Kinetic and potential energy density

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] ... 

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 ... 

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 ... 

The Laplacian is especially telling quantity [17,55], since it is connected to the kinetic and potential energy densities at BCP, G(Fc) and y(Fc), respectively, by the following local-virial theorem expression ... 

To a rough approximation, the kinetic and potential energies of electrons in simple systems vary with density... 

The valence region energy we are looking for, E ", is not simply the sum of the kinetic and potential energies of the electrons in the outer (valence) region, as one would infer from their stationary densities. The valence energy described here accounts for any relaxation that accompanies an acmal removal of the appropriate number of... 

Kohn-Sham orbitals (18)), Vn is the external, nuclear potential, and p is the electronic momentum operator. Hence, the first integral represents the kinetic and potential energy of a model system with the same density but without electron-electron interaction. The second term is the classical Coulomb interaction of the electron density with itself. Exc> the exchange-correlation (XC) energy, and ENR are functionals of the density. The exact functional form for Exc is unknown it is defined through equation 1 (79), and some suitable approximation has to be chosen in any practical application of... 

Invoking the Rayleigh hypothesis by balancing peak kinetic and potential energy dfinsitips (Equations 3.6 and 3.7) gives a relationship between resonant frequency w and surface mass density p, ... 

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].

EQUATIONS FOR BLOWERS AND COMPRESSORS. Because of the change in density during compressible flow, the integral form of the Bernoulli equation is inadequate. Equation (4.32), however, can be written differentially and used to relate the shaft work to the differential change in pressure head. In blowers and compressors the mechanical, kinetic, and potential energies do not change appreciably, and the velocity and static-head terms can be dropped. Also, on the assumption that the compressor is frictionless, / = 1.0 and hf 0. With these simplifications, Eq. (4.32) becomes... 

LFsing the fact that the average kinetic and potential energies are functionals of the electron density, and using (15.108) with i/tq replaced by i/tj, we have for (15.110)... 

The exact kinetic and potential energies are given by the (exact) first- and second-order density matrices, with an implicit summation over electron spin. 

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