Huckel approximations

We start with the oversimplified version of the regular Huckel approximation. This approach considers the complete Hamiltonian of a system of 2n electrons and m nuclear cores, which is given by... 

Heitler-London wave function, 15-16 Helium atom, wave function for, 3 Heterolytic bond cleavage, 46, 51, 47,53 Histidine, structure of, 110 Huckel approximation, 8,9,10,13 Hydrocarbons, force field parameters for, 112... 

The first paper of the frontier-electron theory pointed out that the electrophilic aromatic substitution in aromatic hydrocarbons should take place at the position of the greatest density of electrons in the highest occupied (HO) molecular orbital (MO). The second paper disclosed that the nucleophilic replacement should occur at the carbon atom where the lowest unoccupied (LU) MO exhibited the maximum density of extension. These particular MO s were called "frontier MO s . In homolytic replacements, both HO and LU.were shown to serve as the frontier MO s. In these papers the "partial" density of 2 pn electron, in the HO (or LU) MO, at a certain carbon atom was simply interpreted by the square of the atomic orbital (AO) coefficient in these particular MO s which were represented by a linear combination (LC) of 2 pn AO s in the frame of the Huckel approximation. These partial densities were named frontier-electron densities . 

Polydiacetylene crystals. The enhancement of x because of one-dimensional electron delocalization is strikingly corroborated in the polydiacetylene crystals. Their structure is that of a super alternated chain with four atoms per unit cell and the Huckel approximation yields four bands for the ir-electrons, two valence and two conduction bands. When depicted in the extended Jones zone, each pair can be viewed as arising by a discontinuity at the middle of the Brillouin zone of the polyene chain. The dominant contribution to X(2n 1) comes from the critical point at the edge of the extended Jones zone (initially at the center of the reduced B.Z.). The complete expressions are derived in (4,22) and calculated for different polydiacetylenes. We reproduce the values of x 2 for TCDU and PTS in table IV. The calculated values are in good agreement... 

Fig. 20. 2pir molecular orbital energies in the Huckel approximation for simple radicals (a) and olefins (b). 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

The Huckel approximation is defined by a set of simplicafions to the form of the Hamiltonian in the LCAO-MO description of planar conjugated molecules. Although the Huckel approximations are quite severe, nevertheless they produce results that rationafize qualitatively the resonance energies and spectra of these molecules. 

In the Huckel approximation for hydrocarbons, matrix elements of the Hamiltonian in a basis of p atomic orbitals (pi = apz orbital on atom i) are expressed in terms of two parameters a and / ... 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

What are the assumptions that are needed to obtain the linearized Poisson-Boltzmann (LPB) equation from the Poisson-Boltzmann equation, and under what conditions would you expect the LPB equation to be sufficiently accurate What is the relation between the Debye-Huckel approximation and the LPB equation ... 

Finally, Fei st describes electrostatic interactions and in Debye-Huckel approximation is given by [23, 26] ... 

On the other hand, we should perhaps not be too surprised if values of fi derived from different kinds of measurements do not agree exactly, and indeed this is the case. We have seen above that/ obtained from the vertical resonance energy of benzene is about 37 kcal/mol, whereas Platt first showed that the best overall fit to the spectra of benzene and other unsaturated hydrocarbons was obtained in the framework of the Huckel approximation by taking / to be 55-60 kcal/mol. This high value, which has subsequently been widely adopted to estimate actual differences in energies between MOs, is generally known as the spectroscopic value of / . 

Information on the details of various approximations besides the simple Huckel approximation for organic and heteroorganic molecules and on applications of these calculations will be found in ... 

Figure 6.4. Energy-level diagram for the molecular orbitals of benzene evaluated in the Huckel approximation.

Apart from this approximation the superdelocalizability depends on similar assumptions to the localization energy. There has been some vigorous argument8 over their relative merits. Fukui has shown87-88 that if A(A) is the secular determinant in the Huckel approximation and Arr(A) its rr minor, then the reactivity indices Ft, Jin, Lt, Sr, and Nr can all be expressed in terms of... 

Correct measured values for liquid junction potentials using the Henderson formalism and calculate ion activities according to the Debye-Huckel approximation. 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

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