Molecular isodensity contour concept

The fundamental principle we shall follow in the local shape analysis of functional groups and local molecular moieties is a strict analogy with the shape analysis of complete molecules. Accordingly, instead of molecular isodensity contour (MIDCO) surfaces, the main tool of analysis will be the fragment isodensity contour (FIDCO) surfaces. Some of the ideas and concepts described in this section are illustrated in Figure 1. 

Two important tools of electron density modeling and shape analysis are the concepts of molecular isodensity contour, MIDCO G(K,a) and the associated density domain DD(K,a) [27], Here the nuclear arrangement (also called the nuclear configuration) is denoted by K, whereas the electron density threshold is denoted by a. Each MIDCO G(K,a) is the collection of all those points r of the three-dimensional space where the electron density p(K,r) of the molecule M of conformation K is equal to the threshold a. The density domain DD(K,a) is the collection of all points r where the electron density p(K,r) is greater than or equal to the threshold a. Using formal notations,... 

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... 

The concept of density domains can be introduced in the context of isodensity contour surfaces. A molecular isodensity contour surface, MIDCO G(K,a)y of nuclear configuration K and density threshold a is defined as the collection of all points r of the 3D space where the electronic density is equal to the threshold value a. In the notation for electron density it is useful to specify the nuclear configuration K of the molecule and in the forthcoming discussion the notation p(K, r) will be used for the electron density of nuclear configuration K. Accordingly, the MIDCO G(K,a) is defined as... 

A related concept is the molecular isodensity contour surface, MIDCO (Molecular IsoDensity Contour) G(K,a), defined for the given nuclear configuration K and density threshold a as... 

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