Fixed Points Theorems

V této práci si nejprve řekneme, co je to pevný bod, a následně se podíváme na pevné body zobrazení f:ℝ → ℝ. Poté si nadefinujeme metrické prostory a pojmy potřebné k formulaci a důkazu Banachovy věty o pevném bodě. Následně si ukážeme některé aplikace Banachovy věty (numerický výpočet odmocniny, řešení soustav rovnic, věta o implicitní funkci, věta o existenci a jednoznačnosti řešení Cauchyovy úlohy), kde nám samotná Banachova věta o pevném bodě nabídne i metodu k nalezení řešení, což si ukážeme na příkladech.In this bachelor thesis we first take a look at fixed points and then fixed points of function f:ℝ → ℝ. After that we define metric space and concepts needed to formulation and proof of Banach fixed-point theorem. Then we show some applications of Banach fixed-point theorem (numerical calculation of the square root, solving systems of equations, implicit function theorem, theorem on the existence and uniqueness of solutions of the Cauchy problem), where Banach fixed-point theorem will give us method to find solution which we will show in the examples.470 - Katedra aplikované matematikyvýborn

1 H NMR is not a proof of hydrogen bonds in transition metal complexes

Ministry of Education, Youth and Sports of the Czech Republic Program NPU I [LO1504]; Czech Science FoundationGrant Agency of the Czech Republic [17-07091S]; CESNET [LM2015042]; CERIT Scientific Clouds [LM2015085

Relativity or aromaticity? A first-principles perspective of chemical shifts in osmabenzene and osmapentalene derivatives

We have studied the magnetic response properties and aromaticity of osmium metallacycles by means of scalar-relativistic (1c) and fully relativistic (4c) density functional theory computations. For osmabenzene, whose aromatic character is controversial, a topological analysis of the current density has revealed the presence of a unique σ-type Craig-Möbius magnetic aromaticity. We show that the partially filled osmium valence shell induces a large paratropic current, which may interfere with certain methods commonly used to analyze aromaticity, in particular NICS. Further, we show that the extreme deshielding of the light atoms in the vicinity of the osmium atoms in osmapentalene derivatives is not a consequence of aromaticity but can be explained by paramagnetic couplings between σOs - C bonding orbitals and the π∗Os orbitals. We demonstrate that variations in the orientation of the induced magnetic currents through the molecule dictates the alternating signs of the spin-orbit contribution to the NMR chemical shift. This journal is © the Owner Societies.programme "Projects of Large Research, Development, and Innovations Infrastructures" [CESNET LM2015042]; Research Council of NorwayResearch Council of Norway [LM2015070]; Program I [LO1504]; programme "Projects of Large Research, Development, and Innovations Infrastructures" by CERIT Scientific Cloud [CERIT Scientific Cloud LM2015085]; programme "Projects of Large Research, Development, and Innovations Infrastructures" by IT4 Innovations National Supercomputing Center [LM2015070

Linking the Character of the Metal-Ligand Bond to the Ligand NMR Shielding in Transition-Metal Complexes: NMR Contributions from Spin-Orbit Coupling

Relativistic effects significantly affect various spectroscopic properties of compounds containing heavy elements. Particularly in Nuclear Magnetic Resonance (NMR) spectroscopy, the heavy atoms strongly influence the NMR shielding constants of neighboring light atoms. In this account we analyze paramagnetic contributions to NMR shielding constants and their modulation by relativistic spin-orbit effects in a series of transition-metal complexes of Pt(II), Au(I), Au(III), and Hg(II). We show how the paramagnetic NMR shielding and spin-orbit effects relate to the character of the metal-ligand (M-L) bond. A correlation between the (back)-donation character of the M-L bond in d10 Au(I) complexes and the propagation of the spin-orbit (SO) effects from M to L through the M-L bond influencing the ligand NMR shielding via the Fermi-contact mechanism is found and rationalized by using third-order perturbation theory. The SO effects on the ligand NMR shielding are demonstrated to be driven by both the electronic structure of M and the nature of the trans ligand, sharing the σ-bonding metal orbital with the NMR spectator atom L. The deshielding paramagnetic contribution is linked to the σ-type M-L bonding orbitals, which are notably affected by the trans ligand. The SO deshielding role of σ-type orbitals is enhanced in d10 Hg(II) complexes with the Hg 6p atomic orbital involved in the M-L bonding. In contrast, in d8 Pt(II) complexes, occupied π-type orbitals play a dominant role in the SO-altered magnetic couplings due to the accessibility of vacant antibonding σ-type MOs in formally open 5d-shell (d8). This results in a significant SO shielding at the light atom. The energy- and composition-modulation of σ- vs π-type orbitals by spin-orbit coupling is rationalized and supported by visualizing the SO-induced changes in the electron density around the metal and light atoms (spin-orbit electron deformation density, SO-EDD). © 2017 American Chemical Society.Czech Science Foundation [16-05961S, 15-09381S]; Ministry of Education, Youth and Sports of the Czech Republic [LQ1601, LO1504]; multilateral cooperation project [8X17009]; SASPRO Program [1563/03/02]; European Union; Slovak Academy of Sciences; Grant Agency of the Ministry of Education of the Slovak Republic; Slovak Academy of Sciences VEGA [2/0116/17]; Research Council of Norway [179568]; Norwegian supercomputing program NOTUR [NN4654K

Exact two-component TDDFT with simple two-electron picture-change corrections: X-ray absorption spectra near L- and M-edges of four-component quality at two-component cost

X-ray absorption spectroscopy (XAS) has gained popularity in recent years as it probes matter with high spatial and elemental sensitivities. However, the theoretical modeling of XAS is a challenging task since XAS spectra feature a fine structure due to scalar (SC) and spin-orbit (SO) relativistic effects, in particular near L and M absorption edges. While full four-component (4c) calculations of XAS are nowadays feasible, there is still interest in developing approximate relativistic methods that enable XAS calculations at the two-component (2c) level while maintaining the accuracy of the parent 4c approach. In this article we present theoretical and numerical insights into two simple yet accurate 2c approaches based on an (extended) atomic mean-field exact two-component Hamiltonian framework, (e)amfX2C, for the calculation of XAS using linear eigenvalue and damped response time-dependent density functional theory (TDDFT). In contrast to the commonly used one-electron X2C (1eX2C) Hamiltonian, both amfX2C and eamfX2C account for the SC and SO two-electron and exchange-correlation picture-change (PC) effects that arise from the X2C transformation. As we demonstrate on L- and M-edge XAS spectra of transition metal and actinide compounds, the absence of PC corrections in the 1eX2C approximation results in a substantial overestimation of SO splittings, whereas (e)amfX2C Hamiltonians reproduce all essential spectral features such as shape, position, and SO splitting of the 4c references in excellent agreement, while offering significant computational savings. Therefore, the (e)amfX2C PC correction models presented here constitute reliable relativistic 2c quantum-chemical approaches for modeling XAS. © 2023 The Authors. Published by American Chemical Society.2/0135/21; NN4654K; Ministerstvo Školství, Mládeže a Tělovýchovy, MŠMT: RP/CPS/2022/007; Agentúra na Podporu Výskumu a Vývoja, APVV: APVV-19-0516, APVV-21-0497; Norges Forskningsråd: 262695, 314814, 315822; Horizon 2020: 945478, SASPRO

Structure, solvent, and relativistic effects on the NMR chemical shifts in square-planar transition-metal complexes: assessment of DFT approaches

The role of various factors (structure, solvent, and relativistic treatment) was evaluated for square-planar 4d and 5d transition-metal complexes. The DFT method for calculating the structures was calibrated using a cluster approach and compared to X-ray geometries, with the PBE0 functional (def2-TZVPP basis set) providing the best results, followed closely by the hybrid TPSSH and the MN12SX functionals. Calculations of the NMR chemical shifts using the two-component (2c, Zeroth-Order Regular Approximation as implemented in the ADF package) and four-component (4c, Dirac-Coulomb as implemented in the ReSpect code) relativistic approaches were performed to analyze and demonstrate the importance of solvent corrections (2c) as well as a proper treatment of relativistic effects (4c). The importance of increased exact-exchange admixture in the functional (here PBE0) for reproducing the experimental data using the current implementation of the 2c approach is partly rationalized as a compensation for the missing exchange-correlation response kernel. The kernel contribution was identified to be about 15-20% of the spin-orbit-induced NMR chemical shift, DdSO, which roughly corresponds to an increase in DdSO introduced by the artificially increased exact-exchange admixture in the functional. Finally, the role of individual effects (geometry, solvent, relativity) in the NMR chemical shift is discussed in selected complexes. Although a fully relativistic DFT approach is still awaiting the implementation of GIAOs for hybrid functionals and an implicit solvent model, it nevertheless provides reliable NMR chemical shift data at an affordable computational cost. It is expected to outperform the 2c approach, in particular for the calculation of NMR parameters in heavy-element compounds.Czech Science Foundation [15-09381S, 14-03564S]; European Regional Development Fund [CZ.1.05/1.1.00/02.0068]; Research Council of Norway through a Centre of Excellence [179568, 214095, 177558]; Czech-Norway mobility grant from Norway Funds [NF-CZ07-MOP-3-245-2015]; program Center CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations [CZ.1.05/3.2.00/08.0144

Unwilling U-U bonding in U-2@C-80: cage-driven metal-metal bonds in di-uranium fullerenes

Endohedral actinide fullerenes are rare and a little is known about their molecular properties. Here we characterize the U2@C80 system, which was recently detected experimentally by means of mass spectrometry (Akiyama et al., JACS, 2001, 123, 181). Theoretical calculations predict a stable endohedral system, 7U2@C80, derived from the C80:7 IPR fullerene cage, with six unpaired electrons. Bonding analysis reveals a double ferromagnetic (one-electron-two-center) U-U bond at an rU-U distance of 3.9 Å. This bonding is realized mainly via U(5f) orbitals. The U-U interaction inside the cage is estimated to be about -18 kcal mol-1. U-U bonding is further studied along the U2@Cn (n = 60, 70, 80, 84, 90) series and the U-U bonds are also identified in U2@C70 and U2@C84 systems at rU-U ∼ 4 Å. It is found that the character of U-U bonding depends on the U-U distance, which is dictated by the cage type. A concept of unwilling metal-metal bonding is suggested: uranium atoms are strongly bound to the cage and carry a positive charge. Pushing the U(5f) electron density into the U-U bonding region reduces electrostatic repulsion between enclosed atoms, thus forcing U-U bonds. This journal is © the Owner Societies 2015.Czech Science Foundation [14-03564S]; Czech Academy of Sciences [RVO-61388963]; SoMoPro II program; People Program (Marie Curie action) of the Seventh Framework Program of EU [291782]; South-Moravian Region; European Regional Development Fund [CZ.1.05/1.1.00/02.0068

Enhancing cisplatin anticancer effectivity and migrastatic potential by modulation of molecular weight of oxidized dextran carrier

The molecular weight (Mw) of dextran derivatives, such as regioselectively oxidized dicarboxydextran (DXA), is greatly influencing their faith in an organism, which could be possibly used to improve anticancer drug delivery. Here we present a modified method of sulfonation-induced chain scission allowing direct and accurate control over the Mw of DXA without increasing its polydispersity. Prepared DXA derivatives (Mw = 10–185 kDa) have been conjugated to cisplatin and the Mw of the carrier found to have a significant impact on cisplatin release rates, in vitro cytotoxicity, and migrastatic potential. Conjugates with the high-Mw DXA showed particularly increased anticancer efficacy. The best conjugate was four times more effective against malignant prostatic cell lines than free cisplatin and significantly inhibited the ovarian cancer cell migration. This was traced to the characteristics of spontaneously formed cisplatin-crosslinked DXA nanogels influenced by Mw of DXA and amount of loaded cisplatin. © 2021 Elsevier LtdMinistry of Education, Youth, and Sports of the Czech Republic -DKRVO [RP/CPS/2020/006]; Ministry of Education, Youth and Sports of the Czech RepublicMinistry of Education, Youth & Sports - Czech Republic [CZ.02.1.01/0.0/0.0/15_003/0000444, MUNI/A/1698/2020, MUNI/A/1246/2020, LM2018127]; Masaryk University in Brno [InGA/SUP/08/2020]; Czech Science FoundationGrant Agency of the Czech Republic [19-16861S]; Charles University in Prague [Progress Q26/LF1, Q27/LF1]RP/CPS/2020/006; Univerzita Karlova v Praze, UK: Q27/LF1; Ministerstvo Školství, Mládeže a Tělovýchovy, MŠMT: CZ.02.1.01/0.0/0.0/15_003/0000444, InGA/SUP/08/2020, MUNI/A/1246/2020, MUNI/A/1698/2020; Grantová Agentura České Republiky, GA ČR: 19-16861S, LM201812