Extended Hiickel method applications

Abstract A historical view demystifies the subject. The focus is strongly on chemistry. The application of quantum mechanics (QM) to computational chemistry is shown by explaining the Schrodinger equation and showing how this equation led to the simple Hiickel method, from which the extended Hiickel method followed. This sets the stage well for ab initio theory, in Chapter 5. 

L. A. Donders, F. A. A. M. de Leeuw, and C. Altona, Relationship between proton-proton NMR coupling constants and substituent electronegativities. IV. An extended Karplus equation accounting for interaction between substituents and its application to coupling constant data calculated by the extended Hiickel method, Magn. Reson. Chem., 27 (1989) 556-563. 

The extended Hiickel method (Hoffmann, 1963) might be mentioned here. This is a simple, very fast method, which can be used to gain qualitative insights into problems, but, because of the extensive approximations made, it cannot compete with the other methods mentioned in terms of general reliability and accuracy. It has the attractive feature that it enables reasonably straightforward interpretations to be made of various phenomena, but it is not generally applicable to the kind of problem being discussed here. 

We assume here that the total electronic energy is equal to the sum of the individual electronic energies. This relationship, which has the advantage of being simple, is obtained when the electronic Hamiltonian is written as a sum of monoelectronic Hamiltonians, as in the Htickel and extended Hiickel methods. This approximate formula has, of course, limited application, but it is acceptable for a qualitative analysis of orbital interactions. Further details may be found in Structure ectronique des molecules, by Y. jean and F. Volatron, Volume 2, Chapter 13, Dunod,... 

The first paper to describe the extended Hiickel method in its present form is by R. Hoffmann, J. Chem. Phys. (1963) 39, 1397, but this covers its application to organic molecules. A closely related, very readable blow by blow account for inorganic molecules is by H. D. Bedon, S. M. Horner and S. Y. Tyree Inorg. Chem. (1964) 3, 647. 

Extended Hiickel gives a qualitative view of the valence orbitals. The formulation of extended Hiickel is such that it is only applicable to the valence orbitals. The method reproduces the correct symmetry properties for the valence orbitals. Energetics, such as band gaps, are sometimes reasonable and other times reproduce trends better than absolute values. Extended Hiickel tends to be more useful for examining orbital symmetry and energy than for predicting molecular geometries. It is the method of choice for many band structure calculations due to the very computation-intensive nature of those calculations. 

The methods described in this section were instrumental in the early computational modeling of homogeneous catalysis [34, 37], and are in a number cases the base of more accurate methods described later in this chapter. In any case, the qualitative accuracy they provide makes them of little application in present day research, with the only possible exception of the extended Hiickel approach. 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

The electronic overlap populations in all three cases were calculated from the one electron Extended Hiickel MO s. For the photocyclizations of 1,2-difuryl ethylenes very similar results were obtained also from minimal basis set ab-initio wavefunctions l The possibility of obtaining useful reactivity analyses from wave-functions which are easily available even for large systems could prove to be an important practical consideration for further applications of this method. The dependence on Sri sj ill (5) ensures that electronic overlap populations show the desirable physical characteristics for their use as reactivity measures strong falling-off with increasing interatomic distance and proper directional dependence. This last point is of particular significance for bond formation in polyenes. Thus for two C 2 p atomic... 

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. 

Empirical force-field (EFF) calculations failed to predict the A-ring chair conformation of lanost-8-en-3-one, which was indicated by n.m.r. studies. Combined EFF-extended Hiickel MO calculations were successful, however, giving a preference of 1.14 kcal mol over the boat form, although quantitative accuracy is not claimed for the present method of calculation. A treatment of alcohol conformations by molecular mechanics may have applications in predicting the preferred conformations of hydroxy-steroids. 

Nowadays, the success of the methods proposed by Hoffmann 50> and by Pople and Segal 51> among the chemists tends to promote the use of pure atomic orbital bases for all-valence treatments. The first method is a straightforward application of the Wolfsberg-Helmholz treatment of complexes to organic compounds and is called the Extended Hiickel Theory (EHT), because its matrix elements are parametrized in the same way as the Hiickel method with overlap for n electrons. The other method, known under the abbreviation Complete Neglect of Differential Overlap (CNDO), includes electron repulsion terms by extending to a orbitals the successful approximation of zero-differential overlap postulated for n electrons. 

Historically, there have been two major efforts in the development of semiempiri-cal calculations under the direction of John Pople (Nobel Laureate 1998) and M.J.S. Dewar. The earliest calculations of this nature are, however, the Hiickel (HMO) and extended Hiickel molecular orbital calculations (EHMO). While not mathematically rigorous, or used extensively at the present time, these calculations can provide considerable insight into the methods and in specific applications can be quite accurate. 

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