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molecular physics

(10 points)6. Series 36. Year - S. exciting quanta

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.

Do not worry; the next gift from Mikuláš will not arrive before 5th December.

(10 points)5. Series 36. Year - S. ethanol or methanol?

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. ($3 \mathrm{pts}$)

Alcohol from Mikuláš?!

(10 points)4. Series 36. Year - S. quantum of molecules

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 $+3 \mathrm{pts}$)

Mikuláš keeps on giving even after Christmas.

(10 points)3. Series 36. Year - S. quantum of orbital

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? $(2~b)$

Mikuláš gives presents again, this time at the right time of the year, almost.

(10 points)2. Series 36. Year - P. planetary atmosphere

What parameters does a planet need to have to keep its atmosphere comparable to the Earth? What conditions are essential for the planet to gain such an atmosphere?

(5 points)6. Series 35. Year - 3. wind bubble

Imagine we create a small soap bubble with a bubble blower. How fast does it fall to the ground? The bubble has an outer radius $R$ and an areal density $s$.

Karel was making bubbles in the bathtub.

(10 points)4. Series 35. Year - S. shining

1. How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma1)
1)
The density of plasma $n_e$ is typically expressed as a funciton $n_e = f$\frac {x}{x_c}$$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume
1. that the density of the plasma decreases exponentially with distance from the target,
2. that the density of the plasma decreases linearly with distance from the target.
1. What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.
2. What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?
3. Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

(13 points)2. Series 35. Year - E. light or dense ethanol

Measure the dependance of the density of an alcohol solution in water on its volume concentration in water. Include also the measurement of pure alcohol and pure water for comparison.

Be careful when mixing alcohol and water – remember that the volume of the mixture is not exactly the sum of their original volumes.

Karel was thinking that the participants might have a little sniff.

(10 points)2. Series 35. Year - S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? Note: Assume that the initial plasma is a monatomic ideal gas.

(10 points)1. Series 35. Year - S. commencing fusion

1. Determine the energy gain of the following reactions and the kinetic energy of their products

\begin{align*} {}^{2}\mathrm {D} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {T} + \mathrm {p} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {He} + \mathrm {n} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {n} ,\\ {}^{3}\mathrm {He} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} + \mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + {}^{2}\mathrm {D} ,\\ \mathrm {p} + {}^{11}\mathrm {B} &\rightarrow 3\;{}^{4}\mathrm {He} ,\\ {}^{2}\mathrm {D} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {p} . \end {align*}

1. By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:
1. deuterium - deuterium,
2. proton - boron,
3. deuterium - helium-3.

Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?

1. What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle
1. $E\_k = k\_B T^\alpha$,
2. $E\_k = a T^3 + b T^2 + c T$.

Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?

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