersurface, where two dimensions are treated explicitly and all others are included implicitly by the zero-point correction. The VTST calculations include quantum effects along the reaction coordinate by applying the small curvature tunneling scheme. Even at room temperature (where ruthenium already shows catalytic activity) we find rate enhancement by tunneling by a factor of approximately 70. Inspection of the reaction probabilities shows that the major contribution to reactivity comes from the vibrational ground state of the reactants into vibrationally excited product states. The reaction rates are higher than determined in previous studies, and are compatible with experimental overall rates for ammonia synthesis.

Variational transition state theory calculations for the rate constants of the hydrogen scrambling and the dissociation of BH5 using the multiconfiguration molecular mechanics algorithm

The BH5 molecule contains a weak two-electron-three-center bond and it requires extremely high level of theories to calculate the energy and structure correctly. The potential energy of the hydrogen scrambling in BH5 has been generated by the multiconfiguration molecular mechanics algorithm with 15 high-level Shepard interpolation points, which would be practically impossible to obtain otherwise. The high-level interpolation points were obtained from the multicoefficient correlated quantum mechanical methods. The more high-level points are used, the better the shape of the potential energy surface. The rate constants are calculated using the variational transition state theory including multidimensional tunneling approximation. The potential energy curve for the BH5 dissociation has also been calculated, and the variational transition state was located to obtain the dissociation rate constants. Tunneling is very important in the scrambling, and there is large variational effect on the dissociation. The rate constants for the scrambling and the dissociation are 2.1 x 10(9) and 2.3 x 10(12) s(-1) at 300 K, respectively, which suggests that the dissociation is three orders of magnitude faster than the scrambling.

The ground-state tunneling splitting of various carboxylic acid dimers

Carboxylic acid dimers in gas phase reveal ground-state tunneling splittings due to a double proton transfer between the two subunits. In this study we apply a recently developed accurate semiclassical method to determine the ground-state tunneling splittings of eight different carboxylic acid derivative dimers (formic acid, benzoic acid, carbamic acid, fluoro formic acid, carbonic acid, glyoxylic acid, acrylic acid, and N,N-dimethyl carbamic acid) and their fully deuterated analogs. The calculated splittings range from 5.3e-4 to 0.13 cm(-1) (for the deuterated species from 2.8e-7 to 3.3e-4 cm(-1)), thus indicating a strong substituent dependence of the splitting, which varies by more than two orders of magnitude. One reason for differences in the splittings could be addressed to different barriers heights, which vary from 6.3 to 8.8 kcal/mol, due to different mesomeric stabilization of the various transition states. The calculated splittings were compared to available experimental data and good agreement was found. A correlation could be found between the tunneling splitting and the energy barrier of the double proton transfer, as the splitting increases with increased strength of the hydrogen bonds. From this correlation an empirical formula was derived, which allows the prediction of the ground-state tunneling splitting of carboxylic acid dimers at a very low cost and the tunneling splittings for parahalogen substituted benzoic acid dimers is predicted.

Kinetic isotope effects for concerted multiple proton transfer: a direct dynamics study of an active-site model of carbonic anhydrase II

The rate constant of the reaction catalyzed by the enzyme carbonic anhydrase II, which removes carbon dioxide from body fluids, is calculated for a model of the active site. The rate-determining step is proton transfer from a zinc-bound water molecule to a histidine residue via a bridge of two or more water molecules. The structure of the active site is known from X-ray studies except for the number and location of the water molecules. Model calculations are reported for a system of 58 atoms including a four-coordinated zinc ion connected to a methylimidazole molecule by a chain of two waters, constrained to reproduce the size of the active site. The structure and vibrational force field are calculated by an approximate density functional treatment of the proton-transfer step at the Self-Consistent-Charge Density Functional Tight Binding (SCC-DFTB) level. A single transition state is found indicating concerted triple proton transfer. Direct-dynamics calculations for proton and deuteron transfer and combinations thereof, based on the Approximate Instanton Method and on Variational Transition State Theory with Tunneling Corrections, are in fair agreement and yield rates that are considerably higher and kinetic isotope effects (KIEs) that are somewhat higher than experiment. Classical rate constants obtained from Transition State Theory are smaller than the quantum values but the corresponding KIEs are five times larger. For multiple proton transfer along water bridges classical KIEs are shown to be generally larger than quantum KIEs, which invalidates the standard method to distinguish tunneling and over-barrier transfer. In the present case, a three-way comparison of classical and quantum results with the observed data is necessary to conclude that proton transfer along the bridge proceeds by tunneling. The results suggest that the two-water bridge is present in low concentrations but makes a substantial contribution to proton transport because of its high efficiency. Bridging structures containing more water molecules may have lower energies but are expected to be less efficient. The observed exponential dependence of the KIEs on the deuterium concentration in H(2)O/D(2)O mixtures implies concerted transfer and thus rules out substantial contributions from structures that lead to stepwise transfer via solvated hydronium ions, which presumably dominate proton transfer in less efficient carbonic anhydrase isozymes.

The tightness contribution to the Brønsted alpha for hydride transfer between NAD+ analogues

It has been shown that the rate of symmetrical hydride transfer reaction varies with the hydride affinity of the (identical) donor and acceptor. In that case, Marcus theory of atom and group transfer predicts that the Brønsted alpha depends on the location of the substituent, whether it is in the donor or the acceptor, and the tightness of the critical configuration, as well as the resemblance of the critical configuration to reactants or products. This prediction has now been confirmed for hydride transfer reactions between heterocyclic, nitrogen-containing cations, which can be regarded as analogues of the enzyme cofactor, nicotinamide adenine dinucleotide (NAD+). A series of reactions with substituents in the donor gives Brønsted alpha of 0.67 +/- 0.03 and a tightness parameter, tau, of 0.64 +/- 0.06. With substituents in the acceptor alpha = 0.32 +/- 0.03 and tau = 0.68 +/- 0.08. The reactions are all spontaneous, with equilibrium constants between 0.4 and 3 x 10(4), and the two sets span about the same range of equilibrium constants. The two tau values are essentially identical with an average value of 0.66 +/- 0.05. These results can be semiquantitatively mimicked by rate constants calculated for a linear, triatomic model of the reaction. Variational transition state theory and a physically motivated but empirically calibrated potential function were used. The computed rate constants generate an alpha value of 0.56 if the hydride affinity of the acceptor is varied and an alpha of 0.44 if the hydride affinity of the donor is varied. The calculated kinetic isotope effects are similar to the measured values. A previous error in the Born charging term of the potential function has been corrected. Marcus theory can be successfully fitted to both the experimental and computed rate constants, and appears to be the most compact way to express and compare them. The success of the linear triatomic model in qualitatively reproducing these results encourages the continued use of this easily conceptualized model to think about group, ion, and atom transfer reactions.

Quantum mechanics/molecular mechanics studies of triosephosphate isomerase-catalyzed reactions: effect of geometry and tunneling on proton-transfer rate constants

The role of tunneling for two proton-transfer steps in the reactions catalyzed by triosephosphate isomerase (TIM) has been studied. One step is the rate-limiting proton transfer from Calpha in the substrate to Glu 165, and the other is an intrasubstrate proton transfer proposed for the isomerization of the enediolate intermediate. The latter, which is not important in the wild-type enzyme but is a useful model system because of its simplicity, has also been examined in the gas phase and in solution. Variational transition-state theory with semiclassical ground-state tunneling was used for the calculation with potential energy surface determined by an AM1 method specifically parametrized for the TIM system. The effect of tunneling on the reaction rate was found to be less than a factor of 10 at room temperature; the tunneling becomes more important at lower temperature, as expected. The imaginary frequency (barrier) mode and modes that have large contributions to the reaction path curvature are localized on the atoms in the active site, within 4 A of the substrate. This suggests that only a small number of atoms that are close to the substrate and their motions (e.g., donor-acceptor vibration) directly determine the magnitude of tunneling. Atoms that are farther away influence the effect of tunneling indirectly by modulating the energetics of the proton transfer. For the intramolecular proton transfer, tunneling was found to be most important in the gas phase, to be similar in the enzyme, and to be the smallest in water. The major reason for this trend is that the barrier frequency is substantially lower in solution than in the gas phase and enzyme; the broader solution barrier is caused by the strong electrostatic interaction between the highly charged solute and the polar solvent molecules. Analysis of isotope effects showed that the conventional Arrenhius parameters are more useful as experimental criteria for determining the magnitude of tunneling than the widely used Swain-Schaad exponent (SSE). For the primary SSE, although values larger than the transition-state theory limit (3.3) occur when tunneling is included, there is no clear relationship between the calculated magnitudes of tunneling and the SSE. Also, the temperature dependence of the primary SSE is rather complex; the value of SSE tends to decrease as the temperature is lowered (i.e., when tunneling becomes more significant). For the secondary SSE, the results suggest that it is more relevant for evaluating the "coupled motion" between the secondary hydrogen and the reaction coordinate than the magnitude of tunneling. Although tunneling makes a significant contribution to the rate of proton transfer, it appears not to be a major aspect of the catalysis by TIM at room temperature; i.e., the tunneling factor of 10 is "small" relative to the overall rate acceleration by 10(9). For the intramolecular proton transfer, the tunneling in the enzyme is larger by a factor of 5 than in solution.

Towards the experimental decomposition rate of carbonic acid (H2CO3) in aqueous solution

Dry carbonic acid has recently been shown to be kinetically stable even at room temperature. Addition of water molecules reduces this stability significantly, and the decomposition (H2CO3 + nH2O --> (n+1)H2O + CO2) is extremely accelerated for n = 1, 2, 3. By including two water molecules, a reaction rate that is a factor of 3000 below the experimental one (10 s(-1)) at room temperature was found. In order to further remove the gap between experiment and theory, we increased the number of water molecules involved to 3 and took into consideration different mechanisms for thorough elucidation of the reaction. A mechanism whereby the reaction proceedes via a six-membered transition state turns out to be the most efficient one over the whole examined temperature range. The determined reaction rates approach experimental values in aqueous solution reasonably well; most especially, a significant increase in the rates in comparison to the decomposition reaction with fewer water molecules is found. Further agreement with experiment is found in the kinetic isotope effects (KIE) for the deuterated species. For water-free carbonic acid, the KIE (i.e., kH2CO3/kD2CO3) for the decomposition reaction is predicted to be 220 at 300 K, whereas it amounts to 2.2-3.0 for the investigated mechanisms including three water molecules. This result is therefore reasonably close to the experimental value of 2 (at 300 K). These KIEs are in much better accordance with the experiment than the KIE for decomposition with fewer water entities.