Bond redundant

Figure 2-16. a) The redundant incidence matrix of ethanal can be compressed by b) omitting the zero values and c) omitting the hydrogen atoms, in the non-square matrix, the atoms are listed in columns and the bonds in rows. 

Figure 2-17. a) The redundanl bond malrix of ethanal with ihe zero values omitted, b) It can be compressed by reduction to the top right triangle, c) Omitting the hydrogen atoms provides the simplest non-redundant matrix representation. 

Both tables, the atom and the bond lists, are linked through the atom indices. An alternative coimection table in the form of a redundant CT is shown in Figure 2-21. There, the first two columns give the index of an atom and the corresponding element symbol. The bond list is integrated into a tabular form in which the atoms are defined. Thus, the bond list extends the table behind the first two columns of the atom list. An atom can be bonded to several other atoms the atom with index 1 is connected to the atoms 2, 4, 5, and 6. These can also be written on one line. Then, a given row contains a focused atom in the atom list, followed by the indices of all the atoms to which this atom is bonded. Additionally, the bond orders are inserted directly following the atom in-... 

Figure 2-22. Non-redundant connection table of ethanal. Only non-hydrogen atoms are considered bonds with the lowest indices are counted once (see Figure 2-21).

Almost all chemical information systems work with tlicir own special type of connection table. They often use various formats distinguishing between internal and external connection tables. In most cases, the internal connection tables arc redundant, thus allowing maximum flexibility and increasing the speed of data processing. The external connection tables are usually non-redundant in order to save disk space. Although a connection table can be cprcsented in many different ways, the core remains the same the list of atoms and the list of bonds. Thus, the conversion of one connection table format into another is usually a fairly straightforward task. 

The parameter redundancy is also the reason that care should be exercised when trying to decompose energy differences into individual terms. Although it may be possible to rationalize the preference of one conformation over another by for example increased steric repulsion between certain atom pairs, this is intimately related to the chosen functional form for the non-bonded energy, and the balance between this and the angle bend/torsional terms. The rotational banier in ethane, for example, may be reproduced solely by an HCCH torsional energy term, solely by an H-H van der Waals repulsion or solely by H-H electrostatic repulsion. Different force fields will have (slightly) different balances of these terms, and while one force field may contribute a conformational difference primarily to steric interactions, another may have the... 

The values 1/V(dj dj) are for the atoms i and j, which make up this bond, and the connectivity index, x, is obtained as the sum of the bond connectivities. In molecules containing heteroatoms, the d values were considered to be equal to the difference between the number of valence electrons (E") and the number of hydrogen atoms (hi). Thus, for an alcoholic oxygen atom, d = 1, and d = 5. The valence connectivity-index, y can then be calculated the use of removes redundancies that can occur through the use of y alone. The calculation of connectivity indices and for the case of two isomeric heptanols is as follows. 

Even this brief list may suffice to show that it would be a formidable task to develop a system of factorization free of avoidable redundancies, and that such a system would not be satisfactory even if it avoids arbitrary choices. It would require a rule disqualifying certain centers or lines of stereoisomerism on the basis of their relationships to other potential elements in the same molecule. Such definitions would not be self-contained. Moreover, the products of factorization that would take the place of those dropped cannot be limited to points or lines that are merely differently defined. There would have to be a virtually open-ended proliferation of new elements. This highly undesirable feature would not be offset by a major benefit of the revised system such as a correlation between the numbers of elements and of stereoisomers, because a complete elimination of all redundancies does not seem possible. We conclude that the system of choice is the one based on the principle that the elements of stereoisomerism allow no further factoring. Accordingly we think it best to retain the definitions given in Sects. IV and VI and their strictures that all centers and lines be occupied by atoms or bonds. 

The network equations constitute a set of A a 1 valence sum rule equations (eqn (3.3)) and A b Xa+1 loop equations (eqn (3.4)) where the network contains atoms and A b bonds. Alternatively one can use the equivalent Kirchhoff equations (2.7) and (2.11). One can readily write down equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than Ab — Aa + 1 possible loops in most bond graphs, but only Ab —Aa+ 1 are independent. Equations (3.3) and (3.4) thus constitute a set of Ab equations which is exactly the number needed to solve for the Ab unknown bond valences,. s. 

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