Atom equivalent classes

Figure 2-44, The EC values of the atoins of phenylalanine (without hydrogens) are calculated by considering the class values of the neighboring atoms, After each relaKatlon process, c, the number of equivalent classes (different EC values), is determined.

When all the EC values of the atoms have been calculated, the number of equivalent classes (e) for the first sphere is determined. The number of classes is equivalent to the number of different EC values. 

Many variations of the Morgan Algorithm were introduced, because of problems finding the terminating condition of stage 1 (oscillating number of equivalent classes [80]) or special atoms with isospcctral points [81],... 

The indices are permutable within any interval of indices belonging to constitutionally equivalent atoms, as long as the a-atoms of these atoms do not also belong to constitutionally equivalent classes. In the latter case, additional rules are needed for deciding between equivalent assignment of indices. 

It is further possible to describe a given constitution by mapping the indexed set of atoms onto the independently indexed graph. The indexed set of atoms must be partially symmetrized with regard to the element equivalence classes of the atoms and the indices of the graph with respect to its symmetry. That constitution is defined as the reference constitution in which the atomic indices a and the graph indices g match. 

Such a procedure corresponding to the permutation-nomenclature system of configuration 7> is in accordance with the fact that a chemical constitution is a representative of a product D, X. .. X Da of the double cosets D, of pairs of subgroups symmetric groups Sn,g. The latter correspond to permutations of atoms of the same element and the symmetry of the adjacent subset of graph points whose degree is compatible with the coordination number of the element equivalence class of the atom. 

If the problem were to partition a set of carbon compounds into two equivalence classes, of which one contains only chiral molecules and the other one only achiral ones, it could not be solved with the criterion of asymmetric C-atoms. In the first case, one would assign meso-forms like 9 and compounds with pseudo-asymmetric 22> C-atoms, such as 11, to the chiral equivalence class, and in the second, chiral molecules like 12 would remain in the achiral subset. However, the latter class would be devoid of chiral molecules, if the compounds under consideration have been confined to molecules with free rotation about all C—C bonds. 

Equivalence Class. In the canonicalization of structures that have some element of symmetry, certain atoms that are topologically equivalent may yield the same canonical number. These atoms are considered to be in the same equivalence class. The concept of equivalence class is used, for example, in the Daylight Chemical Information Systems handling of reactions, to examine equivalent atoms when mapping reactant and product atoms. 

Graph vertices are partitioned and ordered into topological equivalence classes, i.e. orbits, according to some special matrices developed for each atom [Bersohn, 1987). These matrices give a representation of the whole molecule as seen from the considered atom. 

The first partition of the vertices depends on their property values P, two atoms X and Y are considered topologically equivalent if (i) they have identical matrices (ii) if X has a neighbour I belonging to a different equivalence class then there must exist a neighbour J of the atom Y in the same equivalence class as the atom I. 

To accoimt for steric effects in molecule-receptor interactions, the weighted information indices by volume have been proposed [Ray et al, 1985]. These molecular descriptors are calculated in the same way as the indices of neighbourhood symmetry defined above using the atomic van der Waals volumes to get the probabilities of the equivalence classes. In other words, the van der Waals voliunes of the atoms belonging to each equivalent class are summed to give a molecule subvolume, then divided by the total molecule volume. For example, the weighted information content by volume is defined as ... 

Molecular descriptors calculated as - information content of molecules. Different criteria are used for defining - equivalence classes, i.e. equivalency of atoms in a molecule such as chemical identity, ways of bonding through space, molecular topology and symmetry, - local vertex invariants [Bonchev, 1983]. 

Bersohn, M. (1987). A Matrix Method for Partitioning the Atoms of a Molecule into Equivalence Classes. Computers Chem., 11,67-72. 

Laidboeur, T, Cabrol-Bass, D. and Ivanciuc, O. (1997). Determination of Topo-Geometrical Equivalence Classes of Atoms. J.Chem.Inf.Comput.Sci., 37,87-91. 

Rucker, G. and Rilcker, C. (1990). Computer Perception of Constitutional (Topological) Symmetry TOPSYM, a Fast Algorithm for Partitioning Atoms and Pairwise Relations among Atoms into Equivalence Classes. J.Chem.lnfComputSci., 30,187-191. 

We will show that for the retrieval of a stored field state only excitations in a certain quasi-particle mode, the dark-state polariton, are of importance. Thus all storage states with the same projection to this mode are equivalent. Due to the existence of these equivalence classes there is no enhanced sensitivity to (individual) decoherence processes in an N-atom system as compared to a single atom system. 

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