Carbon atom, 3-dimensionality ground-state electron

Chapter 3 introduced the concept of a double bond between carbon atoms, represented by C=C, with a length near 1.34 A. The motion of an electron in such a bond can be treated crudely as motion in a one-dimensional box. Calculate the energy of an electron in each of its three lowest allowed states if it is confined to move in a one-dimensional box of length 1.34 A. Calculate the wavelength of light necessary to excite the electron from its ground state to the first excited state. 

Three-quarters localization may be achieved by changing every second carbon atom in PA to a nitrogen atom. Optimization by the one-electron, unrestricted PM3 model leads either to an SDW or a CDW solution. The result is a deformed molecule with a repetition period of four atoms. It is clear that there are two possibilities for the ground state, and that correlation effects have to be included to obtain a correct result. The conclusion is that in the one-dimensional systems, one-half band occupation leads to a trivial Peierls dimerization, while the cases of one-quarter or three-quarters occupation may lead to interesting dynamics. 

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