Molecular orbital method Hiickel approximation

The U s can in the molecular orbital method (Hiickel approximation) be easily expressed in terms of Coulomb integrals a and resonance integrals which characterize the C atoms and the CC bonds.39... 

These integrals are difficult to evaluate exactly and the Hiickel molecular orbital method centres on approximations to them. 

A number of tautomeric forms of some fundamental purines have been studied quantum mechanically with the help of the standard, semiempirical Hiickel approximation of the molecular orbital method.57 58 Limited to ti electrons, these studies nevertheless enabled the determination of a number of electronic differences among the tautomers. Using some simplifying ideas, typical in 77-electron calculations, information has also been obtained, within the limits of this approximation, concerning the relative tendencies of the bases to undergo a given type of tautomerization, and this information has been used successfully for the prediction of the preferential mutagenic sites in the nucleic acids.50... 

The Hiickel approximation of the molecular orbital method (HMO method) can be used to interpret and predict the quantities of Sections IV,B, 1-3 from the energies of the molecular orbitals.64,05 The theoretical data for the quantities of Sections IV, B, 4 and 5 are obtainable from the expansion coefficients of the molecular orbitals. 

Figure 6 (p. 180) shows the relation that has been obtained between the measurements and the calculations which have been carried out up to now. The curve gives the relation between the measured interatomic distances to the bond orders calculated by the molecular orbital method in what is called the L.C.A.O. or Hiickel approximation. ... 

Many approximate molecular orbital theories have been devised. Most of these methods are not in widespread use today in their original form. Nevertheless, the more widely used methods of today are derived from earlier formalisms, which we will therefore consider where appropriate. We will concentrate on the semi-empirical methods developed in the research groups of Pople and Dewar. The former pioneered the CNDO, INDO and NDDO methods, which are now relatively little used in their original form but provided the basis for subsequent work by the Dewar group, whose research resulted in the popular MINDO/3, MNDO and AMI methods. Our aim will be to show how the theory can be applied in a practical way, not only to highlight their successes but also to show where problems were encountered and how these problems were overcome. We will also consider the Hiickel molecular orbital approach and the extended Hiickel method Our discussion of the underlying theoretical background of the approximate molecular orbital methods will be based on the Roothaan-Hall framework we have already developed. This will help us to establish the similarities and the differences with the ab initio approach. 

In Sect. 5.3 that follows, we report on the electronic structural characteristics of the C-based hexagonite structure from the point of view of the extended Hiickel molecular orbital method (EHMO), which is an approximate solid state electronic structure algorithm based upon the tight binding methodology (Hoffmann 1963 ... 

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. 

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. 

Other approximate, more empirical methods are the extended Huckel 31> and hybrid-based Hiickel 32. 3> approaches. In these methods the electron repulsion is not taken into account explicitly. These are extensions of the early Huckel molecular orbitals 4> which have successfully been used in the n electron system of planar molecules. On account of the simplest feature of calculation, the Hiickel method has made possible the first quantum mechanical interpretation of the classical electronic theory of organic chemistry and has given a reasonable explanation for the chemical reactivity of sizable conjugated molecules. 

One of the simplest approaches to comprehensive molecular orbital calculations is the extended Hiickel method. This method was developed by Roald Hoffman in the 1960s, and it was applied to hydrocarbon molecules. From the discussion presented in Chapters 2 and 3, we know that one of the first things that has to be done is to choose the atomic wave functions that will be used in the calculations. One of the most widely used types of wave functions is that known as the Slater wave functions (see Section 2.4). In the extended Hiickel method, the molecular wave functions are approximated as... 

The continued success of the extended Hiickel method in transition metal chemistry, where it was the method of choice until the mid 1980 s is surely related to the problems of other semiempirical methods in this area of chemistry. While methods like MOP AC [21] or AMI [22] have been extremely productive in the field of organic chemistry, they have found little success in transition metal chemistry. These methods are based in equation 2, similar to 1, but with the very significant difference that the Fock matrix F is computed from the molecular orbitals, in an iterative way, though through an approximate formula. 

In order to perform calculations on larger molecules in a reasonable amount of time, approximations are made, which may involve the neglect of certain terms, or the inclusion of experimentally determined parameters. The best known and simplest example of this level of approximation are Hiickel Molecular Orbital (HMO) calculations, which treat only pi-electrons, in conjugated hydrocarbons, with neglect of overlap (1). While obviously limited in use, HMO methods are still used in certain research applications. 

In Eqs. (3) and (4), H stands for the effective one-electron Hamiltonian the integration is over the whole space. In the simple method, a number of simplifying assumptions about the values of the integrals ftjk, j, and Sjk are introduced in the case of molecules containing no heteroatoms these are known as the Hiickel approximations. It seems useful to use this designation also for molecules with heteroatoms, and in the present review this method will be referred to as the Hiickel molecular orbital (HMO) method according to Streitwieser s suggestion.4... 

Several valence-bond (VB) treatments of heterocyclic compounds were reported in the thirties and forties.1, 2 The known difficulty in applying the VB method to complicated molecules has made an overwhelming majority of authors use the molecular orbital (MO) method. In most cases its simplest version, the naive MO LCAO method, has been used. This approximation differs from the well-known Hiickel... 

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. 

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