Electrostatic fields, density functional theory studies

This mathematical theory provides a partition of the space which is analogous to the more familiar partition made in hydrology in river basins delimited by watersheds. It relies on the study of a local function F(r) called the potential function. The potential function carries the physical or chemical information e.g. the electron density, the ELF (see below), or even the electrostatic potential [56-58]. In the cases treated in the present book, the potential function is required to be defined at any point of a manifold which is either for molecules or the unit cell for periodic systems. Moreover the first and second derivatives with respect to the point coordinates must be defined for any point. Its gradient W(r) forms a vector field bounded on the manifold and determines two kinds of points on the one hand are the wandering points corresponding to W(r ) f 0. and on the other hand are the critical points for which VF(rc) = 0. A critical point is characterized by the index Ip, the number of positive eigenvalues of the second derivatives matrix (the Hessian matrix). There are four kinds of critical points in... 

In recent years, the topological analysis of the three-dimensional scalar fields [87-95], such as electron density [55, 67, 92, 95-97], the Laplacian of the electron density [68, 92], the electron localization function (ELF) [94, 98], and molecular electrostatic potential, have been widely used to discern chemical structure and reactivity. This procedure, named quanmm chemical topology (QCT) [99] has been utilized for the study of chemical stmcture and reactivity [100-106]. Since its origins, the well-known approach of the atoms in molecules quantum theory (QTAIM), has evolved to be an invaluable tool for the chemical interpretation of quantum mechanical data, which relies on the properties of the electron density p(r) when atoms interact. Excellent reviews on QTAIM methods have been published elsewhere [69, 96, 107-109]. 

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