Hiickel model/theory

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. 

Effects other than angular distortions can lead to bond fixation. A simple model based on Hiickel aromatic theory accounts for a large number of such cases (e.g., starphenylene and triphenylene). Ideas that electronegative groups in the annelation are responsible for the bond fixation are shown to be inconsistent when tested against a significant sample of data. 

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. 

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. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Most experimental chemists in Germany in those times did not believe that quantum theory beyond the simple Hiickel model had any use for chemistry in the foreseeable future." " So quantum chemistry had to get its main support from the more optimistic international community. Per-Olov Low-din s summer schools held in Uppsala (Sweden) were instrumental in getting young researchers interested in quantum molecular science. Since these schools... 

In dilute electrolyte solutions ion-ion interaction as function of electrolyte concentration is fully explained by the Debye-Hiickel-Onsager theory and its further development. The contribution of ion solvation is noticed, if, for instance, the mobilities at infinite dilution of an ion in different solvent media or as function of ionic radii as considered. Up till now the calculation of that dependence has been only rather approximateAn improvement is quite probable, though, theoretically very involved if the solvent is not regarded as a continuum, but the number and arrangement of solvent molecules in the primary solvation shell of an ion is taken into consideration. Also the lifetime of molecules in the solvation shell must be known. Beyond this region a continuum model of ion-solvent interaction may be sufficient to account for the contributions of solvent molecules in the second or third sphere. 

An analysis of reaction possibilities in the Hiickel-Mobius theory requires us to construct a fully interacting basis set, which is simply a sketch of the transition state, drawn with the maximum possible bonding character (the fewest nodes possible). Next, draw a line to connect the reactant orbitals as they would be connected by conrotatory and disrotatory processes. This gives a complete model of the transition state for butadiene as shown in Fig. 8.49. 

The linearization that leads here to the Debye-Hiickel model is physically consistent in this argument. But the possibility of a model that is unlinearized in this sense is a popular query. More than one response has been offered including the (nonlinear) Poisson-Boltzmann theory and the EXP approximation see (Stell, 1977) also for representative numerical results for the systems discussed here. 

The major feature is a rapid decrease of A at low salt concentrations, followed by a minimum and pronounced increase. At the CP there is a substantial conductance. To interpret this behavior, we first note that the Debye- Hiickel (DH) theory itself predicts an instability regime at low T, but if compared with experiment C is far too low. Taking account for ion association considerably improves thew results. In the presence of ion association, a higher salt concentration is needed to achieve the concentration of free ions to drive phase separation, i.e. C is shifted to higher values. In particular, the Bjerrum model for ion pair association yields ... 

In the present chapter, the properties of electrolyte solutions in water are discussed in detail. Initially the solvation of ions in infinitely dilute solutions is considered on the basis of the Born theory. Then, the Debye-Hiickel model for... 

In conclusion, the MSA provides an excellent description of the properties of electrolyte solutions up to quite high concentrations. In dilute solutions, the most important feature of these systems is the influence of ion-ion interactions, which account for almost all of the departure from ideality. In this concentration region, the MSA theory does not differ significantly from the Debye-Hiickel model. As the ionic strength increases beyond 0.1 M, the finite size of all of the ions must be considered. This is done in the MSA on the basis of the hard-sphere contribution. Further improvement in the model comes from considering the presence of ion pairing and by using the actual dielectric permittivity of the solution rather than that of the pure solvent. 

Osmotic term (Ilion) neglecting of charge-charge interaction as well as counterion condensation Debye-Hiickel model (Hasa-Ilavsky-Dusek theory) Dusek et al. [44]... 

Historically the extended Hiickel model (EHT, Extended Huckel Theory, as it was originally called) is one of the most important schemes developed. Even in its simplest form, orbitals with energies approximating ionization potentials and of the proper nodal structure and symmetries are obtained. The famous Woodward-Hoffmann rules were founded on these simple calculations, and frontier molecular orbital arguments are easily based on EHT orbitals. 

In the simple Debye-Hiickel model, attention is focused on the coulombic interactions as the source of non-ideality. The aim of the Debye-Hiickel theory is to calculate the mean activity coefficient for an electrolyte in terms of these electrostatic interactions between the... 

These points are the main features of the simple Debye-Hiickel model. Other aspects of the Debye-Hiickel theory are illustrated by the mode of approach considered shortly. 

If all sorts of possible combinations of the features suggested above are to be incorporated in the calculations and each result is compared with the others and with experiment, a much deeper and more accurate picture of what is happening at the microscopic level emerges. Consequently, a better theory of electrolyte solutions has been forthcoming. Much highly promising work has been done and modifications both to the Debye-Hiickel model and its... 

Non-ideality has been shown to be due to ionic interactions between the ions and consideration of these led to the concept of the ionic atmosphere (see Sections 10.3 and 10.5). These interactions must be taken into account in any theory of conductance. Most of the theories of electrolyte conduction use the Debye-Hiickel model, but this model has to be modified to take into account extra features resulting from the movement of the ions in the solvent under the applied field. This has proved to be a very difficult task and most of the modern work has attempted many refinements all of which are mathematically very complex. Most of this work has focused on two effects which the existence of the ionic atmosphere imposes on the movement and velocity of the ions in an electrolyte solution. These are the relaxation and electrophoretic effects. 

It is quite obvious that further modification to this conductance theory to take account of the shortcomings of the Debye-Hiickel model as outiined would be even less fintitful than similar attempts on the Debye-Hiickel theory itself. 

Deviations from predicted behaviour are here interpreted in terms of solvation, but other factors such as ion association may also be involved. Ion association leads to deviations in the opposite direction and so compensating effects of solvation and ion association may come into play. The deviations may also be absorbing inadequacies of the Debye-Hiickel model and theory, and so no great reliance can be placed on the actual numerical value of the values emerging. This major method has now been superseded by X-ray diffraction, neutron diffraction, NMR and computer simulation methods. The importance of activity measurements may lie more in the way in which they can point to fundamental difficulties in the theoretical studies on activity coefficients and conductance. The estimates of ion size and hydration studies could well provide a basis for another interpretation of conductance and activity data, or to modify the theoretical equations for mean activity coefficients and molar conductivities. 

Hydration theory was first applied to aqueous electrolytes by Stokes and Robinson (1) in 1948. The approach is to correct for the fact that the actual solute species do not exist as bare ions as they are normally written, but as hydrated aqueous complexes. In the case of a solution of an aqueous nonelectrolyte, this reduces to the assumption that the solution is ideal, or nearly so, if one writes the formula of the solute to include the waters of hydration and computes its concentration and that of the solvent on this basis. In the case of an aqueous electrolyte, it reduces to the assumption that the Debye-Hiickel model becomes adequate (or at least more adequate) to describe the activity coefficients if this treatment is applied. 

It appears worthwhile to attempt to overcome these deficiencies, as the hydration concept has a powerful physical appeal. This does not mean that one must hold that the Debye-Hiickel model is really adequate if one simply corrects for hydration. It has a number of deficiencies, as has been pointed out for example by Pitzer (, 9). Like Pitzer s equations, the kind of model that we are addressing here incorporates some degree of theory as a guide but in essence remains phenomenological. We only expect that the hydration correction will turn out to be a useful part of such a model. 

All carbon-carbon bonds in the skeleton have 50% double bond character. This fact was later confirmed by X-ray diffraction studies. A simple free-electron model calculation shows that there is no energy gap between the valence and conduction bands and that the limit of the first UV-visible transition for an infinite chain is zero. Thus a simple free-electron model correctly reproduces the first UV transition with a metallic extrapolation for the infinite system. Conversely, in the polyene series, CH2=CH-(CH=CH) -CH=CH2, he had to disturb the constant potential using a sinusoidal potential in order to cover the experimental trends. The role of the sinusoidal potential is to take into account the structural bond alternation between bond lengths of single- and double-bond character. When applied to the infinite system, in this type of disturbed free-electron model or Hiickel-type theory, a non-zero energy gap is obtained (about 1.90 eV in Kuhn s calculation), as illustrated in Fig. 36.9. 

Besides these qualitative differences, there also exist quantitative discrepancies between the Hiickel model for polyenes and the experimental observations. The Hiickel theory predicts an order of magnitude larger oscillator strength in the absorption to the lowest dipole allowed state [4]. The bond length alternation required to fit the optical gap in polyenes within a Hiickel model is twice the experimentally observed bond alternation. Thus, the Hiickel model is mainly of pedagogical interest and one needs to go beyond it for dealing accurately with realistic conjugated systems. 

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