Huckel LCAO approximation

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

Actually it is a defect of the simple Huckel LCAO MO approach that spin correlation is not included and is responsible for poor success of these approximate wave functions in calculating nuclear spin-spin interactions and also spin density distributions in nonalternate and odd alternate paramagnetic molecules. 

We have seen so far that MOs resulting from the LCAO approximation are delocalized among the various nuclei in the polyatomic molecule even for the so-called saturated a bonds. The effect of delocalization is even more important when looking to the n electron systems of conjugated and aromatic hydrocarbons, the systems for which the theory was originally developed by Huckel (1930, 1931, 1932). In the following, we shall consider four typical systems with N n electrons, two linear hydrocarbon chains, the allyl radical (N = 3) and the butadiene molecule (N = 4), and two closed hydrocarbon chains (rings), cyclobutadiene (N = 4) and the benzene molecule (N = 6). The case of the ethylene molecule, considered as a two n electron system, will however be considered first since it is the reference basis for the n bond in the theory. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

The Huckel approximation is defined by a set of simplicafions to the form of the Hamiltonian in the LCAO-MO description of planar conjugated molecules. Although the Huckel approximations are quite severe, nevertheless they produce results that rationafize qualitatively the resonance energies and spectra of these molecules. 

Anomeric effect, 82, 310-311, 305 Antarafacial, 163 examples, 164 sigma bonds, 167 Anti-Bredt olefin, 102 Approximations of MO theory Born-Oppenheimer, 22 Hartree-Fock, 222 Huckel, 35, 86 independent electron, 35 LCAO, 229 nonrelativistic, 22 SHMO, 87... 

Huckel s rule (in its original form) stated that monocyclic polyenic molecules are aromatic only if their re-systems contain An + 2) re-el ec-trons, where n is an integer 1>. There have been many advances in LCAO-MO theory since Hiickel s original contributions (although the simplest approximation still bears his name, i.e. HMO), and today a more precise statement of the rule might read as follows. 

Simple HUckel molecular orbital (HMO) calculations on the pyrrolo[2,l-c][l,2,4]triazole (28) suggest that electrophilic attack would occur most readily at C-10, and this prediction was borne out by observations that acid-catalyzed deuteration and bromination by NBS in the dark both occur at this position <85JCR(S)363>. Various reactivity indices have been calculated for a number of pyrrolo[ 1,2-b][, 2,4]triazoles (29) and pyrrolo[2,1 -c][ 1,2,4]triazoles (30) using the MO LCAO method within the semiempirical SCF approximation. These indicate that the 5-position is most susceptible to electrophilic attack, followed by the 7-position <74CHE230>. 

The approximation of the one-electron Hamiltonian is the next step in the framework of the one-electron approximation - the electron-electron interactions are excluded from the Hamiltonian. In solid-state theory the LCAO one-electron Hamiltonian approximation is known as the tight binding method. In molecular quantum chemistry the one-electron Hamiltonians of Huckel or Mulhken-Rtidenberg tjqses (see Chap. 6) were popular in the 1950s and the beginning of the 1960s when the first-principles, Hartree-Fock LCAO calculations were practically impossible. 

An approximate treatment of tt electron systems was introduced in 1931 by Erich Huckel (Figure 15.17) and is called the Huckel approximation of tt orbitals. The first step in a Huckel approximation is to treat the sigma bonds separately from the pi bonds. Therefore, in a Huckel approximation of a molecule, only the tt bonds are considered. The usual assumption is that the <7 bonds are understood in terms of regular molecular orbital theory. The <7 bonds form the overall structure of the molecule, and the tt bonds spread out over, or span, the available carbon atoms. Such 77 bonds are formed from the side-on overlap of the carbon 2p orbitals. If we are assuming that the tt bonds are independent of the cr bonds, then we can assume that the 77 molecular orbitals are linear combinations of only the 2p orbitals of the various carbon atoms. [This is a natural consequence of our earlier linear combination of atomic orbitals—molecular orbitals (LCAO-MO) discussion.] Consider the molecule 1,3-butadiene (Figure 15.18). The tt orbitals are assumed to be combinations of the 2p atomic orbitals of the four carbon atoms involved in the conjugated double bonds ... 

In the case of ethylene the a framework is formed by the carbon sp -orbitals and the rr-bond is formed by the sideways overlap of the remaining two p-orbitals. The two 7r-orbitals have the same symmetry as the ir 2p and 7T 2p orbitals of a homonuclear diatomic molecule (Fig. 1.6), and the sequence of energy levels of these two orbitals is the same (Fig. 1.7). We need to know how such information may be deduced for ethylene and larger conjugated hydrocarbons. In most cases the information required does not provide a searching test of a molecular orbital approximation. Indeed for 7r-orbitals the information can usually be provided by the simple Huckel (1931) molecular orbital method (HMO) which uses the linear combination of atomic orbitals (LCAO), or even by the free electron model (FEM). These methods and the results they give are outlined in the remainder of this chapter. 

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