Topological theory

One further theoretical method that merits consideration at this point is the topological theory of molecular structure exemplified by Bader (1985, 1990). In this method a topological description of the total electron density in the molecule is used. A major advantage of this method is that it allows the total interaction between various centres to be probed. Cremer et al. (1983) used the Bader method to examine the homotropylium cation [12] and concluded that it was indeed homoaromatic. 

The topological theory of atoms in molecules <2003MI190> has been employed to calculate the conformational preference of monosubstituted 1,3-oxathianes. The preferred conformer results from an energy balance between the ring and the substituent. This method has proven to be general and is a new technique for conformational analysis. 

THE LINK BETWEEN THE TOPOLOGICAL THEORY OF RANADA AND TRUEBA, THE SACHS THEORY, AND 0(3) ELECTRODYNAMICS... 

All topological theories are nonlinear, a feature of both the Sachs and Evans theories, and the whole of quantum theory can be replaced by topology [1], which reduces in some circumstances to the Yang-Mills theory [1], of which 0(3) electrodynamics [3] is an example. 0(3) electrodynamics has been developed into an 0(3) symmetry quantum field theory by Evans and Crowell... 

Handling rate equations for complex mechanisms. While steady-state rate equations can be derived easily for the simple cases discussed in the preceding sections, enzymes are often considerably more complex and the derivation of the correct rate equations can be extremely tedious. The topological theory of graphs, widely used in analysis of electrical networks, has been applied to both steady-state and nonsteady-state enzyme kinetics 45-50 The method employs diagrams of the type shown in Eq. 9-50. Here... 

Topaquinone (TPQ) 816, 817s Topoisomerases 219,575, 638, 657,659 Topological theory of graphs 466 Topologies... 

More elaborate electron-counting procedures have been established, and these are usually more appropriate for the rationalisation of high-nuclearity clusters. An approach which was originally applied to complex boranes has been extended to carbonyl clusters and a number of topological theories have been described. Although there is some way to... 

Mermin ND (1979) Topological theory of defects in ordered media. Rev Mod Phys 51(3) 591-648... 

Having reproduced our old proposal, the outcome is now briefly described. According to Aihara s (1988) topological theory of aromaticity, the resonance energy per 7t electron (repe) of 5 is calculated to be 0.0274 / , or about 60% of that of benzene (0.0454 / ). It is thus clear that there is no dramatic increase in the conjugative stability in 5. As far as aromatic stabilization in the football structure is concerned it should be regarded as an extension of two-dimensional aromaticity. Aihara Hosoya (1988) call it spherical aromaticity. 

The existence of Kekule structures in a benzenoid system is the first fundamental problem in the topological theory of benzenoid systems. It was considered as one of the most difficult open problems in this theory. Many investigations have been made in order to find necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. Some fairly simple conditions which are both necessary and sufficient have been given in the last few years. In this chapter we review the main results and give a rigorous proof for some necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. In addition, by using the above results, a construction method of some concealed non-Kekulean benzenoid systems is given. 

A Kekule structure or a 1-factor of a benzenoid system H is an independent edge set in H such that every vertex in H is incident with an edge in the edge set. A benzenoid system is said to be Kekulean if it possesses a Kekule structure, otherwise it is said to be non-Kekulean. It was first pointed out by Clar et al. [1,2] that Kekule structures are of paramount importance for the stability of benzenoid systems. In fact, up to now, no non-Kekulean benzenoid system has been synthesized by chemists. Therefore the existence of Kekule structures in a benzenoid system is a fundamental problem in the topological theory of benzenoid systems. 

After this chemists hoped to find some fairly simple necessary and sufficient conditions. This is why until 1982-1983 I. Gutman and N. Trinajstic still pointed out several times [7-9] that the problem of recognizing Kekulean benzenoid systems was an open problem, and it was thought to be one of the most difficult open problems in the topological theory of benzenoid systems. 

Turro, N.J. (1986) Geometric and topological theory in organic chemistry. Angewandte Chemie, International Edition in English, 25, 882-901. 

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

Angyan JG, Jansen G, Loos M, H attig C, Hess BA (1994) Distributed polarizabilities using the topological theory of atoms in molecules. Chem Phys Lett 219 267-273... 

We wish to conclude this paragraph with a quotation from the author s first paper [2] on benzenoid systems There is no simple recipe to decide by inspection of the molecular graph whether K = 0 or not. In other words, the necessary and sufficient conditions for the existence of Kekule structures seem to be rather complicated". Fifteen years later one may optimistically state that this hard problem of the topological theory of benzenoid hydrocarbons is completely settled. 

The graph-theoretical methods are extensively used in the theory of conjugated systems. It is only natural since in this area of chemistry the HMO method was used most frequently. In the next chapters we shall dwell on some interesting results derived in the topological theory of conjugated and aromatic systems. 

Until now we considered the topology of various organic molecules. In inorganic chemistry the topological methods are not so popular. Therefore, in this chapter we shall confine ourselves to only two topics, the topological theory of borane structure and the discussion of the interrelationship between topology, symmetry, and electronic structure of coordination compounds. 

Lipscomb s topological theorem. In 1954 W. Lipscomb with coworkers suggested a so-called topological theory of boranes which he continues to improve even now. On the basis of that theory it became possible to predict,... 

The results, summarized in Table I, will be essential in the topological theory which will now be outlined. 

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