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 is related to the concept of MIDCO in a simple way. A maximum connected part of an isodensity contour surface G(a) and the corresponding part of the level set F(a) enclosed by it is called a density domain, DDj(a) [109]. Below we shall give a more formal definition and describe the most essential properties of density domains. 

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]. 

The FSGH method (Fused Sphere Guided Homotopy method) [43]. This method has been designed for the construction of approximate, density scalable ("inflatable") isodensity contour surfaces and their dot representations (i.e., for continuous transformations between different isodensity surfaces of a given molecule). 

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. 

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... 

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) ... 

Figure 2 A cross section of an isodensity contour surface G(a) of the molecule shown in Figure 1.

In Figure 3 two approximate isodensity contour surfaces, and in Figure 4 various cross sections of approximate contours of the butadiene molecule... 

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. 

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)...

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