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Molecular Orbital Theory & Group Theory 1) Hückel theory uses the coulomb and exchange integrals  and  to define the energies of the molecular orbitals of conjugated hydrocarbon systems. Explain what these two quantities represent. (4 marks) 2) The  MOs of the cyclopropenyl radical 𝐶̇ 3H3 are formed from combinations of the three carbon p orbitals 1 to 3, shown in figure 1. Each MO is of the form  = c11 + c22 + c33. How many molecular orbitals will be formed from this set of atomic orbitals? (2 marks) 3) State the Hückel approximations and starting from the secular equations for 𝐶̇ 3H3, apply these approximations and hence obtain the secular determinant for 𝐶̇ 3H3. (6 marks) 4) Calculate the energies for the three  orbitals of cyclopropenyl radical 𝐶̇ 3H3 using Hückel molecular orbital theory, expressing the energies in terms of the coulomb and resonance integrals  and Sketch the energy level diagram for the system.See footnote.1 (10 marks) 5) Use the  orbital energies from question 4 to calculate the delocalization energy of the cyclopropenyl cation C3H3 + and compare this with the delocalisation energy of benzene which is 2. What might one therefore conclude about the nature of the chemistry of the C3H3 + cation? (4 marks) 1 You will have to solve a 3  3 determinant: | 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 | = 𝑎 | 𝑒 𝑓 ℎ 𝑖 | − 𝑏 | 𝑑 𝑓 𝑔 𝑖 | + 𝑐 | 𝑑 𝑒 𝑔 ℎ | You might also find the following helpful: x3 – 3x + 2 = (x-1)(x2+x - 2) Figure 1 1 2 3 6) The three carbon p orbitals which make up the  MOs of the cyclopropenyl cation C3H3 +are shown below. Given that the molecule possesses D3h point group symmetry, transform the set of p orbitals (1 + 2 + 3) using the operations of the group to form a reducible representation (1 + 2 + ). The directions of the 3C2 and 3v axes are shown figure 2. (8 marks) 7) Reduce the representation found in question 6 to determine the symmetries of the  MOs formed from the carbon p orbitals. (4 marks) 8) Apply each of the twelve operations of the D3h point group to the atomic orbital 1, in order to complete the table below. E C3 C3 2 C2 (1) C2 (2) C2 (3) h S3 S3 2 v (1) v (2) v (3) 𝑅̂1 1 2 -1 -3 -2 1 (3 marks) 9) Using the projection method, determine the combinations of orbitals 1, 2 and 3 corresponding to each of the irreducible representations identified in question 7. Normalise the orbital coefficients. note: one of the irreducible representations from question 7) is doubly degenerate; the projection method will only find one combination of orbitals corresponding to this representation. This is sufficient for question 9. (9 marks) 1 3 2 C2 (1) , v (1) C2 (2) , v (2) C2 (3) , v (3) Figure

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