Hiickel theory systems

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. 

Theoreticians did little to improve their case by proposing yet more complicated and obviously unreUable parameter schemes. For example, it is usual to call the C2 axis of the water molecule the z-axis. The molecule doesn t care, it must have the same energy, electric dipole moment and enthalpy of formation no matter how we label the axes. I have to tell you that some of the more esoteric versions of extended Hiickel theory did not satisfy this simple criterion. It proved possible to calculate different physical properties depending on the arbitrary choice of coordinate system. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

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 terms of simple Hiickel theory the change from polyenic to aromatic the system experiences on passing from (1) and (2) to (3) was attributed primarily to the lower LVMO energy of the latter (B-73MI52000). A more detailed Hiickel calculation (72T3657) utilizing properly selected models in conjunction with thermochemical data also correctly accounts for the differences in tt-electron makeup between 1/Z-azonine (3a) and oxonin (1), which... 

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). 

Importantly, Hiickel theory affords us certain insights into the allyl system, one in particular being an analysis of the so-called resonance energy arising from electronic delocalization in the n system. By delocalization we refer to the participation of more than two atoms in a... 

Unfortunately, while it is clear that the allyl cation, radical, and anion all enjoy some degree of resonance stabilization, neither experiment, in the form of measured rotational barriers, nor higher levels of theory support the notion that in all three cases the magnitude is the same (see, for instance, Gobbi and Frenking 1994 Mo el al. 1996). So, what aspects of Hiickel theory render it incapable of accurately distinguishing between these three allyl systems ... 

Slater-type orbitals were introduced in Section 5.2 (Eq. (5.2)) as the basis functions used in extended Hiickel theory. As noted in that discussion, STOs have a number of attractive features primarily associated with the degree to which they closely resemble hydrogenic atomic orbitals. In ab initio HF theory, however, they suffer from a fairly significant limitation. There is no analytical solution available for the general four-index integral (Eq. (4.56)) when the basis functions are STOs. The requirement that such integrals be solved by numerical methods severely limits their utility in molecular systems of any significant size. 

In the PPP model, each first-row atom such as carbon and nitrogen contributes a single basis function to the n system. Just as in Hiickel theory, the orbitals y,- are not rigorously defined but we can visualize them as 2pff atomic orbitals. Each first-row atom contributes a certain number of jr-electrons—in the pyridine case, one electron per atom just as in Hiickel 7r-electron theory. 

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