Potential energy densities

The potential energy density, v(r), maps reveal explicitly that the N02-group is the most potential energy-rich area in the molecule (Fig. 12b). This is confirmed by the analysis of the local energies at the bond critical points (see below). The kinetic energy densities, g(r), are also higher for these groups (Fig. 12a). 

In order to reveal the subtle changes in the energy distributions caused by the crystal/molecule formation, we have calculated the deformation kinetic and potential energy densities [34,40] ... 

Here v(rc) is a potential energy density at the critical point, which was calculated as in the previous section. The correlation between the length of the hydrogen bond, p(rc), V2p(rc), and dissociation energy are striking. 

In the preceding discussion and in what follows below we have introduced to the bar symbol to emphasize that we refer to a specific (i.e. per unit mass) quantity. Note that Jk is a mass flux vector although Jk has a strange appearance this quantity is dimensionally correct Jk - pkvk, whence. Jk -pk. vk is the rate of transport of potential energy density. 

The peak potential energy density Up in the TSM resonator occurs at an instant (depicted in Rgure 3.4) when displacement is maximum in the crystal and velocity is zero. From Equation 2.40, this is given by... 

X-, y-, and z-components of displacement kinetic energy density potential energy density... 

The total energy density of a system involves contributions from the internal energy density e, the kinetic energy density JC, and the potential energy density V. 

Averaging of this final operator expression for the virial of V in the manner indicated in eqn (6.69) for the potential energy density TFbWi which is the virial of the Ehrenfest force eqn (6.29), yields... 

The electronic potential energy density (r), the virial of the forces exerted on the electrons (eqn (6.30)), and the electronic kinetic energy density, G(r) (eqn (5.49)), define the electronic energy density. 

Electrostatic potential maps have been used to make predictions similar to these (Scrocco and Tomasi 1978). Such maps, however, do not in general reveal the location of the sites of nucleophilic attack (Politzer et al. 1982), as the maps are determined by only the classical part of the potential. The local virial theorem, eqn (7.4), determines the sign of the Laplacian of the charge density. The potential energy density -f (r) (eqn (6.30)) appearing in eqn (7.4) involves the full quantum potential. It contains the virial of the Ehrenfest force (eqn (6.29)), the force exerted on the electronic charge at a point in space (eqns (6.16) and (6.17)). The classical electrostatic force is one component of this total force. 

The interaction between bonded atoms is characterized by the values of p(r), V-p(r), G r) and V(r) at the bond critical point. G(r) is the positive definite kinetic energy density and V(r) is the potential energy density. At a bond critical point, the kinetic and potential energy densities are related to the Laplacian by the local form of the virial relation ... 

Figure 5 The relationship between D(0-H BCP and oxygen atom separation) and local potential energy density V(rh) with the simple exponential relation V(ri)=A exp(-BD) given for ice X and the antifluorite structure in (a) and (b) respectively, where in (a) A = -11824, B =11.62 and correlation 0.998 and (b) A = -4.90, B = 8.88, correlation 0.999.

Where G(r) is a local kinetic energy density distribution and V(r) is the local potential energy density distribution [31], If H(r) is negative at r, then the local potential energy density distribution V(r) will dominate, and accumulation of electronic charge in the internuclear region will be stabilizing. In this case, one can speak of a covalent bond. 

The local value of the total energy density at a point r, H(r), is another useful topological descriptor that provides supplementary information about the nature of the interaction at r. The total energy density H(r) is the sum of the kinetic energy density G(r), a positive quantity, and the potential energy density F(r), a negative quantity, both densities related with the Laplacian of p(r) through the local expression for the virial theorem [45, 46] ... 

To compare the kinetic and potential energy densities on an equal standing, instead of the 2 1 virial ratio, Cremer and Kraka [110] evaluate the total electronic energy density at the BCP ... 

Figure 3. Dependence of the kinetic (G(r)) and potential energy density (k(r)) at the bond critical point, and the dissociation energy F,., on the O- H distance. Units are kJ/mol/au [3], kJ/mol, and A, respectively. Solid lines correspond to the exponential fitting (18 and 19). Data are from X-ray analyses of 83 (D-H- O, D=C, N, and O) HBs. Reprinted with permission from Ref [93].

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