Implementing Hiickel Theory

Recall the basic assumptions of HMOT (Section 14.2.3). The method is restricted to planar molecules and evaluates only the tt electrons—the t framework is ignored. This is not as drastic an assumption as it may seem, because ct systems and tt systems are of opposite symmetry in a planar molecule, and so o and tt orbitals do not mix. The a electrons can be viewed as simply providing part of the potential field experienced by the tt electrons. 

Putting this all in the context of only two adjacent carbon p orbitals gives the Huckel secular determinant for ethylene shown below. Compare this to the secular determinant for the Two-Orbital Mixing Problem given in Section 14.2.2. 

All the parts of the HMOT analysis of fulvene. A. The secular determinant with numbering as given in the picture of fulvene. B. The secular determinant after dividing by 3 and substituting (a - ) / 3 = -x. C. The energy diagrams derived from solving the sixth-order polynomial from the secular determinant. D. The secular ec uations for fulvene. E. The wavefunctions for the molecular orbitals. F. Pictures of all the Hiickel MOs. 

CHAPTER 14 ADVANCED CONCEPTS IN ELECTRONIC STRUCTURE THEORY 

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The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

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). 

We then present ab initio molecular orbital theory. This is a well-defined approximation to the full quantum mechanical analysis of a molecular system, and also the basis of an array of powerful and popular computational approaches. Molecular orbital theory relies upon the linear combination of atomic orbitals, and we introduce the mathematics and results of such an approach. Then we discuss the implementation of ab initio molecular orbital theory in modern computational chemistry. We also describe a number of more approximate approaches, which derive from ab initio theory, but make numerous simplifications that allow larger systems to be addressed. Next, we provide an overview of the theory of organic TT systems, primarily at the level of Hiickel theory. Despite its dramatic approximations, Hiickel theory provides many useful insights. It lies at the core of our intuition about the electronic structure of organic ir systems, and it will be key to the analysis of pericyclic reactions given in Chapter 15. 

Although the results of HMOT are easy to obtain, and many excellent treatments of the quantitative methodology are available, computer programs should not be treated as "black boxes". Hence, it is useful to understand how HMOT is implemented. Here we will provide a brief overview of the manner in which Hiickel theory is applied, the results obtained from HMOT, and some examples of how it provides useful and general qualitative insights into the electronic structures of organic molecules. 

Let us implement the earlier-described definitions and approach to calculate the equilibrium potential of the Daniell cell at 25°C and 1 bar when concentrations of the cell electrolytes are well defined and relatively low in order to use the Debye-Hiickel theory for calculating the individual activity coefficients of the electrochemically active ions. In this example, the Daniel cell diagram is as follows ... 

The inverse design of molecular structures is a related problem that can make use of the same computational machinery to guide the synthesis or to predict new systems with a desired property. The tight-binding linear combination of atomic potentials (LCAP) is a framework that can be directly applied in combination with the Extended Hiickel Theory. The inverse design by the LCAP method can also be implemented in conjunction with the density functional theory. ... 

Both scalar relativistic effects and spin-orbit coupling have to be considered. Relativistic theories of chemical shifts and of spin-spin coupling constants, as well as their implementation at the semi-empirical extended Hiickel level, have been known for some time (see bibliography given in Ref. 22). However, only very recently have first efforts been made to include relativistic corrections in quantitative calculations of NMR properties. 

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