The Computation of the Localised Molecular Orbitals of Formaldehyde

The localised molecular orbitals of the formaldehyde molecule are computed using a minimum basis set of Slater atomic orbitals. The method of calculation used obtains localised molecular orbitals (l.m.o.s ) directly at the Hartree-Fock level of approximation, rather than the more usual way of obtaining l.m.o.s from the canonical molecular orbitals. The major difficulty in implementing this method is found to lie in satisfying orthogonality conditions, required by the l.m.o. theory, prior to an actual calculation. It is not found possible to satisfy these conditions completely for the formaldehyde molecule. Ways of overcoming this difficulty are discussed. L.m.o.s are calculated using Schmidt and Lowdin orthogonalisation of a suitable set of non-orthogonal starting-point functions. The resulting l.m.o.s are found to give a unique many-electron total wavefunction, which is the same as that obtained by a canonical molecular orbital calculation. The individual l.m.o.s obtained are not unique, their forms depending on the method of orthogonalisation used and on the form of the starting-point functions. Calculations are also made at several stages of approximation, each stage corresponding to ideas of chemical valence theory.Hence, perfectly localised molecular orbitals are computed directly. The results of calculations in which the operator is truncated to include only contributions from electrons and nuclei in the immediate environment of the l.m.o. being calculated are found to be very similar to those using the full Hartree-Fock operator. The chemical significance of the l.m.o.s is examined by calculation of various properties including bond-energies. An examination is also made of the effect of making arbitrary changes in the polarity of one bond on some of the properties of other bonds. Finally, a general study of the electron density given by many l.m.o.s in different molecules is made, and the use of l.m.o.s in describing the formation of a two-electron chemical bond is examined.<p

The trispectrum of the Cosmic Microwave Background on sub-degree angular scales: an analysis of the BOOMERanG data

The trispectrum of the cosmic microwave background can be used to assess the level of non-Gaussianity on cosmological scales. It probes the fourth order moment, as a function of angular scale, of the probability distribution function of fluctuations and has been shown to be sensitive to primordial non-gaussianity, secondary anisotropies (such as the Ostriker-Vishniac effect) and systematic effects (such as astrophysical foregrounds). In this paper we develop a formalism for estimating the trispectrum from high resolution sky maps which incorporates the impact of finite sky coverage. This leads to a series of operations applied to the data set to minimize the effects of contamination due to the Gaussian component and correlations between estimates at different scales. To illustrate the effect of the estimation process, we apply our procedure to the BOOMERanG data set and show that it is consistent with Gaussianity. This work presents the first estimation of the CMB trispectrum on sub-degree scales.Comment: 14 pages, submitted to MNRA

Quantifying galaxy shapes: Sersiclets and beyond

Parametrising galaxy morphologies is a challenging task, e.g., in shear measurements of weak lensing or investigations of galaxy evolution. The huge variety of morphologies requires an approach that is highly flexible, e.g., accounting for azimuthal structure. We revisit the method of sersiclets, where galaxy morphologies are decomposed into basis functions based on the Sersic profile. This approach is justified by the fact that the Sersic profile is the first-order Taylor expansion of any real light profile. We show that sersiclets overcome the modelling failures of shapelets. However, sersiclets implicate an unphysical relation between the steepness of the light profile and the spatial scale of azimuthal structures, which is not obeyed by real galaxy morphologies and can therefore give rise to modelling failures. Moreover, we demonstrate that sersiclets are prone to undersampling, which restricts sersiclet modelling to highly resolved galaxy images. Analysing data from the Great08 challenge, we demonstrate that sersiclets should not be used in weak-lensing studies. We conclude that although the sersiclet approach appears very promising at first glance, it suffers from conceptual and practical problems that severly limit its usefulness. The Sersic profile can be enhanced by higher-order terms in the Taylor expansion, which can drastically improve model reconstructions of galaxy images. If orthonormalised, these higher-order profiles can overcome the problems of sersiclets while preserving their mathematical justification.Comment: 14 pages, 12 figures, 2 tables; accepted by MNRA

Theory, design and application of gradient adaptive lattice filters

SIGLELD:D48933/84 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Combining embedded mean-field theory with linear-scaling density-functional theory

We demonstrate the capability of embedded mean field theory (EMFT) within the linear-scaling density-functional theory code ONETEP, which enables DFT-in-DFT quantum embedding calculations on systems containing thousands of atoms at a fraction of the cost of a full calculation. We perform simulations on a wide range of systems from molecules to complex nanostructures to demonstrate the performance of our implementation with respect to accuracy and efficiency. This work paves the way for the application of this class of quantum embedding method to large-scale systems that are beyond the reach of existing implementations

Dynamical mean-field theory studies on real materials

Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.Numerische Untersuchungen stark korrelierter fermionischer Systeme sind schwierig und beinhalten noch heute essentielle Probleme. Die Hauptgründe dafür sind das exponen- tielle Wachstum des Hilbertraumes der Quantenzustände mit der Systemgröße und das fermionische Vorzeichenproblem bei Monte-Carlo-Rechnungen. Eine der am häufigsten verwendeten Methoden zur Untersuchung zweidimensionaler Gittersysteme sind Cluster- Erweiterungen der dynamische Molekularfeld Theory (DMFT), wie zum Beispiel die dy- namische Cluster Approximation (DCA). Diese Methoden bilden mehrdimensionale Git- tersysteme auf eindimensionale Störstellen-Probleme ab. 2015 wurde gezeigt, dass DMFT auf der imaginären Frequenzachse kombiniert mit der Dichtematrix-Renormierungsgruppe (DMFT+DMRG) Mehrband- und Multisite-Systeme schneller lösen kann, als wenn an- dere Störstellen-Löser verwendet werden. In dieser Arbeit entwickeln wir diesen Ansatz weiter und wenden ihn auf Modelle realer Materialen an. Am Anfang dieser Arbeit besprechen wir relevante Methoden für DMRG+ DMFT, wie zum Beispiel Matrix-Produkt-Zustände, die Dichtematrix-Renormierungs- gruppe und mehrere Zeitentwicklungs-Methoden. In diesem Zusammenhang werden wir auch mehrere Verbesserungen besprechen, die von methodischen Anpassungen von Zeit- entwicklungen bis hin zur Neuordnung des Tensornetzwerkes basierend auf Verschrän- kungs-Eigenschaften reichen. Danach werden wir uns detailliert mit den methodologischen und programmiertechnischen Aspekten von DMFT beschäftigen. Dieses Kapitel dient als Grundlage für andere Forscher, die eigene DMRG+DMFT-Codes programmieren wollen. Abschließend werden wir drei verschiedene Modelle besprechen, um das Ausmaß der Sys- teme zu zeigen, die mit diesem Ansatz gelöst werden können. Wir werden uns im Kon- text des Hubbard-Modells detailliert mit Multisite-DCA beschäftigen und zeigen, dass DMRG+DMFT Ergebnisse für Systeme mit mittleren Wechselwirkungsstärken bei niedri- gen Temperaturen erzeugen kann. Das ist mit anderen Störstellen-Lösern bisher nicht möglich. Im zweiten Fall beschäftigen wir uns mit Strontiumvanadat Sr2VO4 und werden die ersten Zweisite-DCA-Ergebnisse für ein realistisches Dreiband-Modell präsentieren. Im Gegensatz zu bisherigen Erwartungen führt die teilweise Wiedereinführung der Im- pulsabhängigkeit der Selbstenergie nicht zu einer besseren Übereinstimmung von Theorie und Experiment. Das dritte Modell beschreibt Strontiumruthenat Sr2RuO4. In diesem Fall besprechen wir den Einfluss der Spin-Bahn-Kopplung auf DMFT und wie die damit verbundenen Probleme optimal gelöst werden können. Abschließend zeigen wir die ersten Ergebnisse für dieses Modell bei niedrigen Temperaturen

Analysis and Optimisation of Orbit Correction Configurations Using Generalised Response Matrices and its Application to the LHC Injection Transfer Lines TI 2 and TI 8

The LHC injection transfer lines TI 2 and TI 8 will transport intense high-energy beams over considerable distances. In their regular part a FODO lattice is used with 4 bending magnets per half-cell and a half-cell length of 30.3 m, similar to that of the SPS. The relatively tight apertures in these lines require precise trajectory control. Following an earlier study a baseline correction scheme was chosen where two out of every four consecutive quadrupoles are complemented with correctors and beam position monitors ("2-in-4"). With the ordering of the equipment approaching, a further in-depth investigation has been made using a newly developed analytic method. This method evaluates, based on the design specifications, the global performance of an orbit correction system in terms of observability, correctability, correction range and response singularity. In addition, orbit and error envelopes are obtained over the full beam line in an efficient and rigorous manner, providing insights not easily accessible with conventional tools. The cost/performance ratio of a given configuration can be optimised, both analytically through the elimination of structural defects and numerically through fine-tuning. Finally, features for failure mode analysis allow the user to diagnose observed performance anomalies, and features for critical-element analysis enable the user to identify weak spots in the configuration. The method is described in detail to facilitate the interpretation of the results obtained for TI 2 and TI 8, and to allow their application to other orbit correction systems. The new, optimised 2-in-4 scheme permits some hardware economies at comparable performance. Further exploration has identified an alternative scheme with a 1-in-3 corrector and 2-in-3 position-monitor pattern. At an overall cost comparable to the 2-in-4 scheme this latter configuration maintains the possibility of intuitive one-to-one correction, important in the commissioning phase, at a performance slightly above the nominal aperture budget, but allows to reduce, using computer support, the corrected maximum trajectory excursions significantly below those of the 2-in-4 scheme

Embedding high-level quantum mechanical approaches within linear-scaling density functional theory

Advances in computational methods in recent decades have significantly expanded the range of problems in condensed matter physics that can be tackled from first principles. Linear-scaling density functional theory methods enable quantum mechanical calculations to be performed on systems containing tens of thousands of atoms, with modern approaches capable of reproducing the accuracy of plane wave DFT approaches. This opens up the possibility of treating highly complex molecular systems such as doped organic molecular crystals that require the dopant molecule to be contained within a large periodic structure. One example of such a system is pentacene in p-terphenyl, a system that finds use as a room-temperature maser. Understanding the maser mechanism requires both a highly accurate description of the pentacene molecule and a computationally efficient approach that can correctly capture the impact of the p-terphenyl host on the active pentacene subsystem. Quantum embedding allows an accurate but expensive hybrid functional to be embedded within a cheaper semi-local functional, for maximum combination of accuracy and efficiency in a DFT-in-DFT framework. In this dissertation we consider the implementation of embedded mean-field theory (EMFT) in the linear-scaling DFT software package ONETEP, enabling hybrid functionals to be used on selected subsystems within a cheaper DFT environment. This approach is validated for several types of molecular systems, including a crystalline structure containing several thousand atoms, demonstrating the potential of the EMFT approach when combined with linear-scaling and verifying the importance of using a large explicit host environment for accurate calculations.Open Acces

Broadband adaptive beamforming with low complexity and frequency invariant response

This thesis proposes different methods to reduce the computational complexity as well as increasing the adaptation rate of adaptive broadband beamformers. This is performed exemplarily for the generalised sidelobe canceller (GSC) structure. The GSC is an alternative implementation of the linearly constrained minimum variance beamformer, which can utilise well-known adaptive filtering algorithms, such as the least mean square (LMS) or the recursive least squares (RLS) to perform unconstrained adaptive optimisation.A direct DFT implementation, by which broadband signals are decomposed into frequency bins and processed by independent narrowband beamforming algorithms, is thought to be computationally optimum. However, this setup fail to converge to the time domain minimum mean square error (MMSE) if signal components are not aligned to frequency bins, resulting in a large worst case error. To mitigate this problem of the so-called independent frequency bin (IFB) processor, overlap-save based GSC beamforming structures have been explored. This system address the minimisation of the time domain MMSE, with a significant reduction in computational complexity when compared to time-domain implementations, and show a better convergence behaviour than the IFB beamformer. By studying the effects that the blocking matrix has on the adaptive process for the overlap-save beamformer, several modifications are carried out to enhance both the simplicity of the algorithm as well as its convergence speed. These modifications result in the GSC beamformer utilising a significantly lower computational complexity compare to the time domain approach while offering similar convergence characteristics.In certain applications, especially in the areas of acoustics, there is a need to maintain constant resolution across a wide operating spectrum that may extend across several octaves. To attain constant beamwidth is difficult, particularly if uniformly spaced linear sensor array are employed for beamforming, since spatial resolution is reciprocally proportional to both the array aperture and the frequency. A scaled aperture arrangement is introduced for the subband based GSC beamformer to achieve near uniform resolution across a wide spectrum, whereby an octave-invariant design is achieved. This structure can also be operated in conjunction with adaptive beamforming algorithms. Frequency dependent tapering of the sensor signals is proposed in combination with the overlap-save GSC structure in order to achieve an overall frequency-invariant characteristic. An adaptive version is proposed for frequency-invariant overlap-save GSC beamformer. Broadband adaptive beamforming algorithms based on the family of least mean squares (LMS) algorithms are known to exhibit slow convergence if the input signal is correlated. To improve the convergence of the GSC when based on LMS-type algorithms, we propose the use of a broadband eigenvalue decomposition (BEVD) to decorrelate the input of the adaptive algorithm in the spatial dimension, for which an increase in convergence speed can be demonstrated over other decorrelating measures, such as the Karhunen-Loeve transform. In order to address the remaining temporal correlation after BEVD processing, this approach is combined with subband decomposition through the use of oversampled filter banks. The resulting spatially and temporally decorrelated GSC beamformer provides further enhanced convergence speed over spatial or temporal decorrelation methods on their own