Graph theoretical treatment

A thiepin is formally isoelectronic with the 8ic-electron 1,3,5,7-cyclooctatetraene and 1,3,5-cycloheptatrienide ion and, if planar, may actually be antiaromatic. Recently, the question of the antiaromaticity of thiepin has been the subject of interest for both synthetic and theoretical chemists. The apparent instability of the thiepin ring system is in good agreement with theoretical calculations. Dewar and Trinajstic 68) have reported that the thiepin is considered to be weakly antiaromatic (RE = — 1.45 kcal mol-1) based on PPP SCF MO calculations. On the other hand, Hess Jr. and Schaad 69) have found it to be substantially antiaromatic (RE = —0.232 J) by using the Huckel MO method. This result was also supported by a graph-theoretical treatment by Aihara 70). 

Polanski, J. and Rouvray, D.E. (1976a). Graph-Theoretical Treatment of Aromatic Hydrocarbons. I The Formal Graph-Theoretical Description. MATCH (Comm.Math.Comp.Chem.), 2, 63-90. 

Application of graph-theoretical treatment to the experimental data... 

Several years before the formulation of the Tensor Surface Harmonic Theory, King presented a graph theoretical treatment of bonding in borane clusters563. It is instructive to compare this with Stone s methodology. 

Polansky and Rouvray (1976) started their series of pioneering papers, entitled "Graph-Theoretical Treatment of Aromatic Hydrocarbons", at a time when no coronoid hydrocarbon was known chemically. This fact did not prevent the authors to indude corona-condensed systems (coronoids) in their first paper in the series (Polansky and Rouvray... 

Polansky OE, Rouvray DH (1976) Graph—Theoretical Treatment of Aromatic Hydrocarbons I — The Formal Graph—Theoretical Description. Match 2 63... 

Polansky OE, Rouvray DH (1977) Graph-Theoretical Treatment of Aromatic Hydrocarbons III - Corona-Condensed Systems. Match 3 97... 

The most widely used types of topological indices are molecular connectivity indices. They have been widely reported as molecular structure descriptors in SPR studies and in structure-activity (SAR) studies as well. Originally developed by Randic/ and later modified by Kier and Hall, ° these indices have been used in a variety of studies. These indices are based on a graph theoretical treatment of the molecular topology of the compounds, and encode information about the branching and size of the molecules. The general equation for calculating molecular connectivities of the nth order is... 

The total numbers of skeletal bonding orbitals in pyramidal nido systems generated by these three interactions are n — 1,1, and 2, respectively, leading to a total of 77 -I- 2 bonding orbitals holding 2n +4 skeletal electrons. Thus, the graph-theoretical treatment of nonpyramidal and pyramidal nido polyhedra with n vertices leads to the prediction of the same numbers of skeletal bonding orbitals, namely, n + 2, in accord with experimental observations. However,... 

A similar approach was undertaken by Mah et al. (1976) in their attempt to organize the analysis of process data and to systematize the estimation and measurement correction problem. In this work, a simple graph-theoretic procedure for single component flow networks was developed. They then extended their treatment to multicomponent flow networks (Kretsovalis and Mah, 1987), and to generalized process networks, including bilinear energy balances and chemical reactions (Kretsovalis and Mah, 1988a,b). 

The graph-theoretical 4N + 2 Hiickel rule analogy with the aromaticity of two-dimensional polygons requires that N = 0 in all the three-dimensional deltahedra. The Jemmis-Schleyer interstitial electron rule [55], originally introduced for nido half-sandwich species, also relates the 4N + 2 Hiickel rule to the delocalized deltahedra directly In this treatment, N is typically 1. 

Mathematical chemistry, the new challenging discipline of chemistry has established itself in recent years. Its main goal is to develop formal (mathematical) methods for chemical theory and (to some extent) for data analysis. Its history may be traced back to Caley s attempt, more than 100 years ago, to use the graph theoretical representation and interpretation of the chemical constitution of molecules for the enumeration of acyclic chemical structures. Graph theory and related areas of discrete mathematics are the main tools of qualitative theoretical treatment of chemistry [1,2]. However, attempts to contemplate connections between mathematics and chemistry and to predict new chemical facts with the help of formal mathematics have been scarce throughout the entire history of chemistry. 

The values of the enthalpy obtained do not depend on the model chosen for the dimer, as shown by Lussan (19) in a comprehensive summary of the possible theoretical treatments of the NMR data. Lussan demonstrates the form of the equilibrium constant calculation in the case of (1) monomer-dimer (open or cyclic), (2) monomer-cyclic trimer, and (3) monomer-higher acyclic multimers in the two cases of (3a) all K s equal and (3b) ki for monomer-dimer equilibrium unique, k s for higher multimers all equal. He then takes the experimental curves for a number of alcohols in carbon tetrachloride and achieves a reasonable fit to the data up to 0.6 mole fraction by using one or the other of the theoretical relationships. In some cases two sets of theoretical points are plotted on the same graph as the experimental data both are a good fit in the low concentration region, up to 0.1 mole fraction. Above this concentration one or the other of the theoretical curves is much closer to the experimental curve. Lussan implies that hypothesis 3b may be a more accurate fit to the data in the more concentrated solutions. Methanol, ethanol, 2-methyl-2-propanol (tert-butyl alcohol) and 2,2,4-trimethyl-3-pentanol follow the curve for equilibrium 3a, while 2,2,4,4-tetramethyl-3-pentanol fits the monomer-dimer data. Lussan points out that the behavior of the latter alcohol fits in with that of two similar heavily substituted tertiary alcohols which have been found by infrared methods to form only dimers. 

Chemical reactions were studied by exposing small molecules to BH, generated by excimer laser photolysis of B2H6 at 193 nm [31], and monitored by laser-induced fluorescence of BH. With NO or C2H4 in the temperature range 250 to 340 K, graphs of first-order decay rates versus reactant pressure were used to determine second-order rate constants. A theoretical treatment of the BH + NO reaction indicates intermediate formation of HBON [32]. 

The participation of Cd(OH)2 in the deposition of CdS (and other metal chalcogenides) has been demonstrated or suggested on many occasions. Kitaev et al. presented a theoretical thermodynamic treatment of the Cd " /ammonia/ thiourea system to show when Cd(OH)2 should be present as a solid phase in the deposition solution [36]. A graphical representation of this analysis is shown in Eigure 3.1. This graph is based on two equilibria the solubility product of Cd(OH)2 and the stability constant of the ammonia (ammine) complex of Cd. Consider first the former ... 

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