Imagine an electron cloud v a charge distribution4 the carries the charge of N electrons. Unequal a storm cloud, the electron cloud go not change in time (stationary state), however has thickness ρ (r) that alters in an are (similar to the storm cloud). Inside the cloud the nuclei space located. The function ρ (r) exhibits non-analytical behaviour (discontinuity that its gradient) in ~ the location of the nuclei, which outcomes from the poles (−∞) of the potential power at this positions. Remind the form of the 1s wave role for the hydrogen-like atom (see Fig. 4.17), it has a spike at r = 0. In thing 10 that was presented that the correct electronic wave duty has to fulfill the cusp problem in the neighbourhood of every of the nuclei, whereby ρ alters as exp(−2Zr). This condition results in spikes of ρ (r) precisely at the location of the nuclei, Fig. 11.1.a. Exactly how sharp such a spike is5 counts on the charge of the nucleus Z: one infinitesimal deviation from the place of the nucleus (p. 505) has to be attach by together a diminish of the density6 that ∂ρ∂r/ρ=−2Z.
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Fig. 11.1. Electron density ρ for the planar ethylene molecule shown in three cross sections. ∫ ρ(r)d3r = 16, the variety of electrons in the molecule. Fig. (a) shows the cross ar within the molecular plane. The positions of the nuclei can be quickly recognized by the “spikes” the ρ (obviously much an ext pronounced for the carbon atoms than for the hydrogens atoms), your charges can be computed from the steep of ρ. Fig. (b) reflects the cross section along the CC link perpendicular come the molecular plane, therefore, just the maxima in ~ the location of the carbon nuclei are visible. Fig. (c) is the cross ar perpendicular to the molecular airplane and intersecting the CC bond (through that is centre). It is viewed that ρ decays monotonically through the distance from the link centre. Many interesting, however, is that the cross section resembles an ellipse quite than a circle. Note that we execute not see any separate σ or π densities. This is what the principle of π shortcut is all about, just to reflect the link cross ar ellipticity. R.F.W. Bader, T.T. Nguyen-Dang, Y. Tal, Rep. Progr. Phys. 44 (1981) 893, courtesy of institute of Physics Publishing, Bristol, UK.
Thus, since of the Coulombic interactions, the electrons will certainly concentrate close to the nuclei, and therefore us will have maxima the ρ ideal on them.
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It is evident also, the at long ranges from the nuclei the thickness ρ will degeneration to almost zero. Additional details will certainly be of good interest, e.g., are there any concentrations of ρ in the areas between nuclei? If yes, will it occur for every pair of nuclei or for part pairs only? This is of noticeable importance for chemistry, which deals with the ide of chemistry bonds and also a design of the molecule as the nuclei kept together by a chemical bond pattern.