The Electron Localization Function

Wave functions can be calculated rather reliably with quantum-chemical approximations. The sum of the squares of all wave functions of the occupied orbitals at a site x, y. z is the electron density p(x,y,z) =Hwf. It can also be determined experimentally by X-ray diffraction (with high expenditure). The electron density is not very appropriate to visualize chemical bonds. It shows an accumulation of electrons close to the atomic nuclei. The enhanced electron density in the region of chemical bonds can be displayed after the contribution of the inner atomic electrons has been subtracted. But even then it remains difficult to discern and to distinguish the electron pairs. 

Redress can be obtained by the electron localization function (ELF). It decomposes the electron density spatially into regions that correspond to the notion of electron pairs, and its results are compatible with the valence shell electron-pair repulsion theory. An electron has a certain electron density p, (x, y, z) at a site x, y, z this can be calculated with quantum mechanics. Take a small, spherical volume element AV around this site. The product nY(x, y, z) = p, (x, y, z)AV corresponds to the number of electrons in this volume element. For a given number of electrons the size of the sphere AV adapts itself to the electron density. For this given number of electrons one can calculate the probability w(x, y, z) of finding a second electron with the same spin within this very volume element. According to the Pauli principle this electron must belong to another electron pair. The electron localization function is defined with the aid of this probability  

ELF can be visualized with different kinds of images. Colored sections through a molecule are popular, using white for high values of ELF, followed by yellow-red-violet-blue-dark blue for decreasing values simultaneously, the electron density can be depicted by the density of colored points. Contour lines can be used instead of the colors for black and white printing. Another possibility is to draw perspective images with iso surfaces, i.e. surfaces with a constant value of ELF. Fig. 10.2 shows iso surfaces with ELF = 0.8 for some molecules from experience a value of ELF = 0.8 is well suited to reveal the distribution of electron pairs in space. 

On the one hand, Fig. 10.2 exhibits iso surfaces around the fluorine atoms on the other hand the lone electron pairs at the central atoms can be discerned quite well. The space requirement of one lone pair is larger than that of the four electron pairs at one of the more electronegative fluorine atoms. The three lone pairs at the chlorine atom of ClFj add up to a rotation-symmetrical torus. 

10 MOLECULAR ORBITAL THEORY AND CHEMICAL BONDING IN SOLIDS 

Wave functions can be calculated rather reliably with quantum-chemical approximations. The sum of the squares of all wave functions V of the occupied orbitals at a site x, y, z is the electron density p x,y,z) It can also be determined experimentally by X- 

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The location of electrons linking more than three atoms cannot be illustrated as easily. The simple, descriptive models must give way to the theoretical treatment by molecular orbital theory. With its aid, however, certain electron counting rules have been deduced for cluster compounds that set up relations between the structure and the number of valence electrons. A bridge between molecular-orbital theory and vividness is offered by the electron-localization function (cf p. 89). 

According to calculations with the electron localization function (ELF) the electron pairs of the B6Hg cluster are essentially concentrated on top of the octahedron edges and faces (Fig. 13.12). 

In this second edition the text has been revised and new scientific findings have been taken into consideration. For example, many recently discovered modifications of the elements have been included, most of which occur at high pressures. The treatment of symmetry has been shifted to the third chapter and the aspect of symmetry is given more attention in the following chapters. New sections deal with quasicrystals and other not strictly crystalline solids, with phase transitions and with the electron localization function. There is a new chapter on nanostructures. Nearly all figures have been redrawn. 

Distributed Electrostatic Moments Based on the Electron Localization Function Partition... 

Pilme J, Piquemal J-P (2008) Advancing beyond charge analysis using the electronic localization function Chemically intuitive distribution of electrostatic moments. J Comput Chem 29 1440... 

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

Fig. 2.8 The low-barrier hydrogen bond between Lysl6 and an oxygen atom of GTP /1-phosphate group. The electron localization function (ELF) is projected on the plane containing the three atoms involved in the LBHB. The red and yellow areas located between the... 

Kohout [10] used this function as an electron localization indicator (ELI). In the electron localization function (ELF), this function is scaled ... 

In a pericyclic reaction, the electron density is spread among the bonds involved in the rearrangement (the reason for aromatic TSs). On the other hand, pseudopericyclic reactions are characterized by electron accumulations and depletions on different atoms. Hence, the electron distributions in the TSs are not uniform for the bonds involved in the rearrangement. Recently some of us [121,122] showed that since the electron localization function (ELF), which measures the excess of kinetic energy density due to the Pauli repulsion, accounts for the electron distribution, we could expect connected (delocalized) pictures of bonds in pericyclic reactions, while pseudopericyclic reactions would give rise to disconnected (localized) pictures. Thus, ELF proves to be a valuable tool to differentiate between both reaction mechanisms. 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

The question remains whether the nodal planes, essential for the qualitative analysis, remain in the more advanced calculations of wavefunctions. To test this point, the electronic localization function (ELF) as implemented by B. Silvi and A. Savin [24] is used. In Figure 3 we summarize the results. 

Fig. 2. We show the electron localization function (ELF) of (from left to right and from above to below) the Cl-, the AlCLj-, the 12 1 , the A12C17-, and the AI4CI13- species. The purple colored space indicates high values of ELF or electron pairs. Therefore, electron deficiency can be recognized from the half open spheres.

Sn this is not so clear. The 22 valence electrons of the Sn ion could be accommodated in exact agreement with the octet rule according to the formula given in the margin. However, calculations with the electron localization function show that lone electron pairs are also present at the equatorial atoms therefore, only six electron pairs remain for the bonds. This corresponds to the number expected according to the Wade rules, as for bo-ranes ( + 1 multicenter bonds in a closo cluster with n = 5 vertices, cf. p. 144). We will deal with the bonding in such cluster compounds in Section 13.4. 

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