Energy Density Critical Points

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. 

The comparison of the properties at the critical points in the P-NTO electron density with those in the atomic procrystals (Table 6) reveals the contraction of the electron density towards the shared interaction lines during crystal/molecule formation. This contraction is accompanied by a significant enhancement of the local kinetic, potential and electronic energies (Table 7). 

the analysis of energy density distributions should prove to be a useful tool in the study of the fundamental properties of energetic materials. 

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The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

Figure Al.3.19. Simplest possible critical point structure in the joint density of states for a given energy band.

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

Temperature-Controlled Residuiun Oil Supercritical Extraction (ROSE) The Kerr-McCee ROSE process has been used worldwide for over two decades to remove asphaltenes from oil. The extraction step uses a hquid solvent that is recovered at supercritical conditions to save energy as shown in Fig. 20-21. The residuum is contacted with butane or pentane to precipitate the heavy asphaltene fraction. The extract is then passed through a series of heaters, where it goes from the liquid state to a lower-density SCF state. Because the entire process is carried out at conditions near the critical point, a relatively small temperature change is required to produce a fairly large density change. After the light oils have been removed, the solvent is cooled back to the liquid state and recycled. 

Gibbs ensemble. Good for obtaining a few points for subcritical phase coexistence between phases of moderate densities does not provide free energies directly. Primarily used to study fluid (disordered) phases. Is a standalone approach, and requires modest programming and computational effort to set up and equilibrate the multiple simulation boxes. Provides accurate coexistence points at intermediate temperatures below the critical point but with sufficient thermal mobility to equilibrate. 

The theory of polymer adsorption is complicated for most situations, because in general the free energy of adsorption is determined by contributions from each layer i where the segment density is different from that in the bulk solution. However, at the critical point the situation is much simpler since the segment density profile is essentially flat. Only the layer immedia-... 

It can be concluded from the ESP and electron density maps for LiF (as characteristic) that both set of critical points do not coincide (Fig.7). A nuclear potential gives also the picture of CPs differing Ifom one in electron density. These observations reflect a well known theoretical statement that the electron density (and energy) of a many-electron system is not determined fully by the iimer-crystalline electrostatic field. 

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