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The FEMO theory (Problem 11.14) of conjugated molecules is rather crude and better results are obtained with simple Huckel theory.

(a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies Ek of the resulting π molecular orbitals are given by (see also Section 20.9): Use this expression to determine a reasonable empirical estimate of the resonance integral 13 for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that π* ← π ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32900 cm-1, respectively.

(b) Calculate the π -electron delocalization energy, Ede1o, = En-n (a+ /3), of octatetraene, where E π is the total π â€“electron binding energy and n is the total number of π -electrons.

(c) In the context of this Huckel model, the π molecular orbitals are written as linear combinations of the carbon 2p orbitals. The coefficient of the jth atomic orbital in the kth molecular orbital is given by Determine the values of the coefficients of each of the six 2p orbitals in each of the six n molecular orbitals of hexatriene. Match each set of coefficients (that is, each molecular orbital) with a value of the energy calculated with the expression given in part (a) of the molecular orbital. Comment on trends that relate the energy of a molecular orbital with its 'shape', which can be inferred from the magnitudes and signs of the coefficients in the linear combination that describes the molecular orbital.

(a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies Ek of the resulting π molecular orbitals are given by (see also Section 20.9): Use this expression to determine a reasonable empirical estimate of the resonance integral 13 for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that π* ← π ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32900 cm-1, respectively.

(b) Calculate the π -electron delocalization energy, Ede1o, = En-n (a+ /3), of octatetraene, where E π is the total π â€“electron binding energy and n is the total number of π -electrons.

(c) In the context of this Huckel model, the π molecular orbitals are written as linear combinations of the carbon 2p orbitals. The coefficient of the jth atomic orbital in the kth molecular orbital is given by Determine the values of the coefficients of each of the six 2p orbitals in each of the six n molecular orbitals of hexatriene. Match each set of coefficients (that is, each molecular orbital) with a value of the energy calculated with the expression given in part (a) of the molecular orbital. Comment on trends that relate the energy of a molecular orbital with its 'shape', which can be inferred from the magnitudes and signs of the coefficients in the linear combination that describes the molecular orbital.

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