Simple Hiickel molecular orbital theory approximations

Equations (3.3) define the essence of the Hiickel molecular orbital (HMO) theory. Notice that the total energy is just the sum of the energies of the individual electrons. Simple Hiickel molecular orbital (SHMO) theory requires further approximations that we will discuss in due course. 

The simple, or Hiickel based, molecular orbital theory (HMO and PPP methods) frequently provides useful qualitative insights but cannot be used reliably in a quantitative manner. For this purpose it is necessary to use a method which takes account of all the electrons as well as their mutual repulsions. A major bottleneck in such calculations is in the computation and storage of the enormous number of electron-repulsion integrals involved. Early efforts to reduce this problem led Hoffmann to the EH approximation (I.N. Levine, Quantum Chemistry, 4-th ed., 1991, Prentice-Hall, Inc., Ch. 16, 17), and Pople and co-workers to the CNDO, INDO and NDDO-approximations (B-70MI40100). 

It is apparent that the molecular orbital theory is a very useful method of classifying the ground and excited states of small molecules. The transition metal complexes occupy a special place here, and the last chapter is devoted entirely to this subject. We believe that modem inorganic chemists should be acquainted with the methods of the theory, and that they will find approximate one-electron calculations as helpful as the organic chemists have found simple Hiickel calculations. For this reason, we have included a calculation of the permanganate ion in Chapter 8. On the other hand, we have not considered conjugated pi systems because they are excellently discussed in a number of books. 

This book, an account of Coulson s lectures from recordings and notes by Brian O Leary and R. B. Mallion, is far more than a historical document. One can perceive the way in which Coulson thought, his mixture of optimism and pessimism, his sense of rigour and of approximation, and his guesses and analyses. Moreover, one can sense his dedication to the specific problems, and most of all to his communication of an underlying physical and chemical intuition. While the quantitative aspects of molecular-orbital theory have now gone well beyond the Hiickel method, modern computer techniques have not replaced the simple, intuitive ideas of molecular-orbital theory, best exhibited by Hiickel theory as developed in this small volume. 

Molecular-orbital theory has taken many forms and has been dealt with by many approximations. In 1963 Hoffmann S presented a formalism which he referred to as extended Hiickel (EH). In the 1930 s, however, this formalism would simply have been called molecular-orbital, since it is a straightforward application of molecular-orbital (MO) theory, using a one-electron Hamiltonian. Hoffmann referred to it as extended Hiickel because it did not limit itself to 7r-electron systems and was able to deal with saturated molecules by including all overlap integrals. In these respects it did extend the usual, or simple Huckel, method, which was customarily applied to 7T-electrons, and assumed complete tt — a separability. 

Historically the extended Hiickel model (EHT, Extended Huckel Theory, as it was originally called) is one of the most important schemes developed. Even in its simplest form, orbitals with energies approximating ionization potentials and of the proper nodal structure and symmetries are obtained. The famous Woodward-Hoffmann rules were founded on these simple calculations, and frontier molecular orbital arguments are easily based on EHT orbitals. 

From the foregoing discussion it appears that the frontier orbital method is at once a simple, concise, and accurate method for assessing the stereochemical outcome of pericyclic reactions. Furthermore, it is a method that is equally applicable to symmetrical and to unsymmetrical systems. There are some disadvantages in the theory, however. Firstly, it is necessary to derive the general phase characteristics of the HOMO and LUMO levels. Hiickel molecular calculations can be used for tliis purpose, but there are available a number of approximate methods, for example the electron-in-a-box model, which are usually satisfactory even if they are more difficult to apply to more complex systems. Nevertheless, frontier orbital analysis is quicker and more simple than the formalized correlation diagram approach, and with a little practice one can intuitively arrive at the correct relative phase relationsliips in the HOMO and LUMO levels. 

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