Relaxation of caloric curves on complex potential energy surfaces

Time-dependent caloric curves of model systems with complex energy landscapes are calculated by solving master equation kinetics in stepwise heating or cooling protocols. By considering in detail a simple two-state harmonic model, we show that both the transition temperature and the associated latent heat vary significantly if the sampling time is not long enough. Microcanonical characteristics, including possible S-bends in the caloric curve, are also qualitatively affected by insufficient sampling. The geometry of S-bends as a function of the observation time agrees quantitatively with the predictions of catastrophe theory. For two Lennard-Jones clusters with 13 and 31 atoms the relations between the transition temperatures and the sampling time are shown to follow scaling laws, in agreement with the results of molecular dynamics simulations [J. Chem. Phys. 113, 1315 (2000)].     

Electronic structure contributions to electron-transfer reactivity in iron-sulfur active sites: 1. Photoelectron spectroscopic determination of electronic relaxation

Electronic relaxation, the change in molecular electronic structure as a response to oxidation, is investigated in [FeX(4)](2)(-)(,1)(-) (X = Cl, SR) model complexes. Photoelectron spectroscopy, in conjunction with density functional methods, is used to define and evaluate the core and valence electronic relaxation upon ionization of [FeX(4)](2)(-). The presence of intense yet formally forbidden charge-transfer satellite peaks in the PES data is a direct reflection of electronic relaxation. The phenomenon is evaluated as a function of charge redistribution at the metal center (Deltaq(rlx)) resulting from changes in the electronic structure. This charge redistribution is calculated from experimental core and valence PES data using a valence bond configuration interaction (VBCI) model. It is found that electronic relaxation is very large for both core (Fe 2p) and valence (Fe 3d) ionization processes and that it is greater in [Fe(SR)(4)](2)(-) than in [FeCl(4)](2)(-). Similar results are obtained from DFT calculations. The results suggest that, although the lowest-energy valence ionization (from the redox-active molecular orbital) is metal-based, electronic relaxation causes a dramatic redistribution of electron density ( approximately 0.7ē) from the ligands to the metal center corresponding to a generalized increase in covalency over all M-L bonds. The more covalent tetrathiolate achieves a larger Deltaq(rlx) because the LMCT states responsible for relaxation are significantly lower in energy than those in the tetrachloride. The large observed electronic relaxation can make significant contributions to the thermodynamics and kinetics of electron transfer in inorganic systems.     

Critical dynamics of dimers: implications for the glass transition

The Adam-Gibbs view of the glass transition relates the relaxation time to the configurational entropy, which goes continuously to zero at the so-called Kauzmann temperature. We examine this scenario in the context of a dimer model with an entropy-vanishing phase transition and stochastic loop dynamics. We propose a coarse-grained master equation for the order parameter dynamics which is used to compute the time-dependent autocorrelation function and the associated relaxation time. Using a combination of exact results, scaling arguments, and numerical diagonalizations of the master equation, we find nonexponential relaxation and a Vogel-Fulcher divergence of the relaxation time in the vicinity of the phase transition. Since in the dimer model the entropy stays finite all the way to the phase transition point and then jumps discontinuously to zero, we demonstrate a clear departure from the Adam-Gibbs scenario. Dimer coverings are the "inherent structures" of the canonical frustrated system, the triangular Ising antiferromagnet. Therefore, our results provide a new scenario for the glass transition in supercooled liquids in terms of inherent structure dynamics.     

Exploring the similarities between potential smoothing and simulated annealing

Simulated annealing and potential function smoothing are two widely used approaches for global energy optimization of molecular systems. Potential smoothing as implemented in the diffusion equation method has been applied to study partitioning of the potential energy surface (PES) for N‐Acetyl‐Ala‐Ala‐N‐Methylamide (CDAP) and the clustering of conformations on deformed surfaces. A deformable version of the united‐atom OPLS force field is described, and used to locate all local minima and conformational transition states on the CDAP surface. It is shown that the smoothing process clusters conformations in a manner consistent with the inherent structure of the undeformed PES. Smoothing deforms the original surface in three ways: structural shifting of individual minima, merging of adjacent minima, and energy crossings between unrelated minima. A master equation approach and explicit molecular dynamics trajectories are used to uncover similar features in the equilibrium probability distribution of CDAP minima as a function of temperature. Qualitative and quantitative correlations between the simulated annealing and potential smoothing approaches to enhanced conformational sampling are established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 531–552, 2000     

Crystal growth rates and master curves of poly(ethylene succinate) and its copolyesters using a nonisothermal method

The spherulitic growth of poly(ethylene succinate) (PES) and two PES‐rich copolyesters at constant cooling rate was monitored and recorded using a system of polarized light microscope. Individual experiment of these polyesters lasted 40, 60, and 120 min, respectively. A continuous curve of isothermal growth rates between the melting and glass transition temperatures can be obtained after curve fitting procedures. These curves fit very well with those data points determined in the isothermal experiments, which are time consuming. The continuous data of PES was analyzed with the Hoffman (Lauritzen equation. A transition of regime II → III was found at 70.7 °C, which is very close to the literature values. The maximum growth rate was formulated in the Arrhenius and WLF expressions for the molecular transport term. A master curve of crystal growth rate for PES was constructed based on the continuous data of PES. When the reduced growth rates after normalization were plotted against the reduced temperatures, a universal master curve was observed for PES and two PES‐rich copolyesters. This nonisothermal method provides an efficient and reliable way for studying the crystallization kinetics of polymer and for constructing a universal master curve of crystal growth rate of PES. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 932–939, 2010     

Fewest switches adiabatic surface hopping as applied to vibrational energy relaxation

In this contribution quantum/classical surface hopping methodology is applied to vibrational energy relaxation of a quantum oscillator in a classical heat bath. The model of a linearly damped (harmonic) oscillator is chosen which can be mapped onto the Brownian motion (Caldeira-Leggett) Hamiltonian. In the simulations Tully's fewest switches surface hopping scheme is adopted with inclusion of dephasing in the adiabatic basis using a simple decoherence algorithm. The results are compared to the predictions of a Redfield-type quantum master equation modeling using the classical heat bath force correlation function as input. Thereby a link is established between both types of quantum/classical approaches. Viewed from the latter perspective, surface hopping with dephasing may be interpreted as "on-the-fly" stochastic realization of a quantum/classical Pauli master equation.     

Kinetic processes in non-equilibrium carbon monoxide discharges. II. Self-consistent electron energy distribution functions

Self-consistent electron energy distribution functions (EDFs) in carbon monoxide discharges have been calculated by solving the Boltzmann equation coupled to the vibrational master equation and to the plasma chemical kinetics operating in the discharge. The results show a strong coupling between EDF and the vibrational distribution and the chemical composition of the plasma. The consequent temporal evolution of EDFs is analyzed on the basis of the relevant relaxation times characteristic of different processes occurring under plasma conditions.     

A statistical mechanical theory of proton transport kinetics in hydrogen-bonded networks based on population correlation functions with applications to acids and bases

Extraction of relaxation times, lifetimes, and rates associated with the transport of topological charge defects in hydrogen-bonded networks from molecular dynamics simulations is a challenge because proton transfer reactions continually change the identity of the defect core. In this paper, we present a statistical mechanical theory that allows these quantities to be computed in an unbiased manner. The theory employs a set of suitably defined indicator or population functions for locating a defect structure and their associated correlation functions. These functions are then used to develop a chemical master equation framework from which the rates and lifetimes can be determined. Furthermore, we develop an integral equation formalism for connecting various types of population correlation functions and derive an iterative solution to the equation, which is given a graphical interpretation. The chemical master equation framework is applied to the problems of both hydronium and hydroxide transport in bulk water. For each case it is shown that the theory establishes direct links between the defect's dominant solvation structures, the kinetics of charge transfer, and the mechanism of structural diffusion. A detailed analysis is presented for aqueous hydroxide, examining both reorientational time scales and relaxation of the rotational anisotropy, which is correlated with recent experimental results for these quantities. Finally, for OH(-)(aq) it is demonstrated that the "dynamical hypercoordination mechanism" is consistent with available experimental data while other mechanistic proposals are shown to fail. As a means of going beyond the linear rate theory valid from short up to intermediate time scales, a fractional kinetic model is introduced in the Appendix in order to describe the nonexponential long-time behavior of time-correlation functions. Within the mathematical framework of fractional calculus the power law decay ～t(-σ), where σ is a parameter of the model and depends on the dimensionality of the system, is obtained from Mittag-Leffler functions due to their long-time asymptotics, whereas (stretched) exponential behavior is found for short times.     

Comment on "Automatic estimation of pressure-dependent rate coefficients" (J. W. Allen, C. F. Goldsmith, and W. H. Green, Phys. Chem. Chem. Phys., 2011, 14, 1131-1155)

In this comment we discuss briefly the relationship between phenomenological rate constants and the solution to the time-dependent multiple-well master equation. Attention is focused on obtaining rate constants using the CSE (chemically significant eigenmode) method. In particular we describe briefly how to obtain rate constants when one or more of the chemically significant eigenvalues merges with the IEREs (internal energy relaxation eigenvalues).     

Sampling Potential Energy Surfaces in the Condensed Phase with Many-Body Electronic Structure Methods

Sampling potential energy surfaces (PES) is pivotal for understanding chemical structure, energetics and reactivity and is of special importance for complex condensed-phase systems. Until recently such simulations based on electronic structure theory have been performed only by density functional theory and semiempirical methods. Many-body electronic structure methods, almost routinely used for molecules, have been practically unavailable for sampling PES in the condensed-phase. This has changed during the last few years, as efficient algorithms and software implementations for the evaluation of electronic energies and forces on atoms have been developed, allowing for geometry optimization, molecular dynamics and Monte-Carlo simulations, which was previously unthinkable. Herein, we introduce the theory and software developments and overview the applications in the field, the most encouraging results being obtained for aqueous chemistry. Requiring state-of-the-art computer resources PES sampling with many-body electronic structure methods in the condensed phase provides high-quality benchmarks and will gradually become more available due to fast progress in reduced scaling algorithms and computational technologies.     

Kissinger equation versus glass transition phenomenology

The applicability of the Kissinger equation for the evaluation of apparent activation energy corresponding to glass transition kinetics is examined. Theoretically simulated data based on the generally accepted Tool–Narayanaswamy–Moynihan model were used to represent relevant cases of structural relaxation behavior. The values of the apparent activation energy determined by the Kissinger equation were, despite the linearity of the dependencies, in major disagreement with the original values of ?h ^* used for the simulation of the source data. Furthermore, a large dependence of the ?h _Kis ^* evaluation (performed using the Kissinger equation) on the thermal history of the glass was found. The latter represents an unacceptable systematic error in the methodology, implying the incorrectness of the Kissinger equation usage for the evaluation of “glass transition activation energy”. This study addresses the currently widespread (incorrect) usage of the Kissinger equation for the above-mentioned purpose.     

Efficient model chemistries for peptides. I. General framework and a study of the heterolevel approximation in RHF and MP2 with Pople split-valence basis sets

We present an exhaustive study of more than 250 ab initio potential energy surfaces (PESs) of the model dipeptide HCO-L-Ala-NH(2). The model chemistries (MCs) investigated are constructed as homo- and heterolevels involving possibly different RHF and MP2 calculations for the geometry and the energy. The basis sets used belong to a sample of 39 representants from Pople's split-valence families, ranging from the small 3-21G to the large 6-311++G(2df,2pd). The reference PES to which the rest are compared is the MP2/6-311++G(2df,2pd) homolevel, which, as far as we are aware, is the most accurate PES in the literature. All data sets have been analyzed according to a general framework, which can be extended to other complex problems and which captures the nearness concept in the space of MCs. The great number of MCs evaluated has allowed us to significantly explore this space and show that the correlation between accuracy and computational cost of the methods is imperfect, thus justifying a systematic search for the combination of features in a MC that is optimal to deal with peptides. Regarding the particular MCs studied, the most important conclusion is that the potentially very cost-saving heterolevel approximation is a very efficient one to describe the whole PES of HCO-L-Ala-NH(2). Finally, we show that, although RHF may be used to calculate the geometry if a MP2 single-point energy calculation follows, pure RHF//RHF homolevels are not recommendable for this problem.     

Stochastic simulation of catalytic surface reactions in the fast diffusion limit

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques.     

CF3CH3 --> HF + CF2CH2: a non-RRKM reaction?

The experimental shock tube data recently reported by Kiefer et al. [J. Phys. Chem. A 2004, 108, 2443-2450] for the title reaction at temperatures between 1600 and 2400 K have been compared to master equation simulations using three models: (a) standard RRKM theory, (b) RRKM theory modified by local random matrix theory, which introduces dynamical corrections arising from slow intramolecular vibrational energy randomization, and (c) an ad hoc empirical non-RRKM model. Only the third model provides a good fit of the Kiefer et al. unimolecular reaction rate data. In separate simulations, all three models accurately reproduce the experimental 300 K chemical activation data of Marcoux and Setser [J. Phys. Chem. 1978, 82, 97-108] when the energy transfer parameters are freely varied to fit the data. When experimental energy transfer parameters for a geometrical isomer (1,1,2-trifluoroethane) are used, the standard RRKM model fits the chemical activation data better than the other models, but if energy transfer in the 1,1,1-trifluoroethane is significantly reduced in comparison to the 1,1,2 isomer, then the empirical ad hoc non-RRKM model also gives a good fit. While the ad hoc empirical non-RRKM model can be made to fit the data, it is not based on theory, and we argue that it is physically unrealistic. We also show that the master equation simulations can mimic the Kiefer et al. vibrational relaxation data, which was the first shock tube observation of double-exponential relaxation. We conclude that, until more data on the trifluoroethanes become available, the current evidence is insufficient to decide with confidence whether non-RRKM effects are important in this reaction, or whether the Kiefer et al. data can be explained in some other way.     

Temperature dependence of hole growth kinetics in aluminum-phthalocyanine-tetrasulfonate in hyperquenched glassy water

The temperature (T) dependence of hole growth kinetics (HGK) data that span more than four decades of burn fluence are reported for aluminum-phthalocyanine tetrasulfonate (APT) in fresh and annealed hyperquenched glassy water (HGW) for temperatures between 5 and 20 K. The highly dispersive HGK data are modeled by using the "master" equation based on the two level system (TLS) model described in 2000 by Reinot and Small [J. Chem. Phys. 2000, 113, 10207]. We have demonstrated that thermal line broadening is not enough to account for temperature-dependent HGK for temperatures greater than 10 K. To overcome the discrepancy, the hole growth model must account for thermal hole filling (THF) processes. For the first time, the "master" equation used for HGK simulations is modified to take into account both the temperature dependence of the (single site) absorption spectrum and THF processes, effectively turning off those TLS which do not participate in the hole burning process at higher temperatures. A single set of parameters, some of which were determined directly from the hole spectra, was found to provide satisfactory fits to the HGK data for APT in fresh and annealed HGW for holes burned in the 679.7-676.9 nm range from the high to low energy sides of the Qx absorption band. Furthermore, we propose that HGK modeling at high burn fluences requires that the TLS model be further modified to take into account the existence of extrinsic multiple level systems.     

Exact Generator and its High Order Expansions in Time-Convolutionless Generalized Master Equation: Applications to Spin-Boson Model and Exictation Energy Transfer?

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.     

Molecular dynamics of hydrogen dissociation on an oxygen covered Pt(111) surface

The dissociation of hydrogen on a Pt(111) surface covered with a p(2 x 2) oxygen phase was investigated using quasiclassical, six-dimensional molecular dynamics. The potential energy surface (PES) used in these simulations was obtained by an iterative novelty sampling algorithm. Compared to molecular beam experiments performed under similar conditions, the simulations give an accurate prediction of the reaction probability via a direct dissociation pathway. When compared to previously reported reaction probability curves for the clean Pt(111) surface, we find that the presence of an oxygen overlayer inhibits the direct pathway to hydrogen dissociation. This inhibition is a function of incident energy and cannot be described by a simple site blocking model. An indirect pathway to dissociation, which was observed in experiments, is not properly captured by the PES. Spatially resolved "reaction maps" indicate that the preferred site for hydrogen dissociation on an oxygen covered Pt surface is the top site of the Pt atom farthest from the adsorbed oxygen atom.     

Numerical solution methods for large,difficult kinetic master equations

The kinetics of gas-phase reactions, including pressure-dependent weak collision and non-equilibrium effects, can be modelled using a master equation. In this paper, we address the practical computational problem of finding solutions to such kinetic master equations. The mathematical structure of the master equation can be utilised to develop a number of specialised numerical techniques that are capable of solving the master equation in the presence of difficult numerics and for large problems. The former is important for modelling low temperature and pressure systems, and the latter is important for modelling the large networks of isomerising species common in combustion chemistry applications. We focus on numerical methods that exhibit particular practical use because of their robust nature or scalability to many isomers, or both. Recent developments in linear-scaling methods are highlighted.     

Multiple‐Well,multiple‐path unimolecular reaction systems. I. MultiWell computer program suite

Unimolecular reaction systems in which multiple isomers undergo simultaneous reactions via multiple decomposition reactions and multiple isomerization reactions are of fundamental interest in chemical kinetics. The computer program suite described here can be used to treat such coupled systems, including the effects of collisional energy transfer (weak collisions). The program suite consists of MultiWell, which solves the internal energy master equation for complex unimolecular reactions systems; DenSum, which calculates sums and densities of states by an exact‐count method; MomInert, which calculates external principal moments of inertia and internal rotation reduced moments of inertia; and Thermo, which calculates equilibrium constants and other thermodynamics quantities. MultiWell utilizes a hybrid master equation approach, which performs like an energy‐grained master equation at low energies and a continuum master equation in the vibrational quasicontinuum. An adaptation of Gillespie's exact stochastic method is used for the solution. The codes are designed for ease of use. Details are presented of various methods for treating weak collisions with virtually any desired collision step‐size distribution and for utilizing RRKM theory for specific unimolecular rate constants. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 232–245, 2001