Isodensity contours molecular, bodies

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 ... 

FIGURE 1 The fuzzy body of the electron density of a bovine insulin molecule is represented by three molecular isodensity contour surfaces (MlDCOs), for the density thresholds of 0.1, 0.01, and 0.001 a.u. (atomic unit), respectively, as computed using the MEDLA method. Bovine insulin was among the proteins selected for the first ab initio quality electron density computations for macromolecules. ... 

Figure 1.2 The three-dimensional, fuzzy "body" of the charge density distribution of allyl alcohol can be represented by a series of "nested" molecular isodensity contours (MIDCO s). Along each MIDCO the electronic density is a constant value. Three such MIDCO s are shown for the constant electron density values of 0.2, 0.1, and 0.01 (in atomic units), respectively. A contour surface of lower density encloses surfaces of higher density. These MIDCO s are analogous to a series of Russian wooden dolls, each larger doll enclosing a smaller one. These ab initio MIDCO s have been calculated for the minimum energy conformation of allyl alcohol using a 6-31C basis set.

Level sets F(a) [as well as the closely related density domains DD(a), as we shall see in the next section] provide a representation of formal molecular bodies. A similar definition gives a useful concept of a formal molecular surface the concept molecular isodensity contour surface (MIDCO). For any formal nuclear configuration K, it is possible to define a surface by choosing a small value a for the electronic density, and by selecting all those points r in the 3D space where the density p(r) happens to be equal to this value a, that is, where equation (2.3) is fulfilled. For an appropriate small value a, this contour surface may be regarded as the surface of the essential part of the molecule and, in short, it may be referred to as the molecular surface. These surfaces, the molecular isodensity contour surfaces, or MIDCO s, are denoted by G(a) and are defined as... 

For a continuous function, such as the electronic density p(r), all points r fulfilling equation (2.3) do form a continuous surface. Consequently, the terms contour surface and isodensity surface are appropriate for G(a). For the study of the 3D shape properties of molecular bodies, represented by level sets F(a) of electronic charge densities, it is sufficient to study the shape of their boundaries these boundaries are the MIDCO s G(a). 

According to a recent proposal [4,8,9], chemical bonding within formal molecular bodies can be described by molecular isodensity contour (MIDCO) surfaces and by density domains (DD) that are the formal bodies enclosed by... 

The shape group method (SGM), reviewed in ref. [2], has been proposed for the analysis of three-dimensional shape properties of formal molecular bodies. For example, by choosing the electronic charge isodensity contours G(a) (of various density values a) as the physical property P for shape representation, and by taking the family of Betti numbers b as the topological tool for shape description [2], the similarity of the geometrical shapes of two molecules, A and B, is transformed into an equivalence of their topological shape, expressed as... 

The 3D shape properties of molecular bodies, represented by level sets F a) of electronic charge densities, can be described by isodensity contour surfaces, denoted by G(a) ... 

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