Rate Constant Matrix Research Articles

Differentiating the Yield of Chemical Reactions Using Parameters in First-Order Kinetic Equations to Identify Elementary Steps That Control the Reactivity from Complicated Reaction Path Networks.

The yield of a chemical reaction is obtained by solving its rate equation. This study introduces an approach for differentiating yields by utilizing the parameters of the rate equation, which is expressed as a first-order linear differential equation. The yield derivative for a specific pair of reactants and products is derived by mathematically expressing the rate constant matrix contraction method, which is a simple kinetic analysis method. The parameters of the rate equation are the Gibbs energies of the intermediates and transition states in the reaction path network used to formulate the rate equation. Thus, our approach for differentiating the yield allows a numerical evaluation of the contribution of energy variation to the yield for each intermediate and transition state in the reaction path network. In other words, a comparison of these values automatically extracts the factors affecting the yield from a complicated reaction path network consisting of numerous reaction paths and intermediates. This study verifies the behavior of the proposed approach through numerical experiments on the reaction path networks of a model system and the Rh-catalyzed hydroformylation reaction. Moreover, the possibility of using this approach for designing ligands in organometallic catalysts is discussed.

Toward Ab Initio Reaction Discovery Using the Artificial Force Induced Reaction Method.

Predicting the whole process of a chemical reaction while solving kinetic equations presents an opportunity to realize an on-the-fly kinetic simulation that directly discovers chemical reactions with their product yields. Such a simulation avoids the combinatorial explosion of reaction patterns to be examined by narrowing the search space based on the kinetic analysis of the reaction path network, and would open a new paradigm beyond the conventional two-step approach, which requires a reaction path network prior to performing a kinetic simulation. The authors addressed this issue and developed a practical method by combining the artificial force induced reaction method with the rate constant matrix contraction method. Two algorithms are available for this purpose: a forward mode with reactants as the input and a backward mode with products as the input. This article first numerically verifies these modes for known reactions and then demonstrates their application to the actual reaction discovery. Finally, the challenges of this method and the prospects for ab initio reaction discovery are discussed. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 74 is April 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

A reaction route network for methanol decomposition on a Pt(111) surface.

A reaction route network for the decomposition reaction of methanol on a Pt(111) surface was constructed by using the artificial force-induced reaction (AFIR) method, which can search for reaction paths automatically and systematically. Then, the network was kinetically analyzed by applying the rate constant matrix contraction (RCMC) method. Specifically, the time hierarchy of the network, the time evolution of the population initially given to CH3 OH to the other species on the network, and the most favorable route from CH3 OH to major and minor products were investigated by the RCMC method. Consistently to previous studies, the major product on the network was CO+4H, and the most favorable route proceeded through the following steps: CH3 OH → CH2 OH+H → HCOH+2H → HCO+3H → CO+4H. Furthermore, paths to byproducts found on the network and their kinetic importance were discussed. The present procedure combining AFIR and RCMC was thus successful in explaining the title reaction without using any information on its product or the reaction mechanism.

Rate Constant Matrix Contraction Method for Systematic Analysis of Reaction Path Networks

This paper reviews the rate constant matrix contraction (RCMC) method, a kinetics-based graph clustering approach developed for the network of paths of chemical reaction elementary steps. First, fu...

Understanding the Acetalization Reaction Based on its Reaction Path Network

AbstractWe have theoretically investigated the mechanism of the acetalization reaction of acetaldehyde using the artificial force induced reaction (AFIR) method and the rate constant matrix contraction (RCMC) method. A systematic search for reaction paths by the AFIR method resulted in a reaction path network including 162 local minima and 599 transition states. By applying the RCMC method to the network, the time hierarchy of the network was revealed. Using the network and the RCMC method, the overall rate constants and the equilibrium constant including conformational entropy contributions were evaluated. The discussion demonstrates how multicomponent reactions that proceed through multiple steps are understood from their reaction path networks.

A Reaction Path Network for Wöhler's Urea Synthesis

We theoretically studied reaction paths of Wohler's urea synthesis using the artificial force induced reaction method and the rate constant matrix contraction method. The systematic search resulted...

Understanding CO oxidation on the Pt(111) surface based on a reaction route network.

Analysis of a reaction on a solid surface is an important task for understanding the catalytic reaction mechanism. In this study, we studied CO oxidation on the Pt(111) surface by using the artificial force induced reaction (AFIR) method. A systematic reaction path search was done, and the reaction route network was created. This network included not only bond rearrangement paths but also migration paths of adsorbed species. Then, the obtained network was analyzed using a kinetics method called rate constant matrix contraction (RCMC). It is found that the bottleneck of the overall reaction is the CO2 generation step from an adsorbed CO molecule and an O atom. This result is consistent with the Langmuir-Hinshelwood (LH) mechanism with O2 dissociation discussed in previous studies. The present procedure, i.e., construction of the reaction route network by the AFIR method followed by application of the RCMC kinetics method to the resultant reaction route network, was fully systematic and uncovered two aspects: the impact of the existence of multiple paths in each bond rearrangement step and an entropic contribution arising from short-range migration of adsorbed species.

Designing the Backbone of Hexasilabenzene Derivatives with a High Unimolecular Kinetic Stability.

It is an important subject to theoretically predict the kinetic stability of transient species. In this study, we have studied the kinetic stability of hexasilabenzene Si6 H6 and its derivatives, that is, decasilanaphthalene Si10 H8 and Li-substituted hexasilabenzene Si6 Li6 , theoretically by the artificial force induced reaction (AFIR) method combined with the rate constant matrix contraction (RCMC) method. Molecular design was further conducted to extend the unimolecular lifetime of hexasilabenzene derivatives. Although both Si10 H8 and Si6 Li6 were shown to possess shorter lifetimes than Si6 H6 , we found that the lifetimes of Si6 Li6 changed depending on arrangements of Li atoms around the monocyclic Si6 backbone. Based on this knowledge, we found that a compound of an atomic composition Si6 H4 Li2 with a planar, monocyclic Si6 backbone has a relatively long unimolecular lifetime. Moreover, substitution of the two Li atoms by Na atoms further increased the lifetime.

Full rate constant matrix contraction method for obtaining branching ratio of unimolecular decomposition.

The branching ratio of unimolecular decomposition can be evaluated by solving the rate equations. Recent advances in automated reaction path search methods have enabled efficient construction of the rate equations based on quantum chemical calculations. However, it is still difficult to solve the rate equations composed of hundreds or more elementary steps. This problem is especially serious when elementary steps that occur in highly different timescales coexist. In this article, we introduce an efficient approach to obtain the branching ratio from a given set of rate equations. It has been derived from a recently proposed rate constant matrix contraction (RCMC) method, and termed full-RCMC (f-RCMC). The f-RCMC gives the branching ratio without solving the rate equations. Its performance was tested numerically for unimolecular decomposition of C3 H5 and C4 H5 . Branching ratios obtained by the f-RCMC precisely reproduced the values obtained by numerically solving the rate equations. It took about 95 h to solve the rate equations of C4 H5 consisting of 234 elementary steps. In contrast, the f-RCMC gave the branching ratio in less than 1 s. The f-RCMC would thus be an efficient alternative of the conventional kinetic simulation approach. © 2016 Wiley Periodicals, Inc.

On solving the master equation in spatially periodic systems

We present a new method for solving the master equation for a system evolving on a spatially periodic network of states. The network contains 2(ν) images of a "unit cell" of n states, arranged along one direction with periodic boundary conditions at the ends. We analyze the structure of the symmetrized (2(ν)n) × (2(ν)n) rate constant matrix for this system and derive a recursive scheme for determining its eigenvalues and eigenvectors, and therefore analytically expressing the time-dependent probabilities of all states in the network, based on diagonalizations of n × n matrices formed by consideration of a single unit cell. We apply our new method to the problem of low-temperature, low-occupancy diffusion of xenon in the zeolite silicalite-1 using the states, interstate transitions, and transition state theory-based rate constants previously derived by June et al. [J. Phys. Chem. 95, 8866 (1991)]. The new method yields a diffusion tensor for this system which differs by less than 3% from the values derived previously via kinetic Monte Carlo (KMC) simulations and confirmed by new KMC simulations conducted in the present work. The computational requirements of the new method are compared against those of KMC, numerical solution of the master equation by the Euler method, and direct molecular dynamics. In the problem of diffusion of xenon in silicalite-1, the new method is shown to be faster than these alternative methods by factors of about 3.177 × 10(4), 4.237 × 10(3), and 1.75 × 10(7), respectively. The computational savings and ease of setting up calculations afforded by the new method of master equation solution by recursive reduction of dimensionality in diagonalizing the rate constant matrix make it attractive as a means of predicting long-time dynamical phenomena in spatially periodic systems from atomic-level information.

Effects of Nucleotide Analogs at the P2X3 Receptor and Its Mutants Identify the Agonist Binding Pouch

In this study, we investigated the effects of single alanine substitutions of amino acid residues in the supposed ATP binding site of the human P2X3 receptor on the agonistic effect of nucleotide analogs. The wild-type and mutant receptors were expressed in HEK293 cells, and the nucleotide effects were measured by means of the whole-cell patch-clamp method. Modifications in the receptor binding site changed the concentration-response relationship, the current kinetics, and the recovery from desensitization during fast, pulsed, local agonist applications. On the basis of this fact, we were able to distinguish binding from other effects, such as gating/desensitization, by using a hidden Markov model that describes the complete channel behavior with a matrix of rate constants. The binding energies of the nucleotide analogs were calculated and compared with the binding energies of ATP at both the wild-type receptor and its alanine mutants. Changes in the binding energies caused by alterations in the receptor and/or agonist structures were concluded to be attributable to the preferential binding of certain structural constituents of ATP to certain amino acid moieties of the receptor. The results were also checked for consistency with a P2X3 homology model that we developed from the known zebrafish P2X4 crystal structure in the closed state. The functional data correlated well with the predictions of the agonist dockings to the structural model.

Multiple state transition interface sampling of alanine dipeptide in explicit solvent

We have applied the recently developed multiple state transition interface sampling approach to alanine dipeptide in explicit water. We extract the rate constant matrix for configurational changes between each pair of metastable states. The results are comparable with values from previous literature and show that the method is applicable to biomolecular systems.

Direct determination of state-to-state rotational energy transfer rate constants via a Raman-Raman double resonance technique: ortho-acetylene in v2=1 at 155 K

A new technique for the direct determination of state-to-state rotational energy transfer rate constants in the gas phase is presented. It is based on two sequential stimulated Raman processes: the first one prepares the sample in a single rotational state of an excited vibrational level, and the second one, using the high resolution quasi-continuous stimulated Raman-loss technique, monitors the transfer of population to other rotational states of the same vibrational level as a function of the delay between the pump and the probe stages. The technique is applied to the odd-J rotational states of v(2)=1 acetylene at 155 K. The experimental layout, data acquisition, retrieval procedures, and numerical treatment are described. The quantity and quality of the data are high enough to allow a direct determination of the state-to-state rate constant matrix from a fit of the experimental data, with the only conditions of detailed balance and of a closed number of states. The matrix obtained from this direct fit is also compared with those obtained using some common fitting and scaling laws.

Dynamical integration of a Markovian web: A first passage time approach

In this work we address the dynamics of Markovian systems by tracking the evolution of the probability distribution, utilizing mean first passage time theory to augment the set of states considered. The method is validated on a lattice system and is applied, in conjunction with landscape analysis (saddle point searches) and multidimensional transition-state theory, to an atomistic model of glassy atactic polystyrene, in order to follow its time evolution over more than ten orders of magnitude on the time scale, from less than 10(-15) up to 10(-5) s. Frequencies extracted from the eigenvalues of the rate constant matrix are in favorable agreement with experimental measurements of subglass relaxation transitions at 250 K.

Predicting Aggregation Kinetics of DU 145 Prostate Cancer Cells in Liquid-Overlay Culture

The predictive capacity of a novel population-balance model to simulate aggregation kinetics of attachment-dependent cells at the resolution of one-cell increments has been evaluated. Using spheroid assembly of DU 145 human prostate cancer cells as a representative system, the mathematical model proved to be robust in simulating aggregation over a 5-fold range of surface densities from 5 x 10(3) to 2.5 x 10(4) cells/cm(2) with a single matrix of rate constants. For cultures at 1 x 10(5) cells/cm(2), more than 75% of simulated aggregate concentrations are within the standard deviation of measured concentrations. For the two extreme densities, at least two-thirds of model predictions are within 35% of the mean for experimental data. Error in model predictions is attributed to uncertainty in measurements and intrinsic changes in aggregation. The model has application to the rational design of spheroids in tissue engineering and bioseparation processes in pharmaceutical manufacturing.

Some symmetry-induced isotope effects in the kinetics of recombination reactions.

Symmetry-induced isotope effects in recombination and collision-induced dissociation reactions are discussed. Progress on understanding the anomalous isotope effects in ozone is reviewed. Then, calculations are performed for the simpler reaction xNe+yNe+H<-->xNeyNe+H, where x and y label either identical or different isotopes. The atomic masses in the model are chosen so that symmetry is the only difference between the systems. Starting from a single potential energy surface, the properties of the bound, quasibound, and continuum states of the neon dimer are calculated. Then, the vibration rotation infinite order sudden approximation is used to calculate cross sections for all possible inelastic and dissociative processes. A rate constant matrix that exactly satisfies detailed balance is constructed. It allows recombination to occur both via direct three-body collisions and via tunneling into the quasibound states of the energy transfer mechanism. The eigenvalue rate coefficients are determined. Significant isotope effects are clearly found, and their behavior depends on the pressure, temperature, and mechanism of the reaction. Both spin statistics and symmetry breaking produce isotope effects. Under most conditions the breaking of symmetry enhances the rates, but a wide spectrum of effects is observed; they range from isotope effects with a normal mass dependence to huge, mass-independent isotope effects to cancellation and even to reversal of the isotope effects. This is the first calculation of symmetry-induced isotope effects in recombination rates from first principles. The relevance of the present effects to ozone recombination is discussed.

State-to-state rotational relaxation rate constants for CO+Ne from IR-IR double-resonance experiments: comparing theory to experiment.

IR-IR double-resonance experiments were used to study the state-to-state rotational relaxation of CO with Ne as a collision partner. Rotational levels in the range Ji=2-9 were excited and collisional energy transfer of population to the levels Jf=2-8 was monitored. The resulting data set was analyzed by fitting to numerical solutions of the master equation. State-to-state rate constant matrices were generated using fitting law functions. Fitting laws based on the modified exponential gap (MEG) and statistical power exponential gap (SPEG) models were used; the MEG model performed better than the SPEG model. A rate constant matrix was also generated from scattering calculations that employed the ab initio potential energy surface of McBane and Cybulski [J. Chem. Phys. 110, 11 734 (1999)]. This theoretical rate constant matrix yielded kinetic simulations that agreed with the data nearly as well as the fitted MEG model and was unique in its ability to reproduce both the rotational energy transfer and pressure broadening data for Ne-CO. The theoretical rate coefficients varied more slowly with the energy gap than coefficients from either of the fitting laws.

Rotational energy transfer in vibrationally excited acetylene X̃ 1Σg(ν2″=1,J″):ΔJ propensities

A complete set of state-to-state rotational energy transfer rate constants has been measured for acetylene–acetylene collisions at room temperature under single collision conditions. Initial rotational states (Ji=5,7,...,25) were prepared and final states (Jf=1,...,25) interrogated. The measurements were carried out in a typical gas phase pump and probe arrangement. The initial vibrationally excited state was prepared by stimulated Raman pumping using strong Q-branch transitions. State preparation via this branch produces an isotropic spatial distribution of the excited state which is important for data analysis. Narrow bandwidth lasers ensure single rotational state selectivity. The rotational distribution after collisions is monitored by time-delayed laser-induced fluorescence via the à 1Au(ν3′ = 1)←X̃ 1Σg(ν2″ = 1) transition. In general, the rate constants decrease exponentially with the transferred rotational energy. The complete rate constant matrix can consistently be described by a simple parameter set within the dynamical infinite order sudden power approximation. In addition to this general behavior a significant ΔJ propensity of the rate constants is observed. Using the energy corrected sudden approximation with a power law basis an excellent match, reproducing the ΔJ propensities, to the rate constant matrix is obtained, again with a single set of parameters.

On the transformation of compartmental systems and the indeterminacy of the volume of distribution.

A general compartmental system with multiple-point elimination is transformable to a single-point elimination system. Transformation is achieved by a similarity transformation of the rate constant matrix, A, with a diagonal matrix, D. The elements of D (1, d1, d2,...) are equivalent to the "first-pass" effect between compartments and compartment 1. Application of the derived transformation demonstrates that the volume of distribution as defined by the normalized first-moment function is a minimal volume of distribution when there is no "first-pass" effect between the drug input compartment and the observation compartment. In all other cases, the volume of distribution is a meaningless metric since it may be a minimal or maximal metric and the exact status is indeterminate. Theorems on the non-negativity of the elements of (-A)-1 are derived.

A microcomputer algorithm for solving first-order compartmental models involving recycling.

A general algorithm for solving first-order compartmental models including recycling systems has been developed and its implementation on a microcomputer is described. Matrix algebra is used to obtain for any compartmental model an analytical solution, which is expressed as the exponential of a matrix of rate constants. A special technique is used in the algorithm to enable this exponential to be evaluated with a rapidly converging series. Truncation errors incurred in this process are estimated automatically. Thus, in an extreme case, where these errors may be significant, the appropriate action can be taken. Given a particular model, the user enters the model parameters into a rate matrix according to a simple rule. The algorithm then uses this matrix to solve the model, and thus no specialized mathematical knowledge is needed. The algorithm is given in a short BASIC program (60 lines) listed in an appendix. No additional software is required. By running this program on a standard microcomputer, the user can solve models of any complexity: those up to 15 compartments in seconds and those up to 30 compartments within a minute. The algorithm is thus ideally suited to solve kinetic models describing the transport of radionuclides in the environment or the translocation of elements in biological systems such as the metabolic models recommended by the International Commission on Radiological Protection (ICRP). Given the initial amount of material in each compartment at time t = 0, together with its radioactive decay constant, the algorithm gives both the amount in each compartment at any future time t and the number of disintegrations that will have occurred in each compartment up to time t. The computer program, shown in an appendix, could easily be used to calculate disintegrations over any time interval of interest, or to predict the quantities or fractions of an intake expected to be present in any in vivo or excretion compartments of interest. Thus, the algorithm can be useful in both the design and conduct of bioassay and internal dose assessment procedures.