Formulation Of Transition State Theory Research Articles

Dependence of Copolymer Composition in Radical Polymerization on Solution Properties: a Quantitative Thermodynamic Interpretation

The thermodynamic formulation of transition state theory is applied to compute the effect of the medium (solvent type and concentration) on the copolymer composition obtained in radical copolymeriz...

Migration mechanisms and diffusion barriers of vacancies in Ga2O3

We employ the nudged elastic band and the dimer methods within the standard density functional theory (DFT) formalism to study the migration of the oxygen and gallium vacancies in the monoclinic structure of $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$. We identify all the first nearest neighbor paths and calculate the migration barriers for the diffusion of the oxygen and gallium vacancies. We also identify the metastable sites of the gallium vacancies which are critical for the diffusion of the gallium atoms. The migration barriers for the diffusion of the gallium vacancies are lower than the migration barriers for oxygen vacancies by 1 eV on average, suggesting that the gallium vacancies are mobile at lower temperatures. Using the calculated migration barriers we estimate the annealing temperature of these defects within the harmonic transition state theory formalism, finding excellent agreement with the observed experimental annealing temperatures. Finally, we suggest the existence of percolation paths which enable the migration of the species without utilizing all the migration paths of the crystal.

Catalytic routes to fuels from C1 and oxygenate molecules.

This account illustrates concepts in chemical kinetics underpinned by the formalism of transition state theory using catalytic processes that enable the synthesis of molecules suitable as fuels from C1 and oxygenate reactants. Such feedstocks provide an essential bridge towards a carbon-free energy future, but their volatility and low energy density require the formation of new C-C bonds and the removal of oxygen. These transformations are described here through recent advances in our understanding of the mechanisms and site requirements in catalysis by surfaces, with emphasis on enabling concepts that tackle ubiquitous reactivity and selectivity challenges. The hurdles in forming the first C-C bond from C1 molecules are illustrated by the oxidative coupling of methane, in which surface O-atoms form OH radicals from O2 and H2O molecules. These gaseous OH species act as strong H-abstractors and activate C-H bonds with earlier transition states than oxide surfaces, thus rendering activation rates less sensitive to the weaker C-H bonds in larger alkane products than in CH4 reactants. Anhydrous carbonylation of dimethyl ether forms a single C-C bond on protons residing within inorganic voids that preferentially stabilize the kinetically-relevant transition state through van der Waals interactions that compensate for the weak CO nucleophile. Similar solvation effects, but by intrapore liquids instead of inorganic hosts, also become evident as alkenes condense within MCM-41 channels containing isolated Ni2+ active sites during dimerization reactions. Intrapore liquids preferentially stabilize transition states for C-C bond formation and product desorption, leading to unprecedented reactivity and site stability at sub-ambient temperatures and to 1-alkene dimer selectivities previously achieved only on organometallic systems with co-catalysts or activators. C1 homologation selectively forms C4 and C7 chains with a specific backbone (isobutane, triptane) on solid acids, because of methylative growth and hydride transfer rates that reflect the stability of their carbenium ion transition states and are unperturbed by side reactions at low temperatures. Aldol condensation of carbonyl compounds and ketonization of carboxylic acids form new C-C bonds concurrently with O-removal. These reactions involve analogous elementary steps and occur on acid-base site pairs on TiO2 and ZrO2 catalysts. Condensations are limited by α-H abstraction to form enolates via concerted interactions with predominantly unoccupied acid-base pairs. Ketonization is mediated instead by C-C bond formation between hydroxy-enolates and monodentate carboxylates on site pairs nearly saturated by carboxylates. Both reactions are rendered practical through bifunctional strategies, in which H2 and a Cu catalyst function scavenge unreactive intermediates, prevent sequential reactions and concomitant deactivation, and remove thermodynamic bottlenecks. Alkanal-alkene Prins condensations on solid acids occur concurrently with alkene dimerization and form molecules with new C-C bonds as skeletal isomers unattainable by other routes. Their respective transition states are of similar size, leading to selectivities that cannot sense the presence of a confining host. Prins condensation reactions benefit from weaker acid sites because their transition states are less charged than those for oligomerization and consequently less sensitive to conjugate anions that become less stable as acids weaken.

Microcanonical rates, gap times, and phase space dividing surfaces

The general approach to classical unimolecular reaction rates due to Thiele is revisited in light of recent advances in the phase space formulation of transition state theory for multidimensional systems. Key concepts, such as the phase space dividing surface separating reactants from products, the average gap time, and the volume of phase space associated with reactive trajectories, are both rigorously defined and readily computed within the phase space approach. We analyze in detail the gap time distribution and associated reactant lifetime distribution for the isomerization reaction HCN <==> CNH, previously studied using the methods of phase space transition state theory. Both algebraic (power law) and exponential decay regimes have been identified. Statistical estimates of the isomerization rate are compared with the numerically determined decay rate. Correcting the RRKM estimate to account for the measure of the reactant phase space region occupied by trapped trajectories results in a drastic overestimate of the isomerization rate. Compensating but as yet not fully understood trapping mechanisms in the reactant region serve to slow the escape rate sufficiently that the uncorrected RRKM estimate turns out to be reasonably accurate, at least at the particular energy studied. Examination of the decay properties of subensembles of trajectories that exit the HCN well through either of two available symmetry related product channels shows that the complete trajectory ensemble effectively attains the full symmetry of the system phase space on a short time scale t approximately < 0.5 ps, after which the product branching ratio is 1:1, the "statistical" value. At intermediate times, this statistical product ratio is accompanied by nonexponential (nonstatistical) decay. We point out close parallels between the dynamical behavior inferred from the gap time distribution for HCN and nonstatistical behavior recently identified in reactions of some organic molecules.

Molecular modeling of diffusion on a crystalline pentaerythritol tetranitrate surface

Surface diffusion on a pentaerythritol tetranitrate (PETN) crystal was investigated by treating the surface diffusion as an activated process in the formalism of transition state theory. In particular, surface diffusion processes on the (110) and (101) facets, as well as diffusion between these facets, were considered. The authors were able to obtain the potential energy barriers required for PETN surface diffusion. Their results show that the (110) surface is more thermally active than the (101) surface and PETN molecules mainly diffuse from the (110) to (101) facet. These results are in good agreement with experimental observations and previous simulations.

Efficient Computation of Transition State Resonances and Reaction Rates from a Quantum Normal Form

A quantum version of a recent formulation of transition state theory in phase space is presented. The theory developed provides an algorithm to compute quantum reaction rates and the associated Gamov-Siegert resonances with very high accuracy. The algorithm is especially efficient for multi-degree-of-freedom systems where other approaches are no longer feasible.

A formula to compute the microcanonical volume of reactive initial conditions in transition state theory

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.

Reaction rate theory: What it was, where is it today, and where is it going?

A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.

Thermal Rate Coefficients via Variational Transition State Theory for the Unimolecular Decomposition/Isomerization of 1-Pentyl Radical: Ab Initio and Direct Dynamics Calculations

Calculations of thermal rates for the reactions of the isomeric pentyl radicals involving (1,2), (1,3), (1,4), and (1,5) intramolecular H-atom transfer, C−C bond scission, and H-atom elimination have been carried out. Potential energy surfaces and associated properties for these reactions have been used for direct dynamics studies within conventional and variational transition state theory formalism including nonclassical effects, using the dual-level technique (PUMP-SAC2/6-311G**///AM1). We found that for C−C scission, the barrier is broad, and a significant tightening of the loose transition state reduces the rate coefficients across a wide temperature range. Converse behavior is predicted for the isomerization reactions where the optimal combination of a low effective mass with a narrow barrier opens the best tunneling paths. High-pressure limiting rate coefficients and kinetic parameters obtained in this study show good agreement with experimental measurements and previous theoretical work.

Reaction mechanism and isotope effects derived from centroid transition state theory in intramolecular proton transfer reactions

In this article the tautomerization reaction of the enol form of malonaldehyde is used to investigate the magnitude and origin of changes in centroid transition state theory proton transfer reaction rate predictions caused by the quantum dispersion of heavy nuclei. Using an empirical valence bond method to construct the potential energy surface, it is found that quantization of the nuclear degrees of freedom of the carbon atoms significantly influences the centroid potential of mean force used to describe the proton transfer reaction. In contrast, an ab initio simulation carried out using a recently developed molecular mechanics based importance sampling method [J. Chem. Phys. 114, 6763 (2001)] in combination with an accurate density functional theory evaluation of the electronic energies shows a substantially smaller influence of the quantum nuclear degrees of freedom of the secondary atoms on the centroid potential of mean force. A detailed analysis of the different influence of quantization of the nuclear degrees of freedom of secondary atoms observed in the ab initio and empirical valence bond centroid potential of mean force was carried out. It is shown that for the empirical valence bond potential, a significant decrease of the centroid potential of mean force arises through the quantum tunneling of carbon atoms in the molecular backbone. Furthermore, it is demonstrated that in molecular mechanics potentials aimed to describe intramolecular proton transfer reactions, the functional form of the potential energy terms coupling the primary and secondary atom motions as the reaction proceeds as well as the mass of the primary particle can significantly influence the centroid transition state theory predictions of secondary kinetic isotope effects. Finally, the dependence of the reaction rate predictions and isotope effects on the choice of reaction coordinate is investigated and the validity of calculating kinetic isotope effects using the centroid transition state theory formalism is discussed.

Prediction of Absolute Rate Constants for the Reactions of NH2 with Alkanes from ab Initio G2M/TST Calculations

Systematic ab initio calculations of potential energy surfaces for the reactions of NH{sub 2} with various alkanes (CH{sub 4}, C{sub 2}H{sub 6}, C{sub 3}H{sub 8}, and i-C{sub 4}H{sub 10}) which involve abstraction of a hydrogen atom from primary, secondary, and tertiary C-H bonds have been performed using the G2M method. The calculated activation barrier for the NH{sub 2} + CH{sub 4} reaction, 15.2 kcal/mol, is higher than those for the H-abstraction from a primary C-H bond in C{sub 2}H{sub 6}, C{sub 3}H{sub 8}, and i-C{sub 4}H{sub 10}, 11--12 kcal/mol. The barrier height decreases to 8.4 and 8.3 kcal/mol for the abstraction from a secondary C-H bond in C{sub 3}H{sub 8} and a tertiary C-H bond in i-C{sub 4}H{sub 10}, respectively, in line with the weakening strength of the C-H bond and the increase of the reaction exothermicity. The G2M energies and the molecular and transition-state parameters are used to compute thermal reaction rate constants within the transition-state theory formalism with tunneling corrections. A good agreement of the theoretical rate constants with the experimental is found if the computed barriers are adjusted by 0.5--2 kcal/mol, which is within the accuracy of the G2M method. The H-abstraction from the tertiary C-H bondmore » is shown to be faster than the other considered reactions at T {le} 2000 K, while the secondary H-abstraction is the second fastest reaction at T {le} 1600 K. The rate of the primary H-abstraction decreases with the increase of the alkane size, from ethane to propane and to isobutane. The calculated rate constants for the H-abstraction by NH{sub 2} from primary, secondary, and tertiary C-H bonds can serve as models for the reactions of the amino radical with various saturated hydrocarbons.« less

Pressure dependence of amide hydrogen-deuterium exchange rates for individual sites in T4 lysozyme.

We report measurements of the pressure dependence of rate constants for the exchange of amide residue protons with solvent deuterium for T4 lysozyme. Data obtained at nine pressures from 0.1 to 200 MPa are analyzed using an elementary kinetic model and the formalism of transition state theory which yield activation volumes for the exchange process. Resolution of individual amide sites was accomplished using the HSQC two-dimensional (2D) NMR experiment on uniformly (15)N-labeled protein. The observed activation volumes span the range from 2.75 to -25.1 mL/mol at 22 degreesC and pH* 7.5. When corrected for the pressure dependence of the ionic product for water and for the reported activation volume for the amide exchange reaction in model compounds, the portion of the activation volume associated with the accessibility of the solvent or catalyst to the amide sites ranges from -15.1 to 12.8 mL/mol. There is no simple correlation between the activation volumes and the protection factors for amide hydrogen exchange. The activation volumes for residues in close proximity in either the primary sequence or the folded structure may differ considerably. There is no trivial correlation between the activation volume and the secondary structural unit in which a residue is located, and activation volumes for residues that are apparently structurally coupled may be very different. The modest sizes of the activation volumes obtained under these conditions are in contrast to large values reported for bovine pancreatic trypsin inhibitor at more extreme conditions of 60 degreesC and pH* 8 where major unfolding events or structural rearrangements may dominate the mechanism [Wagner, G. (1983) Q. Rev. Biophys. 16, 1-57].

Ab initio study of solvent kinetic isotope effects in microhydrated S N2 reactions

The solvent kinetic isotope effects in the prototype S N2 reaction of methyl chloride with a chloride ion in the presence of each of one, two, three or four water molecule(s) were examined within transition-state theory formalism on the basis of ab initio calculations. The rate ratios k H k D were found to be less than 1 for the monohydrated system and greater than 1 for the system with two waters and to increase with an increase in the number of the microsolvated waters due to the breaking of the hydrogen bonds at the transition state.

Thermal and vibrational-state selected rates of the CH4+Cl↔HCl+CH3 reaction

We present direct ab initio dynamics studies of thermal and vibrational-state selected rates of the hydrogen abstraction CH4+Cl↔CH3+HCl reaction. Rate constants were calculated within the canonical variational transition state theory formalism augmented by multidimensional semiclassical tunneling corrections. A vibrational diabatic model was used for vibrational-state selected rate calculations, particularly for exciting the CH4 symmetric stretching and umbrella bending modes. The potential energy information was calculated by a combined density functional and molecular orbital approach. Becke’s half-and-half (BH&amp;H) nonlocal exchange and Lee–Yang–Parr (LYP) nonlocal correlation functionals (BH&amp;HLYP) were used with the 6-311G(d,p) basis set for determining structures and frequencies at the stationary points and along the minimum energy path (MEP). Energetics information was further improved by a series of single point spin-projected fourth-order Mo/ller–Plesset perturbation theory (PMP4(SDTQ)) calculations using the 6-311+G(2df,2pd) basis set. We found that the calculated thermal rate constants have reasonable agreement with experimental results for both the forward and reverse reactions. Our results also predict that exciting the CH4symmetric stretching mode will greatly enhance the hydrogen atom transfer rate. Surprisingly, exciting the CH4 umbrella bend mode is also predicted to have a noticeable enhancement factor at room temperature.

Pairwise solute–solute interaction parameters and their application to solvent and salt effects on the kinetics of chemical reactions in aqueous solutions

Dependences on the nature and molalities of added solutes of rate constants for reactions in aqueous solutions have been examined, using the formalism of transition-state theory, in terms of the corresponding dependences of activity coefficients for solutes. Activity coefficients for neutral reactants and transition states are discussed in terms of pairwise solute–solute interactions and then in terms of group interaction parameters using the Savage–Wood additivity of groups (SWAG) procedures. The treatment is extended to a consideration of trends in enthalpies of activation in terms of enthalpic interaction parameters. Systems are grouped into four classes on the basis of changes in hydrophobic and hydrophilic character of substrates on proceeding from initial to transition states and on the basis of the hydrophobic and hydrophylic character of added solutes. In the case of reactions between hydroxide and bromophenol blue anions the dependence of kinetic parameters on added salt is understood, using Pitzer's equations, in terms of charge-charge and pairwise ion-ion interactions, the latter accounting for specific salt effects.

A semi-empirical method for the estimation of activation barriers and kinetic parameters of atom transfer reactions

The activation barriers of atom transfer reactions AB + C → A + BC have been estimated using the empirical Morse functions of the bonds AB and BC and the structural parameters of the activated complex, obtained at various approximation levels. Further, the rate constants and activation energies of a series of hydrogen transfer reactions have been calculated using the transition state theory formalism, including an Eckart tunnelling correction, and ab initio or standard structural parameters and vibrational frequencies. In general, the results obtained are fairly satisfactory. Moreover, it was shown that the barrier heights of hydrogen abstraction reactions linearly depend on the corresponding energy changes. Thus, these processes appear to be essentially under thermodynamic control. Finally, the effects of substituents on the barrier heights of hydrogen abstraction reactions from methane derivatives were rationalized. It appears that, whatever the attacking species (C) may be, the lowest barriers correspond always to high stabilized carbon centred radicals (A) and (or) very destabilized molecules (AB).

From rate constants to potential energy hypersurfaces through the transition state theory formalism. Evaluation of the activation barriers of some hydrogen abstraction reactions

A procedure for extracting various properties of potential energy hypersurfaces, such as the activation barrier ▵ E ≠ and the imaginary frequency ν Im ≠ from experimental rate constants, is described using the formalism of the transition state theory including an Eckart tunnelling correction. The procedure is applied to three hydrogen transfer reactions. It is found that reliable activation barriers can be ontained provided that accurate experimental rate constants are available. Moreover the non-Arrhenius behaviour of the hydrogen abstraction reactions considered in this work can, at least partly, be interpreted in terms of tunnelling of H atom through the potential barrier.

Étude théorique des réactions d'abstraction d'hydrogène , avec R, X≡H, CH3, NH2,OH et F

Hydrogen abstraction reactions [Formula: see text] with R, X≡H, CH3, NH2,OH, and F have been studied at the abinitio 6-31G – UHF level. However, energetic properties were computed at the CI level. Rate constants and Arrhenius parameters have been obtained using the transition state theory formalism with Eckart's tunneling correction. The discrepancy between theoretical and experimental results led us to elaborate a semi-empirical procedure to calculate activation barriers, in which the bonds R—H and X—H are represented by Morse curves. Thus, the agreement between theory and experiment is much better. Moreover, the results obtained by this procedure demonstrate the non-Arrhenius behavior of all the reactions under consideration and allow us to rationalize a large number of experimental facts.

On the correlation between diffusion and self-diffusion processes of adsorbed molecules in a simple microkinetic model

Combining the formalism of transition state theory with an appropriate collisional term, a general relation between diffusion and self-diffusion coefficients ( D and D ∗ respectively) in zeolites is derived. For low pore filling factors, this relation reduces to Darken's equation, which is shown to coincide in this case with the simple equation D = ( d dc )(D ∗c) , where c denotes the concentration of adsorbate. Concerning the intracrystalline self-diffusion of ethane in NaCaA zeolites, it is shown, that up to pore filling factors θ ≈ 0.8, Darken's equation should be valid, whereas for higher concentrations considerable deviations from this simple relation may occur and the general formula, as derived in this paper, should be applied.

Formulation of transition state theory for a reaction proceeding by simultaneous mechanisms with a common transition state. II. Propriety of the analysis

Formulation of transition state theory for a reaction proceeding by simultaneous mechanisms with a common transition state. II. Propriety of the analysis