Electron localization function isosurfaces

Figure 5 equally displays the localization domains of aniline determined using the electron localization function technique (ELF)12,64, which defines the electronic basins within a molecular system. The basins correspond to the domains where electrons are localized either in paired bonds or in nonbonding lone pairs. In such a way, the number and nature of bonds between atoms could be determined. The ELF r)(r) value varies from 0 to 1 (0 < rj(r) < 1), which is chosen to determine the isosurfaces of the domains. Basins can be classified according to their synaptic order, that is, monosynaptic basins belong to the cores (denoted by C) whereas disynaptic basins describe the valence regions (denoted by V). 

FIGURE 20. Aspects of the electronic distribution in the vertical triplet aniline (triplet state at the singlet ground-state geometry) (a) Differential density Pink represents the Ap(r) = 0.005 an, and blue represents the Ap(r) = —0.005 an isosurface of the differential density, (b) ELF = 0.87 isosurfaces of the electron localization function. Values correspond to basin populations and integrated spin densities (s values)... 

FIGURE 21. Summary of electronic distribution in triplet aniline (a) Bond distances (A), NBO charges [bracket, in au] and Wiberg indices (parentheses, in au). (b) Topology of the electron density determined from atom-in-molecule calculations p(r) = electron density, L = Laplacian of the density defined as Z,(r) = —V2p(r) and = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Isosurfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins, (e) Spin densities in the molecular (CCN) plane... 

Thus one resorts to contour maps or isosurface plots to represent them. Unfortunately, they show only a part of the information contained in the function, since they depend on the contour (or isosurface) value choice one decides to plot. In order to have a more imambiguous way to analyze a three-dimensional (or higher dimension) function, one could use the framework of the topological analysis. In theoretical chemistiy, this has already been done in the pioneer works of Bader, which originated the Quantum Theory of Atoms in Molecules (QTAIM) [1]. Later this topological analysis was applied to interpret the Electron Localization Function [2-4], and lately it has been applied to the study of the Fukui function [5-7], which is namely the object of this chapter. 

Fig. 6 Isosurface maps for the Berlin local function (see text), electronic / r) and nuclear Fukui functions for formadehyde optimized at the B3LYP/6-311+G(d,p) level of theory both atthe singlet ground state and the triplet excited state. The isosurface value for/." "(r) is 0.5. For the remaining maps, a value of 0.005 for the isosurface has been used. The positive regions are presented in blue and the negative regions in red. (Reproduced with permission from J. Chem. Phys., 2005, 123, 084104. Copyright 2005 American Institute of Physics).

Among the quantities which have proven of value as graphical models are the molecular orbitals, the electron density, the spin density (for radicals and other molecules with unpaired electrons), the electrostatic potential and the local ionization potential. These may all be expressed as three-dimensional functions of the coordinates. One way to display them on a two-dimensional video screen (or on a printed page) is to define a surface of constant value, a so-called isovalue surface or, more simply, isosurface. ... 

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