Quantum instanton evaluation of the thermal rate constants and kinetic isotope effects for SiH4+H-->SiH3+H2 reaction in full Cartesian space

The quantum instanton calculations of thermal rate constants for the gas-phase reaction SiH4+H-->SiH3+H2 and its deuterated analogs are presented, using an analytical potential energy surface. The quantum instanton approximation is manipulated by full dimensionality in Cartesian coordinate path integral Monte Carlo approach, thereby taking explicitly into account the effects of the whole rotation, the vibrotational coupling, and anharmonicity of the reaction system. The rates and kinetic isotope effects obtained for the temperature range of 200-1000 K show good agreements with available experimental data, which give support to the accuracy of the underlying potential surface used. In order to investigate the sole quantum effect to the rates, the authors also derive the classical limit of the quantum instanton and find that it can be exactly expressed as the classical variation transition state theory. Comparing the quantum quantities with their classical analogs in the quantum instanton formula, the authors demonstrate that the quantum correction of the prefactor is more important than that of the activation energy at the transition state.     

Path integral calculation of thermal rate constants within the quantum instanton approximation: application to the H + CH4 --> H2 + CH3 hydrogen abstraction reaction in full Cartesian space

The quantum instanton approximation for thermal rate constants of chemical reactions [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)], which is modeled after the earlier semiclassical instanton approach, is applied to the hydrogen abstraction reaction from methane by a hydrogen atom, H + CH4 --> H2 + CH3, using a modified and recalibrated version of the Jordan-Gilbert potential surface. The quantum instanton rate is evaluated using path integral Monte Carlo approaches based on the recently proposed implementation schemes [Yamamoto and Miller, J. Chem. Phys. 120, 3086 (2004)]. The calculations were carried out using the Cartesian coordinates of all the atoms (thus involving 18 degrees of freedom), thereby taking explicit account of rotational effects of the whole system and also allowing the equivalent treatment of the four methane hydrogens. To achieve such a treatment, we present extended forms of the path integral estimators for relevant quantities that may be used for general N-atom systems with any generalized reaction coordinates. The quantum instanton rates thus obtained for the temperature range T = 200-2000 K show good agreement with available experimental data, which gives support to the accuracy of the underlying potential surface used.     

Quantum instanton calculation of rate constants for the C2H6 + H → C2H5 + H2 reaction: anharmonicity and kinetic isotope effects

Thermal rate constants and kinetic isotope effects for the title reaction are calculated by using the quantum instanton approximation within the full dimensional Cartesian coordinates. The obtained results are in good agreement with experimental measurements at high temperatures. The detailed investigation reveals that the anharmonicity of the hindered internal rotation motion does not influence the rate too much compared to its harmonic oscillator approximation. However, the motion of the nonreactive methyl group in C(2)H(6) significantly enhances the rates compared to its rigid case, which makes conventional reduced-dimensionality calculations a challenge. In addition, the temperature dependence of kinetic isotope effects is also revealed.     

A semiclassical initial-value representation for quantum propagator and boltzmann operator

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.     

On the efficient path integral evaluation of thermal rate constants within the quantum instanton approximation

We present an efficient path integral approach for evaluating thermal rate constants within the quantum instanton (QI) approximation that was recently introduced to overcome the quantitative deficiencies of the earlier semiclassical instanton approach [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)]. Since the QI rate constant is determined solely by properties of the (quantum) Boltzmann operator (specifically, by the zero time properties of the flux-flux and delta-delta correlation functions), it can be evaluated by well-established techniques of imaginary time path integrals even for quite complex chemical reactions. Here we present a series of statistical estimators for relevant quantities which can be evaluated straightforwardly with any nonlinear reaction coordinates and general Hamiltonians in Cartesian space. To facilitate the search for the optimal dividing surfaces required by the QI approximation, we introduce a two-dimensional quantum free energy surface associated with the delta-delta correlation function and describe how an adaptive umbrella sampling can be used effectively to construct such a free energy surface. The overall computational procedure is illustrated by the application to a hydrogen exchange reaction in gas phase, which shows excellent agreement of the QI rates with those obtained from quantum scattering calculations.     

Efficient estimators for quantum instanton evaluation of the kinetic isotope effects: application to the intramolecular hydrogen transfer in pentadiene

The quantum instanton approximation is used to compute kinetic isotope effects for intramolecular hydrogen transfer in cis-1,3-pentadiene. Due to the importance of skeleton motions, this system with 13 atoms is a simple prototype for hydrogen transfer in enzymatic reactions. The calculation is carried out using thermodynamic integration with respect to the mass of the isotopes and a path integral Monte Carlo evaluation of relevant thermodynamic quantities. Efficient "virial" estimators are derived for the logarithmic derivatives of the partition function and the delta-delta correlation functions. These estimators require significantly fewer Monte Carlo samples since their statistical error does not increase with the number of discrete time slices in the path integral. The calculation treats all 39 degrees of freedom quantum mechanically and uses an empirical valence bond potential based on a molecular mechanics force field.     

Vibrational energy relaxation rates via the linearized semiclassical approximation: applications to neat diatomic liquids and atomic-diatomic liquid mixtures

We report the results obtained from the application of our previously proposed linearized semiclassical method for computing vibrational energy relaxation (VER) rates (J. Phys. Chem. A 2003, 107, 9059, 9070) to neat liquid oxygen, neat liquid nitrogen, and liquid mixtures of oxygen and argon. Our calculations are based on a semiclassical approximation for the quantum-mechanical force-force correlation function, which puts it in terms of the Wigner transforms of the force and the product of the Boltzmann operator and the force. The calculation of the multidimensional Wigner integrals is made feasible by the introduction of a local harmonic approximation. A systematic analysis has been performed of the temperature and mole-fraction dependences of the VER rate constant, as well as the relative contributions of centrifugal and potential forces, and of different types of quantum effects. The results were found to be in very good quantitative agreement with experiment, and they suggest that this semiclassical approximation can capture the quantum enhancement, by many orders of magnitude, of the experimentally observed VER rate constants over the corresponding classical predictions.     

The normal range for secondary Swain-Schaad exponents without tunneling or kinetic complexity

An analysis is presented of the range of secondary Swain-Schaad exponents to be expected at 25 degrees C in the absence of tunneling or kinetic complexity. From 15 996 sets of exact harmonic semiclassical equilibrium isotope effects for simple C-H/D/T exchange reactions and 954 sets of exact harmonic semiclassical secondary H/D/T kinetic isotope effects for C-H positions in simple organic reactions, the distribution of Swain-Schaad exponents versus magnitude of the isotope effect is determined. This distribution defines when a secondary Swain-Schaad exponent may be considered to implicate nonsemiclassical behavior, revises the expected Swain-Schaad exponent for extrapolation of secondary isotope effects, and serves as a guide to the uncertainty in such extrapolations.     

Semiclassical instanton approach to calculation of reaction rate constants in multidimensional chemical systems

The semiclassical instanton approximation is revisited in the context of its application to the calculation of chemical reaction rate constants. An analytical expression for the quantum canonical reaction rate constants of multidimensional systems is derived for all temperatures from the deep tunneling to high-temperature regimes. The connection of the derived semiclassical instanton theory with several previously developed reaction rate theories is shown and the numerical procedure for the search of instanton trajectories is provided. The theory is tested on seven different collinear symmetric and asymmetric atom transfer reactions including heavy-light-heavy, light-heavy-light and light-light-heavy systems. The obtained thermal rate constants agree within a factor of 1.5-2 with the exact quantum results in the wide range of temperatures from 200 to 1500 K.     

A+BC共线碰撞能量传递的动力学李代数结合中间绘景研究

在半经典近似下将动力学李代数方法和中间绘景结合应用于原子-双原子分子(非谐振子)共线碰撞中的平动-振动传能的研究。在群参量的一级近似下求解群参量的运动方程,进而确定时间演化算符。并以散射体系H₂+He为例,计算了H₂分子的振动跃迁几率,计算结果与精确量子力学计算结果符合得较好。     

Quantum effects in ab-initio calculations of rate constants for chemical reactions occuring in the condensed phase

An overview of recent advances in the development of methods designed to calculate rate constants for chemical reactions obeying mass action kinetic equations in condensed phases is presented. A general framework addressing mixed quantum-classical systems is elaborated that enables quantum features such as tunneling effects, zero-point vibrations, dynamic quantum coherence, and non-adiabatic effects to be calculated. An efficient Monte Carlo sampling method for performing ab-initio calculations of rate constants and isotope effects in chemical processes in condensed phases is outlined, and the connection of isotope effects to reaction mechanism is explored     

Path-integral calculations of heavy atom kinetic isotope effects in condensed phase reactions using higher-order Trotter factorizations

A convenient approach to compute kinetic isotope effects (KIEs) in condensed phase chemical reactions is via path integrals (PIs). Usually, the primitive approximation is used in PI simulations, although such quantum simulations are computationally demanding. The efficiency of PI simulations may be greatly improved, if higher-order Trotter factorizations of the density matrix operator are used. In this study, we use a higher-order PI method, in conjunction with mass-perturbation, to compute heavy-atom KIE in the decarboxylation of orotic acid in explicit sulfolane solvent. The results are in good agreement with experiment and show that the mass-perturbation higher-order Trotter factorization provides a practical approach for computing condensed phase heavy-atom KIE.     

Tunneling splitting and decay of metastable states in polyatomic molecules: invariant instanton theory

This paper gives an overview of recently developed instanton theory of multidimensional tunneling and demonstrates its applicability to real polyatomic systems. One of the key features of the present formulation is rigorous solution of the multidimensional Hamilton-Jacoby and transport equations which constitutes the basis of the semiclassical theory accurate up to the first order in the Planck constant variant Planck's over 2pi. Apart from this fundamental assumption of the semiclassical dynamics the present instanton theory is exact, i.e. it neither involves any further approximation, nor relies on any models. For practical application, the theory is supplemented by numerical methods to construct a multi-dimensional tunneling path (instanton) and to combine the semiclassical theory with high-quality quantum chemical calculations. Emphasis is put on the instanton theory of tunneling splitting in polyatomic molecules. Importance of accurate potential energy surface (PES) information and multidimensional effects is demonstrated by applications to real molecules. The theory of life time of metastable states is also briefly explained.     

Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent

The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of approximately 2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.     

Examining the importance of dynamics,barrier compression and hydrogen tunnelling in enzyme catalysed reactions

Nuclear quantum mechanical tunnelling is important in enzyme-catalysed H-transfer reactions. This viewpoint has arisen after a number of experimental studies have described enzymatic reactions with kinetic isotope effects that are significantly larger than the semiclassical limit. Other experimental evidence for tunnelling, and the potential role of promoting vibrations that transiently compress the reaction barrier, is more indirect, being derived from the interpretation of e.g. mutational analyses of enzyme systems and temperature perturbation studies of reaction rates/kinetic isotope effects. Computational simulations have, in some cases, determined exalted kinetic isotope effects and tunnelling contributions, and identified putative promoting vibrations. In this review, we present the available evidence – both experimental and computational – for environmentally-coupled Htunnelling in several enzyme systems, namely aromatic amine dehydrogenase and members of the Old Yellow Enzyme family. We then consider the relative importance of tunnelling contributions to these reactions. We find that the tunnelling contribution to these reactions confers a rate enhancement of ～1000-fold. Without tunnelling, a 1000-fold reduction in activity would seriously impair cellular metabolism. We therefore infer that tunnelling is crucial to host organism viability thereby emphasising the general importance of tunnelling in biology.     

Kinetic isotope effects calculated with the instanton method

The ring-opening reaction of the cyclopropylcarbinyl radical proceeds via heavy-atom tunneling at low temperature. We used instanton theory to calculate tunneling rates and kinetic isotope effects with on-the-fly calculation of energies by density functional theory (B3LYP). The accuracy was verified by explicitly correlated coupled-cluster calculations (UCCSD(T)-F12). At cryogenic temperatures, we found protium/deuterium KIEs up to 13 and inverse KIEs down to 0.2. We also studied an intramolecular tautomerization reaction. A simple and computationally efficient method is proposed to calculate KIEs with the instanton method: the instanton path is assumed to be independent of the atomic masses. This results in surprisingly good estimates of the KIEs for the cyclopropylcarbinyl radical and for the secondary KIEs of the tautomerization. Challenges and capabilities of the instanton method for calculating KIEs are discussed.     

Thermochemistry and accurate quantum reaction rate calculations for H2/HD/D2 + CH3

Accurate quantum-mechanical results for thermodynamic data, cumulative reaction probabilities (for J = 0), thermal rate constants, and kinetic isotope effects for the three isotopic reactions H2 + CH3 --> CH4 + H, HD + CH3 --> CH4 + D, and D2 + CH3 --> CH(3)D + D are presented. The calculations are performed using flux correlation functions and the multiconfigurational time-dependent Hartree (MCTDH) method to propagate wave packets employing a Shephard interpolated potential energy surface based on high-level ab initio calculations. The calculated exothermicity for the H2 + CH3 --> CH4 + H reaction agrees to within 0.2 kcal/mol with experimentally deduced values. For the H2 + CH3 --> CH4 + H and D2 + CH3 --> CH(3)D + D reactions, experimental rate constants from several groups are available. In comparing to these, we typically find agreement to within a factor of 2 or better. The kinetic isotope effect for the rate of the H2 + CH3 --> CH4 + H reaction compared to those for the HD + CH3 --> CH4 + D and D2 + CH3 --> CH(3)D + D reactions agree with experimental results to within 25% for all data points. Transition state theory is found to predict the kinetic isotope effect accurately when the mass of the transferred atom is unchanged. On the other hand, if the mass of the transferred atom differs between the isotopic reactions, transition state theory fails in the low-temperature regime (T < 400 K), due to the neglect of the tunneling effect.     

High-precision quantum thermochemistry on nonquasiharmonic potentials: converged path-integral free energies and a systematically convergent family of generalized Pitzer-Gwinn approximations

Accurate quantum mechanical (QM) vibrational-rotational partition functions for HOOD, D(2)O(2), H(18)OOH, H(2)(18)O(2), D(18)OOH, and H(18)OOD are determined using a realistic potential energy surface for temperatures ranging from 300 to 2400 K by using the TT-FPI-ESPE path-integral Monte Carlo method. These data, together with our prior results for H(2)O(2), provide benchmarks for testing approximate methods of estimating isotope effects for systems with torsional motions. Harmonic approximations yield poor accuracy for these systems, and although the well-known Pitzer-Gwinn (PG) approximation provides better results for absolute partition functions, it yields the same results as the harmonic approximation for isotope effects because these are intrinsically quantal phenomena. We present QM generalizations of the PG approximation that can provide high accuracy for both isotope effects and absolute partition functions. These approximations can be systematically improved until they approach the accurate result and converge rapidly. These methods can also be used to obtain affordable estimates of zero-point energies from accurate partition functions-even those at relatively high temperatures.     

Boundary conditions for the Swain-Schaad relationship as a criterion for hydrogen tunneling

Hydrogen quantum mechanical tunneling has been suggested to play a role in a wide variety of hydrogen-transfer reactions in chemistry and enzymology. An important experimental criterion for tunneling is based on the breakdown of the semiclassical prediction for the relationship among the rates of the three isotopes of hydrogen (hydrogen -H, deuterium -D, and tritium -T). This is denoted the Swain-Schaad relationship. This study examines the breakdown of the Swain-Schaad relationship as criterion for tunneling. The semiclassical (no tunneling) limit used hereto (e.g., 3.34, for H/T to D/T kinetic isotope effects), was based on simple theoretical considerations of a diatomic cleavage of a stable covalent bond, for example, a C-H bond. Yet, most experimental evidence for a tunneling contribution has come from breakdown of those relationship for a secondary hydrogen, that is, not the hydrogen whose bond is being cleaved but its geminal neighbor. Furthermore, many of the reported experiments have been mixed-labeling experiments, in which a secondary H/T kinetic isotope effect was measured for C-H cleavage, while the D/T secondary effect accompanied C-D cleavage. In experiments of this type, the breakdown of the Swain-Schaad relationship indicates both tunneling and the degree of coupled motion between the primary and secondary hydrogens. We found a new semiclassical limit (e.g., 4.8 for H/T to D/T kinetic isotope effects), whose breakdown can serve as a more reliable experimental evidence for tunneling in this common mixed-labeling experiment. We study the tunneling contribution to C-H bond activation, for which many relevant experimental and theoretical data are available. However, these studies can be applied to any hydrogen-transfer reaction. First, an extension of the original approach was applied, and then vibrational analysis studies were carried out for a model system (the enzyme alcohol dehydrogenase). Finally, the effect of complex kinetics on the observed Swain-Schaad relationship was examined. All three methods yield a new semiclassical limit (4.8), above which tunneling must be considered. Yet, it was found that for many cases the original, localized limit (3.34), holds fairly well. For experimental results that are between the original and new limits (within statistical errors), several methods are suggested that can support or exclude tunneling. These new and clearer criteria provide a basis for future applications of the Swain-Schaad relationship to demonstrate tunneling in complex systems.