Huckel methods

The logical order in which to present molecular orbital calculations is ab initio, with no approximations, through semiempirical calculations with a restricted number of approximations, to Huckel molecular orbital calculations in which the approximations are numerous and severe. Mathematically, however, the best order of presentation is just the reverse, with the progression from simple to difficult methods being from Huckel methods to ab initio calculations. We shall take this order in the following pages so that the mathematical steps can be presented in a graded way. 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Because of its severe approximations, in using the Huckel method (1932) one ignores most of the real problems of molecular orbital theory. This is not because Huckel, a first-rate mathematician, did not see them clearly they were simply beyond the power of primitive mechanical calculators of his day. Huckel theory provided the foundation and stimulus for a generation s research, most notably in organic chemistry. Then, about 1960, digital computers became widely available to the scientific community. 

In the Huckel theory of simple hydrocarbons, one assumes that the election density on a carbon atom and the order of bonds connected to it (which is an election density between atoms) are uninfluenced by election densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to election repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method. 

The Huckel method and is one of the earliest and simplest semiempirical methods. A Huckel calculation models only the 7t valence electrons in a planar conjugated hydrocarbon. A parameter is used to describe the interaction between bonded atoms. There are no second atom affects. Huckel calculations do reflect orbital symmetry and qualitatively predict orbital coefficients. Huckel calculations can give crude quantitative information or qualitative insight into conjugated compounds, but are seldom used today. The primary use of Huckel calculations now is as a class exercise because it is a calculation that can be done by hand. 

The Pariser-Parr-Pople (PPP) method is an extension of the Huckel method that allows heteroatoms other than hydrogen. It is still occasionally used when... 

The first empirical and qualitative approach to the electronic structure of thiazole appeared in 1931 in a paper entitled Aspects of the chemistry of the thiazole group (115). In this historical review. Hunter showed the technical importance of the group, especially of the benzothiazole derivatives, and correlated the observed reactivity with the mobility of the electronic system. In 1943, Jensen et al. (116) explained the low value observed for the dipole moment of thiazole (1.64D in benzene) by the small contribution of the polar-limiting structures and thus by an essentially dienic character of the v system of thiazole. The first theoretical calculation of the electronic structure of thiazole. benzothiazole, and their methyl derivatives was performed by Pullman and Metzger using the Huckel method (5, 6, 8). 

TheHiickel constant (k) scales the interaction energy between two atomic orbitals (see Extended Huckel Method on page 125). HyperChem uses the default value of 1.75 (see the second part of this book. Theory and Methods). You should use this value for most cases. A suggested range for experimental adjustment of this constant is 1.6-2.0. ... 

The Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

The conformational properties of various 1,1 -diheteroferrocenes (7-10) have been the subject of three computational studies using extended Huckel methods.19,46 471,1 -Diphosphaferrocene has also been studied using the Fenske-Hall approach.48 and an MS Xa method.46 Where they overlap, the four treatments are in reasonable qualitative agreement. 

As a final exercise for the reader, consider the naphthalene module (symmetry 02h) as shown in Fig. 10. Application of the HUcKel method leads to a lOx 10 secular determinant (see problem 30). However, with the application... 

Table 5. Average values of bonds order and effective charges (Zeff) in pertechnetates and binuclear technetium acetates calculated by the extended Huckel method [58]...

The extended Huckel method [13] is an extension of the traditional Huckel method [14] expanding its range of applicability beyond planar conjugate systems. From a mathematical point of view, it consists simply in solving the matricial equation 1, where H is the hamiltonian matrix, C are... 

Ligand Li forms infinite chain by the coordination to Cu(I). Ligand L2 coordinates to Cu(I) as pendant The t and t values were calculated by the extended Huckel method... 

The photocyclodehydrogenation of thienyl ethylenes is well-defined when both thiophene rings are bound via a C(2) atom to the ethylenic bond as in (70). In other cases, however, more cyclization products are possible. To predict the photocyclization mode for heterohelicenes the F s rule fails in many cases, because correction factors for the hetero atoms in the Huckel MO calculation have to be introduced and the systems are not well comparable with carbocyclic diaryl ethylenes. A better reaction parameter in these cases is the Mulliken overlap population (nrs)51), introduced by Muszkat52) for these cases. The overlap populations of the atoms r and s in ground and excited state (nIiS and n s), are calculated using the extended Huckel method. Cyclizations should not occur when nr>s and An s (= nr>s — n s) have negative values. (This method can also be used for diaryl olefins, but in these cases calculation of F s is more simple.). 

We have emphasized that the Diels-Alder reaction generally takes place rapidly and conveniently. In sharp contrast, the apparently similar dimerization of olefins to cyclobutanes (5-49) gives very poor results in most cases, except when photochemically induced. Fukui, Woodward, and Hoffmann have shown that these contrasting results can be explained by the principle of conservation of orbital symmetry,895 which predicts that certain reactions are allowed and others forbidden. The orbital-symmetry rules (also called the Woodward-Hoffmann rules) apply only to concerted reactions, e.g., mechanism a, and are based on the principle that reactions take place in such a way as to maintain maximum bonding throughout the course of the reaction. There are several ways of applying the orbital-symmetry principle to cycloaddition reactions, three of which are used more frequently than others.896 Of these three we will discuss two the frontier-orbital method and the Mobius-Huckel method. The third, called the correlation diagram method,897 is less convenient to apply than the other two. 

As expected, the Mobius-Huckel method leads to the same predictions, Here we look at the basis set of orbitals shown in F and G for [1,3] and [1,5] rearrangements, respectively. 

The distances found between platinum centers in these molecules have been correlated with the resonating valence bond theory of metals introduced by Pauling. The experimentally characterized partially oxidized one-dimensional platinum complexes fit a correlation of bond number vs. metal-metal distances, and evidence is presented that Pt—Pt bond formation in the one-dimensional chains is resonance stabilized to produce equivalent Pt—Pt distances.297 The band structure of the Pt(CN)2- chain has also been studied by the extended Huckel method. From the band structure and the density of states it is possible to derive an expression for the total energy per unit cell as a function of partial oxidation of the polymer. The equilibrium Pt-Pt separation estimated from this calculation decreases to less than 3 A for a loss of 0.3 electrons per platinum.298... 

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