P9E.11 (a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies E, of the resulting A molecular orbitals are given by: E, =a+2B cos- N+1 k=1, 2,.,N Use this expression to make a reasonable empirical estimate of the resonance integral B for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that t-n ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32 900 cm", respectively. (b) Calculate the T-electron delocalization energy, Egdo:= E, - n(a+ B), of octatetraene, where E, is the total T-electron binding energy and n is the total number of T-electrons. (c) In the context of this Hückel model, the 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: cN sin j=1,2.N jkn j=1, 2,.,N Evaluate the coefficients of each of the six 2p orbitals in each of the six T 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.

P9E.11 (a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies E, of the resulting A molecular orbitals are given by: E, =a+2B cos- N+1 k=1, 2,.,N Use this expression to make a reasonable empirical estimate of the resonance integral B for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that t-n ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32 900 cm", respectively. (b) Calculate the T-electron delocalization energy, Egdo:= E, - n(a+ B), of octatetraene, where E, is the total T-electron binding energy and n is the total number of T-electrons. (c) In the context of this Hückel model, the 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: cN sin j=1,2.N jkn j=1, 2,.,N Evaluate the coefficients of each of the six 2p orbitals in each of the six T 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.