Toward a General Yet Effective Computational Approach for Diffusive Problems: Variable Diffusion Tensor and DVR Solution of the Smoluchowski Equation along a General One-Dimensional Coordinate

Abstract

A generalization to arbitrary large amplitude motions of a recent approach to the evaluation of diffusion tensors [ J. Comput. Chem. , 2009 , 30 , 2 - 13 ] is presented and implemented in a widely available package for electronic structure computations. A fully black-box tool is obtained, which, starting from the generation of geometric structures along different kinds of paths, proceeds toward the evaluation of an effective diffusion tensor and to the solution of one-dimensional Smoluchowski equations by a robust numerical approach rooted in the discrete variable representation (DVR). Application to a number of case studies shows that the results issuing from our approach are identical to those delivered by previous software (in particular DiTe) for rigid scans along a dihedral angle, but can be improved by employing relaxed scans (i.e., constrained geometry optimizations) or even more general large amplitude paths. The theoretical and numerical background is robust and general enough to allow quite straightforward extensions in several directions (e.g., inclusion of reactive paths, solution of Fokker-Planck or stochastic Liouville equations, multidimensional problems, free-energy rather than electronic-energy driven processes).

DiTe2: Calculating the diffusion tensor for flexible molecules

“…Some of the authors of this article introduced the former DiTe software package as a general tool for the computation of the generalized friction tensor of flexible molecules, with flexibility given in terms of dihedral angles only. Finally, Piserchia et al developed a module for the Gaussian quantum chemistry software package that extends the modeling introduced by DiTe to internal flexibility in terms of delocalized internal coordinates.…”

General Approach to Coupled Reactive Smoluchowski Equations: Integration and Application of Discrete Variable Representation and Generalized Coordinate Methods to Diffusive Problems

“…For these two different views a set of Cartesian velocities and forces for the unconstrained system and a set of external velocities and forces for the constrained system are considered. A laboratory frame (LF) and a molecular frame (MF) (fixed in the Eckart orientation on the molecular center of mass) are chosen and after some classical mechanics passages17,24 we can express the Cartesian velocities v i of the i -th atom for the unconstrained system as boldvi=υ+boldEtr(Ω)⋅(ω×boldci+∂boldci∂xx˙) where υ , ω , ẋ are the set of external velocities, respectively the translational velocity, the angular velocity and the generalized coordinate moment. E ( Ω ) is the Euler matrix dependent on the set of Euler angles Ω that brings the LF into the MF and c i are the Cartesian coordinates expressed in the MF of the i -th atom.…”

“…All the technical details about the diffusion tensor model here omitted for sake of brevity can be found in our previous work, ref. 17, to which the interested reader is addressed.…”

“…We have merged in a modular code all our previous developments on the solution of the one-dimensional Smoluchowski equation and molecular diffusion field. These are respectively, the robust numerical approach rooted in the discrete variable representation (DVR) in order to solve the equation16 and the variable diffusion tensor along a one-dimensional generalized coordinate considered as a diffusive process 17…”

“…These are respectively, the robust numerical approach rooted in the discrete variable representation (DVR) in order to solve the equation 16 and the variable diffusion tensor along a one-dimensional generalized coordinate considered as a diffusive process. 17 We briefly recall that the advantage in using DVR, instead of classical numerical methods such as finite difference methods and/or spectral methods, is the higher performances in terms of rapid eigenvalues convergence with the use of few grid points.…”