Twin-states

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

Scheme 19. A VB Mixing Diagram Showing the Splitting of the Kekule Structures into the Twin States, , and P a...

The symmetry assignment of the twin states can be made using the information in Scheme 33c. For... 

Scheme 33. Generation of the Twin States by Mixing of the Kekule Structures for Benzene (a) and Cyclobutadiene (b) as Archetypal Aromatic and Antiaromatic Species, (c) Transformation Properties of Ki and K2 with Respect to Symmetry Operations of the Dgh and Dih Point Groups...

Figure 17. Calculated (refs 34 and 164a) geometries for the twin states of 37. The diagrams at the bottom show the o jt-interplay in the two states. The ground state exhibits an inverse Mills—Nixon effect since, due to repulsion of the substituents with the jt. bonds of Ki, the jt. curve is displaced in the AR > 0 direction.

Parts b and c of Figure 18 show the results of a recent computational study by Zilberg and Haas188 210 aimed at probing the properties of the ground Big and excited 1 Aig twin states of cyclobutadiene and... 

Figure 18. (a) jt—a-Interplay diagrams for the twin states of /7-electron—/7-center antiaromatic species. (b,c) Calculated geometries for the twin states of cyclobutadiene and cyclooctatetraene (from refs 164a, 177, 188, and 219). 

As we are about to tell now, the theme of the Kekule modes involves stories within stories. The longer and fascinating one is that of the Kekule modes of benzene and aromatics.1 3 209 The complementary story deals with Kekule modes in antiaromatic molecules188 210 and is likely to continue in the future with further advances in spectroscopic techniques.218-222 Finally, another off-shoot of the main theme relates to the twin states in the case of aromatic transition states.225... 

Figure 19. Computed (full-jrCI/STO-3G data from refs 164a and 209) a-curve and total energy curves for the twin states of benzene along the b2U mode.

The frequency exaltation of the Kekule mode is mirrored by the structural manifestations in the twin states, discussed with reference to Figures 16 and 17. Thus, the repulsive jr-curve in the ground state softens the potential and thereby enables the ground-state molecule to distort along the Kekule mode when angular strain is exerted. In contrast, the attractive jr-curve in the twin excited state stiffens the potential and restores the local Deh symmetry of the benzene nucleus. The two physical effects are in perfect harmony and find a natural reflection in the VB model. 

Scheme 36. (a) Kekule Structures for Naphthalene, (b) A VB Mixing Diagram Which Shows the Twin States Generated from the Ag and B2u Combinations of the Kekule Structures... 

Scheme 38. (a) Four Kekule Structures for Anthracene and (b) Their Symmetry Adapted Combinations, (c) Generation of the Twin States 0 o, I ) from These Combinations... 

C. The Kekule Mode in the Twin States of Antiaromatic Molecules... 

The Kekule Modes in Twin States of Chemical Transition States... 

Scheme 41. (a) Transition State for the Degenerate Semibullvalene Rearrangement, (b) Avoided Crossing of the Corresponding Kekule Structures Along the Reaction Coordinate Mode (t>2), and the Generation of the Twin States by Avoided Crossing, (c) Frequencies of the b2 Mode in the Two States (Ref 225). Delocalized Bonds in c Are Removed for Clarity... 

---