Hiickel approximations

Most correlations of rates with localisation energies have used values for the latter derived from the Hiickel approximation. More advanced methods of m.o. theory can, of course, be used, and fig. 7.1 illustrates plots correlating data for the nitration of polynuclear hydrocarbons in acetic anhydride -" with localisation energies derived from self-... 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

In the Extended Hiickel approximation, the charges in the unselected part are treated like classical point charges. The correction of these classical charges to the diagonal elements of the Hamiltonian matrix may be written as ... 

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

A simple qualitative model of the three-electron hemibond in [X.. X], based on the Hiickel approximation, has been proposed by Gill and Radom [122]. This qualitative model predicts that the strength of the hemibond should vary in proportion to the Hiickel parameter a, which can be replaced by the HOMO energy in X because a good correlation is found between Eho-Mo(X) and De(X-X ). This model readily rationahzes the marked substituent effect on the strength of the hemibond. In particular, electron-withdrawing substituents are found to have a strengthening effect. 

If the full molecular symmetry is assumed, the ground states of the cation radical of fulvalene and the anion radical of heptafulvalene are both predicted to be of symmetry by using the semiempirical open-shell SCF MO method The lowest excited states of both radicals are of symmetry and are predicted to be very close to the ground states in the framework of the Hiickel approximation these states are degenerate in both cases (Fig. 4). Therefore, it is expected that in both these radicals the ground state interacts strongly with the lowest excited state through the nuclear deformation of symmetry ( — with the result that the initially assumed molecular... 

On the other hand, in the anion radical of fulvalene and the cation radical of heptafulvalene, the energy gaps between the ground and lowest excited state (which is in both cases doubly degenerate in the Hiickel approximation (Fig. 4)) are predicted to be reasonably large (1.4 and 1.7 eV, respectively), so that these radicals would not suffer a symmetry reduction. 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

The Hiickel approximation (3.136) is equivalent to neglect of fi and p" (the tight-binding approximation), leading to the simpler Hiickel-type matrix h(HMO) ... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

The easiest way to calculate the terms Hj and Sij is within the simple Hiickel approximation, where it is assumed that ... 

For example, the Hiickel approximation to the secular equation for benzene is ... 

Problem 11-13. Write down and solve the Hiickel approximation to the secular equation for the vr system of ethylene. Noting that the energy of two electrons on non-interacting carbon atoms is 2o , show that the binding energy of the two vr electrons is —2/3. 

Problem 11-15. Verify that the molecular orbitals are orthogonal, within the framework of the Hiickel approximation. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

Here aH is the hyperfine splitting, QH is the value of aH for CH3 (—23 G) and p is the spin density. McLachlan (60MI20400) has used the Hiickel approximation (which gives just as reliable results as SCF theory in this case) to calculate the electron spin distribution in 7r-electron radicals, and this method has frequently been used by subsequent workers, as we shall see. 

We introduce the first of the Debye-Hiickel approximations by considering only those situations for which < kBT). In this case the exponentials in Equation (28) may be expanded (see Appendix A) as a power series. If only first-order terms in z,eyp/kBT) are... 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

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