Local canonical curvatures

The local canonical curvatures can be compared to a reference curvature parameter b [156,199]. For each point r of the molecular surface G(a) a number X = x(r,b) is defined as the number of local canonical curvatures [the number of eigenvalues of the local Hessian matrix H(r) that are less than this reference value b. The special case of b=0 allows one to relate this cla.ssification of points to the concept of ordinary convexity. If b=0, then p is the number of negative eigenvalues, also called the index of critical point r. As mentioned previously, in this special case the values 0, 1, or 2 for p(r,0) indicate that at the point r the molecular surface G(a) is locally concave, saddle-type, or convex, respectively [199]. 

By generalizing the idea of local convexity for any reference curvature value b [199], the number p(r,b) is the tool used for a classification of points r of the contour G(a) into various domains. For any fixed b, each point r of the contour surface G(a) belongs to one of three disjoint subsets of G(a), denoted by Aq, A, or A2, depending on whether at point r none, one, or both, respeetively, of the local canonical curvatures h and h2 are smaller than the reference value b [156]. The union of the three sets Aq, A, and A2 generates the entire contour surface, that is. 

In the above discussion we have assumed that the molecular contour surface G(a) is twice differentiable. This condition is required for gradients and local Hessian matrices of the local elevation function at all points along the surface, and for the local canonical curvatures of G(a) at each point r of G(a), needed for their classification into shape domains. 

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. 

For instance, we can truncate all points in G(a, K) whose local canonical curvatures are both smaller than b. The case 6 = 0 corresponds to the removal of convex regions. For 6 0, domains on the surface are classified according to a generalized notion of convexity.Por each pair of values a and b, we can compute a set of topological invariants. Note that these descriptors will vary in discrete fashion only some particular values of a, b) will result in a change of values for the invariants. 

Under this assumption, the bending energy Eg can be represented in terms of the membrane s curvature. For this reason. Eg is also referred to as the curvature elastic energy. The curvature of smooth surfaces is characterized by two functions that depend on the local canonical curvatures, h t) and h lr), in a surface element dS centered at r. These functions are the mean curvature, H = ( 1 + hi) , and the Gaussian curvature, K = hih2- In general, H and K change with the point r. 

A small neighborhood of a point r on a closed surface embedded in is locally a 2D domain. Its basic geometric properties are its two canonical curvatures, h r) and These curvatures are the eigenvalues of the local Hes-... 

Except for some in special cases, the canonical curvatures are finite properties that can be computed everywhere on the surface. Suppose we now group all the points on G a,K) that satisfy a curvature criterion, say those for which the two curvatures /7j(r) and h2 i) are negative. Such a domain on the surface can be indicated as Dc(a, K), where C indicates the criterion followed for classification. With the criterion of two negative curvatures, Dda, K) [e.g., Diia, K)] corresponds to the regions on the surface that are locally convex. [Note that Dcia, K) can be empty or composed by several disjoint pieces.] Finally, if we now remove (i.e., cut away) this region, we derive a truncated surface from the original one ... 

If the critical point K(X,i) has a single negative canonical curvature, that is, if the local Hessian matrix has precisely one negative eigenvalue, X=l, then K(, i) is called a simple saddle point and the corresponding catchment region C(X,i) represents a transition structure (a "transition state", that is not a state at all). 

The result of the preceding subsection is not very convenient for practical use, because it requires the solution of the differential equation Equation (8.21) that contains the Hessian of the potential in terms of the transverse local coordinates together with the corresponding curvatures. One can obtain a more useful canonically invariant form that does not rely on any local coordinates. To do this, we first rewrite the first integral in Equation (8.27) in terms of the time x along the instanton trajectory. Using the transformation... 

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