By Regitz M., et al. (eds.)
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Extra resources for Houben-Weyl Methoden der organischen Chemie vol.E1 Organic Phosphorus Compounds I
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1]propellane (b). Also shown are contour maps overlaid with bond paths of the same function for the same planes. 0 au. Dashed contours denote negative values of V^p, regions of charge concentration, while solid contours denote positive values, regions of charge depletion. The values of V^p at the bond critical points are given in Table 1. 40 au. What appears to be a corresponding non-bonded maximum on each carbon in cyclopropane is in fact not a local maximum (three negative curvatures) but rather a (3, — 1) critical point (two negative curvatures and one positive curvature) in the negative of the Laplacian enables one to predict the positions of attack within a molecule and the relative geometry of approach of the reactants.
Bader (a) F I G U R E 7. Relief map, (a) contour map (b) and associated gradient vector field map (c) of the charge density in the C — C interatomic surface of the ethylene molecule, the symmetry plane bisecting the C — C internuclear axis. The contour map illustrates the elliptical nature of the ethylenic charge distribution with the major axis perpendicular to the plane of the nuclei. This figure again emphasizes that the charge density is a maximum in the interatomic surface at the position of the bond critical point.
The Laplacian appears in t h e local expression for the virial theorem: (/i74m)VV(r) = r{X) + 2G(r) (10) The quantity I^{X) when integrated over a n a t o m or over all space yields the corresponding value of the potential energy and is, therefore, the potential energy density. Correspondingly, G(r) is the kinetic energy density since its integration over a n a t o m or over all space yields t h e kinetic energy T. Since the integral of the Laplacian of P over an atom Q or over the total system vanishes, corresponding integrations of equation 10 yield t h e virial theorem for a n atom, ir(Q) + 2T(Q) = 0, or for the total system, + 2 r = 0.