Molecular isodensity contour surface

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. ... 

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 formal definition of a molecular isodensity contour surface, MIDCO G(K, a) of nuclear configuration K and density threshold a is given as the collection of points of the 3D space where the electronic density p(K,r) is equal to the... 

The shape of molecules is the shape of their three-dimensional 3D electron density clouds. In Figure 1, nine molecular isodensity contour surfaces (MIDCOs) of the electron density cloud of the water molecule are shown, at density levels ranging from 0.001 au to 0.45 au, where the atomic unit of charge density is used, 1 au = e bohr e is the electronic charge). Molecular shape has a fundamental effect on both physical and chemical properties of molecules. The development and application of computational approaches to molecular shape analysis are among the main roles of computational chemistry. 

Figure 1 Molecular isodensity contour surfaces, MIDCOs, of water shown at nine density levels ranging from 0.001 au to 0.45 au (electron density in atomic units, 1 au = e bohr where e is the electronic charge)...

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. 

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.

For example, if the shape domains are defined in terms of local convexity, and if we select the locally convex domains, then the shape groups of G(a) are the homology groups of the truncated isodensity contour surface G(a,2), obtained from the molecular contour surface G(a) by eliminating all domains of index p = 2. This family of shape groups, obtained by cutting out all locally convex domains of G(a), has been studied in most detail for several molecules [192,262,263,342]. 

A general method, proposed by Politzer and coworkers, to estimate physico-chemical properties depending on noncovalent interactions [Brinck et ai, 1993 Murray et al., 1993 Politzer etal., 1993 Murray et al., 1994]. This is based on molecular surface area in conjunction with some statistically-based quantities related to the - molecular electrostatic potential (MEP) at the - molecular surface. The electron isodensity contour surface [0.001 a.u. contour of Q(r)j is taken as the molecular surface model. 

An alternative approach to domain subdivisions of the molecular surface is based on local curvature properties. It is applicable only to differentiable molecular surfaces such as contour surfaces, e.g. the electron isodensity contour surface G(m), m being the threshold value defining the contour surface. 

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... 

IV. 1 Molecular shape characterization of an isodensity contour surface of water using the program GSHAPE. 

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) ... 

The union surface inherits many of the features of the molecular surfaces used for its calculation. For example, if isodensity surfaces are used then the union surface is likely to be closely related to some isodensity contour surface of the cavity region. However, some of the properties of union surfaces derive from the superposition procedure, and these features may be quite prominent if only a few drug molecules are used. For example, the superposition may result in foldings and the occurrence of break lines on the union surface that are, clearly, not inherent properties of any of the isodensity contours of the cavity region. In particular, even if the union surface is generated by the superposition of differentiable contour surfaces, such as isodensity contours, the resulting union surface is not necessarily differentiable. 

A typical discrete set of chemical importance is a nuclear arrangement in the clamped nucleus version of the Bom-Opp-enheimer approximation. A single molecular isodensity contour (MIDCO) surface is a crisp continuum set. A simple generalization of the CSM approach from finite, discrete point sets to continua is provided by the crisp average of sets. 

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). 

Shape codes [43,109,196,351,408]. The simplest topological shape codes derived from the shape group approach are the (a,b) parameter maps, where a is the isodensity contour value and b is a reference curvature against which the molecular contour surface is compared. Alternative shape codes and local shape codes are derived from shape matrices and the Density Domain Approach to functional groups [262], as well as from Shape Globe Invariance Maps (SGIM). 

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