Curvature electron density

We can measure the extent electronic charge is preferentially accumulated by a quantity called the ellipticity e. At the bond critical point it is defined in terms of the negative eigenvalues (or curvatures), Aj and A2 as e = (A1/A2) — I. As A1 < A2 < 0, we have that A ]/A2 > 1, and therefore the ellipticity is always positive. Tf e = 0 then we have a circularly symmetric electron density, which is typically found at bond critical points in linear molecules. 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

Fig. 7.1 The electron density p(t) is displayed in the and

The topological characteristics of CPs on bonds gives a quantitative explanation of the known effect that the formation of a Ge crystal is accompanied by shifting the electron density towards the Ge-Ge bonding line. This is can be seen by comparing the parameters of the curvature of the electron density at the critical point (3,-1) with analogous parameters for a procrystal (a set of noninteracting spherical atoms placed at the same... 

Figure 11. Laplacian of electron density for Ge - V2 p (r) for small fragment in Ge curvature of electron density along Ge-Ge bond shows only small shift of electrons to the...

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

For the covalent bonds in the diatomic molecules studied by Bader and Essen (1984), values of ky and k2 vary from —25 to —45 eA-s, while /3 is positive in the range of 0-45 eA-5. The sum of the curvatures, V2p, is invariably negative, indicating the concentration of electron density in the internuclear region. But for second-row atoms, the larger positive value of A3 may dominate the Laplacian. 

In contrast, methyl-for-chlorine substitution is decidedly nonlinear, a feature also displayed by the lighter Group 14 compounds. This curvature is not an artifact of the BAC-MP4 predictions, since it is observable in the (admittedly limited) experimental data for these compounds (Fig. 10). In fact, the deviations from linearity are even greater in the experimental data. Such behavior is also observed in the analogous Si compounds and is related to the negative hypercongugation (anomeric) effect, in which electron density from... 

If the second derivative, and hence the curvature of/, is negative at x, then / at x will be larger than the average of / at all neighbouring points, i.e. / concentrates at point x73. Therefore - V2p(r), which is the second derivative of a function depending on three coordinates x, y and z, has been called the Laplace concentration of the electron density distribution. Furthermore, the Laplacian of pir) provides the link between electron density p(r) and energy density Hir) via a local virial theorem (equation 8)67,... 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

The other aspect of fullerene electronic structure which relates to their ability to accept electron density is the rehybridization of the carbon n atomic orbitals as a result of the curvature which is imposed on the conjugated carbon atoms by the shape of the molecules (Haddon et al. 1986a, b). The discussion of rehybridization effects is deferred to 6, and we begin with a treatment of the topological contribution to fullerene electronegativity within simple hmo theory. The presence of 12 conjugated 5MR in the fullerenes suggests that they will be biased towards reduction in their redox chemistry. 

Bonds with r < dl < d[ become possible because of nuclear screening (increased bond order), which causes concentration of the bonding pair directly between the nuclei. The exclusion limit is reached at d = t and appears as a geometrical property of space. The distribution of molecular electron density is dictated by the local geometry of space-time. Model functions, such as VSEPR or minimum orbital angular momentum [65], that correctly describe this distribution, do so without dictating the result. The template is provided by the curvature of space-time which appears to be related to the three fundamental constants tt, t and e. 

For both types of FIDCO surfaces, the usual Shape Group method [2] of electron density shape analysis is applicable. The additional formal domain boundaries AD i(Ga b(3)) and AD i(GA(B)(a)) introduce one additional index -1, which can be treated the same way as relative curvature indices. The one-dimensional homology groups obtained by truncations using all possible index combinations are the shape groups of FIDCO surfaces. The (a,b)-parameter maps and shape codes are generated the same way as for complete molecules [2],... 

In most interactions between two reactants, local shape complementarity of functional groups is of importance. A local shape complementarity of molecular electron densities represented by FIDCOs implies complementary curvatures for complementary values of the charge density threshold parameters a. For various curvature domains of a FIDCO, we shall use the notations originally proposed for complete molecues [2], For example, the symbol D2(b),i(a, Fj) stands for the i-th locally convex domain of a FIDCO G(a) of functional group Fj, where local convexity, denoted by subscript 2(b), is interpreted relative to a reference curvature b. For locally saddle type and locally concave domains relative to curvature b, the analogous subscripts 1(b) and 0(b) are used, respectively. 

In general, a locally convex domain D2(b),j(a> Fj) of a functional group F, relative to a reference curvature b, shows local shape complementarity with a locally concave domain Do( b),j(a, F2) of a complementing functional group F2, relative to a reference curvature of -b. The threshold values a and a are also likely to complement each other the shape complementarity between the higher electron density contours of one functional group and the lower electron density contours of the other functional group is relevant. 

Then, analyzing the electron density topology requires the calculation of Vp and of the hessian matrix. After diagonalization one can find the critical points in a covalent bond characterized by a (3, -1) critical point, the positive curvature X3 is associated with the direction joining the two atoms covalently bonded, and the X2, curvatures characterize the ellipticity of the bond by ... 

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