Quantum Mechanical Transition State Theory Research Articles

Chemical kinetics study through observation of individual reaction events with atomic-resolution electron microscopy.

Single-molecule atomic-resolution real-time electron microscopic movie imaging is an emerging new tool for obtaining dynamic structural information on molecules and molecular assemblies. This method provides a hitherto inaccessible possibility to in situ observe the time evolution of chemical events at various temperatures from the beginning till the end, as demonstrated for the kinetics study of [2 + 2] cycloaddition of [60]fullerene molecules, which was found to occur via an excited state or via radical cation depending on the temperature. One unique feature of this methodology is that, by observing directly the reaction events, one can obtain information on the frequency of events unperturbed by molecular diffusion. With the obtained experimental data set, we provided the first experimental proof of what the quantum mechanical transition state theory predicted, in that isolated molecules behave as if all their accessible states were occupied in a random order. We also found that, under the 1-D reaction conditions, molecular-level information on a few hundred molecules suffices to deduce statistically meaningful kinetics data that match with those obtained by bulk experiments.

Atmospheric Chemistry of Ethyl Propionate

Ethyl propionate is a model for fatty acid ethyl esters used as first-generation biodiesel. The atmospheric chemistry of ethyl propionate was investigated at 980 mbar total pressure. Relative rate measurements in 980 mbar N(2) at 293 ± 0.5 K were used to determine rate constants of k(C(2)H(5)C(O)OC(2)H(5) + Cl) = (3.11 ± 0.35) × 10(-11), k(CH(3)CHClC(O)OC(2)H(5) + Cl) = (7.43 ± 0.83) × 10(-12), and k(C(2)H(5)C(O)OC(2)H(5) + OH) = (2.14 ± 0.21) × 10(-12) cm(3) molecule(-1) s(-1). At 273-313 K, a negative Arrhenius activation energy of -3 kJ mol(-1) is observed.. The chlorine atom-initiated oxidation of ethyl propionate in 980 mbar N(2) gave the following products (stoichiometric yields): ClCH(2)CH(2)C(O)OC(2)H(5) (0.204 ± 0.031), CH(3)CHClC(O)OC(2)H(5) (0.251 ± 0.040), and C(2)H(5)C(O)OCHClCH(3) (0.481 ± 0.088). The chlorine atom-initiated oxidation of ethyl propionate in 980 mbar of N(2)/O(2) (with and without NO(x)) gave the following products: ethyl pyruvate (CH(3)C(O)C(O)OC(2)H(5)), propionic acid (C(2)H(5)C(O)OH), formaldehyde (HCHO), and, in the presence of NO(x), PAN (CH(3)C(O)OONO(2)). The lack of acetaldehyde as a product suggests that the CH(3)CH(O)C(O)OC(2)H(5) radical favors isomerization over decomposition. From the observed product yields, we conclude that H-abstraction by chlorine atoms from ethyl propionate occurs 20.4 ± 3.1%, 25.1 ± 4.0%, and 48.1 ± 8.8% from the CH(3)-, -CH(2)-, and -OCH(2)- groups, respectively. The rate constant and branching ratios for the reaction between ethyl propionate and the OH radical were investigated theoretically using quantum mechanical calculations and transition state theory. The stationary points along the reaction path were optimized using the CCSD(T)-F12/VDZ-F12//BH&HLYP/aug-cc-pVTZ level of theory; this model showed that OH radicals abstract hydrogen atoms primarily from the -OCH(2)- group (80%).

A Molecular-Level Reaction Algorithm for Proton Transport Consistent with Quantum Mechanical Transition State Theory and Macroscopic Conductivity

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Quantum mechanical transition state theory and tunneling corrections

An efficient implementation of the quantum mechanical transition state theory recently proposed by Hansen and Andersen [J. Chem. Phys. 101, 6032 (1994); J. Phys. Chem. 100, 1137 (1996)] is presented. Their method approximates the flux–flux autocorrelation function by using short-time information to fit an assumed functional form (with physically correct properties). The approach described here exploits the low rank of the half-Boltzmannized flux operator, thereby facilitating application to reactions involving many degrees of freedom. In addition, we show how the quantum transition state theory can be used to obtain tunneling corrections within the framework of more traditional transition state theory approaches, i.e., those making an assumption of separability. Directions for possible improvements of the theory are discussed.

The symmetrized quantum thermal flux operator

Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L2 integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.

Comment on ‘‘Comparison of positive flux operators for transition state theory using a solvable model’’ [J. Chem. Phys. 104, 7015 (1996)

The Weyl definition of the positive component of the flux operator is discussed. Specifically, it is noted that the imaginary part of the operator makes no contribution to the quantum mechanical transition state theory rate constant, and this is why it was omitted in previous work on the topic.

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.

The quantum dynamics of hydrogen and deuterium on the Pd(111) surface: A path integral transition state theory study

The surface diffusion constant for hydrogen and deuterium on the palladium(111) surface is calculated using quantum mechanical transition state theory. The rate constants for diffusion into the subsurface layer are also calculated. Quantum effects are seen to be most important for the surface/subsurface transition and cause an inverse isotope effect in which the rate for deuterium is greater than the rate for hydrogen. The results of ground and excited state wave function calculations show localized hydrogenic states, despite large zero point energies, and that the preferred binding site can vary with isotope between surface and subsurface sites. In addition, estimates of the tunneling rate between the surface and subsurface are in qualitative agreement with the low temperature transition state results.

Resonance state approach to quantum mechanical variational transition state theory

Variational transition state theory is reformulated in terms of compound state resonances associated with the transition state region. The use of the formalism is illustrated for two-dimensional model problems, which are treated by the adiabatic theory of reactions and by perturbation theory. The limitations of the latter are delineated for the case of a biquadratic (quartic) coupling of the reaction coordinate to a transverse vibration. 51 refs.

Calculation of thermal rate coefficients from the quantum flux autocorrelation function: Converged results and variational quantum transition state theory for O+HD↔OD+H and O+HD↔OH+D

The thermal rate coefficients for the title reactions with zero total angular momentum are calculated by an ℒ 2 basis-set evaluation of the flux autocorrelation function. We use three dividing surfaces, each one separating one arrangement from the other two. The rate coefficients for O+HD↔OD+H or OH+D are converged to within 1.2% over the temperature range 300–1000 K and to within 2.4% at 200 K. The rate coefficients for the reactions H+OD↔D+OH are converged to within 22% at 1000 K and upper bounds to these rate coefficients are given for the temperature range 200–800 K. Flux autocorrelation functions and rate coefficients are calculated for different positions of the dividing surface to show that the computed rate coefficient is stable with respect to varying this position. Rate coefficients are also calculated by a variational version of quantum mechanical transition state theory and the results are compared to the accurate rate coefficients obtained from flux autocorrelation function theory.

Some New Approaches to Semiclassical and Quantum Transition State Theory

AbstractSemiclassical and quantum mechanical transition state theory is reviewed and two new approaches described. One is a general implementation of a semiclassical rate expression [Miller, Faraday Discuss. Chem. Soc. 62, 40 (1977)] that involves the “good” action‐angle variables associated with the saddle point (i.e., transition state) of a potential energy surface. The other is an evaluation of a formally exact quantum expression for the rate [Miller, Schwartz, and Tromp, J. Chem. Phys. 79, 4889 (1983)] in terms of Siegert eigenvalues associated with the transition state. Siegert eigenvalues are usually associated with scattering resonances, so their identification with the saddle point of a potential surface, and the expression for the reaction rate in terms of them, is quite an unexpected and novel development.

Analytic expression for the transmission coefficient in quantum mechanical transition state theory

An analytic expression for the transmission coefficient in a recently formulated quantum mechanical transition state theory is derived. This formula is developed for a reaction coordinate having a general nonlinear potential energy barrier coupled to a linearly responding environment or “bath”. The final result has the same compact form as the classical Grote-Hynes transmission coefficient, but it also reflects important properties of the quantum mechanical tunneling. The present quantum transmission coefficient may be evaluated from a classical molecular dynamics simulation in conjunction with a quantum statistical mechanical variational principle.

A nonseparable quantum mechanical transition state theory

A quantal transition state bound on the rate of chemical reactions in any dimension is derived. The magnitude of the bound is dependent on the location of a dividing surface and thus is variational. Analysis of the quality of the bound in one and two dimensions shows that the quality of the bound improves with increasing dimensionality of the problem. The new theory requires only the computation of diagonal density matrix elements and so should prove to be computationally useful.

An exact quantum mechanical transition state theory. 1. An overview

Starting with the collision theory expression for the rate constant k of a thermal gas phase bimolecular reaction in terms of reaction cross sections, we derive a modified exact expression for k which has a mathematical form identical with that provided by transition state theory (TST). No approximations are introduced in this derivation and the only difference between this exact expression and the TST one is in the transmission coefficient kappa. The exact one, kappa/sup ex/, is expressed in terms of the correct reaction probabilities, whereas the TST one, kappa/sup TST/, involves transmission probabilities across appropriately defined one-mathematical-dimensional adiabatic barriers. It is shown that kappa/sup ex/ can be larger or smaller than unity, and need not approach one as the temperture of the reaction tends to infinity. Comparison of the expressions for kappa/sup ex/ and kappa/sup TST/ leads to a necessary and sufficient set of conditions for validity of TST. These involve only dynamical assumptions regarding reaction probabilities, and do not require any additional equilibrium condition among activated complexes or between them and reagents; the only equilibrium assumption made is the one among reagents, which is inherent in the definition of a thermal rate constant. A comparison of usual TSTmore » rate constants with the exact one for the H + H/sub 2/, F + H/sub 2/, and F + D/sub 2/ systems is given, the validity of the dynamical assumptions of that theory is examined, and suggestions for its improvement are presented and tested. 11 figures, 59 references.« less

Importance of nonseparability in quantum mechanical transition-state theory

ADVERTISEMENT RETURN TO ISSUEPREVArticleImportance of nonseparability in quantum mechanical transition-state theoryWilliam H. MillerCite this: Acc. Chem. Res. 1976, 9, 8, 306–312Publication Date (Print):August 1, 1976Publication History Published online1 May 2002Published inissue 1 August 1976https://doi.org/10.1021/ar50104a005RIGHTS & PERMISSIONSArticle Views470Altmetric-Citations131LEARN 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 (885 KB) Get e-Alerts Get e-Alerts

Classical dynamics and Miller’s collinear quantum mechanical transition state theory

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Semiclassical transition state theory for nonseparable systems: Application to the collinear H+H2 reaction

Two different kinds of semiclassical approximations are used to evaluate a previously obtained quantum mechanical transition state theory rate expression. No assumptions, however, such as separability of the Hamiltonian, vibrationally adiabatic motion along a reaction coordinate, etc., are incorporated. Application is made to the collinear H+H2 reaction, and agreement with accurate quantum scattering calculations is found to be reasonably good. The results indicate that transition state theory—provided no assumptions of separability are included—is probably as accurate quantum mechanically as it has been found to be classically for describing the threshold of chemical reactions with an activation barrier.

Path integral representation of the reaction rate constant in quantum mechanical transition state theory

Feynman’s path integral representation of the Boltzmann operator e−βH is used to express the rate constant of a recently formulated quantum mechanical version of transition state theory. By evaluating the path integral in two separate stages, one is able to interpret the result as a generalization of a model suggested several years ago by Johnston and Rapp for handling the nonseparable aspect of tunneling in transition state theory. A Fourier series expansion of the path integral is also developed, and this approach has promise for direct numerical evaluation of the quantum rate expression.

Semiclassical limit of quantum mechanical transition state theory for nonseparable systems

The semiclassical limit of quantum mechanical transition state theory is derived by invoking the classical path approximation for the Boltzmann density operator and making use of the stationary phase approximation; separability of motion along a reaction coordinate is not assumed. The resulting expression for the rate constant bears an interesting similarity to that of conventional transition state theory, although all quantities in it refer to the full classical dynamics on the potential energy surface. In place of the vibrational frequencies of the ’’activated complex’’ which appear in the conventional theory, for example, the semiclassical expression contains characteristic frequencies related to the stability properties of a periodic classical trajectory. Conservation of total angular momentum is easily accounted for in a rigorous manner so that the semiclassical model can be applied to three−dimensional dynamical systems.

Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants

An ``exact'' quantum mechanical transition state theory is defined, i.e., a model which invokes the basic transition state idea to calculate the rate of a chemical reaction but which is free of any auxiliary approximations. Most importantly, for example, it is not necessary to assume that the Hamiltonian is separable about the saddle point. It is argued that this model should provide an accurate description of the threshold region of the reaction where quantum effects are most significant. Finally, an even more general model, a new kind of semiclassical approximation, is presented which is essentially a synthesis of this quantum mechanical transition state model and the completely classical trajectory procedure for determining the rate constant; at sufficiently high temperatures, quantum effects become negligible, so that the correct rate constant is obtained; while at low temperature, the correct result is obtained because the transition state model becomes valid.