Absolute shape domains

For the characterization of the shapes of molecular contour surfaces, such as MIDCO s and MEPCO s, one may subdivide the surface into domains fulfilling some local shape criteria. One can distinguish two types of criteria, relative, and absolute, leading to a relative shape domain or an absolute shape domain subdivision of the molecular contour surface. 

An absolute shape characterization is obtained if a molecular contour surface is compared to some standard surface, such as a plane, or a sphere, or an ellipsoid, or any other clo.sed surface selected as standard. For example, if the contour surface is compared to a plane, then the plane can be moved along the contour as a tangent plane, and the local curvature properties of the molecular surface can be compared to the plane. This leads to a subdivision of the molecular contour surface into locally convex, locally concave, and locally saddle-type shape domains. These shape domains are absolute in the above sense, since they are compared to a selected standard, to the plane. A similar technique can be applied when using a different standard. By a topological analysis and characterization of these absolute shape domains, an absolute shape characterization of the molecular surface is obtained. 

Several topological methods of shape analysis of molecular contour surfaces have been designed to take advantage of such relative and absolute shape domain subdivisions of the contours, according to. some physical or geometrical conditions [155-158,199]. 

As an example of absolute shape criteria, the local curvature properties of a MIDCO can be used for defining absolute shape domains on it [156], and for a subsequent global shape characterization. In Figure 5.1 a MIDCO G(a) is shown as an illustration of some of the concepts discussed. The simplest method [155] is based on comparisons to a reference of a tangent plane what leads to the identification of locally convex, concave, and saddle-type domains, as mentioned previously, although much finer characterizations are also possible [156,199]. 

A more general family of methods for absolute shape domain subdivision of molecular surfaces with reference to regular standard objects, such as plane, spheres, and ellipsoids, can be described within the common framework of generalized convexity [199]. These techniques are applicable for smooth (differentiable) molecular surfaces. 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

Fig. 5.3.6 Absolute values of the frequency responses to 10 ms long 90° pulses with rectangular (r) and Gaussian (G) shape in the time domain. The response to the Gaussian pulse is negligible beyond an offset of 300 Hz, while that to a rectangular pulse extends to about 1000 Hz. Adapted from [Baul] with permission from Wiley-VCH.

Figure 9.12. Absolute value frequency domain excitation profiles for (a) a rectangular pulse and (b) a Gaussian shaped pulse.

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