Questions 45 - 50

Q45 Eigenvectors & eigenvalues

Write down the characteristic equations and find the eigenvectors and normalized eigenvalues of the following two matrices. Check whether the eigenvectors are orthogonal.

\[\begin{split}(a) \displaystyle \begin{bmatrix}4 & -i\\i & 2 \end{bmatrix} , \qquad (b) \begin{bmatrix} 1& -i\\ -i & 1 \end{bmatrix}\end{split}\]

Strategy: Expand the determinants of the matrices to find the characteristic equations, then use Python/Sympy, as shown in the text, or, if you wish, do the calculation by hand. The normalization term is \(\sqrt{v\cdot v}\). The numpy/Sympy instruction for a dot product is a.dot(b)

Q46 Eigenvalues

Find the eigenvalues of the matrix by hand

\[\begin{split}\displaystyle \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 4 & 0 & 0 & 0 \\ 0 & 5 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 0 \\ 0 & 0 & 0 & 3 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ \end{bmatrix} \end{split}\]

and confirm using a computer.

Strategy: This is a block diagonal matrix so use this property to make smaller matrices.

Q47 Solve equations

Find the characteristic equation, eigenvalues, and eigenvectors of \(\displaystyle \begin{bmatrix}0 & -u & v\\u & 0 & 0 \\ -v & 0 & 0 \end{bmatrix}\) Simplify your answers using \(z^2 = u^2 + v^2\).

Strategy: Use python/Sympy to solve the characteristic equation. The pattern of cofactors are shown in eqn2.

Q48 Butadiene MO energies

In Section 2.5(iii) the MO energies of butadiene were calculated by the Huckel method, using as a basis set the atomic wavefunctions \((\psi_1, \psi_2, \psi_3, \psi_4)\), where the subscript labels the \(n = 4\) atoms. The Huckel matrix is

\[\begin{split}\displaystyle \begin{bmatrix} x & 1 & 0 & 0 \\ 1 & x & 1 & 0 \\ 0 & 1 & x & 1 \\ 0 & 0 & 1 & x \\ \end{bmatrix} =0, \qquad \text{where} \qquad x\frac{\alpha-E}{\beta}\end{split}\]

(a) Using the eigenvalue - eigenvector method, calculate not only the energies, but also the orbital coefficients, which are the eigenvectors.

(b) Calculate the delocalization energy, which is the Huckel energy less \(n(\alpha + \beta)\) where \(\alpha\) is the Coulomb self-energy of a \(\pi\) electron and \(\beta\) the overlap energy.

(c) Calculate the bond order, charge density, and dipole moment. The bond order is

\[\displaystyle \rho_{ab}=\sum_i^n m_ic_{ai}c_{bi}\]

where \(c_{ai}\) is the coefficient on carbon atom \(a\) and of orbital \(i\) and \(m_i = 0, 1, 2 \) and is the number of electrons in orbital \(i\). The total bond order is larger by \(1\) when the \(\sigma\) bond order is added.

The charge density is

\[\displaystyle q_a = \sum_i m_i|c_{ia}|^2\]

and dipole moment

\[\displaystyle d_\pi=\sum_a(1-q_a)r_a\]

where \(r_a\) is the coordinate of atom \(a\). The dipole moment of a CH bond is 0.3 D.

Strategy: As this is a largish matrix and not block diagonal, use python/Sympy to perform the calculation. Each MO with index \(i\) is the wavefunction \(\Psi_i = c_{1i}\psi_1 + c_{2i}\psi_2 + c_{3i}\psi_1 + c_{4i}\psi_1\) where the \(c\)’s are the elements of the \(i^{th}\) eigenvector; the second (column) index identifies the eigenvalue i, the first the atom.

The full spatial dependence of the orbitals would involve calculating the \(\psi\) in three dimensions; instead, and just as effectively, the coefficients \(c\) are used to represent the \(\pi\) electron density on each atom which allows us to find the MO’s pattern and hence the number of nodes. The node pattern can be used to determine the energy ordering; as a rule of thumb, the larger the number of nodes the higher the energy.

Q49 Fulvalene MO energies

Repeat the calculation of the previous question, (Q48), for fulvalene; see Q8 for the numbering of the atoms. Confirm that the dipole is \(-0.711eL\) or \(0.48\) D where \(e\) is the charge on the electron (in Coulombs) and \(L\) is the bond length, which is \(\approx 140\) pm. The experimentally measured dipole is \(0.4\) D. Take atom 2 to be at the origin (\(x=y=0\)) Confirm also that the \(\pi\) bond order between atoms is as shown below:

\[\begin{split}\displaystyle \begin{array}{c|cc} \text{bond number} & \text{Bond order}\\ \hline 1 \to 2 & 0.759\\ 2 \to 3 & 0.499\\ 3 \to 4 & 0.788\\ 4 \to 5 & 0.520 \\ \hline \end{array}\end{split}\]

The remainder of the bond orders follow by symmetry.

Q50 Benzene MO energies

Repeat the Huckel MO calculation for benzene; the matrix is worked out in Q7. Calculate the eigenvectors and plot out the MO coefficients.

(a) Is the pattern what you expect? You should find that some MOs are simply rotations of others. What distinguishes these?

(b) Make linear combinations of these MOs to form new MOs with the usual orbital shapes. Draw out the results you obtain.

Strategy: for the first part, use python/Sympy to obtain the eigenvectors.