Bimolecular reaction rates from ring polymer molecular dynamics: application to H + CH4 → H2 + CH3

In a recent paper, we have developed an efficient implementation of the ring polymer molecular dynamics (RPMD) method for calculating bimolecular chemical reaction rates in the gas phase, and illustrated it with applications to some benchmark atom-diatom reactions. In this paper, we show that the same methodology can readily be used to treat more complex polyatomic reactions in their full dimensionality, such as the hydrogen abstraction reaction from methane, H + CH(4) → H(2) + CH(3). The present calculations were carried out using a modified and recalibrated version of the Jordan-Gilbert potential energy surface. The thermal rate coefficients obtained between 200 and 2000 K are presented and compared with previous results for the same potential energy surface. Throughout the temperature range that is available for comparison, the RPMD approximation gives better agreement with accurate quantum mechanical (multiconfigurational time-dependent Hartree) calculations than do either the centroid density version of quantum transition state theory (QTST) or the quantum instanton (QI) model. The RPMD rate coefficients are within a factor of 2 of the exact quantum mechanical rate coefficients at temperatures in the deep tunneling regime. These results indicate that our previous assessment of the accuracy of the RPMD approximation for atom-diatom reactions remains valid for more complex polyatomic reactions. They also suggest that the sensitivity of the QTST and QI rate coefficients to the choice of the transition state dividing surface becomes more of an issue as the dimensionality of the reaction increases.     

Accurate quantum-mechanical rate constants for a linear response Azzouz-Borgis proton transfer model employing the multilayer multiconfiguration time-dependent Hartree approach

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is applied to simulate the quantum dynamics and thermal rate constant of the Azzouz-Borgis model of proton transfer in a polar solvent. To this end, the original atomistic potential is mapped to a system-bath model. Employing the flux correlation function formalism and importance sampling techniques, accurate quantum mechanical rate constants are obtained, which provide a benchmark for evaluating approximate approaches to study the quantum dynamics of condensed-phase chemical reactions. Furthermore, the validity of the mapping procedure is discussed based on the comparison of the classical dynamics of the original atomistic Azzouz-Borgis model and the mapped system-bath model.     

Proton transfer in a polar solvent from ring polymer reaction rate theory

We have used the ring polymer molecular dynamics method to study the Azzouz-Borgis model for proton transfer between phenol (AH) and trimethylamine (B) in liquid methyl chloride. When the A-H distance is used as the reaction coordinate, the ring polymer trajectories are found to exhibit multiple recrossings of the transition state dividing surface and to give a rate coefficient that is smaller than the quantum transition state theory value by an order of magnitude. This is to be expected on kinematic grounds for a heavy-light-heavy reaction when the light atom transfer coordinate is used as the reaction coordinate, and it clearly precludes the use of transition state theory with this reaction coordinate. As has been shown previously for this problem, a solvent polarization coordinate defined in terms of the expectation value of the proton transfer distance in the ground adiabatic quantum state provides a better reaction coordinate with less recrossing. These results are discussed in light of the wide body of earlier theoretical work on the Azzouz-Borgis model and the considerable range of previously reported values for its proton and deuteron transfer rate coefficients.     

Influences of molecular packing on the charge mobility of organic semiconductors: from quantum charge transfer rate theory beyond the first-order perturbation

The electronic coupling between adjacent molecules is an important parameter for the charge transport properties of organic semiconductors. In a previous paper, a semiclassical generalized nonadiabatic transition state theory was used to investigate the nonperturbative effect of the electronic coupling on the charge transport properties, but it is not applicable at low temperatures due to the presence of high-frequency modes from the intramolecular conjugated carbon-carbon stretching vibrations [G. J. Nan et al., J. Chem. Phys., 2009, 130, 024704]. In the present paper, we apply a quantum charge transfer rate formula based on the imaginary-time flux-flux correlation function without the weak electronic coupling approximation. The imaginary-time flux-flux correlation function is then expressed in terms of the vibrational-mode path average and is evaluated by the path integral approach. All parameters are computed by quantum chemical approaches, and the mobility is obtained by kinetic Monte-Carlo simulation. We evaluate the intra-layer mobility of sexithiophene crystal structures in high- and low-temperature phases for a wide range of temperatures. In the case of strong coupling, the quantum charge transfer rates were found to be significantly smaller than those calculated using the weak electronic coupling approximation, which leads to reduced mobility especially at low temperatures. As a consequence, the mobility becomes less dependent on temperature when the molecular packing leads to strong electronic coupling in some charge transport directions. The temperature-independent charge mobility in organic thin-film transistors from experimental measurements may be explained from the present model with the grain boundaries considered. In addition, we point out that the widely used Marcus equation is invalid in calculating charge carrier transfer rates in sexithiophene crystals.     

Quantum transition state theory for the full three-dimensional H+H2 reaction

A recently developed quantum transition state theory (QTST) [E. Pollak and J. L. Liao, J. Chem. Phys. 108, 2733 (1998)] for calculating thermal rate constants of chemical reactions is applied to the full three-dimensional hydrogen exchange reaction. Results are compared with other numerical results, for temperatures ranging from T=300 K to T=1500 K. The QTST rate is almost exact at high temperature and is 20% greater than the exact rate at T=300 K, where there is extensive tunneling.     

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.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

Test of the quantum instanton approximation for thermal rate constants for some collinear reactions

Two variants of the recently developed quantum instanton (QI) model for calculating thermal rate constants of chemical reactions are applied to several collinear atom-diatom reactions with various skew angles. The results show that the original QI version of the model is consistently more accurate than the "simplest" quantum instanton version (both being applied here with one "dividing surface") and thus to be preferred. Also, for these examples (as with other earlier applications) the QI results agree well with the correct quantum rates (to within approximately 20% or better) for all temperatures >200 K, except for situations where dynamical corrections to transition state theory (i.e., "re-crossing" dynamics) are evident. (Since re-crossing effects are substantially reduced in higher dimensionality, this is not a cause for serious concern.) A procedure is also described which facilitates use of the METROPOLIS algorithm for evaluating all quantities that appear in the QI rate expression by Monte Carlo path integral methods.     

Nuclear quantum effects on an enzyme-catalyzed reaction with reaction path potential: proton transfer in triosephosphate isomerase

Nuclear quantum mechanical effects have been examined for the proton transfer reaction catalyzed by triosephosphate isomerase, with the normal mode centroid path integral molecular dynamics based on the potential energy surface from the recently developed reaction path potential method. In the simulation, the primary and secondary hydrogens and the C and O atoms involving bond forming and bond breaking were treated quantum mechanically, while all other atoms were dealt classical mechanically. The quantum mechanical activation free energy and the primary kinetic isotope effects were examined. Because of the quantum mechanical effects in the proton transfer, the activation free energy was reduced by 2.3 kcal/mol in comparison with the classical one, which accelerates the rate of proton transfer by a factor of 47.5. The primary kinetic isotope effects of kH/kD and kH/kT were estimated to be 4.65 and 9.97, respectively, which are in agreement with the experimental value of 4+/-0.3 and 9. The corresponding Swain-Schadd exponent was predicted to be 3.01, less than the semiclassical limit value of 3.34, indicating that the quantum mechanical effects mainly arise from quantum vibrational motion rather than tunneling. The reaction path potential, in conjunction with the normal mode centroid molecular dynamics, is shown to be an efficient computational tool for investigating the quantum effects on enzymatic reactions involving proton transfer.     

Distance dependent coherence variation in DNA charge-transfer processes

We explore distance dependent variation of the coherence length relevant to DNA charge-transfer processes within 5'-GAnG3-3' DNA sequences. Recently developed on-the-fly filtered propagator functional path integral approach was employed to sort out transport trajectories with significant contribution and to analyze correlation between electronic states. In particular, the coherence length was quantitatively determined through characteristics of off-diagonal quantum trajectories. Simulated coherence lengths and experimentally observed rate constants [Nature 2001, 412, 318] were found to be consistent such that, up to n = 2, the exponential decrease of the rate constants is associated with the donor-acceptor coherence driven charge transfer. In contrast, the rate constants become insensitive to the distance for n > or = 3 in which donor and acceptor are no longer significantly correlated. It was also found that the coherence within a collective state governs the overall charge transfer, which is composed of a part of a sequence within the coherence length from the donor.     

Temperature dependence for the rate of hole transfer in DNA: Nonadiabatic regime

The positive charge transfer in DNA is investigated, using the first principle treatment of the electron-vibrational interaction. We show that rearrangements of atoms belonging to base pairs induced by charge transfer are essentially quantum mechanical in nature. Particularly at room temperature, around half of the rearrangements occur via quantum tunneling, while the other half takes place via thermally activated transitions. This effect reduces activation energies for charge transfer between both AT and GC pairs by a factor of two compared to their classical values. These behaviors are described within small polaron theory for the non-adiabatic charge transfer and compared to the experimental data and previous theoretical studies.     

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.     

Combined QM/MM and path integral simulations of kinetic isotope effects in the proton transfer reaction between nitroethane and acetate ion in water

An integrated Feynman path integral-free energy perturbation and umbrella sampling (PI-FEP/UM) method has been used to investigate the kinetic isotope effects (KIEs) in the proton transfer reaction between nitroethane and acetate ion in water. In the present study, both nuclear and electronic quantum effects are explicitly treated for the reacting system. The nuclear quantum effects are represented by bisection sampling centroid path integral simulations, while the potential energy surface is described by a combined quantum mechanical and molecular mechanical (QM/MM) potential. The accuracy essential for computing KIEs is achieved by a FEP technique that transforms the mass of a light isotope into a heavy one, which is equivalent to the perturbation of the coordinates for the path integral quasiparticle in the bisection sampling scheme. The PI-FEP/UM method is applied to the proton abstraction of nitroethane by acetate ion in water through molecular dynamics simulations. The rule of the geometric mean and the Swain-Schaad exponents for various isotopic substitutions at the primary and secondary sites have been examined. The computed total deuterium KIEs are in accord with experiments. It is found that the mixed isotopic Swain-Schaad exponents are very close to the semiclassical limits, suggesting that tunneling effects do not significantly affect this property for the reaction between nitroethane and acetate ion in aqueous solution.     

Mediation of resonance energy transfer by a third molecule

The influence of a third molecule on the rate of resonance energy transfer is studied using diagrammatic perturbation theory within the framework of molecular quantum electrodynamics. Two distinct mechanisms are identified. One corresponds to direct transfer between donor and acceptor while the other involves relay of energy by the third species. Fermi Golden rule transition rates valid for all separation distances beyond wave function overlap are evaluated for these two processes as well as for the interference term between direct and indirect exchange, thereby extending previous work which was limited to the near-zone only. Short- and long-range limits are also obtained in each case. It is found that in the near-zone the indirect rate contribution exhibits inverse sixth power dependence on relative distances of emitter and absorber relative to the third body, in contrast to its far-zone counterpart, which exhibits inverse square behavior. The interference term, however, displays inverse cubic dependence on all three distance vectors at short-range and inverse behavior in the far-zone. Interestingly, for a collinear arrangement of the three molecules in the near-zone, the interference term is negative, reducing the overall rate of energy transfer. The results obtained are interpreted in terms of microscopic and macroscopic pictures of transfer occurring within a surrounding medium.     

Path Integral Molecular Dynamics of Liquid Water in a Mean-Field Particle Reservoir

We present a simulation scheme for path integral simulation of molecular liquids where a small open region is embedded in a large reservoir of non interacting point-particles. The scheme is based on the latest development of the adaptive resolution technique AdResS and allows for the space-dependent change of molecular resolution from a path integral representation with 120 degrees of freedom to a point particle that does not interact with other molecules and vice versa. The method is applied to liquid water and implies a sizable gain regarding the request of computational resources compared to full path integral simulations. Given the role of water as universal solvent with a specific hydrogen bonding network, the path integral treatment of water molecules is important to describe the quantum effects of hydrogen atoms’ delocalization in space on the hydrogen bonding network. The method presented here implies feasible computational efforts compared to full path integral simulations of liquid water which, on large scales, are often prohibitive.     

A quantum generalization of intrinsic reaction coordinate using path integral centroid coordinates

We propose a generalization of the intrinsic reaction coordinate (IRC) for quantum many-body systems described in terms of the mass-weighted ring polymer centroids in the imaginary-time path integral theory. This novel kind of reaction coordinate, which may be called the "centroid IRC," corresponds to the minimum free energy path connecting reactant and product states with a least amount of reversible work applied to the center of masses of the quantum nuclei, i.e., the centroids. We provide a numerical procedure to obtain the centroid IRC based on first principles by combining ab initio path integral simulation with the string method. This approach is applied to NH(3) molecule and N(2)H(5) (-) ion as well as their deuterated isotopomers to study the importance of nuclear quantum effects in the intramolecular and intermolecular proton transfer reactions. We find that, in the intramolecular proton transfer (inversion) of NH(3), the free energy barrier for the centroid variables decreases with an amount of about 20% compared to the classical one at the room temperature. In the intermolecular proton transfer of N(2)H(5) (-), the centroid IRC is largely deviated from the "classical" IRC, and the free energy barrier is reduced by the quantum effects even more drastically.     

Finite temperature application of the corrected propagator method to reactive dynamics in a condensed-phase environment

The recently proposed mixed quantum-classical method is extended to applications at finite temperatures. The method is designed to treat complex systems consisting of a low-dimensional quantum part (the primary system) coupled to a dissipative bath described classically. The method is based on a formalism showing how to systematically correct the approximate zeroth-order evolution rule. The corrections are defined in terms of the total quantum Hamiltonian and are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The evolution of the primary system is governed by the corrected propagator yielding the exact quantum dynamics. The method has been tested on a standard model system describing proton transfer in a condensed-phase environment: a symmetric double-well potential bilinearly coupled to a bath of harmonic oscillators. Flux correlation functions and thermal rate constants have been calculated at two different temperatures for a range of coupling strengths. The results have been compared to the fully quantum simulations of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] with the real path integral method.     

An efficient ring polymer contraction scheme for imaginary time path integral simulations

A quantum simulation of an imaginary time path integral typically requires around n times more computational effort than the corresponding classical simulation, where n is the number of ring polymer beads (or imaginary time slices) used in the calculation. However, this estimate neglects the fact that the potential energies of many systems can be decomposed into a sum of rapidly varying short-range and slowly varying long-range contributions. For such systems, the computational effort of the path integral simulation can be reduced considerably by evaluating the long-range forces on a contracted ring polymer with fewer beads than are needed to evaluate the short-range forces. This idea is developed and then illustrated with an application to a flexible model of liquid water in which the intramolecular forces are evaluated with 32 beads, the oxygen-oxygen Lennard-Jones forces with seven, and the intermolecular electrostatic forces with just five. The resulting static and dynamic properties are within a few percent of those of a full 32-bead calculation, and yet they are obtained with a computational effort less than six times (rather than 32 times) that of a classical simulation. We hope that this development will encourage future studies of quantum mechanical fluctuations in liquid water and aqueous solutions and in many other systems with similar interaction potentials.     

Mechanism of Cations Suppressing Proton Diffusion Kinetics for Electrocatalysis

Proton transfer is crucial for electrocatalysis. Accumulating cations at electrochemical interfaces can alter the proton transfer rate and then tune electrocatalytic performance. However, the mechanism for regulating proton transfer remains ambiguous. Here, we quantify the cation effect on proton diffusion in solution by hydrogen evolution on microelectrodes, revealing the rate can be suppressed by more than 10 times. Different from the prevalent opinions that proton transport is slowed down by modified electric field, we found water structure imposes a more evident effect on kinetics. FTIR test and path integral molecular dynamics simulation indicate that proton prefers to wander within the hydration shell of cations rather than to hop rapidly along water wires. Low connectivity of water networks disrupted by cations corrupts the fast-moving path in bulk water. This study highlights the promising way for regulating proton kinetics via a modified water structure.