Topology of the electron density

The theory of atoms in molecules defines chemical properties such as bonds between atoms and atomic charges on the basis of the topology of the electron density p, characterized in terms of p itself, its gradient Vp, and the Laplacian of the electron density V p. The theory defines an atom as the region of space enclosed by a zero-/lMx surface the surface such that Vp n=0, indicating that there is no component of the gradient of the electron density perpendicular to the surface (n is a normal vector). The nucleus within the atom is a local maximum of the electron density. 

The preceding analysis of the topology of the electron density in a molecule enables us to to define both the atoms and the bonds in a molecule. 

The atoms defined in the quantum theory of atoms in molecules (QTAIM) satisfy these requirements [1], The atoms of theory are regions of real space bounded by a particular surface defined by the topology of the electron density and they have all the properties essential to their role as building blocks ... 

The key to investigating the topology of the electron density p is the gradient vector V p, which is perpendicular to a constant electron density snrface and points in the direction of steepest ascent. Then, a sequence of infinitesimal gradient vectors corresponds to a gradient path. Since gradient vectors are directed, gradient paths also have a direction They can go uphill or downhill. 

An interaction can be termed a hydrogen bond when it corresponds to modem criteria based on the topology of the electron density. 

Interactions between two hydrogen atoms separated by a distance of less than 2.4 A can be formulated as dihydrogen bonds if they correspond to criteria based on the topology of the electron density in the H- H directions The pc values should be small, and the V pc values should be small and positive. 

Classification of Bonds Based on the Topology of the Electron Density... 

Luana, V., Costales, A., and Pendas, A. M. (1997). Ions in crystals The topology of the electron density in ionic materials II. The cubic halide per-ovskites. Phys. Rev. B 55, 4285-97. 

The topology of the electron density also leads to the identification of a chemical bond with a line linking neighboring nuclei along which the electron density is a maximum.190 This identification leads to a definition of molecular structure that is remarkable in its ability to recover all chemical structures. The dynamics of the density, as occasioned by nuclear displacements and analyzed by means of the mathematics of qualitative dynamics, leads to a complete theory of structural stability, one that clarifies the meaning of the making and breaking of a chemical bond. 

The Laplacian of the electron density plays a dominant role throughout the theory.191 In addition, Bader has shown that the topology of the Laplacian recovers the Lewis model of the electron pair, a model that is not evident in the topology of the electron density itself. The Laplacian of the density thus provides a physical valence-shell electron pair repulsion (VSEPR) basis for the model of molecular geometry and for the prediction of the reaction sites and their relative alignment in acid-base reactions. This work is closely tied to earlier studies by Bader of the electron pair density, demonstrating that the spatial localization of electrons is a result of a corresponding localization of the Fermi correlation hole. 

It is found that multiple bonds do not appear as such in the topology of the electron density. However, the value of the charge density at a bond critical point reflects the bond multiplicity and can indeed be empirically correlated with bond orders (refs. 93 and 94). As expected, it is also found that the charge distribution in the CC interatomic surface in ethylene has an elliptical nature associated with the presence of a tt bond. 

Espinosa E, Molins E (2000) Retrieving interaction potentials from the topology of the electron density distribution The case of hydrogen bonds. J ChemPhys 113 5686-5694 Falconer K (1990) Fractal Geometry Mathematical Foundations and Applications. New York John Wiley and Sons... 

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... 

A contemporary calculation [168] verified the importance of electrostatics in this complex but argued that this interaction was indeed a tme H-bond based on the topology of the electron density. This conclusion confirmed Novoa et al. [169] who claimed that even the weak interaction between CH4 and formaldehyde constitutes a H-bond, based again on the properties of the electron density and its bond critical points. Similar treatments further buttressed this conclusion for other donor-acceptor combinations [170,171]. Finally, a combination of at initio calculations and statistical analysis of crystal stmctures [172] suggested that while the CH- -O interaction polarizes its partner proton acceptor molecule less than would a traditional OH donor, one should nonetheless categorize both as H-bonds. 

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