Escape from a cavity through a small window: turnover of the rate as a function of friction constant

To escape from a cavity through a small window the particle has to overcome a high entropy barrier to find the exit. As a consequence, its survival probability in the cavity decays as a single exponential and is characterized by the only parameter, the rate constant. We use simulations to study escape of Langevin particles from a cubic cavity through a small round window in the center of one of the cavity walls with the goal of analyzing the friction dependence of the escape rate. We find that the rate constant shows the turnover behavior as a function of the friction constant, zeta: The rate constant grows at very small zeta, reaches a maximum value which is given by the transition-state theory (TST), and then decreases approaching zero as zeta-->infinity. Based on the results found in simulations and some general arguments we suggest a formula for the rate constant that predicts a turnover of the escape rate for ergodic cavities in which collisions of the particle with the cavity walls are defocusing. At intermediate-to-high friction the formula describes transition between two known results for the rate constant: the TST estimation and the high friction limiting behavior that characterizes escape of diffusing particles. In this range of friction the rate constants predicted by the formula are in good agreement with those found in simulations. At very low friction the rate constants found in simulations are noticeably smaller than those predicted by the formula. This happens because the simulations were run in the cubic cavity which is not ergodic.     

Barrier crossing dynamics of a charged particle in the presence of a magnetic field: a new turnover phenomenon

In this paper we have investigated the effect of a magnetic field on the barrier crossing rate of a charged particle. At the low friction regime we have observed a new turnover phenomenon for the variation of rate as a function of field strength. Thus although the force due to the magnetic field is not dissipative in nature, it plays a role in the steady state barrier crossing rate similar to that of a dissipative force in the weak damping regime. For appreciable damping strength, the rate monotonically decreases with the increase of field strength. We have demonstrated an interesting resonance effect due to the variation of frequency of the harmonic oscillator associated with the y-component motion at low damping and magnetic field strength.     

Polymer escape from a metastable Kramers potential: path integral hyperdynamics study

We study the dynamics of flexible, semiflexible, and self-avoiding polymer chains moving under a Kramers metastable potential. Due to thermal noise, the polymers, initially placed in the metastable well, can cross the potential barrier, but these events are extremely rare if the barrier is much larger than thermal energy. To speed up the slow rate processes in computer simulations, we extend the recently proposed path integral hyperdynamics method to the cases of polymers. We consider the cases where the polymers' radii of gyration are comparable to the distance between the well bottom and the barrier top. We find that, for a flexible polymers, the crossing rate (R) monotonically decreases with chain contour length (L), but with the magnitude much larger than the Kramers rate in the globular limit. For a semiflexible polymer, the crossing rate decreases with L but becomes nearly constant for large L. For a fixed L, the crossing rate becomes maximum at an intermediate bending stiffness. For the self-avoiding chain, the rate is a nonmonotonic function of L, first decreasing with L, and then, above a certain length, increasing with L. These findings can be instrumental for efficient separation of biopolymers.     

Temperature dependence of excited state proton transfer in ice

We have studied the excited-state proton-transfer rate of four photoacids in ice as a function of temperature. For all four photoacids, we have found a non Arrhenius behavior of the proton-transfer rate constant, k(PT). d(ln k(PT))/d(1/T) decreases as the temperature decreases. The average slope of ln(k(PT))versus 1/T depends on the photoacid strength (pK*). The stronger the photoacid is, the smaller the slope. For the strongest photoacid 2-naphthol-6,8-disulfonate (2N68DS) the largest slope is 35 kJ/mol at about 270 K, and the smallest measured slope is about 8 kJ/mol at about 215 K. We propose that the temperature dependence of k(PT) in ice at the temperature range 270 > T > 200 K can be explained as arising from contributions of two proton-transfer mechanisms over the barrier and tunneling under the barrier. At very low temperatures T < 200 K, the slope of ln(k(PT)) versus 1/T increases again. At about 170 K, the proton-transfer rate is much slower than the radiative rate, and the deprotonated form of the photoacid cannot be detected in the steady-state emission spectrum. At lower temperatures, T < 200 K, the rate further decreases because of a limitation on the reaction caused by the restrictions on the H2O hydrogen reorientations.     

Effects of external electric fields on double proton transfer kinetics in the formic acid dimer

Molecules can be exposed to strong local electric fields of the order of 10(8)-10(10) V m(-1) in the biological milieu. The effects of such fields on the rate constant (k) of a model reaction, the double-proton transfer reaction in the formic acid dimer (FAD), are investigated. The barrier heights and shapes are calculated in the absence and presence of several static homogenous external fields ranging from 5.14 × 10(8) to 5.14 × 10(9) V m(-1) using density functional theory (DFT/B3LYP) and second order M?ller-Plesset perturbation theory (MP2) in conjunction with the 6-311++G(d,p) Pople basis set. Conventional transition state theory (CTST) followed by Wigner tunneling correction is then applied to estimate the rate constants at 25 °C. It is found that electric fields parallel to the long axis of the dimer (the line joining the two carbon atoms) lower the uncorrected barrier height, and hence increase the raw k. These fields also flatten the potential energy surface near the transition state region and, hence, decrease the multiplicative tunneling correction factor. The net result of these two opposing effects is that fields increase k(corrected) by a factor of ca. 3-4 (DFT-MP2, respectively) compared to the field-free k. Field strengths of ～3 × 10(9) V m(-1) are found to be sufficient to double the tunneling-corrected double proton transfer rate constant at 25 °C. Field strengths of similar orders of magnitudes are encountered in the scanning tunneling microscope (STM), in the microenvironment of a DNA base-pair, in an enzyme active site, and in intense laser radiation fields. It is shown that the net (tunneling corrected) effect of the field on k can be closely fitted to an exponential relationship of the form k = aexp(bE), where a and b are constants and E the electric field strength.     

Study of the Reaction Rate,in the Critical Regime,of a Solute Embedded in a Lennard-Jones Crystal

In a recent study, we showed that the Grote-Hynes theory for the chemical reaction rate exhibits critical phenomena near a critical coupling strength and large solvent response times. Therefore it is possible to write (he rate in a scaling form with a universal scaling function that depends only on the short-time behavior of the memory friction kernel. This study involved a friction kernel involving only one relaxation time. In this paper, we apply the same techniques to a crystal friction involving multiple timescales. We show that the rate shows critical behavior and obtain the universal scaling function. We find the asymptotic behavior of the rate in different regimes of interest. In this way, we uncover the rich mathematical structure of the reaction rate associated with the crystal friction with multiple timescales.     

Eliminating chaos in the Belousov–Zhabotinsky reaction by no-delay feedback and delayed feedback

We present numerical simulations of the three-variable model of the Belousov–Zhabotinsky reaction developed by Gyorgyi and Field, where chaos can occur in the free-running system. As the rate of in-flow in the continuous-flow stirred-tank reactor is controlled by the concentration of one of the reactor species through a feedback loop, we find two simple ways to bring the chaotic behavior to periodic behavior. One is to modify the feedback strength without time delay in the feedback loop, and the other is to modify the delay time of the feedback at constant feedback strength. The possible mechanisms for the two ways are discussed.     

The quantum solvation, adiabatic versus nonadiabatic, and Markovian versus non-Markovian nature of electron-transfer rate processes

In this work, we revisit the electron-transfer rate theory, with particular interests in the distinct quantum solvation effect and the characterizations of adiabatic/nonadiabatic and Markovian/non-Markovian rate processes. We first present a full account for the quantum solvation effect on the electron transfer in Debye solvents, addressed previously in J. Theor. Comput. Chem. 2006, 5, 685. Distinct reaction mechanisms, including the quantum solvation-induced transitions from barrier crossing to tunneling and from barrierless to quantum barrier crossing rate processes, are shown in the fast modulation or low viscosity regime. This regime is also found in favor of nonadiabatic rate processes. We further propose to use Kubo's motional narrowing line shape function to describe the Markovian character of the reaction. It is found that a non-Markovian rate process is most likely to occur in a symmetric system in the fast modulation regime, where the electron transfer is dominant by tunneling due to the Fermi resonance.     

Controlling activated processes of nonadiabatically, periodically driven dynamical systems: a multiple scale perturbation approach

We arrive at the escape rate from a metastable state for a system of Brownian particles driven periodically by a space dependent, rapidly oscillating external perturbation (with frequency ω) in one dimension (one of the most important class of nonequilibrium system). Though the problem may seem to be time-dependent, and is poised on the extreme opposite side of adiabaticity, there exists a multiple scale perturbation theory ("Kapitza window") by means of which the dynamics can be treated in terms of an effective time-independent potential that is derived as an expansion in orders of 1/ω to the order ω(-3). The resulting time-independent equation is then used to calculate the escape rate of physical systems from a metastable state induced by external monochromatic field in the moderate-to-large damping limit and to investigate the effect of ω on the resulting rate in conjunction with the thermal energy. With large value of ω, we find that the environment with moderate-to-large damping impedes the escape process of the particle while high amplitude of the periodic driving force allows the particle to cross the barrier with a large escape rate. A comparison of our theoretical expression with numerical simulation gives a satisfactory agreement.     

Dynamic docking and electron-transfer between cytochrome b5 and a suite of myoglobin surface-charge mutants. Introduction of a functional-docking algorithm for protein-protein complexes

Horse myoglobin (Mb) provides a convenient "workbench" for probing the effects of electrostatics on binding and reactivity in the dynamic [Mb, cytochrome b(5)] electron-transfer (ET) complex. We have combined mutagenesis and heme neutralization to prepare a suite of six Mb surface-charge variants: the [S92D]Mb and [V67R]Mb mutants introduce additional charges on the "front" face, and incorporation of the heme di-ester into each of these neutralizes the charge on the heme propionates which further increases the positive charge on the "front" face. For this set of mutants, the nominal charge of Mb changes by -1 to +3 units relative to that for native Mb. For each member of this set, we have measured the bimolecular quenching rate constant (k(2)) for the photoinitiated (3)ZnDMb --> Fe(3+)b(5) ET reaction as a function of ionic strength. We find: (i) a dramatic decoupling of binding and reactivity, in which k(2) varies approximately 10(3)-fold within the suite of Mbs without a significant change in binding affinity; (ii) the ET reaction occurs within the "thermodynamic" or "rapid exchange" limit of the "Dynamic Docking" model, in which a large ensemble of weakly bound protein-protein configurations contribute to binding, but only a few are reactive, as shown by the fact that the zero-ionic-strength bimolecular rate constant varies exponentially with the net charge on Mb; (iii) Brownian dynamic docking profiles allow us to visualize the microscopic basis of dynamic docking. To describe these results we present a new theoretical approach which mathematically combines PATHWAY donor/acceptor coupling calculations with Poisson-Boltzmann-based electrostatics estimates of the docking energetics in a Monte Carlo (MC) sampling framework that is thus specially tailored to the intermolecular ET problem. This procedure is extremely efficient because it targets only the functionally active complex geometries by introducing a "reactivity filter" into the computations themselves, rather than as a subsequent step. This efficiency allows us to employ more computationally expensive and accurate methods to describe the relevant intermolecular interaction energies and the protein-mediated donor/acceptor coupling interactions. It is employed here to compute the changes in the bimolecular rate constant for ET between Mb and cyt b(5) upon variations in the myoglobin surface charge, pH, and ionic strength.     

Dependence of the energies of fusion on the intermembrane separation: optimal and constrained

We calculate the characteristic energies of fusion between planar bilayers as a function of the distance between them, measured from the hydrophobic/hydrophilic interface of one of the two nearest, cis, leaves to the other. The two leaves of each bilayer are of equal composition: 0.6 volume fraction of a lamellar-forming amphiphile, such as dioleoylphosphatidylcholine, and 0.4 volume fraction of a hexagonal-forming amphiphile, such as dioleoylphosphatidylethanolamine. Self-consistent field theory is employed to solve the model. We find that the largest barrier to fusion is that to create the metastable stalk. This barrier is the smallest, about 14.6k(B)T, when the bilayers are at a distance about 20% greater than the thickness of a single leaf, a distance which would correspond to between 2 and 3 nm for typical bilayers. The very size of the protein machinery which brings the membranes together can prevent them from reaching this optimum separation. For even modestly larger separations, we find a linear rate of increase of the free energy with distance between bilayers for the metastable stalk itself and for the barrier to the creation of this stalk. We estimate these rates for biological membranes to be about 7.1k(B)Tnm and 16.7 k(B)Tnm, respectively. The major contribution to this rate comes from the increased packing energy associated with the hydrophobic tails. From this we estimate, for the case of hemagglutinin, a free energy of 38k(B)T for the metastable stalk itself and a barrier to create it of 73 k(B)T. Such a large barrier would require that more than a single hemagglutinin molecule be involved in the fusion process, as is observed.     

Influence of temperature inhomogeneity on product profile of reactions occurring within zeolites

In zeolites, diffusion is often accompanied by a reaction or sorption which in turn can induce temperature inhomogeneities. Monte Carlo simulations of Lennard-Jones atoms in zeolite NaCaA are reported for the presence of a hot zone presumed to be created by a reaction or chemi- or physi-sorption site. These simulations show that the presence of localized hot regions can alter both kinetic and transport properties such as diffusion. Further, we show that enhancement of diffusion constant is greater for systems with larger barrier height, a surprising result that may be of considerable significance in many chemical and biological processes. We find an unanticipated coupling between reaction and diffusion due to the presence of a hot zone in addition to that which normally exists via concentration. Implications of this coupling for the product profile of a reaction are discussed. We also propose a mechanism by which mobility of ions or diffusion of molecular species within biomembranes may take place. Dedicated to Professor C N R Rao on his 70th birthday     

Distance dependent quenching effect in nanoparticle dimers

In this paper, we investigate the emission characteristics of a molecule placed in the gap of a nanoparticle dimer configuration. The emission process is described in terms of a local field enhancement factor and the overall quantum yield of the system. The molecule is represented as a dipolar source, with fixed length and fed by a constant current. We first describe the coupled dimer-molecule system and compare these results to a single sphere-molecule system. Next, the effect of dimer size is investigated by changing the radius of the nanoparticles. We find that when the radius increases, a saturation effect occurs that trends towards the case of a radiating dipole between two flat interfaces, which we refer to as a parallel plate waveguide geometry. An analytical solution for the parallel plate waveguide geometry is presented and compared to the results for the spherical dimer configuration. We use this approximation as a reference solution, and also, it provides useful guidelines to understand the physical mechanism behind the energy transfer between the molecule and the dimer. We find that the emission intensity undergoes a quenching effect only when the inter-nanoparticle gap distance of the dimer is very small, meaning that strong coupling prevails over energy engaged in the heating process unless the molecule is extremely close to the metal surface.     

Semiclassical quantum unimolecular reaction rate theory revisited

We describe a semiclassical quantum unimolecular reaction rate theory derived from the corresponding classical theory developed by Davis, Gray, Rice and Zhao (DGRZ). The analysis retains the intuitively useful mechanistic distinctions between intramolecular energy transfer and reaction, with the consequence that the semiclassical quantum theory version neglects some interference effects in the reaction dynamics. In the limiting case that intramolecular energy transfer is very fast compared to the rate of reaction we show that the DGRZ representation of the rate constant can be transformed, using the Weyl correspondence between quantum operators and classical variables, to the quantum flux–flux correlation function representation of the rate constant. In the more general case that the rate of intramolecular energy transfer influences the reaction dynamics, the semiclassical representation of the Wigner function for a classical system with both quasiperiodic and chaotic motion is used to obtain the reaction rate constant. Our analysis identifies the quantum analogue of the classical bottleneck to intramolecular energy transfer with the scars of unstable periodic orbits; it leads to a flux–flux correlation function representation of the rate constant for intramolecular energy transfer.     

Kinetik der unkatalysierten Urethanbildung

Summary The kinetics of the reaction of phenylisocyanate with 1-butanol have been studied at low concentrations in carbon tetrachloride solution at three temperatures. The present paper describes the relation between the reaction rate and the concentration of the alcohol dimer and the 1:1 alcoholurethane associate. Therefrom we conclude that the monomeric alcohol does not play any significant part in the reaction. We explain the observed rate equation with the second-order term of the reactants with the equality of the productsk _AA K _AA andk _AU K _AU (rate constant and association constant of alcohol dimer and 1:1 alcohol-urethane associate).For the reaction of alcohol dimer H =45.4 kJ mol^–1 and S =– 142J mol^–1K^–1. We interpret this result in terms of a six-cyclic transition state.

     

Temperature effects, arrhenius activation parameters and rate constants for the photochemical reactivity of cyano, boronato, trifluoromethyl and methoxy substituted toluenes in the excited singlet state

Total rate constants of decay (k(t)) as a function of temperature from -45 to +65 degrees C for the compounds 1 and 2 in AN and TFE and 3 and 4 in AN have been determined by fluorescence lifetime measurements. The data have been fit to an equation that assumes that the rate constants of fluorescence (k(f)) and intersystem crossing (k(isc)) are temperature independent, that k(ic) = 0 and that the rate constant of reaction (k(r)) is activated according to the Arrhenius expression. For compounds 1-3, values of k(f) and k(isc) were found to be independent of solvent for any given compound, but k(r) was consistently greater in TFE than AN. For the anisoles 4, the temperature effect was very small, indicating that k(r) did not compete with k(f) + k(isc) and suggesting that an activated intersystem crossing was the dominant temperature-dependent process. The k(r), A and E(a) values obtained for compounds 1-3 were rationalized in terms of their known photochemistry, phototransposition reactions in AN and photoadditions in TFE. The critical reactive intermediate in all cases is a bicycle[3.1.0]hexenyl biradical/zwitterion that is formed in an activated process from S1. This reactive intermediate returns to starting material faster than it rearranges, and therefore an activated internal conversion is a major pathway for deactivation of S1.     

Quantum equilibrium structure for strong nonadiabatic coupling: Reaction rate enhancement

The quantum reactive flux correlation function is computed for a two-level system using an expression for the quantum equilibrium structure appropriate for strong nonadiabatic coupling, in conjunction with quantum–classical Liouville dynamics. The magnitude of the quantum mechanical enhancement of the reaction rate as a result of strong nonadiabatic coupling is studied. The reaction rate is found to increase strongly with an increase in the nonadiabatic coupling strength as well as with a decrease in the temperature. Equilibrium quantum effects increase the ground-state contribution to the rate constant but these effects decrease the excited-state contribution.     

Correlating DFT-Calculated Energy Barriers to Experiments in Nonheme Octahedral Fe(IV) O Species

The experimentally measured bimolecular reaction rate constant, k(2) , should in principle correlate with the theoretically calculated rate-limiting free energy barrier, ΔG(≠) , through the Eyring equation, but it fails quite often to do so due to the inability of current computational methods to account in a precise manner for all the factors contributing to ΔG(≠) . This is further aggravated by the exponential sensitivity of the Eyring equation to these factors. We have taken herein a pragmatic approach for C?H activation reactions of 1,4-cyclohexadiene with a variety of octahedral nonheme Fe(IV) O complexes. The approach consists of empirically determining two constants that would aid in predicting experimental k(2) values uniformly from theoretically calculated electronic energy (ΔE(≠) ) values. Shown in this study is the predictive power as well as insights into energy relationships in Fe(IV) O C?H activation reactions. We also find that the difference between ΔG(≠) and ΔE(≠) converges at slow reactions, in a manner suggestive of changes in the importance of the triplet spin state weight in the overall reaction.     

A laser flash photolysis and quantum chemical study of the fluorinated derivatives of singlet phenylnitrene.

Laser flash photolysis (LFP, Nd:YAG laser, 35 ps, 266 nm, 10 mJ or KrF excimer laser, 10 ns, 249 nm, 50 mJ) of 2-fluoro, 4-fluoro, 3,5-difluoro, 2,6-difluoro, and 2,3,4,5,6-pentafluorophenyl azides produces the corresponding singlet nitrenes. The singlet nitrenes were detected by transient absorption spectroscopy, and their spectra are characterized by sharp absorption bands with maxima in the range of 300-365 nm. The kinetics of their decay were analyzed as a function of temperature to yield observed decay rate constants, k(OBS). The observed rate constant in inert solvents is the sum of k(R) + k(ISC) where k(R) is the absolute rate constant of rearrangement of singlet nitrene to an azirine and k(ISC) is the absolute rate constant of nitrene intersystem crossing (ISC). Values of k(R) and k(ISC) were deduced after assuming that k(ISC) is independent of temperature. Barriers to cyclization of 4-fluoro-, 3,5-difluoro-, 2-fluoro-, 2,6-difluoro-, and 2,3,4,5,6-pentafluorophenylnitrene in inert solvents are 5.3 +/- 0.3, 5.5 +/- 0.3, 6.7 +/- 0.3, 8.0 +/- 1.5, and 8.8 +/- 0.4 kcal/mol, respectively. The barrier to cyclization of parent singlet phenylnitrene is 5.6 +/- 0.3 kcal/mol. All of these values are in good quantitative agreement with CASPT2 calculations of the relative barrier heights for the conversion of fluoro-substituted singlet aryl nitrenes to benzazirines (Karney, W. L. and Borden, W. T. J. Am. Chem. Soc. 1997, 119, 3347). A single ortho-fluorine substituent exerts a small but significant bystander effect on remote cyclization that is not steric in origin. The influence of two ortho-fluorine substituents on the cyclization is pronounced. In the case of the singlet 2-fluorophenylnitrene system, evidence is presented that the benzazirine is an intermediate and that the corresponding singlet nitrene and benzazirine interconvert. Ab initio calculations at different levels of theory on a series of benzazirines, their isomeric ketenimines, and the transition states converting the benzazirines to ketenimines were performed. The computational results are in good qualitative and quantitative agreement with the experimental findings.     

Effect of anisotropic diffusion and external electric field on the rate of diffusion-controlled reactions

In this paper we investigate theoretically the effect of an external electric field on the rate constant of steady-state bulk diffusion-controlled reactions. We generalize previously derived results for isotropic diffusion in the absence of interparticle interaction [J. Chem. Phys. 87, 4622 (1987)] to the case where translational diffusion is anisotropic. A frequently occurring situation of transverse isotropy where D(x)=D(y) not equal to D(z) is considered in detail. We derive the first-order expansion for the reaction rate constant in terms of the electric field strength E, k(E)=k(0) (1+1/2epsilongamma), where gamma=k(0)/4piRD( perpendicular ), epsilon=qER/k(B)T, q is the charge, R is the contact distance, and D( perpendicular ) is the transverse diffusion coefficient. Numerical calculations show that this first-order expansion works well in the whole range of applicability of the Nernst-Einstein relation, i.e., for epsilon<1.