# Free-Electron Model for Linear Polyenes

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The simplest nontrivial application of the Schrödinger equation is the one-dimensional particle in a box. Remarkably, this simple system can provide a useful model for a significant chemical problem—the structure of linear polyene molecules. A polyene is a hydrocarbon with alternating single and double carbon-carbon bonds, the simplest example being butadiene . The series continues with hexatriene, octatetrene, and so on, with the generic structure , a chain of conjugated carbon atoms. After all the single bonds are accounted for, each carbon atom contributes one -electron. Linear combinations of these atomic -orbitals can form -molecular orbitals (MOs), delocalized over the entire length of the molecule.

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Contributed by: S. M. Blinder (April 2014)

Open content licensed under CC BY-NC-SA

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References

[1] S. M. Blinder, *Introduction to Quantum Mechanics: In Chemistry, Materials Science, and Biology*, Amsterdam: Elsevier Academic Press, 2004, pp. 37–39.

[2] S. Huzinaga and T. Hasino, "Electronic Energy Levels of Polyene Chains," *Progress of Theoretical Physics,* 18(6), 1957 pp. 649–658. doi:10.1143/PTP.18.649.

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