Comparison of classical reaction paths and tunneling paths studied with the semiclassical instanton theory
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Atom tunneling in the hydrogen atom transfer reaction of the 2,4,6-tri-tert-butylphenyl radical to 3,5-di-tert-butylneophyl, which has a short but strongly curved reaction path, was investigated using instanton theory. We found the tunneling path to deviate qualitatively from the classical intrinsic reaction coordinate, the steepest-descent path in mass-weighted Cartesian coordinates. To perform that comparison, we implemented a new variant of the predictor-corrector algorithm for the calculation of the intrinsic reaction coordinate. We used the reaction force analysis method as a means to decompose the reaction barrier into structural and electronic components. Due to the narrow energy barrier, atom tunneling is important in the abovementioned reaction, even above room temperature. Our calculated rate constants between 350 K and 100 K agree well with experimental values. We found a H/D kinetic isotope effect of almost 106 at 100 K. Tunneling dominates the protium transfer below 400 K and the deuterium transfer below 300 K. We compared the lengths of the tunneling path and the classical path for the hydrogen atom transfer in the reaction HCl + Cl and quantified the corner cutting in this reaction. At low temperature, the tunneling path is about 40% shorter than the classical path.Keywords:
Semiclassical physics
Instanton
Kinetic isotope effect
Reaction coordinate
Hydrogen atom
Rectangular potential barrier
Reaction rate
Isotope effects (IEs) on chemical reaction rates due to tunneling can be classified as three kinds: primary, secondary, and reaction-path curvature effects. The primary IE originates from the effective mass for the reaction coordinate and is seen when an atom in a bond that is formed or broken is replaced with an isotope. The secondary IE originates from variation in the zero-point energy (ZPE) along the reaction coordinate and is seen when a “spectator” atom is replaced with an isotope. This effect is usually small as compared to the primary IE and often shows an inverse effect, i.e., an isotopic substitution reduces the rate constant. The reaction-path curvature effect originates from the corner cutting of the tunneling path. The curvature of the reaction path depends only on the mass combination of the system; for example, it is large for a heavy-light-heavy system and small for a light-heavy-light system.
Kinetic isotope effect
Reaction coordinate
Elementary reaction
Primary (astronomy)
Hydrogen atom abstraction
Reaction rate
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A theoretical study of primary kinetic isotope effects (KIEs) is presented for proton transfer (PT) reactions in a polar environment in the nonadiabatic, i.e., tunneling, regime. This treatment differs from traditional descriptions for PT most notably in the identification of a solvent coordinate as the reaction coordinate. The theory explicitly addresses KIE features that are extremely sensitive to the proton donor−proton acceptor mode dynamics. Besides KIE behaviors that are not consistent with nontunneling PT, individual KIE aspects in some cases, such as magnitude, temperature dependence, variation with reaction asymmetry, and Swain−Schaad behavior can yield results consistent with nontunneling PT. However, a combination of KIE aspectswith particular emphasis on KIE variation with reaction asymmetry or temperaturecan clearly identify tunneling in PT systems. In addition, PT via excited proton vibrational states is shown to significantly contribute to the reaction rate and KIEs, especially for extremely asymmetric reactions, where it can dominate.
Kinetic isotope effect
Reaction coordinate
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The relationship between the two hydrogen isotope effects, kH/kT and kD/kT, can provide a probe for the role of tunneling and coupled motion in enzyme-catalyzed reactions.1,2 Using vibrational analysis and the Bigeleisen−Mayer equation, we have developed a simple computational model to explain the unusual exponential relationships that have been experimentally observed in the yeast alcohol dehydrogenase (YADH)3-catalyzed oxidation of benzyl alcohol.2 The experimental results are fitted by a model that has both substantial hydrogen tunneling and coupling between the reaction coordinate and a large number of vibrational modes. We show that the secondary kD/kT isotope effect is expected to be the most sensitive parameter to changes in reaction coordinate properties. A high degree of coupled motion leads to an unexpected suppression of the semiclassical secondary isotope effects, resulting in secondary isotope effects which are primarily manifestations of tunneling. This has implications for the use of secondary hydrogen isotope effects as probes of transition state position. During the course of these computational studies, primary carbon isotope effects were shown to undergo a consistent increase in magnitude upon substitution of the transferring protium with deuterium. We suggest that this effect can be explained using simple semiclassical principles and will apply to a broad range of reactions. Inference of reaction mechanism from the observation of a change in a heavy-atom isotope effects upon substrate deuteration should be interpreted with caution when the position of heavy atom and hydrogen isotope substitution are the same.
Kinetic isotope effect
Reaction coordinate
Hydrogen atom
Semiclassical physics
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Abstract Reactions occurring at a carbon atom through the Walden inversion mechanism are one of the most important and useful classes of reactions in chemistry. Here we report an accurate theoretical study of the simplest reaction of that type: the H+CH 4 substitution reaction and its isotope analogues. It is found that the reaction threshold versus collision energy is considerably higher than the barrier height. The reaction exhibits a strong normal secondary isotope effect on the cross-sections measured above the reaction threshold, and a small but reverse secondary kinetic isotope effect at room temperature. Detailed analysis reveals that the reaction proceeds along a path with a higher barrier height instead of the minimum-energy path because the umbrella angle of the non-reacting methyl group cannot change synchronously with the other reaction coordinates during the reaction due to insufficient energy transfer from the translational motion to the umbrella mode.
Kinetic isotope effect
Reaction coordinate
Transition state
Elementary reaction
Reaction rate
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Primary H/D kinetic isotope effects for the [1,5] hydrogen shift reaction in 13-atomic 1,3-pentadiene and [1,7] hydrogen shift reaction in 23-atomic 7-methylocta-1,3,5-triene are calculated using the semiclassical instanton approach. All 33 and 63 internal degrees of freedom, respectively, are treated quantum mechanically with multidimensional tunneling automatically accounted for by the instanton approach. Reactive potential energy surfaces are calculated on-the-fly using mPW1K/6-31+G(d,p) and mPWB1K/6-31+G(d,p) electronic structure methods. The calculated kinetic isotope effects agree well with the previously reported experimental measurements. The analytical expressions of the semiclassical instanton approach allow one to determine quantitative contributions of various physical mechanisms to the calculated kinetic isotope effects. Multidimensional tunneling is found to play important role in both studied hydrogen shift reactions.
Instanton
Semiclassical physics
Kinetic isotope effect
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Rectangular potential barrier
Maxima and minima
Reaction coordinate
Mode (computer interface)
Tunnel effect
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The factors affecting kinetic isotope effects in barrierless recombination reactions are considered from the perspective of variational transition state theory (VTST). Despite the broad application of VTST methods, a general consideration of kinetic isotope effect predictions of the theory has not previously been undertaken, especially for cases where changes in the internal structure and vibrational frequencies of the fragments (i.e., the conserved modes) can be assumed to be negligible. Use of the center-of-mass separation as the reaction coordinate in such a case entails some restriction on the range of kinetic isotope effects which can be accommodated. Larger effects are possible within a variable reaction coordinate implementation of transition state theory, and the predicted kinetic isotope effects are shown to be strongly dependent on the location of the pivot point. Illustrative model calculations demonstrate the feasibility of reproducing the experimentally observed kinetic isotope effects for the CH + O2, HCC + O2, CH + C2H2, and CH + C2H4 reactions with realistic deviations of the pivot points from the center-of-mass. In contrast, calculations restricted to center-of-mass pivot points predict isotope effects that are even inverted. For the CH + CH4 reaction, the isotope effects appear too large to be explained by the reaction coordinate variations, and changes in the conserved modes play a key role in the observed isotope effects, as demonstrated with ab initio based TST simulations. Overall, the experimentally observed kinetic isotope effects in CH addition reactions are strongly suggestive of an optimum reaction coordinate corresponding to a pivot point located near the center of the radical orbital.
Kinetic isotope effect
Reaction coordinate
Reaction rate
Transition state theory
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Quantum tunneling is mostly discussed in the Euclidean path integral formalism using instantons. On the other hand, it is difficult to understand quantum tunneling based on the real-time path integral due to its oscillatory nature, which causes the notorious sign problem. We show that recent development of the Lefschetz thimble method enables us to investigate this issue numerically. In particular, we find that quantum tunneling occurs due to complex trajectories, which are actually observable experimentally by using the so-called weak measurement.
Instanton
Formalism (music)
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Thermal decomposition of (CH3)2C(OH)CD3 in a static reactor and in a shock tube reactor led to intramolecular symmetry-corrected kH/kD kinetic isotope effects for eliminations of HOH and HOD of 1.80 ± 0.08 at 436 to 481 °C and 1.54 ± 0.12 at 813 to 883 °C. Calculations with B3LYP/6-31G* theory defined the transition structure for the 1,2-elimination reaction, the internal reaction coordinate path, and kH/kD predictions. The reaction takes place through a four-centered transition structure approached through very different progressions of bond length changes along the reaction coordinate.
Kinetic isotope effect
Transition state
Elimination reaction
Reaction coordinate
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Abstract The symmetric dependence of experimental primary kinetic isotope effects on the free energy change, obtained for reaction series with zero equilibrium isotope effects, can be inverted to obtain the free energy profile along the reaction coordinate. Trends in the kinetic isotope effects for the general case of non‐vanishing equilibrium effects can be understood by an extension of the same theory. Comparison is made with experiments for reactions of halogen atoms with hydrogen/deuterium molecules.
Kinetic isotope effect
Reaction coordinate
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