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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

Ion-Pair Dissociation on α-MoO 3 Surfaces: Focus on the Electrolyte–Cathode Compatibility Issue in Mg Batteries

Nanosizing has been explored as one option to enhance typically sluggish Mg ion mobility in cathode materials. Here, we explore the effects of the surfaces of α-MoO3 in facilitating or hindering Mg ion desolvation at the electrolyte–cathode interface. It is well-known in the Mg battery community that Mg–anion contact ion pairs may form in the electrolyte and thus the insertion of Mg into the cathode must involve desolvation and dissociation, where Mg is detached from the counterion at the cathode surface. Here we compare two representative Mg–anion pairs, i.e., (Mg–Cl)+ and (Mg–TFSI)+ (TFSI–, bis(trifluoromethanesulfonyl)imide), differentiated according to the size and polarizability of the anion, and examine their dissociations on different α-MoO3 surfaces. We find that (Mg–Cl)+ requires a higher dissociation energy than (Mg–TFSI)+, which implies that the Mg–TFSI-type electrolyte is more suitable to be integrated with the MoO3 cathode when creating an electrochemical cell. The dissociation of (Mg–TFSI)+ occurs, however, not on the lowest energy (010) surface where the van der Waals (vdW) gap opens, but on the hydroxyl-terminated (001) surface. Therefore, to optimize the performance of α-MoO3 toward (Mg–TFSI)+ dissociation, preferential growth of the hydroxyl-terminated (001) surface is required.

Wan, Liwen F.↗
DOE OSTI2015

Direct Dynamics Simulation of Dissociation of the [CH3--I--OH]- Ion-Molecule Complex

Direct dynamics simulations were used to study dissociation of the [CH3--I--OH]- complex ion, which was observed in a previous study of the OH- + CH3I gas phase reaction (J. Phys. Chem. A 2013, 117, 7162). Restricted B97-1 simulations were performed to study dissociation at 65, 75 and 100 kcal/mol and the [CH3--I--OH]- ion dissociated exponentially, in accord with RRKM theory. For these energies the major dissociation products are CH3I + OH-, CH2I- + H2O, and CH3OH + I-. Unrestricted B97-1 and restricted and unrestricted CAM-B3LYP simulations were also performed at 100 kcal/mol to compare with the restricted B97-1 results. The {CH3I + OH-}:{CH2I- + H2O}:{CH3OH + I-} product ratio is 0.72 : 0.15 : 0.13, 0.81 : 0.05 : 0.14, 0.71 : 0.19 : 0.10 , and 0.83 : 0.13 : 0.04 for the restricted B97-1, unrestricted B97-1, restricted CAM-B3LYP, and unrestricted CAM-B3LYP simulations, respectively. Other product channels found are CH2 + I- + H2O, CH2 + I-(H2O), CH4 + IO-, CH3 - + IOH, and CH3 + IOH-. The CH3 - + IOH singlet products are only given by the restricted B97-1 simulation and the lower energy CH3 + IOH- doublet products are only formed by the unrestricted B97-1 simulation. Also studied were the direct and indirect atomic-level mechanisms for forming CH3I + OH-, CH2I- + H2O, and CH3OH + I-. The majority of CH3I + OH- were formed through a direct mechanism. For both CH2I- + H2O and CH3OH + I-, the direct mechanism is overall more important than the indirect mechanisms, with the round-about like mechanism the most important indirect mechanism at high excitation energies. Mechanism comparisons between the B97-1 and CAM-B3LYP simulations showed that formation of the CH3OH---I- complex is favored for the B97-1 simulations, while formation of the HO----HCH2I complex is favored for the CAM-B3LYP simulations. The unrestricted simulations give a higher percentage of indirect mechanisms than the restricted simulations. The possible role of the self-interaction error in the simulations is also discussed. The work presented here gives a detailed picture of the [CH3--I--OH]- dissociation dynamics, and is very important for unraveling the role of [CH3--I--OH]- in the dynamics of the OH-(H2O)n=1,2 + CH3I reactions.

Environmental Molecular Sciences Laboratory↗
DOE OSTI2019

Beyond the Coulson–Fischer point: characterizing single excitation CI and TDDFT for excited states in single bond dissociations

Linear response time dependent density functional theory (TDDFT), which builds upon configuration interaction singles (CIS) and TD-Hartree–Fock (TDHF), is the most widely used class of excited state quantum chemistry methods and is often employed to study photochemical processes. This paper studies the behavior of the resulting excited state potential energy surfaces beyond the Coulson–Fischer (CF) point in single bond dissociations, when the optimal reference determinant is spin-polarized. Many excited states exhibit sharp kinks at the CF point, and connect to different dissociation limits via a zone of unphysical concave curvature. In particular, the unrestricted M S = 0 lowest triplet T 1 state changes character, and does not dissociate into ground state fragments. The unrestricted M S = ±1 T 1 CIS states better approximate the physical dissociation limit, but their degeneracy is broken beyond the CF point for most single bond dissociations. On the other hand, the M S = ±1 T 1 TDHF states reach the asymptote too soon, by merging with the ground state from the CF point onwards. Use of local exchange–correlation functionals causes M S = ±1 T 1 TDDFT states to resemble their unphysical M S = 0 counterpart. The 2 orbital, 2-electron model system of minimal basis H 2 is analytically treated to understand the origin of these issues, revealing that the lack of double excitations is at the root of these remarkable observations. In conclusion, the behavior of excited state surfaces is also numerically examined for species like H 2 , NH 3 , C 2 H 6 and LiH in extended basis sets.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Methane dissociative chemisorption and detailed balance on Pt(111): Dynamical constraints and the modest influence of tunneling

A dynamically biased (d-) precursor mediated microcanonical trapping (PMMT) model of the activated dissociative chemisorption of methane on Pt(111) is applied to a wide range of dissociative sticking experiments, and, by detailed balance, to the methane product state distributions from the thermal associative desorption of adsorbed hydrogen with coadsorbed methyl radicals. Tunneling pathways were incorporated into the d-PMMT model to better replicate the translational energy distribution of the desorbing methane product from the laser induced thermal reaction of coadsorbed hydrogen and methyl radicals occurring near T{sub s} = 395 K. Although tunneling is predicted to be inconsequential to the thermal dissociative chemisorption of CH{sub 4} on Pt(111) at the high temperatures of catalytic interest, once the temperature drops to 395 K the tunneling fraction of the reactive thermal flux reaches 15%, and as temperatures drop below 275 K the tunneling fraction exceeds 50%. The d-PMMT model parameters of (E{sub 0} = 58.9 kJ/mol, s = 2, η{sub v} = 0.40) describe the apparent threshold energy for CH{sub 4}/Pt(111) dissociative chemisorption, the number of surface oscillators involved in the precursor complex, and the efficacy of molecular vibrational energy to promote reaction, relative to translational energy directed along the surface normal. Molecular translations parallel to the surface and rotations are treated as spectator degrees of freedom. Transition state vibrational frequencies are derived from generalized gradient approximation-density functional theory electronic structure calculations. The d-PMMT model replicates the diverse range of experimental data available with good fidelity, including some new effusive molecular beam and ambient gas dissociative sticking measurements. Nevertheless, there are some indications that closer agreement between theory and experiments could be achieved if a surface efficacy less than one was introduced into the modeling as an additional dynamical constraint.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Dissociative diffusion mechanism in vacancy-rich materials according to mass action kinetics

We conducted two sets of diffusion-reaction numerical simulations using a finite difference method (FDM) in order to investigate fast impurity diffusion via interstitial sites in vacancy-rich materials such as Cu(In,Ga)Se 2 (CIGS) and Cu 2 ZnSn(S, Se) 4 (CZTSSe or CZTS) via the dissociative diffusion mechanism where the interstitial diffuser ultimately reacts with a vacancy to produce a substitutional. The first set of simulations extends the standard interstitial-limited dissociative diffusion theory to vacancy-rich material conditions where vacancies are annihilated in large amounts, introducing non-equilibrium vacancy concentration profiles. The second simulation set explores the vacancy-limited dissociative diffusion where impurity incorporation increases the equilibrium vacancy concentration. In addition to diffusion profiles of varying concentrations and shapes that were obtained in all simulations, some of the profiles can be fitted with the constant- and limited-source solutions of Fick’s second law despite the non-equilibrium condition induced by the interstitial-vacancy reaction. The first set of simulations reveals that the dissociative diffusion coefficient in vacancy-rich materials is inversely proportional to the initial vacancy concentration. In the second set of numerical simulations, impurity-induced changes in the vacancy concentration lead to distinctive diffusion profile shapes. The simulation results are also compared with published data of impurity diffusion in CIGS. And according to the characteristic properties of diffusion profiles from the two set of simulations, experimental detection of the dissociative diffusion mechanism in vacancy-rich materials may be possible.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Water dissociation on Ni(100), Ni(110), and Ni(111) surfaces: Reaction path approach to mode selectivity

We performed a comparative study of mode-selectivity of water dissociation on Ni(100), Ni(110), and Ni(111) surfaces at the same level of theory using a fully quantum approach based on the reaction path Hamiltonian. Calculations show that the barrier to water dissociation on the Ni(110) surface is significantly lower compared to its close-packed counterparts. Transition states for this reaction on all three surfaces involve the elongation of one of the O–H bonds. Furthermore, a significant decrease in the symmetric stretching and bending mode frequencies near the transition state is observed in all three cases and in the vibrational adiabatic approximation, excitation of these softened modes results in a significant enhancement in reactivity. Inclusion of non-adiabatic couplings between modes results in the asymmetric stretching mode showing a similar enhancement of reactivity as the symmetric stretching mode. Dissociation probabilities calculated at a surface temperature of 300 K showed higher reactivity at lower collision energies compared to that of the static surface case, underlining the importance of lattice motion in enhancing reactivity. Mode selective behavior is similar on all the surfaces. Molecules with one-quantum of vibrational excitation in the symmetric stretch, at lower energies (up to 0.45 eV), are more reactive on Ni(110) than the Ni(100) and Ni(111) surfaces. But, the dissociation probabilities approach saturation on all the surfaces at higher incident energy values. Ultimately, Ni(110) is found to be highly reactive toward water dissociation among the low-index nickel surfaces owing to a low reaction barrier resulting from the openness and corrugation of the surface. These results show that the mode-selective behavior does not vary with different crystal facets of Ni qualitatively, but there is a significant quantitative effect.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Equation of state for a chemically dissociative, polyatomic system: Carbon dioxide

A notorious challenge in high-pressure science is to develop an equation of state (EOS) that explicitly treats chemical reactions. For instance, many materials tend to dissociate at high pressures and temperatures where the chemical bonds that hold them together break down. We present an EOS for carbon dioxide (CO 2 ) that allows for dissociation and captures the key material behavior in a wide range of pressure– temperature conditions. Carbon dioxide is an ideal prototype for the development of a wide-ranging EOS that allows for chemical-dissociation equilibria since it is one of the simplest polyatomic systems and because it is of great interest in planetary science and in the study of detonations. Here, we show that taking dissociation into account significantly improves the accuracy of the resulting EOS compared to other EOSs that either neglect chemistry completely or treat CO 2 dissociation in a more rudimentary way.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Dissociative electron attachment to amide bond containing molecules: N-ethylformamide and N-ethylacetamide

To advance our quest to understand the role of low energy electrons in biomolecular systems, we performed investigations on dissociative electron attachment (DEA) to gas-phase N-ethylformamide (NEF) and N-ethylacetamide (NEA) molecules. Both molecules contain the amide bond, which is the linkage between two consecutive amino acid residues in proteins. Thus, their electron-induced dissociation can imitate the resonant behavior of the DEA process in more complex biostructures. Our experimental results indicate that in these two molecules, the dissociation of the amide bond results in a double resonant structure with peaks at ~5 eV and 9 eV. We also determined the energy position of resonant states for several negative ions, i.e., the other dissociation products from NEF and NEA. Our predictions of dissociation channels were supported by density functional theory calculations of the corresponding threshold energies. Our results and those previously reported for small amides and peptides imply the fundamental nature for breakage of the amide bond through the DEA process.

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

Photofragment imaging differentiates between one- and two-photon dissociation pathways in MgI+

The bond strength and photodissociation dynamics of MgI+ are determined by a combination of theory, photodissociation spectroscopy, and photofragment velocity map imaging. From 17 000 to 21 500 cm−1, the photodissociation spectrum of MgI+ is broad and unstructured; photofragment images in this region show perpendicular anisotropy, which is consistent with absorption to the repulsive wall of the (1) Ω = 1 or (2) Ω = 1 states followed by direct dissociation to ground state products Mg+ (2S) + I (2P3/2). Analysis of photofragment images taken at photon energies near the threshold gives a bond dissociation energy D0(Mg+-I) = 203.0 ± 1.8 kJ/mol (2.10 ± 0.02 eV; 17 000 ± 150 cm−1). At photon energies of 33 000–41 000 cm−1, exclusively I+ fragments are formed. Over most of this region, the formation of I+ is not energetically allowed via one-photon absorption from the ground state of MgI+. Images show the observed product is due to resonance enhanced two-photon dissociation. The photodissociation spectrum from 33 000 to 38 500 cm−1 shows vibrational structure, giving an average excited state vibrational spacing of 227 cm−1. This is consistent with absorption to the (3) Ω = 0+ state from ν = 0, 1 of the (1) Ω = 0+ ground state; from the (3) Ω = 0+ state, absorption of a second photon results in dissociation to Mg* (3P°J) + I+ (3PJ). From 38 500 to 41 000 cm−1, the spectrum is broad and unstructured. We attribute this region of the spectrum to one-photon dissociation of vibrationally hot MgI+ at low energy and ground state MgI+ at higher energy to form Mg (1S) + I+ (3PJ) products.

Chemistry↗
DOE OSTI2019

Infrared laser-induced isotope-selective dissociation of mixed (CF{sub 3}Br){sub m}Ar{sub n} van der Waals clusters

The paper presents the results of studies of IR laser-induced bromine-isotope-selective dissociation of small mixed (CF{sub 3}Br){sub m}Ar{sub n} van der Waals clusters (m = 1, 2 and 1 ≤ n ≤ 5 is the number of molecules and atoms in the clusters, respectively). The experiments used a pulsed CO{sub 2} laser to excite clusters and a quadrupole mass spectrometer to detect a molecular cluster beam. The research method is based on the selective vibrational excitation of clusters by IR laser radiation, leading to their predissociation. The possibilities of implementing bromine-isotope-selective dissociation of clusters are considered. The experimental setup and research method are briefly described. The results of determining the cluster dissociation efficiency and selectivity as functions of the energy and frequency of laser radiation are presented. It is shown that resonant vibrational excitation of clusters makes it possible to induce their isotope-selective dissociation. Thus, using a CF{sub 3}Br/Ar mixture with a pressure ratio of 1/200, the enrichment factors of the {sup 79}Br isotope are found to be K{sub enr}({sup 79}Br) = 1.15 ± 0.04 and 0.95 ± 0.03 under cluster irradiation conditions on the 9R(30) (ν = 1084.635 cm{sup −1}) and 9R(24) (ν = 1081.087 cm{sup −1}) lines, respectively The achieved dissociation selectivities for the clusters are, respectively, 1.16 ± 0.05 and 0.95 ± 0.04. The results obtained suggest that this method can be used to separate isotopes in molecules containing isotopes of heavy elements, which have a slight isotopic shift in the IR absorption spectra. (laser isotope separation)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

X-ray scattering signatures of early-time accelerations in iodine dissociation

Ultrafast time-resolved x-ray scattering (TRXS) from a photoexcited molecular ensemble measures a distribution $S(Q, τ)$ of the x-ray momentum transfer Q and pump-probe delay τ in which all modes of motion induced by the excitation overlap. Frequency-resolved x-ray scattering (FRXS) based on $\tilde {S}\left(Q,\omega \right)$ separates each oscillation and dissociation channel in the TRXS data, enabling measurements of vibrational frequencies and phases, and dissociation velocities and time shifts. In this work, we extend FRXS analysis to study early-time accelerations as well. We show how these appear as diffuse scattering patterns with characteristic phase evolution in FXRS and we develop a set of transformations that isolate individual channels to measure the early-time accelerated motion. This procedure is used to analyze diatomic iodine x-ray scattering data with multiple dissociations, and the ability of this technique to characterize early-time accelerations of one dissociation channel even in the presence of another dissociation is demonstrated.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Dissociation of gas-phase anisole induced by low-energy electron interactions: understanding patterns of aromatic bond cleavage

Abstract In view of elucidating the fragmentation patterns of aromatic systems induced by low-energy electron interactions, dissociative electron attachment (DEA) to gas-phase anisole was performed. Anionic fragments resulting from this DEA process were detected by a quadrupole mass spectrometer, and ion yields of those fragments as a function of incident electron energy were rendered. Our study showed the formation of CH 3 − , HCC − , and OCH 3 − fragments, suggesting that various dissociation channels proceed out of DEA to anisole. We employed density functional theory to compute thermodynamic threshold energies for each potential dissociation channel. Those theoretical calculations supported the prediction that the CH 3 − and OCH 3 − fragments form via mechanisms of single-bond cleavage; the HCC − fragments may form through two-, three-, or four-body dissociation channels that entail hydrogen transfers and the cleavage of multiple aromatic bonds. The experimental resonance energies that form the CH 3 − , HCC − , and OCH 3 − fragments were 6.0 eV, 5.8 and 9.7 eV, and 9.8 eV, respectively. Given the classification of anisole as a monosubstituted aromatic species, our results explain generalizable patterns of electron-mediated dissociation in aromatic systems.

Finley, Jacob (ORCID:0009000606963352)↗
DOE OSTI2025

Irreversible oxygen poisoning: Modeling the initial water dissociation kinetics on δ-Pu(111) and δ-Pu(100) through density functional theory

In this work, the initial kinetics of water dissociation on two facets of δ-plutonium, δ-Pu(111) and δ-Pu(100), are explored through density functional theory in order to understand how water dissociation occurs on these facets. We explored the dissociation of water via the formation of hydroxyls, atomic hydrogen, and atomic oxygen species on each facet. We calculate low energetic barriers for water to split to adsorbed hydrogen and hydroxyl species at 0.19 eV for δ-Pu(111) and 0.07 eV for δ-Pu(100). The hydroxyl has a barrier of 0.64 and 0.37 eV to cleave the hydrogen–oxygen bond on δ-Pu(111) and δ-Pu(100), respectively. Due to the highly exergonic adsorption free energy of atomic oxygen of −2.10 eV, the metallic surfaces are found to be fully covered in oxygen, even with the inclusion of oxygen lateral interactions. When combined with the reaction thermodynamics, this free energy of adsorption forms a molecular oxygen desorption barrier greater than 9.51 eV (918 kJ/mol). These results, combined with simulated temperature programed desorption spectra, indicate that oxygen formed via water dissociation on δ-Pu (111) and (100) facets induces an irreversible poison that prevents further reaction of water directly on metallic plutonium surfaces, most likely due to the strong hybridization of the Pu and O valence states. Therefore, these results imply that another mechanism is responsible for the continuously experimentally measured water dissociation, which is most likely due to the Pu oxide.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Controlling Chemical Reactions in Confined Environments: Water Dissociation in MOF-74

The confined porous environment of metal organic frameworks (MOFs) is an attractive system for studying reaction mechanisms. Compared to flat oxide surfaces, MOFs have the key advantage that they exhibit a well-defined structure and present significantly fewer challenges in experimental characterization. As an example of an important reaction, we study here the dissociation of water—which plays a critical role in biology, chemistry, and materials science—in MOFs and show how the knowledge of the structure in this confined environment allows for an unprecedented level of understanding and control. In particular, combining in-situ infrared spectroscopy and first-principles calculations, we show that the water dissociation reaction can be selectively controlled inside Zn-MOF-74 by alcohol, through both chemical and physical interactions. Methanol is observed to speed up water dissociation by 25% to 100%, depending on the alcohol partial pressure. On the other hand, co-adsorption of isopropanol reduces the speed of the water reaction, due mostly to steric interactions. In addition, we also investigate the stability of the product state after the water dissociation has occurred and find that the presence of additional water significantly stabilizes the dissociated state. With this being said, our results show that precise control of reactions within nano-porous materials is possible, opening the way for advances in fields ranging from catalysis to electrochemistry and sensors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1977

Dissociative excitation of H2, HD, and D2 by electron impact

Time-of-flight techniques have been used to investigate the electron-impact dissociation of H2, HD, and D2 in order to determine the effect of isotopic mass variation in the target molecule on the dissociative excitation process. At incident electron energies near 100 eV, the time-of-flight spectrum produced from each molecule consists of atoms in the metastable 2s state and in high-lying long-lived Rydberg levels. The individual time-of-flight distributions, kinetic-energy spectra, and relative differential cross sections for these two species resulting from each molecule have been measured. The kinetic-energy spectrum of the Rydberg atoms produced from dissociative excitation of H2 was notably dissimilar in shape from the corresponding distributions produced from HD and D2. Also the 2s and Rydberg production cross sections differed between the three molecules. In the dissociation of the heteronuclear HD molecule, the ratio of fast H(2s) atoms to D(2s) atoms was about 1 to 1, while the same ratio comparing the Rydberg atoms was nearly 2 to 1. These differences indicate the influence of the mass variation on the position of the Franck-Condon region in the production of 2s atoms and on the competition between autoionization and dissociation in the formation of Rydberg fragments.

Carnahan, B. L.↗
NASA NTRS1977

Collisionless dissociation of SF6 using two resonant frequency CO2 laser fields

The collisionless dissociation of SF6 has been studied using simultaneous irradiation by two frequencies from a CO2 laser which are both nearly resonant with the SF6nu3 absorption band. It was found that the dissociation was enhanced, and occurred over a wider frequency range, than for single frequency dissociation. No threshold effect was observed for a weak resonant and a much higher energy field pumping slightly off-resonance. For such two frequency irradiation, the peak in the dissociation curve was found to be shifted to lower frequencies with respect to that for single frequency dissociation.

Gower, M. C.↗
NASA NTRS1980

Emission line shapes produced by dissociative excitation of atmospheric gases

The spectral line shapes of the radiation emitted from O atoms produced by the dissociative excitation of O2, CO, CO2 and NO are investigated. Doppler line shapes are derived from time-of-flight spectra of O (5S0) atoms decaying by the emission of 1356-A radiation after being produced in electron impact experiments at incident electron energies from 25 to 300 eV. It is shown that the effective line width of the radiation is large compared with the Doppler absorption widths of ambient O atoms in both photoelectron and auroral excitation, and thus the dissociatively excited component of the O I 1304-A airglow will behave as though it were optically thin, exhibiting pronounced limb brightening effects and a scale height characteristic of the initial, local source function. It is found that the average kinetic energy of the dissociation fragments inferred from O I (5S) time-of-flight spectra is in good agreement with that of O I (3S) atoms in the electron impact dissociation of CO2, although not for O2. Finally, it is suggested that although electron impact dissociation of CO and CO2 contributes to the 1304-A emission in the upper atmosphere of Venus, it cannot be the dominant source of this radiation since the absolute cross sections for the reaction are too small.

Zipf, E. C.↗
NASA NTRS1991

Dissociation of diatomic gases

The Landau-Zener theory of reactive cross sections has been applied to diatomic molecules dissociating from a ladder of rotational and vibrational states. Although the preexponential factor of the Arrhenius rate expression is shown to be a complex function of the dimensionless activation energy, the average over all states in the ladder is well represented by a single factor that varies about as T exp (-n), where the coefficient n is the order of unity. This relation agrees very well with experimental data for dissociation of O2 and N2, for example. The results validate previous empirical assignment of a single preexponential factor in the Arrhenius expression and justify the extrapolation of the expression well beyond the range of data. The theory is then used to calculate the effect of vibrational nonequilibrium on dissociation rate. For Morse oscillators the results are about the same as for harmonic oscillators, and the dissociation from a ladder of equilibrium rotational and nonequilibrium vibrational states is close to an analytic approximation provided by Hammerling, Kivel, and Teare for harmonic oscillators all dissociating from the ground rotational state.

Hansen, C. F.↗