Kinetic isotope effects calculated with the instanton method

Heavy-Atom Quantum Tunnelling in Spin Crossovers of Nitrenes

We simulate two recent matrix-isolation experiments at cryogenic temperatures, in which a nitrene undergoes spin crossover from its triplet state to a singlet state via quantum tunnelling. We detail the failure of the commonly applied weak-coupling method (based on a linear approximation of the potentials) in describing these deep-tunnelling reactions. The more rigorous approach of semiclassical golden-rule instanton theory in conjunction with double-hybrid density-functional theory and multireference perturbation theory does, however, provide rate constants and kinetic isotope effects in good agreement with experiment. In addition, these calculations locate the optimal tunnelling pathways, which provide a molecular picture of the reaction mechanism. The reactions involve substantial heavy-atom quantum tunnelling of carbon, nitrogen and oxygen atoms, which unexpectedly even continues to play a role at room temperature.

Instanton theory for Fermi's golden rule and beyond

Instanton theory provides a semiclassical approximation for computing quantum tunnelling effects in complex molecular systems. It is typically applied to proton-transfer reactions for which the Born-Oppenheimer approximation is valid. However, many processes in physics, chemistry and biology, such as electron transfers, are non-adiabatic and are correctly described instead using Fermi's golden rule. In this work, we discuss how instanton theory can be generalized to treat these reactions in the golden-rule limit. We then extend the theory to treat fourth-order processes such as bridge-mediated electron transfer and apply the method to simulate an electron moving through a model system of three coupled quantum dots. By comparison with benchmark quantum calculations, we demonstrate that the instanton results are much more reliable than alternative approximations based on superexchange-mediated effective coupling or a classical sequential mechanism. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Determination of concerted or stepwise mechanism of hydrogen tunneling from isotope effects: Departure between experiment and theory

Isotope substitution is an important experimental technique that offers deep insight into reaction mechanisms, as the measured kinetic isotope effects (KIEs) can be directly compared with theory. For multiple proton transfer processes, there are two types of mechanisms: stepwise transfer and concerted transfer. The Bell-Limbach model provides a simple theory to determine whether the proton transfer mechanism is stepwise or concerted from KIEs. Recent STM experiments have studied the proton switching process in water tetramers on NaCl(001). Theoretical studies predict that this process occurs via a concerted mechanism, however, the experimental KIEs resemble the Bell-Limbach model for stepwise tunneling, raising question on the underlying mechanism or the validity of the model. We study this system using ab initio instanton theory, and in addition to thermal rates, we also considered microcanonical rates, as well as tunneling splittings. Instanton theory predicts a concerted mechanism, and the KIEs for tunneling rates (both thermal and microcanonical) upon deuteration are consistent with the Bell-Limbach model for concerted tunneling, but could not explain the experiments. For tunneling splittings, partial and full deuteration changes the size of it in a similar fashion to how it changes the rates. We further examined the Bell-Limbach model in another system, porphycene, which has both stepwise and concerted tunneling pathways. The KIEs predicted by instanton theory are again consistent with the Bell-Limbach model. This study highlights differences between KIEs in stepwise and concerted tunneling, and the discrepancy between theory and recent STM experiments. New theory/experiments are desired to settle this problem.

Calculations of quantum tunnelling rates for muonium reactions with methane, ethane and propane

Thermal rate constants for Mu + CH₄, Mu + C₂H₆ and Mu + C₃H₈ and their equivalent reactions with H were evaluated with ab initio instanton rate theory. The potential-energy surfaces are fitted using Gaussian process regression to high-level electronic-structure calculations evaluated around the tunnelling pathway. This method was able to successfully reproduce various experimental measurements for the rate constant of these reactions. However, it was not able to reproduce the faster-than-expected rate of Mu + C₃H₈ at 300 K reported by Fleming et al. [Phys. Chem. Chem. Phys., 2015, 17, 19901 and Phys. Chem. Chem. Phys., 2020, 22, 6326]. Analysis of our results indicates that the kinetic isotope effect at this temperature is not significantly influenced by quantum tunnelling. We consider many possible factors for the discrepancy between theory and experiment but conclude that in each case, the instanton approximation is unlikely to be the cause of the error. This is in part based on the good agreement we find between the instanton predictions and new multiconfigurational time-dependent Hartree (MCTDH) calculations for Mu + CH₄ using the same potential-energy surface. Further experiments will therefore be needed to resolve this issue.

Atom tunnelling in the reaction NH3+ + H2 → NH4+ + H and its astrochemical relevance

The title reaction is involved in the formation of ammonia in the interstellar medium. We have calculated thermal rates including atom tunnelling using different rate theories. Canonical variational theory with microcanonically optimised multidimensional tunnelling was used for bimolecular rates, modelling the gas-phase reaction and also a surface-catalysed reaction of the Eley-Rideal type. Instanton theory provided unimolecular rates, which model the Langmuir-Hinshelwood type surface reaction. The potential energy was calculated on the CCSD(T)-F12 level of theory on the fly. We report thermal rates and H/D kinetic isotope effects. The latter have implications for observed H/D fractionation in molecular clouds. Tunnelling causes rate constants to be sufficient for the reaction to play a role in interstellar chemistry even at cryogenic temperature. We also discuss intricacies and limitations of the different tunnelling approximations to treat this reaction, including its pre-reactive minimum.

Quantum tunneling during interstellar surface-catalyzed formation of water: the reaction H + H2O2 → H2O + OH

The final step of the water formation network on interstellar grain surfaces starting from the H + O₂ route is the reaction between H and H₂O₂. This reaction is known to have a high activation energy and therefore at low temperatures it can only proceed via tunneling. To date, however, no rate constants are available at temperatures below 200 K. In this work, we use instanton theory to compute rate constants for the title reaction with and without isotopic substitutions down to temperatures of 50 K. The calculations are based on density functional theory, with additional benchmarks for the activation energy using unrestricted single-reference and multireference coupled-cluster single-point energies. Gas-phase bimolecular rate constants are calculated and compared with available experimental data not only for H + H₂O₂ → H₂O + OH, but also for H + H₂O₂ → H₂ + HO₂. We find a branching ratio where the title reaction is favored by at least two orders of magnitude at 114 K. In the interstellar medium this reaction predominantly occurs on water surfaces, which increases the probability that the two reactants meet. To mimic this, one, two, or three spectator H₂O molecules are added to the system. Eley-Rideal bimolecular and Langmuir-Hinshelwood unimolecular rate constants are presented here. The kinetic isotope effects for the various cases are compared to experimental data as well as to expressions commonly used in astrochemical models. Both the rectangular barrier and the Eckart approximations lead to errors of about an order of magnitude. Finally, fits of the rate constants are provided as input for astrochemical models.

Kinetic isotope effects and how to describe them

We review several methods for computing kinetic isotope effects in chemical reactions including semiclassical and quantum instanton theory. These methods describe both the quantization of vibrational modes as well as tunneling and are applied to the ⋅H + H2 and ⋅H + CH4 reactions. The absolute rate constants computed with the semiclassical instanton method both using on-the-fly electronic structure calculations and fitted potential-energy surfaces are also compared directly with exact quantum dynamics results. The error inherent in the instanton approximation is found to be relatively small and similar in magnitude to that introduced by using fitted surfaces. The kinetic isotope effect computed by the quantum instanton is even more accurate, and although it is computationally more expensive, the efficiency can be improved by path-integral acceleration techniques. We also test a simple approach for designing potential-energy surfaces for the example of proton transfer in malonaldehyde. The tunneling splittings are computed, and although they are found to deviate from experimental results, the ratio of the splitting to that of an isotopically substituted form is in much better agreement. We discuss the strengths and limitations of the potential-energy surface and based on our findings suggest ways in which it can be improved.

Comparison of classical reaction paths and tunneling paths studied with the semiclassical instanton theory

Comparison of classical reaction paths and semiclassical instanton paths for a proton transfer reaction mechanism.

Quantum Tunneling Rates of Gas-Phase Reactions from On-the-Fly Instanton Calculations

The instanton method obtains approximate tunneling rates from the minimum-action path (known as the instanton) linking reactants to the products at a given temperature. An efficient way to find the instanton is to search for saddle-points on the ring-polymer potential surface, which is obtained by expressing the quantum Boltzmann operator as a discrete path-integral. Here we report a practical implementation of this ring-polymer form of instanton theory into the Molpro electronic-structure package, which allows the rates to be computed on-the-fly, without the need for a fitted analytic potential-energy surface. As a test case, we compute tunneling rates for the benchmark H + CH₄ reaction, showing how the efficiency of the instanton method allows the user systematically to converge the tunneling rate with respect to the level of electronic-structure theory.

Formation of the prebiotic molecule NH2CHO on astronomical amorphous solid water surfaces: accurate tunneling rate calculations

Investigating how formamide forms in the interstellar medium is a hot topic in astrochemistry, which can contribute to our understanding of the origin of life on Earth. We have constructed a QM/MM model to simulate the hydrogenation of isocyanic acid on amorphous solid water surfaces to form formamide. The binding energy of HNCO on the ASW surface varies significantly between different binding sites, we found values between ∼0 and 100 kJ mol-1. The barrier for the hydrogenation reaction is almost independent of the binding energy, though. We calculated tunneling rate constants of H + HNCO → NH2CO at temperatures down to 103 K combining QM/MM with instanton theory. Tunneling dominates the reaction at such low temperatures. The tunneling reaction is hardly accelerated by the amorphous solid water surface compared to the gas phase for this system, even though the activation energy of the surface reaction is lower than the one of the gas-phase reaction. Both the height and width of the barrier affect the tunneling rate in practice. Strong kinetic isotope effects were observed by comparing to rate constants of D + HNCO → NHDCO. At 103 K we found a KIE of 231 on the surface and 146 in the gas phase. Furthermore, we investigated the gas-phase reaction NH2 + H2CO → NH2CHO + H and found it unlikely to occur at cryogenic temperatures. The data of our tunneling rate constants are expected to significantly influence astrochemical models.

Kinetic Isotope Effects in Multipath VTST: Application to a Hydrogen Abstraction Reaction

In this work we apply multipath canonical variational transition state theory with small-tunneling corrections (MP-CVT/SCT) to the hydrogen abstraction reaction from ethanol by atomic hydrogen in aqueous solution at room temperature. This reaction presents two transition states which can interconvert by internal rotations about single bonds and another two transition states that are non-interconvertible enantiomers to the former structures. The study also includes another three reactions with isotopically substituted species for which there are experimental values of thermal rate constants and kinetic isotope effects (KIEs). The agreement between the MP-CVT/SCT thermal rate constants and the experimental data is good. The KIEs obtained by the MP-CVT/SCT methodology are factorized in terms of individual transition state contributions to facilitate the analysis. It was found that the percentage contribution of each transition state to the total KIE is independent of the isotopic substitution.

Microcanonical and thermal instanton rate theory for chemical reactions at all temperatures

Semiclassical instanton theory is used to study the quantum effects of tunnelling and delocalization in molecular systems. An analysis of the approximations involved in the method is presented based on a recent first-principles derivation of instanton rate theory [J. Chem. Phys., 2016,144, 114106]. It is known that the standard instanton method is unable to accurately compute thermal rates near the crossover temperature. The causes of this problem are identified and an improved method is proposed, whereby an instanton approximation to the microcanonical rate is defined and integrated numerically to obtain a thermal rate at any temperature. No new computational algorithms are required, but only data analysis of a number of standard instanton calculations.

On the tunneling instability of a hypercoordinated carbocation

C(CH₃)₅⁺ was predicted to be a “fleeting” molecule. But will it stand the tunneling test?

Role of tunneling in the enzyme glutamate mutase

The role of quantum mechanical atom tunneling during the conversion of glutamate to methylaspartate catalyzed by glutamate mutase is investigated by quantum mechanical/molecular mechanical (QM/MM) simulations based on coupled cluster and density functional calculations. The use of instanton theory allows us to calculate the tunneling contributions of up to 78 atoms in the active site. We calculate kinetic isotope effects (KIEs) and compare them to experimental data. The simulations lead to deuterium KIEs of 10 for the hydrogen abstraction from glutamate substrate and 16 for the hydrogen abstraction from methylaspartate substrate, which are consistent with the experimental results. The hydrogen abstraction from methylaspartate has higher primary deuterium and tritium (46.1) KIEs than the abstraction from glutamate. The tunneling effect increases the reaction rate by a factor of 12.3 for the hydrogen abstraction from methylaspartate at 0. Tunneling is supported by the environment by preparing the enzyme through classical motions. Consideraton of the tunneling contributions of more and more atoms around the active center shows that the motions at the ribose ring play a central role during the tunneling enhancement of the hydrogen transfers. Our simulations give new insight into the catalytic process in glutamate mutase and the way enzymes use tunneling effects for a successful catalysis.