Hiickel treatment

In the case of benzene, Hiickel treatment of the six 2p orbitals on the carbon atoms and perpendicular to the plane of the ring leads to the secular determinant... 

In the Hiickel treatment, atomic orbitals on nonadjacent atoms are assumed to have no interacticai. They are neither bonding nor antibonding. The concept of homoconjugation suggests that such orbitals may interact, especially in rigid structures which direct orbitals toward one another. Consider, for example, bicyclo[2.2.1]hepta-2,S-diene ... 

More rigorous Debye-Hiickel treatment of the activity coefficient... 

All this suggests a further simplification, which has proved to be eminently successful in many cases. It is known that independent electron treatments, such as the Hiickel (HMO) treatment2 or the extended Hiickel treatment (EHT)172, which do not take the electron-electron interaction explicitly into account, yield—by and large—orbitals derived from sophisticated SCF calculations. In particular, the HMO and ETH molecular orbitals reflect faithfully the symmetry and nodal properties of their counterparts obtained from SCF treatments. 

Hiickel and extended Hiickel treatments that consider many fundamental ideas of orbitals and bonding but neglect electron repulsions in molecules. Treats systems up to 200 atoms. 

Anderson modification of extended Hiickel treatment, 34 136 -MO, 37 151-153 Anderson structure heteropolyacids, 41 123 Andrews-Beams turbine spinner, 33 209 Aniline... 

ASED, see Anderson modification of extended Hiickel treatment Ash, 27 317... 

Effective medium theory, 37 154 Eggshell catalysts, 39 231 EH method, 37 153 EHT, see Extended Hiickel treatment Eigenberger model, oscillatory reactions, 39 80-81, 83... 

Exothermic reaction parameters of, 27 63 temperature oscillations, 27 65-67 Explosion, petovskite preparation, 36 250 Extended Hiickel treatment, 34 136, 147, 154, 156, 166, 173... 

This equation contains the activity coefficients 71, 72, and 7. Recall from the Debye-Hiickel treatment of ionic interactions in dilute solutions that the magnitude of these coefficients shows the following dependence on ionic strength fi for a solution of electrolytes ... 

The most extensive correlation of ESR results for trimethylsilyl-substituted aromatic radical anions with Hiickel MO calculations to date is the work of Sipe and West (118, 119). The optimum heteroatom parameters for both the simple Hiickel treatment and the Hiickel-McLachlan treatment, using a Qm = 28 G, for Me3Si were as follows Hiickel, hsi = -2.00, kcS1 = 0.70 Hiickel-McLachlan, hsi = -1.80, kcsi = 0.70, X = 0.30 The values for Me3Ge were as follows Hiickel, hoe = -1.45, kCGe = 0.40 Hiickel-Me Lachlan, = -1.40, kCGe = 0.40 X =... 

The same theoretical methods have been applied to ethylidyne absorbed in the Rh(100) surface, with the conclusion, as experimentally found, that this species will be stable on the 4-fold site (237). However, the extended-Hiickel treatment, as well as the atom-superposition and electron-delocalization molecular orbital (ASED-MO) method, predict that the CC bond length will be shorter, and hence the vCC mode higher in wavenumber on the 4-fold site, whereas experimentally... 

In his classic paper in the Journal of Chemical Education entitled Why do some molecules have symmetry different from that expected [9], Heilbronner pointed out that the notion of 77-distortivity is already implicit in the Hiickel treatment of benzene. 

Consider a reaction between two molecules A and B. For simplicity, we assume that each molecule has only one MO 0FA° of energy EA° and T," of energy EB°, respectively). During the reaction, the reagents evolve to produce the supermolecule (A B). As we saw in the previous section, the MOs of (A B) can be calculated by a perturbation approach which is entirely analogous to the Hiickel treatment of a diatomic molecule. In fact, we only need to take the MOs of the diatomic and replace ... 

Calculated reactivity indices are at one with qualitative electronic theory in accounting for ready nucleophilic attack at C-6 in the phenanthridine molecule. However, the limited data available on positional reactivities in electrophilic substitutions is not accounted for satisfactorily by any of the available treatments (see later) and it has been pointed out that the simple Hiickel treatment used by some authors,223, 220 is generally inapplicable in heteroaromatic systems.228... 

The methods for ji-electrons have been also applied to u-orbitals. Sandorfy and Daudel (1954), and later Sandorfy (1955), have applied the Hiickel treatment to saturated hydrocarbons, using sp3 hybrid orbitals for the carbon atoms and Is orbitals for the hydrogen atoms. These methods, improved by Yoshizumi (1957), have been used by Fukui, Kato, and Yonezawa (1960—1961) and Klopman (1962, 1963). More complete treatments have been developed by Hoffman (1963) and Pople and Santry (1963). 

It is true that in some few cases appreciable differences can be detected in both the total -electron charges and the bond orders, reflecting the importance of the choice of parameters. In general it can be observed, however, that most of the values collected are strikingly similar the various methods of integral evaluation do not seem to influence decisively the results. It may be concluded that a Hiickel treatment, carried out carefully, may yield results as satisfactory as a more complicated calculation. On the other hand one could just as well arrive at the conclusion that all these treatments are equally meaningless. 

The conclusion to be drawn from these considerations is simple enough. If a qualitative description of the reactivity of the different centers in the bases is the purpose of the calculations, a simple Hiickel treatment is sufficient. A quantitative discussion of the problems of reactivity and specificity requires, however, more accurate calculations not hindered by the above approximations. 

We now turn to a different type of photochemistry and different mechanistic questions, namely photochemical pericyclic reactions and the utility of the Mobius-Hiickel treatment of these transformations. 

This equation is precisely equal to that in the Debye-Hiickel treatment of ionic interaction for dilute electrolyte solutions14, only that the distance x refers to a central ion (point charge) and not to an electrode. In the Debye-Hiickel case, since the central ion is small and 0A small we can make the approximation (elinear approximation is not valid. 

Semiempirical quantum-chemical methods can be subdivided into two groups. The first one covers so-called simple MO LCAO methods, of which extended Hiickel treatment (EHT) (7) and its modification by Anderson (ASED) (5), additionally taking into account core repulsion, are most widely used in chemisorption computations. Anderson s improvement of EHT was aimed at obtaining more reliable values of the total energy and at gaining an opportunity to optimize the geometry of chemisorption structures. The method was mainly used in calculations of chemisorption and catalysis on metals. Its validity for oxide systems, with their rather highly ionic bonds, is formally less justified. 

The coefficients of the atomic orbitals in the retype molecular orbitals of the extended Hiickel treatment are reasonably close to those of both the Hiickel and Pople-SCF 7r-electron calculations on NF2, cis- and ran -N2F2. The deviations tend to increase with increasing orbital energy. 

In this research we have found that for general descriptions of bonding in N, F compounds an extended Hiickel treatment leads to results consistent with the properties and behavior of known N, F compounds. Furthermore, certain other, as yet uninterpreted properties of these systems can be gleaned from the calculational results. 

However, for higher concentrations the model is no longer valid, and further, the approximations, i ZlUkT < 1, cannot be valid close to the ion i [Eq. (XV.7.2)]. Bjerrum made the proposal that any pair of ions whose interaction is of the order of 2kT or more should be considered as an ion pair, not as independent ions, and that the Debye-Hiickel treatment should be reserved only for the free ions separated by distances sufficiently large that their interaction is less than this. If we call this distance tb and neglect the ion atmosphere around such an ion pair, then for the ion pair and... 

This does not mean that the Debye-Htickel theory gives the right answer when there is ion-pair formation. The extent of ion-pair formation decides the value of the concentration to be used in the ionic-cloud model. By removing a fraction 0 of the total number of ions, only a fraction 1 - 0 of the ions remain for the Debye-Hiickel treatment, which interests itself only in the free charges. Thus, the Debye-Htickel expression for the activity coefficient [Eq. (3.120)] is valid for the free ions, with two important modifications (1) Instead of there being a concentration c of ions, there is only (1 - 0)c the remainder Oc is not reckoned with owing to association. (2) The distance of closest approach of free ions is q and not a. These modifications yield... 

Attention should be drawn to the fact that the activity coefficients given by the Debye-Hiickel treatment are the so-called rational coefficients (p. 135) to express the values in the form of the practical activity coefficients, it is necessary to make use of equation (26). If the solvent... 

The Osmotic Coefficient.—Instead of calculating activity coefficients from freezing-point and other so-called osmotic measurements, the data may be used directly to test the validity of the Debye-Hiickel treatment. If 6 is the depression of the freezing point of a solution of molality m of an electrolyte which dissociates into v ions, and X is the molal freezing-point depression, viz., 1.858° for water, a quantity , called the osmotic coefficient, may be defined by the expression... 

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