Curvature domains

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. 

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 ... 

The next step involves a truncation of MIDCO surfaces according to the various curvature domains identified on them. For each (a,b) pair of values, all curvature domains D fb) of a specified type p (usually, the type p = 2) are... 

Investigation of the Great Curvature Domain of the Response Surface Sequential Experimental Planning... 

The example shown above, introduces the necessity for a statistical investigation of the response surface near its great curvature domain. We can establish the proximity of the great curvature domain of the response surface by means of more complementary experiments in the centre of the experimental plan (xj = 0,X2 = 0,...Xij = 0). In these conditions, we can compute y, which, together with Pq (computed by the expression recommended for a factorial experiment... 

Figure 5.11 Representation of the displacement to the great curvature domain. A, according to the greatest slope method B, according to the regular simplex method.

If p denotes the number of negative eigenvalues of the local Hessian matrix H(r), then point r is said to belong to a domain Djj of the contour surface G(a). A local curvature analysis along the surface generates a subdivision into various curvature domains. For the three possible p values of 0, I, and 2, one obtains... 

The relative curvature domains Do(bG)- Dl(bG)> D2(bG). specified in terms of the scaled reference curvature be provide a size-independent shape characterization of the object G(a) for all curvatures. 

The resulting curvature domains Do(bK)> D (bK). and D2(bK) are not invariant with respect to the size of the G(a) objects (this size is dependent on the contour parameter a), nevertheless, the scaling is specific for the size of the nuclear arrangement K, hence these shape domains provide a valid shape comparison of MIDCO s or other molecular surfaces of molecules of different sizes. This approach is simpler than the fully size-invariant approach using the reference curvature be, where a new scaling factor r(G(a)) is required for each new MIDCO G(a). 

For the special case of reference curvature b = 0 (i.e., for the tangent plane T of ordinary convexity), the pattern of the original curvature domains Do(0), D](0), and D2(0) is already size-invariant. 

One of the most useful shape codes is based on shape matrices. As we have seen in Chapter 5, the N-neighbor relation N(D j, D j ) of various curvature domains and, given by Equation (5.8), leads to a shape matrix... 

The assignment of the index i to the curvature domains D j can be used to encode additional information for example, some indication of relative sizes of the shape domains. In one implementation [109], the index i follows the ordering of all the shape domains according to the decreasing size of their surface areas on... 

Considering a finite number of threshold values m, a set of contour surfaces G(m) is studied for each molecule combined with a set of reference curvature values b. Therefore, for each pair (m, b) of parameters, the curvature domains Do(m, b), b) and T>2 m, b) are computed and the truncation of contour surfaces G m) is performed by removing all curvature domains of specified type p (in most applications p = 2) from the contour surface, thus obtaining a truncated surface G(m, p) for each m, b) pair. For most small changes of the parameter values, the truncated surfaces remain topologically equivalent, and only a finite number of equivalence classes is obtained for the entire range of m and b values. 

STEP 1. For each contour value a within a range of values for 3D property P(r), the IPCOs G(a) are partitioned into local curvature domains relative to each value b of a range of reference curvatures. 

STEP 2. For each pair of values of IPCO threshold parameter a and reference curvature parameter b, all curvature domains D (b) of a specified type )i are formally removedfrom the corresponding IPCO G(a). 

The shape group method is a technique of shape characterization based on local geometrical curvature features analyzed topologically in terms of the patterns and topological invariants that various curvature domains of the electron density generate. Each MIDCO G(K, a) of the electron density is partitioned into domains based on local curvature thresholds b. For the entire electron density there are infinitely many... 

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