Setting up the Huckel Determinant

A Identifying the Basis Atomic Orbitais and Constructing a Determinant 

The allyl radical, C3H5, is a planar molecule with three unsaturated carbon centers (see Fig. 8-1). The minimal basis set of AOs for this molecule consists of a Is AO on each hydrogen and Is, 2s, 2p c, 2pj , and 2p AOs on each carbon. Of all these AOs only the 2p AOs at the three carbons are antisymmetric for reflection through the molecular plane. 

Following Huckel, we ignore all the a-type AOs and take the three 2pz AOs as our set of basis functions. Notice that this restricts us to the carbon atoms the hydrogens are not treated explicitly in the simple HMO method. We label our three basis functions Xu X2, X3 as indicated in Fig. 8-2. We will assume these AOs to be normalized. 

Suppose that we now perform a linear variation calculation using this basis set. We know this will lead to a 3 x 3 determinant having roots that are MO energies which can be used to obtain MO coefficients. The determinantal equation is 

We have already indicated that there is no way to write an explicit expression for Hjj that is both consistent with our separability assumptions and physically correct. But, without an expression for how can we evaluate the integrals Hip. The HMO method sidesteps this problem by carrying certain of the Hij integrals along as symbols until they can be evaluated empirically by matching theory with experiment. 

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