Rate processes with non-Markovian dynamical disorder

Rate processes with dynamical disorder are investigated within a simple framework provided by unidirectional electron transfer (ET) with fluctuating transfer rate. The rate fluctuations are assumed to be described by a non-Markovian stochastic jump process which reflects conformational dynamics of an electron transferring donor-acceptor molecular complex. A tractable analytical expression is obtained for the relaxation of the donor population (in the Laplace-transformed time domain) averaged over the stationary conformational fluctuations. The corresponding mean transfer time is also obtained in an analytical form. The case of two-state fluctuations is studied in detail for a model incorporating substate diffusion within one of the conformations. It is shown that an increase of the conformational diffusion time results in a gradual transition from the regime of fast modulation characterized by the averaged ET rate to the regime of quasistatic disorder. This transition occurs at the conformational mean residence time intervals fixed much less than the inverse of the corresponding ET rates. An explanation of this paradoxical effect is provided. Moreover, its presence is also manifested for the simplest, exactly solvable non-Markovian model with a biexponential distribution of the residence times in one of the conformations. The nontrivial conditions for this phenomenon to occur are found.     

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.     

Maximum Caliber: a variational approach applied to two-state dynamics

We show how to apply a general theoretical approach to nonequilibrium statistical mechanics, called Maximum Caliber, originally suggested by E. T. Jaynes [Annu. Rev. Phys. Chem. 31, 579 (1980)], to a problem of two-state dynamics. Maximum Caliber is a variational principle for dynamics in the same spirit that Maximum Entropy is a variational principle for equilibrium statistical mechanics. The central idea is to compute a dynamical partition function, a sum of weights over all microscopic paths, rather than over microstates. We illustrate the method on the simple problem of two-state dynamics, A<-->B, first for a single particle, then for M particles. Maximum Caliber gives a unified framework for deriving all the relevant dynamical properties, including the microtrajectories and all the moments of the time-dependent probability density. While it can readily be used to derive the traditional master equation and the Langevin results, it goes beyond them in also giving trajectory information. For example, we derive the Langevin noise distribution rather than assuming it. As a general approach to solving nonequilibrium statistical mechanics dynamical problems, Maximum Caliber has some advantages: (1) It is partition-function-based, so we can draw insights from similarities to equilibrium statistical mechanics. (2) It is trajectory-based, so it gives more dynamical information than population-based approaches like master equations; this is particularly important for few-particle and single-molecule systems. (3) It gives an unambiguous way to relate flows to forces, which has traditionally posed challenges. (4) Like Maximum Entropy, it may be useful for data analysis, specifically for time-dependent phenomena.     

Separable approximation for diatomic transition operators: a variational approach

A general finite-rank separable approximation for the two-body transition operator has been obtained from the Lippmann--Schwinger variational principle. The method involves expansion of the off-shell wavefunction in a finite set of square integrable functions, and is applied to ¹Σ and ³Σ potentials of H₂ for thermal energies and momenta.     

A one-dimensional energy diffusion approach to multidimensional dynamical processes in the condensed phase

We propose a generalized one-dimensional energy diffusion approach for describing the dynamics of multidimensional dynamical processes in the condensed phase. On the basis of a formalism originally due to Zwanzig, we obtain a one-dimensional kinetic equation for a properly selected relevant dynamical quantity and derive new analytical results for the dynamics of a multidimensional electron-transfer process, nonequilibrium solvation, and diffusive escape from a potential well. The calculated results for electron-transfer reactions in solvent-separated and contact ion pair systems are found to be in good agreement with the experimental results. We are able to explain the rate of the electron-transfer reaction using much smaller and reasonable values of the solvent reorganization energy in contrast to earlier works that had to use a much larger value. The proposed theory is not only conceptually simpler than the conventional approaches but is also free from many of their limitations. More importantly, it provides a single theoretical framework for describing a wide class of dynamical phenomena.     

On a variational approach to the Soret coefficient

In this paper, a variational approach to the Soret coefficient initiated by Kempers [J. Chem. Phys. 90, 6541 (1989)] is critically revisited. We show that the physical coherence of the whole procedure leads to a very peculiar choice of the type of constraint one can select to mimic the nonequilibrium stationary state. However, we demonstrate that its precise definition would require a statistical evaluation of the heat of transfer, or a variational approach based on more microscopic ingredients.     

Transition state theory: variational formulation, dynamical corrections, and error estimates

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).     

A variational density matrix approach with nonlocal effective potential

We show that using the Colle–Salvetti correlation-energy functional (Colle and Salvetti in Theoret Chim Acta 37:329, 1975) in the Hartree–Fock-type procedure suggested by Kohn and Sham (Phys Rev 140:A1133PR, 1965), one can calculate quite accurately electronic properties of systems in which the “dynamical” correlation energy is dominant. We compare our results with those obtained by Grabo and Gross (Chem Phys Lett 240:141, 1995) using the optimized effective potential method, and we discuss characteristics and advantages of our procedure.     

Ultrafast UV spectroscopy: from a local to a global view of dynamical processes in macromolecules

The aim of this Perspective article is to cover recent developments in the application of femtosecond UV spectroscopy to understand molecular dynamics, and outlining potential future directions in this area. With several examples from recent literature the added-value of these techniques and their capability to study in real time changes in structure, dynamics and electrostatic fields of macromolecules in a site-specific fashion, as well as to uncover concerted dynamics in biomolecules, will be shown and discussed. The emerging fields of UV pulse-shaping techniques and UV optical nonlinear spectroscopies will be discussed to outline their potential to generate a novel family of coherent nonlinear spectroscopies for spectroscopic and microscopic applications.     

Development of a new variational approach for thermal density matrices

A McLachlan-type variational principle is derived for thermal density matrices. In this approach, the trace of the mean square of the differences between the derivatives of the exact and model density matrices is minimized with respect to the parameters in the model Hamiltonian. Applications to model anharmonic systems in the independent particle model show that the method can provide thermodynamic state functions accurately (within 5% of the converged basis set results) and at the same level of accuracy as the results using Feynman-Gibbs-Bogoliubov variational principle at this level of approximation.     

Temperature-induced dynamical conformational disorder in 4-vinyl benzoic acid molecular crystals: a molecular simulation study

Extensive molecular simulations are carried out as a function of temperature to understand and quantify the conformational disorder in molecular crystals of 4-vinyl benzoic acid. The conformational disorder is found to be dynamic and associated with a flip-flop motion of vinyl groups. The population of minor conformer is less than 3% up to 300 K and is 13.2% at 350 K and these results are consistent with the experimental observations. At still higher temperatures, the population of minor conformer increases up to 25%. The evolution of structure at both molecular and unit-cell level of the molecular crystal as a function of temperature has been characterized by various quantities such as radial distribution functions, average cell parameters, volume, and interaction energies. The van't Hoff plot shows a nonlinear behavior at lower temperatures as it has been reported recently by Ogawa and co-workers in the case of stilbene, azobenzene, and N-(4-methylbenzylidene)-4-methylaniline molecular crystals. A set of rigid body simulations were also carried out to quantify the effect of conformational disorder on structural quantities such as unit-cell volume and interaction energy. The anomalous shrinkage of vinyl C=C bond length as a function of temperature has been explained by combining the results of simulations and a set of constrained optimizations using ab initio electronic structure calculations for various molecular structures differing in torsional angle.     

Self-consistent-field variational approach to the interaction between a polymer and a small molecule

A variational SCF treatment based on a perturbational concept is developed and applied to the interaction between trans-polyacetylene and a small molecule. The validity of the present method is examined by comparing the results with those from the conventional tight-binding SCF crystal orbital method. The interaction energies and charge distributions obtained are in good agreement between the two methods. This result suggests that the present variational approach is promising for application to complicated interactions between a polymer and impurities.     

A variational approach for a class of nonlocal elliptic boundary value problems

The aim of this paper is to apply the variational iteration method to a class of nonlinear, nonlocal, elliptic boundary value problems. The uniform convergence of the scheme is presented and the work is illustrated by considering a number of test examples that confirm the accuracy and efficacy of the iterative process. The computational results show that the scheme is reliable, converges fast and compares very well with the existing analytic solutions.     

Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime: a variational approach

We consider the issue of optimizing linear-regime nonequilibrium simulations to estimate free-energy differences. In particular, we focus on the problem of finding the best-possible driving function lambda(t) that, for a given thermodynamic path, simulation algorithm, and amount of computational effort, minimizes dissipation. From the fluctuation-dissipation theorem it follows that, in the linear-response regime, the dissipation is controlled by the magnitude and characteristic correlation time of the equilibrium fluctuations in the driving force. As a result, the problem of finding the optimal switching scheme involves the solution of a standard problem in variational calculus: the minimization of a functional with respect to the switching function. In practice, the minimization involves solving the associated Euler-Lagrange equation subject to a set of boundary conditions. As a demonstration we apply the approach to the simple, yet illustrative problem of computing the free-energy difference between two classical harmonic oscillators with very different characteristic frequencies.     

A concise review of dynamical processes in polymorphic environments of a block copolymer: Rotational diffusion and photoisomerization

This article describes our ongoing efforts to understand dynamical processes such as rotational diffusion and photoisomerization in polymorphic environments of a block copolymer. The objective is to explore how the typical properties of a block copolymer solution such as critical micelle temperature (CMT) and temperature-induced sol-gel transition influence the rotational diffusion of hydrophobic solute molecules. Rotational diffusion of solute molecules differs significantly below and above the CMT of a block copolymer solution, while there is no influence of sol-gel transition on solute rotation. This is rationalized on the basis of the site of solubilization of the solute molecules which is the palisade layer of the micelles in both phases and unaffected by gelation. A similar result has been obtained in case of photoisomerization studies carried out with a carbocyanine derivative in the sol and gel phases of the block copolymer. The isomerization studies have been extended to the reverse phases (sol and gel phases) of the block copolymer to explore the nature of the water present in the cores of the reverse micelles. Our results provide evidence for the existence of water droplets with properties resembling bulk water. In essence, we show that despite having vastly differing bulk properties, both the solution and gel phases (normal as well as reverse) offer identical microscopic environment.     

The clerodane ring system: investigating the viability of a direct Diels-Alder approach

A direct synthetic approach to the spiro-γ-lactone clerodane ring system has been investigated. This work builds on that of Jung and highlights the inherent difficulties associated with the otherwise obvious Diels-Alder approach.