Hiickel model

In 1930, the German theoretical chemist Erich Hiickel proposed a quantum mechanical model that can be applied to the aromatic % electrons. Aromatic systems are also referred to as x-systems. % electrons in a system of conjugated double bonds interact in a very special way. The Hiickel model is a very simple model. Nevertheless, it is behind a lot of photophysics and ET. The results have proven to be surprisingly accurate. The model is perfect for understanding, but nowadays quantum chemical calculations provide more accurate results, particularly for spectra. 

FIGURE 3.5 Carbon skeleton numbering in ethylene (ethene), hexatriene, and benzene. 

A matrix eigenvalue problem is derived in the ordinary way (H - e)c = 0, where H is the Hamiltonian matrix and c is the eigenvector. The following simplifications are done in the Hamilton matrix  

Hjiv = J Xn HXvdv = p 0, if 0, and v are neighboring carbon atoms Hjiv = if carbon atoms 0, and v are not neighbors 

As an example, we may consider ethylene (or ethene, C2H4) and benzene (CgHg) with the following (H - e) matrices  

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The neglect of electron-electron interactions in the Extended Hiickel model has several consequences. For example, the atomic orbital binding energies are fixed and do not depend on charge density. With the more accurate NDO semi-empirical treatments, these energies are appropriately sensitive to the surrounding molecular environment. 

A Hiickel model used for calculating aromaticity indicated that the isoxazole nucleus is considerably less aromatic than other five-membered heterocycles, including oxazole and furan. SCF calculations predicted that isoxazole is similar to oxazole. Experimental findings are somewhat difficult to correlate with calculations (79AHC(25)147). PRDDO calculations were compared with ab initio values and good agreement for the MO values was reported... 

We will now discuss the electronic structure of single-shell carbon nanotubes in a progression of more sophisticated models. We shall begin with perhaps the simplest model for the electronic structure of the nanotubes a Hiickel model for a single graphite sheet with periodic boundary conditions analogous to those im-... 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

Hiickel models of molecular electronic structure enjoyed many years of popularity, particularly the r-electron variants. Authors sought to extract the last possible amount of information from these models, perhaps because nothing more refined was technically feasible at the time. Thus, for example, the inductive effect was studied. The inductive effect is a key concept in organic chemistry a group R should show a - -1 or a —I effect (according to the nature of the group R) when it is substituted into a benzene ring. 

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. 

Electrocyclic reactions are examples of cases where n-electron bonds transform to sigma ones [32,49,55]. A prototype is the cyclization of butadiene to cyclobutene (Fig. 8, lower panel). In this four electron system, phase inversion occurs if no new nodes are formed along the reaction coordinate. Therefore, when the ring closure is disrotatory, the system is Hiickel type, and the reaction a phase-inverting one. If, however, the motion is conrotatory, a new node is formed along the reaction coordinate just as in the HC1 + H system. The reaction is now Mobius type, and phase preserving. This result, which is in line with the Woodward-Hoffmann rules and with Zimmerman s Mobius-Hiickel model [20], was obtained without consideration of nuclear symmetry. This conclusion was previously reached by Goddard [22,39]. 

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. 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Bearing in mind these limitations on the Debye—Hiickel model of electrolytes, the influence of ionic concentration on the rate coefficient for reaction of ions was solved numerically by Logan [54, 93] who evaluated the integral of eqn. (56) with the potential of eqn. (55). He compared these numerical values with the predictions of the Bronsted— Bjerrum correction to the rate of a reaction occurring between ions surrounded by equilibrated ionic atmospheres, where the reaction of encounter pairs is rate-limiting... 

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... 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

Figure 2 shows y for a few electrolytes as a function of m, as extracted from Robinson and Stokes (2002), as well as their calculated values using the Debye-Hiickel model integrated with a linear function of m to extend its accuracy beyond the dilute region ... 

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... 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

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