Electron pair localization function

To understand chemical processes, it is useful to have information besides total energies. Electron localization methods provide insight on the behavior of electrons in molecules. Properties, such as electron density, spin density, and the electron pair localization function (EPLF) [33], can routinely be computed by post-processing. The EPLF provides a quantitative description of electron pairing in molecular systems and has similarities to the electron localization function (ELF) of Becke and Edgecombe [34]. The QMC method is a particularly well-suited approach for obtaining such information because the simple and general definition of EPLF is easily evaluated in QMC. 

Fig. 15.1 Plot of the z=0 plane of the electron pair localization function (EPLF) function values for Spo in the ground state singlet (top), and the triplet state (bottom). The triplet state shows spin-polarized regions (dark gray) that are absent in the singlet state. The EPLF domain for both plots is [-0.1 0. ].

A. Scemama, P. Chaquin, M. Caffarel, Electron pair localization function A practical tool to visualize electron localization in molecules from quantum Monte Carlo data. J. Chem. Phys. 121, 1725-1735 (2004)... 

An interesting approach analyzing the pairing of electrons was presented in 2(X)4 by Scemama et al. [69]. There, the basic ingredients are the average distances d (r) and daa r) between an electron located at (r) and the closest same-spin and opposite-spin electron, respectively (computed from quantum Monte Carlo approach). The electron pair localization function (EPLF) is defined as follows ... 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

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 ... 

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). 

Figure 6-4. Electron localization function domains (concentration of electrons) in glycine. Lone pair domains are displayed in red...

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. 

By considering the extreme case of a crystal completely covered by a layer of foreign atoms, we have already seen in Sec. III,B that, if chemisorption involves the formation of localized electron pair bonds, some interesting interaction effects are to be expected. In this section, we approach the problem from the other extreme by considering just two atoms chemisorbed on a crystal surface. If the localized level formed by the interaction does not lie too far below the normal crystal band (or any surface band), the wave function for the localized level is damped only slowly in the crystal. Therefore, two chemisorbed atoms will be in interaction at distances when the interaction between the isolated atoms would be entirely negligible. To investigate this effect, we take the simplest model which may be expected to yield useful results 11). The crystal is represented by a straight... 

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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