Hiickel theory allyl radical

Unfortunately, while it is clear that the allyl cation, radical, and anion all enjoy some degree of resonance stabilization, neither experiment, in the form of measured rotational barriers, nor higher levels of theory support the notion that in all three cases the magnitude is the same (see, for instance, Gobbi and Frenking 1994 Mo el al. 1996). So, what aspects of Hiickel theory render it incapable of accurately distinguishing between these three allyl systems ... 

We have seen so far that MOs resulting from the LCAO approximation are delocalized among the various nuclei in the polyatomic molecule even for the so-called saturated a bonds. The effect of delocalization is even more important when looking to the n electron systems of conjugated and aromatic hydrocarbons, the systems for which the theory was originally developed by Hiickel (1930, 1931, 1932). In the following, we shall consider four typical systems with N n electrons, two linear hydrocarbon chains, the allyl radical (N = 3) and the butadiene molecule (N = 4), and two closed hydrocarbon chains (rings), cyclobutadiene (N = 4) and the benzene molecule (N = 6). The case of the ethylene molecule, considered as a two n electron system, will however be considered first since it is the reference basis for the n bond in the theory. 

Problem 1.3. Hiickel LCAO-MO theory 1 the allyl radical (See e.g. Lionel Salem, The Molecular Orbital Theory of Conjugated Systems, W. A. Benjamin, Inc. (1974), Chap. 1, or Peter W. Atkins, Physical Chemistry, Wiley-VCH (1988).)... 

For the vinyl substituent, we can analyze the stabilization in terms of simple Hiickel MO theory. The interaction of a p orbital with an adjacent vinyl group creates the allyl radical. In Hiickel calculations, the resulting orbitals have energies... 

As shown by the data, the barrier depends upon the metal ion. Explain the basis for the metal ion dependence. Data on the rotational barrier of the allyl cation ( 18 kcal/mol) and allyl radical (>17 kcal/mol) indicate that the barrier for rotation in all three species is similar. According to simple Hiickel MO theory, should the barrier to rotation in the allyl system increase, decrease, or remain the same as electrons are added to the tt system in the progression CH2=CHCH2, CH2=CHCH2, and CH2=CHCH2 ... 

Although satisfactory for allyl cation. Figure 10.1 is insufficient for species with more than two tt electrons because the tt orbital in (c) can accommodate only two electrons. Molecular orbital (MO) theory, however, offers an alternative to resonance and valence-bond theory for understanding the structure and reactions of not only allylic cations, but radicals (three rr electrons) and anions (four tt electrons) as well. In a simplification known as the Hiickel, or ir-electron, approximation the tt MOs are considered as separate from... 

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