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Hückel methodThe Hückel method or Hückel molecular orbital method (HMO) proposed by Erich Hückel in 1930, is a very simple LCAO MO Method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene, benzene and butadiene. [1] [2] It is the theoretical basis for the Hückel's rule; the extended Hückel method developed by Roald Hoffmann is the basis of the Woodward-Hoffmann rules [3]. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms. [4] It is a very powerful educational tool and details appear in many chemistry textbooks. Additional recommended knowledge
Hückel characteristicsThe method has several characteristics:
Hückel resultsThe results for a few simple molecules are tabulated below:
The theory predicts two energy levels for ethylene with its two pi electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 pi electrons occupy 2 low energy MO's out of a total of 4 and for benzene 6 energy levels are predicted two of them degenerate. For linear and cyclic systems (with n atoms), general solutions exist [6]. Linear: Cyclic: Many predictions have been experimentally verified:
Mathematics behind the Hückel MethodThe Hückel method can be derived from the Ritz method with a few further assumptions concerning the overlap matrix S and the Hamiltonian matrix H. It is assumed that the overlap matrix S is the identity Matrix. This means that overlap between the orbitals is neglected and the orbitals are considered orthogonal. Then the generalised eigenvalue problem of the Ritz method turns into an eigenvalue problem. The Hamiltonian matrix H = (Hij) is parametrised in the following way: Hii = α for C atoms and α + hA β for other atoms A. Hij = β if the two atoms are next to each other and both C, and kAB β for other neighbouring atoms A and B. Hij = 0 in any other case The orbitals are the eigenvectors and the energies are the eigenvalues of the Hamiltonian matrix. If the substance is a pure hydrocarbon the problem can be solved without any knowledge about the parameters. For heteroatom systems, such as pyridine, values of hA and kAB have to be specified. Hückel solution for ethyleneIn the Hückel treatment for ethylene [9], the molecular orbital is a linear combination of the 2p atomic orbitals at carbon with their ratio's : This equation is substituted in the Schrödinger equation: with the Hamiltonian and the energy corresponding to the molecular orbital to give: This equation is multiplied by and integrated to give new set of equations:
where: All diagonal Hamiltonian integrals are called coulomb integrals and those of type , where atoms i and j are connected, are called resonance integrals with these relationships: Other assumptions are that the overlap integral between the two atomic orbitals is 0 leading to these two homogeneous equations: with a total of five variables. After converting this set to matrix notation: the trivial solution gives both wavefunction coefficients c equal to zero which is not useful so the other (non-trivial) solution is : which can be solved by expanding its determinant: or and After normalization the coefficients are obtained: The constant β in the energy term is negative and therefore α + β is the lower energy corresponding to the HOMO and is α - β the LUMO energy. Further reading
References
Categories: Molecular physics | Semiempirical quantum chemistry methods |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Hückel_method". A list of authors is available in Wikipedia. |