FIGURE 9. Perspective drawing of the calculated electron density distribution p (r) in the plane of the cyclopropane ring [HF/6-31 G(d,p) calculations]. Point p denotes the position of the bond critical point between two neighbouring C atoms. For better presentation, density values above 14 e A 3 are cut off [Pg.61]

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 [Pg.376]

As mentioned in Chapter 8 (page 172), the double bond is associated with an elliptical distribution of electronic charge in the plane perpendicular to the CC nuclear axis and containing its mid-point where the electronic density has a local maximum (critical point in the theory of Bader). The relief diagram and the contour plots of Fig. 9.7 taken from the work of Bader et al. (ref. 92) show the distribution of the electronic charge density in the nuclear plane of the molecule. [Pg.216]

FIGURE 13. Topological CC bond orders n of homocyclopropenium (44) and homotropenylium cation (45) calculated from the MP2 electron density distribution p(r) at the bond critical points. Note that n values for C1,C3 of 44 and C1,C7 of 45 correspond to interaction indices. MP2 bond lengths (in A) are also given44,56 [Pg.377]

On the basis of these definitions one can describe chemical bonding in molecules containing noble gas elements with the aid of the properties of p(r). One starts by searching for the bond paths 2uid their associated bond critical points Tg in the molecular electron density distribution. If all bond paths are found, then the properties of p(r) along the bond paths will be used to characterize the chemical bonds. For example, the value of can be used to determine a bond order, the anisotropy of Pp can be related to the n character of a bond, the position of the bond critical point is a measure of the bond polarity and the curvature of the bond path reveals the bent-bond character of a bond [17, 19]. [Pg.26]

The quantum theory of atoms in molecules (QTAIM) [25, 26] is based on analyses of the electron density distribution. The electron density of such systems such as simple molecules or ions, and also complexes, complex molecular and ionic aggregates, as well as crystals may be analyzed using this approach. QTAIM is a powerful tool that allows characterizing of various interactions covalent bonds, ionic bonds, van der Waals interactions and, what is the most important for this review, also HBs. The analysis of critical points of the electron density is very useful. For the critical points (CPs), the gradient of electron density, p(r), vanishes [Pg.262]

The Coulombic potential becomes infinitely negative when an electron and a nucleus coalesce and, because of this, the state function for an atom or molecule must exhibit a cusp at a nuclear position. That is, as shown by Kato (1957), the first derivative of the function is discontinuous at the position of a nucleus. Thus, while the charge density is a maximum at the position of a nucleus, this point is not a true critical point because Vp, like is discontinuous there. However, as discussed in Election E2.1, this is not a problem of practical import and the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution and hereafter they will be referred to as such. [Pg.19]

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