Huckel LCAO

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. 

X) is most likely, and the observation of two different vicinal fluorine-fluorine coupling constants from the analysis of the variable temperature F NMR spectrum (Section XI), indicated that one rotamer must predominate (143). However semiempirical and extended Huckel LCAO-MO calculations suggest that the stability of the various conformations for P2F4 decreases in the order gauche > cis > trans- (69a). 

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. 

Returning to Huckel theory for ethylene, and substituting the first LCAO [Eq. (6-15a)] for v /, we have... 

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. 

The origin of cyclopropenone chemistry goes back to the successful preparation of stable derivatives of the cyclopropenium cation <5 3), the first member of a series of Huckel-aromatic monocyclic carbo-cations possessing a delocalized system of (4n + 2)-7r-electrons. This experimental confirmation of LCAO-MO theory stimulated efforts to prepare other species formally related to cyclopropenium cation by a simple resonance description of electron distribution, namely cyclopropenone 7 and methylene cyclopropene (triafulvene) 8 ... 

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. 

To further illuminate the LCAO variational process, we will carry out the steps outlined above for a specific example. To keep things simple (and conceptual), we consider a flavor of molecular orbital theory developed in the 1930s by Erich Huckel to explain some of the unique properties of unsaluralcd and aromatic hydrocarbons (Huckel 1931 for historical... 

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 (p. 328) was a pioneer in the field of molecular orbital theory. He developed the LCAO method in its simplest form, yet Htickel molecular orbitals have proved enormously successful in dealing with organic molecules. Htickel proposed the 4/i + 2 rule in 1931. It has been tested in many ways since then, and it works. Now, what is the theoretical basis for this rule ... 

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. 

The obvious and logical extension to the discussion so far outlined in this chapter is to ask whether, in the case of, say, a general, non-alternant hydrocarbon, it is possible and legitimate to perform an iterative calculation in which both Coulomb integrals ( 7.2 and 7.3) and resonance integrals ( 7.4) are varied simultaneously. In such a scheme, calculations based on relations (7-6) and (7-8) would be carried out in an iterative fashion such that the one-electron Hamiltonian-matrix [ar, 0rJ furnished qr and p identical with those which had served to calculate the particular ar - and / elements in question such LCAO-MO coefficients and energy-levels as were derived from this process would then in principle be truly self-consistent , in the sense implied. This approach has been called1122 the self-consistent Huckel-method or the / a/co" method . 

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. 

H was the matrix-component of the Hiickel effective-Hamiltonian operator, effective between two basis atomic-orbitals, 4>r and 4>s, Srs was the overlap integral between 4>r and s, and H was set equal to a, H to / . This is how we developed the simple HMO-approach in Chapter Two. What Roothaan did was to show that a formally similar determinant is obtained in a full treatment of the re-electrons, but that it involves a somewhat more complicated expression for the matrix-elements, H . Furthermore, he showed that this more-complicated expression somehow had to take into account interactions between any one re-electron and all the other re-electrons. We do not go into the details of this here, except to say that, in order to find the LCAO-MO coefficients for one molecular orbital, it is necessary to know all the others, because all the others appear in the expressions for the equivalent terms, Hrs. This is a very familiar situation which mathematicians have long known how to deal with and which we encountered during our discussion of the self-consistent" Huckel-methods in 7.2—7.5 it is necessary to use an iterative scheme. An initial guess is made of all the orbitals except one and these are used to calculate the H -terms for the one orbital which has not yet... 

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>. 

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