Computational Electrochemistry of 3d Transition Metal Complexes

The topic of this thesis is the computational quantum chemical (QC) description of homogeneous first-row (3d) transition metal (TM) electrocatalysis. This branch of chemistry holds great potential for employing Earth-abundant 3d TMs in renewable energy concepts. Therefore, routine predictions for the reactivity of 3d TM electrocatalysts are desirable, but due to numerous challenges, they are only possible to a limited extent. The thesis describes the development, assessment, and application of QC methods with the aim of such predictions. In the first Chapter, an introduction to the QC treatment of 3d TM electrocatalysis is given, followed by a brief overview of the different QC methods in the second Chapter. The computationally most affordable methods are the semiempirical quantum mechanical (SQM) methods, which are the topic of the third Chapter, where the inclusion of spin-polarization in the extended tight-binding Hamiltonian (xTB) is elaborated. The next higher QC level is density functional theory (DFT), which is the topic of Chapter four. Here, the extension of the non-empirical r²SCAN density functional approximation (DFA) to the hybrid level, resulting in the r2SCANh, r2SCAN0, and r2SCAN50 DFAs, is described. At the highest DFT level are the double-hybrids (DHs), which are the subject of Chapter five. Their applicability is extended with the domain-based local pair natural orbital (DLPNO) approximation for second-order Møller–Plesset perturbation theory (MP2). The highest level belongs to the wave function theory (WFT) methods. Their application can face severe difficulties in 3d TM electrocatalysis due to multireference (MR) character, which is the subject of the sixth Chapter. Here, the recognition of MR systems and the calculation of their ionization potentials (IPs) is studied at the highest feasible WFT level. For this purpose, a new benchmark set of electrocatalysts, termed 3dTMV, is compiled, and coupled cluster calculations (CCSD(T)) as well as quantum Monte Carlo (ph-AFQMC) calculations were conducted. Chapter seven deals with the application of SQM and DFT methods for the elucidation of electrocatalytic cycles with three-legged piano-stool iron complexes. An efficient workflow is presented for the calculation of Gibbs free energies yielding a free energy map that is used to propose an initial catalytic cycle. The extension of the free energy map to also include kinetics by transition state theory is shown in Chapter eight. Finally, in the ninth Chapter, the findings of this work are summarized, and their impact on the theoretical description of 3d TM electrocatalysis and 3d TM chemistry in general, are evaluated. Novel QC workflows can benefit from the methods and findings presented in this work and accelerate the discovery of efficient (electro-)catalysts employing Earth-abundant 3d transition metals

Mass and angular-momentum inequalities for axi-symmetric initial data sets I. Positivity of mass

We extend the validity of Brill's axisymmetric positive energy theorem to all asymptotically flat initial data sets with positive scalar curvature on simply connected manifolds.Comment: 33 pages in A

The Ernst equation and ergosurfaces

We show that analytic solutions \mcE of the Ernst equation with non-empty zero-level-set of \Re \mcE lead to smooth ergosurfaces in space-time. In fact, the space-time metric is smooth near a "Ernst ergosurface" $E_f$ if and only if \mcE is smooth near $E_f$ and does not have zeros of infinite order there.Comment: 23 pages, 4 figures; misprints correcte

Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a Physical Review Letter, and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulae for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest non-static solutions (which are of genus two) to which the rigidly rotating dust disk belongs.Comment: 32 pages, 2 figures, uses pstricks.sty, updated version (October 7, 1998), to appear in Phys. Rev.

Time-Independent Gravitational Fields

This article reviews, from a global point of view, rigorous results on time independent spacetimes. Throughout attention is confined to isolated bodies at rest or in uniform rotation in an otherwise empty universe. The discussion starts from first principles and is, as much as possible, self-contained.Comment: 47 pages, LaTeX, uses Springer cl2emult styl

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.Comment: 56 pages, 21 figures, 1 tabl

Stationary Black Holes: Uniqueness and Beyond

The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.Comment: Major update of the original version by Markus Heusler from 1998. Piotr T. Chru\'sciel and Jo\~ao Lopes Costa succeeded to this review's authorship. Significantly restructured and updated all sections; changes are too numerous to be usefully described here. The number of references increased from 186 to 32

WEBT and XMM-Newton observations of 3C 454.3 during the post-outburst phase. Detection of the little and big blue bumps

The blazar 3C 454.3 underwent an unprecedented optical outburst in spring 2005. This was first followed by a mm and then by a cm radio outburst, which peaked in February 2006. We report on follow-up observations by the WEBT to study the multiwavelength emission in the post-outburst phase. XMM-Newton observations on July and December 2006 added information on the X-ray and UV fluxes. The source was in a faint state. The radio flux at the higher frequencies showed a fast decreasing trend, which represents the tail of the big radio outburst. It was followed by a quiescent state, common at all radio frequencies. In contrast, moderate activity characterized the NIR and optical light curves, with a progressive increase of the variability amplitude with increasing wavelength. We ascribe this redder-when-brighter behaviour to the presence of a "little blue bump" due to line emission from the broad line region, which is clearly visible in the source SED during faint states. Moreover, the data from the XMM-Newton OM reveal a rise of the SED in the UV, suggesting the existence of a "big blue bump" due to thermal emission from the accretion disc. The X-ray spectra are well fitted with a power-law model with photoelectric absorption, possibly larger than the Galactic one. However, the comparison with previous X-ray observations would imply that the amount of absorbing matter is variable. Alternatively, the intrinsic X-ray spectrum presents a curvature, which may depend on the X-ray brightness. In this case, two scenarios are possible.Comment: 9 pages, 7 figures, accepted for publication in A&

Dust and molecular shells in asymptotic giant branch stars - Mid-infrared interferometric observations of R Aql, R Aqr, R Hya, W Hya and V Hya

Mid-IR (8 - 13 micron) interferometric data of four oxygen-rich AGB stars (R Aql, R Aqr, R Hya, and W Hya) and one carbon-rich AGB star (V Hya) were obtained with MIDI/VLTI between April 2007 and September 2009. The spectrally dispersed visibility data are analyzed by fitting a circular fully limb-darkened disk (FDD). Results. The FDD diameter as function of wavelength is similar for all oxygen-rich stars. The apparent size is almost constant between 8 and 10 micron and gradually increases at wavelengths longer than 10 micron. The apparent FDD diameter in the carbon-rich star V Hya essentially decreases from 8 to 12 micron. The FDD diameters are about 2.2 times larger than the photospheric diameters estimated from K-band observations found in the literature. The silicate dust shells of R Aql, R Hya and W Hya are located fairly far away from the star, while the silicate dust shell of R Aqr and the amorphous carbon (AMC) and SiC dust shell of V Hya are found to be closer to the star at around 8 photospheric radii. Phase-to-phase variations of the diameters of the oxygen-rich stars could be measured and are on the order of 15% but with large uncertainties. From a comparison of the diameter trend with the trends in RR Sco and S Ori it can be concluded that in oxygen-rich stars the overall larger diameter originates from a warm molecular layer of H2O, and the gradual increase longward of 10 micron can be most likely attributed to the contribution of a close Al2O3 dust shell. The chromatic trend of the Gaussian FWHM in V Hya can be explained with the presence of AMC and SiC dust. The observations suggest that the formation of amorphous Al2O3 in oxygen- rich stars occurs mainly around or after visual minimum. However, no firm conclusions can be drawn concerning the mass-loss mechanism.Comment: 32 pages (including 7 pages appendix), 10 figure