Physical Chemistry: The Simple Hückel Method (Part II)

In the previous blog, we have learn how to set up the Hückel determinant for an aromatic molecule based on the topology of the pi-bonds. In this second part, we are going to learn how to calculate from the determinantal equation both the eigenvalues and the eigenvectors, corresponding to the orbital energy and orbital functions of the molecular system.

EIGENVALUES FROM THE HÜCKEL DETERMINANT

The calculation of the determinat gives a so-called characteristic equation. Namely a polynomial equation whose roots (x_i) are the eigenvalues of the system. As described in the previous article, the eigenvalues are related to the energy of the system by the relation x=\frac{\alpha-E}{\beta}. Let work out the determinant for the allyl molecule. The determinant is given by

that is solved as

The polynomial (x^3-2x)=0 has three root that can be easily found by rearranging it in x(x^2-2)=0 giving x_1=0 and x_{2,3}=\pm\sqrt{2}.

We can now use these value to calculate the energy of the Hückel orbitals using the relation x=\frac{\alpha-E}{\beta}. Therefore for

x_1=0=\frac{\alpha-E_1}{\beta}, we obtain E_1=\alpha, and

x_{2,3}=\pm\sqrt{2}=\frac{\alpha-E_{2,3}}{\beta}, we obtain E_{2,3}=\alpha \mp\sqrt{2}\beta , and

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REFERENCES

J.P. Lowe Quantum Chemistry. 1993, Academic Press.

About Danilo Roccatano

I have a Doctorate in chemistry at the University of Roma “La Sapienza”. I led educational and research activities at different universities in Italy, The Netherlands, Germany and now in the UK. I am fascinated by the study of nature with theoretical models and computational. For years, my scientific research is focused on the study of molecular systems of biological interest using the technique of Molecular Dynamics simulation. I have developed a server (the link is in one of my post) for statistical analysis at the amino acid level of the effect of random mutations induced by random mutagenesis methods. I am also very active in the didactic activity in physical chemistry, computational chemistry, and molecular modeling. I have several other interests and hobbies as video/photography, robotics, computer vision, electronics, programming, microscopy, entomology, recreational mathematics and computational linguistics.
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