Barrierless Recombination Research Articles

Comprehensive computational automated search of barrierless reactions leading to the formation of benzene and other C6-membered rings.

We present the systematic exploration of various potential energy surfaces for systems with C6H6-x (x = 0, 1, 2, or 3) empirical formula using an automatic search approach. The primary objective of this study is to identify reaction pathways that lead to the creation of benzene, o-benzyne, and other rings. These pathways initiate with a barrierless recombination reaction and involve subsequent isomerization reactions with submerged transition states until the final product is reached. The reported reaction profiles are consistent with the existing conditions in the interstellar medium if the hot complex formed can cool down through radiative relaxation. Recent studies on the deactivation of polyaromatic hydrocarbons (PAHs) support the possibility of these reactions taking place. The C6-membered rings are considered precursors of PAHs, and our focus is on identifying pathways originating from the barrierless recombination of reactive molecules known to exist in the interstellar medium, with potential implications in other environments.

Theoretical and kinetic study of the thermal decomposition mechanism of long chain aldehydes

The mechanism of decomposition of the butanal and pentanal long chain aldehydes was investigated using the ab initio transition state theory-based master equation approach. The rate constants of the homolytic bond scission reactions of butanal and pentanol leading to the formation of CHO+C3H7, CH2CHO+C2H5, C2H4CHO+CH3, C3H7CO+H, C2H5ĊHCHO+H and CH2CHO+C3H7, C2H4CHO+C2H5, C3H6CHO+CH3, C4H9CO+H, C3H7ĊHCHO+H, respectively, as well as of the reverse barrierless recombination channels, were determined using Variable Reaction Coordinate Transition State Theory. The potential energy surfaces were determined at the multireference CASPT2/aug-cc-pVTZ level on ωB97X-D/jun-cc-pVTZ structures, while stochastic samplings were performed at the CASPT2 level using a (4e,4o) active space. Rate laws for the alkyl+R-CHO, H+R-ĊHCHO, and H+R-ĊO classes of reactions were then determined. Rate constants of H-abstraction reactions and of the three body butanal →CO+H2+C3H6 channel were determined using ωB97X-D/jun-cc-pVTZ structures and Hessians and CCSD(T) energies, extrapolated to the complete basis set. It was found that the calculated homolytic bond scission reactions are faster by a factor of 2–3 with respect to literature previous estimates. The calculated reaction rates were inserted in the CRECK kinetic scheme and used to simulate literature speciation data of butanal and pentanal pyrolysis in shock tubes and ignition delay times, finding that their inclusion into the kinetic scheme enhances the predictive capabilities of the mechanism. The aldehyde decomposition mechanism and the rate constants determined in this study are suitable to be used to interpret and predict the reactivity of aldehydes in a wide range of temperatures and pressures.

Formation of methylglyoxal (CH3C(O)CHO) in interstellar analog ices - a key intermediate in cellular metabolism.

Ketoaldehydes are key intermediates in biochemical processes including carbohydrate, lipid, and amino acid metabolism. Despite their crucial role in the interstellar synthesis of essential biomolecules necessary for the Origins of Life, their formation mechanisms have largely remained elusive. Here, we report the first bottom-up formation of methylglyoxal (CH3C(O)CHO)-the simplest ketoaldehyde-through the barrierless recombination of the formyl (HĊO) radical with the acetyl (CH3ĊO) radical in low-temperature interstellar ice analogs upon exposure to energetic irradiation as proxies of galactic cosmic rays. Utilizing vacuum ultraviolet photoionization reflectron time-of-flight mass spectrometry and isotopic substitution studies, methylglyoxal and its enol tautomer 2-hydroxypropenone (CH3C(OH)CO) were identified in the gas phase during the temperature-programmed desorption of irradiated carbon monoxide-acetaldehyde (CO-CH3CHO) ices, suggesting their potential as promising candidates for future astronomical searches. Once synthesized in cold molecular clouds, methylglyoxal can serve as a key precursor to sugars, sugar acids, and amino acids. Furthermore, this work provides the first experimental evidence for tautomerization of a ketoaldehyde in interstellar ice analogs, advancing our fundamental knowledge of how ketoaldehydes and their enol tautomers can be synthesized in deep space.

Spin-Forbidden Carbon-Carbon Bond Formation in Vibrationally Excited α-CO.

Fourier transform infrared spectroscopy of laser-irradiated cryogenic crystals shows that vibrational excitation of CO leads to the production of equal amounts of CO2 and C3O2. The reaction mechanism is explored using electronic structure calculations, demonstrating that the lowest-energy pathway involves a spin-forbidden reaction of (CO)2 yielding C(3P) + CO2. C(3P) then undergoes barrierless recombination with two other CO molecules forming C3O2. Calculated intersystem crossing rates support the spin-forbidden mechanism, showing subpicosecond spin-flipping time scales for a (CO)2 geometry that is energetically consistent with states accessed through vibrational energy pooling. This spin-flip occurs with an estimated ∼4% efficiency; on the singlet surface, (CO)2 reconverts back to CO monomers, releasing heat which induces CO desorption. The discovery that vibrational excitation of condensed-phase CO leads to spin-forbidden C–C bond formation may be important to the development of accurate models of interstellar chemistry.

Unconventional excited-state dynamics in the concerted benzyl (C7H7) radical self-reaction to anthracene (C14H10)

Polycyclic aromatic hydrocarbons (PAHs) are prevalent in deep space and on Earth as products in combustion processes bearing direct relevance to energy efficiency and environmental remediation. Reactions between hydrocarbon radicals in particular have been invoked as critical molecular mass growth processes toward cyclization leading to these PAHs. However, the mechanism of the formation of PAHs through radical – radical reactions are largely elusive. Here, we report on a combined computational and experimental study of the benzyl (C7H7) radical self-reaction to phenanthrene and anthracene (C14H10) through unconventional, isomer-selective excited state dynamics. Whereas phenanthrene formation is initiated via a barrierless recombination of two benzyl radicals on the singlet ground state surface, formation of anthracene commences through an exotic transition state on the excited state triplet surface through cycloaddition. Our findings challenge conventional wisdom that PAH formation via radical-radical reactions solely operates on electronic ground state surfaces and open up a previously overlooked avenue for a more “rapid” synthesis of aromatic, multi-ringed structures via excited state dynamics in the gas phase.

Formation of Phenanthrene via Recombination of Indenyl and Cyclopentadienyl Radicals: A Theoretical Study.

This work presents quantum chemical G3(MP2,CC)//B2PLYPD3/6-311G(d,p) calculations of the potential energy surface for the indenyl (C9H7) + cyclopentadienyl (C5H5) reaction followed by unimolecular decomposition of the C14H11 radicals formed as the primary products, as well as the Rice-Ramsperger-Kassel-Marcus master equation (RRKM-ME) calculations to predict temperature- and pressure-dependent reaction rate constants and product branching ratios. The reaction begins with the barrierless recombination of indenyl and cyclopentadienyl forming a C14H12 molecule with a new C-C bond connecting two five-membered rings, which subsequently dissociates to C14H11 radicals by H losses. The primary products of the C9H7 + C5H5 → C14H11 + H reaction can directly decompose by another H loss to benzofulvalene, and this pathway is most favorable in terms of the entropy factor and hence is preferable at higher temperatures. Otherwise, the initial C14H11 isomers can undergo significant structural rearrangements before eliminating an H atom and producing phenanthrene, anthracene, or benzoazulenes, among which the formation of phenanthrene via the "spiran" pathway is clearly preferred. The calculated barriers along the computed favorable dissociation pathways are relatively low, in the ∼30-40 kcal/mol range, making the C14H11 radicals unstable at temperatures above 1000-1250 K at 1 atm. The results of RRKM-ME calculations show that, under typical combustion conditions, the decomposition of the C14H11 radicals predominantly leads to benzofulvalene. However, the latter can be rapidly converted to phenanthrene via H-assisted isomerization with the rate constant for the benzofulvalene + H → phenanthrene + H reaction being close to 10-11 cm3 molecule-1 s-1 at 1000-1500 K and 1 atm. The results provide further support for the hypothesis that recombination of two π radicals containing five-membered rings can lead to a growth of PAH with the formation of two fused six-membered rings, but the reaction mechanism may not be direct and is likely to involve two consecutive H atom losses leading to a fulvalene-like product, with subsequent H-assisted isomerization of the latter to a benzenoid PAH.

Interstellar Formation of Biorelevant Pyruvic Acid (CH3COCOOH)

<h2>Summary</h2> Pyruvic acid represents a key molecule in prebiotic chemistry to form metabolites and amino acids. Without liquid water on the early Earth, endogenous formation of pyruvic acid is unlikely, and an exogenous delivery constitutes an appealing alternative. However, despite the detection of more than 200 molecules in space, pyruvic acid is elusive. Here, we describe its formation by barrierless recombination of hydroxycarbonyl (HOCO⋅) and acetyl (CH₃CO⋅) radicals in ices of acetaldehyde (CH₃CHO) and carbon dioxide (CO₂) modeling interstellar conditions driven by cosmic rays. Exploiting isotopically labeled ices and photoionization reflectron time-of-flight mass spectrometry, the reaction products were selectively photoionized in the temperature-programmed desorption phase and isomers discriminated based on their ionization energies. This reveals a key reaction pathway for pyruvic acid synthesis through non-equilibrium reactions in interstellar cold molecular clouds and star-forming regions, thus offering a unique entry point to abiotic organic synthesis in deep space.

Interstellar Formation of Biorelevant Pyruvic Acid (CH &lt;sub&gt;3&lt;/sub&gt;COCOOH)

Pyruvic acid represents a key molecular building block in prebiotic chemistry to form metabolites and amino acids. As conditions on early Earth were not favorable for an endogenous formation of pyruvic acid, an exogenous delivery by comets and meteorites constitutes an appealing alternative. However, despite the detection of more than 200 molecules in interstellar and circumstellar environments, pyruvic acid has yet remained elusive. Here we report on the formation of pyruvic acid by barrierless recombination of hydroxycarbonyl (HOCO) and acetyl (CH3CO) radicals in interstellar ices modelling interstellar conditions of acetaldehyde (CH3CHO) and carbon dioxide (CO2) driven by galactic cosmic rays. Exploiting isotopically labelled ices and photoionization reflectron time-of-flight mass spectrometry, the subliming neutral reaction products were selectively photoionized in the temperature programmed desorption phase and isomers discriminated based on their distinct ionization energies. These results reveal a key reaction pathway for the synthesis of pyruvic acid through non-equilibrium reactions in interstellar ices in cold molecular clouds like TMC-1 and star forming regions such as the Orion Molecular Cloud complex (OMC-1), thus offering a unique entry point to abiotic organic synthesis in deep space.

Formation of Glyoxylic Acid in Interstellar Ices: A Key Entry Point for Prebiotic Chemistry.

With nearly 200 molecules detected in interstellar and circumstellar environments, the identification of the biologically relevant α-keto carboxylic acid, glyoxylic acid (HCOCOOH), is still elusive. Herein, the formation of glyoxylic acid via cosmic-ray driven, non-equilibrium chemistry in polar interstellar ices of carbon monoxide (CO) and water (H2 O) at 5 K via barrierless recombination of formyl (HCO) and hydroxycarbonyl radicals (HOCO) is reported. In temperature-programmed desorption experiments, the subliming neutral molecules were selectively photoionized and identified based on the ionization energy and distinct mass-to-charge ratios in combination with isotopically labeled experiments exploiting reflectron time-of-flight mass spectrometry. These studies unravel a key reaction path to glyoxylic acid, an organic molecule formed in interstellar ices before subliming in star-forming regions like SgrB2(N), thus providing a critical entry point to prebiotic organic synthesis.

Formation of Glyoxylic Acid in Interstellar Ices: A Key Entry Point for Prebiotic Chemistry

AbstractWith nearly 200 molecules detected in interstellar and circumstellar environments, the identification of the biologically relevant α‐keto carboxylic acid, glyoxylic acid (HCOCOOH), is still elusive. Herein, the formation of glyoxylic acid via cosmic‐ray driven, non‐equilibrium chemistry in polar interstellar ices of carbon monoxide (CO) and water (H2O) at 5 K via barrierless recombination of formyl (HCO) and hydroxycarbonyl radicals (HOCO) is reported. In temperature‐programmed desorption experiments, the subliming neutral molecules were selectively photoionized and identified based on the ionization energy and distinct mass‐to‐charge ratios in combination with isotopically labeled experiments exploiting reflectron time‐of‐flight mass spectrometry. These studies unravel a key reaction path to glyoxylic acid, an organic molecule formed in interstellar ices before subliming in star‐forming regions like SgrB2(N), thus providing a critical entry point to prebiotic organic synthesis.

Can the C5H5 + C5H5 → C10H10 → C10H9 + H/C10H8 + H2 Reaction Produce Naphthalene? An Ab Initio/RRKM Study

Ab initio and density functional calculations using a variety of theoretical methods (CASSCF, B3LYP, CASPT2, CCSD(T), and G3(MP2,CC)) have been carried out to unravel the mechanism of unimolecular isomerization and dissociation of 9,10-dihydrofulvalene C(10)H(10) (S0) formed by barrierless recombination of two cyclopentadienyl radicals. Different reaction pathways on the C(10)H(10) potential energy surface (PES) are found to lead to the production of 9-H-fulvalenyl radical + H, 9-H-naphthyl radical (a naphthalene precursor) + H, and naphthalene + H(2). RRKM calculations of thermal rate constants and product branching ratios at the high pressure limit show that at temperatures relevant to combustion the 9-H-fulvalenyl radical formed by a direct H loss from S0 with endothermicity of 76.3 kcal/mol is expected to be the dominant reaction product. The naphthalene precursor 9,10-dihydronaphthalene (D3) can be produced from the initial S0 adduct by a multistep diradical mechanism involving the formation of a metastable tricyclic diradical intermediate, followed by its three-step opening to a 10-member ring structure, which then undergoes ring contraction producing the naphthalene core structure in D3, with the highest barrier on this pathway being 70.3 kcal/mol. D3 can lose molecular hydrogen producing naphthalene via a barrier of 77.7 kcal/mol relative to the initial adduct. Another possibility is a hydrogen atom elimination in D3 giving rise to the 9-H-naphthyl radical without exit barrier and with overall endothermicity of 59.2 kcal/mol. The pathway to 9-H-naphthyl appears to be preferable as compared to the direct route to 9-H-fulvalenyl at temperatures below 600 K, but the rate constants at these temperatures are too slow for the reaction to be significant. The naphthalene + H(2) channel is not viable at any temperature. The following reaction sequence is suggested for kinetic models to account for the recombination of two cyclopentadienyl radicals: c-C5H5+c-C5H5-->9,10-dihydrofulvalene-->9-H-fulvalenyl+H(C10H10PES), 9-H-fulvalenyl-->naphthalene+H/fulvalene+H(C10H9PES). We conclude that naphthalene can be produced from the recombination of two cyclopentadienyl radicals and is expected to be a favorable product of this reaction sequence at T < 1000 K, but this molecule would be formed through isomerizations and H atom loss on the C(10)H(9) PES (after the initial H elimination from C(10)H(10) S0) and not in conjunction with molecular hydrogen. The alternative product, fulvalene, can potentially contribute to the growth of cyclopentafused polycyclic aromatic hydrocarbons.

Global analysis of reaction pathways on the potential energy surface of cyanoacetylene by the scaled hypersphere search method

The potential energy surfaces involving 10 singlet minima and 28 transition states for the isomerization pathways of the HC 3N isomers are explored by the scaled hypersphere search method at ab initio levels. In addition to seven known species, three new cyclic isomers, bicyclic-CCCN–H, C–cyclic-CCN–H, and cyclic-CCCN–H, are predicted at the CCSD(T)/cc-pVTZ//B3LYP/6-311++G(d, p)+ZPE level. They can interconvert to each other, isomerize to other low-lying isomers, or decompose into other fragments easily. All isomers can be considered as adducts via barrierless recombination channels between various fragments detected in interstellar space or favorable hydrogen-adding processes.

Prediction of Reduced Falloff Curves for Recombination Reactions at Low Temperatures

AbstractStrong collision falloff curves for barrierless recombination reactions at low temperatures are calculated taking into account transitional modes only. Specific rate constantsk(E,J) from statistical adiabatic channel/classical trajectory (SACM/CT) calculations are used. Broadening factors of the falloff curves are found to depend only weakly on the temperature. A systematic analysis of the influence of the centrifugal barriersE0(J), which are governed by the potential energy surface of the bond energiesE0, and of the number of transitional modes is made. Guidelines for estimating center broadening factors and shape functions for the broadening factors are given.

A theoretical study on the mechanism of camphor hydroxylation by compound I of cytochrome P450

Mechanistic and energetic aspects for the conversion of camphor to 5-exo-hydroxycamphor by the compound I iron-oxo species of cytochrome P450 are discussed from B3LYP DFT calculations. This reaction occurs in a two-step manner along the lines that the oxygen rebound mechanism suggests. The activation energy for the first transition state of the H atom abstraction at the C5 atom of camphor is computed to be more than 20 kcal/mol. This H atom abstraction is the rate-determining step in this hydroxylation reaction, leading to a reaction intermediate that involves a carbon radical species and the iron-hydroxo species. The second transition state of the rebound step that connects the reaction intermediate and the product alcohol complex lies a few kcal/mol below that for the H atom abstraction on the doublet and quartet potential energy surfaces. This energetic feature allows the virtually barrierless recombination in both spin states, being consistent with experimentally observed high stereoselectivity and br...

A theoretical study on the mechanism of camphor hydroxylation by compound I of cytochrome p450.

Mechanistic and energetic aspects for the conversion of camphor to 5-exo-hydroxycamphor by the compound I iron-oxo species of cytochrome P450 are discussed from B3LYP DFT calculations. This reaction occurs in a two-step manner along the lines that the oxygen rebound mechanism suggests. The activation energy for the first transition state of the H atom abstraction at the C5 atom of camphor is computed to be more than 20 kcal/mol. This H atom abstraction is the rate-determining step in this hydroxylation reaction, leading to a reaction intermediate that involves a carbon radical species and the iron-hydroxo species. The second transition state of the rebound step that connects the reaction intermediate and the product alcohol complex lies a few kcal/mol below that for the H atom abstraction on the doublet and quartet potential energy surfaces. This energetic feature allows the virtually barrierless recombination in both spin states, being consistent with experimentally observed high stereoselectivity and brief lifetimes of the reaction intermediate. The overall energetic profile of the catalytic mechanism of camphor hydroxylation particularly with respect to why the high activation energy for the H atom abstraction is accessible under physiological conditions is also considered and calculated. According to a proton source model involving Thr252, Asp251, and two solvent water molecules (Biochemistry 1998, 37, 9211), the energetics for the conversion of the iron-peroxo species to compound I is studied. A significant energy over 50 kcal/mol is released in the course of this dioxygen activation process. The energy released in this chemical process is an important driving force in alkane hydroxylation by cytochrome P450. This energy is used for the access to the high activation energy for the H atom abstraction.

Flexible Transition State Theory for a Variable Reaction Coordinate: Analytical Expressions and an Application

Completely general canonical and microcanonical (energy-resolved) flexible transition state theory (FTST) rate constant expressions for an arbitrary choice of reaction coordinate have recently been derived [Robertson et al. J. Chem. Phys. 2000, 113, 2648.] by the present authors. The rate expressions apply to any definition of the separation distance between fragments in a barrierless recombination (or dissociation) that is held fixed during hindered rotations at the transition state, and to any combination of fragment structure (atom, linear top, nonlinear top). The minimization of the rate constant with respect to this definition can be regarded as optimizing the reaction coordinate within a canonical or microcanonical framework. The expressions are analytic, with the exception of a configuration integral whose evaluation generally requires numerical integration over an integrand which depends on internal angles (from one to five depending on the fragment structures). The primary component of the integrand is the determinant of the inverse G-matrix associated with the external rotations and the relative internal rotation of the fragments. In this paper, we derive closed-form, analytic expressions for the inverse G-matrix determinant for all combinations of fragment top types for an arbitrary reaction coordinate definition entirely in terms of kinetic energy matrix elements for a centers-of-mass reaction coordinate. For a model potential for CFH2 + H, the effect of optimizing the reaction coordinate definition is displayed, and the optimized coordinate is compared to the traditional center-of-mass definition at the canonical level. The associated rate constant is about a factor of 20% to 45% lower than that obtained using a centers-of-mass reaction coordinate.

The challenges of combustion for chemical theory

A variety of studies drawn from the thermochemistry and kinetics of combustion are used to assess both recent advances and continuing challenges in theoretical chemistry. In all studies examined, the underlying molecular interactions as expressed by the potential energy surface are emphasized. In thermochemistry, very high level calculations on H2O and OH show the absolute limits of accuracy (0.1–0.2 kcal/mol) that can be currently achieved by ab initio techniques. On the other hand, high-level semiempirical Gn calculations, whose parameters are specifically turned to only a few members of a family of molecules, can in some circumstances achieve errors only a few times larger for a long sequence of family members. In reaction kinetics, for reactions with barriers, the tuning of high-level semiempirical theories to the barriers of reaction families are just beginning and several successes and failures will be discussed. For barrierless recombination reactions, detailed theoretical studies of six reactions (CH3+CH3, H+CH3, H+C2H3, H+C2H5, H+C3H3 and H+NO2) during the past few years are compared with each other and with experiment. The residual disagreements are driving more detailed analysis of the limitations of theoretical and experimental techniques. The recent experimental/theoretical efforts on a seventh reaction (CH3+O) confirm that, for a variety of addition-elimination reactions, substantial amounts of reaction products can not be described in terms of reaction paths. This discovery and its implications are discussed. In inelastic, collisional energy transfer kinetics, recent experimental and theoretical studies on H+O2 recombination with H2O as the bath gas emphasize the importance of electrostatic interactions in enhancing collisional energy transfer with the bath gas.

Kinetic Isotope Effects and Variable Reaction Coordinates in Barrierless Recombination Reactions

The factors affecting kinetic isotope effects in barrierless recombination reactions are considered from the perspective of variational transition state theory (VTST). Despite the broad application of VTST methods, a general consideration of kinetic isotope effect predictions of the theory has not previously been undertaken, especially for cases where changes in the internal structure and vibrational frequencies of the fragments (i.e., the conserved modes) can be assumed to be negligible. Use of the center-of-mass separation as the reaction coordinate in such a case entails some restriction on the range of kinetic isotope effects which can be accommodated. Larger effects are possible within a variable reaction coordinate implementation of transition state theory, and the predicted kinetic isotope effects are shown to be strongly dependent on the location of the pivot point. Illustrative model calculations demonstrate the feasibility of reproducing the experimentally observed kinetic isotope effects for the CH + O2, HCC + O2, CH + C2H2, and CH + C2H4 reactions with realistic deviations of the pivot points from the center-of-mass. In contrast, calculations restricted to center-of-mass pivot points predict isotope effects that are even inverted. For the CH + CH4 reaction, the isotope effects appear too large to be explained by the reaction coordinate variations, and changes in the conserved modes play a key role in the observed isotope effects, as demonstrated with ab initio based TST simulations. Overall, the experimentally observed kinetic isotope effects in CH addition reactions are strongly suggestive of an optimum reaction coordinate corresponding to a pivot point located near the center of the radical orbital.

Flexible transition state theory for a variable reaction coordinate: Derivation of canonical and microcanonical forms

A completely general canonical and microcanonical (energy-resolved) flexible transition state theory (FTST) expression for the rate constant is derived for an arbitrary choice of reaction coordinate. The derivation is thorough and rigorous within the framework of FTST and replaces our previous treatments [Robertson et al., J. Chem. Phys. 103, 2917 (1995); Robertson et al., Faraday Discuss. Chem. Soc. 102, 65 (1995)] which implicitly involved some significant assumptions. The canonical rate expressions obtained here agree with our earlier results. The corresponding microcanonical results are new. The rate expressions apply to any definition of the separation distance between fragments in a barrierless recombination (or dissociation) that is held fixed during hindered rotations at the transition state, and to any combination of fragment structure (atom, linear top, nonlinear top). The minimization of the rate constant with respect to this definition can be regarded as optimizing the reaction coordinate within a canonical or microcanonical framework. The expression is analytic except for a configuration integral whose evaluation generally requires numerical integration over internal angles (from one to five depending on the fragment structures). The form of the integrand in this integral has important conceptual and computational implications. The primary component of the integrand is the determinant of the inverse G-matrix associated with the external rotations and the relative internal motion of the fragments.

Flux Factors in Variational Transition-State Theory

The role of the flux, or velocity, factor in variational transition-state theory (VTST) with a generalized reaction coordinate is examined within the context of barrierless recombination reactions. It is found that well-behaved approximations to the rate coefficient can be obtained by allowing more flexibility in the choice of the flux factor. Transition-state theory traditionally prescribes a flux factor which is the velocity perpendicular to the dividing surface. We examine two models for the application of VTST to barrierless recombination reactions with a bond-length reaction coordinate: one in which the flux factor is the rate of change of the bond length (i.e., perpendicular to the dividing surface) and the other in which the flux factor is the rate of change of the separation between the fragment centers of mass. The bond-length flux factor produces results in agreement with those of Klippenstein's model [J. Chem. Phys. 1991, 94, 6469], reducing the computation time by approximately 2 orders of magnitude. The center-of-mass flux factor, which can be implemented variationally, also produces excellent results and has the additional advantage of correlating correctly to the number of asymptotically open channels predicted by phase space theory (PST) at large separation of the fragments, where the variational transition state typically lies for energies close to threshold.