Simple Hiickel molecular orbital theory

In the simple Hiickel molecular orbital theory of w-electronic systems, molecular orbitals are derived by combinations of 2p orbitals, one from each bonded atom. All of the exchange integrals for bonded atoms are taken as equal and interactions between non-adjacent atoms are ignored. The energy of each molecular orbital then has the form,... 

For an assemblage of two identical molecules spaced d nm apart, the HOMO and LUMO energies split into four levels, each split by 2t eV apart ("dimer splitting") [26] here t is akin to the Hiickel69 resonance integral (i of Section 3.15 Indeed, chemists will remember Eq. (8.6.10) from the simple Hiickel molecular orbital theory for aromatic 7r-electron systems. As the number of molecules N increases, the energy levels become spaced more closely, until they form a quasi-continuous band of bandwidth W, where... 

Figure 11 Energy level diagram for the HOMO and first two LUMOs of the 60jr-electron system of Ceo, calculated by simple Hiickel molecular orbital theory...

For the benefit of the more mathematically-inclined reader we can conveniently illustrate here the relevance of abstract Graph Theory (a branch of Pure Mathematics) to simple Hiickel molecular-orbital theory. 

The important energies in this Hamiltonian are the on-site energy U (akin, in a very simple way, to the Hiickel on-site integral a in simple Hiickel molecular orbital theory) and the ofF-site energy t (same integral as in the bandwidth 41 discussed above) These on-site (U) and off-site (t) energies are often used in discussions of organic metals. 

In order to explain the observed rate effects of substituents, Burrows and Carpenter used simple Hiickel molecular orbital theory to evaluate the influence of a substituent on the transition state and the reactants (Scheme 11.8) [16]. Their... 

Physical organic chemistry, the study of the basic physical principles of organic reactions, is not a new field in 1940, Hammett had already written a book with this title. This area has developed during the last 20 years mainly because of the explosive growth of sophisticated analytical instrumentation and computational techniques, going from the simple Hiickel molecular orbital theory to ab initio calculations of increasing accuracy enabled by the advent of fast supercomputers. 

A problem in simple Hiickel molecular orbital theory requires putting in diagonal form the matrix describing the set of secular equations, an example of which is... 

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. 

In this chapter, simple Hiickel molecular orbital (SHMO) theory is developed. The reference energy, a, and the energy scale in units of fi are introduced. 

In Chapter 5, conventional simple Hiickel molecular orbital (SHMO) theory is introduced. The Hiickel a is suggested as a reference energy, and use of as a unit of energy is advocated. Parameters for heteroatoms and hybridized orbitals are given. An interactive computer program, SHMO, which uses the conventions introduced in this chapter, is available on the Web [12]. 

At present, aromaticity is usually expressed in MO terminology. Structures that have a particularly stable series of occupied ir-molecular orbitals are called aromatic. A simple expression of the relationship between a MO description of structure and aromaticity is known as the Hiickel rule. It is derived from simple Hiickel molecular orbital (HMO) theory, and states that planar monocyclic completely conjugated hydrocarbons will be aromatic when the ring contains 4n+2 v-electrons. HMO... 

The Kronig-Penney model, although rather crude, has been used extensively to generate a substantial amount of useful solid-state theory [73]. Simple free-electron models have likewise been used to provide logical descriptions of a variety of molecular systems, by a method known in modified form as the Hiickel Molecular Orbital (HMO) procedure [74]. 

In the early years of quantum theory, Hiickel developed a remarkably simple form of MO theory that retains great influence on the concepts of organic chemistry to this day. The Hiickel molecular orbital (HMO) picture for a planar conjugated pi network is based on the assumption of a minimal basis of orthonormal p-type AOs pr and an effective pi-Hamiltonian h(ctT) with matrix elements... 

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). 

To further illuminate the LCAO variational process, we will carry out the steps outlined above for a specific example. To keep things simple (and conceptual), we consider a flavor of molecular orbital theory developed in the 1930s by Erich Hiickel to explain some of the unique properties of unsaturated and aromatic hydrocarbons (Hiickel 1931 for historical... 

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. 

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