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The superconducting state in Carbonaceous Sulfur Hydride (C-S-H) system is characterized by the record-high critical temperature of $288$~K experimentally observed at $\sim$267 GPa. Herein, we determined the properties of the \mbox{C-S-H} superconducting phase within the scope of both classical Eliashberg equations (CEE) and the Eliashberg equations with vertex corrections (VCEE). We took into account the scenarios pertinent to either the intermediate or the high value of electron-phonon coupling constant ($\lambda\sim 0.75$ or $\lambda\sim 3.3$, respectively). The scenario for the intermediate value, however, cannot be actually realized due to the anomally high value of logarithmic phonon frequency ($\omega_{\rm ln}/k_{B}=7150$~K) it would require. On the other hand, we found it possible to reproduce correctly the value of $T_{C}$ and other thermodynamic quantities in the case of strong coupling. However, the vertex corrections lower the order parameter values within the range from $\sim 50$~K to $\sim275$~K. For the upper critical field $H_{C2}\sim 27$~T, the Ginzburg-Landau parameter $\kappa$ is of the order of $1.7$. This correlates well with the sharp drop of resistance observed by Hirsch and Marsiglio at the critical temperature. The strong-coupling scenario for C-S-H system is also suggested by the high values of $\lambda$ estimated for ${\rm H_{3}S}$ ($\lambda\sim 2.1$, $\kappa\sim 1.5$), ${\rm LaH_{10}}$ ($\lambda\sim 2.8$-$3.9$, $\kappa\sim 1.6$), and ${\rm YH_{6}}$ ($\lambda\sim 1.7$, $\kappa\sim 1.3$) compounds., Comment: 4 pages

Thermodynamic parameters of the $\rm LaH_{10}$ superconductor were an object of our interest. $\rm LaH_{10}$ is characterised by the highest experimentally observed value of the critical temperature: $T^{a}_{C}=215$~K ($p_{a}=150$~GPa) and $T^{b}_{C}=260$~K ($p_{b}=190$~GPa). It belongs to the group of superconductors with a~strong electron--phonon coupling ($\lambda_{a}\sim 2.2$ and $\lambda_{b}\sim 2.8$). We calculated thermodynamical parameters of this superconductor and found that the values of the order parameter, the thermodynamic critical field, and the specific heat differ significantly from the values predicted by the conventional BCS theory. Due to the specific structure of the Eliashberg function for the hydrogenated compounds, the qualitative analysis suggests that the superconductors of the ${\rm La_{\delta}X_{1-\delta}H_{10}}$--type (LaXH--type) structure, where ${\rm X}\in\{{\rm Sc},{\rm Y}\}$, would exhibit significantly higher critical temperature than $T_{C}$ obtained for $\rm LaH_{10}$. In the case of LaScH we came to the following assessments: $T^{a}_{C}\in\left<220,267\right>$~K and $T^{b}_{C}\in\left<263,294\right>$~K, while the results for LaYH were: $T^{a}_{C}\in\left<218,247\right>$~K and $T^{b}_{C}\in\left<261,274\right>$~K., Comment: 6 pages, 5 figures

In the framework of strong-coupling formalism, we have calculated the thermodynamic parameters of superconducting state in the ${\rm YB_{6}}$ compound. The values of critical temperature ($T_{C}$) are $9.5$~K and $7.9$~K for the Coulomb pseudopotential $\mu^{\star}=0.1$ and $0.2$, respectively. In the paper, we have determined the low temperature values of order parameter ($\Delta(0)$), specific heat jump at the critical temperature ($\Delta C(T_{C})$), and thermodynamic critical field ($H_C(0)$). The dimensionless thermodynamic ratios: $R_{\Delta}=2\Delta\left(0\right)/{k_BT_C}$, $R_C=\Delta C\left(T_C\right)/C^N\left(T_C\right)$, and $R_H=T_CC^N\left(T_C\right)/H_C^2\left(0\right)$ are equal to: $R_{\Delta}\left(\mu^{\star}\right)\in\lbrace 4.48,4.35\rbrace$, $R_{C}\left(\mu^{\star}\right)\in\lbrace 2.62,2.55\rbrace$, and $R_{H}\left(\mu^{\star}\right)\in\lbrace 0.146,0.157\rbrace$. Due to the significant strong-coupling and retardation effects ($k_{B}T_{C} / \omega_{\rm ln}\sim 0.1$) those values highly deviate from the predictions of BCS theory., Comment: 5 pages, 4 figures

We have determined the entropy, the total energy, and the specific heat of the systems consisting of $M\geq 3$ hydrogen molecules. The calculations were conducted in the framework of the nonextensive Tsallis statistics. The relation between $M$ and the entropic index $q$ is given by $q = 1 + 1/M$, which results from the fact that the temperature of the nanosystems fluctuates around the temperature of the reservoir (Wilk and W{\l}odarczyk, Phys. Rev. Lett. {\bf 84}, 2770 (2000)). The electron energy states of the hydrogen molecule have been determined with the help of the Hubbard Hamiltonian, which models all two-body interactions. The Hubbard Hamiltonian integrals have been calculated by using the variational method, whereas the Wannier function has been associated with $1s$ Slater-type orbitals. We have included the contributions to the energy of the hydrogen molecule coming from the oscillatory (either harmonic or anharmonic), rotational and translational degrees of freedom. In addition, we have investigated the impact of the external force ($F$) or the magnetic field ($h$) on the thermodynamic parameters of the systems. In each case the noticeable deviation from the results of the classical statistical physics can be observed for the systems consisting of $M