Solution to the π-Distortivity Problem.

Traditionally, the delocalized π system of benzene is believed to be responsible for its perfectly symmetric D6h geometry. However, it has also been suggested that the π system prefers a distorted D3h geometry. Arguments for this have been based on clever use of VB methods as well as through shifts in the frequency of the distortive b2u mode. Evidence has been provided through different ways of partitioning the total electronic energy between the σ and the π systems. These methods are plagued by the fact that there is no unique way to partition the energy, leading to questions regarding the validity of the conclusions. Here we note that even though energy cannot be partitioned exactly, force acting on a nucleus depends only on the single particle density and can hence be partitioned exactly. Using good-quality wave functions that are numerically found to obey the Hellmann-Feynman theorem to good accuracy, we calculate the σ and π components of the force and provide conclusive evidence of π-distortivity at the HF level. Our approach provides an unambiguous way to approach the problem with wave functions that account for electron correlation. Our calculations suggest that the conclusion is valid at the MP2 level, too.

Diffusing Diffusivity: Survival in a Crowded Rearranging and Bounded Domain.

We consider a particle diffusing in a bounded, crowded, rearranging medium. The rearrangement happens on a time scale longer than the typical time scale of diffusion of the particle; as a result, effectively, the diffusion coefficient of the particle varies as a stochastic function of time. What is the probability that the particle will survive within the bounded region, given that it is absorbed the first time it hits the boundary of the region in which it diffuses? This question is of great interest in a variety of chemical and biological problems. If the diffusion coefficient is a constant, then analytical solutions for a variety of cases are available in the literature. However, there is no solution available for the case in which the diffusion coefficient is a random function of time. We discuss a class of models for which it is possible to find analytical solutions to the problem. We illustrate the method for a circular, two-dimensional region, but our methods are easy to apply to diffusion in arbitrary dimensions and spherical/rectangular regions. Our solution shows that if the dimension of the region is large, then only the average value of the diffusion coefficient determines the survival probability. However, for smaller-sized regions, one would be able to see the effects of the stochasticity of the diffusion coefficient. We also give generalizations of the results to N dimensions.

Diffusion in a Crowded, Rearranging Environment.

It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))(2)⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))(2)⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.