Truncation surface

The atomic structure of a surface is usually not a simple tennination of the bulk structure. A classification exists based on the relation of surface to bulk stnicture. A bulk truncated surface has a structure identical to that of the bulk. A relaxed surface has the synnnetry of the bulk stnicture but different interatomic spacings. With respect to the first and second layers, lateral relaxation refers to shifts in layer registry and vertical relaxation refers to shifts in layer spacings. A reconstructed surface has a synnnetry different from that of the bulk synnnetry. The methods of stnictural analysis will be delineated below. 

Often (adsorption, reconstruction) the periodicity at the surface is larger than expected for a bulk-truncated surface of the given crystal this leads to additional (superstructure) spots in the LEED pattern for which fractional indices are used. The lattice vectors bi and b2 of such superstructures can be expressed as multiples of the (1 X 1) lattice vectors ai and Zx. ... 

A special notation is used to describe surface reconstructions and surface overlayers and is described in books on surface crystallography (Clarke, 1985). The lattice vectors a and b of an overlayer are described in terms of the substrate lattice vectors a and b. If the lengths la I = mlal and Ib l = nibl, the overlayer is described as mXn. Thus, a commensurate layer in register with the underlying atoms is described as 1 X 1. The notation gives the dimension of the two-dimensional unit cell in terms of the dimensions of an ideally truncated surface unit cell. 

Since for the whole range of FIDCO surfaces of a given functional group there are only a finite number of topologically different truncated surfaces, consequently, there are only a finite number of shape groups characterizing the shape of a functional... 

Brownian dynamics by generating trajectories starting from r = b and terminating them when the molecule collides with the reaction surface or the truncation surface at r = q is the fraction of molecules that collide with the... 

The number of holes of a truncated surface G is an easily visualizable topological feature. If the number of holes of G is one or more, then it is equal to bi + 1. 

As a result of the removal of domain E, only the two-dimensional Betti number has changed, from 1 to 0, indicating that the new, truncated surface is no longer a closed surface, it has at least one hole. In fact, the value of b] = 0, in combination with b2 = 0, tells us that there is one hole, which turns out to be in the place of the missing cell E. 

It should be emphasized that the above shape group methods combine the advantages of geometry and topology. The truncation of the MIDCO s is defined in terms of a geometrical classification of points of the surfaces, and the truncated surfaces are characterized topologically by the shape groups. 

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. 

In the next step, the shape groups of the molecular surface, that is, the zero-, one-, and two-dimensional algebraic homology groups of the truncated surfaces, are computed. The zero-, one-, and two-dimensional Betti numbers b (m, b), bj (m, b), andb (m, b) are the ranks of these zero-, one-, and two-dimensional homology groups, that is, the shape groups. They are a list of... 

The curvature-based shape analysis of each MIDCO surface G(K,a) can be repeated for a whole range of reference curvature values b, providing a detailed shape characterization of G(K,a). It is important to point out that for the complete range of chemically relevant reference curvature values b, there exist only a finite number of topologically different truncated MIDCOs G(K,a,p) obtained from G(K,a). When these truncated surfaces are characterized by their topological invariants, then a numerical shape characterization is obtained. 

In this step, a truncated surface is generated for each IPCO G(a) for each pair of parameter values a and b. Although there are infinitely many pairs of parameter values a and b, consequently, there are infinitely many such truncated surfaces, for most small changes of parameters a and b the truncated surfaces remain topologically equivalent. Consequently, there are only a finite number of classes of topologically different truncated surfaces for the entire range of parameter values a and b. 

Step 3. The shape groups of the entire 3D molecular property are calculated by determining the algebraic homology groups for each topological equivalence class of the truncated surfaces. 

By definition, the shape groups of the property P(r) are the algebraic homology groups of the family of topological equivalence classes of the truncated surfaces. The family of all of these equivalence classes involves all property thresholds a as well as all reference curvatures b. The shape groups provide a detailed shape description of the entire 3D property P(r). 

An early approach to the local shape problem was based on the Shape Group Methods as applied to truncated surfaces, where the truncation was selected using essentially arbitrary criteria for deciding which part of the molecule belongs to a local region. The purpose of the study was to analyze the influence of various substituents on the local shape of the rest of the molecule. 

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 ... 

A number of topological invariants can be used to describe the truncated surfaces Gci<, The Betti numbers Bp, which are the ranks or number... 

The shape of isoproperty surfaces (Eq. [25]) can be analyzed in detail by using a 2D map of shape descriptors. The key notion is as follows the truncated surface G a, K) can be specified by two parameters the density value a, and a reference curvature value b, defining the criterion for truncation. 

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