Bond critical point electronic potential energy density

Fig. 5.41 The distribution of the electron density (charge density) p for a homonuclear diatomic molecule X2. One nucleus lies at the origin, the other along the positive z-axis (the z-axis is commonly used as the molecular axis). The xz plane represents a slice through the molecule along the z-axis. The —p = f(x, z) surface is analogous to a potential energy surface E = /(nuclear coordinates), and has minima at the nuclei (maximum value of p) and a saddle point, corresponding to a bond critical point, along the z axis (midway between the two nuclei since the molecule is homonuclear)...

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

Thus, if it is assumed that the local virial theorem is valid for the model electron densities fitted to the experimental structure factors, the kinetic, g(r), and potential, v(r), energy densities may be mapped, as well as the energy characteristics of the (3,-1) bond critical points evaluated [38]. 

In the following, this approach has been used to study the energy features of 3-NTO [29]. The kinetic, g(r), and potential, v(r), energy density maps have been calculated from the experimental electron densities with the WinXPRO program package [39] using the approach described above, as well as their difference with respect to atomic procrystals with no chemical bonds. Critical point characteristics have also been similarly analyzed. 

Cremer and co-workers extended and widely applied the Bader topological analysis of atoms in molecules to a range of (potential) homoaromatic molecules. The recommended bond or interaction indices of item 2 are derived from such an analysis and are based on the electron densities calculated at the bond critical point (for bond homoaromaticity) or at the midpoint of the homoconjugative internuclear gap (for no-bond homoaromaticity where no bond critical point exists). Williams, Kurtz, and Farley developed semiempirical discriminators for the confirmation of homoaromaticity based on two-center energy partitioning terms (a negative value indicates... 

Abstract Based on an investigation of empirical links of the bond valence method to observable quantities, especially the electron density at the bond critical point as well as absolute electronic potential and hardness values in the frame of the hard and soft acids and bases concept, it is ascertained that bond valence can be understood as a functional of valence electron density. Therefrom a systematic approach for deriving bond valence parameters and related quantities such as coordination numbers and bond breaking energies is discussed that together allow for a conversion of the bond valence method to a simple effective atomistic forcefield. 

The first line lists the experimental result the second line- the atomic procrystal p is the electron density V2p is the Laplacian, g, v and he are the kinetic, potential and total electronic energies at the critical point R is the bond path length. 

Figure 6 shows a contour line diagram of the electron density distribution of the van der Waals complex Hcj (He,He distance 2.74 A). The electron density at the midpoint between the two He atoms is just 0.008 e/A, quite different from the values found for a typical covalent bond between first-row elements (1-5 e/A ). Despite the smallness of p(r) in the internuclear region, the He nuclei are linked by a MED path and the midpoint is the position of a (3, — 1) critical point (Fig. 6). As pointed out above this does not imply the existence of a chemical bond. The energy density H(r) is positive at the (3, — 1) critical point, which means that the kinetic energy rather than the potential energy dominates in the internuclear region. There is no chemical bond between the He atoms.

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