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<p>In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case.</p><p>Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.</p> / Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.

In this thesis we study the relativistic multipole moments for stationary asymptotically flat spacetimes as introduced by Geroch and Hansen. These multipole moments give an asymptotic description of the gravitational field in a coordinate independent way. Due to this good description of the spacetimes, it is natural to try to construct a spacetime from only the set of multipole moments. Here we present a simple method to do this for the static axisymmetric case. We also give explicit solutions for the cases where the number of non-zero multipole moments are finite. In addition, for the general stationary axisymmetric case, we present methods to generate solutions. It has been a long standing conjecture that the multipole moments give a complete characterization of the stationary spacetimes. Much progress toward a proof has been made over the years. However, there is one remaining difficult task: to prove that a spacetime exists with an a-priori given arbitrary set of multipole moments subject to some given condition. Here we present such a condition for the axisymmetric case, and prove that it is both necessary and sufficient. We also extend this condition to the general case without axisymmetry, but in this case we only prove the necessity of our condition.

In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series. / <p>Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.</p>

The geometry of a plasmonic nanostructure determines the charge-current distributions of its localized surface plasmon resonances (LSPR), thereby determining the device’s interactions with external electromagnetic fields. To target specific applications, we manipulate the nanostructure geometry to create different electromagnetic multipole moments, from basic electric and magnetic dipoles to more exotic higher order and toroidal multipoles. The nanoscale nature of the resonance phenomena makes electron beam spectromicroscopy techniques uniquely suited to probe LSPRs over a wide spectral range, with nanoscale spatial resolution. We use electron energy loss spectroscopy (EELS) in a monochromated scanning transmission electron microscope and cathodoluminescence spectroscopy (CL) in a scanning electron microscope to probe the near-field and far-field properties of LSPR. Electric dipoles within triangular prisms and apertures in Sierpiński fractals couple as the generation number is advanced, creating predictable spectral bands from hybridized dipole modes of parent generations with hierarchical patterns of high field intensity, as visualized in EELS. A magnetic dipole moment is engineered using a vertical split ring resonator (VSRR), pushing the limits of nanofabrication techniques. On this nanostructure we demonstrate the calculation of spatially resolved Stokes parameters on the emission of the magnetic dipole mode and a series of coupled rim modes. Coupling of the magnetic dipole mode of four VSRRs in a circular array creates an LSPR mode supporting the lesser-known toroidal dipole moment. We further probe the near-field configuration of this 3D array through tilting under the electron beam in EELS, and the far-field emission through CL of higher order rim modes. We also propose further configurations of five and six VSRRs to strengthen the toroidal dipole moment. All of the data presented herein was analyzed using custom Python code, which provides a unique graphical interface to 3D spectromicroscopy datasets, and a parallelized implementation of the Richardson-Lucy deconvolution algorithm. / Thesis / Doctor of Philosophy (PhD) / Certain types of metallic particles are capable of trapping light on a scale far below that which we can see; their light-trapping properties depend on their material and on their geometry. Using these tiny particles, we can manipulate the behaviour of light with greater freedom than is otherwise possible. In this thesis, we study how we can engineer the geometry of these particles to give predictable responses that can then be targeted towards specific applications. We study a fractal structure with predictable self-similar responses useful for high sensitivity detection of disease or hormone biomarkers; a resonating structure emulating a magnetic response which can be used in the design of unique new materials capable of bending light backwards and cloaking objects from sight; and a combination of these resonators in an array to demonstrate exotic electromagnetic behaviour still on the limit of our understanding.

Yes / Key to progress in molecular simulation is the development of advanced models that go beyond the limitations of traditional force fields that employ a fixed, point charge‐based description of electrostatics. Taking water as an example system, the FFLUX framework is shown capable of producing models that are flexible, polarizable and have a multipolar description of the electrostatics. The kriging machine‐learning methods used in FFLUX are able to reproduce the intramolecular potential energy surface and multipole moments of a single water molecule with chemical accuracy using as few as 50 training configurations. Molecular dynamics simulations of water clusters (25–216 molecules) using the new FFLUX model reveal that incorporating charge‐quadrupole, dipole–dipole, and quadrupole–charge interactions into the description of the electrostatics results in significant changes to the intermolecular structuring of the water molecules. / EPSRC. Grant Number: K005472

To accurately solve the general nuclear spin state function in Nuclear Magnetic Resonance (NMR), a rotation wave approach was employed, allowing the reference frame to rotate in sync with the oscillating magnetic field. The spin state system was analogously treated as a Rubik's Cube, ensuring the diagonalization of only the time-dependent part of the state function. Although Gottfried's equation (1966) aligns with transitions between specific spin states m and m′, his second rotation contradicts the conservation of angular momentum, resulting in inaccuracies for spin states with initial phase shifts or entangled states. Contrarily, Schwinger (1937) efficiently computed the coefficients for each spin state in a frequency range opposite to the Larmor frequency, using an unorthodox approach in quantum mechanics, which unfortunately led to the oversight of his work in subsequent citations. This methodology was also applied to derive the general electron spin state function in Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR), enabling the construction of a doubly rotated ground state for time-dependent perturbation theory. This was particularly relevant as the Hamiltonians for magnetic dipole, electric quadrupole, and magnetic octupole moments incorporate powers of I · J terms, necessitating the calculation of sub-state energy levels for perturbation, including those of molecules 14N7 and 7Li3. Furthermore, the study expanded to the general Lie group for 3D rotations along three linearly independent axes, resulting in 12 distinct methods to achieve rotations in any arbitrary direction using these axes, yielding wave function with only one spin operator in each exponent. The ongoing research is now concentrated on generating NMR spectra for 14N7 in amino acids, furthering the understanding of nuclear spin dynamics in complex molecular systems.

Αντικείμενο της παρούσης διδακτορικής διατριβής αποτελεί ο υπολογισμός των ηλεκτρικών πολυπολικών ροπών (διπολική έως και δεκαεξαπολική) ακετυλενικών αλυσίδων οι οποίες υπάγονται στους γενικούς τύπους H-(C≡C)n-H, Χ-(C≡C)n-H και Χ-(C≡C)n-X, n=1–7, X=F, Cl, Br, I, CN, NC, η συστηματική μελέτη των (υπερ)πολωσιμοτήτων των μη υποκατεστημένων ακετυλενικών αλυσίδων Η-(C≡C)n- H, n=3–7 και τέλος ο υπολογισμός των ηλεκτρικών ιδιοτήτων αλληλεπίδρασης συστημάτων του τύπου Η-(C≡C)n-H…Ηe, n=1–7. Σε όλες τις περιπτώσεις χρησιμοποιήθηκαν οι ab initio μέθοδοι SCF και MP2 καθώς και οι ευρέως χρησιμοποιούμενες DFT μέθοδοι B3LYP, B3PW91 και mPW1PW91. Στους υπολογισμούς των ηλεκτρικών πολυπολικών ροπών χρησιμοποιήθηκε η βάση ccpVDZ σε όλες τις περιπτώσεις και ιδιαίτερη έμφαση δόθηκε στον τρόπο μεταβολής των υπό μελέτη ιδιοτήτων με την μεταβολή του υποκαταστάτη στα άκρα της αλυσίδας. Στους υπολογισμούς των ηλεκτρικών (υπερ)πολωσιμοτήτων των μη υποκατεστημένων ακετυλενικών αλυσίδων Η-(C≡C)n-H χρησιμοποιήθηκαν ελάχιστα πολωμένα σύνολα βάσης και πραγματοποιήθηκε σύγκριση με τα αποτελέσματα που προέκυψαν από την χρήση μεγαλύτερων συνόλων βάσης. Για τον υπολογισμό των εν λόγω ιδιοτήτων χρησιμοποιήθηκε η μέθοδος του πεπερασμένου πεδίου. Τέλος πραγματοποιήθηκαν υπολογισμοί των ηλεκτρικών ιδιοτήτων αλληλεπίδρασης συστημάτων Η-(C≡C)n-H…Ηe για δυο συγκεκριμένες διαμορφώσεις ενώ για την αποφυγή του σφάλματος υπέρθεσης συνόλου βάσης χρησιμοποιήθηκε η υπερμοριακή προσέγγιση των Boys-Bernardi. / The purpose of the present doctoral dissertation is the calculation of the electric multipole moments (dipole up to hexadecapole) of acetylenic chains under the general formulas H-(C≡C)n-H, Χ-(C≡C)n-H and Χ-(C≡C)n-X, n=1–7, X=F, Cl, Br, I, CN, NC, the systematic study of the (hyper)polarizabilities of the non substituted acetylenic chains Η-(C≡C)n-H, n=3–7 and finally the calculation of interaction induced electric properties of systems of the general formula Η-(C≡C)n-H…Ηe, n=1– 7. In all cases considered the ab initio SCF and MP2 methods have been used along with the very popular DFT methods B3LYP, B3PW91 and mPW1PW91. For the electric multipole moment calculations the cc-pVDZ basis set has been used in all cases and special attention has been paid in the way the properties of interest change by changing the substituent at the ends of the acetylenic chains. For the electric (hyper)polarizability calculations on the non substituted acetylenic chains H-(C≡C)n- H minimally polarized basis sets have been used and comparison was made with the results obtained by using larger basis sets. In order to calculate the above mentioned properties the finite field method was applied. Finally, calculations were performed in order to calculate the interaction induced electric properties of the systems Η-(C≡C)n- H…Ηe for two particular configurations while in order to avoid the basis set superposition error the Boys-Bernardi counterpoise method has been applied.

Atomistic modeling and simulation methods enable a modern molecular approach to bio-medical research. Issues addressed range from structure-function relationships to structure-based drug design. The ability of these methods to address biologically relevant problems is largely determined by their accurate treatment of electrostatic interactions in the target biomolecular structure. In practical molecular simulations, the electrostatic charge density of molecules is approximated by an arrangement of fractional "point charges" throughout the molecule. While chemically intuitive and straightforward in technical implementation, models based exclusively on atom-centered charge placement, a major workhorse of the biomolecular simulations, do not necessarily provide a sufficiently detailed description of the molecular electrostatic potentials for small systems, and can become prohibitively expensive for large systems with thousands to millions of atoms. In this work, we propose a rigorous and generally applicable approach, Optimal Point Charge Approximation (OPCA), for approximating electrostatic charge distributions of biomolecules with a small number of point charges to best represent the underlying electrostatic potential, regardless of the distance to the charge distribution. OPCA places a given number of point charges so that the lowest order multipole moments of the reference charge distribution are optimally reproduced. We provide a general framework for calculating OPCAs to any order, and introduce closed-form analytical expressions for the 1-charge, 2-charge and 3-charge OPCA. We demonstrate the advantage of OPCA by applying it to a wide range of biomolecules of varied sizes. We use the concept of OPCA to develop a different, novel approach of constructing accurate and simple point charge water models. The proposed approach permits a virtually exhaustive search for optimal model parameters in the sub-space most relevant to electrostatic properties of the water molecule in liquid phase. A novel rigid 4-point Optimal Point Charge (OPC) water model constructed based on the new approach is substantially more accurate than commonly used models in terms of bulk water properties, and delivers critical accuracy improvement in practical atomistic simulations, such as RNA simulations, protein folding, protein-ligand binding and small molecule hydration. We also apply our new approach to construct a 3-point version of the Optimal Point Charge water model, referred to as OPC3. OPCA can be employed to represent large charge distributions with only a few point charges. We use this capability of OPCA to develop a multi-scale, yet fully atomistic, generalized Born approach (GB-HCPO) that can deliver up to 2 orders of magnitude speedup compared to the reference MD simulation. As a practical demonstration, we exploit the new multi-scale approach to gain insight into the structure of million-atom 30-nm chromatin fiber. Our results suggest important structural details consistent with experiment: the linker DNA fills the core region and the H3 histone tails interact with the linker DNA. OPC, OPC3 and GB-HCPO are implemented in AMBER molecular dynamics software package. / Ph. D.