Interaction‐site representation of the Smoluchowski–Vlasov equation: The space–time correlation functions in a molecular liquid

Solvation dynamics on the diffusion timescale elucidated using energy-represented dynamics theory

Photoexcitation of a solute alters the solute-solvent interaction, resulting in the nonequilibrium relaxation of the solvation structure, often called a dynamic Stokes shift or solvation dynamics. Thanks to the local nature of the solute-solvent interaction, the characteristics of the local solvent environment dissolving the solute can be captured by the observation of this process. Recently, we derived the energy-represented Smoluchowski-Vlasov (ERSV) equation, a diffusion equation for molecular liquids, which can be used to analyze the solvation dynamics on the diffusion timescale. This equation expresses the time development for the solvent distribution on the solute-solvent pair interaction energy (energy coordinate). Since the energy coordinate can effectively treat the solvent flexibility in addition to the position and orientation, the ERSV equation can be utilized in various solvent systems. Here, we apply the ERSV equation to the solvation dynamics of 6-propionyl-2-dimethylamino naphthalene (Prodan) in water and different alcohol solvents (methanol, ethanol, and 1-propanol) for clarifying the differences of the relaxation processes among these solvents. Prodan is a solvent-sensitive fluorescent probe and is thus widely utilized for investigating heterogeneous environments. On the long timescale, the ERSV equation satisfactorily reproduces the relaxation time correlation functions obtained from the molecular dynamics (MD) simulations for these solvents. We reveal that the relaxation time coefficient on the diffusion timescale linearly correlates with the inverse of the translational diffusion coefficients for the alcohol solvents because of the Prodan-solvent energy distributions among the alcohols. In the case of water, the time coefficient deviates from the linear relationship for the alcohols due to the difference in the extent of importance of the collective motion between the water and alcohol solvents.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Diffusion theory of molecular liquids in the energy representation and application to solvation dynamics

The generalized Langevin equation (GLE) formalism is a useful theoretical fundament for analyzing dynamical phenomena rigorously. Despite the systematic formulation of dynamics theories with practical approximations, however, the applicability of GLE-based methods is still limited to simple polyatomic liquids due to the approximate treatment of molecular orientations involved in the static molecular liquid theory. Here, we propose an exact framework of dynamics based on the GLE formalism incorporating the energy representation theory of solution, an alternative static molecular liquid theory. A fundamental idea is the projection of the relative positions and orientations of solvents around a solute onto the solute-solvent interaction, namely the energy coordinate, enabling us to describe the dynamics on a one-dimensional coordinate. Introducing systematic approximations, such as the overdamped limit, leads to the molecular diffusion equation in the energy representation that is described in terms of the distribution function of solvents on the energy coordinate and the diffusion coefficients. The present theory is applied to the solvation dynamics triggered by the photoexcitation of benzonitrile. The long-time behavior of the solvation time correlation function is in good agreement with that obtained by the molecular dynamics simulation.

Solvation dynamics in electronically polarizable solvents: Theoretical treatment using solvent-polarizable three-dimensional reference interaction-site model theory combined with time-dependent density functional theory

The theory of solvation structure in an electronically polarizable solvent recently proposed by us, referred to as the "solvent-polarizable three-dimensional reference interaction-site model theory," is extended to dynamics in this study through the combination with time-dependent density functional theory. Test calculations are performed on model charge-transfer systems in water, and the effects of electronic polarizability on solvation dynamics are examined. The electronic polarizability slightly retards the solvation dynamics. This is ascribed to the decrease in the curvature of the nonequilibrium free energy profile along the solvation coordinate. The solvent relaxation is bimodal, and the faster and the slower modes are assigned to the reorientational and the translational modes, respectively, as was already reported by the surrogate theory combined with the site-site Smoluchowski-Vlasov equation. The relaxation path along the solvation coordinate is a little higher than the minimum free energy path because the translational mode is fixed in the time scale of the reorientational relaxation.

Dynamics theory for molecular liquids based on an interaction site model

Dynamics theories for molecular liquids based on an interaction site model have been developed over the past few decades and proved to be powerful tools to investigate various dynamical phenomena.

Structural fluctuation of proteins induced by thermodynamic perturbation

A theory to describe structural fluctuations of protein induced by thermodynamic perturbations, pressure, temperature, and denaturant, is proposed. The theory is formulated based on the three methods in the statistical mechanics: the generalized Langevin theory, the linear response theory, and the three dimensional interaction site model (3D-RISM) theory. The theory clarifies how the change in thermodynamic conditions, or a macroscopic perturbation, induces the conformational fluctuation, which is a microscopic property. The theoretical results are applied, on the conceptual basis, to explain the experimental finding by Akasaka et al., concerning the NMR experiment which states that the conformational change induced by pressure corresponds to structural fluctuations occurring in the ambient condition. A method to evaluate the structural fluctuation induced by pressure is also suggested by means of the 3D-RISM and the site-site Kirkwood-Buff theories.

Water Turns the "Non-biological" Fluctuation of Protein into "Biological" One

Structural fluctuation of protein is not just an mechanical "oscillation," but an event induced by an interplay of mechanical and thermodynamic processes in which water plays crucial role. The chapter is devoted to provide a theoretical description concerning the concept of structural fluctuation of protein, based on methods of the statistical mechanics.

Development of three-dimensional site-site Smoluchowski-Vlasov equation and application to electrolyte solutions

Site-site Smoluchowski-Vlasov (SSSV) equation enables us to directly calculate van Hove time correlation function, which describes diffusion process in molecular liquids. Recently, the theory had been extended to treat solute-solvent system by Iida and Sato [J. Chem. Phys. 137, 034506 (2012)]. Because the original framework of SSSV equation is based on conventional pair correlation function, time evolution of system is expressed in terms of one-dimensional solvation structure. Here, we propose a new SSSV equation to calculate time evolution of solvation structure in three-dimensional space. The proposed theory was applied to analyze diffusion processes in 1M NaCl aqueous solution and in lithium ion battery electrolyte solution. The results demonstrate that these processes are properly described with the theory, and the computed van Hove functions are in good agreement with those in previous works.

Structural fluctuation of protein in water around its native state: a new statistical mechanics formulation

A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-reference interaction site model (RISM)/RISM theory of molecular liquids. The displacement vector of atom positions, and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations, which describe the time evolution of fluctuation concerning the density field of solvent and the conformation of protein around an equilibrium state, which are coupled with each other. The equation concerning the protein dynamics is formally akin to that of the coupled Langevin oscillators, and is a generalization of the latter, to atomic level. The most intriguing feature of the new equation is that it contains the variance-covariance matrix as the "Hessian" term describing the "force" restoring an equilibrium conformation, which is the second moment of the fluctuation of atom positions. The "Hessian" matrix is naturally identified as the second derivative of the free energy surface around the equilibrium. A method to evaluate the Hessian matrix based on the 3D-RISM/RISM theory is proposed. Proposed also is an application of the present formulation to the molecular recognition, in which the conformational fluctuation of protein around its native state becomes an important factor as exemplified by so called "induced fitting."

Dynamics of the solvent around a solute: generalized Langevin theory

The generalized Langevin theory for a solution has been derived as the infinite dilution limit of the theory for a two component mixture. Following a similar formalism, the mode coupling approximations of the memory kernel have been also obtained. We have applied this method for one component bulk liquid of Lennard-Jones spheres and proved this approximation theoretically. The analysis of the space and time pair correlation proposed by Van Hove has been carried out as a function of solute particle sizes. It is found that the size of the solute particle is deeply related to the relaxation process of the solvation structure characterized around a solute particle at equilibrium. We have also investigated the relation between the different thermodynamic environments and relaxation process. From these studies, we have obtained the useful information about the rapidity of the relaxation of the solvation structure around a solute at equilibrium.

Site-site memory equation approach in study of density/pressure dependence of translational diffusion coefficient and rotational relaxation time of polar molecular solutions: acetonitrile in water, methanol in water, and methanol in acetonitrile

We present results of the theoretical study and numerical calculation of the dynamics of molecular liquids based on the combination of the memory equation formalism and the reference interaction site model (RISM). Memory equations for the site-site intermediate scattering functions are studied in the mode-coupling approximation for the first-order memory kernels, while equilibrium properties such as site-site static structure factors are deduced from RISM. The results include the temperature-density (pressure) dependence of translational diffusion coefficients D and orientational relaxation times tau for acetonitrile in water, methanol in water, and methanol in acetonitrile--all in the limit of infinite dilution. Calculations are performed over the range of temperatures and densities employing the extended simple point charge model for water and optimized site-site potentials for acetonitrile and methanol. The theory is able to reproduce qualitatively all main features of temperature and density dependences of D and tau observed in real and computer experiments. In particular, anomalous behavior, i.e, the increase in mobility with density, is observed for D and tau of methanol in water, while acetonitrile in water and methanol in acetonitrile do not show deviations from the ordinary behavior. The variety exhibited by the different solute-solvent systems in the density dependence of the mobility is interpreted in terms of the two competing origins of friction, which interplay with each other as density increases: the collisional and dielectric frictions which, respectively, increase and decrease with increasing density.