### What point symmetry group does the molecule belong

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Problem 1 Consider the molecule furan C4H4O, whose planar carbon framework is shown in the figure: O 1 2 4 3 5 x z y (a) A H¨uckel type π-molecular orbital calculation considering heteroatoms produces the following formulas for the molecular orbitals (important: they are presented in no particular order!!!) φa = 0.27 χ1 - 0.60 χ2 + 0.33 χ3 + 0.33 χ4 - 0.60 χ5 , φb = 0.43 χ1 - 0.23 χ2 - 0.60 χ3 - 0.60 χ4 - 0.23 χ5 , φc = -0.37 χ2 + 0.60 χ3 - 0.60 χ4 + 0.37 χ5 , φd = 0.86 χ1 + 0.30 χ2 + 0.19 χ3 + 0.19 χ4 + 0.30 χ5 , φe = -0.60 χ2 - 0.37 χ3 + 0.37 χ4 + 0.60 χ5 . The H¨uckel parameters for the Oxygen are hO¨ = 2.0 and kC-O¨ = 0.8. Consider all the other Coulomb and resonance integrals as those for unsaturated hydrocarbons. The orbital energies of orbitals φa, φb, φc, and φe are oa = α - 0.95β ; ob = α + 1.33β ; oc = α - 1.61β ; oe = α + 0.61β Calculate the orbital energy o for the π-molecular orbital φd using the formula od = h φd | ˆhπ | φd i , where hˆ π is the effective 1-electron π-Hamiltonian of the H¨uckel approximation. Express your results in terms of the carbon Coulomb integral α and the resonance integral β of the H¨uckel approximation method.

Problem 2 Consider again the system of problem (1).

(a) What is the π-electronic energy of the molecule?

(b) Based on the data and your result for problem (1), order the orbitals according to their energy (that is, call φ1 the molecular orbital with lowest energy, . . ., and call φ5 the molecular orbital with the highest energy). Draw schematically a molecular energy diagram for the π orbitals, and indicate the occupation numbers for each of the orbitals when the molecule is in the ground electronic state.

(c) Which molecular orbital is the HOMO and which molecular orbital is the LUMO?

(d) Calculate the average π-electronic charges q1, q2, and q3 at, respectively, atoms 1, 2, and 3 when the molecule is in the electronic ground state. Calculate the value of q1, q2 and q3 when the molecule is in the first electronic excited state. Based on these results indicate in which direction the π-electronic charge "flows" when the molecule absorbs a photon of energy hν = oLUMO - oHOMO.

Problem 3 Consider again the molecule and data from problem (1).

(a) To what point symmetry group does the molecule belong?

(b) Label each molecular orbital according to its symmetry species. (This means: find like what irreducible representation of the point symmetry group each molecular orbital transforms).

Problem 4 Consider the following molecules (a), (b), and (c), all of which are planar:

(i)To what point symmetry group each of the molecules belongs?

(ii)If you were to apply the simple H¨uckel method to each molecule, for which one(s) would you expect to find degenerate π-molecular orbitals? Explain why.

Problem 5 The equilibrium bond length of H79Br is 1.413 °A. Assume that the atomic molar masses are H= 1.00784 g mol-1 and 79Br = 78.9183 g mol-1 .

(a) Calculate the value of the rotational constant for H79Br. Express your result in wavenumbers.

(b) A microwave spectrum of H79Br is taken. At what wavenumber νe will the J = 2 → J = 3 rotational transition be observed? (Assume that H79Br behaves as a rigid rotator).

(c) Determine the most populated rotational state of H79Br at a temperature of 800 K.

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