Topological analysis of the electron density

Such analysis sufficiently supplements with topological analysis of the electron density proposed earlier by Bader, where the condition... 

Bond critical points represent extremes of electronic density. For this reason, these points are located in space where the gradient vector V p vanishes. Then the two gradient paths, each of which starts at the bond critical point and ends at a nucleus, will be the atomic interaction line. When all the forces on all the nuclei vanish, the atomic interaction line represents a bond path. In practice, this line connects two nuclei which can consequently be called bonded [5]. In terms of topological analysis of the electron density, these critical points and paths of maximum electron density (atomic interaction lines) yield a molecular graph, which is a good representation of the bonding interactions. 

Figures 5.20 illustrates the equilibrium and transition-state structures obtained for these complexes. As shown, the L1H-H20 complex in equilibrium state 1 shows an intramolecular dihydrogen bond with a very short H- H distance of 1.580 A calculated at the MP2/6-311++G(2d,2p) level. The topological analysis of the electron density on the H- H direction has resulted in the small pc and positive V pc values (0.0388 and 0.0453 an, respectively) typical of dihydrogen bonding. In contrast, no dihydrogen bonding was observed in the LiH-H2S molecule (3), where the corresponding hydrogen atoms are strongly remote.

A topological analysis of the electron density in the framework of AIM theory, performed for the systems in Figure 6.2, has completely confirmed their formulation as dihydrogen-bonded complexes. In accord with the AIM criteria, the pc and V pc parameters at the bond critical points found in the H- - -H directions are typical of dihydrogen bonds 0.042 and 0.057 au for complex LiH HF and 0.046 and 0.048 au for complex NaH- - -HF, respectively. The presence of the bond critical points can be well illustrated by the molecular graph in Figure 6.3, obtained for the HCCH H-Li complex by Grabowski and co-workers [8]. 

For comparison, the authors have probed a complex formed by the same proton-donor molecule and molecular hydrogen. In this very weak complex, HCCH- - (H2), the H- - (H2) distance has been calculated as 2.606 A (i.e., significantly larger than the sum of the van der Waals radii of H). It is extremely interesting that a topological analysis of the electron density also leads to the appearance of the bond critical point in the H- - (H2) direction. However, the Pc and V pc values are very small (0.0033 and 0.0115 au, respectively) compared with those in the HCCH- - -HLi complex (0.0112 and 0.0254 au, respectively). The most important conclusion of this comparison is There is no evident borderline between the dihydrogen-bonded complexes and the van der Waals systems. 

NH4-CH4]+ complex in the gas phase [36]. Topological analysis of the electron density performed in the framework of AIM theory shows the bond critical points on the H- H directions with pc values of 0.013 an. It is interesting that the electron density in this complex is larger than that obtained for the BH4 - CH4 dihydrogen-bonded system (pc = 0.007 an), the CH4 molecule of which acts as a proton donor. In accordance with the electronic density, the H- H distances in the BH4 - H4C complex were remarkably longer than 2.4 A (2.797, 2.929,... 

AIM topological analysis of the electron density performed for two complexes and for isolated components is shown in Table 6.15. The bond critical points found in the H H directions are characterized by the small electronic density with Pc = 0.002 and 0.009 au in the CILj- HF and SilLj- HF systems, respectively. The Laplacian, V pc, is also small but takes positive values in accordance with the AIM criteria for dihydrogen bonding. 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

The mathematical theory of topology is the basis of other approaches to understanding inorganic structure. As mentioned in Section 1.4 above, a topological analysis of the electron density in a crystal allows one to define both atoms and the paths that link them, and any description of structure that links pairs of atoms by bonds or bond paths gives rise to a network which can profitably be studied using graph theory. 

Although it is impossible to formulate a definition of molecular geometry that is fully quanturn-mechanical in nature and at the same time universally applicable to all chemical species, topological analysis of the electron density leads to a rigorous statement of the dominant molecular structure for any state, spectroscopic or localized, stationary or time dependent, with zero angular momentum. In this sense, unlike geometry or shape, structure is an observable property of an isolated molecule. 

Several alternatives to the Mulliken population have been presented that attempt to provide more rigorous estimates of the charges on atoms in molecules or clusters although not all have been applied in chemisorption and catalysis. We quote the Natural Bond Order analysis, and the elegant topological analysis of the electron density or of the electron localization function, ELF, introduced by Becke and Edgecombe. " ELF analysis has... 

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