Barrier Curvature Research Articles

Modeling proton mobility in acidic zeolite clusters: II. Room temperature tunneling effects from semiclassical rate theory

We have developed a novel semiclassical transition state theory (SC-TST) for truncated parabolic barriers, based on the formulation of Hernandez and Miller [Chem. Phys. Lett. 214, 129 (1993)]. Our SC-TST rate coefficient has the form kSC-TST=kTST⋅Γ, where Γ depends on the zero point corrected barrier, ΔE0, and the barrier curvature, |ωF‡|. Our rate expression is stable to arbitrarily low temperatures, as opposed to purely harmonic SC-TST, because we identify the maximum possible semiclassical action in the reaction coordinate. For low temperatures, we derive an analytical approximation for Γ that is proportional to eβ ΔE0. We develop a theory for the tunneling crossover temperature, Tx, yielding kBTx≅ℏ|ωF‡|ΔE0/(2π ΔE0−ℏ|ωF‡|ln 2), which generalizes the harmonic theory for systems with large but finite barriers. We have calculated rate coefficients and crossover temperatures for the O(1)→O(4) jump in H–Y and D–Y zeolites, yielding Tx=368 K and 264 K, respectively. These results suggest that tunneling dominates proton transfer in H–Y up to and slightly above room temperature, and that true proton transfer barriers are being underestimated by neglecting tunneling in the interpretation of experimental mobility data.

Nonlocal effects in the nucleus-nucleus fusion cross section.

Effects of the nonlocality of factorizable potentials are taken into account in the calculation of nucleus-nucleus fusion cross section through an effective mass approach. This cross section makes use of the tunneling factor calculated for the nonlocal barrier, without the explicit introduction of any result coming from coupled channel calculation, besides the approximations of Hill-Wheeler and Wong. Its new expression embodies the nonlocal effects in a factor which redefines the local potential barrier curvature. Applications to different systems, namely, $^{16}\mathrm{O}$${+}^{59}$Co, $^{16,18}\mathrm{O}$${+}^{58,60,64}$Ni, and $^{16,18}\mathrm{O}$${+}^{63,65}$Cu are presented, where the nonlocal range is treated as a free parameter.

An effective barrier model for describing quantum mechanical activated rate processes in condensed phases

An analytic variational model is developed to treat quantum activated rate processes in condensed phase systems. An effective parabolic barrier curvature in this model describes both the barrier nonlinearity along the reaction coordinate as well as the influence of dissipation from the fluctuating condensed phase environment. The theory is developed by exploiting the relationship between the statistics of Feynman path integral centroid densities and the rate of quantum activated events. The result is numerically tested on a model activated dynamics problem and applied to estimate the quantum tunneling correction to an SN2 reaction in water.

Reaction dynamics on a fluctuating potential

The steady state rate constant for passage over a fluctuating potential barrier is derived from the Fokker–Planck equation using the stable states picture. The time scale of fluctuations in barrier height and position do not influence the reactive frequency; however, the reactive frequency does depend upon the time scale of fluctuations in the barrier curvature. If these are fast relative to [ω2m+(ξ/2)2]1/2, where ωm is the mean barrier frequency and ξ is the friction for motion along the reaction coordinate, then the crossing takes place on the potential of mean force. If the barrier fluctuations are slow on the same time scale, then the rate is an average over that for the distribution of barriers. In both of these limits the form of the Kramers solutions is recovered, whereas in the intermediate case, the rate also depends explicitly on the relaxation time scale of the fluctuations. A physically reasonable model for the decay of fluctuations in the potential is proposed and used to obtain an analytic expression for the rate that covers the range between these two limits.

Thermally activated escape from a Lennard-Jones potential well

The Kramers theory of chemical kinetics is modified in order to describe the escape of particles from a potential well of the Lennard-Jones type, for which there is no well-defined barrier position or curvature. Approximate analytical methods are used to derive from the Fokker-Planck equation two distinct formulas for the escape rate, each valid in a different regime of the friction constant β. The large-β result is seen to be consistent with that obtained from the Smoluchoeski equation, and it shows that the resistance to escape is dominated by a purely diffusive component that causes nonequilibrium effects to be felt well down into the reactant region. On the other hand, the small-β result describes a situation in which diffusional resistance is of minor importance, and the escape rate is determined largely by the characteristics of the well itself. The approximate formulas give a reasonably good picture of the transition between the two kinds of limiting behavior.

Molecular dynamics of the A+BC reaction in rare gas solution

Molecular dynamics are computed for model atom transfers A+BC→AB+C in rare gas solvents at liquid densities. We find that the reaction dynamics can be understood in terms of a simple picture which consists of three stages: (1) activation of reactants, (2) barrier crossing, and (3) deactivation of products. The effects seen in stages (1) and (3) can be largely interpreted in terms of existing models of energy and phase decay in solution, while the effects seen in stage (2) can be largely interpreted in terms of gas phase A+BC barrier crossing dynamics. We find that transition state theory is in perfect agreement with the simulations for the 20 and 10 kcal/mol barrier reactions and is a very good description for a 5 kcal/mol reaction barrier. At low barrier curvature, dynamical effects due to the solvent are shown to induce some recrossings of the transition state barrier, thus causing rate constants calculated by simple transition state theory to be slightly too high. The Grote–Hynes modification of transition state theory, which considers the effect of the time dependent friction of the solvent on the dynamics at the transition state, predicts corrections to the rate constants in good agreement with the results from the simulations.

Nonequilibrium solvation dynamics in solution reactions

We construct a theoretical framework for the description of nonequilibrium solvation and solvent participation in the reaction coordinate for solution reactions. The framework is illustrated by a model of reactive dipole isomerization. We show that a multidimensional reaction coordinate picture is equivalent to a one dimensional description in which a generalized friction characterizes and quantifies nonequilibrium solvation effects on the reaction rate. The adiabatic regime where equilibrium solvation and mean potential ideas are correct is identified. Several distinct regimes of nonequilibrium solvation are identified and described in molecular terms. In the effective mass regime, equilibrium solvation ideas give the reaction barrier curvature correctly, but solvent inertia modifies the barrier passage rate. In the nonadiabatic regime, the solvent is ‘‘frozen’’ during the barrier passage and cannot provide equilibrium solvation. In the polarization caging regime, the reacting species adjust to the moving solvent, rather than vice versa, and the solvent is heavily involved in the reaction coordinate. The rate constant in each of these regimes is related to reactive and solvent dynamics.

Dynamical polar solvent effects on solution reactions: A simple continuum model

The influence of polar solvent dynamics in solution reactions is investigated for a simple model. In this model, a charge is subject to a chemical free energy barrier, the successful crossing of which constitutes reaction. The retarding influence of the time dependent solvent polarization is described at the continuum Debye relaxation level. The reaction rate constant k is determined as a function of the barrier curvature, the charge-solvent interaction strength, and the solvent polarization relaxation time. The reduction of k with increasingly slower solvent relaxation is found to depend sensitively on the magnitude of charge-solvent interactions. When the latter are weak, the behavior of k differs qualitatively from a standard Kramers prediction. The analogous reaction problem for a dipole is discussed briefly. Directions for a more realistic treatment of polar solvent effects on reaction rates are described.

A semi-numerical approach to the construction and fitting of triatomic potential energy surfaces

The Wall-Porter method of rotating a Morse function to construct potential energy surfaces for collinear atom-diatom chemical reactions is modified by using a numerical spline interpolation technique. This procedure is shown to have great flexibility in giving a potential surface desired characteristics, such as the saddle-point location and the barrier height and curvature at this point, for either symmetrical or unsymmetrical reactions. Extensions of the method to non-reactive and non-collinear potential surfaces are also discussed.

Effects of thermal excitation and quantum-mechanical transmission on photothreshold determination of Schottky barrier height

Quantum-mechanical electron transmission effects on photothreshold determination in Schottky barriers with high electric fields are shown to produce a strong effect on apparent photothresholds without a concomitant large change in shape of the photoresponse curve. Quantum-mechanical electron transmission probabilities are determined by numerical integration of the Schroedinger equation for a Schottky barrier model which includes Thomas-Fermi shielding in the metal, image force lowering in the semiconductor, and conservation of transverse momentum at the metal-semiconductor interface. Results are presented for <111> n-type Si and n-type GaAs. In the case of Si, conservation of transverse momentum at the interface produces a discontinuity in the electron kinetic energy associated with motion normal to the interface but does not greatly affect the transmission probability. Using these results, the difference between the true barrier height, qΦ B , and the apparent barrier height, qΦ A , is calculated as a function of temperature, doping, and electric field at the metal-semiconductor interface. qΦ A is defined as the intercept on the hv axis of the slope of a plot of (photocurrent per absorbed photon) 1 2 versus quantum energy, hv. Effects of barrier curvature due to doping are negligible for impurity densities below about 10 18 cm −3 for Si and 10 17 cm −3 for GaAs. Values of q( Φ B - Φ A ) in excess of 50 meV occur in many cases. The corrections derived are used to reinterpret previously reported measurements of the dependence of Schottky barrier height on temperature and impurity concentration.

Systematics of reaction cross sections and interaction barriers for charged particles

Experimentally determined total reactin cross sections ${\ensuremath{\sigma}}_{R}$ have been used to obtain the systematics of interaction barriers in reactions between charged nuclei. The real potential was assumed to be parabolic at its maximum and hence the characteristic parameters are barrier height, radius, and curvature, ${E}_{0}$, ${R}_{0}$, and $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$, respectively. Barrier heights and radii follow clear systematic trends and therefore these systematics can be used to predict total reaction cross sections for energies close to and greater than the barrier. Experimental data for incident energies below the barrier are sparse and therefore they cannot be well systematized at this time. The cross section patterns for low incident energies seem to reflect individuality of the collision partners.NUCLEAR REACTIONS Total cross sections ${\ensuremath{\sigma}}_{R}$ for charged particle reactions are systematized in terms of a real inverted parabolic barrier. Graphical interpolation and extrapolation of the interaction barrier ${E}_{0}$ permit prediction of ${\ensuremath{\sigma}}_{R}$.

Calculation of heavy-atom kinetic isotope effects: Barrier curvature and internal coordinate effects with three-element reaction coordinates

Effects of barrier curvature and of a restricted method for selection of values for off-diagonal force constants in the transition state were studied by computation for several three-element reaction coordinates assumed for decomposition of formic acid; the temperature independent (TIF) and temperature dependent factors (TDF) of the rate constant ratio k/k′ = k12c/k13c were calculated. As barrier curvature increases, the area in eigenvector space corresponding to valid transition states decreases in extent and alters in shape; these changes depend on the method of selection of certain secondary off-diagonal force constants. TIF is not affected. TDF is affected, particularly in the part of eigenvector space where the extent of the allowed region is being reduced by barrier curvature. While shapes and values associated with the TIF and TDF surfaces in eigenvector space can be related at least approximately to the input F- and G-matrix properties in some cases, such accounting is generally not accurate in the first order high temperature approximation.

Three-element reaction coordinates. III. Barrier curvature effects on intramolecular kinetic isotope effects

A study was made of the influence of curvature of the potential barrier on heavy-atom kinetic isotope effects calculated for reaction coordinates comprising nonzero displacements in three internal coordinates. Increase in barrier curvature may increase or decrease the isotope effect. In addition to the expected dependence on transition state bond force constants and geometry, reaction coordinate composition, and barrier curvature, the results depend in most cases on the specific selection of minor transition state force constants (required to meet the input restrictions of the method). Although chemically and physically reasonable choices of certain transition state parameters result in drastic reduction in redundancy of the output, further reduction depends on the development of criteria for assigning the barrier curvature and of rationale for selection of off-diagonal force constants. (auth)

Effects of variations in reaction coordinate eigenvalue on heavy-atom kinetic isotope effects: Comparison of two methods for curving the potential barrier

The formic acid molecule is used as a model for a computational comparison of the Willi-Wolfsberg curvature parameter (K) method (Type K calculations) and the Johnston-Bonner-Wilson method (Type II calculations), both including bond order (N) and barrier curvature (ν1‡ ≠ 0) effects, for calculation of 13C kinetic isotope effects involving reaction coordinate eigenvectors consisting nominally of two nonzero elements. Results obtained by the two methods cover similar ranges of value of calculated k12C/k13C. Reaction coordinate eigenvectors (uncontaminated for Type II results) may involve as much as 20% extraneous displacement in Type K results when ν1‡ ≅ 800 i cm−1; however, the effect on (ν1‡/ν′1‡), the temperature independent factor (TIF) in k12C/k13C, is so small as to be of limited practical significance. The remainder (TDF) of the isotopic rate constant ratio for various combinations of K and N (Type K) or of ν1‡ and N (Type II) can be arranged well into magnitude sequence by invocation of details of the input F and G matrices in the first order high temperature approximation. But, Type K and Type II results similar only with respect to barrier curvature (as indicated by value of ν1‡) may be very different. The Type K approach (at least for two-element reaction coordinates) seems the more serviceable for data matching and may be conceptually the simpler. The Type II approach, however, seems the more instructive concerning the influence of specific internal coordinate displacements on values of the kinetic isotope effect.

Effects of variations in reaction coordinate eigenvalue on heavy-atom kinetic isotope effects: Eigenvectors consisting of one and two nonzero elements

The pentatomic formic acid molecule is used as a model for the survey of the influence on calculated heavy atom kinetic isotope effects of curvature of the potential energy surface (ν1‡ = 0–800i cm−1) along the reaction coordinate at the transition state. Restrictions employed are: invariant geometry and basis diagonal force field; light species reaction coordinate eigenvector consisting of one or two nonzero elements only. The temperature independent factor TIF = (ν1‡/ν1′ ‡) in the rate constant ratio k/k′ generally falls as ν1‡ is increased, but the effect is small. The effect for two-element reaction coordinates of this restricted type is likely the largest such, therefore the influence of barrier curvature on ν1‡/ν1′ ‡ is negligible for practical reaction modeling purposes. The remaining part of k/k′, the temperature dependent factor TDF, arising in isotopic shifts of genuine vibrations, can be strongly affected by barrier curvature. In a few cases, TDF rises slightly as ν1‡ is increased; usually it falls, sometimes so rapidly that the sign of ln(k/k′) is reversed. Since ν1‡ is not an experimental variable, its significance to reaction modeling lies in the fact that graphs of ln(k/k′) vs (1/T) for various reaction coordinates with various ν1‡ do not often intersect at large angles; adjustment of ν1‡ can therefore assist in reducing the redundancy of results of calculations based on a variety of models being compared with experimental results. The barrier curvature and reaction coordinate complexity problem are discussed in terms of the first-order high temperature approximation, ``cut-off procedures,'' and other approaches to the simplification and rationalization of ab initio calculations of kinetic isotope effects.

Effects of variations in reaction coordinate eigenvalue and eigenvector on the rate and hydrogen isotope effects of the reaction H+Cl2 → HCl+Cl

Effects on the rate constant k, and on the temperature independent and temperature dependent factors of kh/kD, of variations in the eigenvalue (λ1) and eigenvector (L1) associated with the reaction coordinate have been studied by computation for the reaction H+Cl2 → HCl+Cl. A slightly bent transition state was assumed and its diagonal force field maintained invariant at partial bond levels. The properties of the transition state were varied by systematic manipulation of the three off-diagonal force constants. Values of the several temperature dependent quantities calculated were controlled largely by the interaction force constant between the H···Cl stretch and the H···Cl···Cl bend. Curving the potential barrier affects drastically the range of reaction coordinate ``compositions'' which correspond to acceptable transition states, but neither k nor kH/kD is sufficiently sensitive to barrier curvature to allow an estimate of a ``best value'' for λ1 by comparison of calculated and experimental values. Eigenvector variations have similarly modest effects on k, but the sensitivity of kH/kD seems enough greater that it could be used for model refinement if experimental results of moderate precision were available. Both these sensitivities would be enhanced if the transition state diagonal force field were treated as a parameter in concert with L1. The hypothesized general inverse correlation between isotope effects and reaction rates is shown to be limited by temperature.

Structure of the Fission Transition NucleusU235

Fission-fragment angular distributions for the ${\mathrm{U}}^{234}$ ($n, f$) reaction were measured for incident neutron energies of 200, 300, 400, 500, 600, 700, 843, 998, and 1184 keV. A novel $2\ensuremath{\pi}$ geometry Lexan (polycarbonate) detector was used to detect the fission fragments. The angular distributions show intermediate-angle peaks and change shape rapidly with changes in neutron energy. Based on a sophisticated Hauser-Feshbach analysis, which includes level-width fluctuation corrections of the energy variation of the fission cross section and angular distributions, a description of the highly deformed fission transition nucleus ${\mathrm{U}}^{235*}$ is given. Assignments of the quantum numbers ($K$, $\ensuremath{\pi}$) are made for three low-energy single-particle states, and they are \textonehalf{}+, $\frac{3}{2}+$, and $\frac{3}{2}\ensuremath{-}$. In addition, the presence of another $K=\frac{1}{2}$ state is demonstrated. Assignments of the energy $E$ and the barrier curvature $\ensuremath{\hbar}\ensuremath{\omega}$ of these states are also given.