Molecular method: Hiickel theory

The theoretical methods used commonly can be divided into three main categories, semi-empirical MO theory, DFT and ab-initio MO theory. Although it is no longer applied often, Hiickel molecular orbital (HMO) theory will be employed to introduce some of the principles used by the more modem techniques. 

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. 

The molecular orbital methods which have been employed for such studies include extended Hiickel theory (EHT), CNDO, and ab initio LCAO-SCF. 

One of the most used approaches for predicting homoaromaticity has been the perturbational molecular orbital (PMO) theory of Dewar (1969) as developed by Haddon (1975). This method is based on perturbations in the Hiickel MO theory based on reducing the resonance integral (/3) of one bond. This bond represents the homoaromatic linkage. The main advantage of this method is its simplicity. PMO theory predicted many potential homoaromatic species and gave rise to several experimental investigations. 

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. 

We have seen three broad techniques for calculating the geometries and energies of molecules molecular mechanics (Chapter 3), ab initio methods (Chapter 5), and semiempirical methods (Chapters 4 and 6). Molecular mechanics is based on a balls-and-springs model of molecules. Ab initio methods are based on the subtler model of the quantum mechanical molecule, which we treat mathematically starting with the Schrodinger equation. Semiempirical methods, from simpler ones like the Hiickel and extended Hiickel theories (Chapter 4) to the more complex SCF semiempirical theories (Chapter 6), are also based on the Schrodinger equation, and in fact their empirical aspect comes from the desire to avoid the mathematical... 

The most widely used semiempirical quantum chemistry technique for theoretical chemisorption studies is the Extended Hiickel Theory (EHT). The method was first proposed by Hoffmann/95/ in its nonrelativistic form, and by Lohr and Pyykko/96/ and also Messmer/97/ in its relativistic form, based on the molecular orbital theory for calculating molecular electronic and geometric properties. For a cluster the molecular orbitals are expanded as linear combinations of atomic orbitals... 

London orbitals were introduced by Fritz London, who in 1937 used Hiickel theory to calculate the contribution to the magnetizability from the ring currents in the 7r-orbital backbone of some aromatic molecules [13]. The great virtue of London s approach is that each individual AO—the building blocks of molecular wave functions—has been harnessed to respond correctly (to first order at least) to the application of an external magnetic field, irrespective of the choice of the external gauge origin. Moreover, since, in London s approach, only the atomic orbitals are modified, this method is fully transparent to the treatment of the electronic structure otherwise. 

By far, the theoretical approaches that experimental inorganic chemists are most familiar with and in fact nse to solve questions qnickly and qnalitatively are the simple Huckel method and Hoffinann s extended Hiickel theory. These approaches are nsed in concert with the application of symmetry principles in the bnUding of syimnetry adapted linear combinations (SALCs) or gronp orbitals. The ab initio and other SCF procednres ontlined above prodnce MOs that are treated by gronp theory as well, bnt that type of rigor is not usually necessary to achieve good qnahtative pictures of the character aud relative orderiugs of the molecular orbitals. 

The band structure of a three-dimensional solid, such as a semiconductor crystal, can be obtained in a similar fashion to that of a polyene. Localized molecular orbitals are constructed based on an appropriate set of valence atomic orbitals, and the effects of delocalization are then incorporated into the molecnlar orbital as the number of repeat units in the crystal lattice is increased to infinity. This process is widely known to the chemical conununity as extended Hiickel theory (see Extended Hiickel Molecular Orbital Theory). It is also called tight binding theory by physicists who apply these methods to calcnlate the band structures of semiconducting and metallic solids. 

This method involves no assumptions beyond the wave function itself. It s an objective method, maximizing the repulsion of two electrons when they were in the same orbital and minimizing their repulsion when they were in different orbitals. This method didn t have to give three-center bonds, but it did. In a sense we used known theory to transform molecular orbitals to localized bonds, which immediately tested the way that we drew bonds in 1954. That was soon after my beginning in theory. We then went on to much more complicated molecules. Shortly after I came to Harvard, I proposed a computerized version of an extended Hiickel theory. 

Calculations have been made of the rr-electron distribution in the pyrazine ring by many methods (127-134) and the effects of protonation on the total electron densities have been calculated using the extended Hiickel theory (135). A good correlation was obtained between total carbon electron densities and both proton and carbon-13 chemical shifts. A recent molecular orbital study has been made of protonation in pyrazine (and other diazines) (135a). 

Extended Hiickel Theory (EHT) uses the highest degree of approximation of any of the approaches we have already considered. The Hamiltonian operator is the least complex and the basis set of orbitals includes only pure outer atomic orbitals for each atom in the molecule. Many of the interactions that would be considered in semi-empirical MO theory are ignored in EHT. EHT calculations are the least computationally expensive of all, which means that the method is often used as a quick and dirty means of obtaining electronic information about a molecule. EHT is suitable for all elements in the periodic table, so it may be applied to organometallic chemistry. Although molecular orbital energy values and thermodynamic information about a molecule are not accessible from EHT calculations, the method does provide useful information about the shape and contour of molecular orbitals. 

This is a semiempirical all-valence electron quantum mechanical method, apart from the Tr-approximation and the neglect of overlap integrals, as those of Hiickel molecular orbital (HMO) theory. The method reproduces, relatively well, the shapes and the order of the energy levels of molecular orbitals. To consider the overlapping, it is possible to describe the net destabilization caused by the interaction of the two doubly occupied orbitals, the effect of which is not reproduced by HMO theory. 

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. 

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