Homework 1 Part 2

Homework 1 Part 2 - Given the two eigenvalue equations Âφ...

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Homework 1 Part 2 (Due Jan. 18 th ) 1. As seen in class, the Huckel theory may be used to calculate the energy levels of conjugated pi systems, like that of benzene. We will calculate the energy levels, as well as the wavefunctions for butadiene, shown below. Before we move on, a little silliness is in order: Down to business: A. Write out the Secular Determinate for butadiene, with Huckel approximations. B. Find the determinate of this matrix and calculate all the energy levels for butadiene in terms of α and β. (It may help to use the substitution and solve for x first) C. Using your calculated energy levels from the previous question, draw out the energy level diagram for butadiene with its electrons filled. 2. In the following problem, you will prove that the eigenvalues of Hermitian operators are real and their eigenfunctions are orthogonal.
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Unformatted text preview: Given the two eigenvalue equations Âφ n = a n φ n and Âφ m = a m φ m Multiply the first equation by φ m and integrate over all space. Then, take the complex conjugate of the second equation and multiply it by φ n , and integrate over all space. Subtract the two equations from each other and label this equation as (4 – 1). Given that by the definition of a Hermitian operator, discuss the two possibilities n=m and n≠m. In your explanation, make sure you prove that the eigenvalues are real numbers and that the two eigenfunctions are orthogonal. 3. The following are the three normalized sp 2 hybrid orbitals. Show that Ψ 1 is normalized and that Ψ 1 is orthogonal to Ψ 2 . Which orbital is missing from these equations and why is it missing?...
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Homework 1 Part 2 - Given the two eigenvalue equations Âφ...

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