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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: III. Vibrational excitation via electronic excitation and radiative decay

We present state-resolved cross sections for vibrational excitation via electronic excitation followed by radiative decay (ERD), for electrons scattering on all bound vibrational levels of the ground electronic state ($X\hspace{1.5 mm} ^1Σ_g^+$) of molecular hydrogen and its isotopologues (H 2 , HD, HT, D 2 , DT and T 2 ). We consider excitation of the singlet $\textit{n}$ = 2–3 states (where n refers to the united-atoms-limit principle quantum number) and account for all possible decay pathways back to the bound vibrational levels of the ground electronic state ($X\hspace{1.5 mm} ^1Σ_g^+$) to produce an estimate for the ERD cross sections. A selection of the results are presented in graphical form and a full data set is provided as both numerical values and analytic fit functions in supplementary data files. The uncertainty in the cross sections is estimated to be 11%, except for scattering on the highest bound vibrational level of HD, HT, D 2 , and DT, where the uncertainty is 21%. The data can be downloaded from the MCCC database at mccc-db.org.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: IV. Vibrationally-resolved ionization of the ground and excited electronic states

Here, we present a comprehensive set of vibrationally-resolved cross sections for electron-impact ionization of molecular hydrogen and its isotopologues (H 2 , D 2 , T 2 , HD, HT, and DT) in both the ground and excited electronic states. We apply the adiabatic-nuclei molecular convergent close-coupling (MCCC) method to calculate cross sections from threshold to 1000 eV for ionization of the ground and excited vibrational levels of the X 1 Σ$^{+}_{g}$, B 1 Σ$^{+}_{u}$, C 1 π u , EF 1 Σ$^{+}_{g}$, a 3 Σ$^{+}_{g}$, and c 3 π u electronic states, representing all states with united-atoms-limit principle quantum number n=1–2. The cross sections are presented in graphical form and provided as both numerical values and analytic fit functions in supplementary data files. The data can also be downloaded from the MCCC database at mccc-db.org.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2024

Resistive Coatings for High-performance, Low-background MCPs Operating Across Broad Temperature Ranges and at Cryogenic Temperatures

Microchannel plate with improved thermo-electric properties are a high risk, high payoff development undertaken by a consortium of effort that links the Argonne National Lab, the Space-Science Lab at UC Berkeley and the small businesses (Incom Inc.) that will commercialize the advanced technology in open MCPs and LAPPDs. This development will satisfy the needs for new instrumentation for homeland security (non-proliferation) sensors to screen vehicles and cargo for Special Nuclear Materials (SNMs) and scientific detectors for astrophysics, electron microscopy, time-of-flight mass spectrometry, molecular and atomic collision studies, and fluorescence imaging applications in biotechnology and medical imaging products including positron emission tomography (PET scanning).

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2025

Competing ionization and dissociation: Extension of the energy-dependent frame transformation to the gerade symmetry of H 2

This article solves two major tasks that frequently arise in the theory of electron collisions with a target molecular cation. First, it extends the energy-dependent frame transformation (EDFT) treatment, which is needed to map fixed-nuclei electron-molecule scattering matrices into an energy-dependent laboratory-frame scattering matrix with vibrational channel indices. The EDFT mapping can now be carried out even when the target molecule possesses multiple low-energy potential curves, significantly transcending previous applications. Second, it implements a method to extract the rest of the full laboratory-frame scattering matrix, i.e., the columns and rows describing input and/or output dissociation channels. The treatment is benchmarked in this article against the essentially exact solution of a refined two-dimensional model of the singlet gerade Σ symmetry of H 2 . Our tests demonstrate that the theory accurately maps fixed-nuclei scattering information, of the type provided by existing electron-molecule computer codes, into a laboratory-frame scattering matrix that includes both ionization and dissociation. Furthermore, this treatment can provide a general framework applicable to a broad class of electron collision processes involving diatomic target ions, suitable for an accurate description of challenging processes such as dissociative recombination.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2026

Formation of 1 H -Phenalene (C 13 H 10 ) in the Taurus Molecular Cloud via Methylidyne Addition-Cyclization-Aromatization (MACA)

The formation of 1H-phenalene (C 13 H 10 ) in cold molecular clouds, such as the Taurus Molecular Cloud-1 (TMC-1), presents a significant challenge to traditional astrochemical models, which predominantly suggest high-temperature pathways for polycyclic aromatic hydrocarbon (PAH) formation. In this study, we explore computationally the Methylidyne Addition-Cyclization-Aromatization (MACA) mechanism as a viable, barrierless pathway for phenalene synthesis under low-temperature conditions. Through electronic structure calculations and Rice–Ramsperger–Kassel–Marcus (RRKM) statistical methods, we demonstrate that the reaction of 1-vinylnaphthalene (C 10 H 7 C 2 H 3 ) with the methylidyne radical (CH) leads to the formation of 1H-phenalene via a bimolecular reaction, a process that is exoergic and without entrance barrier. The MACA mechanism facilitates the growth of the aromatic carbon backbone via a [5 + 1] ring annulation, providing a new insight into PAH formation in cold molecular clouds. Notably, the MACA mechanism has previously been shown to form indene (C9H8), which was detected in TMC-1 as well, via a [4 + 1] annulation, demonstrating its potential to produce a variety of complex PAHs by addition of a five- and six-membered ring to a benzene moiety via [4 + 1] and [5 + 1] annulation, respectively. As a result, this work highlights the importance of barrierless, exoergic reactions involving MACA in the synthesis of complex aromatic molecules in space, expanding our physicochemical understanding of carbon-rich chemistry in cold molecular clouds.

Aromatic compounds↗
DOE OSTI2023

Atomistic insight into the effects of electrostatic fields on hydrocarbon reaction kinetics

Reactive Molecular Dynamics (MD) and Density Functional Theory (DFT) computations are performed to provide insight into the effects of external electrostatic fields on hydrocarbon reaction kinetics. By comparing the results from MD and DFT, the suitability of the MD method in modeling electrodynamics is first assessed. Results show that the electric field-induced polarization predicted by the MD charge equilibration method is in good agreement with various DFT charge partitioning schemes. Then, the effects of oriented external electric fields on the transition pathways of non-redox reactions are investigated. Results on the minimum energy path suggest that electric fields can cause catalysis or inhibition of oxidation reactions, whereas pyrolysis reactions are not affected due to the weaker electronegativity of the hydrogen and carbon atoms. MD simulations of isolated reactions show that the reaction kinetics is also affected by applied external Lorentz forces and interatomic Coulomb forces since they can increase or decrease the energy of collision depending on the molecular conformation. In addition, electric fields can affect the kinetics of polar species and force them to align in the direction of field lines. These effects are attributed to energy transfer via intermolecular collisions and stabilization under the external Lorentz force. The kinetics of apolar species is not significantly affected mainly due to the weak induced dipole moment even under strong electric fields. The dynamics and reaction rates of species are studied by means of large-scale combustion simulations of n-dodecane and oxygen mixtures. Results show that under strong electric fields, the fuel, oxidizer, and most product molecules experience translational and rotational acceleration mainly due to close charge transfer along with a reduction in their vibrational energy due to stabilization. This study will serve as a basis to improve the current methods used in MD and to develop novel methodologies for the modeling of macroscale reacting flows under external electrostatic fields.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

A combined experimental and computational study on the reaction dynamics of the 1-propynyl (CH 3 CC, X 2 A 1 ) – propylene (CH 3 CHCH 2 , X 1 A') system: formation of 1,3-dimethylvinylacetylene (CH 3 CCCHCHCH 3 , X 1 A') under single collision conditions

The reaction of the 1-propynyl radical (CH 3 CC; X 2 A 1 ) with propylene (CH 3 CHCH 2 ; X 1 A') was studied in a crossed molecular beam machine at a collision energy of 37 ± 1 kJ mol –1 . Experimental data combined with high-level electronic structure (CCSD(T)-F12/cc-pVTZ-F12//ωB97X-D/6-311G(d,p)) and RRKM calculations reveal the reaction mechanism. Here, the overall barrierless and exoergic reaction involves indirect reaction dynamics and commences preferentially with addition of 1-propynyl with its radical centre to the carbon–carbon double bond at the terminal carbon atom of propylene. This work focuses on molecular mass growth process (hydrogen loss channels) although theory suggests methyl loss as a prevalent channel. In these processes, the C6H9 collision complexes either emit atomic hydrogen or undergo isomerisation followed by atomic hydrogen loss to preferentially yield the cis/trans isomers of 1,3-dimethylvinylacetylene (2-hexen-4-yne) as the primary product. Analysis of reaction dynamics of 1-propynyl and ethynyl radicals with propylene along with their fractional abundance in deep space suggests formation of methyl- and dimethyl derivatives of vinylacetylene in cold molecular clouds. Once formed they may engage in fundamental molecular mass growth processes via the barrierless Hydrogen Abstraction Vinylacetylene Addition mechanism that leads to the formation of methyl- and dimethylnaphthalenes thus providing a versatile route to methyl-substituted PAHs in interstellar medium.

1-propynyl↗
DOE OSTI2021

Searching for Low-mass Population III Stars Disguised as White Dwarfs

It is uncertain whether or not low-mass Population III stars ever existed. While limits on the number density of Population III stars with M {sub *} ≈ 0.8 M {sub ⊙} have been derived, using Sloan Digital Sky Survey (SDSS) data, little is known about the occurrence of Population III stars at lower masses. In the absence of reliable parallaxes, the spectra of metal-poor main-sequence (MPMS) stars with M {sub *} ≲ 0.8 M {sub ⊙} can easily be confused with those of cool white dwarfs. To resolve this ambiguity, in this paper we present a classifier that differentiates between MPMS stars and white dwarfs, based on photometry and/or spectroscopy without the use of parallax information. We build and train our classifier using state-of-the-art theoretical spectra, and evaluate it on existing SDSS-based classifications for objects with reliable Gaia DR2 parallaxes. We then apply our classifier to a large catalog of objects with SDSS photometry and spectroscopy to search for MPMS candidates. We discover several previously unknown, extremely metal-poor (EMP) candidate stars, and recover numerous confirmed EMP stars already noted in the literature. We conclude that archival SDSS spectroscopy has already been exhaustively searched for EMP stars. We predict that the lowest-mass stars of primordial composition will have redder optical-to-infrared colors than cool white dwarfs at constant effective temperature, due to surface gravity-dependent collision-induced absorption from molecular hydrogen. We suggest that the application of our classifier to data produced by next-generation spectroscopic surveys will set stronger constraints on the number density of low-mass Population III stars in the Milky Way.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Know the enemy: 2D Fermi liquids

Highlights: • Regime of validity of the Fermi liquid theory in 2D interacting Fermi systems has been investigated. • Phase diagram for hydrodynamic and ballistic regimes in the frequency–temperature space has been obtained. • Temperature correction to quasiparticle effective mass has been calculated. • 2D Wiedemann–Franz law and 2D Kadawoki–Woods relation have been discussed. We describe an analytical theory investigating the regime of validity of the Fermi liquid theory in interacting, via the long-range Coulomb coupling, two-dimensional Fermi systems comparing it with the corresponding 3D systems. We find that the 2D Fermi liquid theory and 2D quasiparticles are robust up to high energies and temperatures of the order of Fermi energy above the Fermi surface, very similar to the corresponding three-dimensional situation. We calculate the phase diagram in the frequency–temperature space separating the collisionless ballistic regime and the collision-dominated hydrodynamic regime for 2D and 3D interacting electron systems. We also provide the temperature corrections up to third order for the renormalized effective mass, and comment on the validity of 2D Wiedemann–Franz law and 2D Kadawoki–Woods relation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Statistical Learning for Nonlinear Model Reduction from Local Simulations of Stochastic and Particle- and Agent-Based Systems

Stochastic physical systems across the sciences that have very high-dimensional state spaces, with a large number of fast degrees of freedom that force direct simulators to proceed by integration steps that are orders of magnitude smaller than events of interests (e.g., particle collisions). Examples range from molecular motion to dynamics of large populations of cells. A grand challenge in the simulation and understanding of such systems is the systematic construction of accurate, interpretable, reduced models, enabling faster simulations, revealing fundamental properties of the dynamics, and predicting phenomena of interest that the original simulator could not reached with sufficient accuracy or within a given computational budget. In this projected we developed novel statistical estimation/machine learning techniques for analyzing and building empirical reduced models for important families of high-dimensional stochastic systems, in particular: - we developed techniques for estimating interaction kernels in interacting particle- and agent-based systems, which are ubiquitous in Physics, Biology and many other sciences, given observed trajectories of the system; - we developed techniques for nonlinear model reduction for high-dimensional stochastic systems that have a small number of unknown, nonlinear slow variables, and a large number of fast modes, that are possibly of large magnitude, given observed short trajectories of the system in the form of bursts of trajectories from different initial conditions; - we developed novel techniques for estimating linear dynamical systems on graphs when both the dynamics and the underlying graph are unknown, and we have a sparse set of space-time observations; - we considered the problem of estimating an unknown nonlinear observation function of a standard process (e.g. Brownian motion), so that we can recognized if an observed dynamics is "just" a nonlinear version of a known dynamics; we also developed benchmarks for learning algorithms aimed at learning and classifying diffusion processes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Dynamics of collision-induced energy transfer

Collisions are of fundamental importance in many relevant physico-chemical phenomena in the gas phase and at the gas-surface interface. Collision-induced energy transfer not only offers a sensitive probe of interatomic interactions, but also impacts an array of processes ranging from transport to reactivity. This perspective highlights some recent advances on collision-induced energy transfer dynamics. In the gas phase, cold collisions revealed remarkable quantum effects and stereodynamics of bimolecular inelastic scattering, which motivated development of new state-to-state quantum scattering methods. For surface processes, experiments of hydrogen atom scattering from various surfaces combined with first-principles molecular dynamics simulations shed valuable light on the relative importance of adiabatic energy transfer to surface phonons and nonadiabatic energy transfer to surface electrons. We outline further challenges in these fields and expect more fruitful interplay between experiment and theory.

Cold collision↗
DOE OSTI2022

Directed gas phase preparation of ethynylallene (H 2 CCCHCCH; X 1 A') via the crossed molecular beam reaction of the methylidyne radical (CH; X 2 Π) with vinylacetylene (H 2 CCHCCH; X 1 A')

The gas-phase bimolecular reaction of the methylidyne (CH; X 2 Π) radical with vinylacetylene (H 2 CCHCCH; X 1 A') was conducted at a collision energy of 20.3 kJ mol –1 under single collision conditions exploiting the crossed molecular beam experimental results merged with ab initio electronic structure calculations and ab initio molecular dynamics (AIMD) simulations. The laboratory data reveal that the bimolecular reaction proceeds barrierlessly via indirect scattering dynamics through long-lived C 5 H 5 reaction intermediate(s) ultimately dissociating to C 5 H 4 isomers along with atomic hydrogen with the latter predominantly originating from the vinylacetylene reactant as confirmed by the isotopic substitution experiments in the D1-methylidyne–vinylacetylene reaction. Combined with ab initio calculations of the potential energy surface (PES) and statistical Rice–Ramsperger–Kassel–Marcus (RRKM) calculations, the experimental determined reaction energy of –146 ± 26 kJ mol –1 along with the distribution minimum of T(θ) at 90° and isotopic substitution experiments suggest ethynylallene (p1; ΔrG = –230 ± 4 kJ mol –1 ) as the dominant product. The ethynylallene (p1) may be formed with extensive rovibrational excitation, which would result in a lower maximum translational energy. Further, AIMD simulations reveal that the reaction dynamics leads to p1 (ethynylallene, 75%) plus atomic hydrogen with the dominant initial complex being i1 formed by methylidyne radical addition to the double C[double bond, length as m-dash]C bond in vinylacetylene. Altogether, combining the crossed molecular beam experimental results with ab initio electronic structure calculations and ab initio molecular dynamics (AIMD) simulations, ethynylallene (p1) is expected to represent the dominant product in the reaction of the methylidyne (CH; X 2 Π) radical with vinylacetylene (H 2 CCHCCH; X 1 A').

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Gas-phase synthesis of naphthalene through an unconventional thermal alkyne–alkene [2 + 2] cycloaddition mechanism

Exotic cycloaddition entrance channels were discovered for the bimolecular gas-phase reactions of the phenylethynyl radical (C 6 H 5 CC, X 2 A 1 ) with ethylene-d 4 (C 2 D 4 ) and propylene (C 3 H 6 ) as explored under single-collision conditions utilizing the crossed molecular beams technique combined with electronic structure and statistical calculations. Connecting the concepts of barrierless entrance channels, excited states, and facile non-photochemically activated cycloadditions, the reaction pathway features an unconventional thermal [2 + 2] cycloaddition forming a four-membered ring collision complex followed by multiple isomerizations prior to unimolecular decomposition via atomic hydrogen loss to (un)substituted naphthalenes—naphthalene-d 4 (C 10 H 4 D 4 ) and 1-/2-methylnaphthalene (C 11 H 10 ). The small energy gap between the singly-occupied a 1 highest occupied molecular orbital (HOMO) with a σ-character and the underlying doubly-occupied b 1 molecular orbital with a π-character allows a facile promotion of an electron. This in turn enables a versatile low-temperature reactivity of phenylethynyl, where the end-on and side-on barrierless approaches of ethylene are due to its interaction with the σ and π orbitals, respectively, thus suggesting this mechanism as a possible method for tuning substituents in polycyclic aromatic hydrocarbon (PAH) formation and highlighting its versatility as a probe of fundamental carbon chemistry via counterintuitive cycloaddition reactions under single-collision conditions.

Goettl, Shane J. [University of Hawaii at Manoa, H↗
DOE OSTI2023

Collisional-energy-cascade model for nonthermal velocity distributions of neutral atoms in plasmas

Nonthermal velocity distributions with much greater tails than a Maxwellian have been observed for radical atoms in plasmas for a long time. Historically, such velocity distributions have been modeled by a two-temperature Maxwell distribution. In this paper, I propose a model based on collisional energy cascade, which has been studied in the field of granular materials. In the collisional energy cascade, a particle ensemble undergoes energy input at the high-energy region, entropy production by elastic collisions among particles, and energy dissipation. For radical atoms, energy input may be caused by the Franck-Condon energy of molecular dissociation or charge-exchange collision with hot ions, and the input energy is eventually dissipated by collisions with the walls. Here, I show that the steady-state velocity distribution in the collisional energy cascade is approximated by the generalized Mittag-Leffler distribution, which is a one-parameter extension of the Maxwell distribution. This parameter indicates the degree of the nonthermality and is related to the relative importance of energy dissipation over entropy production. This model is compared with a direct molecular dynamics simulation for simplified gaseous systems with energy input, as well as some experimentally observed velocity distributions of light radicals in plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Eisenbud–Wigner–Smith time delay in atom–laser interactions

We report recent development of ultrafast measurement techniques on the sub-femtosecond time scale has enabled us to see the atom–laser interaction in real time. One of the important observables to study the temporally resolved dynamics is the various time delay in the process. Time delays is an experimentally measurable quantity, where the time is not. Time delays in quantum collisions and in photoionization/photodetachment of atomic and molecular systems is reviewed. The formalisms of time delay in the context of quantum collisions by Eisenbud and Wigner that by Smith and their equivalence is discussed. The time-reversal symmetry between solutions with outgoing and ingoing wave boundary conditions allow us to interpret photoionization/photodetachment as half-scattering. Subsequently, the formalism of Eisenbud–Wigner–Smith time delay is extended in photoionization/photodetachment from collisions. We discuss that the time delay can be written as a self-adjoint quantum operator which depicts its measurability. On the experimental side, the measurement protocols are reviewed for attosecond streaking and RABBITT. Along with the EWS time delay, the measurement introduces an additional (which is negative) time delay. From the measurement of total delay and an analytical/numerical estimate of the measurement induced part, one gets the EWS time delay in the process. A few illustrative examples of studies on time delay are given to get a flavour of the outstanding advances made in this field in the last two decades.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

Thermalization of nuclear matter in heavy-ion collisions at Fermi energies

We analyze the time evolution of the kinetic properties of nuclear matter produced in heavy-ion collisions at Fermi energies. The collision system is simulated using Constrained Molecular Dynamics (CoMD) transport calculations whose output is the isospin, position, and momentum of the nucleons. Focusing on central 35 A·MeV 40 Ca+ 40 Ca collisions we utilize this information to extract localized momentum distributions in volume elements of 8 fm 3 and time steps of 5 fm/$c$. We then parameterize the single-particle momentum distributions with thermally motivated fit functions in the local rest frame of each cell. While the transverse-momentum distributions are well reproduced by thermal ones, the longitudinal ones carry a marked imprint of the initial nuclear motion which we capture by introducing a centroid motion into our fit functions. In particular, we find that Fermi distributions yield significantly better fits than Boltzmann ones, a consequence of the Pauli blocking implemented in CoMD. From the fits we extract the time dependence of the thermodynamic and collective properties of the excited nuclear medium. We find that the transverse temperature gradually rises to about 6 MeV, which is accompanied by a dissipation of the initial centroid motion of the incoming nuclei which vanishes at about 100 fm/c after initial impact. In conclusion, we are therefore able to track the transition of beam energy into random kinetic energy for nucleons, suggesting a three-dimensional equilibration of energy in the late stages of the collision.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Linear polarization fractions of Fulcher- α fluorescence in electron collisions with H 2

We apply the molecular convergent close-coupling method to the calculation of linear polarization fractions for Fulcher-band fluorescence following electron-impact excitation of the H 2 d 3 Π u state. Furthermore, the results exhibit the opposite threshold behavior compared to the only previous calculations, but are in agreement with the most recent measurements for the Q(1), R(1), and Q(3) transitions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Gas-Phase Study of the Elementary Reaction of the D1-Ethynyl Radical (C 2 D; X 2 Σ + ) with Propylene (C 3 H 6 ; X 1 A') under Single-Collision Conditions

The bimolecular gas-phase reactions of the D1-ethynyl radical (C 2 D; X 2 Σ + ) with propylene (C 3 H 6 ; X 1 A’) and partially substituted D3-3,3,3-propylene (C 2 H 3 CD 3 ; X 1 A’) were studied under single collision conditions utilizing the crossed molecular beams technique. Combining our laboratory data with electronic structure and statistical calculations, the D1-ethynyl radical is found to add without barrier to the C1 and C2 carbons of the propylene reactant, resulting in doublet C 5 H 6 D intermediate(s) with lifetime(s) longer than their rotational period(s). These intermediates undergo isomerization and unimolecular decomposition via atomic hydrogen loss through tight exit transition states forming predominantly cis/trans-3-penten-1-yne ((HCC)CH=CH(CH 3 )) and to a minor amount 3-methyl-3-buten-1-yne ((HCC)C(CH 3 )=CH 2 ) via overall exoergic reactions. Although the title reaction does not lead to the cyclopentadiene molecule (c-C 5 H 6 , X 1 A 1 ), high temperature environments can convert the identified acyclic C 5 H 6 isomers through hydrogen atom assisted isomerization to cyclopentadiene (c-C 5 H 6 , X 1 A 1 ). Since both the ethynyl radical and propylene reactants have been observed in cold interstellar environments such as TMC-1 and the reaction is exoergic and all barriers lie below the energy of the separated reactants, the these C 5 H 6 product isomers are predicted to form in those low temperature regions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗