Hiickel theory, extended approximations

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. 

Hiickel molecular-orbital calculation, 34 227 Hiickel theory, extended, 25 2, 3 approximations, 25 6... 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

The off-diagonal elements of Extended Hiickel theory, (fi v) represent the effects of bonding between the atoms and are assumed to be proportional to the overlap, Sj y. An approximation for differential overlap referred to as the Mulliken approximation... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

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... 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

The thermodynamic model of Krissmann [53] was used in the calculations of these experiments, though this was limited by the phase equilibrium (Eq. (3)) and the reaction equilibrium (Eq. (4)). Calculation of the activity coefficients of the H+ ions and HSOj was performed according to the extended Debye-Hiickel theory, using the approximation of Pitzer... 

The extended Hiickel theory calculations, used in this work and discussed below, are based on the approaches of Hoffmann Although VSIP values given by Cusachs, Reynolds and Barnard were explored for use as the Coulomb integrals, the VSIP values obtained from a Hartree-Fock-Slater approximation by Herman and Skillman were consistently used in the present EHT calculations by this author. Both the geometric mean formula due to Mulliken and Cusachs formula ) were considered for the Hamiltonian construction, but the Mulliken-Wolfsberg-Helmholtz arithmetic mean formula was chosen for use. 

Pure atomic orbitals Extended Htickel theory 40,50) Iterative extended Hiickel theories80 81 82) Quasi SCF diagonal element method85) Kinetic-energy Hiickel theory 58) Zero-differential overlap approximation 61,69) Standard SCF method73 74 75 76,77,78) SCF group function method 83) Random-phase... 

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. 

Computations on biomolecules have been performed by practically all the available all-valence electrons prcedures. The simplest is the extended Hiickel theory (EHT) in the version developed by Hoffmann 2) which is an extension of the well-known Hiickel approximation for it electrons in the sense that the molecular orbitals are obtained.as the eigenvalues of an effective Hamiltonian H that is not explicit, the matrix elements of H being treated as empirical input characteristics of the atoms involved. 

Extended Hiickel Theory (EHT) uses the highest degree of approximation of any of the approaches we have already considered. The Hamiltonian operator is the least complex and the basis set of orbitals includes only pure outer atomic orbitals for each atom in the molecule. Many of the interactions that would be considered in semi-empirical MO theory are ignored in EHT. EHT calculations are the least computationally expensive of all, which means that the method is often used as a quick and dirty means of obtaining electronic information about a molecule. EHT is suitable for all elements in the periodic table, so it may be applied to organometallic chemistry. Although molecular orbital energy values and thermodynamic information about a molecule are not accessible from EHT calculations, the method does provide useful information about the shape and contour of molecular orbitals. 

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]... 

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]. 

Bromley, L.A., "Approximate individual ion values of 3 (or B) in extended Debye-Hiickel theory for uni-univalent aqueous solutions at 298.15K", J. Chem. Thermo., v4. pp669-673 (1972)... 

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. 

Now, if we assume that S = 0, then lAE l = lAEI and closed-shell repulsion is undone. This may seem an unreasonable situation, but in fact many levels of theory make this approximation. As noted earlier in this chapter, most semi-empirical methods, such as AMI and MNDO, neglect overlap. As such, there is no closed-shell repulsion at these levels of theory, a serious deficiency in some situations. In contrast, for all its limitations, extended Hiickel theory (EHT) does not neglect overlap, and so closed-sheU repulsion survives. As discussed, classical Hiickel theory does neglect overlap. In fact, as previously noted, HMOT can be viewed as perturbation theory with S = 0, Haa = Hbb = a, and Hab = P. 

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