Morgan

Morgan J D III and Simon B 1980 Behavior of molecular potential energy curves for large nuclear separations/nf. J. Quantum Chem. 17 1143... 

Deverell C, Morgan R E and Strange J FI 1970 Studies of ohemioal exohange by nuolear magnetio relaxation in the rotating frame Mol. Phys. 18 553-9... 

Various methods have been developed for a unique and unambiguous numbering of the atoms of a molecule and thus for deriving a canonical code for this molecule [76]. Besides eigenvalues of adjacency matrices [77], it is mainly the Morgan Algorithm that is used [79]. 

The Morgan Algorithm classifies all the congeneric atoms of a compound and selects invariant-labeled atoms (see Section 2.5.3.1). The classification uses the concept of considering the number of neighbors of an atom (connectivity), and does so in an iterative manner (extended connectivity, EC). On the basis of certain rules. 

Figure 2-42. The Morgan Algorithm generates an unambiguous and unique numbering of phenylalanine (see Tutorial, Section 2,S,J,1).

Two aspects of tlie Morgan Algorithm play a dominant role for structure coding ... 

Consideration of stereochem-iitry. The parity or handedness - R/S or chjirans - of a stcreoccnter can be obtained by considering the sequence of the Morgan numbers of die atoms, similarly to CIP. Then the number of pairwise interclianges is counted until the numbers arc in ascending order (see Section 2,8,5). 

However, it should not be concealed that in some cases the Morgan Algorithm has problems. In some structures the numbering show oscillatoiy behavior (for an example, sec Figure 2-44). Moreover, although equivalent atoms obtain the same extended connectivity value, it cannot always be concluded that atoms with the same extended connectivity value are equivalent. 

In the fir.st iteration proces.s, the class values of the atoms of the structure. show information already known (the degree of the nodes). Hence Morgan takes the neighboring atoms into account. He considers the environment of an atom by summing class values of all directly adjacent atoms. This process results in a new class value called the extended connectivity (EC) value of the atom. The new EC value expresses indirectly the neighborhood of the adjacent atoms in a second sphere (Figure 2-43). 

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],... 

Figure 2-77. Determination of parity value, a) First the structure is canonicalized. Only the Morgan numbers at the stereocenter are displayed here, b) The listing starts with the Morgan numbers of the atoms next to the stereocenter (1), according to certain rules. Then the parity value is determined by counting the number of permutations (odd = 1, even = 2).

First, both the skeleton and the ligands at a stereocenter have to be numbered independently of each other. The sites of the skeleton can be numbered arbitrarily but then this numbering has to remain fixed all the time in any further operations. The atoms directly bonded to the stereocenter have to be numbered according to rules such as the CIP rules or the Morgan Algorithm (Figure 2-79). 

The constitution can be represented in an unambiguous and unique manner by canonicalization (Morgan Algorithm). 

R. S. Michalski, R. E. Stepp. Learning from Observation Ganceptual Clustering, in Machine Learning An Artificial Intelligence Approach, R.S. Michalski, f.G. CarboneU, T.M. Mitchell (Eds.), Morgan Kauffmatm, San Mateo, GA, 1983, pp. 331-363. 

U. Fayyad, G. Grinstein, A. Wierse, Information Visualization in Data Mining and Knowledge Discovery, Morgan Kaufman Publishers, San Frandsco, USA, 2002. 

The one-to-one transformation between the global and local coordinate systems can be established using a variety of techniques (Zienkiewicz and Morgan,... 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

Zienkiewicz, O. C. and Morgan, K., 1983. Finite Elements and Approximation, Wiley, New York,... 

Carley, A. F. Morgan, P. H., 1989. Computational Methods in the Chemical Sciences. Halsted Press (Wiley) New York. 

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