Reactive Flux Correlation Function Research Articles

Forward–backward initial value representation for semiclassical time correlation functions

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

Semiclassical theory of electronically nonadiabatic dynamics: Results of a linearized approximation to the initial value representation

A linearized approximation to the semiclassical initial value representation (SC-IVR), referred to herein as the LSC-IVR, was used by us in a recent paper [J. Chem. Phys. 108, 9726 (1998)] to calculate reactive flux correlation functions for a model of a chemical reaction on a single potential energy surface. This paper shows how the LSC-IVR—which is much easier to apply than the full SC-IVR because it linearizes the phase difference between interfering classical trajectories—can be applied to electronically nonadiabatic processes, i.e., those involving transitions between different potential-energy surfaces. Applications to several model problems are presented to show its usefulness: These are the nonadiabatic scattering problems used by Tully to test surface-hopping models, and also the spin–boson model of coupled electronic states in a condensed phase environment. Though not as accurate as the full SC-IVR, the LSC-IVR does a reasonably good job for all these applications, even describing correctly Stuckelberg oscillations (interference between nonadiabatic transitions) and the transition between coherent and incoherent behavior in the spin–boson example.

On the semiclassical description of quantum coherence in thermal rate constants

An earlier paper of ours [J. Chem. Phys. 108, 9726 (1998)] used an approximate (linearized) version of the semiclassical initial value representation (SC-IVR) to calculate reactive flux correlation functions for a model of unimolecular isomerization, namely a 1-d double well potential coupled to an infinite set of harmonic oscillators, obtaining excellent agreement with accurate quantum results for this system. Here we analyze this linearized approximation (LA) further, however, and show that it is not capable of describing quantum interference/coherence effects in the longer time recrossing behavior of the isomerization dynamics. (The recrossing effects seen in our earlier work were due to classical mechanics.) To accentuate quantum effects in the recrossing dynamics, the present article considers the double well potential without the harmonic bath, using both the LA and the full SC-IVR. The results of the calculations show that the flux correlation functions given by the LA agrees well with the exact quantum correlation function for times up to ≈ℏβ, meaning that it describes quantum effects in the direct or transition state theory like dynamics accurately. For the longer time recrossing dynamics, however, it agrees with the classical rather than the quantum correlation function, i.e., it does not describe quantum coherence effects on this time scale. The full SC-IVR calculations, however, are in reasonably good agreement with the quantum correlation function for these longer times.

Transition path sampling and the calculation of rate constants

We have developed a method to study transition pathways for rare events in complex systems. The method can be used to determine rate constants for transitions between stable states by turning the calculation of reactive flux correlation functions into the computation of an isomorphic reversible work. In contrast to previous dynamical approaches, the method relies neither on prior knowledge nor on explicit specification of transition states. Rather, it provides an importance sampling from which transition states can be characterized statistically. A simple model is analyzed to illustrate the methodology.

Solvation effects on a model SN2 reaction in water clusters

We present a series of molecular dynamics experiments for the nucleophilic substitution reaction Cl−+CH3Cl→ClCH3+Cl− taking place in liquid simple point charge water nanoclusters containing 6, 16, and 32 solvent molecules at temperatures close to 200 K. A three-dimensional potential energy for the reagent interatomic interactions is employed. Equilibrium and dynamical aspects of the reactive process are investigated. Solvation effects lead to significant enhancements of the computed free energy barriers even in aggregates containing only six water molecules. The equilibrium spatial and orientational correlations describing the changes in the solvation structure along the reaction path are also presented. The reactive/product states are characterized by a fully solvated Cl− ion embedded within the cluster while the CH3Cl remains on the surface; at the transition state, the complex lies at the cluster surface adopting a linear geometry tangential to the cluster boundary. We have also monitored the time relaxation of the solvation structures as the system evolves from the transition to the stable product states. Our results show the reaction proceeds by a series of highly coordinated motions involving the different components of the reagent. Estimates for the rate constants in clusters of different sizes are also computed using the reactive flux correlation function formalism.

Molecular dynamics of adiabatic and nonadiabatic electron transfer at the metal–water interface

A molecular model for an electron transfer reaction at a solution–electrode interface is developed. The solvent diabatic free energy curves for the reaction Fe+3+e−→Fe+2 are calculated using an umbrella sampling procedure. These are used to calculate the rate of electron transfer as a function of the electrode–solution potential difference in the electronically nonadiabatic region. A model Hamiltonian for the adiabatic case is also developed and used to calculate the adiabatic free energy surface. Reactive flux correlation functions are used to determine the solvent dynamical corrections to the rate. A comparison between the molecular dynamics and the Kramers and Grote–Hynes theories is made.

Centroid-density quantum rate theory: Dynamical treatment of classical recrossing

A new method is presented for the calculation of quantum mechanical rate constants for activated processes. This method is a hybrid approach involving Feynman path integrals and classical dynamics that is an extension of previous work of Messina, Schenter, and Garrett [J. Chem. Phys. 98, 8525 (1993)]. We make an ansatz for the quantum mechanical analog to the classical flux correlation function expression for the rate constant. This expression involves an imaginary-time, phase-space Feynman path integral, with the dividing surface and characteristic function expressed as a function of the phase-space centroid variables. The reactive flux correlation function is obtained from a classical-like expression in which the characteristic function is evaluated by evolving the phase-space centroid variables as if they were classical dynamical variables. We show that the theory gives exact analytic results in the high temperature and harmonic limits. The theory is further tested on a model anharmonic two-dimensional system of an Eckart barrier coupled to a harmonic oscillator. The results of the theory compare favorably to accurate numerical calculations.

Quasi-adiabatic propagator path integral methods. Exact quantum rate constants for condensed phase reactions

We describe an exact quantum-mechanical methodology for calculating Boltzmann-averaged rate constants for a system coupled to a harmonic bath of arbitrary dimensionality. Reactive flux correlation functions are evaluated using the quasi-adiabatic propagator path integral (QUAPI) method. Extremely rapid convergence is achieved: i.e. exact results are obtained with three to five time slices even at temperatures as low as 100 K, and the numerical effort required is essentially independent of the number of bath degrees of freedom. Thus, the proposed method provides an accurate and efficient scheme for studying quantum effects in the low-temperature kinetics of condensed phase reactions.

Structure and hydrogen bond dynamics of water–dimethyl sulfoxide mixtures by computer simulations

We have used two different force field models to study concentrated dimethyl sulfoxide (DMSO)–water solutions by molecular dynamics. The results of these simulations are shown to compare well with recent neutron diffraction experiments using H/D isotope substitution [A. K. Soper and A. Luzar, J. Chem. Phys. 97, 1320 (1992)]. Even for the highly concentrated 1 DMSO : 2 H2O solution, the water hydrogen–hydrogen radial distribution function, gHH(r), exhibits the characteristic tetrahedral ordering of water–water hydrogen bonds. Structural information is further obtained from various partial atom–atom distribution functions, not accessible experimentally. The behavior of water radial distribution functions, gOO(r) and gOH(r) indicate that the nearest neighbor correlations among remaining water molecules in the mixture increase with increasing DMSO concentration. No preferential association of methyl groups on DMSO is detected. The pattern of hydrogen bonding and the distribution of hydrogen bond lifetimes in the simulated mixtures is further investigated. Molecular dynamics results show that DMSO typically forms two hydrogen bonds with water molecules. Hydrogen bonds between DMSO and water molecules are longer lived than water–water hydrogen bonds. The hydrogen bond lifetimes determined by reactive flux correlation function approach are about 5 and 3 ps for water–DMSO and water–water pairs, respectively, in 1 DMSO : 2 H2O mixture. In contrast, for pure water, the hydrogen bond lifetime is about 1 ps. We discuss these times in light of experimentally determined rotational relaxation times. The relative values of the hydrogen bond lifetimes are consistent with a statistical (i.e., transition state theory) interpretation.

Isomerization reaction dynamics and equilibrium at the liquid-vapor interface of water. A molecular-dynamics study.

The gauche-trans isomerization reaction of 1,2-dichloroethane at the liquid-vapor interface of water is studied using molecular-dynamics computer simulations. The solvent bulk and surface effects on the torsional potential of mean force and on barrier recrossing dynamics are computed. The isomerization reaction involves a large change in the electric dipole moment, and as a result the trans/gauche ratio is considerably affected by the transition from the bulk solvent to the surface. Reactive flux correlation function calculations of the reaction rate reveal that deviation from the transition-state theory due to barrier recrossing is greater at the surface than in the bulk water. This suggests that the system exhibits non-Rice-Ramsperger-Kassel-Marcus behavior due to the weak solvent-solute coupling at the water liquid-vapor interface.

Studies on the influence of nonlinearity in classical activated rate processes

Computational studies are presented which examine the accuracy of two approximate theories for activated rate processes in condensed matter classical systems. One theory is based on the generalized Langevin equation and the other on multidimensional transition state theory. The specific focus is on studies of effective Hamiltonians which contain nonlinear coupling terms between the reaction coordinate and bath coordinates. Two of these systems phenomenologically describe the activated dynamics of realistic physical problems. The reactive flux correlation function method is used to calculate the numerically exact rate constant and, in turn, compared to the value of the rate constant calculated from approximate analytic theories. In all cases, the value of the rate constant exhibits a dependence on the nonlinearities in the equations of motion. The results suggest that the generalized Langevin equation model and multidimensional harmonic transition state theory may give different predictions for the activated rate constant in nonlinear systems and that both theories should be applied with some care.

Vacancy migration rates by molecular dynamics with constraints

Rate constants for vacancy migration in a Lennard-Jones crystal are evaluated within the frame of the reactive flux correlation function formalism by molecular dynamics simulation. In the authors' calculation a key role is played by a holonomic constraint acting on the reaction coordinate. The definition of a suitable reaction coordinate is a subtle matter as becomes evident when using their techniques. They show how the ambiguities in the definition of the reaction coordinate can be circumvented. The authors' results are two orders of magnitude more precise than previous 'exact' calculations.

Molecular dynamics study of cyclohexane interconversion

Classical molecular dynamics calculations are reported for one C 6H 12 molecule in a bath of 250 CS 2 molecules at roomtemperature and liquid densities of 1.0, 1.3, 1.4 and 1.5 g/cm 3. The solvent contribution to the free energy of activation for the chair-boat isomerization has been determined to high accuracy. The transmission coefficient and reactive flux correlation functions have also been computed. The results obtained agree with earlier conclusions drawn from RISM integral equation calculations and stochastic molecular dynamics calculations. Namely, the solvent effect on the rate manifests a qualitative breakdown of transition state theory and the RRKM picture of unimolecular kinetics. Analysis of the activated trajectories indicate a significant degree of quasiperiodicity.

Exponential power series expansion for the quantum time evolution operator

The coordinate matrix element of the time evolution operator, exp[−iĤt/ℏ], is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ‘‘time slices’’ in the matrix product exp[(−i/ℏ)ΔtĤ]N, Δt=t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods [Chem. Phys. Lett. 151, 1 (1988)].

The reactive flux correlation function for collinear reactions H + H2, Cl + HCl and F + H2

The thermal rate constant for a chemical reaction is given, in its full quantum-mechanical form and without approximation, as the time integral of the flux-flux autocorrelation function. This provides an efficient way for computing rate constants because the correlation function decays to zero in times much shorter than that required for the full state-to-state reactive scattering process, thus requiring one to determine the quantum dynamics of the system only for short times. Reactive flux correlation functions have been calculated for the three title collinear reactions in order to illustrate the various generic behaviour expected for reactive flux correlation functions. H + H2 is an example of simple ‘direct’ dynamics, Cl + HCl has more complex dynamics because of the heavy + light-heavy mass combination, and F + H2 has effects due to the formation of a short-lived collision complex. The reactive flux correlation functions have also been computed using classical mechanics to show the nature of the classical–quantum comparison for them.

New approach to quantum mechanical transition-state theory

Classical transition-state theory is based on the direct, short-time (actually, zero time) classical dynamics through a dividing surface that separates reactants and products. This paper defines a new quantum mechanical version of transition-state theory by using the reactive flux correlation function of Miller, Schwartz, and Tromp (J. Chem. Phys. 1983, 79, 4889) to identify the short-time quantum dynamics through the dividing surface. It is seen that short time quantum mechanically is not zero time, as it is classically, but rather a time of order Planck's constant..beta... This new quantum version of transition-state theory is also seen to have some of the variational character of the classical case. Finally, it is shown how the reactive flux correlation function of Miller, Schwartz, and Tromp can be generalized by choosing various forms of the projector onto reactants, and this is seen to have important consequences for the accuracy of this new quantum transition-state theory.

Roles of classical dynamics and quantum dynamics on activated processes occurring in liquids

This paper reviews the reactive flux correlation function approach to studying the classical dynamics of activated processes in liquids. The possibilities and consequences of nonadiabatic electronic transitions in affecting this dynamics is also considered. We emphasize the feasibility of quantitative trajectory studies and the fact that these studies have yet to be fully exploited in the development of approximate theories of activated processes.

‘‘Direct’’ calculation of quantum mechanical rate constants via path integral methods: Application to the reaction path Hamiltonian, with numerical test for the H+H2 reaction in 3D

The method recently proposed by Miller, Schwartz, and Tromp for determining Boltzmann rate constants ‘‘directly’’—by the path integral evaluation of a reactive flux correlation function—is developed within the framework of the reaction path Hamiltonian model for a general polyatomic reaction. The expression for the correlation function, the time integral of which is the rate constant, is reduced to a single path integral over only one degree of freedom (the reaction coordinate). Effects of tunneling, ‘‘frictional’’ effects on the reaction coordinate due to coupling to other degrees of freedom, and the effects of recrossing the transition state dividing surface are all correctly accounted for in the approach. Numerical tests of the formulas for the 3D version of the H+H2 reaction (on the Porter–Karplus potential surface) gives excellent agreement with the (known) accurate results for this system.

Stochastic molecular dynamics study of t r a n s–g a u c h e isomerization processes in simple chain molecules

T r a n s–gauche isomerization rates for n-butane, n-pentane, and n-decane are determined from a stochastic molecular dynamics calculation of the reactive fluxes. Analysis of the deviations of these rates from the transition state theory predictions shows that the transmission coefficients differ markedly from unity for all physically realizable conditions. The dependence of the transmission coefficient on the collision frequency or friction constant parameterizing the stochastic element is discussed. A comparison between the stochastic dynamics and a true Newtonian molecular dynamics simulation of n-butane in a simple liquid solvent indicates that the reactive flux correlation function is significantly different than would be predicted by the high friction noninertial theory first discussed long ago by Kramers. For longer chain molecules, the isomerization rates depend upon which particular torsional angle is considered, and upon the configurational state of the neighboring angles. This dependence is analyzed. Finally, calculations are presented which address certain questions concerning the use of holonomic constraints in the dynamics of alkane chains.