Graph of reaction distances

For a fixed family 3 (A) of isomeric synthons we construct the so-called graph of reaction distances [18, 21, 16, 25] denoted by RD(A). The vertex set of this graph is formally identical with the family A) without forbidden synthons, its two distinct vertices v and v, assigned to the synthons S(d) and S (/l), are connected by an edge [u, v ] if such an elementary transformation i = a, p exists so that the synthon S(/l) is transformed into the synthon i.e. 

Fig. 7. An example of subfamily J (A), where A = landO < s 3, it contains 14 synthons indexed by 1 to 14. The corresponding graph of reaction distances 9R0(A) is composed of 14 vertices (synthons)...

The notion of a stable neighborhood of a synthon will be illustrated by using the subfamily J fA) displayed in Fig. 7, the corresponding graph of reaction distances is given in Fig. 8. Now we will assume that the synthons S3(A), S9(A), and Si 3(A) are the only stable synthons (they are denoted by encircled vertices), see... 

Fig. 11. Modified graph of reaction distances RD(A) from Fig. 7, now the synthons indexed by 3, 9, and 13 are declared as the only stable ones (encircled vertices)...

In a similar way as was done in Sect. 2.3, we construct for a fixed family 3 pq of isomeric graphs the so-called graph of reaction distances. denoted Qpq Its vertex set is again identified with the family 3,... 

This means that in the graph > the vertex vj (assigned to Gj) is adjacent to the vertices V2i V3, vg, and vg. Applying a similar procedure for all graphs from the family 3 3 we arrive at the following graph of reaction distances... 

When the number of types of elementary transformations is extended, other edges corresponding to the application of these new elementary transformations will be added to the graph of reaction distances... 

The graph of reaction distances constructed over the family 3Fp can be substantially reduced by deleting all vertices that are represented by forbidden S-graphs, i.e. the resulting subgraph, denoted 5p is induced by the union of subfamilies 3Fp u this subgraph of will be called reduced graph of reaction distances. 

Definition 3.12. The reduced reaction distance D(0, 02) between two S-graphs from 3fp is the graph distance determined on the reduced graph of reaction distances. 

The graph of reaction distances (cf. Sect. 3.4) will be constructed for all synthons taken from the same family FIS(A), and it will be denoted by number of vertices need not be specified as it is equal to the cardinality of the atomic set). In the present approach, two distinct vertices [corresponding to synthons from FIS(A)] are connected by an edge iff there exists an elementary operator I-a, B, y, S that transforms one synthon into another. In Scheme 5.1 we show a small part of the graph constructed on the atomic set As(C,C,N>. 

An estimate of the numbers of vertices in several atomic sets has been made in our recent communication [5]. The smallest reaction-distance graphs exist for atomic sets composed of one element, i.e. card(A) l. These graphs were studied for 17 elements [10] (H, Li, Be, B, C, N, 0, F, Na, Mg, Al, Si, P, S, Cl, Br, I). We found a general scheme [8,10,13] for an arbitrary chemical element, which, when filled out, provides the graph of reaction distances for that element in the framework of octet chemistry. Scheme 5.2 illustrates such a graph for A= C). 

Since the graph of reaction distances is connected [18], the reaction distance for synthons from the family FISreaction distance RD, may be formally treated as a metric space. The evaluation of the reaction distance between two synthons from the same family... 

The above outlined method can serve as an almost exact approach for the evaluation of reaction distance between two isomeric synthons S(A) and S (A). Its almost exactness follows from the fact that there can be no previously prescribed mapping of vertices and than the reaction graph is not unique. There can be constructed corresponding reaction graphs for every mapping and the reaction distance should be obtained as a minimum of minimal coverings of those reaction graphs. 

Based on the graph GECVs we define the notion of reaction distance, an important heuristic in our approach to CAOS. 

It should be mentioned that the notion of reaction distance, as defined for valence states of atoms, can be used also for synthons. It means that RD(S(A), S (A)) is defined as the length of the shortest path between S(A) and S (A) in the graph Gfis[Pg.160]

All these properties immediately follow from the definition of reaction distance as a graph distance in the graph 4 q ... 

As a by-product of this bilateral approach for the evaluation of reaction distances between isomeric graphs and G2> is the possibility to construct from the sets Xi, f and 11], ... 

In this chapter we have described for molecular graphs the monolateral and bilateral approaches for the evaluation of reaction distances and the construction of precursors and successors. Both these techniques can be simply modified for S-graphs and/or synthons, and moreover, the concept of reduced reaction distance introduced here may be easily applied. 

The notion of reaction distance can be similarly interpreted in the framework of graph-theory model of synthons. Let Gjj be the SR-graph of isomerization S(A) — S (A) it can be written as follows [15,16]... 

The reaction distance [16,18,21,25] between two isomeric synthons Si (A) and S2(A) will be used as a proper tool for the construction of reaction graphs [29]. The reaction graph obtained corresponds to the minimal number of the so-called elementary chemical transformations, the number of which determines the reaction distance between the synthons SjfA) and S2(A). 

Sn(A). Applying the similar procedure for all synthons from the subfamily of A) we arrive at 9 XD(A) given in Fig. 7. From this graph one can simply evaluate the reaction distance for an arbitrary pair of synthons, e.g. S4(A)) = 6... 

Reaction distance of two valence states, v and v, is the length of the shortest path between v and v in the graph GECVS. In other words, RD is the minimal number of ECVS which are needed for the conversion v - v or back. 

Figure B3.4.1. The potential surface for the collinear D + H2 DH + H reaction (this potential is the same as for H + H2 H2 + H, but to make the products and reactants identification clearer the isotopically substituted reaction is used). The D + H2 reactant arrangement and the DH + H product arrangement are denoted. The coordinates are r, the H2 distance, and R, the distance between the D and the H2 centre of mass. Distances are measured in angstroms the potential contours shown are 4.7 eV,-4.55 eV,.. ., -3.8 eV. (The potential energy is zero when the particles are far from each other. Only the first few contours are shown.) For reference, the zero-point energy for H2 is -4.47 eV, i.e. 0.27 eV above the H2 potential minimum (-4.74 eV) the room-temperature thermal kinetic energy is approximately 0.03 eV. The graph uses the accurate Liu-Seigbahn-Truhlar-Horowitz (LSTH) potential surface [195).

Figure 5.2. (a) Schematic contour plot showing the energy of the A-B-C cluster as a function of the distances between the atoms. ABC is the transition state, (b) Graph of the energy of the cluster as a function of its location along the reaction coordinate, which is the dashed line in (a). 

This requirement is automatically satisfied due to the definition of reaction graph based on the notion of the maximal common subgraph. Ugi et al. [12,16] summarized the above property as the principle of minimal chemical distance, a very effective heuristic rule for the construction of reaction matrices. 

Example 2.10. The matrix ID = (D(Gj, Gj)) of the reaction distances between graphs from 2,2 simply constructed from... 

Example 2.11. Apply the algorithm 2.1 for the evaluation of the reaction distance between graphs G4 and Giq from example 2.4.1. 

The method of construction of all possible successors of the given graph G with the specified maximum reaction distance is schematically illustrated as follows... 

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