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.     

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.     

Ring-Polymer Molecular Dynamics Studies of Thermal Rate Coefficients for Reaction F+H2O→HF+OH

The prototype tetra-atomic reaction F+H₂O→HF+OH plays a significant role in both atmospheric and astronomical chemistry. In this work, thermal rate coefficients of this reaction are determined with the ring polymer molecular dynamics (RPMD) method on a full-dimensional potential energy surface (PES). This PES is the most accurate one for the title reaction, as demonstrated by the correct barrier height and reaction energy, compared to the benchmark calculations by the focal point analysis and the high accuracy extrapolated ab initio thermochemistry methods. The RPMD rate coefficients are in excellent agreement with those calculated by the semiclassical transition state theory and a two-dimensional master equation technique, and some experimental measurements. As has been found in many RPMD applications, quantum effects, including tunneling and zeropoint energy effects, can be efficiently and effectively captured by the RPMD method. In addition, the convergence of the results with respect to the number of beads is rapid, which is also consistent with previous RPMD applications.     

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.     

Rate Coefficients of Roaming Reaction H+MgH Using Ring Polymer Molecular Dynamics

The ring-polymer molecular dynamics (RPMD) was used to calculate the thermal rate coefficients of the multi-channel roaming reaction H+MgH→Mg+H₂. Two reaction channels, tight and roaming, are explicitly considered. This is a pioneering attempt of exerting RPMD method to multi-channel reactions. With the help of a newly developed optimization-interpolation protocol for preparing the initial structures and adaptive protocol for choosing the force constants, we have successfully obtained the thermal rate coefficients. The results are consistent with those from other theoretical methods, such as variational transition state theory and quantum dynamics. Especially, RPMD results exhibit negative temperature dependence, which is similar to the results from variational transition state theory but different from the ones from ground state quantum dynamics calculations.     

Observation of the marcus inverted region of electron transfer reactions at a liquid/liquid interface

Scanning electrochemical microscopy was used to probe the influence of a driving force on the heterogeneous electron transfer (ET) processes at the externally polarized water/1,2-dichloroethane interface. Being a part of the driving force, the Galvani potential difference at the interface, Deltaowphi, can be quantitatively controlled in a wide range, allowing the precise measurements of the rate constants of the ET reactions. Two opposite systems were chosen in this work. One was 5,10,15,20-tetraphenyl 21H,23H-porphyrin zinc (ZnPor, O)/Fe(CN)64- (W), and the other was TCNQ (O)/Fe(CN)63- (W). For both systems studied, the relations between the rate constant and the Deltaowphi were of parabolic shape; that is, the rate constants increased initially with the Deltaowphi until reaching a maximum and then decreased steadily as the Deltaowphi increased further. This is in accordance with the prediction of the Marcus theory. To our knowledge, this is the first report that the Marcus inverted region can be observed electrochemically at an unmodified liquid/liquid interface with only one redox couple at each phase. The effect of the concentrations of the organic supporting electrolyte has also been discussed in detail.     

Non-Condon效应对电子转移速率影响的理论模拟及其在有机半导体中的应用

随着Condon近似下各种电子转移理论的不断发展与完善和人们对non-Condon效应在电子转移过程中重要作用认识的逐步深入,已建立了几个理论模型来研究这种效应对电子转移速率的影响.本文主要总结了近两年来我们在non-Condon效应电子转移理论方面的工作,首先阐述了指数型、高斯型以及直线型non-Condon电子转移速率的全量子表达式,然后运用该理论模型以及分子动力学模拟计算了二噻吩四硫富瓦烯（DT-TTF）有机半导体的迁移率.此外,还进一步利用数值模拟详细研究了这三种线型的non-Condon效应在量子尺度上对电子转移速率的影响.     

Comparison of approximate quantum simulation methods applied to normal liquid helium at 4 K

The Feynman-Kleinert linearized path integral molecular dynamics (FK-LPI), ring polymer molecular dynamics (RPMD), and centroid molecular dynamics (CMD) methods are applied to the simulation of normal liquid helium. Comparisons of the simulation results at the T = 4 K and rho = 0.01873 A-3 state point are presented. The calculated quantum correlation functions for the three methods show significant differences, both in the short time and in the intermediate regions of the spectrum. Our simulation results are also compared to the recent results of other approximate quantum simulation methods. We find that FK-LPI qualitatively agrees with other approximate quantum simulation results while CMD and RPMD predict a qualitatively different impulsive rebound in the velocity autocorrelation function. Frequency space analysis reveals that RPMD exhibits a broad high-frequency tail similar to that from quantum mode coupling theory and numerical analytic continuation approaches, while FK-LPI provides a somewhat more rapid decay at high frequency than any of these three methods. CMD manifests a high-frequency component that is greatly reduced compared with the other methods.     

Investigation of Electrochemical Properties of Metalloporphyrin Species at the Liquid/Liquid Interface by Switching Substitutes on the Porphyrin Ring

In order to investigate the electrochemical properties of porphyrin complexes species in biological systems, metalloporphyrin with different substitutes was applied to observe the process of heterogeneous electron transfer (ET) at the interface between two immiscible electrolyte solutions (ITIES) by scanning electrochemical microscopy (SECM). Experimental results demonstrated that the process of electron transfer was affected dramatically by the presence of different substitutes. Our results also show that the rate constant follows Bulter? Volmer kinetics where the rate increases with increasing force at the low driving force, and Marcus inverted region kinetics at the high driving force where the rate decreases.     

Semiclassical treatment of thermally activated electron transfer in the inverted region under the fast dielectric relaxation

The previously formulated semiclassical theory (Zhao, Liang, and Nakamura, J. Phys. Chem. A 2006, 110, 8204) is used to study electron transfer in the Marcus inverted case by considering multidimensional potential energy surfaces of donor and acceptor. The Zhu-Nakamura formulas of nonadiabatic transition in the case of Landau-Zener type are incorporated into the approach. The theory properly takes into account the nonadiabatic transition coupled with the nuclear tunneling and can cover the whole range from weak to strong coupling regime uniformly under the assumption of fast solvent relaxation. The numerical calculations are performed for the 12-dimensional model of shifted harmonic oscillators and demonstrate that the reaction rate with respect to the electronic coupling shows a maximum, confirming the adiabatic suppression in the strong coupling limit. The adiabatic suppression is dramatically reduced by the effect of nuclear tunneling compared to the case that the Landau-Zener formula is used. The possible extension and applications to the case of the slow solvent dynamics are discussed.     

Dynamics of nonadiabatic chemical reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.     

Semiclassical quantum unimolecular reaction rate theory revisited

We describe a semiclassical quantum unimolecular reaction rate theory derived from the corresponding classical theory developed by Davis, Gray, Rice and Zhao (DGRZ). The analysis retains the intuitively useful mechanistic distinctions between intramolecular energy transfer and reaction, with the consequence that the semiclassical quantum theory version neglects some interference effects in the reaction dynamics. In the limiting case that intramolecular energy transfer is very fast compared to the rate of reaction we show that the DGRZ representation of the rate constant can be transformed, using the Weyl correspondence between quantum operators and classical variables, to the quantum flux–flux correlation function representation of the rate constant. In the more general case that the rate of intramolecular energy transfer influences the reaction dynamics, the semiclassical representation of the Wigner function for a classical system with both quasiperiodic and chaotic motion is used to obtain the reaction rate constant. Our analysis identifies the quantum analogue of the classical bottleneck to intramolecular energy transfer with the scars of unstable periodic orbits; it leads to a flux–flux correlation function representation of the rate constant for intramolecular energy transfer.     

Modulating charge separation and charge recombination dynamics in porphyrin-fullerene linked dyads and triads: Marcus-normal versus inverted region.

Photoinduced charge separation (CS) and charge recombination (CR) processes have been examined in various porphyrin-fullerene linked systems (i.e., dyads and triads) by means of time-resolved transient absorption spectroscopy and fluorescence lifetime measurements. The investigated compounds comprise a homologous series of rigidly linked, linear donor-acceptor arrays with different donor-acceptor separations and diversified donor strength: freebase porphyrin-C60 dyad (H2P-C60), zincporphyrin-C60 dyad (ZnP-C60), ferrocene-zincporphyrin-C60 triad (Fc-ZnP-C60), ferrocene-freebase porphyrin-C60 triad (Fc-H2P-C60), and zincporphyrin-freebase porphyrin-C60 triad (ZnP-H2P-C60). Most importantly, the lowest lying charge-separated state of all the investigated systems, namely, that of ferrocenium ion (Fc+) and the C60 radical anion (C60.-) pair in the Fc-ZnP-C60 triad, has been generated with the highest quantum yields (close to unity) and reveals a lifetime as long as 16 micros. Determination of CS and CR rate constants, together with the one-electron redox potentials of the donor and acceptor moieties in different solvents, has allowed us to examine the driving force dependence (-DeltaG0ET) of the electron-transfer rate constants (kET). Hereby, the semilogarithmic plots (i.e., log kET versus -DeltaG0ET) lead to the evaluation of the reorganization energy (lambda) and the electronic coupling matrix element (V) in light of the Marcus theory of electron-transfer reactions: lambda = 0.66 eV and V = 3.9 cm(-1) for ZnP-C60 dyad and lambda = 1.09 eV and V = 0.019 cm(-1) for Fc-ZnP-C60, Fc-H2P-C60, and ZnP-H2P-C60 triads. Interestingly, the Marcus plot in Fc-ZnP-C60, Fc-H2P-C60, and ZnP-H2P-C60 has provided clear evidence for intramolecular CR located in both the normal and inverted regions of the Marcus parabola. The coefficient for the distance dependence of V (damping factor: betaCR = 0.58 A(-1) is deduced which depends primarily on the nature of the bridging molecule.     

Rate Coefficients and Kinetic Isotope Effects of Cl+XCl→XCl+Cl (X=H,D, Mu) Reactions from Ring Polymer Molecular Dynamics

The ring-polymer molecular dynamics (RPMD) was used to calculate the thermal rate coefficients and kinetic isotope effects of the heavy-light-heavy abstract reaction Cl+XCl\begin{document}$ \rightarrow $\end{document}XCl+Cl (X = H, D, Mu). For the Cl+HCl reaction, the excellent agreement between the RPMD and experimental values provides a strong proof for the accuracy of the RPMD theory. And the RPMD results are also consistent with results from other theoretical methods including improved-canonical-variational-theory and quantum dynamics. The most novel finding is that there is a double peak in Cl+MuCl reaction near the transition state, leaving a free energy well. It comes from the mode softening of the reaction system at the peak of the potential energy surface. Such an explicit free energy well suggests strongly there is an observable resonance. And for the Cl+DCl reaction, the RPMD rate coefficient again gives very accurate results compared with experimental values. The only exception is at the temperature of 312.5 K, results from RPMD and all other theoretical methods are close to each other but slightly lower than the experimental value, which indicates experimental or potential energy surface deficiency.     

Ab initio molecular dynamics simulation of the aqueous Ru2+/Ru3+ redox reaction: the Marcus perspective

A well-behaved (low spin) transition metal aqua ion, Ru(aq)(2+), is used as a model system in an ab initio molecular dynamics study of a redox half reaction to which the Marcus theory of electron transfer is assumed to apply. Using constraint methods, we show that aqueous Ru(2+) can be reversibly transformed to Ru(3+) under the control of the classical solvent electrostatic potential as order parameter. The mean force is found to vary linearly with the order parameter in accordance with the Marcus theory. As can be expected for a half reaction, the slope in the oxidized and reduced states are asymmetric differing by approximately a factor of two. As a further test, we verify that the corresponding quadratic potential of mean force is in excellent agreement with the free energy profile obtained from the Gaussian distribution of potential fluctuations sampled from free (unconstrained) runs of the reduced and oxidized system. Similar to experimental electrochemical methods, our simulation scheme enables us to manipulate the effective thermodynamic driving force and align the free energy minima of product and reactant state. The activation energy and reaction entropy computed under these conditions are discussed and analyzed from the Marcus perspective.     

Semiclassical treatments of electron transfer rate from weak to strong electronic coupling regime

Electron transfer (ET) rate is a fundamental parameter to characterize ET processes in physical, chemical, material and biologic sciences. It is affected by a number of quantum phenomena, such as nuclear tunneling, curve crossing, quantum interference, and the coupling to the environment. It is thus a challenge to accurately evaluate the ET rate since one has to incorporate both quantum effects and dissipation. In this review article, we present several semiclassical theories proposed in our group to cover the regime from weak to strong electronic coupling. Their applications to some concrete systems are also shown.     

Quantum-instanton evaluation of the kinetic isotope effects

A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis-Monte Carlo evaluation of the Boltzmann operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction H + H2 --> H2 + H. In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than approximately 10%.     

Measurement of the free-energy dependence of interfacial charge-transfer rate constants using ZnO/H2O semiconductor/liquid contacts

The dependence of electron-transfer rate constants on the driving force for interfacial charge transfer has been investigated using n-type ZnO electrodes in aqueous solutions. Differential capacitance versus potential and current density versus potential measurements were used to determine the energetics and kinetics, respectively, of the interfacial electron-transfer processes. A series of nonadsorbing, one-electron, outer-sphere redox couples with formal reduction potentials that spanned approximately 900 mV allowed evaluation of both the normal and Marcus inverted regions of interfacial electron-transfer processes. All rate processes were observed to be kinetically first-order in the concentration of surface electrons and first-order in the concentration of dissolved redox acceptors. The band-edge positions of the ZnO were essentially independent of the Nernstian potential of the solution over the range 0.106-1.001 V vs SCE. The rate constant at optimal exoergicity was observed to be approximately 10(-)(16) cm(4) s(-)(1). The rate constant versus driving force dependence at n-type ZnO electrodes exhibited both normal and inverted regions, and the data were well-fit by a parabola generated using classical Marcus theory with a reorganization energy of 0.67 eV. NMR line broadening measurements of the self-exchange rate constants indicated that the redox couples had reorganization energies of 0.64-0.69 eV. The agreement between the reorganization energy of the ions in solution and the reorganization energy for the interfacial electron-transfer processes indicated that the reorganization energy was dominated by the redox species in the electrolyte, as expected from an application of Marcus theory to semiconductor electrodes.