Serial of year 36

Find the Rydberg constant's value and determine which hydrogen spectral lines belong to the visible spectrum. These lines are the only ones Rydberg could use to discover his formula, as neither UV nor IR spectra could yet be measured. What color are they, and which transitions in the Bohr model do they correspond to? (3pts)

Calculate de Broglie wavelength of your body. How does this value compare to the size of an atom or atomic nucleus? (3pts)

Assume you have a cuvette with $10 \mathrm{ml}$ of fluorescein water solution. Then, you point an argon laser at the cuvette. The laser is characterized by a wavelength of $488 \mathrm{nm}$ and a power of $10 \mathrm{W}$. At the same time, the fluorescein molecule fluoresces at a wavelength of $521 \mathrm{nm}$ with a quantum yield (proportion of absorbed photons that are emitted back) of $95 \mathrm{\%}$. If the initial temperature of the cuvette is $20 \mathrm{\C }$, how long will it take for its contents to start boiling? Assume that the cuvette is perfectly thermally insulated, the laser beam is fully absorbed in it, and the amount of fluorescein is negligible in terms of heat capacity. (4pts)

1. Find a beta-carotene molecule and calculate what color should it have or rather what wavelength it absorbs. Use a simple model of an infinite potential well in which $\pi $ electrons from double bonds are „trapped“ (i.e., two electrons for each double bond). The absorption then corresponds to such a transition that an electron jumps from the highest occupied level to the first unoccupied level.
   Compare the calculated value with the experimental one. Why doesn't the value obtained by our model come out the way we would expect? (5b)
   
2. Let's try to improve our model. When studying some substances, especially metals or semiconductors, we introduce the effective mass of the electron. Instead of describing the environment in which the electrons move in a complex way, we pretend that the electrons are lighter or heavier than in reality. What mass would they need to have to give us the correct experimental value? Give the result in multiples of the electron's mass. (2b)
   
3. If we produce microscopic spheres (nanoparticles) of cadmium selenide ($\ce {CdSe}$) with a size of $2{,}34 \mathrm{nm}$, they will glow bright green when irradiated by UV light with a wavelength of $536 \mathrm{nm}$. When enlarged to a size of $2{,}52 \mathrm{nm}$, the wavelength of the emitted light shifts to the yellow region with a wavelength of $570 \mathrm{nm}$. What would the size of spheres need to be to make them emit orange with a wavelength of $590 \mathrm{nm}$? (3b)
   
   Hint: $\ce {CdSe}$ is a semiconductor, so it has a fully occupied electron band, then a (narrow!) forbidden band, and finally an empty conduction band. Thus, we must consider that the emitted photon corresponds to a jump from the conduction band (where such states are as in the infinite potential well) to the occupied band. Therefore, all the energies of the emitted photons will be shifted by an unknown constant value corresponding to the width of the forbidden band.
   

Finally, a bonus for those who would be disappointed if they didn't integrate – the 1s orbital of the hydrogen atom has a spherically symmetric wave function with radial progression $\psi (r) = \frac {e^{-r/a_0}}{\sqrt {\pi }a_0^{3/2}}$, where $a_0=\frac {4\pi \epsilon _0\hbar ^2}{me^2}$ is the Bohr radius. Since the orbitals as functions of three spatial variables would be hard to plot, we prefer to show the region where the electron is most likely to occur. What is the radius of the sphere centered on the nucleus in which the electron will occur with a probability of $95 \mathrm{\%}$? (+2b)

1. Similarly to the series, use the Hückel method to create the Hamiltonian matrix for the cyclobutadiene molecule and verify that its eigenvalues are $\alpha +2\beta $, $\alpha $, $\alpha $, $\alpha -2\beta $. Sketch the diagram of the final energies in the resulting orbitals. And show how the electrons will occupy them. $(4~b)$
   Bonus: What is the main difference in the characterics of these orbitals and their occupancy compared to a benzene molecule we showed in the series? What are the consequences for the cyclobutadiene molecule? $(2~b)$
   
2. Try going back to the beta-carotene molecule and calculate again at what wavelength it should absorb using the Hückel method. What should the value of the parameter $\beta $ be equal to in order to be consistent with the experimental results
   Alternative: If you encounter a problem with the diagonalisation of the hamiltonian, solve the problem statement with the hexa-1,3,5-triene molecule. The experimentally determined absorption value in this case is at a wavelength of $250 \mathrm{nm}$. $(4~b)$
   
3. What happens to a molecule (a molecule with only simple bonds is sufficient) if we use UV light to excite an electron from the $\sigma $ to the $\sigma ^\ast $ orbital? $(2~b)$

English version of the serial will be released soon.

1. At the beginning of the series, we mentioned a couple of approximations we made – fixing the nuclei and also neglecting relativistic effects. Which chemical elements would you expect to have the strongest mutual interaction between the electrons and the motion of the nuclei, and why? In which part of the periodic table do you think relativistic effects will be most apparent? What is the reason? $\(2 \mathrm{pts}\)$
2. The total energy of a water molecule, obtained from a quantum chemical calculation, is approximatelly $-75 \mathrm{Ha}$. The energy released by the fusion of hydrogen and oxygen into water is $242 \mathrm{kJ\cdot mol^{-1}}$. If we calculate the energy of both the reactants and products with an error of $1 \mathrm{\%}$, how big will the error be in the determination of the reaction energy? Also, try to find some analogy to real-life measurements. (For example: “I would weigh myself with a five-crown coin and without it to determine its weight.“) $\(3 \mathrm{pts}\)$
3. Install the program Psi4 and try to calculate the difference of energies of the chair and (twist-)boat conformations of cyclohexane. You can use the attached input files, where the geometry is already optimized. How much does the result differ from the experimental value $21 \mathrm{kJ\cdot mol^{-1}}$? $\(2 \mathrm{pts}\)$ $\\$ Note: If you encounter a problem with Psi4, please feel free to contact me at ${\href{mailto:mikulas@fykos.cz}{mikulas@fykos.cz}}$
   
4. Try calculating the reaction energy for the chlorination of benzene $\ce{C}_{6}\ce{H}_{6} + \ce{Cl}_{2} \Rightarrow \ce{C}_{6}\ce{H}_{5}\ce{Cl} + \ce{HCl}$. Compare it with the experimental value of $-134 \mathrm{kJ\cdot mol^{-1}}$. You can use the included geometry of the benzene molecule. $\(3 \mathrm{pts}\)$ $\\$ Bonus: Choose your favorite (or any other) chemical reaction and calculate its energy. (up to $+3 \mathrm{pts}$)

The binding energy of a fluorine molecule is approximately $37 \mathrm{kcal/mol}$. Assuming the range of binding interactions to be approximately $3 \mathrm{\AA }$ from the optimum distance, what (average) force do we have to exert to break the molecule? Calculate the „stiffness“ of the fluorine molecule if such an average force was applied in the middle of this range. What would be the vibrational frequency of this molecule? Compare this with the experimental value of $916{,}6 \mathrm{cm^{-1}}$. ($4 \mathrm{pts}$)

Using Psi4, calculate the dissociation curve $\mathrm {F_2}$ and fit a parabola around the minimum. What value will you get for the energy of the vibrational transitions this time? ($3 \mathrm{pts}$)

You are given two bottles of alcohol that you found suspicious, to say the least. After taking them to the lab, you obtain the following Raman spectra from them. Using the Psi4 program, calculate the frequencies at which the vibrational transitions of both the methanol and ethanol molecules occur. Use this to determine which bottle contains methanol and which one contains ethanol. You can use the approximate geometries of ethanol and methanol, which are included in the problem statement on the web. ($3 \mathrm{pts}$)

The lowest-lying excited singlet state of beta-carotene has an energy $1{,}8 \mathrm{eV}$, which is higher than the ground state energy. However, the transition between this state and the ground state is prohibited, so the molecule does not absorb photons at this energy. On the other hand, the transition to the second lowest-lying singlet state with energy $2{,}4 \mathrm{eV}$ is allowed and responsible for the bright orange color of the molecule. The lowest-lying triplet level is at $0{,}9 \mathrm{eV}$ energy. Draw a Jablonski diagram and use it to explain why beta-carotene does not fluoresce even though it significantly absorbs visible light. $\(3 \mathrm{pts}\)$

Bonus:: Why is it so important for life on earth that oxygen is a triplet in the ground state? $\(+1 \mathrm{pts}\)$

Try to calculate the approximate limit on the number of orbitals in the active space with the CASSCF method. Consider that you have as many electrons as orbitals in the active space (which corresponds to the fact that half of them in $\ce {HF}$ will be occupied) and that the most of today's supercomputers have at most $1 \mathrm{TB}$ of RAM for computing, in which you need to fit a Hamiltonian. $\(3 \mathrm{pts}\)$

For lithographic manufactured modern semiconductor chips, so-called excimer lasers are used to glow with the spectrum far into UV region. They are based on so-called excimers, which are molecules that are stable only in the excited state, while in the ground state, they decay. As a result, the molecule decays after the photon is emitted, ensuring that a larger fraction of the molecules are in the higher state than in the lower state. That is the necessary condition for the laser to work. Try using Psi4 for the helium dimer ($\ce{He}_{2}^*$) to calculate and plot the dissociation curves of the ground and lowest-laying excited states. ($\ce{He}_{2}^*$) is not yet used for lasers, but for example $\ce{Ar}_{2}^*$ or $\ce{Kr}_{2}^*$ are.) At what wavelength would the laser work? Compare it with the experimental wavelength $66 \mathrm{nm}$. $\(4 \mathrm{pts}\)$

Note:: In the problem statement on the website, you will find a prepared input file for one geometry. Do not be surprised that it has a total of three states set up. It needs to have those because we have two excited states close to each other. If we were to include only one of them in the calculations for some internuclear distances, this would lead to problems with convergence.