Density distribution definition

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

As different sources are considered, the statistical properties of the emitted field changes. A random variable x is usually characterized by its probability density distribution function, P x). This function allows for the definition of the various statistical moments such as the average. 

The definition of the radius of an ion in a crystal as the distance along the bond to the point of minimum electron density is identical with the definition of the radius of an atom in a crystal or molecule that we discuss in the analysis of electron density distributions in Chapter 6. The radius defined in this way does not depend on any assumption about whether the bond is ionic or covalent and is therefore applicable to any atom in a molecule or crystal independently of the covalent or ionic nature of the bond, but it is not constant from one molecule or crystal to another. The almost perfectly circular form of the contours in Figure... 

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

Ei,mono and thus reproduce the exact electric field outside the atom as determined by the distribution of electrons within the atom. To obtain the correct field inside the atom requires the addition of the term fjjocai which can be calculated if the electron density distribution is known. For present purposes it is sufficient to note that f,local is, by definition, zero outside the atom and therefore can have no long-range effect on the structure. All the long-range influences are therefore carried by fj mono and, to a much lesser extent, , niuit-... 

When the distance between each A reactant is very large compared with that between each pair of B reactants, at a point about midway between a pair of A reactants, the concentration of B reactants is unlikely to be significantly affected by the presence of the A reactants. Smoluchowski suggested that such B reactants are effectively an infinite distance from the A reactants under discussion. By effectively an infinite distance is meant perhaps 1000 times the molecular diameter or encounter distance R. In this region, the concentration of B reactants at any time during the reaction is very close to the initial concentration, i.e. [B](1000iZ) [B]0 for all time (t > 0). From the definition of the density distribution, eqn. (2), this boundary condition as r - °° is... 

The second equation, with the third (constant) term discarded, is the standard result [compare Eqs. (5), (6) and (Al)]. However, the first expression appears more natural, and is retained, within the combinatorial derivation it embodies the intuitively reasonable prescription that entropy is best measured relative to the parent distribution (°)( t). This was also an essential ingredient of the projection method, as described in Section n.A, where the prior in the entropy expression was likewise identified with the parent. Retaining the parent as prior also avoids subtleties with the definition of the integrals in Eq. (36) in the case where the phases contain monodisperse components, corresponding to 5-peaks in the density distributions [8]. 

At the onset of phase coexistence, one of the coexisting phases is by definition the parent p (cr) the lever rule does not yet play any role because the daughters p (a) (a = 1. ..p) occupy an infinitesimal fraction of the total volume. It then follows from Eq. (48) that all daughters lie within the family (39) of density distributions. As shown in Eq. (49), the condition for equality of chemical potentials p(a) between any two of the coexisting phases (parent and daughters) then becomes... 

As pointed out in Section III.A, the definition of the moment free energy depends on a prior R(p) and represents the properties of systems with density distributions p(a) in the corresponding maximum entropy family (7). Instead of identifying R(o) = p (cr), we now allow a general prior R(ff). Concep-... 

Further studies are required to unravel this mystery of how the methoxy substitutions and the a, p-unsaturated p-diketone moiety actually influence conformational changes, lipophillicity, electron density distribution, and redox properties of curcuminoids. Correlating these physicochemical properties with the unique pleiotropic effects of curcuminoids is a rewarding exercise. Such studies would definitely provide proper reasoning in understanding these markedly different antioxidant, antitumor, and anti-inflammatory activities of natural curcuminoids from turmeric. 

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

Several approximations that allow simple estimates of bond parameters are presented as a demonstration that predictions based on quantum potentials are of correct order, and not as an alternative to well-established methods of quantum chemistry. In the same spirit it is demonstrated that the fundamental thermodynamic definition of chemical equilibrium can be derived directly from known quantum potentials. The main advantage of the quantum potential route is that it offers a logical scheme in terms of which to understand the physics of chemical binding. It is only with respect to electron-density distributions in bonds that its predictions deviate from conventional interpretations in a way that can be tested experimentally. 

It was in connection with the results obtained for electron density distribution in the aza[3.1.1]propellane 86 that the question when is a propellane a propellane could be posed. The length of the conjoining bond (the basis for the definition of propellanes ... 

Whereas the molecular center of mass is of importance in both dynamics and spectroscopy, a formal center of the electronic density distribution has direct significance in shape characterization. A suitable definition of this latter center may differ from the molecular center of mass. The fuzzy set model of electron densities is represented by the... 

Figure 7-1 Definition of the adjoint gravity field (jy. The field (/] is generated by the sources located in F. The density distribution J) Q within F is a mirror image of the density distribution /> (,) in [ /7 (j) -- piC)-...

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y)... 

In Fig. 3.1 two possible temporal density distributions (B2 > Bi) are represented, whereby the form of the curve for 02 permits different definitions of a relative deviation from homogeneity [205] ... 

On the basis of these definitions one can describe chemical bonding in molecules containing noble gas elements with the aid of the properties of p(r). One starts by searching for the bond paths 2uid their associated bond critical points Tg in the molecular electron density distribution. If all bond paths are found, then the properties of p(r) along the bond paths will be used to characterize the chemical bonds. For example, the value of can be used to determine a bond order, the anisotropy of Pp can be related to the n character of a bond, the position of the bond critical point is a measure of the bond polarity and the curvature of the bond path reveals the bent-bond character of a bond [17, 19]. 

The theory of AIM allows one to study the concept of chemical bond and the bond strength in terms of electron density distribution function [6, 193]. It exploits the topological features of electron density and thereby a definition of chemical bonding through bond path and bond critical point (BCP). A BCP (it is a point at which gradient vector vanishes, Vp(r) = 0) is found between the... 

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