The Hiickel Model

One of the earliest models for treating conjugated molecules is afforded by the Hiickel rr-electron model. This dates from the work of E. Hiickel in 1931. The ideas are simple and appealing, and the model enjoyed many years of successful application to individual molecules, molecular clusters and solids. 

There are 16 electrons in total, 14 cr and 2 n and so our total wavefunction will be some complicated function of their spatial (ri, rj. rie) and spin variables (si, S2. sie). The electronic state wavefimction for the molecule can therefore be written 

Don t confuse the state wavefunction with a molecular orbital we might well want to build the state wavefunction, which describes all the 16 electrons, from molecular orbitals each of which describe a single electron. But the two are not the same. We would have to find some suitable one-electron wavefunctions and then combine them into a slater determinant in order to take account of the Pauli principle. 

Where might these one-electron wavefunctions come from I explained the basic ideas of HF and HF-LCAO theory in Chapter 6 we could find the molecular orbitals as linear combinations of appropriate atomic orbitals by solving the HF eigenvalue problem 

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 and write 

The most drastic approximation is to fix the positions of the nuclei and to neglect the electron-electron interactions. Noninteracting electrons with fixed nuclei geometry are described by the Hiickel model (Hiickel 1931, 1932), defined as 

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A great failing of the Hiickel models is their treatment of electron repulsion. Electron repulsion is not treated explicitly it is somehow averaged within the spirit of Hartree-Fock theory. 1 gave you a Hiickel jr-electron treatment of pyridine in Chapter 7. Orbital energies are shown in Table 8.1. 

Polarography and ESR data provide important information about the energies and electron distribution of the excited states of annelated benzenes. " By incorporating rehybridization effects into the Hiickel model of electron densities, a correlation between ring strain, experimental spin densities, and redox potentials is obtained for a series of naphthalenes and naphthoquinones. These studies provide further support for ring-strain induced rehybridization. 

The formal similarity with the Hiickel model is obvious The expressions for the n double bond polyene in the Simpson model are entirely equivalent to the expressions for the n ir-electron system in the Hiickel model. 

Using the empirical parameters given below for B and H (taken from Appendix F and "The HMO Model and its Applications" by E. Heilbronner and H. Bock, Wiley-Interscience, NY, 1976), apply the Hiickel model to borane (BH3) in order to determine the valence electronic structure of this system. 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Hiickel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be intuitive. One such model that continues to see extensive use today is the so-called extended Hiickel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

Many of the above deficiencies were removed in further refinements of the Hiickel method, which was anyway made obsolete by the development of ab initio and DFT techniques. Still, organic chemists adhere to the original Hiickel description, which is often sufficient to make qualitative predictions about the nature of n-conjugated systems. In particular, the Hiickel model finds widespread use in porphyrinoid chemistry. The so-called annulene model, which will be used throughout this review, is outlined below for the parent porphyrin macrocycle. 

Theoreticians call any non-hydrogen atom a heavy atom, and any heavy atom other than carbon a heteroatom. In the Hiickel model, all carbon atoms are assumed to be the same. Consequently, their Coulomb and resonance integrals never change from a and If respectively. However, heteroatom X and carbon have different electronegativities, so we have to set ccx = a. Equally, the C-X and C-C bond strengths are different, so that Pcx X p. Thus, for heteroatoms, we employ the modified parameters... 

The two n electrons in ethylene occupy the f(n) orbital. Hence the energy for a n bond in the Hiickel model is... 

The first term is essentially the total 7r-electron bond order for bonded sites, since v+ is the Hiickel model (1) for a ring with e = 0 and t = -1 between neighbors. 

The generalization of the Hiickel model (1) to a donor-acceptor stack is through a site energy, ep = (—1)PA, that lowers the energy of electrons on odd-numbered D sites for A > 0. In principle we have different on-site U in (3) for donors and acceptors, but a single U is usually assumed. The Coulomb terms in (4) now have zp = 2 and 0 at odd and even sites, respectively, because neutral D and A require two and no electrons. The complete basis for a stack is identical to the PPP basis... 

After the discovery of the metallic behaviour of PAc (Shirakawa et al., 1977) most of the work on theoretical models concentrated on this polymer. This built on the earlier studies of Longuet-Higgins and Salem (1959) and others who used the Hiickel model to demonstrate that the backbone of PAc in its ground state has a bond alternated structure, Fig. 9.8(b), rather than one with equal length C-C bonds, Fig. 9.8(a). In 1979 a Hiickel tight binding model was introduced that provided the basis for much of the subsequent discussion of themolecular and electronic structure of PAc (Su, Schrieffer and Heeger, 1979 and 1980). It is now usually referred to as the SSH model. In the adiabatic... 

In Chapter 3, the Hiickel model of linear and closed polyene chains is used to explain the origin of band structure in the one-dimensional crystal, outlining the importance of the nature of the electronic bands in determining the different properties of insulators, conductors, semiconductors and superconductors. 

In the Hiickel model, the result of minimizing the energy of the appropriately occupied n molecular orbitals results in the following set of linear equations in the unknown atomic orbital coefficients, tv, together with the molecular orbital energy, e ... 

The magnitude of the nonlinearity in this case is determined by the absorption coefficient of the semiconductor, a0, as well as the electron effective mass, me. The effective mass is determined by the transfer (or resonance) integral between atoms or molecules in the framework of the Hiickel model [15]. Stronger interatomic interaction gives a larger transfer integral, smaller effective mass, and steeper curvature in the energy band... 

Coulson free valences (F) (see Table III) calculated from the Hiickel model indicate that, when there are phenyl groups substituted at positions 1 and 4 and methyl groups at positions 5 and 7, electrophilic substitution will take place at the ortho and para positions of the phenyl groups as well as at position 6 of the dihydrodiazepinium ring.34 Similar results are suggested for 5,7-phenyl-disubstituted dihydrodiazepinium salts, with substitution occurring preferentially at position 6. Bromination and nitration of the 5,7-dimethyl-l, 4-diphenyldihydrodiazepinium cation does indeed take place at the predicted sites. Bromination of 5,7-di-... 

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