Nodal characteristics

The two antibond NBOs t(a) and T(7t) have distinctive shapes and nodal characteristics that may be characterized as follows. 

FIGURE 7. Nodal characteristics of the five occupied tt MOs of naphthalene... 

Fig. 14. Nodal characteristics of the lowest-lying ir -antibonding level of butadiene.

As discussed at the end of Chapter 3, one group orbital of a methyl or methylene group will always have the correct nodal characteristics to interact with an adjacent n orbital or with an adjacent spn orbital in fashion. The degree of interaction may be inferred from the energies of the orbitals, which may in turn be obtained by measurements of ionization potentials and application of Koopmans theorem. Thus, the methyl groups adjacent to the n bond in (Z)-2-butene (ionization potential IP = 9.12 eY [63]) raise the energy of the n orbital by 1.39 eV relative to that of ethylene (IP = 10.51 eV [87]). A similar effect is observed in cyclohexene [64]. 

What should be correlated In an orbital correlation diagram, the shapes and energies of orbitals are examined to see if the electronic structure of the reactants could be smoothly converted into the electronic structure of the products, each defined by the structures and occupancies of their respective orbitals. The nodal characteristics of orbitals are very resistant even to rather large perturbations and will tend to be conserved in chemical reactions. If an element of symmetry, for example, a mirror plane, is maintained during the course of the reaction, the nodal characteristics separate the orbitals into two sets, the members of one set being symmetric with respect to reflection... 

The shapes of the seven 4f orbitals in the general set are illustrated in Fig. 8.11.1, and their nodal characteristics are shown in Fig. 8.11.2. The number of vertical nodal planes varies from 0 to 3. The z3, yz2, and xz2 orbitals each has two nodes that are the curved surfaces of a pair of cones with a common vertex at the origin. 

Nodal characteristics of the general set of 4f orbitals. The positive and negative lobes in each orbital are shaded and un-shaded, respectively. 

The closest forerunner to TSH theory was due to Hoffmann, Ruedenberg, and Verkade. These authors used spherical harmonics at the center of the cluster to generate linear combinations of orbitals with particular symmetry and nodal characteristics. The energy of these orbitals could then be estimated by the number of angular nodes in the parent spherical harmonic. For aUcah metal clusters, one could actually produce the same answers for the symmetries of... 

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