Cumulative Reaction Probability Research Articles

Algorithms for Obtaining Cumulative Reaction Probabilities for Chemical Reactions

Miller and coworkers have proposed calculating the cumulative reaction probabilities of a chemical reaction through the computation of the few lowest eigenvalues of a matrixZ. In this study, the possibility of computing these eigenvalues using the iterative IRLM approach of Sorensen is investigated. The application to collinear H + H2→ H + H2reactions shows that it is possible to compute the physical eigenvalues ofZiteratively using the IRLM formulation with a Chebychev preconditionedZmatrix as the primary matrix. Because the IRLM formulation only requires simple matrix–vector operations, one never needs to assemble and store the wholeZmatrix. Moreover, the multiplication ofZby a vector takes full advantage of the underlying sparseness of the matrix. This study presents several new and successful strategies to improve the convergence speed of IRLM, as well as strategies for obtaining the eigenvalues ofZwhen the eigenvalue span ofZis very large (1–1010).

Coupled potential-energy surfaces and quantum reactive scattering for the Cl(2P)+HCl→ClH+Cl(2P) reaction

We construct global multiple-coupled potential-energy surfaces for the three-dimensional Cl( 2 P) + HCl → ClH + Cl( 2 P) reaction which we use to generate results from quantum reactive scattering calculations. The three potential surfaces (1 2 A′, 2 2 A′ and 1 2 A″) are derived from multi-configuration self-consistent field (MCSCF) electronic structure calculations for geometries where all atoms are strongly interacting. The MCSCF energies are used to construct diabatic potentials which are fitted with rotated Morse cubic spline functions. The diabatic surfaces are then combined with empirical long-range potentials due to Dubernet and Hutson, which are also used to determine empirical coupling surfaces. Spin–orbit interaction is added perturbatively to the resulting surfaces. Saddle-point properties on these fitted surfaces are scaled to reproduce experimental thermal rate coefficient data based on the quantum scattering calculations. The quantum scattering calculations are based on a coupled-channel hyperspherical coordinate (CCH) method. The basis expansion used in the scattering calculation contains diatomic vibration and rotation, and the atomic fine-structure states, including full angular momentum coupling of rotation with electronic spin and orbital angular momentum. All terms in the combined rovibronic Hamiltonian are evaluated explicitly, including spin–orbit coupling, Coriolis and electrostatic interactions. State-selected and total cumulative reaction probabilities and thermal rate coefficients are calculated using a J-shifting approximation, which starts from converged CCH results for total angular momentum quantum number J = 1/2. The calculated rate coefficients indicate that spin-state change collisions account for a few per cent of the thermal rate coefficient at 300 K. Transition-state ClHCl resonances are lower in energy and broader than those on earlier (single) surfaces, and there is a more extensive progression in the symmetric stretch mode.

Centrifugal-sudden hyperspherical study of Cl + HCl → ClH + Cl reaction dynamics on ‘tight-bend’ and ‘loose-bend’ potential energy surfaces

A centrifugal-sudden hyperspherical (CSH) quantum calculation has been carried out for the three-dimensional reaction CI + HCl → ClH + Cl, using the extended London -Eyring-Polanyi-Sato (LEPS) potential energy surface no. 3 of Persky and Komweitz (PK3), which is an example of a 'tight-bend' surface. Quantities calculated include: state-to-state and cumulative reaction probabilities, and differential and integral cross sections. Comparison is made with CSH results from the ‘loose-bend’ LEPS surface of Bondi-Connor-Manz-Römelt (BCMR). Three simple models are used to understand qualitatively trends in the computed results from the PK3 and BCMR surfaces: (1) a classical hard-sphere model, which describes collisions of the reactants, (2) a J shift approximation for the reaction probabilities, where J is the total angular momentum quantum number, (3) a semi-classical optical model for the reactive differential cross sections. In addition, the angular distributions are analysed by an (asymptotic) nears ide-farside (NF) theory in order to extract their physical content. The NF analysis shows that high frequency oscillations in the forward-angle scattering are an interference effect caused by a resonance.

Resonances in the cumulative reaction probability for a model electronically nonadiabatic reaction

The cumulative reaction probability, flux–flux correlation function, and rate constant are calculated for a model, two-state, electronically nonadiabatic reaction, given by Shin and Light [S. Shin and J. C. Light, J. Chem. Phys. 101, 2836 (1994)]. We apply straightforward generalizations of the flux matrix/absorbing boundary condition approach of Miller and co-workers to obtain these quantities. The upper adiabatic electronic potential supports bound states, and these manifest themselves as ‘‘recrossing’’ resonances in the cumulative reaction probability, at total energies above the barrier to reaction on the lower adiabatic potential. At energies below the barrier, the cumulative reaction probability for the coupled system is shifted to higher energies relative to the one obtained for the ground state potential. This is due to the effect of an additional effective barrier caused by the nuclear kinetic operator acting on the ground state, adiabatic electronic wave function, as discussed earlier by Shin and Light. Calculations are reported for five sets of electronically nonadiabatic coupling parameters.

Cumulative reaction probability via transition state wave packets

A new time-dependent approach to the cumulative reaction probability, N(E), has been developed based on the famous formulation given by Miller and co-workers [J. Chem. Phys. 79, 4889 (1983)], N(E)=[(2π)2/2] tr[δ(E−H)Fδ(E−H)F]. Taking advantage of the fact that the flux operator has only two nonzero eigenvalues, we evaluate the trace efficiently in a direct product basis of the first flux operator eigenstates and the Hamiltonian eigenstates on the dividing surface (internal states). Because the microcanonical density operator, δ(E−H), will eliminate contributions to N(E) from an internal state with the energy much higher than the total energy E, we can minimize the number of internal states required by choosing a dividing surface with the lowest density of internal states. If the dividing surface is located in an asymptotic region, one just needs to include all the open channels, i.e., with internal energy lower than the total energy. Utilizing the Fourier transform for δ(E−H), we can obtain the information for all the energies desired by propagating these wave packets once. Thus the present approach will be much more efficient than the initial state selected wave packet (ISSWP) approach to N(E) for systems with many rotation degrees of freedom because the density of states in asymptotic region for such systems is much higher than that in the transition state region. With the present method one can also calculate the cumulative reaction probability from an initial state (or to a final state) by locating the second flux operator in the corresponding asymptotic region. This provides an alternative to the ISSWP approach which may be more efficient if the reaction probabilities from a large number of initial states are desired. The method is applied to the 3D H + H2 (even rotation) reaction for J=0 by locating the first dividing surface in the transition state region. The demonstration also shows an aspect less than ideal; the contribution to N(E) from a wave packet may be slightly larger than 1 or slightly smaller than 0, making it improper to interpret the contribution as a probability.

Dynamics of the photodissociation of triplet ketene

Calculations of the microcanonical dissociation rate for vibrationally excited ketene on the first excited triplet surface (T1) are presented. The calculations utilize the quantum reactive scattering methodology of absorbing boundary conditions (ABC) with a discrete variable representation (DVR) to obtain the cumulative reaction probability for dissociation over the barrier. Model 1- and 2-degree of freedom potential energy surfaces for the T1 surface were obtained by fitting to the best available ab initio structures, energies, and frequencies. The dissociation rates in these reduced-dimensionality calculations give good overall agreement with the experimentally measured rates, although the steplike features seen in the experiments are washed out by the tunneling through the narrow barrier predicted in the ab initio calculations. Further model calculations reveal that a barrier frequency of approximately 50–100i cm−1 is required to recover the step structure seen experimentally, which suggests that there is either another transition state region on the T1 surface farther out towards the product channel, or that there is surface-hopping dynamics taking place between the T1 and S0 ketene potential energy surfaces, or that the ab initio barrier frequency is simply too large.

Quantum Functional Sensitivity Analysis of the D + H2 Reaction Rate Coefficient via the Separable Rotation Approximation

Sensitivities of the D + H{sub 2} cumulative reaction probability (CRP) are calculated for total angular momentum J = 0 and 6 to test the applicability of the separable rotation approximation. The CRPs and their sensitivities are then thermally averaged to obtain sensitivities of the log-normalized rate coefficient using a single J. This approach has great promise of being a simple, yet accurate, method of providing reliable feedback information to quantum chemists without resorting to full-dimensionality quantal calculations. On the basis of our sensitivity analysis, we propose three potential energy surface (PES) modifications which should remove the remaining discrepancies between the experimental and theoretical rate coefficients, especially at the high temperatures. Interestingly, none of these potential modifications lie at a particularly `high`-energy region of the PES as was previously thought. 34 refs., 15 figs., 3 tabs.

Semiclassical calculation of cumulative reaction probabilities

It is shown how the rigorous quantum mechanical expression for the cumulative reaction probability (CRP) obtained by Seideman and Miller [J. Chem. Phys. 96, 4412; 97, 2499 (1992)], N(E)=4 tr[ε̂r⋅Ĝ*(E)⋅ε̂p⋅Ĝ(E)], which has been the basis for quantum calculations of the CRP for simple chemical reactions, can also be utilized with a semiclassical approximation for the Green’s function, Ĝ(E)≡(E+iε̂−Ĥ)−1=(iℏ)−1∫∞0 exp(iEt/ℏ)exp(−i(Ĥ−iε̂)t/ℏ). Specifically, a modified Filinov transformation of an initial value representation of the semiclassical propagator has been used to approximate the Green’s function. Numerical application of this trajectory-based semiclassical approximation to a simple one-dimensional (barrier transmission) test problem shows the approach to be an accurate description of the reaction probability, even some ways into the tunneling regime.

Quantum Mechanical Rate Coefficients for the Cl + H2 Reaction

We present quantal reaction rate coefficients obtained from well-converged cumulative reaction probabilities calculated by an algebraic variational method for total angular momentum J in the range 0−6 and obtained from the separable-rotation approximation for J ≠ 6. These results are used to test variational transition state theory calculations with optimized multidimensional tunneling contributions and are compared to experiment.

Resonant features in the energy dependence of the rate of ketene isomerization

Calculations of the microcanonical isomerization rates for vibrationally excited ketene are presented. The calculations utilize the quantum reactive scattering methodology of absorbing boundary conditions with a discrete variable representation to obtain the cumulative reaction probability for one form of ketene to isomerize via the oxirene intermediate, and were carried out with model 1-, 2-, and 3-degree-of-freedom potential energy surfaces constructed using ab initio data. Significant differences are seen in the energy dependent features of the microcanonical rate for the single mode and multi-mode potentials; e.g., the single mode potential exhibits tunneling resonances with widths of around 1 cm−1, while the calculations involving more than one degree of freedom have additional resonant features that have widths around 10 cm−1 and also exhibit non-Breit–Wigner resonant line shapes. This suggests that many of the resonance features are best described as Feshbach (energy transfer, or dynamical) resonances that result because of a strongly bent region on the multi-mode potential energy surfaces. The calculated rates show reasonable qualitative agreement with the experimental results of Lovejoy and Moore [J. Chem. Phys. 98, 7846 (1993)].

Comparisons between statistics, dynamics, and experiment for the H+O2→OH+O reaction

The accuracy of the variable reaction coordinate (VRC) implementation of transition state theory (TST) is investigated for the bimolecular reaction of H with O2 via direct comparisons with quantum scattering theory for J=0, classical trajectory simulations for a wide range of J, and experimental canonical rate constants. The DMBE IV potential energy surface of Varandas and co-workers is employed in each of the theoretical calculations. The first two comparisons indicate that the VRC-TST approach overestimates the cumulative reaction probability (CRP) for this reaction by a factor of 2.3, roughly independent of E and J for moderate energies. The trajectory simulations further indicate that this failure of TST is primarily the result of the rapid redissociation of a large fraction of the initially formed HO2. An estimate for the quantum CRP on the basis of the combined dynamical and statistical results is seen to provide a useful alternative to the more standard quasiclassical trajectory estimates. A thermal averaging over the E and J-dependence of the TST estimates for the CRP provides canonical rate constants, k(T), which, when corrected for the above-mentioned overestimate, are still a factor of 1.7–2.0 times greater than the experimental data. This discrepancy is most likely the result of either (i) inaccuracies in the DMBE IV surface and/or (ii) an overestimate of the contribution to the reactive flux from the nearly degenerate first excited state in the exit channel region.

The quasi-minimal residual algorithm applied to complex symmetric linear systems in quantum reactive scattering

The solution of systems of linear equations Ax=b with complex symmetric coefficient matrix A of size N, typically appearing in quantum-reactive scattering problems, is discussed. The quasiminimal residual (QMR) method is introduced to solve the complex symmetric linear system and is compared to the generalized minimal residual (GMRES) method. The methods are applied to two different chemical problems: the initial state-selected reaction probability for the H2+OH→H +H2O reaction, and the cumulative reaction probability for the isomerization of ketene, both with N≳104. It is shown that the QMR method behaves more favorably, i.e., converges faster, than the GMRES for large N, especially when high accuracy is needed.

Cumulative reaction probability from time independent scattering theory

The formally exact expression for the cumulative reaction probability, N(E), was first derived by Miller, Schwartz and Tromp (MST) [J. Chem. Phys. 79, 4889 (1983)] from the flux–flux time-dependent expression for the thermal rate constant. We derive the MST formula for N(E) from the time-independent expression for the state-to-state transition probability. Our proof shows that the flux can be calculated at a dividing surface which can be located at any point, including a point where bound states have nonzero probability density.

Cumulative Reaction Probabilities of Cl+H2 from Quantum Scattering Calculations and Evaluation of Thermal Rate Constant

We report the cumulative reaction probabilities (CRP) of the three-dimensional Cl+H2 → H+HCl calculated on the Stern-Persky-Klein potential energy surface, using the generalized Newton variational principle, for total angular momentum J = 0, over a total reaction energy range from 9 to 19 kcal/mol. The relation between CRP and reaction rate constants, as well as the calculation for complete reaction rates (summed over all J), bypassing the direct calculation of the CRP's for J>0, are also discussed.

Collision induced isomerization of a semirigid bender hydrogen cyanide

The isomerization of a hydrogen cyanide molecule from a localized HCN state to a localized HNC state is formulated as a bimolecular scattering process. The scattering partner is Ar, and the intermolecular interaction potential is written as a pairwise summation of Lennard-Jones potentials. The parameters of this potential are optimized to yield the correct Ar–hydrogen cyanide van der Waals structure. The intramolecular isomerization potential is based on an approximate reaction path through a three-dimensional ab initio potential, and HCN/HNC is treated as a semirigid bender. Eighty molecular eigenstates are calculated, and used as a coupled-channel basis in a scattering calculation in which the hydrogen cyanide rotation is treated in the infinite order sudden approximation. Transition probabilities and cumulative reaction probabilities to HNC states from initial HCN states are focused on; however, some results to final delocalized states are also presented. The bimolecular isomerization rate constant is presented over a wide temperature range. The energy transfer between Ar and initial HCN states is also briefly examined, as a function of the initial relative translational energy.

Quantum mechanical calculations of the rate constant for the H2+OH→H+H2O reaction: Full-dimensional results and comparison to reduced dimensionality models

The cumulative reaction probability is calculated for the H2+OH→H+H2O reaction in its full (six) dimensionality for total angular momentum J=0. The calculation, which should give the (numerically) exact result for the assumed potential energy surface, yields the cumulative reaction probability directly, without having to solve the complete state-to-state reactive scattering problem. Higher angular momenta (J≳0) were taken into account approximately to obtain the thermal rate constant k(T) over the range 300°<T<700°. The result deviates significantly from the experimental rate constant, suggesting that the potential energy surface needs to be improved. A systematic series of reduced dimensionality calculations is carried out in order to characterize the behavior and reliability of these more approximate treatments; a comparison of the full dimensional results with previous reduced dimensionality calculations is also made.

New Reduced Dimensionality Calculations of Cumulative Reaction Probabilities and Rate Constants for the H + H2 and D + H2 Reactions

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTNew Reduced Dimensionality Calculations of Cumulative Reaction Probabilities and Rate Constants for the H + H2 and D + H2 ReactionsDesheng Wang and Joel M. BowmanCite this: J. Phys. Chem. 1994, 98, 33, 7994–7999Publication Date (Print):August 1, 1994Publication History Published online1 May 2002Published inissue 1 August 1994https://doi.org/10.1021/j100084a013RIGHTS & PERMISSIONSArticle Views74Altmetric-Citations24LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (693 KB) Get e-Alerts

Reaction rates of BrH+Cl→Br+HCl using semiclassical transition state theory

The semiclassical transition state theory, introduced by the authors with Miller and Hernandez, for the calculation of cumulative reaction probabilities and thermal reaction rates, is used to investigate the reaction HBr+Cl→Br+HCl. The reaction rates are compared with experimental values and to quantum mechanical and variational transition state theory calculations.

Comparison of transition state theory with quantum scattering theory for the reaction Li+HF→LiF+H

The validity of transition state theory is examined for the bimolecular reaction of Li+HF→LiF+H. Accurate three-dimensional quantum scattering theory calculations of the cumulative reaction probability are reported for energies ranging from threshold (0.255 eV) up to 0.600 eV and a total angular momentum J of 0. Transition state theory estimates of the effect of both the entrance and exit channels on the cumulative reaction probability are reported for the same energy range and J value. The transition state theory results are found to provide an accurate description of the smoothed energy dependence of the cumulative reaction probabilities with a maximum disagreement between the two calculations of about 25% arising at the highest energy considered of 0.6 eV.

A coupled channel hyperspherical scattering study of the Cl + HCl → ClH + Cl reaction: cumulative and state-selected probabilities, integral cross sections, and product rotational distributions

We present accurate quantum results for the Cl + HCl → ClH + Cl reaction using a coupled channel hyperspherical coordinate method (denoted CCH). A semiempirical extended London–Eyring–Polanyi–Sato potential energy surface is employed. In our calculations, the body-fixed z projection quantum number Ω in the rotational basis set has been truncated at Ωmax = 2, since this is sufficient to give converged results near to, and above, the reaction threshold (total energies E in the range 0.40–0.60 eV). We compare our results with earlier work using the centrifugal sudden hyperspherical (CSH) method, in which only Ω = 0 is included in the basis; other features of the calculations are identical. At E = 0.40 eV, where reactivity is low, the CCH(Ωmax = 2) and CSH(Ω = 0) degeneracy averaged reaction probabilities, integral cross sections and product rotational distributions are mostly in excellent agreement. This agreement deteriorates with increasing E, such that at E = 0.60 eV, the CCH(Ωmax = 2) cumulative reaction probability is approximately twice its CSH(Ω = 0) counterpart. Even at E = 0.40 eV there are some important differences between CCH(Ωmax = 2) and CSH(Ω = 0) concerning mj-resolved reaction probabilities. In addition, there are differences in the relative contributions of "tight bend" and "nearly free-rotor" mechanisms to the reaction probabilities and product rotational distributions, with the "nearly free-rotor" mechanism dominant in all cases.