Computational aspects of Laplace transform inversion of thermal unimolecular rate constant

The thermal limiting high-pressure unimolecular rate constant k∞ represents, operationally, the Laplace transform of the product of microcanonical rate constant for decomposition of molecules having specified energy E [k(E)] and the density of states [N(E)]. By inversion, it is possible to recover k(E)N(E), from which one can obtain the energy dependence of k(E) and the pressure dependence of k_uni, the thermal general-pressure unimolecular rate constant. This article examines numerical aspects of three methods of inversion, their reliability and dependence on sampling, i.e., on the number of available experimental data points, by comparing exact k(E) and k_uni with those obtained by inversion. It turns out that the method of steepest descents is the best all-round performer.     

Determination of the Glass Electrode Parameters by Means of Potentiometric Titration of Acetic Acid in Aqueous Sodium or Potassium Chloride Solutions at 25°C

Single-ion activity coefficient equations are presented for the calculation of stoichiometric (molality scale) dissociation constants K _m for acetic acid in aqueous NaCl or KCl solutions at 25°C. These equations are of the Pitzer or Hückel type and apply to the case where the inert electrolyte alone determines the ionic strength of the acetic acid solution considered. K _m for a certain ionic strength can be calculated from the thermodynamic dissociation constant K _a by means of the equations for ionic activity coefficients. The data used in the estimation of the parameters for the activity coefficient equations were taken from the literature. In these data were included results of measurements on galvanic cells without a liquid junction (i.e., on cells of the Harned type). Despite the theoretical difficulties associated with the single-ion activity coefficients, K _m can be calculated for acetic acid in NaCl or KCl solutions by the Pitzer or Hückel method (the two methods give practically identical K _m values) almost within experimental error at least up to ionic strengths of about 1 mol-kg^–1. Potentiometric acetic acid titrations with base solutions (NaOH or KOH) were performed in a glass electrode cell at constant ionic strengths adjusted by NaCl or KCl. These titrations were analyzed by equation E = E _o + k(RT/F) ln[m(H⁺)], where m(H⁺) is the molality of protons, and E is the electromotive force measured. m(H⁺) was calculated for each titration point from the volume of the base solution added by using the stoichiometric dissociation constant K _m obtained by the Pitzer or Hückel method. During each base titration at a constant ionic strength, E _o and k in this equation were observed to be constants and were determined by linear regression analysis. The use of this equation in the analysis of potentiometric glass electrode data represents an improvement when compared to the common methods in use for two reasons. No activity coefficients are needed and problems associated with liquid junction potentials have been eliminated.     

Kinetic study of the reaction of chlorine atoms with CF3I and the reactions of CF3 radicals with O2, Cl2 and NO at 296 K

The kinetics of the gas-phase reaction of Cl atoms with CF₃I have been studied relative to the reaction of Cl atoms with CH₄ over the temperature range 271–363 K. Using k(Cl + CH₄) = 9.6 × 10^?12 exp(?2680/RT) cm³ molecule^?1 s^?1, we derive k(Cl + CF₃I) = 6.25 × 10^?11 exp(?2970/RT) in which E_a has units of cal mol^?1. CF₃ radicals are produced from the reaction of Cl with CF₃I in a yield which was indistinguishable from 100%. Other relative rate constant ratios measured at 296 K during these experiments were k(Cl + C₂F₅I)/k(Cl + CF₃I) = 11.0 ± 0.6 and k(Cl + C₂F₅I)/k(Cl + C₂H₅Cl) = 0.49 ± 0.02. The reaction of CF₃ radicals with Cl₂ was studied relative to that with O₂ at pressures from 4 to 700 torr of N₂ diluent. By using the published absolute rate constants for k(CF₃ + O₂) at 1–10 torr to calibrate the pressure dependence of these relative rate constants, values of the low- and high-pressure limiting rate constants have been determined at 296 K using a Troe expression: k₀(CF₃ + O₂) = (4.8 ± 1.2) × 10^?29 cm⁶ molecule^?2 s^?1; k_∞(CF₃ + O₂) = (3.95 ± 0.25) × 10^?12 cm³ molecule^?1 s^?1; F_c = 0.46. The value of the rate constant k(CF₃ + Cl₂) was determined to be (3.5 ± 0.4) × 10^?14 cm³ molecule^?1 s^?1 at 296 K. The reaction of Cl atoms with CF₃I is a convenient way to prepare CF₃ radicals for laboratory study. © 1995 John Wiley & Sons, Inc.     

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions r ⁿⁱ exp(−ζ _i r). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments <r ^k > (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F ^′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54. Received: 11 July 1997 / Accepted: 27 August 1997     

Calculation of strong-collision dissociation rate constants from NASA thermodynamic polynomials

A new method for calculating low-pressure strong-collision rate constants of dissociation and recombination reactions was proposed. The method is based on determining the density of states of the internal degrees of freedom of the reactant molecule by applying the inverse Laplace transform to the respective partition function, which, in turn, is calculated from the thermodynamic properties in the form of NASA polynomials. The proposed model is universal in the sense that the required NASA polynomials can be calculated using molecular properties obtained by various methods, both theoretical and experimental or a combination thereof. In the present study, the NASA polynomials were taken from the available databases or calculated from thermodynamic functions, either tabulated or determined by statistical mechanics in the rigid-rotor harmonic-oscillator approximation, with a simple anharmonicity correction introduced when necessary. In addition, a model for calculating the rotational factor is developed and tested. It is based on determination of the centrifugal barrier as a function of the dissociating bond length at a given rotational energy through calculating the principal moments of inertia of the molecule at each step of elongation of the bond. The proposed approach is exemplified for the dissociation of the H₂O, HO₂, and H₂O₂ molecules and the corresponding reverse reactions. A comparison with the available experimental data made it possible to estimate the weak-collision efficiency. At high temperatures, for all the reactions studied, the weak collision efficiency appears to be quite reasonable, whereas, at low temperatures, the situation is unsatisfactory, except perhaps for H₂O dissociation. Given that the energy threshold E₀ of dissociation reactions is typically well known, the calculated and measured dissociation rate constants are represented and handled in terms of the preexponential factor A(T) in the expression k(T) = A(T)exp(−E₀/RT). A new formula for fitting A(T) was proposed: log A(T) = a + b(1000/T) + c(1000/T)^p, which turned out to be a good approximation for the preexponential factors of not only the rate constants but also the equilibrium constants.     

Potential energy surfaces and nonadiabatic transitions in the asymptotic regions of ICN photodissociation to study the interference effects in the F1 and F2 spin-rotation levels of the CN products

One of the most spectacular yet unsolved problems for the ICN -band photodissociation is the non-statistical spin-rotation F₁ = N + 1/2 and F₂ = N − 1/2 populations for each rotation level N of the CN fragment. The F₁/F₂ population difference function f(N) exhibits strong N and λ dependences with an oscillatory behavior. Such details were found to critically depend on the number of open-channel product states, namely, whether both I (²P_3/2) and I (²P_1/2) are energetically available or not as the dissociation partner. First, in the asymptotic region, the exchange and dipole-quadrupole inter-fragment interactions were studied in detail. Then, as the diabatic basis, we took the appropriate symmetry adapted products of the electronic and rotational wavefunctions for the F₁ and F₂ levels at the dissociation limits. We found that the adiabatic Hamiltonian exhibits Rosen–Zener–Demkov type nonadiabatic transitions reflecting the switch between the exchange interaction and the small but finite spin-rotation interaction within CN at the asymptotic region. This non-crossing type nonadiabatic transition occurs with the probability 1/2, that is, at the diabatic limit through a sudden switch of the quantization axis for CN spin S from the dissociation axis to the CN rotation axis N . We have derived semiclassical formulae for f(N) and the orientation parameters with a two-state model including the 3A′ and 4A′ electronic states, and with a four-state model including the 3A′ through 6A′ electronic states. These two kinds of interfering models explain general features of the F₁ and F₂ level populations observed by Zare's group and Hall's group, respectively. © 2018 Wiley Periodicals, Inc.     

Comment on Dual-Hard-Sphere Models for Liquid Semiconductors Si And Ge

Abstract

Various generalized dual-hard-sphere (DHS) models are reviewed on calculating the liquid structure factor for semiconductor elements Si and Ge. It is found that the model generalized by Canessa, Mariani and Vignolo gives the best fitting of experimental structure factor S _exp(k) in the range k > 2k_F , (k_F , the Fermi wave vector), and all previous models including a new generalized model by the author fail to reproduce the experimental structure factor S _exp(k) of Si and Ge in the whole range of k vector.     

Relation between the inverse Laplace transforms of I(tβ) and I(t): Application to the Mittag-Leffler and asymptotic inverse power law relaxation functions

The relation between H(k), inverse Laplace transform of a relaxation function I(t), and H_β(k), inverse Laplace transform of I(t^β), is obtained. It is shown that for β < 1 the function H_β(k) can be expressed in terms of H(k) and of the Lévy one-sided distribution L_β(k). The obtained results are applied to the Mittag-Leffler and asymptotic inverse power law relaxation functions. A simple integral representation for the Lévy one-sided density function L_1/4(k) is also obtained.AMS (MOS) classification: 33E12 Mittag-Leffler functions and generalizations, 44A10 Laplace transform, 60E07 Infinitely divisible distributions; stable distributions     

Influence of heating rate on kinetic quantities of solid phase thermal decomposition

Thermogravimetric analyses of thermal decomposition (pyrolysis, thermal dissociation and combustion) of 9 different samples were carried out in dynamic conditions at different heating rates. The kinetic parameters (E, A and k_m) of thermal decomposition were determined and interrelations between the parameters and heating rate q were analyzed. There were also relations between Arrhenius and Eyring equations analyzed for thermal decomposition of solid phase. It was concluded that Eyring theory is an element, which interconnects used thermokinetic equations containing Arrhenius law and suggests considering kinetic quantities in way relative to 3 kinetic constants (E, A and k_m). Analysis of quantities other than km (i.e. E, A, Δ⁺H, Δ⁺S) in relation to heating rate is an incomplete method and does not lead to unambiguous conclusions. It was ascertained that in ideal case, assuming constant values of kinetic parameters (E and A) towards heating rate and satisfying both Kissinger equations, reaction rate constant k_m should take on values intermediate between constants (k_m)₁ and (k_m)₂ determined from these equations. Whereas behavior of parameters E and A towards q were not subjected to any rule, then plotting relation k_m vs. q in the background of (k_m)₁ and (k_m)₂ made possible classification of differences between thermal decomposition processes taking place in oxidizing and oxygen-free atmosphere.     

Rapid Screening of Drug Absorption Potential Using the Immobilized Artificial Membrane Phosphatidylcholine Column and Molar Volume

The immobilized artificial membrane phosphatidylcholine (IAMPC) chromatography was evaluated for the predictability of oral absorption potential of 40 structurally unrelated drugs. The chromatographic capacity factors (k_IAM) were determined as a function the pH and composition of the mobile phase, and were corrected for the molar volume of the solutes (k_IAM/MWⁿ). The correlation between k_IAM/MWⁿ and the human fraction of intestinal absorption (F_a) was highest when measured at 20% acetonitrile (pH 5.5) with the power function n = 2.5. The best-fit equation for the sigmoid relationship between k_IAM/MWⁿ and F_a was obtained: F_a (%) = 94.3 × {1-exp[-17.9 × (k_IAM/MW^2.5) × 10⁶]}^2.1 (r = 0.925). This in vitro prediction method may be useful in a rapid screening of drug candidates with high oral absorption potential in humans.     

A generalized formula for the energies of alternant molecular orbitals. II. Heteronuclear molecules

Using the method of alternant molecular orbitals (AMO ), it is shown that the energies of AMOS (E_kσ) for an arbitrary heteronuclear alternant system, having a singlet ground state, are connected with the energies of MOS (e_{k(k )}) obtained by means of the conventional Hartree–Fock (HF ) method (SCF -LCAO -MO -PPP ) via the formula: In the general case, the determination of the correlation corrections δ_i,kσ is connected with the solving of a complicated system of integral equations, which is considerably simplified if the Hubbard approximation is accepted for the electron interaction. The energy spectrum of a chain with two atoms in the elementary cell (AB)_n is considered as an example. It is shown that if nontrivial solutions exist (δ_i,kσ ≠ 0), the correlation correction for AMOS of different spin are different (δ_i,kσ ≠ δ_i,kβ), from which it follows, that the width of the energy gap ΔE_∞ for AMOS with different spin is different: ΔE_∞,α ≠ ΔE_∞,β.     

Phenomenological kinetics of irreversible electrochemical dissolution of metal-oxide microparticles

The electrochemical dissolution of metal oxides and other stable solid phases composed of nano- to micro-crystalline particles is generally a complex process. It can be simplified by distinguishing two main contributions to the reactivity of the solid: the potential-dependent rate coefficient k(E), and the conversion-dependent function f(y). These contributions can be evaluated by a combination of potentiostatic and potentiodynamic experiments. Both k(E) and f(y) were obtained experimentally for the dissolution of iron and chromium oxides, and theoretical consequences of their particular forms are discussed. A peak-shaped function k(E) was observed in the case of γ-Fe₂O₃, whereas, for α-Cr₂O₃ and CrO₂, a different model based on intermediate surface complexes is proposed. This model also explains the complete electrochemical dissolution of metal oxides regardless of their low intrinsic electric conductivity. Received: 2 July 1997 / Accepted: 3 November 1997     

Very-low-pressure pyrolysis of nitroso- and pentafluoronitrosobenzene C?NO bond dissociation energies

The high-pressure absolute rate constants for the decomposition of nitrosobenzene and pentafluoronitrosobenzene were determined using the very-low-pressure pyrolysis (VLPP) technique. Bond dissociation energies of DH⁰(C₆H₅? NO) = 51.5 ± 1 kcal/mole and DH⁰ (C₆F₅? NO) = 50.5 ± 1 kcal/mole could be deduced if the radical combination rate constant is set at log k_r(M^?1·sec^?1) = 10.0 ± 0.5 for both systems and the activation energy for combination is taken as 0 kcal/mole at 298°K. δH_f⁰(C₆H₅NO), δH_f⁰(C₆F₅NO), and δH_f⁰(C₆F₅) could be estimated from our kinetic data and group additivity. The values are 48.1 ± 1, –160 ± 2, and – 130.9 ± 2 kcal/mole, respectively. C–X bond dissociation energies of several perfluorinated phenyl compounds, DH⁰(C₆F₅–X), were obtained from the reported values of δH_f⁰(C₆F₅X) and our estimated δH_f⁰(C₆F₅) [X = H, CH₃, NO, Cl, F, CF₃, I, and OH].     

Properties of the atomic form factor: Applications and shell structure

A relationship between the atomic form factor, F(k), and its first derivative, both at the origin, is presented. A new function f(k), related in a simple way to F(k), has been studied and some applications have been performed. They led us to find lower bounds to F(k) for all k and to other quantities such as the charge density at the origin, ρ(0), and radial expectation values. Finally, interesting effects on the Laplacian of f(k) due to the atomic shell structure were found. © 1995 John Wiley & Sons, Inc.     

Coordination polymers and discrete complexes of Ag(I)-N-(pyridylmethylene)anilines: synthesis,crystal structures and photophysical properties

Reactions of AgO₂C₂F₃ with (E)-N-(pyridylmethylene)aniline in which the pyridyl N is in the p- or m-position yielded two 1-D coordination polymers, [(AgO₂C₂F₃)₂(L_a)₂]_n (L_a = (E)-2,6-diisopropyl-N-(pyrid-3-ylmethylene)aniline) (1) and [(AgOC₂F₃)₂(L_d)₂]_n (L_d = (E)-2,6-diisopropyl-N-(pyrid-4-ylmethylene)aniline) (5), and three discrete complexes, [(AgO₂C₂F₃)₂(L_a)₄] (2), [AgO₂C₂F₃(L_b)₂] (L_b = (E)-N-(pyrid-4-ylmethylene)aniline) (3) and [(AgOC₂F₃)₂(L_c)₄] (L_c = (E)-2,6-dimethyl-N-(pyrid-4-ylmethylene)aniline) (4). The structures were determined by MS, FT-IR and NMR spectroscopies, elemental analysis and single crystal XRD. 1 is an organometallic coordination polymer with silver in η¹-arene coordination, but is a discrete dimeric complex 2 when crystallized from warm diethylether. The geometries around silver(I) in 1 and 4 are tetrahedral, ‘inverted seesaw’ in 2 and T-shaped in 3 and in all the anion seems to play a role. Ag(I) centers in 5 have distorted trigonal bipyramid and inverted seesaw geometries. The trifluoroacetate anions in these complexes display variable monodentate and short bridging coordination patterns. All complexes absorb and strongly emit UV-Vis radiation at room temperature.     

Development of dissociative force field for all-atomistic molecular dynamics calculation of fracture of polymers

A dissociative force field for all-atomistic molecular dynamics calculations has been developed to investigate impact fracture of polymers accompanying dissociation of chemical bonds of polymer main chain. Energy of dimer molecules was evaluated as a function of both bond-length b and bond-angle θ by CASPT2 calculations, whose quality is enough to describe dissociation of chemical bonds. Because we found that the bond dissociation energy D decreases with increasing bond-angle, we employed the Morse-type function V_Bond(b, θ) = {D − V_Angle(θ)}[1 − exp{−α(b − b₀) − β(b − b₀)²}] where a quartic function V_Angle(θ) = k₁(θ − θ₀) + k₂(θ − θ₀)² + k₃(θ − θ₀)³ + k₄(θ − θ₀)⁴ . This function reproduced well the CASPT2 potential energy surface in a wide range of b and θ. The parameters have been obtained for four popular glassy polymers, polyethylene, poly(methyl methacrylate), poly(styrene), and polycarbonate. © 2019 Wiley Periodicals, Inc.     

Determination of the energy distribution in unimolecular reactions under soft collision conditions

The effective energy distribution of activated molecules at the time of reaction under soft collision conditions ?′(E, ω) is pressure dependent and therefore difficult to recover from unimolecular decomposition data obtained at different pressures. We show in this work that the part of this function restricted to the condition that the collision frequency ω has to be equal to the microscopic rate constant k(E) at the energy given ?″[E,ω = k(E)] is a reasonable approximation to the input energy distribution ?(E) for quite soft collisions. This function is not pressure dependent and then recoverable at least in principle and as a matter of fact is not conceptually far from the function that the already reported deconvolution methods based in physical approximations attempt to recover. The deconvolution methods have been checked under soft collision conditions. We have found that the input energy distribution is recovered with reasonable accuracy for energies transferred by collision 〈ΔE〉 above 5 kcal mol^?1, conditions common in polyatomic systems.     

Relative Stability of Peptide Sequence Ions Generated by Tandem Mass Spectrometry

We report the use of unimolecular dissociation by infrared radiation for gaseous multiphoton energy transfer to determine relative activation energy (E_a,laser) for dissociation of peptide sequence ions. The sequence ions of interest are mass-isolated; the entire ion cloud is then irradiated with a continuous wave CO₂ laser, and the first order rate constant, k_d, is determined for each of a series of laser powers. Provided these conditions are met, a plot of the natural logarithm of k_d versus the natural logarithm of laser power yields a straight line, whose slope provides a measure of E_a,laser. This method reproduces the E_a values from blackbody radiative dissociation (BIRD) for the comparatively large, singly and doubly protonated bradykinin ions (nominally y ₉ and y ₉ ²⁺ ). The comparatively small sequence ion systems produce E_a,laser values that are systematic underestimates of theoretical barriers calculated with density functional theory (DFT). However, the relative E_a,laser values are in qualitative agreement with the mobile proton model and available theory. Additionally, novel protonated cyclic-dipeptide (diketopiperazine) fragmentation reactions are analyzed with DFT. FT-ICR MS provides access to sequence ions generated by electron capture dissociation, infrared multiphoton dissociation, and collisional activation methods (i.e., b _n , y _m , c _n , z _m ^• ions).     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Relative rate constants for reactions of HFC 152a, 143, 143a, 134a,and HCFC 124 with F or Cl atoms and for CF2CH3, CF2HCH2, and CF3CFH radicals with F2, Cl2, and O2

Relative rate experiments using UV photolysis of F₂ or Cl₂ have been used to determine rate constant ratios for several hydrofluorocarbon (HFC) reactions with Cl or F atoms and for HFC alkyl radicals with molecular halogens. For mixtures with F₂ present, dark reactions are, also, observed which are attributed to thermal dissociation of the F₂ to form F atoms. At 296 K, the rate of reaction (1a) [CF₂HCH₃ + F → CF₂CH₃ + HF] relative to (1b) [CF₂HCH₃ + F → CF₂HCH₂ + HF] is k_1a/k_1b = 0.73 (±0.13) and is independent of T (= 262–348 K). At 296 K, the ratio of reaction (2a) [CF₂HCH₂F + F → products] to that of (k_1a + k_1b) is (k_1a + k_1b)/k_2a = 2.7 (±0.4), and for reaction (2b) [CF₃CH₃ + F → products] (k_1a + k_1b)/k_2b = 22 ± 12. The temperature dependence (263–365 K) of the rate constant of reaction (3) [CF₃CFH₂ + Cl → products] relative to reaction (4) [CF₃CFClH + Cl → products] is k₃/k₄(±10%) = 1.55 exp(?300 K/T). For the alkyl radicals formed from HFC 152a (CF₂HCH₂ and CF₂CH₃) and from HFC 134a (CF₃CFH), rate constants for the reactions with F₂ and Cl₂ were measured relative to their reactions with O₂. The rate constant of reaction (5cl) [CF₂CH₃ + Cl₂ → CF₂ClCH₃ + Cl] relative to (5o) [CF₂CH₃ + O₂ → CF₂(O₂)CH₃] is k_5cl/k_5o(±15%) = 0.3 exp(200 K/T). For reaction (5f) [CF₂CH₃ + F₂ → CF₃CH₃ + F], k_5f/k_5o(±35%) = 0.23. The ratio for reaction (6f) [CF₂HCH₂ + F₂ → CF₂HCH₂F + F] relative to (6o) [CF₂HCH₂ + O₂ → CF₂HCH₂O₂] is k_6f/k_6o(±40%) = 1.23 exp(?730 K/T). The rate constant ratio for reaction (8cl) [CF₃CFH + Cl₂ → CF₃CFClH + Cl] relative to reaction (8o) [CF₃CFH + O₂ → CF₃CFHO₂] is k_8cl/k_8o(±18%) = 0.16 exp(?940 K/T). For reaction (8f) [CF₃CFH + F₂ → CF₃CF₂H + F], k_8f/k_8o(±35%) = 0.6 exp(?860 K/T). © 1993 John Wiley & Sons, Inc.