On the theory of the migration of incoherent excitons

The probability of a localized exciton jump between molecules with allowance for exciton-exciton interaction is obtained in the adiabatic approximation. Above the Debye temperature, an analytic dependence of this probability on the exciton-exciton interaction is found and the role of the interaction in the annihilation process is discussed.     

Diffusiophoresis of a cylindrical colloidal particle oriented parallel to an electrolyte concentration gradient field

We derive the general expression for the diffusiophoretic mobility of a cylindrical particle oriented parallel to an applied electrolyte concentration gradient field in a symmetrical electrolyte solution. From the general mobility expression as combined with an approximate analytic expression with negligible error for the electric potential distribution around a cylinder, an accurate analytic mobility expression is obtained, which is applicable for arbitrary values of the particle zeta potential and the electrical double layer thickness. It is also found that the low zeta potential approximation is an excellent approximation for low-to-moderate values of the particle zeta potential.     

Thermogenesis: Identification by means of modulating functions

This paper describes how to obtain an analytic approximation to the transfer function of a conduction calorimeter, namely a procedure to identify the calorimetric system. In this case modulating functions are used directly on the thermogram. The method is used twice: to obtain the time constants and the amplitudes. Its feasibility is tested on two models which span the frequency range usually attained by actual calorimeters. The influence of random noise and baseline drift have also been analyzed. The results show that three or four time constants are correctly obtained.     

The residence probability: single molecule fluorescence correlation spectroscopy and reversible geminate recombination

The residence probability of a freely diffusing particle within an open d-dimensional ball is calculated as a function of time, for an initial distribution that is either a spherical delta function or uniform within the sphere. The latter is equivalent to the autocorrelation function (ACF) of fluorescence correlation spectroscopy (FCS) when utilizing near-field scanning optical microscopy (NSOM) probes. Starting from the general equation for the Laplace transform of the residence probability, we solve it in Laplace space for any dimensionality, inverting it into the time domain in one- and three-dimensions. The short- and long-time asymptotic behaviors of the residence probability are derived and compared with the exact results. Approximations for the two-dimensional ACF are discussed, and a new approximation is derived for the NSOM-FCS ACF. Also of interest is an analytic expression for a three-dimensional ACF, which could be useful for two-photon FCS. Analogy with the binding probability for reversible geminate recombination suggests that more information could be extracted from the long-time tails in FCS experiments.     

Ionization probability of the hydrogen atom suddenly released from confinement

The problem of the stability of a confined atom when it is extracted from the confining cavity has been investigated, modeled by a spherical hard wall potential. The ionization probability when the atom is released from confinement has been obtained. The dependence of the ionization probability on the confinement radius and on the quantum numbers of the initial confined state has been studied. The probability density function of the ionization energy of the ejected electron has been obtained for the different cases considered. The oscillatory structure of this distribution function, with a principal maximum located in the neighborhood of the energy of the initial state and minima very close to zero has been elucidated. The sudden approximation has been applied and the analytic continuation method has been used to calculate the different stationary states.     

Solution of quantum Langevin equation: approximations, theoretical and numerical aspects

Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.     

Thermogenesis: Identification by means of pade approximants

The paper describes how to obtain an analytic approximation to the transfer function of a conduction calorimeter, namely, a procedure to identify the calorimetric system. In this case Pade approximants are used on the Laplace transform of the thermogram. The feasibility of the method is tested on two models which span the frequency range usually attained by actual calorimeters. The influence of random noise and baseline drift have also been analyzed. The results show that three or four time constants are correctly obtained.     

Toward making the mean spherical approximation of primitive model electrolytes analytic: an analytic approximation of the MSA screening parameter

The mean spherical approximation (MSA) for the primitive model of electrolytes provides reasonable estimates of thermodynamic quantities such as the excess chemical potential and screening length. It is especially widely used because of its explicit formulas so that numerically solving equations is minimized. As originally formulated, the MSA screening parameter Γ (akin to the reciprocal of the Debye screening length) does not have an explicit analytic formula; an equation for Γ must be solved numerically. Here, an analytic approximation for Γ is presented whose relative error is generally ?10(-5). If more accuracy is desired, one step of an iterative procedure (which also produces an explicit formula for Γ) is shown to give relative errors within machine precision in many cases. Even when ion diameter ratios are ～10 and ion valences are ～10, the relative error for the analytic approximation is still ?10(-3) and for the single iterative substitution it is ?10(-9).     

On the applicability of the Born-Oppenheimer approximation in a theory of nonadiabatic charge transfer reactions: An exactly solvable model

The question concerning the applicability of the Born-Oppenheimer approximation (BOA) for the calculation of the transition probability for a nonadiabatic process of charge transfer in a polar environment with allowance made for temperature effects is investigated theoretically. Considered is the transfer of a quantum particle (proton) that interacts with a local vibration mode in a model of bound harmonic oscillators. The model admits an exact solution for wavefunctions of the initial and final states. A calculation shows that BOA is applicable even for very large distances of the proton transfer. At the same time, the exact result and the BOA result severely differ from a probability calculated in a crude Condon approximation. It is demonstrated that the non-Condon effects are in a general case temperature-dependent and may substantially influence the calculated values of the transition probability.     

Accuracy of finite‐difference harmonic frequencies in density functional theory

Analytic Hessians are often viewed as essential for the calculation of accurate harmonic frequencies, but the implementation of analytic second derivatives is nontrivial and solution of the requisite coupled‐perturbed equations engenders a sizable memory footprint for large systems, given that these equations are not required for energy and gradient calculations in density functional theory. Here, we benchmark the alternative approach to harmonic frequencies based on finite differences of analytic first derivatives, a procedure that is amenable to large‐scale parallelization. Not only for absolute frequencies but also for isotopic and conformer‐dependent frequency shifts in flexible molecules, we find that the finite‐difference approach exhibits mean errors < 0.1 cm⁻¹ as compared to results based on an analytic Hessian. For very small frequencies corresponding to nonbonded vibrations in noncovalent complexes (for which the harmonic approximation is questionable anyway), the finite‐difference error can be larger, but even in these cases the errors can be reduced below 0.1 cm⁻¹ by judicious choice of the displacement step size and a higher‐order finite‐difference approach. The surprising accuracy and robustness of the finite‐difference results suggests that availability of the analytic Hessian is not so important in today's era of commodity processors that are readily available in large numbers. © 2017 Wiley Periodicals, Inc.     

Path integral approach to Brownian motion driven with an ac force

Brownian motion in a periodic potential driven by an ac (oscillatory) force is investigated for the full range of damping constant from the overdamped limit to the underdamped limit. The path (functional) integral approach is advanced to produce formulas for the probability distribution function and for the current of the Brownian particle in response to an ac driving force. The negative friction Langevin dynamics technique is employed to evaluate the dc current for various parameters without invoking the overdamped or the underdamped approximation. The dc current is found to have nonlinear dependence upon the damping constant, the potential parameter, and the ac force magnitude and frequency.     

Correlation energy of many-electron systems: a modified Colle-Salvetti approach

The Colle and Salvetti approach [Theo. Chim. Acta 37, 329 (1975)] to the calculation of the correlation energy of a system is modified in order to explicitly include into the theory the kinetic contribution to the correlation energy. This is achieved by deducing from a many electrons wave function, including the correlation effects via a Jastrow factor, an approximate expression of the one-electron reduced density matrix. Applying the latter to the homogeneous electron gas, an analytic expression of the correlation kinetic energy is derived. The total correlation energy of such a system is then deduced from its kinetic contribution inverting a standard procedure. At variance of the original Colle-Salvetti theory, the parameters entering in both the kinetic correlation and the total correlation energies are determined analytically, leading to a satisfactory agreement with the results of Perdew and Wang [Phys. Rev. B 45, 13244 (1992)]. The resulting (parameter-free) expressions give rise to a modified-local-density approximation that can be used in self-consistent density-functional calculations. We have performed such calculations for a large set of atoms and ions and we have found results for the correlation energies and for the ionization potentials which improve those of the standard local-density approximation.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

The surface charge density/surface potential relationship and the Poisson-Boltzmann equation

A procedure is described which provides an approximate analytic expression for the relationship between the surface charge density and the surface potential of a spherical or cylindrical colloidal particle in a general type of electrolyte. The first approximation is found and the approximation error is given. Also derived are the fourth approximation for a symmetric z-z type electrolyte and the second approximation for a symmetric z-z, 2z-2z type electrolyte.     

Nanofiltration theory: good co-ion exclusion approximation for single salts

We applied an approximate analytic method, the good co-ion exclusion (GCE) approximation, to the hindered electrotransport theory describing salt and solution transport across charged nanofiltration membranes. This approximation, which should be valid at sufficiently low feed electrolyte concentration, leads to a considerable simplification of the exact parametrized equations obtained previously for single salt nanofiltration parameters (salt rejection, electric filtration potential, and volume flux density) and therefore provides further insight into ion transfer in nanoporous membranes. We also established the domain of validity of the GCE approximation as a function of the salt type for 1:1, 2:1, 1:2, and 2:2 salts. Our results for the volume flux density, obtained within an extended GCE approximation, confirm that the global osmotic reflection coefficient in the solution flux equation is not equal to the limiting salt rejection.     

Theory of kinetic transport coefficients for organic semi-conductors

A generalization of the small-times approximation is considered for an electron-phonon system having a fairly narrow conduction band. The result differs from the semiclassical approximation in giving an analytic expression for the generating function of the semi-invariants. Edgworth's expansion is used as the interpolation function for the electrical-conductivity tensor. More rigorous criteria for the application of the semiclassical approximation are derived for the high-temperature range. Allowance is made for temperature corrections to the generating function, and trial estimates show that this revised function may be applicable down to temperatures well below normal.     

Nonadiabatic proton-coupled electron transfer reactions: impact of donor-acceptor vibrations, reorganization energies, and couplings on dynamics and rates

Fundamental aspects of proton-coupled electron transfer (PCET) reactions in solution are analyzed with molecular dynamics simulations for a series of model systems. The analysis addresses the impact of the solvent reorganization energy, the proton donor-acceptor mode vibrational frequency, and the distance dependence of the nonadiabatic coupling on the dynamics of the reaction and the magnitude of the rate. The rate for nonadiabatic PCET is expressed in terms of a time-dependent probability flux correlation function. The time dependence of the probability flux correlation function is determined mainly by the solvent reorganization energy and is not significantly influenced by the proton donor-acceptor frequency or the distance dependence of the nonadiabatic coupling. The magnitude of the PCET rate becomes greater as the solvent reorganization energy decreases, the proton donor-acceptor frequency decreases, and the distance dependence of the nonadiabatic coupling increases. The approximations underlying a previously derived analytical PCET rate expression are also investigated. The short-time approximation for the solvent is valid for these types of systems. In addition, solvent damping effects on the proton donor-acceptor motion are not significant on the time scale of the probability flux. The rates calculated from the molecular dynamics simulations agree well with those calculated from the analytical rate expression.     

Generalized Born model with a simple, robust molecular volume correction

Generalized Born (GB) models provide a computationally efficient means of representing the electrostatic effects of solvent and are widely used, especially in molecular dynamics (MD). A class of particularly fast GB models is based on integration over an interior volume approximated as a pairwise union of atom spheres-effectively, the interior is defined by a van der Waals rather than Lee-Richards molecular surface. The approximation is computationally efficient, but if uncorrected, allows for high dielectric (water) regions smaller than a water molecule between atoms, leading to decreased accuracy. Here, an earlier pairwise GB model is extended by a simple analytic correction term that largely alleviates the problem by correctly describing the solvent-excluded volume of each pair of atoms. The correction term introduces a free energy barrier to the separation of non-bonded atoms. This free energy barrier is seen in explicit solvent and Lee-Richards molecular surface implicit solvent calculations, but has been absent from earlier pairwise GB models. When used in MD, the correction term yields protein hydrogen bond length distributions and polypeptide conformational ensembles that are in better agreement with explicit solvent results than earlier pairwise models. The robustness and simplicity of the correction preserves the efficiency of the pairwise GB models while making them a better approximation to reality.     

Gaussian approximation for the temperature dependence of a continuous electronic spectrum

Exact expressions for the temperature dependence of the mean frequency and width of an absorption spectrum for an allowed transition between a harmonic lower state and a linear repulsive upper state potential are derived. This is sufficient information to obtain a Gaussian approximation for the absorption spectrum. The present results for the first two spectral moments are exact, and therefore represent an improvement over the Sulzer-Wieland equation in its original or modified forms.