Electron density threshold

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 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) ... 

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 ... 

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. 

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... 

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. 

Consider a macromolecule M of some nuclear configuration K. For an electron density threshold a, the family of functional groups... 

F(a) and G(a), are now considered as a single family, denoted by DD(a). A DD(a) set can be regarded as the molecular body at an electronic density threshold a. Indeed, if we imagine that we can actually see molecules, and the sensitivity of our eyes is adjustable to notice only electronic densities that are equal to or greater than the value a, we would then see these DD(a) sets as the molecular bodies. 

For example, consider a given electron density threshold and reference curvature parameter pair (a,b), and assume that the associated truncated contour surface falls into seven pieces. Furthermore, assume that a shape group analysis gives the sequence... 

For the given mutual arrangement of the two molecules M] and M2, the contact density ag corresponds to the unique electron density threshold value of the MIDCO s of case 2. 

The corresponding overlap volume is the sum of the elementary volumes of all space pixels in which the electron density of both molecules is above a given threshold. The case is illustrated in Table 1.5 for the succinic anhydride crystal. As expected, the overlap integral correlates with the distance between molecular centers of mass. Coherently with the results shown in Table 1.4, the overlap volumes are non-zero only if the electron density threshold is less than 0.02 e A . In Section 12.2 it will be shown that the overlap integral correlates with the repulsion energy between neighbor molecules, and that the outer electrons play a crucial role in defining the details of the directional intermolecular interaction in crystals. 

Procrystal electron density threshold for a migration pathway Bond valence site energy threshold for a migration pathway Absolute bond valence sum mismatch Bond valence sit energy of ion M Density functional theory... 

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