Dynamical theory of proton tunneling transfer rates in solution: general formulation

A general dynamical theory is presented for the rate constant of weak coupling, nonadiabatic proton-tunneling reactions in solution. The theory incorporates the critical role of the solvent and the vibration of the separation of the heavy particles between which the proton transfers, including their dynamics. The formulation which allows the computation of the quantum rate constant k via classical molecular dynamics simulation techniques is presented, as are a number of approximate analytic results for k in a variety of different important regimes. The frequent appearance of (nearly) classical Arrhenius behavior for k — even though the intrinsic reactive event is quantum proton tunneling — is discussed, together with the solvent and vibrational contributions to the apparent activation energy. In certain weak solvation limits, however, non-Arrhenius behavior for k is found and is related to vibrational Franck-Condon features in the reaction.     

Toward an accurate and efficient semiclassical surface hopping procedure for nonadiabatic problems

The derivation of a semiclassical surface hopping procedure from a formally exact solution of the Schrodinger equation is discussed. The fact that the derivation proceeds from an exact solution guarantees that all phase terms are completely and accurately included. Numerical evidence shows the method to be highly accurate. A Monte Carlo implementation of this method is considered, and recent work to significantly improve the statistical accuracy of the Monte Carlo approach is discussed.     

Polymer coupling and theory of on-random crosslinking: an analytical solution

Gel formation is an important feature in free-radical polymer coupling. Due to the different possible combination reactivities of each polymer backbone radical, polymer chains are crosslinked in a non-random manner. Equations of the moments have been derived to predict the pregel molecular weight development and the crosslink density at gel point. This work provides an analytical solution for the differential equations. The model agrees with the Flory-Stockmayer gelation theory under the condition of random crosslinking. The magnitude of deviations from the classical theory for non-random crosslinking depends on the product of the radical termination reactivity ratios (r₁r₂), the ratio of the rate constants of backbone radical generation (k), the ratio of the weight-average chain lengths of primary polymers (y), and the polymer weight fractions (w₂).     

Extended spin-boson model for nonadiabatic hydrogen tunneling in the condensed phase

A nonadiabatic rate expression for hydrogen tunneling reactions in the condensed phase is derived for a model system described by a modified spin-boson Hamiltonian with a tunneling matrix element exponentially dependent on the hydrogen donor-acceptor distance. In this model, the two-level system representing the localized hydrogen vibrational states is linearly coupled to the donor-acceptor vibrational mode and the harmonic bath. The Hamiltonian also includes bilinear coupling between the donor-acceptor mode and the bath oscillators. This coupling provides a mechanism for energy exchange between the two-level system and the bath through the donor-acceptor mode, thereby facilitating convergence of the time integral of the probability flux correlation function for the case of weak coupling between the two-level system and the bath. The dependence of the rate constant on the model parameters and the temperature is analyzed in various regimes. Anomalous behavior of the rate constant is observed in the weak solvation regime for model systems that lack an effective mechanism for energy exchange between the two-level system and the bath. This theoretical formulation is applicable to a wide range of chemical and biological processes, including neutral hydrogen transfer reactions with small solvent reorganization energies.     

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.     

Trajectory-based solution of the nonadiabatic quantum dynamics equations: an on-the-fly approach for molecular dynamics simulations

The non-relativistic quantum dynamics of nuclei and electrons is solved within the framework of quantum hydrodynamics using the adiabatic representation of the electronic states. An on-the-fly trajectory-based nonadiabatic molecular dynamics algorithm is derived, which is also able to capture nuclear quantum effects that are missing in the traditional trajectory surface hopping approach based on the independent trajectory approximation. The use of correlated trajectories produces quantum dynamics, which is in principle exact and computationally very efficient. The method is first tested on a series of model potentials and then applied to study the molecular collision of H with H(2) using on-the-fly TDDFT potential energy surfaces and nonadiabatic coupling vectors.     

Toward a statistical mechanical theory for water: analytical theory for a short-ranged reference system

Starting from a realistic Hamiltonian and making use of recent findings that the properties of associating fluids are determined primarily by short-ranged interactions, this methodology has been implemented using statistical mechanical approaches and thermodynamic perturbation theory for the TIP4P model of water. We focus on the short-range reference system for which an analytic expression for the Helmholtz free energy is derived. It is found that the model (reference system) exhibits, in addition to a faithful representation of the structure of water, the same features that are characteristic for real water, namely, (i) the temperature of the density maximum and its pressure dependence, including the inflection point at high pressures and (ii) the temperature minima of the constant pressure heat capacity and the coefficient of isothermal compressibility.     

A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator

A justification is given for the validity of a nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method. The method is based on a propagator that combines the single surface HK SC-IVR method [J. Chem. Phys. 84, 326 (1986)] and Herman's nonadiabatic semiclassical surface hopping theory [J. Chem. Phys. 103, 8081 (1995)], which was originally developed using the primitive semiclassical Van Vleck propagator. We show that the nonadiabatic HK SC-IVR propagator satisfies the time-dependent Schrodinger equation to the first order of variant Planck's over 2pi and the error is O(variant Planck's over 2pi(2)). As a required lemma, we show that the stationary phase approximation, under current assumptions, has an error term variant Planck's over 2pi(1) order higher than the leading term. Our derivation suggests some changes to the previous development, and it is shown that the numerical accuracy in applications to Tully's three model systems in low energies is improved.     

Collision-induced absorption in molecular gases: an analytical solution

An analytic solution for the translational two-body contribution to the quadrupole-induced dipole correlation function is presented. It allows one to evaluate the far-infrared collision-induced absorption spectra of molecular systems, composed of quasi-spherical molecules at low density from the knowledge of few parameters derived from different experiments. The comparison with a translational correlation function extracted from the absorption spectrum of gaseous N₂, under the same approximations, is also discussed.     

Simplified semiclassical theory for the electronic shell structure of metallic clusters

We present a simplified theory for the electronic shell structure of large metallic clusters by extending the semiclassical analysis by Gutzwiller, Balian and Bloch. Thus analytical results are given for the shell effect in the Fermi energy and cohesive energy. In particular, results for the temperature dependence of the shell structure are given. New results are presented for the shell structure in the ionization potential and photoyield and for the existence of electronic- versus atomic shell structure as a function of cluster size and temperature. We obtain good quantitative agreement with previous numerical calculations and with experiments on Na clusters.     

Importance of anharmonicity, recrossing effects, and quantum mechanical tunneling in transition state theory with semiclassical tunneling. A test case: the H2 + Cl hydrogen abstraction reaction

The hydrogen abstraction reaction from H2 by the Cl atom is studied by means of the variational transition state theory with semiclassical tunneling coefficients on the BW2 potential energy surface. Vibrational anharmonicity and coupling between the bending modes are taken into account. The occurrence of trajectories that recross the transition state is estimated by means of the canonical unified statistical method and by classical trajectories calculations. Different semiclassical methods for tunneling calculations are tested. Our results show that anharmonicity has a small but nonnegligible effect on the thermal rate constants, recrossing can be neglected, and tunneling is adequately described by the least-action approximation, and less successfully by the large-curvature version 3 approximation. However, the large-curvature version 4 and small-curvature approximations lead to a severe underestimation of tunneling. Thermal rate constants calculated using transition state theory including anharmonicity and tunneling agree very well with accurate quantal thermal rate constants over a wide temperature range, although the improvement over the harmonic transition state theory with the microcanonically optimized semiclassical tunneling approximation (based on version 3 of the large-curvature tunneling method) used in a previous study of this reaction is only marginal.     

Exact semiclassical solution for the time evolution of a quantum-mechanical system in a circularly polarized monochromatic driving field

A closed-form exact solution is presented for the time evolution operator of a nonrelativistic hydrogen atom driven by a circularly polarized monochromatic light beam. The solution has the form U(t, 0) = exp(?itF)exp(?itG), where F and G are time independent operators. All temporal effects of the oscillating field are included. Extension to other systems is easily made. A generalization of the rotating wave approximation to multilevel systems is also presented.     

CMMSE-17: general analytical laws for metabolic pathways

In this paper a general formulation for the kinetics of multi-step enzymatic reactions is presented. The optimal enzyme and metabolite concentrations are studied for the problem of minimizing the operation time in which the substrate is converted into the product. We give an analytic solution for three different kinetic models for both the unbranched and branched cases. Sufficient conditions for the optimality of the solution are studied. Several examples are presented.     

Catalytic mechanism in cyclic voltammetry at disc electrodes: an analytical solution

The theory of cyclic voltammetry at disc electrodes and microelectrodes is developed for a system where the electroactive reactant is regenerated in solution using a catalyst. This catalytic process is of wide importance, not least in chemical sensing, and it can be characterized by the resulting peak current which is always larger than that of a simple electrochemical reaction; in contrast the reverse peak is always relatively diminished in size. From the theoretical point of view, the problem involves a complex physical situation with two-dimensional mass transport and non-uniform surface gradients. Because of this complexity, hitherto the treatment of this problem has been tackled mainly by means of numerical methods and so no analytical expression was available for the transient response of the catalytic mechanism in cyclic voltammetry when disc electrodes, the most popular practical geometry, are used. In this work, this gap is filled by presenting an analytical solution for the application of any sequence of potential pulses and, in particular, for cyclic voltammetry. The induction principle is applied to demonstrate mathematically that the superposition principle applies whatever the geometry of the electrode, which enabled us to obtain an analytical equation valid whatever the electrode size and the kinetics of the catalytic reaction. The theoretical results obtained are applied to the experimental study of the electrocatalytic Fenton reaction, determining the rate constant of the reduction of hydrogen peroxide by iron(II).     

Semiclassical theory of electronically nonadiabatic chemical dynamics: incorporation of the Zhu-Nakamura theory into the frozen Gaussian propagation method

The title theory is developed by combining the Herman-Kluk semiclassical theory for adiabatic propagation on single potential-energy surface and the semiclassical Zhu-Nakamura theory for nonadiabatic transition. The formulation with use of natural mathematical principles leads to a quite simple expression for the propagator based on classical trajectories and simple formulas are derived for overall adiabatic and nonadiabatic processes. The theory is applied to electronically nonadiabatic photodissociation processes: a one-dimensional problem of H2+ in a cw (continuous wave) laser field and a two-dimensional model problem of H2O in a cw laser field. The theory is found to work well for the propagation duration of several molecular vibrational periods and wide energy range. Although the formulation is made for the case of laser induced nonadiabatic processes, it is straightforwardly applicable to ordinary electronically nonadiabatic chemical dynamics.     

Temperature and pressure dependences of tunneling rate constant: density-functional theory potential-energy surface for H-atom transfer in the fluorene-acridine system

Temperature and pressure dependences of rate constants for solid phase tunneling reactions are analytically considered within the framework of modified theory of radiationless transitions, taking into account the intermolecular and soft intramolecular promotive vibrations of reagents. This treatment allows us to describe theoretically the process of atomic tunneling and the effect of temperature on the potential barrier and reorganization of the reagents. The influence of external pressure appears in our treatment as a static reduction of widths and heights of the potential barrier with hydrostatic compression of the matrix, and also as an increase of frequencies of promotive vibrational modes owing to anharmonicity. The theoretical results are used to interpret experimental data concerning the effect of temperature and pressure on the hydrogen-atom tunneling in the fluorene-acridine reaction system. It has been shown that by taking into account the contributions from reorganization of the reagents, which statically reduce the tunneling barrier and are related to four types of promotive vibrations (translational, librational, and two low-frequency intramolecular modes at 95 and 238 cm(-1)), one can reproduce the experimental data available in the literature. The parameters of the reaction system required for this analysis are calculated from two-dimensional potential-energy surfaces generated at the DFT-B3LYP/6-31G* level.     

Severe hyperlipidemia,an analytical problem: Enzymic clearing,a simple solution

Biological samples, in this case human blood and plasma samples, pose unique challenges to the analytical chemist. These samples are not only inherently unstable but also vary widely in their composition. The latter phenomenon makes our necessary reliance on high-throughput automated instrumentation difficult. This review describes both the deleterious effects that elevated levels of serum solids, particularly triglycerides, have on analytical measurements and the possible remedial actions available to us.     

Collisional energy transfer with statistical energy exchange: an analytical solution in the statistical limit

Collisional energy transfer between highly vibrationally excited molecules and bath gas is considered with a statistical kernel, describing energy exchange in complex-forming collisions. Knowledge of the bilinear formula for the Laguerre polynomials offers a means for determining eigenvalues and eigenfunctions of the kernel. An exact solution to master equation for the conditional probability is given as an expansion in terms of these eigenfunctions. The bulk averages of internal energy moments and energy transfer moments are calculated analytically.     

Direct interchange reaction in a homopolymer melt. Evolution of the MWD: an analytical solution

Kinetics of a direct interchange reaction in a homopolymer melt is studied theoretically. Relaxation of the molecular weight distribution to its most probable (Flory) stationary form^[1] is considered. To this end, the time‐dependent generating function of the transient distribution is calculated analytically. Peculiarities of the relaxation process are investigated for two kinds of the initial distribution, namely, the sum of two Flory distributions with different number averages N₁ and N₁, and the delta‐function. The former case describes a blend of two polydisperse polymers whereas the latter corresponds to a monodisperse melt. In each case, the dependencies of the differential molecular weight distribution and the weight‐ and z‐average polymerization degrees on the number of interchanges per number average chain are obtained in the explicit form. Earlier studies of this problem are briefly discussed.     

Instanton theory for the tunneling splitting of low vibrationally excited states

We develop the instanton theory for calculating the tunneling splitting of excited states. For the case of low vibrational quantum states we derive a canonically invariant formula which is applicable to a multidimensional system of arbitrary Riemannian metric. The effect of multidimensionality in relation to the vibrational excitation is explained in terms of the effective frequencies along the instanton trajectory. The theory is demonstrated to work well by taking HO2 molecule as an example.