<p>a.) Assume the H$_2$O molecule lies completely in the yz-plane as shown in Fig. 1. It belongs to the $C_{2v}$ symmetry point group, which includes the identity element $E$, a $180°$ degree rotation around the z-axis $C_2$, and the two mirror planes $\sigma_v$ and $\sigma_v'$. Complete the character table below, showing the four irreducible representations of this symmetry group.</p>
$$
\begin{array}{c|cccc}
C_{2v} & E & C_2 & \sigma_{v} & \sigma_{v'} \\
\hline
A_1 & 1 & 1 & 1 & \\
A_2 & 1 & 1 & & -1 \\
B_1 & 1 & & 1 & -1 \\
B_2 & & -1 & -1 & 1 \\
\end{array}
$$<p>$\textit{Hint: Use that the rows are orthogonal and normalized(norm$^2$ = $|C_{2v}|$ = 4, where $|C_{2v}|$ is the number of elements in the group) with respect to the usual scalar product, and that the identity column is the dimension of the representation.}$</p>
\end{align*}<p>with $m_{\alpha}$ being the mass of molecule $\alpha$ and $(x_{\alpha}, y_{\alpha},z_{\alpha})$ and $(x_{\alpha, e}, y_{\alpha, e}, z_{\alpha, e})$ being its position and equilibrium position respectively.</p>
<p>Write the character table for the physical 9 dimensional representation $\Gamma
<p>c.) Divide $\Gamma$(H$_2$O) into irreducible representations.</p>
<p>$\textit{Hint: Again use that the rows corresponding to the irreducible representations are orthogonal and normalized with respect to the usual scalar product.}$</p>
<p>The remaining three irreducible representations will correspond to the three normal vibrational modes, which we will study next. Departing from general symmetry considerations (which work for any potential energy $V$) we choose the following explicit potential energy</p>
$$V = \frac{1}{2}k_r (r_1 - r_{e})^2 + \frac{1}{2}k_r (r_2-r_{e})^2 + \frac{1}{2}k_\theta(\theta - \theta_e)^2,$$<p>where $r_1 = |\vec{r}_{H1}-\vec{r}_O|$, $r_2 = |\vec{r}_{O}-\vec{r}_{H2}|$ and $\theta$ is the angle between the H$_1$O and OH$_{2}$ bonds. The interatomic distance $r_e$ as well as the radial and angular force constants $k_r$ and $k_\theta$ are assumed to be constant.</p>
<span class="n">phl</span> <span class="o">=</span> <span class="c1">## photon wavelength vector, choose the three eigenmodes which correspond to the vibrations</span>
<p>i.) Sketch the eigenmodes of the hydrogen molecule. Are the corresponding irreducible representations in accordance with exercises c.), d.) and e.)</p>
<span class="n">dy</span> <span class="o">=</span> <span class="n">evecs</span><span class="p">[::</span><span class="mi">2</span><span class="p">,</span><span class="c1">##]/4 #displacements y-direction (the division by four is simply to make the arrows smaller) </span>
<span class="n">dz</span> <span class="o">=</span> <span class="n">evecs</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span><span class="c1">##]/4 #displacements z-direction (the division by four is simply to make the arrows smaller) </span>
In the 1950s, azulene drew significant interest from the scientific community. As one of the first examples, theoretical (computer-assisted) calculations succeeded to quantitatively determine the excited state energies (1956, Pariser) and allowed for a reasonable assignment of the transitions measured by electronic spectroscopy.
In this exercise, we’ll follow a simplified version of that early research and use the (basic) Hückel model to calculate the electronic states.</i>
<p>a) Using the Hückel model (Serie 7, ex 3), numerically calculate the electronic states and energies of azulene. What is the HOMO-LUMO gap and the corresponding photon wavelength?
$\newline$</p>
<p>$\textit{Remark: You can use your old code.}$</p>
<p>b) Use the code below to generate the color corresponding to the photon wavelength. Which color do you think azulene will have? Compare to a quick web search.</p>
<p>$\textit{The wavelength calculated in a.) is the adsorbed one. The color seen is the transmitted/reflected part of the spectrum that manifests as the complementary color.}$</p>
<p>$\textit{Remark: For those interested you may look up Lactarius indigo, a surprsingly edible mushroom containing the azulene pigment.}$</p>
<p>c.) The transitions calculated in a.) corresponds to the $S_0 \to S_1$ (ground state to first excited state) absorption. Find the energy and photon wavelength corresponding to the $S_0 \to S_2$ absorption.</p>
<p>$\textit{Hint: The 4 molecular orbitals that normally need to be considered are the two below and two above the Fermi energy.}$
$\textit{They are named HOMO-1, HOMO, LUMO and LUMO + 1 with ascending energy.}$
$\textit{Remark: The actual photon wavelength of the $S_0 \to S_2$ transition is slightly smaller than calculated (around 350 nm) and hence does not have an effect on the color of azulene.}$</p>
<p>In the following we will verify our results using symmetry considerations.</p>
<p>Assume that the azulene molecule lies in the $yz$-plane, with $z$ pointing along the long axis of the molecule. We first note that the symmetry group of azulene is $C_{2v}$ (see exercise 1) and recall that it consists of four elements $id$ (the identity), $C_2$ ($180^°$ rotation around the z-axis), $\sigma_{v}$ (reflection with respect to the $xz$-plane) and $\sigma_{v'}$ (reflection with respect to the $yz$ plane).</p>
<p>e) We know from experiment that the first excited state $S_1$ has $B_2$ symmetry, while the second excited stated $S_2$ has $A_1$ symmetry. Do you arrive at the same result using the Hückel model?</p>
<p>$\textit{Hint:}$ Generally the representation of $n$ electrons is given as $\otimes_i \Gamma_i$, where $\Gamma_i$ are the representations of electron $i$. The characters of a tensor representation are the products of the characters of the individual representations.</p>