Effective dynamics along given reaction coordinates, and reaction rate theory

2016 ◽  
Vol 195 ◽  
pp. 365-394 ◽  
Author(s):  
Wei Zhang ◽  
Carsten Hartmann ◽  
Christof Schütte
Keyword(s):  
Reaction Rates ◽  
Coordinate Space ◽  
Rate Theory ◽  
Ergodic Diffusion Process ◽  
Reaction Coordinates ◽  
Effective Dynamics ◽  
Special Cases ◽  
Heterogeneous Multiscale ◽  
Algorithmic Techniques ◽  
Computational Resources

In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: we first show that if we start with an ergodic diffusion process whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Mori–Zwanzig, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the “equation-free” approach and the “heterogeneous multiscale method” can be seen as special cases of our approach.

scholarly journals Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Andreas Bittracher ◽  
Stefan Klus ◽  
Boumediene Hamzi ◽  
Péter Koltai ◽  
Christof Schütte
Keyword(s):  
Stochastic Systems ◽  
Reproducing Kernel ◽  
Learning Algorithm ◽  
Reproducing Kernel Hilbert Space ◽  
Mathematical Framework ◽  
Reaction Coordinates ◽  
Effective Dynamics ◽  
Distortion Bounds ◽  
Low Dimensional ◽  
Metastable Systems

AbstractWe present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches.

Temperature, Dynamics, and Enzyme-Catalyzed Reaction Rates

2020 ◽  
Vol 49 (1) ◽  
pp. 163-180 ◽  
Author(s):  
Vickery L. Arcus ◽  
Adrian J. Mulholland
Keyword(s):  
Molecular Mechanisms ◽  
Activation Parameters ◽  
Reaction Rates ◽  
Rate Theory ◽  
Different Temperatures ◽  
State Ensemble ◽  
Catalyzed Reactions ◽  
Enzyme Catalyzed Reactions ◽  
Transition State Ensemble ◽  
Rate Limiting

We review the adaptations of enzyme activity to different temperatures. Psychrophilic (cold-adapted) enzymes show significantly different activation parameters (lower activation enthalpies and entropies) from their mesophilic counterparts. Furthermore, there is increasing evidence that the temperature dependence of many enzyme-catalyzed reactions is more complex than is widely believed. Many enzymes show curvature in plots of activity versus temperature that is not accounted for by denaturation or unfolding. This is explained by macromolecular rate theory: A negative activation heat capacity for the rate-limiting chemical step leads directly to predictions of temperature optima; both entropy and enthalpy are temperature dependent. Fluctuations in the transition state ensemble are reduced compared to the ground state. We show how investigations combining experiment with molecular simulation are revealing fundamental details of enzyme thermoadaptation that are relevant for understanding aspects of enzyme evolution. Simulations can calculate relevant thermodynamic properties (such as activation enthalpies, entropies, and heat capacities) and reveal the molecular mechanisms underlying experimentally observed behavior.

scholarly journals Measuring reaction rates at equilibrium with the isotope doping method

2019 ◽  
Vol 98 ◽  
pp. 13003
Author(s):  
Chen Zhu ◽  
Yilun Zhang ◽  
J Donald Rimstidt ◽  
Honglin Yuan
Keyword(s):  
Chemical Engineering ◽  
Experimental Testing ◽  
Detailed Balance ◽  
Reaction Rates ◽  
Irreversible Thermodynamics ◽  
Experimental Conditions ◽  
Special Cases ◽  
Wide Range ◽  
Long Time ◽  
Dissolution And Precipitation

Since the time of J. H. van’t Hoff [1], it has been known that chemical equilibrium is dynamic, meaning that at equilibrium, chemical reactions do not cease, but instead the forward and backward reaction rates are equal. The constant concentrations at equilibrium preclude the use of concentrations to measure reaction rates at equilibrium. Therefore, with the exception of a few special cases, no reaction rates at equilibrium have been published in the literature of chemistry, physics, or chemical engineering. Here we report dissolution and precipitation rates at equilibrium for quartz and barite with the isotope-doping method. Experimental data show that dissolution and precipitation rates are equal at equilibrium, indicating the principle of detailed balance (PDB) appear to be applicable at these experimental conditions. The PDB has been a cornerstone for irreversible thermodynamics and chemical kinetics for a long time, and its wide application in geochemistry has mostly been implicit and without experimental testing of its applicability. Nevertheless, many extrapolations based on PDB without experimental validation have far reaching impacts on society’s mega environmental enterprises. The isotope doping method appears to able to test its applicability for a variety of minerals at a wide range of conditions.

scholarly journals Quantum dynamics of wave packets in a non-stationary parabolic potential and the Kramers escape rate theory

2016 ◽  
Vol 01 (02) ◽  
pp. 1650010 ◽  
Author(s):  
Vladimir I. Dubinko ◽  
Alexander S. Mazmanishvili ◽  
Denis V. Laptev ◽  
Juan F. R. Archilla
Keyword(s):  
Quantum Dynamics ◽  
Thermal Fluctuations ◽  
Reaction Rates ◽  
Strong Increase ◽  
Rate Theory ◽  
Driving Frequency ◽  
Rate Of Escape ◽  
System Temperature ◽  
Analytical Expressions ◽  
Time Periodic

At sufficiently low temperatures, the reaction rates in solids are controlled by quantum rather than by thermal fluctuations. We solve the Schrödinger equation for a Gaussian wave packet in a non-stationary harmonic oscillator and derive simple analytical expressions for the increase of its mean energy with time induced by the time-periodic modulation. Applying these expressions to the modified Kramers theory, we demonstrate a strong increase of the rate of escape out of a potential well under the time-periodic driving, when the driving frequency of the well position equals its eigenfrequency, or when the driving frequency of the well width exceeds its eigenfrequency by a factor of [Formula: see text]. Such regimes can be realized near localized anharmonic vibrations (LAVs), in which the amplitude of atomic oscillations greatly exceeds that of harmonic oscillations (phonons) that determine the system temperature. LAVs can be excited either thermally or by external triggering, which can result in strong catalytic effects due to amplification of the Kramers rate.

Free energies

2021 ◽  
pp. 139-178
Author(s):  
James P. Sethna
Keyword(s):  
Molecular Motors ◽  
Degrees Of Freedom ◽  
Reaction Rates ◽  
Coarse Grained ◽  
Rate Theory ◽  
Free Energies ◽  
Reaction Rate Theory ◽  
The World ◽  
The Cost ◽  
Grand Canonical

Free energies ignore most of a system, to provide the emergent statistical ensemble describing things we care about. Free energies can ignore the external world. The cost of borrowing energy from the world is measured by the temperature, giving us the canonical ensemble and Helmholtz free energy. Similarly, borrowing particles and volume from the world gives us the grand canonical and Gibbs ensembles. Free energies can ignore unimportant internal degrees of freedom. These lead to friction and noise, and theories of chemical reactions and reaction rates. Free energies can be coarse-grained, removing short distances and times. Exercises apply free energies to molecular motors, thermodynamic relations, reaction rate theory, Zipf’s law for word frequencies, zombie outbreaks, and nucleosynthesis.

Classical variational rate theory portraits of the dynamical stereochemistry of the F + H2 —> FH + H reaction

1999 ◽  
Vol 77 (5-6) ◽  
pp. 695-708 ◽  
Author(s):  
Irina Rutenburg ◽  
Gerald W Koeppl
Keyword(s):  
Power Series ◽  
Cross Section ◽  
Reaction Cross Section ◽  
Orientation Angle ◽  
Reaction Rates ◽  
Reaction Cross ◽  
Rate Theory ◽  
Variational Theory ◽  
Dividing Surface ◽  
Energy Dependent

A general classical variational theory of reaction rates is applied to the F + H2 →> FH + H reaction. The variational theory gives the rate as the equilibrium flux of phase points through a trial surface which divides reactants from products and is varied to obtain a least upper bound for the rate. This dividing surface (DS) is defined by a power-series expansion of the H-H internuclear separation (r) in internal coordinates R and θ, i.e., r = F(R, θ) where R is the approach coordinate and θ is the orientation angle. The downhill simplex algorithm is used to search the space of 6 and 10 variational parameters of second- and third-order expansions of the DS and obtain minimum values for the canonical rate constant or, in the microcanonical formulation of the theory, the energy-dependent mean reaction cross section. The presence of angle-dependent terms in the DS makes it possible to describe the dynamical stereochemistry of atom-diatom reactions in a new and useful manner. Portraits of the dynamical stereochemistry are obtained by plotting contours of the density of reaction systems on the DS; such plots are reactivity relief maps of the DS. Reactivity relief maps show how the field of reactivity which surrounds the diatomic reactant molecule expands with increasing temperature and energy. Results are presented here for a new power series formulation of the DS which obeys a condition: δF(R, θ)/δθ = 0 at θ = π/2 which is appropriate for reaction of a homonuclear diatomic molecule. The relationship between reactivity relief maps obtained using quadratic and cubic formulations of the new DS and the locations of angle-dependent energy barriers for reaction is described. Variational and classical mechanical trajectory results are used to show how energy-dependent factors, which correct the variational mean reaction cross section for trajectories which cross and recross the DS, depend on the orientation angle. Key words: variational, transition, rate, dividing, surface.

Unified Rate Theory of Electrochemistry and Electrocatalysis: Fixed Potential Formulation for General, Electron Transfer, and Proton-Coupled Electron Transfer Reactions

2019 ◽  
Author(s):  
Marko Melander
Keyword(s):  
Electron Transfer ◽  
Electrode Potential ◽  
Canonical Ensemble ◽  
Reaction Rates ◽  
Transfer Reactions ◽  
Rate Theory ◽  
Electron Transfer Reactions ◽  
Proton Coupled Electron Transfer ◽  
Fixed Electrode ◽  
Grand Canonical

<div>Atomistic modeling of electrocatalytic reactions is most naturally conducted within the grand canonical ensemble (GCE) which enables fixed chemical potential calculations. While GCE has been widely adopted for modeling electrochemical and electrocatalytic thermodynamics, the electrochemical reaction rate theory within GCE is lacking. Molecular and condensed phase rate theories are formulated within microcanonical and canonical ensembles, respectively, but electrocatalytic systems described within the GCE require extension of the conventionally used rate theories for computation reaction rates at fixed electrode potentials. In this work, rate theories from (micro)canonical ensemble are generalized to the GCE providing the theoretical basis for the computation reaction rates in electrochemical and electrocatalytic systems. It is shown that all canonical rate theories can be extended to the GCE. From the generalized grand canonical rate theory developed herein, fixed electrode potential rate equations are derived for i) general reactions within the GCE transition state theory (GCE-TST), ii) adiabatic curve-crossing rate theory within the empirical valence bond theory (GCE-EVB), and iii) (non-)adiabatic electron and proton-coupled electron transfer reactions. The rate expressions can be readily combined with ab initio methods to study reaction kinetics reactions at complex electrochemical interfaces as a function of the electrode potential. The theoretical work herein provides a single, unified approach for electrochemical and electrocatalytic kinetics and the inclusion of non-adiabatic and tunneling effects in electrochemical environments widening the scope of reactions amenable to computational studies.</div>

Rate Theory for Electrocatalytic Systems: Fixed Potential Formulation for General, Electron Transfer, and Proton-Coupled Electron Transfer Reactions

2019 ◽  
Author(s):  
Marko Melander
Keyword(s):  
Electron Transfer ◽  
Electrode Potential ◽  
Canonical Ensemble ◽  
Reaction Rates ◽  
Transfer Reactions ◽  
Rate Theory ◽  
Electron Transfer Reactions ◽  
Proton Coupled Electron Transfer ◽  
Fixed Electrode ◽  
Grand Canonical

<div>Atomistic modeling of electrocatalytic reactions is most naturally conducted within the grand canonical ensemble (GCE) which enables fixed chemical potential calculations. While GCE has been widely adopted for modeling electrochemical and electrocatalytic thermodynamics, the electrochemical reaction rate theory within GCE is lacking. Molecular and condensed phase rate theories are formulated within microcanonical and canonical ensembles, respectively, but electrocatalytic systems described within the GCE require extension of the conventionally used rate theories for computation reaction rates at fixed electrode potentials. In this work, rate theories from (micro) canonical ensemble are generalized to the GCE providing the theoretical basis for the computation reaction rates in electrochemical systems. It is shown that all canonical rate theories can be extended to the GCE. From the generalized grand canonical rate theory developed herein, fixed electrode potential rate equations are derived for i) general reactions within the GCE transition state theory (GCE-TST), ii) adiabatic curve-crossing rate theory within the empirical valence bond theory (GCE-EVB), and iii) (non-) adiabatic electron and proton-coupled electron transfer reactions. The rate expressions can be readily combined with ab initio methods to study reaction kinetics reactions at complex electrochemical interfaces as a function of the electrode potential. The theoretical work herein provides the basis for treating electrochemical kinetics and the inclusion of non-adiabatic and tunneling effects in electrochemical environments widening the scope of reactions amenable to computational studies.</div>

scholarly journals Bridging the gap between ab initio simulations and experiments through EIS

2018 ◽  
Author(s):  
Mohit Mehta ◽  
Vamsci Venkat Bevara ◽  
Petru Andrei
Keyword(s):  
Analytical Solution ◽  
Ab Initio ◽  
Electrochemical Impedance ◽  
Energy Levels ◽  
Reorganization Energy ◽  
Reaction Rates ◽  
Electron Transfer Reaction ◽  
Rate Theory ◽  
Impedance Model ◽  
Ab Initio Simulations

We have developed an analytical solution to compute the impedance spectra using ab initio parameters such as total Gibbs free energy change for the electron transfer reaction, reorganization energy, activation energy, and occupied energy levels in the electrode. The impedance model is developed to relate DFT computed parameters to experiments using Electrochemical Impedance Spectroscopy (EIS) using the analytical solution of the Marcus-Hush-Chidsey (MHC) reaction rate theory developed by Zeng et al. (Y. Zeng, R.B. Smith, P. Bai, M.Z. Bazant, “Simple formula for Marcus–Hush–Chidsey kinetics”, J. Electroanal. Chem. 735 (2014) 77–83.). Next, we compare the reaction rates computed by Butler-Volmer and MHC theory and impedance responses using the two reaction models. Finally, we provide a few analytical predictions of our analytical model using the MHC theory.

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