The Huckel Method

In the late 1920s, it was shown that the chemical bond existing between two identical hydrogen atoms in H2 can be described mathematically by taking a linear combination of the Is orbitals j)i and 2 of the two H atoms that are partners in the molecule (Heitler and London, 1927). When this is done, the combination 

It is a property of linear, homogeneous differential equations, of which the Schroedinger equation is one. that a solution multiplied by a constant is a solution and a solution added to or subtracted from a solution is also a solution. If the solutions Pi and p2 in Eq. set (6-13) were exact molecular orbitals, id v would also be exact. Orbitals p[ and p2 are not exact molecular orbitals they are exact atomic orbitals therefore. j is not exact for the ethylene molecule. 

Huckel (properly, Huckel) molecular orbital theory is the simplest of the semiempirical methods and it entails the most severe approximations. In Huckel theory, we take the core to be frozen so that in the Huckel treatment of ethene, only the two unbound electrons in the pz orbitals of the carbon atoms are considered. These are the electrons that will collaborate to form a n bond. The three remaining valence electrons on each carbon are already engaged in bonding to the other carbon and to two hydrogens. Most of the molecule, which consists of nuclei, nonvalence electrons on the carbons and electrons participating in the cr 

We now carry out a deceptively simple piece of algebra and make use of the variational principle. The time independent Schroedinger equation (3.6) can be multiplied by F(r) on both sides to give 

We divide both sides by the integral of the product E(r) F(r), also over all space, to get 

Expansion in a linear combination of atomic orbitals % leads to a set of integrals that are given the symbols 

If the basis functions have been selected so that they are normal, their integral over all space is S =1.0 for ju = v. In the lowest level approximation, it is common to set S equal to zero for // v, even though we know that the basis functions do not constitute a complete 

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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). 

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... 

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... 

Fig. 3. Distribution of formal charge in the anionic intermediate of the Birch reduction of benzene calculated (a) by the Huckel method, (b) by the Pople method.

Within this approximation, known as the Huckel method, the orbital energies are... 

The 7r-electron wave functions in the Huckel method are given by... 

Maximum simplicity. So let us return to the simplest version of the Huckel method. We have not as yet mentioned still another simplifying assumption of this method most of the matrix elements in Eq. (6) are assumed... 

We avoid using here the terms bonding , antibonding , and nonbonding MO because in the framework of the Huckel method the HOMO of a ribbon of a certain... 

Failures of the Huckel Method An Outline of More-Sophisticated Theories... 

We now work through in detail an example of the application of the Huckel method to a specific molecule in its simplest form, the calculation will be characterised by just two empirical parameters, a and / . Butadiene is selected as the example and the molecular-orbital energies and LCAO combinatorial coefficients calculated for it here will be used in later discussions of other quantities derivable from them (such as charge, bond order and free valence discussed in Chapter Four). Butadiene has four carbon atoms, the cr-bond connectivity of which may conveniently be depicted schematically as in Fig. 2-6. 

We will now continue the discussion using the Extended-Huckel related tight-binding method, which is identical to the Huckel method as it is called by organic chemists. In it, the overlap matrix elements are approximated as ... 

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