Density Evaluations

Many atomic and molecular properties depend on the electron density, and some depend on the gradient of the electron density evaluated at certain positions in space. 

Consider constant flow into protected equipment (blocked outlet). For the steady-state design scenario with a constant flow of fluid Win (kg/s) from a pressure source that is above the maximum allowed pressure in the protected equipment, volume is being generated within the equipment at a rate of Win/pin, where pin (kg/m3) is the incoming fluid density evaluated at the maximum allowed pressure. Denoting pout (kg/m3) as the vent stream fluid density, Eq. (23-89)... 

In summary, the original Thomas-Fermi-Dirac DFT was unable to give binding in molecules. This was corrected by Kohn-Sham, [11] who chose to use an orbital rather than density evaluation of the kinetic energy. By the virial theorem, = —E, so this was a necessity to obtain realistic results for energies. Next, it was shown that the exact exchange requires an orbital-dependent form, too. [47,48] The future seems to demand an orbital-dependent form for the correlation. 

Expectation Values of the Li-H Internuclear Distance, Its Square, and Electron-Positron Contact Densities Evaluated at 3200-Function Basis Size [122] ... 

P9 Density evaluated at 90% of the saturation (vapour) pressure at inlet temperature, T), after a flash calculation. For a multicomponent mixture, use the bubble point pressure at T). The flash calculation should preferably be carried out isentropically, but an isenthalpic flash is sufficient CO E D)... 

Particle density can alternatively be evaluated from a set of retention measurements made in carriers of different density (1J5). Such density modifications are typically accomplished by successive additions of sucrose to the carrier. The observed retention values yield particle size and density through Equation 2. A rearrangement of this equation leads to a graphical procedure for size/density evaluation. The rearranged form is... 

Using equation (5) we cannot proceed much further except to note the negative sign of the second term. This sign implies that a higher electron density (evaluated with a 1/p weighting factor) would produce a shift toward lower fields, that is, toward lower shielding. 

Figure 2.12. Nodal properties of the transition densities of the first four transitions in benzene, a) Representation of the complex LCAO coefficients of HOMOs 01 and 0, as well as LUMOs nd by means of a phase polygon. Each coefficient has the absolute magnitude n and the complex phase shown by a dot in the complex plane of which the real and imaginary axes are abscissa and ordinate, b) Representation of the overlap densities evaluated from the complex coefficients, and c) values of the overlap densities at the vertices of the perimeter and the resulting nodal properties.

Mass-transport-corrected currents at T = 30 °C, that is, the kinetic current densities evaluated from the measured T = 30 °C... 

All Ng,C bonds of cations NgCCNg ( Z ) investigated so far are covalent. This is suggested by the properties of the electron and the energy density evaluated at the bond critical points (Table 14). The density data also reflect the fact that the Ne,C bond is actually weaker than the He,C bond due to 2pn-n electron repulsion absent in the He compound. 

Yu. A. Abramov, On the possibility of kinetic energy density evaluation from the experimental electron-density distribution, Acta Crystallogr. A 53, 264-271 (1997). 

As the dihedral angle is increased from cf) = 0° to the critical value beyond which the bond path vanishes (27°), the H-H BCP and the corresponding 6-MR RCP come closer and closer. The annihilation of the H-H BCP at ca. 27° entails a sudden change in structure , a catastrophe [130], known as the fold catastrophe [63] (see change in structure in Fig. 6). As the BCP and the RCP approach one another, the property densities evaluated at each CP assume closer and closer values until the two points eventually merge and annihilate each other [57]. The plot in Fig. 7a shows the change in the distance... 

For gases, the dependence of density upon pressure and temperature has a significant effect on the flow behavior when the change in pressure is significant (e.g., 20% or more) relative to the absolute pressure in the system. At low velocities (relative to the speed of sound), the relative change in pressure and associated effects are often small, and the assumption of constant density (evaluated... 

How much error would be introduced in the answer to Prob. 2.2 if the equation for hydrostatic equilibrium [Eq. (2.5)] were used, with the density evaluated at 0°C and an arithmetic average pressure ... 

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