Minimum energy density

The minimum energy density )3 is then I (3fe33 + fe22)g Q kqx i... 

Minimum Energy Density Necessary for the Photochemical Effect... 

Figure Cl. 1.6. Minimum energy stmctures for neutral Si clusters ( = 12-20) calculated using density functional theory witli tire local density approximation. Cohesive energies per atom are indicated. Note tire two nearly degenerate stmctures of Si g. Ho K M, Shvartsburg A A, Pan B, Lu Z Y, Wang C Z, Wacher J G, Fye J L and Jarrold M F 1998 Nature 392 582, figure 2.

The atoms and molecules at the interface between a Hquid (or soHd) and a vacuum are attracted more strongly toward the interior than toward the vacuum. The material parameter used to characterize this imbalance is the interfacial energy density y, usually called surface tension. It is highest for metals (<1 J/m ) (1 J/m = N/m), moderate for metal oxides (<0.1 J/m ), and lowest for hydrocarbons and fluorocarbons (0.02 J /m minimum) (4). The International Standards Organization describes weU-estabHshed methods for determining surface tension, eg, ISO 304 for Hquids containing surfactants and ISO 6889 for two-Hquid systems containing surfactants. 

MO calculations can provide the minimum-energy structure, total energy, and overall electron density of a given molecule. However, this information is in the form of the sum of the individual MOs and cannot be easily dissected into contributions by specific atoms or groups. How can the properties described by the MOs be related to our concept of molecules as a colleetion of atoms or functional groups held together by chemical bonds ... 

D p) Path of A -particle states each state on the path corresponds to a density p and is the minimum energy state for that density. 

Figure 2. A schematic of the free energy density of an aperiodic lattice as a function of the effective Einstein oscillator force constant a (a is also an inverse square of the locahzation length used as input in the density functional of the liquid). Specifically, the curves shown characterize the system near the dynamical transition at Ta, when a secondary, metastable minimum in F a) begins to appear as the temperature is lowered. Taken from Ref. [47] with permission.

In terms of the two-phase system which comprises dispersions of solids in liquids, the minimum energy requirement is met if the total interfacial energy of the system has been minimized. If this requirement has been met, chemically, the fine state of subdivision is the most stable state, and the dispersion will thus avoid changing physically with time, except for the tendency to settle manifest by all dispersions whose phases have different densities. A suspension can be stable and yet undergo sedimentation, if a true equilibrium exists at the solid-liquid interface. If sedimentation were to be cited as evidence of instability, no dispersion would fit the requirements except by accident—e.g., if densities of the phases were identical, or if the dispersed particles were sufficiently small to be buoyed up by Brownian movement. 

As H2 approaches the methyl group of the Hf complex, electron density is donated from the dnrc bond into the ohh antibond while the cthh bond donates into the Hf metal center. Somewhat surprisingly, however, the primary metal acceptor is not an empty metal d orbital but rather a aum antibonding orbital. The reason is revealed by the transition-state structure, as displayed in Fig. 4.68. The approach found in this reaction trajectory (which is not necessarily the minimum-energy... 

The conclusion that the local hardness is given entirely by the variable parts of the kinetic energy is very logical. It is the kinetic energy increase which limits the distribution of electron density in all systems with fixed nuclei. Since the equilibrium state of atoms and molecules is characterized by minimum energy, they will also be marked by maximum kinetic energy because of the virial theorem. This will put them in agreement with the principles of maximum hardness, for which much evidence exists. 

Since Eo, the minimum energy required for E in order to react, is usually quite large compared to the vibrational spacing in the reactant, the spacing between the active levels in A is small and it makes sense to employ the density of states function p(Ev ) for A (i.e. instead of attempting to sum discrete individual states). For... 

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