Polyenes Hiickel molecular orbitals

In this section, we first discuss the bonding in two linear triatomic molecules BeH2 with only a bonds and C02 with both a and n bonds. Then we go on to treat other polyatomic molecules with the hybridization theory. Next we discuss the derivation of a self-consistent set of covalent radii for the atoms. Finally, we study the bonding and reactivity of conjugated polyenes by applying Hiickel molecular orbital theory. 

Hiickel molecular orbital theory for conjugated polyenes... 

In this section, we first study the jt bonding in conjugated polyenes by means of the Hiickel molecular orbital theory. Then we will see how the wavefunctions obtained control the course of the reaction for these molecules. 

The band structure of a three-dimensional solid, such as a semiconductor crystal, can be obtained in a similar fashion to that of a polyene. Localized molecular orbitals are constructed based on an appropriate set of valence atomic orbitals, and the effects of delocalization are then incorporated into the molecnlar orbital as the number of repeat units in the crystal lattice is increased to infinity. This process is widely known to the chemical conununity as extended Hiickel theory (see Extended Hiickel Molecular Orbital Theory). It is also called tight binding theory by physicists who apply these methods to calcnlate the band structures of semiconducting and metallic solids. 

The concept of the Mobius strip was explained earlier (see p. 55). The basis of the Zimmerman analysis is an extension of this idea. A cyclic polyene is defined as a Hiickel system if its basis molecular orbital (i.e. the lowest filled TT-level as in the case of benzene, for example) contains zero or an even number of phase dislocations. Mbbius systems possess an odd number of phase dislocations in the basis molecular orbitals. In accordance with the rules predicting aromaticity for these systems, which results from the application of the Hiickel molecular orbital theory, it may be inferred that since cyclic conjugation also arises in the transition states of pericyclic reactions, the foDowing conclusions apply ... 

One of molecular- orbital theories early successes came in 1931 when Erich Hiickel discovered an interesting pattern in the tt orbital energy levels of benzene, cyclobutadiene, and cyclooctatetraene. By limiting his analysis to monocyclic conjugated polyenes and restricting the structures to planar- geometries, Hiickel found that whether a hydrocar bon of this type was aromatic depended on its number of tt electrons. He set forth what we now call Hiickel s rule ... 

In the 1930 s HiickeP proposed, on the basis of molecular-orbital calculations, a theoretical criterion for aromaticity of cyclic polyenes, known as Hiickers rule, which states that cyclic polyenes should be aromatic if, and only if, they contain 4n- -2 Jt-electrons. At that time only two of such cyclic polyenes were known benzene and cyclo-pentadienyl anion, each having six rc-electrons and satisfying Huckel s rule. Since then, the validity of Hiickel s rule had not been challenged... 

We have outlined above the procedure for the construction of orthonormal molecular-orbital and atomic-orbital Gel fand states and for the conversion of the latter to the non-orthogonal valence bond states. We require, in addition, a freeon Hamiltonian to compute the spectra of the several polyene systems. For this we employ the freeon, reduced Hiickel-Hubbard Hamiltonian which has the following form ... 

These ideas were introduced by Hiickel, who used a molecular orbital approach to calculate the energy of cyclic polyenes. According to this theory, the molecular yr-orbitals of a regular m-sided polygon of carbon atoms may be classified by a quantum number j which may take any of the values = 0,... 

The electrical properties of an infinite polyene have intrigued theoretical chemists for more than fifty years. Shortly after Hiickel introduced his 7i-electron theory for unsaturated systems in 1931, speculations began to appear about the carbon-carbon bond lengths to be expected in an infinite polyene. The question was interesting because, if the bonds were of equal length, 7i-electron theory predicted that the 7i-molecular orbitals would form a continuous band which would be half-filled, whereas alternating carbon-... 

These especially stable molecular orbital systems are found in planar, cyclic, fully conjugated polyenes that contain 4 -I- 2 7t electrons (Hiickel s rule). 

The effects of different types of electron distribution are of much interest. The most effective bridges, i.e., those with the smallest values of f) (0.14—0.56 A ), are those in which the bridging group contains mainly tt-orbitals. Correspondingly, theoretical molecular-orbital treatments by extended Hiickel methods indicate that aromatic or polyene bridges should be better conductors than purely aliphatic ones (cf. Section 9.2.2.5 below) [27], Peptide bridges, made up of a combination of saturated and unsaturated units, provide a fairly good medium (P 0.7 A ) and likewise proteins = 0.7—1.0 A ). A weak dependence of p on the ionization potential of D has been noted [28]. All this evidence links conduction properties with other indications of electronic mobility. 

The FEMO theory Problem 10.32) of conjugated molecules is rather crude and better results are obtained with simple Hiickel theory, (a) For a linear conjugated polyene with each of Nq carbon atoms contributing an electron in a 2p orbital, the energies of the resulting % molecular orbitals are given by... 

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