Example exam questions

1. Answer part (a) and either part (b) or part (c).

(a) The statements below relate to the Hückel molecular orbital method, and the approximations made in it. Assess each of the following statements as either true or false.

In Hückel Theory:

Only p electrons are considered (1 mark)

Each molecular orbital is made up of a linear combination of atomic orbitals (1 mark)

The overlap integrals for orbitals on adjacent atoms are given the value b. (1 mark)

All Coulomb integrals (H_rr) are given the value a

(1 mark)

The Coulomb integral, a, and the resonance integral, b, are positive energy quantities. (1 mark)

The value of the resonance integral for adjacent carbon atoms is zero. (1 mark)

The value of the resonance integral for non-adjacent carbon atoms is b. (1 mark)

In finding the molecular orbitals, the secular determinant must be zero for a non-trivial solution. (1 mark)

(b) (i) Consideration of orbital symmetry can often be useful in predicting whether cycloaddition reactions will proceed thermally. Briefly describe the process which would be applied in such an analysis of a reaction. Identify the role of ‘correlation diagrams’. It is not necessary to give a detailed example. (5 marks)

(ii) Use the principles of orbital symmetry conservation to predict whether or not the cycloaddition reaction of two ethene molecules to form cyclobutane is allowed thermally. Illustrate the answer clearly, indicating the relevant symmetry elements. Without giving a full description, indicate briefly how the analysis would differ for the photochemical reaction. (11 marks)

(i) Write down the Hückel secular determinant for this molecule, using the numbering scheme shown, and the convenient substitution x = (a – E)/b. (2 marks)

(ii) The secular determinant for trimethylenemethane can be expanded as

y² – 3y = 0

where y = x² (x = (a – E)/b).

Find the energies of the molecular orbitals. Draw a molecular orbital diagram showing the energies of the molecular orbitals, and how the four p electrons occupy these orbitals. Comment on the electronic structure of this compound.

(10 marks)

(iii) Calculate the delocalization energy of trimethylenemethane. Butadiene (CH₂-CH-CH-CH₂)and trimethylenemethane are structural isomers. The delocalization energy of butadiene is found to be 0.472b. Which compound is expected to be more stable, and why? (4 marks)

2. Answer 3 of parts (a-d). All parts carry equal marks.

(a) Write down the Hückel secular determinant for 1,3-butadiene (CH₂CHCHCH₂), using the convenient substitution x = (a – E)/b.

The determinant can be solved as

y² – 3y +1 = 0

where y = x². Calculate the energies of the Hückel molecular orbitals of 1,3-butadiene, in terms of a and b. Also calculate the delocalization energy of 1,3-butadiene (hint: the energy of the lowest Hückel molecular orbital in ethene is a + b)

(b) The thermal ring opening of cyclobutene (C₄H₆) to give 1,3-butadiene (CH₂CHCHCH₂) occurs in a conrotatory fashion. Illustrate what is meant by the terms conrotatory and disrotatory for this reaction. Indicate clearly separate symmetry elements for the conrotatory and disrotatory paths that could be used to analyse and explain the observed behaviour. State the symmetry of the lowest energy molecular orbital (only) of 1,3-butadiene, and of the lowest energy molecular orbital corresponding to the breaking s bond in cyclobutene, with respect to each of these symmetry elements.

y_a = 0.371f₁ + 0.602f₂ + 0.602f₃ + 0.371f₄

y_b = 0.602f₁ + 0.371f₂ – 0.371f₃ – 0.602f₄

Using the Hückel approximations, show that y_a is normalized. Sketch both y_a and y_b. State, with a reason, which orbital is lower in energy. Calculate the Hückel p bond order between the central carbon atoms (atoms 2 and 3) and between two other adjacent atoms (e.g. atoms 1 and 2). Comment on the values obtained in relation to a simple drawing of the structure of 1,3-butadiene.

(d) Draw the frontier orbitals involved in (i) the dimerization of ethene and (ii) the Diels-Alder reaction of 1,3-butadiene with ethene. By considering their interactions, show which of these reactions is allowed and which is forbidden. For ethene dimerization, show similarly how photochemical excitation can be used to change the situation.

Determinants

For this course, you need to be able to calculate determinants. A determinant is written like a matrix, but with straight lines down each side instead of brackets. For a two by two matrix,

the determinant is:

For example:

To solve bigger determinants, you need to break them down into two by two determinants. For a three by three determinant:

Notice what is happening here: the numbers along the top row are ‘taken out’, one at a time, and each multiplies a smaller (two by two) determinant (made from the numbers remaining when the row and column containing the number taken out are excluded). For the first number (a), the sign of this term is positive, for the second (b), it is negative, for the 3^rd it is positive, and so on, alternating positive and negative. For example:

The biggest determinant you would need to solve is a four by four determinant (see the example of the secular determinant of 1,3-butadiene in your lecture notes).