Benzene Hiickel approximation

For example, the Hiickel approximation to the secular equation for benzene is ... 

As an illustration let us summarize the indirect % bonds between carbon atoms in benzene and butadiene [58] in the Hiickel approximation. For the consecutive numbering of carbons in the ring and chain, the relevant CBO matrix elements in benzene read = 1, Yw+i = 2/3, Y ,i+2 = 0, Yi.i+3 = -1/3, while the relevant off-diagonal part of the CBO matrix in butadiene is fuUy characterized by the elements... 

The condition (17.38) determining the tt-MO coefficients for benzene was derived solely from symmetry considerations, without use of the Hiickel approximations. Thus the MOs (17.41) are (except for normalization constants) the correct minimal-basis-set SCF TT-electron MOs for benzene. (The Hiickel energies ei,..., e are, however, not the true SCF orbital energies. The Hiickel method ignores electron repulsions and takes the total TT-electron energy as the sum of orbital energies. The SCF MO method takes electron... 

The Hiickel approximation is especially useful in understanding the chemical stability of benzene, and by extension other aromatic compounds. Recall that benzene (Figure 15.21) is more stable than expected for a cyclohexatriene, and its chemistry is representative of an entire class of aromatic hydrocarbons as opposed to the nonaromatic aliphatic hydrocarbons. The Hiickel approximation provides some clues for benzene s distinctions. 

It was seen above that the Hiickel approximation, applied to benzene, led to the energies of the molecular orbit s... 

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. 

In a molecule with electrons in n orbitals, such as formaldehyde, ethylene, buta-1,3-diene and benzene, if we are concerned only with the ground state, or excited states obtained by electron promotion within 7i-type MOs, an approximate MO method due to Hiickel may be useM. 

Comparison with (3.156) shows that F(NBO) is intrinsically of significantly higher accuracy than h(HMO) for describing the actual pi interactions of benzene. Because F(nbo) js tjje fundamental starting point for localized NBO analysis of conjugafive interactions, we can conclude that the NBO donor-acceptor picture is inherently more accurate than that based on the Hiickel tight-binding approximation. 

Although the Hiickel An+ 2 rule is rigorously derived for monocyclic systems, it is also applied in an approximate way to fused-ring compounds. Since two fused rings must share a pair of ir electrons, the aromaticity and the delocalization energy per ring is less than that of benzene itself. Decreased aromaticity of polynuclear aromatics is also revealed by the different C—C bond lengths. 

When the chain is closed. Using the Hiickel method and the jt-electron approximation, let us consider now the most typical aromatic compound, benzene. Repeating the arguments and the calculations presented above, we obtain the secular equation for benzene in the form... 

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. 

In 1927, Heitler and London used valence bond theory to treat the H2 molecule but to treat larger molecules, further simplifications were needed. In 1931, Erich Hiickel introduced an extremely simple approximation which could be used to treat the 7i-electrons in flat organic molecules such as benzene, napthaline, and so on. This approximation yielded matrices to be diagonalized, and it is a measure of the state of computers at that time to remember that during World War II, Alberte Pullman sat in a basement room in Paris diagonalizing Hiickel matrices with a mechanical desk calculator, while her husband-to-be Bernard drove a tank with the Free French Forces in North Africa. Alberte s hand-work led to the publication of the Pullmans early book Quantum Biochemistry. ... 

If one considers only the conjugated Jt system of benzene, it contains 6 electrons. If one considers the entire benzene molecule, it contains 42 electrons. The 6-electron problem could be dealt with by hand calculation to some reasonable extent, whereas the 42-electron problem was hopelessly complicated. In the Hiickel method the interactions of each of the electrons with each of the nuclei are accounted for, but the interactions of the electrons with one another are simply ignored. The results are then empirically adjusted in a systematic way so as to approximately allow for this. Although the method is based on a really severe approximation, it nonetheless yielded a great many interesting and suggestive results regarding molecular structure. ... 

---