Saddle type domain

Shape complementarity of functional groups involves matches between locally concave and locally convex domains, and also matches between properly aligned saddle-type domains, that is, between curvature domain pairs of the following combinations ... 

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]. 

On the local level, shape complementarity implies matches between locally concave and locally convex domains, as well as matches between properly placed saddle-type domains, where a directional convex-concave match is important. Replacing the simple D (K,a) notation, the more elaborate notation D b),i(K,a) is used sometimes when studying the complementarity of local shape domains, where the notation includes the relative convexity specification fi(b). This quantity takes values... 

As an illustration, here we shall outline only one of the simplest of the nonvisual, topological methods for shape characterization, applicable for smooth (differentiable) molecular surfaces. This method is based on the classification of the points of a molecular surface into convex, concave, and saddle-type domains using local curvature properties, and on the representation of the mutual arrangements of these domains by a matrix or by an equivalent graph. One of the advantages of the method is the fact that the generation... 

For two molecules identical shape matrices indicate a rather strong similarity of their shapes. If the two matrices are not identical, but if they can be converted into identical forms by simultaneous row and column permutations, then the essential shape patterns of the distribution of convex, concave, and saddle-type domains agree, but the sizes of these domains are sufficiently different, so they do not follow the same order. This latter case implies a somewhat weaker similarity between the two molecules. The complexity (e.g., the number of inversions) of the actual permutation required serves as a further qualification of their similarity. 

The most natural, and informative choice for reference curvature is b = 0. However, one may be interested in much finer details of shape than the simplest classification of surface domains into concave, convex, and saddle-type domains. A more detailed description is possible by utilizing the fact that for a different reference curvature value b the shape matrix may be different, although it is invariant within certain intervals of b values. By considering an entire range of curvature parameter b, and by listing the finite number of distinct shape matrices and the curvature values b, where a change in the shape matrix occurs, a detailed, numerical shape characterization of the molecular surface G a) can be given. 

In most interactions between two reactants, local shape complementarity of functional groups is of importance. A local shape complementarity of molecular electron densities represented by FIDCOs implies complementary curvatures for complementary values of the charge density threshold parameters a. For various curvature domains of a FIDCO, we shall use the notations originally proposed for complete molecues [2], For example, the symbol D2(b),i(a, Fj) stands for the i-th locally convex domain of a FIDCO G(a) of functional group Fj, where local convexity, denoted by subscript 2(b), is interpreted relative to a reference curvature b. For locally saddle type and locally concave domains relative to curvature b, the analogous subscripts 1(b) and 0(b) are used, respectively. 

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. 

Figure 5.1 The shape domains of local convexity of a MIDCO surface G(a) are shown. A geometrical interpretation of the classification of points r of G(a) into locally concave Dp. locally saddle-type D, and locally convex D2 domains is given when comparing local neighborhoods of the surface to a tangent plane T. Each point r of G(a) is classified into domains Dp. D, and D2 depending on whether at point r a local neighborhood of point r on the tangent plane (r not included) falls within the interior of the surface G(a), or it cuts into the surface G(a) within any small neighborhood of point r, or it falls on the outside of G(a).

The more general case of b >t0 corresponds to a generalization of the concept of convexity [156,199]. This concept of relative local convexity has a u.seful geometrical interpretation. Eor a fixed value of parameter b, relative local convexity classifies the points r of G(a) into domains D x(b), depending whether at point r the surface G(a) is curved more in all directions, more in. some and less in some other directions, or less in all directions, than a test sphere T of radius 1/b. The corresponding three types of domains are denoted by Do(b), D (b), and D2(b), where the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b. 

The local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

Figure M-2. A subdivision of a molecular contour surface G(m), based on local curvature properties. The contour surface is subdivided into locally concave (Do, ), saddle-type (Di, ), and locally convex (D2, ) domains with respect to the local tangent plane in each point r. The second index k refers to an ordering of these domains according to decreasing surface area.

In the most common applications of shape groups, the local shape properties are specified in terms of shape domains for example, in terms of the locally convex, concave, or saddle-type regions of MIDCOs, relative to some curvature reference parameter b. 

The Saddle Point Features of the 2-D Gratings For an ideal 2-D sinewave the saddle point features should appear to have 4-fold symmetry when viewed in LEEM images. From the sketch of figure 13 it can be seen that the hypocycloid shaped terrace at the saddle point has the same type of monoatomic step on all four sides due to the difference in the free energies of the two steps, Sa and Sb, on Si(OOl) there should be a strong preference for Sa steps and hence each maximum would prefer to be flanked by two white domains and two "black ones as is the case in figure 10. (A similar conclusion follows if the the saddle point terrace is surrounded by two Sa steps and two double steps of Db type[31]). 

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