Conjugation—Hiickel Theory

In the Hiickel theory, the 7r-electron energy of a conjugated molecule can be expressed by the following equation ... 

Extended conjugated alkenes, with Pd re-complexes, 8, 334 Extended Hiickel theory, in semi-empirical quantum mechanics, 1, 655... 

Hiickel theory [1], the oldest quantum-mechanical approach to calculating the properties of organic molecules, has been outshined by the more rigorous semi-empirical and ab initio techniques for at least two decades. Even if the ab initio total energies of conjugated systems were found [2, 3] to parallel the Hiickel tt-electron energies, the Hiickel method is certainly too crude to be of substantial value for quantitative considerations. 

Anti-aromaticity was predicted by the Hiickel approach for conjugated cyclic planar structures with 4n 7i electrons due to the presence of two electrons in antibonding orbitals, such as in the cydopropenyl anion, cydobutadiene, and the cydopentadienyl cation (n = 1), and in the cydoheptatrienyl anion and cydooctatetraene (n = 2). It has been argued that a simple definition of an anti-aromatic molecule is one for which the 1H NMR shifts reveal a paramagnetic ring current, but the subject is controversial. The power of the Hiickel theory indeed resides not only in the aromatic stabilization of cydic 4n + 2 electron systems, but also in the destabilization of those with An electrons [22, 27, 42]. 

In the simplest application of Hiickel theory to the n electrons of planar conjugated hydrocarbons, a is taken to be the same for all C atoms, and ft to be the same for all bonded pairs of C atoms it is then customary to write the Hiickel secular determinant in terms of the dimensionless parameter x. 

This separation of the cr framework and the re bond is the essence of Hiickel theory. Because the re bond in ethylene in this treatment is self-contained, we may treat the electrons in it in the same way as we do for the fundamental quantum mechanical picture of an electron in a box. We look at each molecular wave function as one of a series of sine waves, with the limits of the box one bond length out from the atoms at the end of the conjugated system, and then inscribe sine waves so that a node always comes at the edge of the box. With two orbitals to consider for the re bond of ethylene, we only need the 180° sine curve for re and the 360° sine curve for re. These curves can be inscribed over the orbitals as they are on the left of Fig. 1.23, and we can see on the right how the vertical lines above and below the atoms duplicate the pattern of the coefficients, with both c and c2 positive in the re orbital, and c positive and c2 negative in re. ... 

Hiickel theory and it was introduced in order to describe the n electrons in planar conjugated molecules, a and / are the Coulomb and resonance integrals, respectively, of some effective one-electron Hamiltonian operator 65>. 

We would also like to mention here another important point. Since the Hiickel and topological matrices are closely related for a particular conjugated molecule, all properties of a molecule (i.e. energy, MOs, bond orders, charge densities) which may be derived from the topological matrix by mathematical treatment must be dependent on the molecular topology 23>. This may be one reason why the predictive power of elementary Hiickel theory is in many cases (e.g. for alternant hydrocarbons) as good as that of any more elaborate approach 58>. 

The interaction of atomic orbitals giving rise to molecular orbitals is the simplest type of conjugation. Thus in ethylene the two p orbitals can be described as being conjugated with each other to make the n bond. The simplest extension to make longer conjugated systems is to add one p orbital at a time to the n bond to make successively the n components of the allyl system with three carbon atoms, of butadiene with four, of the pentadienyl system with five, and so on. Hiickel theory applies, because in each case we separate completely the n system from the a framework, and we can continue to use the electron-in-the-box model. 

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