Density threshold

Diameter Density Threshold Pressure Detonation Velocity ... 

Kugler R, Bouloussa O, Rondelez F (2005) Evidence of a charge-density threshold for optimum efficiency of biocidal cationic surfaces. Microbiology 151 1341-1348... 

Current density thresholds noticed in various disciplines. 

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

For any continuous electronic density function p(r) of a molecule, the set of all points r fulfilling equation (2) must form a set of a finite number of continuous surfaces. For low values of the electron density threshold a, G(a) is usually a single, closed surface, whereas for high values of threshold a, G(a) is usually a collection of several closed surfaces, each surrounding some of the nuclei of the molecule. 

A pictorial analogy between macroscopic clouds of various densities and molecular charge densities can be used here. A density domain DD(a) is analogous to a cloud we could see if our eyes were adjusted to notice only densities equal to or higher than the threshold a. By readjusting our eyes, different "density domains" of clouds of higher or lower density threshold values could be observed. 

A density domain DD(a,K) represents a formal molecular body at an electronic density threshold a and nuclear configuration K. A body DD(a,K) may be a single piece or it may be a collection of several disconnected pieces, the maximum connected components DD (a,K) of DD(a,K) ... 

One of the exceptions, that offers an alternative to the conventional bond diagrams is the density domain approach [4,5] to chemical bonding. This approach is based on the following observation for a given molecule with a specified nuclear configuration K, the infinite family (DD(a,K) of density domains for the range (0, amax] of density thresholds,... 

At some medium density threshold a2, symmetry implies that electronic density clouds appear simultaneously around each of the four hydrogen nuclei one finds that there are five separate density domains. That is, at an intermediate density threshold a2, the set of density domains has five elements,... 

At some low enough density threshold a3, the methane molecule has a single density domain that contains all five nuclei C, H, H, H, and H. At this density threshold a3 the set of density domains has again only a single element,... 

The shapes of these density domains are characteristic to the set of nuclei enclosed by them, to the nuclear geometry, and also to the location of these density domains within the molecule, collectively represented by the configuration variable K, as well as to the actual density threshold a. The sequence of density domains as a function of density threshold a, augmented with the results of a local shape analysis of these density domains [2], provides a detailed description of chemical bonding within the methane molecule. 

Atomic range. [amax, af) amax, is defined above, af is the lowest density threshold where two density domains join. Only individual nuclear neighborhoods appear as disconnected density domains, that is, there is precisely one nucleus within each density domain which appears. 

Strictly atomic range, [amax, ap) amax, is defined above, ap is the lowest density threshold where at least one density domain is no longer convex, as it "reaches out" to join a neighboring density domain. Note that within the strictly atomic range all density domains are convex, and each density domain contains precisely one nucleus. 

Quasi-spherical molecular range, [aq, amjn) both aq and amjn are defined above. For any threshold value within the quasi-spherical range the density domain representing the molecule is convex. If amjn is small enough, then in the strict mathematical sense, for very low density thresholds all molecules have convex density domains. 

The gradual decrease of the electron density threshold value reveals many interesting trends. One such trend is called the "Late - Early Rule" [2]. 

The trend described by the Late - Early rule can be phrased in terms of the expected inclusion relations of those electron density threshold intervals where the density domains of individual nuclei exist as separate entities. A given set of the nuclei of type A is ordered according to the increasing effective electronegativity of the neighbor nuclei BOO involved in the first merger of each of the density domains of nuclei of type A. This ordering is represented by the index k ... 

The interval of density thresholds where a separate density domain exists for nucleus AOO is denoted by (ak, a k). Using these notations, the trend expressed by the "Late - Early Rule" is equivalent to the following, ideal sequence of inclusion relations ... 

One example that has been studied in some detail is the ethanol molecule [2]. Let us choose H as nucleus A. The density domain of the OH proton appears relatively late in the process of gradually decreasing the electron density threshold. This observation can be justified by the high electronegativity of oxygen, resulting in a depletion of the electron density at the nearby proton, that has a chance for the formation of a density domain of its own only at a somewhat lower density threshold. 

One of the main advantages of the density domain approach is the introduction of a natural model for a quantum chemical representation of formal functional groups [1-3]. Consider the simplest case a single connected density domain DD(a,K) and all the nuclei contained within DD(a,K). The boundary MIDCO G(a,K) of the density domain DD(a,K) separates this subset of the nuclei of the molecule from the rest of the nuclei. This fact indicates that the nuclei embedded within DD(a,K), together with a local electronic density cloud surrounding them, represent a sub-entity of the molecule. This sub-entity has an individual identity, since for a range of density threshold values including the value a, the local electron density cloud is separable from the density cloud of the rest of the molecule. 

We may consider some chemical examples. Several alcohols, including ethanol and ally alcohol, have been studied using the density domain shape analysis approach [2,3], and in all these cases a whole range [a, a"] of density threshold values have been found within which the O and H nuclei of the OH group are completely surrounded by MIDCO s, separating these nuclei from all the other nuclei of the molecule. This criterion, the existence of a MIDCO that separates a group of nuclei from all other nuclei of a molecule, is used for the identification and a detailed characterization of chemical functional groups [1-3]. 

With minor modifications, the fuzzy electron density membership function formalism of molecular families can also be applied to a family of functional groups within a molecule. Consider a molecule X and some electron density threshold a within the functional group range of density. Consider the functional groups appearing as separate density domains... 

In most interactions between two reactants, local shape complementarity of functional groups is of importance. A local shape complementarity of molecular electron densities represented by FIDCOs implies complementary curvatures for complementary values of the charge density threshold parameters a. For various curvature domains of a FIDCO, we shall use the notations originally proposed for complete molecues [2], For example, the symbol D2(b),i(a, Fj) stands for the i-th locally convex domain of a FIDCO G(a) of functional group Fj, where local convexity, denoted by subscript 2(b), is interpreted relative to a reference curvature b. For locally saddle type and locally concave domains relative to curvature b, the analogous subscripts 1(b) and 0(b) are used, respectively. 

We define the contact density a0 for a given mutual arrangement of two functional groups Fi and F2 as the threshold value that corresponds to the unique electron density threshold of the FIDCOs of case 2. 

If a contact density threshold ao can be chosen for a given interaction between two functional groups, then the local shape complementarity between G(ao, Fj) and G(a<), F2) is clearly of importance. However, complementarity should also manifest itself within a whole range of density thresholds. One may consider the local shape complementarity of FIDCOs G(ao-a, F[) and G(ao+a, F2) in a density interval containing the contact density threshold ao,... 

Complementarity of the curvature types for truncation is not sufficient for a direct comparison of the two (a.b)-maps, since one must also take into account the required complementarity of density thresholds a and reference curvatures b. This... 

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