Topology of the ELF

The purpose of the topological analysis of the electron localization function is to provide a sound mathematical model of the Lewis , and theories which 

The ELF defined in equation 14 is a local function which describes to what extent the Pauli repulsion is efficient at a given point of the molecular space. Originally, the ELF was derived from the Laplacian of the conditional probability [V Pcondiri, r2)]ri=r2- An 

From a quantitative point of view a localization basin (core or valence) is characterized by its population, i.e. the integrated one-electron density p(r) over the basin (equation 15) 

In equation 17, N i2i)N i2j) is the number of the electron pairs classically expected from the basin population whereas N(Qi, f2 ) is the actual number of pairs obtained by integration of the pair-electron function over the basins and i2j. The variance is then a measure of the quantum mechanical uncertainty of the basin population which can be interpreted as a consequence of the electron delocalization, whereas the pair covariance indicates how much the population flucmations of two given basins are correlated. Within the AIM framework, the atomic locahzation and delocalization indices X A) and 5(A, B) have been introduced and defined by equations 18 and 19  

The AIM delocalization indices are sometimes referred to as bond orders . The above notation can be generalized to any partition in the direct space and therefore is adopted in the present work. Within the ELF approach, the core population variance and 

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Finally, some of the unexpected results revealed for iodophenols warn against the use of large core pseudopotentials in the ELF analysis. It is noteworthy that the analysis of the topology of the ELF enables us to predict favoured protonation sites with the help of a least topological change principle which will be discussed in a following section. 

As seen above, the topology of the ELF suggests that the most favoured protonation site can be found by using a least topological change principle which states that ... 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

The ELF was proposed by Becke and Edgecombe [7] in 1990 and very soon extensively applied to a variety of systems ranging from atoms to inorganic and organic molecules to solids [9]. In 1994, a topological analysis of the ELF was... 

The Bonding Evolution Theory, briefly presented in Appendix B, provides a description of the bonding features of a system, along with their evolution accompanying a reaction path. It relies on the variation of the ELF topological profile as a function of nuclear coordinates. The ELF makes a partition of the molecular space into open sets having a... 

Figure 5. Tridimensional visualisation of the ELF function for TiH3+ for the D3h and C3V symmetry, The ML3 topology of the C3v species is clearly seen in the side view. The orientation of the core basins vs. the Ti-H dissynaptic basins appears in the top view.

FIGURE 5. Summary of electronic distribution in 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 L(r) = —V2p(r) and e = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Iso-surfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins... 

In the next section, the principal ingredients involved in the ELF will be explained, and their relation with chemical concepts will be clarified. Then, a brief comparison of the ELF with other theoretical related tools, like the atoms in molecules model of Bader, will be done. Next, some elementary concepts from the mathematical theory of topological analysis will be in a rather crude way presented. After that, some applications, extensions and results will be discussed, focusing in particular on applications developed at our group. 

This good property of the ELF can be at first glance explained looking at the qualitative behaviour of the total electron density, which is also very independent of the level of calculation. Of course the numerical values can be different, so are the expectation values based on the density. But the maxima and minima are all almost at the same positions. More precise, all the critical points are almost at the same positions. Next, it will be shown that mathematically this means that the function is topologically invariant to the level of calculation. 

Another useful concept of topological theory, which can help in quantifying the information carried on the basins, is the concept of bifurcation. At very low values of the ELF, you have only one f-domain which is reducible (it contains more than one... 

Topological analysis of the ELF constructed from density components has also been evaluated. Separations of the a-j3 spin41,51 and the (t-tt52 electron contributions to density have been recently reported. Although the total ELF is not recovered by the addition of the individual components, the separation is a useful tool to evaluate some important electronic aspects of different classes of chemical systems as radicals or aromatic species. 

In the same way as the a, -separation has been performed, one can proceed to a cr, rr-separation.52 This separation has been used to evaluate the aromaticity of organic molecules and clusters. An index of aromaticity was proposed using a scale based on the bifurcation analysis of the ELF constructed from the separated densities. In principle, the total ELF has no information about tt and cr bonds, it depends only on the total density. Hence, the ELF does not show clear differences between both kinds of bonds. However, the topological analysis over separated densities, ones formed by the rr-orbitals and the other ones formed by the cr-orbitals, yields the necessary information.52 Of course, this is possible only for the molecules which present the cr, tt symmetries, i.e. planar molecules. The bifurcation analysis of the news ELFW and ELFCT can be interpreted as a measure of the interaction among the different basins and chemically, as a measure of electron delocalization.45 In this way, the tt and a aromaticity for the set of planar molecules described in the Scheme 1 has been characterized.52... 

The topological analysis of the ELF and the calculation of integrated properties were performed with the TOPMOD package [83]. A step interval of 0.07 bohr was chosen in each space direction to set 3D grids of ELF basins. For the systems in Fig. 1, this gives place to a number of points per geometry between 3.8 X 10 and 9.4 x 10 depending on the intermolecular distance. 

The topological analysis of the density and of the ELF funetion provides new information to understand the nature of the Si-0 bond. On the one hand, the atomic population and the bond ellipticities tell us that the SiO bond is partly ionic and also that there is no evidence for a partial double bond character. This latter point is confirmed by the analysis of the ELF function since there is only one attractor between the oxygen and silicon cores and that its basin population is always less than or equal to 2 electrons. Moreover, it appears more important to consider the oxygen than the silicon to discuss the bonding in silica. The Si-0 bond is found to belong to the electron shared interaction by the ELF analysis. 

ELF. - In order to understand depleted homopolar bonds better Llusar et performed an ELF study of the N-N, 0-0 and F-F bonds. Deformation density maps are known to show a depletion of p between formally covalent bonds. However a topological analysis of the ELF function demonstrates that bond strength appears to be correlated with the disynaptic basin populations. From a qualitative point of view, the splitting of a disynaptic basin into two monosynaptic ones upon bond stretching is the signature of the covalent bond. 

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