- A Mapped Fourier Grid method for solving the radial Schrödinger equation is improved. It is observed that a discrete sine and cosine transform algorithm allows to compute a Hamiltonian matrix the spectrum of which is free of spurious eigenvalues. - The energies of the highest, least bound vibrational states of the Na - Na2 van der Waals complex are computed using a hyperspherical diabatic-by-sector method. The computed levels are analyzed using quantum defect theory.
A Mapped Fourier Grid method for solving the radial Schrödinger equation is improved. It is observed that a discrete sine and cosine transform algorithm allows to compute a Hamiltonian matrix the spectrum of which is free of spurious eigenvalues. - The energies of the highest, least bound vibrational states of the Na - Na2 van der Waals complex are computed using a hyperspherical diabatic-by-sector method. The computed levels are analyzed using quantum defect theory.
Cette thèse, rédigée en anglais, est une étude théorique de collisions lentes atome-diatome et d'états vibrationnels du trimère près du seuil de dissociation. Elle est composée des chapitres suivants: I. Introduction. II. Mapped Fourier Grid. III. Quantum dynamics of a triatomic system. IV. Potential energy surface of the sodium trimer. V. Loosely bound vibrational states of the sodium trimer. VI. Prospects for the photoassociation of Na3. VII. Conclusion. Les parties originales du travail sont présentées dans les chapitres II et V. Le chapitre II concerne la méthode de grille de Fourier à pas variable. Cette méthode numérique permet de calculer des états vibrationnels de dimère proches des seuils de dissociation, mais souffre de l'apparition d'un petit nombre d'états fantômes. Ce problème est résolu en remplaçant l'algorithme de transformation de Fourier discrète par un algorithme de transformations sinus et cosinus discrètes [K. Willner, J. Chem. Phys. 120, 548 (2004)]. Le chapitre V réalise l'étude des derniers états vibrationnels très peu liés du trimère Na3. Le calcul repose sur une adaptation du code de J. M. Launay sur la diffusion réactive atom-diatome. Le trimère Na3 dans l'état électronique 1 4A2' est traité pour le cas d'un moment angulaire total nul et en utilisant la surface de potentiel construite comme une somme de potentiels de paires. La méthode dite diabatique- par- secteur est adaptée au calcul des valeurs propres correspondant aux états vibrationnels de Na3. Dans une deuxième approche, la théorie de la diffusion stationnaire écrite en coordonnées de Jacobi est adaptée au calcul et à l'analyse des niveaux vibrationnels proche du seuil de dissociation en Na et Na2.
We discuss the bound states of weakly bound van der Waals trimers within the framework of hyperspherical coordinates. The wave function is expanded in terms of hyperspherical harmonics, which form a complete basis set in the angular variables. The resulting set of coupled second-order differential equations in the hyperradius is solved exactly. Our method gives a value for the zero-point energy of H+3 which is in excellent agreement with previous calculations. For (H2)3 and Ne3, however, our results show some discrepancy with earlier work.
We study theoretically the ground and excited bound states of the bosonic rare gas van der Waals trimer Ne(3). A slow variable discretization approach is adopted to solve the nuclear Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed finite-element methods discrete variable representation hyper-radii. We consider not only zero total nuclear orbital angular momentum, J = 0, states but also J > 0 states. By using the best empirical neon dimer interaction potentials, all the bound state energy levels of Ne(3) will be calculated for total angular momenta up to J = 6, as well as their average root-mean-square radii. We also analyze the wave functions in hyperspherical coordinates for several selected bound states.
We study the structure of triatomic molecules containing two helium atoms and one alkali-metal atom X (X = Li, Na). The three-body Schr\"odinger equation is solved in hyperspherical coordinates using the He-He and He-X pair van der Waals potentials available in the literature. We calculate the structural properties of the He2Li and He2Na bound states, and analyze one-dimensional pair and angle distribution functions as well as two-dimensional pair-pair and angle-angle distribution functions. These bound states are characterized by so peculiar weakly bound structures that classifying them into particular sizes and geometrical shapes appears to be elusive. Particularly, the ${J}^{\mathrm{\ensuremath{\Pi}}}={0}^{+}$ excited states of 4He26Li and 4He27Li are found to constitute giant triatomic molecules with their size amounting to several hundred bohrs.
When studying the structure formation and fragmentation of complex atomic and nuclear systems it is preferable to start with simple systems where all details can be explored. Some of the knowledge gained from studies of atomic dimers can be generalised to more complex systems. Adding a third atom to an atomic dimer gives a first chance to study how the binding between two atoms is affected by a third. Few-body physics is an intermediate area which helps us to understand some but not all phenomena in many-body physics.Very weakly bound, spatially very extended quantum systems with a wave function reaching far beyond the classical forbidden region and with low angular momentum are characterized as halo systems. These unusual quantum systems, first discovered in nuclear physics may also exist in systems of neutral atoms.Since the first clear theoretical prediction in 1977, of a halo system possessing an Efimov state, manifested in the excited state of the bosonic van der Waals helium trimer 42He3, small helium and different spin-polarised halo hydrogen clusters and their corresponding isotopologues have been intensively studied the last three decades.In the work presented here, the existence of the spin-polarized tritium trimer ground state, 31H3, is demonstrated, verifying earlier predictions, and the system's properties elucidated. Detailed analysis has found no found evidence for other bound states and shape resonances in this system. The properties of the halo helium trimers, 42He3 and 42He2-32He have been investigated. Earlier predictions concerning the ground state energies and structural properties of these systems are validated using our three-dimensional finite element method. In the last part of this work we present results on the bound states and structural properties of the van der Waals bosonic atomic trimers Ne3 and Ar3. We believe to be the first to find evidence of a possible shape resonance just above the three-body dissociation limit of the neon trimer.
We calculate the binding energy and geometry of the weakly bound e(+)Li and e(+)Na systems within the framework of hyperspherical coordinates. The Schrodinger equation in hyperangular coordinates is solved at a series of fixed hyper-radii using B-splines and the resulting coupled hyper-radial equation is solved using the slow variable discretization method developed by Tolstikhin et al. [J. Phys. B 29, L389 (1996)]. Great efforts are made in optimizing the distribution of B-splines to overcome the slow convergence of the binding energy and geometrical quantities. This approach allows us to obtain the results with improved convergence that are in good agreement with the best values reported to date. In addition, an analysis of the structure of the two systems is also made and the e(+)Na system is seen to exhibit quantum halo features.