Hiickel theory, extended equations

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. 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

Equation (3.43) resembles the formula [equation (3.44)] adopted by Wolfsberg and Helmholtz [59] for the interaction matrix elements. This form, with the empirical factor — 1.75, was adopted by Hoffmann in his derivation of extended Hiickel theory [60], namely... 

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... 

We have seen three broad techniques for calculating the geometries and energies of molecules molecular mechanics (Chapter 3), ab initio methods (Chapter 5), and semiempirical methods (Chapters 4 and 6). Molecular mechanics is based on a balls-and-springs model of molecules. Ab initio methods are based on the subtler model of the quantum mechanical molecule, which we treat mathematically starting with the Schrodinger equation. Semiempirical methods, from simpler ones like the Hiickel and extended Hiickel theories (Chapter 4) to the more complex SCF semiempirical theories (Chapter 6), are also based on the Schrodinger equation, and in fact their empirical aspect comes from the desire to avoid the mathematical... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation. The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Hiickel theory because the model includes the finite size of the solute molecules. 

In the previous chapters we sketched an elementary model of the chemical bond occurring between atoms in terms of a simple Hiickel theory mostly involving solution of 2 x 2 secular equations. The theory, first concerned with cr-bonding in H,. H2, He,. He2, was next extended to a- and 71-bonding in first-row homonuclear diatomics and to the study of multiple bonds, the fundamental quantity being a bond integral /3, whose form is... 

Equation (26.41) predicts to within approximately 10% mean molal activity coefficients for salt concentrations up to 0.1 molal. The more accurate form of the activity coefficient equation [Equation (26.40)] allows the model to be extended to salt concentrations up to 0.5 molal. To expand the applicability of the Debye-Hiickel theory to higher concentrations, additional terms are added to Equation (26.40), such as [4]... 

Most of the useful versions of extended Hiickel theories have been developed on heuristic grounds, but they can be developed easily from invoking the equations developed in an earlier section. 

Hiickel theory is clearly limited, in part because it is restricted to tt systems. The extended Huckel method is a molecular orbital theory that takes account of all the valence electrons in the molecule [Hoffmann 1963]. It is largely associated with R Hoffmann, who received the Nobel Prize for his contributions The equation to be solved is FC = SCE, with the... 

In these equations, n and v are two atomic orbitals (e.g. Slater type orbitals), is the ionisation potential of the orbital and fC is a constant, which was originally set to 1.75. The formula for the off-diagonal elements (where fj, and u are on different atoms) was originally suggested by R S Mulliken. These off-diagonal matrix elements are calculated between all pairs of valence orbitals and so extended Hiickel theory is not limited to tt systems. 

However, the solution given by Eq. (48) is based on the form of effective independent-electron Hamiltonians that can be empirically constructed as in Extended Hiickel Theory [168]. Such arbitrariness can be nevertheless avoided by the so-called self-consistent field (SCF), in which the one-electron effective Hamiltonian is considered to depend on the solution of Eq. (40) itself, i.e., the matrix of coefficients (C). The resulting Hamiltonian is called the Fock operator, while the associated eigen-problem is the Hartree-Fock equation ... 

In Eq. (10.9), is the overlap energy and S the corresponding overlap between adsorbate and surface atomic orbitals. Approximate Eq. (10.9) applies when ad molecule orbitals are s-symmetric and the metal electrons are also described by s-atomic orbital. Equation (10.9) is a familiar expression within molecular orbital theory and is deduced from tight binding theory including overlap of the atomic orbitals as in extended Hiickel theory (Hoffmann) is used [9]. 

With increasing electrolyte concentration, the short-range interactions become more and more dominating. Therefore, in activity coefficient models the Debye-Hiickel term, which describes the long-range interactions, has to be extended by a term describing the short-range interactions. A well-known empirical extension of the Debye-Hiickel theory is the Bromley equation [5] ... 

We now turn our attention to a method of deriving molecular orbitals that does not require evaluation of two-electron integrals. This is the method used with the approximate techniques known as extended Hiickel and Hiickel theory. It is also used in perturbational molecular orbital theory, which we present later in this chapter. This method takes advantage of secular determinants, a way in which to represent the Schrbdinger equation as a di-agonalizable matrix. 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Abstract A historical view demystifies the subject. The focus is strongly on chemistry. The application of quantum mechanics (QM) to computational chemistry is shown by explaining the Schrodinger equation and showing how this equation led to the simple Hiickel method, from which the extended Hiickel method followed. This sets the stage well for ab initio theory, in Chapter 5. 

We have already seen examples of semiempirical methods, in Chapter 4 the simple Hiickel method (SHM, Erich Hiickel, ca. 1931) and the extended Hiickel method (EHM, Roald Hoffmann, 1963). These are semiempirical ( semi-experimental ) because they combine physical theory with experiment. Both methods start with the Schrodinger equation (theory) and derive from this a set of secular equations which may be solved for energy levels and molecular orbital coefficients (most efficiently... 

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