Topologically equivalent orbitals

Most clusters, however, caimot be described adequately in terms of two-center, two-electron bonds because the coimectivity of the vertices exceeds the number of valence orbitals that are available for bonding. Early efforts to rationalize such systems, such as Lipscomb s styx approach and Kettle s Topological Equivalent Orbital Method, are described by Mingos and Johnston. In these more difi cult cases, the simple valence-bond picture is inappropriate examples are deltahedral clusters composed of B-H vertices or conical M(CO)3 fragments, both of which usually have only three orbitals available for skeletal bonding. Theoretical models for describing these systems will be discussed in the next section. 

Kettle attempted to circumvent the resonance problems inherent in earlier localised orbital methods by allowing for interaction between the edge or face-localised cluster bonding orbitals. His Topological Equivalent Orbital methodology applied group theoreti-... 

Using [MoeClg] as an example, apply Stone s surface tensor harmonic model to the species [Ta6Cli2]. To what extent does the topological equivalent orbital model facilitate the application ... 

The importance of the arguments we have outlined lies in the fact that they provide a theoretical foundation both for aromaticity-antiaromaticity and for pericyclic selection rules. They furthermore demonstrate the relationship between the two The topological equivalence between an array of p orbitals in a w system of a carbon chain or ring and a pericyclic transition state, composed of an... 

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. 

Topologically equivalent vertices constitute disjoint subsets of vertices called orbits. 

Intuitively, two phase portraits are topologically equivalent if one is a distorted version of the other. Bending and warping are allowed, but not ripping, so closed orbits must remain closed, trajectories connecting saddle points must not be broken, etc. 

Assume the hypotheses of Dulac s criterion, except now suppose that R is topologically equivalent to an annulus, i.e., it has exactly one hole in it. Using Green s theorem, show that there exists at most one closed orbit in R. (This result can be useful sometimes as a way of proving that a closed orbit is unique.)... 

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

In the past 15 years, the synthesis of a series of 3-connected electron precise hydrocarbon clusters [CnHn]5 (see Fig. 2 for examples) has renewed interest in localised bonding schemes, since the bonding orbitals in these clusters are topologically equivalent to edge-localised 2 c-2 e bond pairs in a Valence Bond sense. Schulman et al. performed ab initio calculations on cubane [CgH8] utilising a basis set of 2c-2e localised C-C and C-H... 

Furthermore, the chemical identity of the nuclei plays no direct part in determining the nodal patterns of the orbitals that they generate. The important feature is the number and kind of atomic orbitals that they contribute. Changing nuclei distorts and displaces the nodes, but they will still be between certain atoms in each orbital. Thus the MO s formed by two sets of different nuclei will be topologically equivalent if the same atomic orbitals are used. 

Figure 13.21 Ring current in the transition structure of the Diels-Alder reaction (ethy-leneH-butadiene cyclohexene), (a) the six basis orbitals exhibit a cyclic overlap that is topologically equivalent with the % system of delocalized electrons In benzene.

Figure 13.26 The orbital overlap of the orbital basis of the coarctate transition state of the cyclopropyl carbene fragmentation is topologically equivalent to the % system of...

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