Hiickel approximation, application

The Debye-Hiickel approximation is strictly applicable only in the case of low potentials. Nevertheless, there are several reasons why the significance of Equation (37) should be fully appreciated ... 

The surface potential on the particle is low enough that the Debye-Hiickel approximation is applicable. 

An alternative equation, the extended DeBye-Hiickel approximation of the DeBye-Hiickel limiting law, which is applicable for ionic strength of less than approximately 0.1, is... 

When O = (zef)lkT is sufficiently small to ensure the applicability of the Debye-Hiickel approximation to the electrolyte solution, eqn. 5.2.31 is capable of representing the experimental data very closely up to concentrations such that Ka < 0.2-0.5. ... 

The mathematical descriptions outlined so far for analyzing cases 1 and 2 implicitly assume the validity of the celebrated Debye-Hiickel linearization principle, as described earlier. However, for higher pH values (such as pH > 8), the surface potential may be such that the value of eij/lksT cannot be taken to be small at all locations. A limiting condition that constraints the applicability of the Debye-Hiickel approximation occurs for e j/lk T 1, which, for standard temperatures. 

At room tenperature k T je 30 mV. However, for silica substrates ( 50— 100 mV in typical applications. Thus, the Debye-Hiickel approximation is often not strictly valid. Nevertheless, it is a very useful approximation because it enormously simplifies mathematical investigations related to the Debye layer. 

As noted above, we find that this kinetics formulation is applicable to cases where the salt concentration is low, and deposition extensive, so that the "smeared charge approximation holds. Assuming that depositing DNA-SWCNTs form a homogenous layer on the surface implies that the surface charge will increase with deposited density Ps (in p.m of nanotubes per p.m of surface). As surface charge increases, surface potential relative to bulk solution will also increase. Within the linearized Debye-Hiickel approximation, the surface potential is related to surface charge as ... 

It is well known that Hund s rule is applicable to atoms, but hardly so to the exchange coupling between two singly occupied molecular orbitals (SOMOs) of a diradical with small overlap integrals. Several MO-based approaches were then developed. Diradicals were featured by a pair of non-bonding molecular orbitals (NBMOs), which are occupied by two electrons [65-67]. Within the framework of Hiickel MO approximation, the relationship between the number of NBMOs,... 

The second-order changes, in terms of which polarizability coefficients may be defined, are much more difficult to discuss because they involve essentially a change in the wave function (made in such a way as to preserve self-consistency)—unlike the first-order changes, which involve the Mwperturbed wave function only. Approximate formulae for the polarizabilities were first obtained (McWeeny, 1956) using a steepest descent method to minimize the energy, a useful result being the establishment of a connection between tt,, and F, valid for systems of any kind (non-alternant or heteroaromatic included) and applicable either in Hiickel theory or in a more complete theory. 

In order to perform calculations on larger molecules in a reasonable amount of time, approximations are made, which may involve the neglect of certain terms, or the inclusion of experimentally determined parameters. The best known and simplest example of this level of approximation are Hiickel Molecular Orbital (HMO) calculations, which treat only pi-electrons, in conjugated hydrocarbons, with neglect of overlap (1). While obviously limited in use, HMO methods are still used in certain research applications. 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

The widespread application of MO theory to systems containing a bonds was sparked in large part by the development of extended Hiickel (EH) theory by Hoffmann (I) in 1963. At that time, 7r MO theory was practiced widely by chemists, but only a few treatments of a bonding had been undertaken. Hoffmann s theory changed this because of its conceptual simplicity and ease of applicability to almost any system. It has been criticized on various theoretical grounds but remains in widespread use today. A second approximate MO theory with which we are concerned was developed by Pople and co-workers (2) in 1965 who simplified the exact Hartree-Fock equations for a molecule. It has a variety of names, such as complete neglect of differential overlap (CNDO) or intermediate neglect of differential overlap (INDO). This theory is also widely used today. 

For soil solutions, the Davies equation, which is a modification of the Debye-Hiickel equation, is commonly used, and is applicable to solutions up to approximately I — 0.7 mol (see also Section 3.2.1) ... 

The application of Blum s theory to experiment is unexpectedly impressive it can even represent conductance up to 1 mol dm . Figure 4.96 shows experimental data and both theories—Blum s theory and the Debye-Hiickel-Onsager first approximation. What is so remarkable is that the Blum equations are able to show excellent agreement with experiment without taking into account the solvated state of the ion, as in Lee and Wheaton s model. However, it is noteworthy that Blum stops his comparison with experimental data at 1.0 M. 

A solution to these difficulties is a blend of the chemical picture in which clustered ion configurations are described by the mass action law, while the interactions between the various entities are treated by methods applying the high-temperature approximations of the /-functions, e.g. by the MSA. The Debye-Hiickel (DH) theory [26], although derived from classical electrostatics, is also a high-temperature approximation, whose range of applicability can be extended by supplementing a mass action law for ion pair formation [27],... 

Nowadays, the success of the methods proposed by Hoffmann 50> and by Pople and Segal 51> among the chemists tends to promote the use of pure atomic orbital bases for all-valence treatments. The first method is a straightforward application of the Wolfsberg-Helmholz treatment of complexes to organic compounds and is called the Extended Hiickel Theory (EHT), because its matrix elements are parametrized in the same way as the Hiickel method with overlap for n electrons. The other method, known under the abbreviation Complete Neglect of Differential Overlap (CNDO), includes electron repulsion terms by extending to a orbitals the successful approximation of zero-differential overlap postulated for n electrons. 

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