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

Abstract

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 approximated 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 (ER) 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 energy coordinate, enabling us to describe the dynamics on one-dimensional coordinate. Introducing the systematic approximations such as the overdamped limit leads to 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 (MD) simulation.

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

“…In addition, S ( t ) in water shows the damped oscillation at Dt < 0.2 Å 2 . In our previous study, 50 a similar oscillation is observed for the solvation dynamics of benzonitrile in water due to the rotational motion of the water molecules hydrogen bonded with benzonitrile. The rotational motion of the water molecules would also bring the oscillation observed in the present water system.…”

“…The formulation for multi-component solvents is available in ref. 50. Furthermore, we assume that the solute molecule is fixed in space.…”

“…To solve the ERSV and ERS equations, we used the scheme developed in our previous study. 50 We used the finite volume method (FVM) to discretize the energy coordinate. For numerical efficiency and accuracy, we used non-uniform grids on the energy coordinate which are fine around e = 0.…”

“…48,49 Recently, we have derived an alternative expression for the molecular diffusion based on the energy representation (ER) which is amenable to MD simulations. 50 In this theory, the configuration of a solvent molecule is projected onto the solutesolvent interaction energy, namely the energy coordinate. [51][52][53] It should be noted that the intramolecular degree of freedom is naturally taken into account using the energy coordinate.…”

“…By applying the Zwanzig-Mori projection operator method to the solvent distribution on the energy coordinate, the energy-represented generalized Langevin equation (ERGLE) can be derived in an exact way. 50 The systematic approximations about the dynamics such as the overdamped limit give the energy-represented Smoluchowski-Vlasov (ERSV) equation, a diffusion equation of solution that describes the self-diffusion, drift motion, and collective motion of solvents on the energy coordinate. Once we calculate the static correlation functions and diffusion coefficient of solvents involved in the ERSV equation, the time development of the solvent distribution functions can be obtained by solving this equation.…”