京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20880号 / 理博第4332号 / 新制||理||1622(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 上 正明, 教授 宍倉 光広 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We discuss the physical problem of a molecule interacting with an electromagnetic field pulse and model the problem using a time-dependent perturbation of the Born-Oppenheimer approximation to the Schrodinger equation. Using previous results that develop asymptotic series solutions in the Born-Oppenheimer parameter ε, we derive a formal Dyson series expansion in the perturbation parameter μ, which is proportional to the electromagnetic field strength. We then prove that this series is asymptotically accurate in both parameters, provided that the Hamiltonian for the electrons has purely discrete spectrum. Under more general hypotheses, we show that the series is accurate to first order in μ, and that it is accurate to one higher order if we place conditions on the abruptness of the EM pulse. We also show how this series development provides a justification for the Franck-Condon factors in the case of a diatomic molecule. / Ph. D.

The scope of this thesis is in the time-dependent formulation of the two dimensional model of resonant electron-diatomic molecule collisions in the range of low energies. In its time independent form the model was previously numerically solved without the Born-Oppenheimer approximation with use of modern tools such as the finite element method with discrete variable representation (FEM-DVR) or exterior complex scaling (ECS). Within the scope of this model we numerically solve the evolution problem, with use of the Crank-Nicolson method and the Padé approximation. Later we evaluate the cross section of the elastic and some inelastic processes with the correlation function approach. At last we make a comparison of the evolution and the cross sections to time dependent formulation of the local complex potential approximation of the electron-molecule collisions.

The accuracy of a quantum chemical calculation inherently depends on the ability to account for the completeness of the one- and n-particle spaces. The size of the basis set used can be systematically increased until it reaches the complete one-particle basis set limit (CBS) while the n-particle space approaches its exact full configuration interaction (FCI) limit by following a hierarchy of electron correlation methods developed over the last seventy years. If extremely high accuracy is desired, properly correcting for very small effects such as those resulting the Born-Oppenheimer approximation and the neglect of relativistic effects becomes indispensable. For a series of chemically interesting and challenging systems, we identify the limits of conventional approaches and use state-of-the-art quantum chemical methods along with large basis sets to get the “right answer for the right reasons.” First, we quantify the importance of small effects that are ignored in conventional quantum chemical calculations and manage to achieve spectroscopic accuracy (agreement of 1 cm−1 or less with experimental harmonic vibrational frequencies) for BH, CH+ and NH. We then definitively resolve the global minimum structure for Li₆ , Li₆⁺ , and Li₆- using high accuracy calculations of the binding energies, ionization potentials, electron affinities and vertical excitation spectra for the competing isomers. The same rigorous approach is used to study a series of hydrogen transfer reactions and validate the necessary parameters for the hydrogen abstraction and donation steps in the mechanosynthesis of diamondoids. Finally, in an effort to overcome the steep computational scaling of most high-level methods, a new hybrid methodology which scales as O(N⁵) but performs comparably to O(N⁶) methods is benchmarked for its performance in the equilibrium and dissociation regimes.

Carbon nanotubes possess various unique and interesting properties. They have very high thermal and electrical conductivities, high stiffness, mechanical strength, and optical properties. Due to these properties, CNTs are widely used materials in a variety of fields. It is used for biotechnological and biomedical applications, as chemical and biosensor, in energy storage and field emission transistor. Experimentally synthesized CNTs are generally found in bundle form due to the strong vander Waals (vdW) at-traction between the individual tubes. To use CNTs in real life applications, we often require specific nanotubes with particular characteristics. The nanotube bundle is a mixture of various chirality, diameters and electronic properties (metallic and semiconducting). Only thermal energy is not sufficient to disperse nanotubes from the bundle geometry overcoming the strong vdW attraction between nanotubes. The hydrophobic and insoluble nature of CNTs in the aqueous medium makes the dispersion of CNTs even more difficult. So, it is a big challenge to get single pristine nanotube from the bundle geometry. Many experimental and theoretical studies have addressed the problem of nanotube dispersion from the bundle geometry. Ultrasonic dispersing method is a widely used technique for this purpose where ultrasonic sound is applied to agitate particles in a system. Other methods include using different organic and inorganic solutions, various surfactant molecules, different polymers as dispersing agents. In this study we extend our e orts to develop some better methods and improved dispersing agents. In this thesis, we address the problem of CNT dispersion. To address this issue, we rst give a quantitative estimation of the effective interaction between nanotubes. Next, we introduce different polymers (ssDNA and dendrimers) as external agents and show that they help to overcome the strong adhesive interaction between CNTs and make nanotube dispersion possible from the bundle geometry. For all of the works presented in this thesis, we have used fully atomistic MD simulation and DFT level calculations. We study ssDNA-CNT complex using all-atom MD simulation and calculate various structural quantities to show the stability of ssDNA-CNT complex in aqueous medium. The adsorption of ssDNA bases on CNT surface is driven by - interaction between nucleic bases and CNT. Using the potential of mean forces (PMF) calculation, we study the binding strength of the polynucleotide ssDNA for poly A, T, G, and C with CNT of chirality (6,5). From the PMF calculation, we show the binding sequence to be A > T > C > G. Except for poly G, our result is in good agreement with earlier reported single molecule force spectroscopy results where the sequence of binding interaction was reported to be A > G > T > C. To explore how the interaction between two CNTs mod-i ed in presence of ssDNA between them, we perform PMF calculation between the two ssDNA-wrapped CNTs. The PMF shows the sequence of interaction strength between two ssDNA-wrapped CNTs for different nucleic bases to be T > A > C > G. Thus, from PMF calculations we show the poly T to have the highest dispersion efficiency, which is consistent with earlier reported experimental study. Our PMF calculation shows that poly C and poly G reduce the attraction between two CNTs drastically, whereas poly A and poly T make the interaction fully repulsive in nature. We also present microscopic pictures of the various binding conformations for ssDNA adsorbed on CNT surface. We also study the dendrimer-CNT complex for both the PAMAM and PETIM dendrimers of different generations at various protonation states and present microscopic pictures of the complex. We calculate PMF between two dendrimer wrapped CNTs and show that protonated and higher generations (G3, G4, and so forth) non-protonated PAMAM dendrimers can be used as e ective agents to disperse CNTs from bundle geometry. We also study the chirality dependence of PMF respectively. Finally, we study the interaction of mannose dendrimer with CNTs and show that the wrapping of mannose dendrimer can drive a metal to semiconducting transition in a metallic CNT. We attribute the carbon-carbon bond length assymetry in CNT due to the wrapping of mannose dendrimer as the reason for this band gap opening which leads to metal-semiconductor transition in CNT. Thus, the wrapping of mannose dendrimer on CNT can change its electronic properties and can be used in the band gap engineering of CNT in future nanotechnology. Thus, the works carried out here in this dissertation will help to address the problem of nanotube dispersion from the bundle geometry which will in turn help to use CNT for various applications in diverse fields.

La thèse consiste en l’étude spectrale d’opérateurs partiellement semi-classiques. Quand la géométrie du problème suggère une localisation anisotrope des fonctions propres associées aux basses énergies (bord du domaine, lieu d’annulation du champs magnétique), le développement local de l’opérateur amène naturellement à une structure à double échelle. Il s'agit, via un schéma de réduction "à la Born-Oppenheimer", utilisant le formalisme du calcul pseudodifférentiel pour des symboles à valeur opérateur, de montrer l’existence d’un opérateur effectif à symbole scalaire. On en déduit ensuite des formules de Weyl pour le comptage des basses valeurs propres. Cette stratégie est appliquée : au Laplacien de Robin sur un domaine borné, en dimension quelconque et au Laplacien magnétique dans R², dans le cas où le champ magnétique s’annule sur une courbe fermée. / The thesis consists in the spectral study of partially semiclassical operators. When the geometry of the problem suggests an anisotropic localization of the eigenfunctions associated to low energies (boundary of the domain, vanishing magnetic field), the local expansion of the operator naturally brings to a doublescale structure. Via a reduction scheme "à la Born-Oppenheimer", using the formalism of pseudodifferential calculus for operator-valued symbols, we can show the existence of an effective operator, with scalar symbol. Then, we deduce Weyl formulae for the number of low-lying eigenvalues. This strategy is applied : to the Robin Laplacian on a bounded domain, in any dimension and to the magnetic Laplacian in R², in the case where the magnetic field vanishes on a closed curve.

Cette thèse est consacrée à l'étude du spectre de l'opérateur de Laplace avec conditions de Dirichlet dans différents domaines du plan ou de l'espace. Dans un premier temps on s'intéresse à des triangles asymptotiquement plats et des cônes de petite ouverture. Ces problèmes admettent une reformulation semi-classique et nous donnons des développements asymptotiques à tout ordre des premières valeurs et fonctions propres. Ce type de résultat est déjà connu pour des domaines minces à profil régulier. Pour les triangles et les cônes, on prouve que le problème admet maintenant deux échelles. Dans un second temps, on étudie une famille de couches coniques indexées par leur ouverture. Là encore, on s'intéresse à la limite semi-classique quand l'ouverture tend vers zéro: on donne un développement asymptotique à deux termes des premières valeurs propres et on démontre un résultat de localisation des fonctions propres associées. Nous donnons également, à ouverture fixée, un équivalent du nombre de valeurs propres sous le seuil du spectre essentiel. / This thesis deals with the spectrum of the Dirichlet Laplacian in various two or three dimensional domains. First, we consider asymptotically flat triangles and cones with small aperture. These problems admit a semi-classical formulation and we provide asymptotic expansions at any order for the first eigenvalues and the associated eigenfunctions. These type of results is already known for thin domains with smooth profiles. For triangles and cones, we show that the problem admits now two different scales. Second, we study a family of conical layers parametrized by their aperture. Again, we consider the semi-classical limit when the aperture tends to zero: We provide a two-term asymptotics of the first eigenvalues and we prove a localization result about the associated eigenfunctions. We also estimate, for each chosen aperture, the number of eigenvalues below the threshold of the essential spectrum.

Diese Arbeit befasst sich mit der theoretischen Beschreibung nichtlinearer optischer Phänomene in Hinblick auf das (numerische) unstetige Galerkin-Zeitraumverfahren. Insbesondere werden zwei Materialmodelle behandelt: das hydrodynamische Modell für Metalle und das Modell für Raman-aktive Materialien. Im ersten Teil der Arbeit wird das hydordynamische Modell für Metalle unter Verwendung eines störungstheoretischen Ansatzes behandelt. Insbesondere wird dieser Ansatz genutzt, um die nichtlinearen optischen Effekte, Erzeugung zweiter Harmonischer und Summenfrequenzerzeugung, mit Hilfe des unstetigen Galerkin-Verfahrens zu studieren. In diesem Zusammenhang wird demonstriert, wie das optische Signal zweiter Ordnung von Nanoantennen optimiert werden kann. Hierzu wird ein hier erarbeitetes Schema für die Abstimmung des eingestrahten Lichtes angewandt. Zudem führt eine intelligente Wahl des Antennendesigns zu einem optimierten Signal. Im zweiten Teil dieser Arbeit wird das Modell für Raman-aktive Dielektrika behandelt. Genauer wird die nichtlineare Antwort dritter Ordnung für stimulierte Raman-Streuung hergeleitet. Diese wird dazu genutzt, um ein System aus Hilfsdifferentialgleichungen für das unstetige Galerkin-Verfahren zu konstruieren. Die Ergebnisse des erweiterten numerischen Verfahrens werden im Anschluss gezeigt und diskutiert. / This thesis is concerned with the theoretical description of nonlinear optical phenomena with regards to the (numerical) discontinuous Galerkin time-domain (DGTD) method. It deals with two different material models: the hydrodynamic model for metals and the model for Raman-active dielectrics. In the first part, we review the hydrodynamic model for metals, where we apply a perturbative approach to the model. We use this approach to calculate the second-order nonlinear optical effects of second-harmonic generation and sum-frequency generation using the DGTD method. In this context, we will see how to optimize the second-order response of plasmonic nanoantennas by applying a deliberate tuning scheme for the optical excitations as well as by choosing an intelligent nanoantenna design. In the second part, we examine the material model for Raman-active dielectrics. In particular, we see how to derive the third-order nonlinear response by which one can describe the process of stimulated Raman scattering. We show how to incorporate this third-order response into the DGTD scheme yielding a novel set of auxiliary differential equations. Finally, we demonstrate the workings of the modified numerical scheme.