Plot of the electron density distribution

Figure 5-20 Plots of the electron density distributions associated with s orbitals. For any s orbital, this plot is the same in any direction (spherically symmetrical). The sketch below each plot shows a cross-section, in the plane of the atomic nucleus, of the electron cloud associated with that orbital. Electron density is proportional to r ip. ...

Figure 1. Contour plot of the electron density distribution in the BF3 molecule. The lines connecting the nuclei are the bond paths along which the electron density is greater than along any other line connecting the two nuclei. The curved lines between the atoms are the lines along which the interatomic (zero-flux) surfaces cut the molecular plane.

The calculated enthalpies for silica in the quartz and stishovite phases are shown in Figure 3 as a funetion ofpressure. The stishovite structure beeomes more stable than the quartz strueture at 3.5 GPa with the distorted ion model, and at 21 GPa with the spherical ion model. In comparison, the experimental zero temperature transition pressure for the quartz to stishovite phase transition is estimated to be 5.5 GPa from thermodynamic data [53], and the transition pressure for the similar cristobalite to stishovite phase transition is caleulated to be 6 GPa by periodie Hartree-Fock methods [54]. The non-spherical distortions improve the modeling of this phase transition by stabilizing stishovite with respeet to quartz the greater stabilization ofstishovite occurs because the distortions strengthen three bonds per anion in stishovite, and only two bonds per anion in quartz (the bonds are significantly covalent in both structures, as shown above in the plots of the electron density distributions). 

FIG. 17. A schematic overview of Cso represented by a stick model, 3D and 2D contour plots of the electron density. In the 3D plot in the middle the single contour has been chosen to show how the electrons are distributed in the bonds. The 2D contour plot shows the electron density in a plane that includes the center of the molecule. We clearly see that there is a void, which means that Ceo constitutes a spherical shell. 

All theoretical studies agree with this second picture [13-17,39,99]. This is shown by the plots of the electron density maps and in particular by density difference maps which show very clearly the electron localization at the center of the vacancy [39,55]. This is true not only for the bulk but also for the surface of MgO, Fig. 5. The localization of the electrons in the center of the vacancy is an indirect proof of the highly ionic nature of MgO. In fact, the electrons are trapped in the cavity by the crystalline Madelung potential. Calculations performed on cluster models have shown that in absence of the external field the electrons tend to distribute more over the 3s levels of the Mg ions around the vacancy [38]. The localization of the electron in the center of the vacancy is... 

In Figure 1.6, the atomic scattering factors f(s) for hydrogen, carbon, and oxygen are plotted against s = 2(sin 6)/X. In the forward direction (s = 0) the x-ray waves scattered from different parts of the electron cloud in an atom are all in phase, and the wave amplitudes simply add up, rendering /(0) equal to the atomic number Z. As s increases, the waves from different parts of the atom develop more phase differences, and the overall amplitude begins to decrease. The exact shape of the curve f(s) reflects the shape of the electron density distribution in the atom. The... 

Figure 2. A summary of the properties of the eleetron density distribution for the skeletal SiOSi dimers in a set of H6Si207 moleeules with geometries fixed at those observed for the Si207 dimers in eoesite. In (a), (b) and (e), respectively, the individual SiO bond lengths, R(SiO), observed for eoesite are plotted against Ai, A2 and A.i, the curvatures of the electron density distribution calculated for the molecules at their saddle points Fc. In (d), R(SiO) is plotted against the magnitude of the electron density, p(rc). In (e) R(SiO) is plotted against G r /p tc) where G(rc)/p(rc) is the kinetic energy density and in (f) R(SiO) is plotted vs. ellipticity, f, of the bonds.

There is one further descriptor that can be used to characterize double bonds. This is bond ellipticity. Figure 10.48(a) and (b) show an uneven distribution of the electron density along the N=N bond in HN=NH. This is more apparent in Figure 10.51, which shows a plot of the electron density in a plane perpendicular to the AIL at the BCP. The plot is elhptical. In the standard valence-bond picture a double bond consists of a rotational symmetry of electron density about the line linking the two bonded nuclei) and a it bond of two overlapping p orbitals perpendicular to this line. The preferential accumulation of electronic charge, or the amount of deviation from a circular symmetric distribution of p(r), is called... 

Using the combination of main-frame CDC 6400 and Tektronix computations, a number of phenomena were studied with electron density functions, and especially with projection plots. Particularly useful were plots of difference functions in which the electron distributions of isoelectronic systems were compared directly. In such applications, we noted that a corresponding difference plot of the electron density itself in any given plane is not meaningful since the number of electrons may change that is, from one compound to the next the electron density can shift from one plane to elsewhere. In the projection plot the total number of electrons remains the same for both species and the integral of an isoelectronic difference function must sum to zero. Some examples of the kinds of problems studied are the vm transition of formaldehyde, substituent effects in substituted benzenes, and polarization... 

The first series of plots represent the limiting and perfectly balanced cases for the distribution of the electron density (positive values only are shown). These spin density plots show the excess density perfectly balanced between the two terminal heavy atoms for allyl radical, drawn toward the substituent for Be and pushed away from the substituent for acetyl radical. 

Not only is hybridization an artificial simulation without scientific foundation, but even the assumed "orbital shapes" that it relies upon, are gross distortions of actual electron density distributions. The density plot shown above, like all textbook caricatures of atomic orbitals, is a misrepresentation of the spherical surface harmonics that describe normal excitation modes of atomic charge distributions. These functions are defined in the surface of the charge-density function, as in Fig. 2.13, and not at r = 0, as shown in Figure 2.16. 

The electron density distribution in the three-carbon plane of cyclopropane is shown in Figure 1(a). The total densities rarely are very informative because the high local concentrations near the nuclei dominate the distribution. However, we may subtract the density which would be appropriate for spherically averaged carbon atoms placed at the coordinates of the cyclopropane carbons. This will show how the electron density distribution has been affected by bond formation (Figure 1(b)). It has been called a deformation density plot . 

The electron density distribution is a four-dimensional function (the number of elearons at a given point (x,y,z)), which is difficult to visually represent. Figures 1 and 2, respectively, show a three-dimensional isoelectronic surface of benzene and a contour plot of the elearon density p(r) in the molecular plane of benzene. Both representations show only gross features of the density. In particular, the total electron density distribution is dominated by the core electrons and appears simply as an aggregate of slightly distorted spheres... 

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