New integrated numerical approaches to the Smoluchowski equation for the interpretation of molecular properties in solution phase chemistry

The necessity of a proper modelization of molecular diffusion in solution phase chemistry in order to rationalize properties and observables of the same give rise to the question \which tools from the theoretical chemistry field can be used?". The most commonly used approaches in order to answer this question relies on the usage of stochastic models. Then, depending on the particular conformational dynamics of the studied systems and the time scale on which molecular relaxation phenomena occurs, different models and tools can fit the desired level of description. Among all these approaches I focused on the Smoluchowski equation since its wide application in the theoretical chemistry community and since it well adapts to many diffusion problems in solution phase chemistry. Smoluchowski equation is a partial derivative equation that describes how the probability density of a set of coordinates evolves along time. Once the equation is solved, the probability profiles enclose informations upon specific properties and observables. My research activity in this field focused on new and more general numerical approaches to the solution of Smoluchowski equation with the application to specific case studies of chemical interest. Initially I have studied a class of methods apt to solve partial derivative differential equations. In particular, the chosen methodology is known as Discrete Variable Representation (DVR). Successively I have used this methodology in order to solve the one-dimensional Smoluchowski equation, in the stochastic processes framework applied to molecular systems. Comparing this method with preexisting ones I have validated this novel approach. This method has been implemented on a FORTRAN code with the aim of having an integrated approach for the solution of the one-dimensional Smoluchowski equation. Then I focused my research activity on the enhancement of the several physico- chemical ingredients that enter into the one-dimensional Smoluchowski equation. In par- ticular I extended an earlier diffusion tensor model including the dependence of the same on a generalized coordinate. Then, applying DVR theory to one-dimensional Smoluchowski equation I gave a more complete and general theoretical scenario of the same. In the framework of stochastic processes applied to molecular systems I have implemented and validated this novel approach. At last, for what concerns the Smoluchowski equation solved with DVR, I extended the same formalism to coupled one-dimensional Smoluchowski equations along the same generalized coordinate where there is the possibility of reactive exchanges of population and/or sinking terms between different coupled states. This is of great interest for example in the context of photoexcitations, where one has a population evolving through a ground and an excited state, along a specific coordinate, e.g. a twisting coordinate. From the temporal evolution of the probability density of the excited state one can retrieve lifetimes and/or compute the time resolved spectra at different times, etc. This last implementation merge the use of DVR basis with product approximation and diffusion tensor calculation along a generalized coordinate. The result is an integrated computational \black-box" tool in the framework of Gaussian software that give access to the generic user the possibility to study these specific systems of interest.