The Laplacian of the Electron Density and the Lewis and VSEPR Models
While the topology of the electron density "" provides a faithful mapping of the concepts of atoms, bonds and structure, it provides no indication of maxima in corresponding to the electron pairs of the Lewis model. This model is secondary only to the atomic hypothesis itself in understanding chemical bonding and reactivity and the geometry of molecules, the latter as predicted in terms of the localized electron pairs assumed in the VSEPR model. The physical basis of this most important model is one level of abstraction above the visible topology of the electron density, appearing in the topology of the Laplacian of the density. This function is the scalar derivative of the gradient vector field of the electron density, the quantity 2, and it determines where electronic charge is locally concentrated, 2 < 0, and depleted, 2 > 0, the local charge concentrations providing a mapping of the electron pairs of the Lewis and VSEPR models.
The Laplacian of the electron density recovers the shell structure of an atom by displaying a corresponding number of alternating shells of charge concentration and charge depletion. The uniform sphere of charge concentration present in the valence shell of a free atom is distorted upon chemical combination to form local maxima and minima. The number of local maxima in -2 in the valence shell, the local valence charge concentrations, together with their relative positions and magnitudes, coincide with the number and corresponding properties of the localized electron pairs assumed to exist in the VSEPR model of molecular geometry.
Figure 11. A relief map of the Laplacian of the electron density for the ClF3 molecule in the equatorial plane (left) and in the plane containing all four nuclei (right).
The realization of the VSEPR model in terms of the Laplacian of is illustrated in Figure 11 showing a relief map of the Laplacian for the ClF3 molecule. The chlorine atom exhibits three shells of charge concentration, the fluorine atom two such shells, corresponding to the presence of three and two quantum shells respectively. The VSEPR model predicts a T-shaped geometry for ClF . This geometry enables the two non-bonded electron pairs, which are assumed to be larger than the bonded pairs, to occupy the least crowded equatorial positions. The equatorial plane (top diagram) shows the presence of two non-bonded and one smaller bonded charge concentrations. In the axial plane there are three bonded charge concentrations and a fourth apparent maximum which is actually another view of the (3,-1) critical point between the two non-bonded maxima, that is, a critical point with one positive curvature. Thus the valence shell charge concentration of the chlorine atom possesses two non-bonded and three bonded charge concentrations in agreement with the five electron pairs assumed in the Lewis model and in this geometry, they maximally avoid one another, in agreement with the VSEPR model. The sizes of the charge concentrations decrease in the order, nonbonded > equatorial bonded > axial bonded, as assumed in the VSEPR model, the bonded sizes reflecting a greater net charge on the axial fluorines.
The Lewis model is also used to rationalize chemical reactivity. In addition to a local charge concentration in the valence shell that behaves as Lewis base or nucleophile, there are also local charge depletions, and such charge depletions behave as Lewis acids or electrophiles. A chemical reaction corresponds to the combination of a charge concentration in the valence shell charge concentration of the base with a charge depletion in the valence shell charge concentration of the acid, the reactivity paralleling the magnitude of the charge concentration and the depth of the charge depletion. The geometry of approach of the acid and base molecules is predicted through the alignment of the corresponding "lumps" and "holes" in their Laplacian distributions, as illustrated for the approach of the non-bonded charge concentration on carbon of the CO molecule to the hole on the boron atom in BH3 , Figure 12. This predictive property of the Laplacian has been illustrated in many different reactions including those as diverse as the approach of methane to the oxygen atom of a metal oxide surface and the geometries of hydrogen bonded complexes.
The final Figure illustrates the charge concentrations present in the outer core of the barium atom in BaH2, a bent molecule. The use of the Laplacian of the electron density to account for the bent geometries of the hydride, halide and methylide molecules of calcium, strontium and barium, in terms of a distortion of the outer core of the electron density of the metal atom is discussed in reference 12.
Figure 13. The zero envelope of the Laplacian distribution for the barium core in BaH2 showing the presence of four charge concentrations in the outer core of the shell and the spherical inner core. The two charge concentrations at the bottom edges of the diagram are portions of the charge concentrations associated with the protons.
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