Ethylene, Huckel approximation

Naturally, we expect a relation between the bond order and the bond length for carbon-carbon bonds. The simplest assumption is a linear relation = a + bp where a and b are constants. The rr-electron MO coefficients for ethylene and for benzene are determined completely by symmetry and are independent of the Huckel approximations. We therefore use the bond orders and bond lengths of ethylene (2 1.335 A) and benzene (5/3 1.397 A) to find a and b we get... 

There will be no change in the Huckel approximation of ethylene because the hydrogen atom (any isotope) does not contribute to the TT orbitals. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

We now turn to the localized description of polyenes. As the zeroth-order approximation, we assume that the polyene consists of n noninteracting double bonds or ethylenic units. To obtain the energy corresponding to this description we note that the Huckel matrix for ethylene is... 

The simplest approximation to the delocalization energy is on the level of the Huckel method. For benzene the energy of the delocalized n electron system can be described as the sum of the orbital energies of the three doubly occupied orbitals. For the localized system, the n electron energy of three ethylene molecules is chosen. 

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 Huckel (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. 

In the case of ethylene the a framework is formed by the carbon sp -orbitals and the rr-bond is formed by the sideways overlap of the remaining two p-orbitals. The two 7r-orbitals have the same symmetry as the ir 2p and 7T 2p orbitals of a homonuclear diatomic molecule (Fig. 1.6), and the sequence of energy levels of these two orbitals is the same (Fig. 1.7). We need to know how such information may be deduced for ethylene and larger conjugated hydrocarbons. In most cases the information required does not provide a searching test of a molecular orbital approximation. Indeed for 7r-orbitals the information can usually be provided by the simple Huckel (1931) molecular orbital method (HMO) which uses the linear combination of atomic orbitals (LCAO), or even by the free electron model (FEM). These methods and the results they give are outlined in the remainder of this chapter. 

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