A computer-algebraic approach to the study of the symmetry properties of molecules and clusters

This thesis work is dedicated to use the computer-algebraic approach for dealing with the group symmetries and studying the symmetry properties of molecules and clusters. The Maple package Bethe, created to extract and manipulate the group-theoretical data and to simplify some of the symmetry applications, is introduced. First of all the advantages of using Bethe to generate the group theoretical data are demonstrated. In the current version, the data of 72 frequently applied point groups can be used, together with the data for all of the corresponding double groups. The emphasize of this work is placed to the applications of this package in physics of molecules and clusters. Apart from the analysis of the spectral activity of molecules with point-group symmetry, it is demonstrated how Bethe can be used to understand the field splitting in crystals or to construct the corresponding wave functions. Several examples are worked out to display (some of) the present features of the Bethe program. While we cannot show all the details explicitly, these examples certainly demonstrate the great potential in applying computer algebraic techniques to study the symmetry properties of molecules and clusters. A special attention is placed in this thesis work on the flexibility of the Bethe package, which makes it possible to implement another applications of symmetry. This implementation is very reasonable, because some of the most complicated steps of the possible future applications are already realized within the Bethe. For instance, the vibrational coordinates in terms of the internal displacement vectors for the Wilson's method and the same coordinates in terms of cartesian displacement vectors as well as the Clebsch-Gordan coefficients for the Jahn-Teller problem are generated in the present version of the program. For the Jahn-Teller problem, moreover, use of the computer-algebraic tool seems to be even inevitable, because this problem demands an analytical access to the adiabatic potential and, therefore, can not be realized by the numerical algorithm. However, the ability of the Bethe package is not exhausted by applications, mentioned in this thesis work. There are various directions in which the Bethe program could be developed in the future. Apart from (i) studying of the magnetic properties of materials and (ii) optical transitions, interest can be pointed out for (iii) the vibronic spectroscopy, and many others. Implementation of these applications into the package can make Bethe a much more powerful tool.

Symmetry properties of one- and two- electron molecular integrals

The maximum numbers of distinct one- and two-electron integrals that arise in calculating the electronic energy of a molecule are discussed. It is shown that these may be calculated easily using the character table of the symmetry group of the set of basis functions used to express the wave function. Complications arising from complex group representations and from a conflict of symmetry between the basis set and the nuclear configuration are considered and illustrated by examples.

Fast assembly of fock matrices utilising symmetry properties of the basis set

A method of assembling the elements of the Fock matrix is described which is a modification of that due to Dacre. Lists of symmetry equivalent one-electron integrals are used as pointers to abbreviate the process of collecting two-electron integrals into the Fock matrix.

Properties and algorithms of the hyper-star graph and its related graphs

The hyper-star interconnection network was proposed in 2002 to overcome the drawbacks of the hypercube and its variations concerning the network cost, which is defined by the product of the degree and the diameter. Some properties of the graph such as connectivity, symmetry properties, embedding properties have been studied by other researchers, routing and broadcasting algorithms have also been designed. This thesis studies the hyper-star graph from both the topological and algorithmic point of view. For the topological properties, we try to establish relationships between hyper-star graphs with other known graphs. We also give a formal equation for the surface area of the graph. Another topological property we are interested in is the Hamiltonicity problem of this graph. For the algorithms, we design an all-port broadcasting algorithm and a single-port neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs. These algorithms are both optimal time-wise. Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be maixmally fault-tolerant.

Evaluating perceptual maps of asymmetries for gait symmetry quantification and pathology detection

Le mouvement de la marche est un processus essentiel de l'activité humaine et aussi le résultat de nombreuses interactions collaboratives entre les systèmes neurologiques, articulaires et musculo-squelettiques fonctionnant ensemble efficacement. Ceci explique pourquoi une analyse de la marche est aujourd'hui de plus en plus utilisée pour le diagnostic (et aussi la prévention) de différents types de maladies (neurologiques, musculaires, orthopédique, etc.). Ce rapport présente une nouvelle méthode pour visualiser rapidement les différentes parties du corps humain liées à une possible asymétrie (temporellement invariante par translation) existant dans la démarche d'un patient pour une possible utilisation clinique quotidienne. L'objectif est de fournir une méthode à la fois facile et peu dispendieuse permettant la mesure et l'affichage visuel, d'une manière intuitive et perceptive, des différentes parties asymétriques d'une démarche. La méthode proposée repose sur l'utilisation d'un capteur de profondeur peu dispendieux (la Kinect) qui est très bien adaptée pour un diagnostique rapide effectué dans de petites salles médicales car ce capteur est d'une part facile à installer et ne nécessitant aucun marqueur. L'algorithme que nous allons présenter est basé sur le fait que la marche saine possède des propriétés de symétrie (relativement à une invariance temporelle) dans le plan coronal.

Local attractors, degeneracy and analyticity: Symmetry effects on the locally coupled Kuramoto model

In this work we study the local coupled Kuramoto model with periodic boundary conditions. Our main objective is to show how analytical solutions may be obtained from symmetry assumptions, and while we proceed on our endeavor we show apart from the existence of local attractors, some unexpected features resulting from the symmetry properties, such as intermittent and chaotic period phase slips, degeneracy of stable solutions and double bifurcation composition. As a result of our analysis, we show that stable fixed points in the synchronized region may be obtained with just a small amount of the existent solutions, and for a class of natural frequencies configuration we show analytical expressions for the critical synchronization coupling as a function of the number of oscillators, both exact and asymptotic. © 2013 Elsevier Ltd. All rights reserved.

Atividade óptica de um meio dielétrico diluído: Pasteur e as simetrias moleculares

Em 1848 Pasteur conjeturou que a rotação do plano de polarização da luz em um meio diluído é gerada pelas propriedades de simetria das moléculas do meio no qual a luz se propaga. O objetivo do nosso artigo é de mostrar que Pasteur estava correto usando conhecimentos de eletromagnetismo e mecânica quântica de um curso de graduação em física. Faremos um breve retrospecto das ideias básicas da teoria eletromagnética necessárias para o estudo da atividade óptica. A seguir, usando a teoria de perturbações em mecânica quântica e levando em conta as simetrias das moléculas calcularemos a atividade óptica do meio. Mostraremos que as previsões teóricas, que estão plenamente de acordo com os resultados experimentais, comprovam a hipótese de Pasteur.

Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices

The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.

Extended integrability regime for the supersymmetric U model

An extension of the supersymmetric U model for correlated electrons is given and integrability is established by demonstrating that the model can he constructed through the quantum inverse scattering method using an R-matrix without the difference property. Some general symmetry properties of the model are discussed and from the Bethe ansatz solution an expression for the energies is presented.

Tight open knots.

The tightest open conformations of a few prime knots (3(1), 4(1), 5(1), 5(2), and 6(3)) found with the use of the SONO algorithm are presented and discussed. The conformations are obtained from the tightest closed conformations of the knots by cutting and opening them at different locations. The length of the rope engaged in each of the open conformations is calculated. The curvature and torsion profiles of the unique tightest conformations of the 3(1) and 4(1) open knots are presented and discussed. In particular, symmetry properties of the knots reflected within their curvature and torsion profiles are analysed. Connections with the physics of polymers are discussed.

Effective eigenvalue for the intensity correlations of single-mode and two-mode lasers

Exact formulas for the effective eigenvalue characterizing the initial decay of intensity correlation functions are given in terms of stationary moments of the intensity. Spontaneous emission noise and nonwhite pump noise are considered. Our results are discussed in connection with earlier calculations, simulations, and experimental results for single-mode dye lasers, two-mode inhomogeneously broadened lasers, and two-mode dye ring lasers. The effective eigenvalue is seen to depend sensitively on noise characteristics and symmetry properties of the system. In particular, the effective eigenvalue associated with cross correlations of two-mode lasers is seen to vanish in the absence of pump noise as a consequence of detailed balance. In the presence of pump noise, the vanishing of this eigenvalue requires equal pump parameters for the two modes and statistical independence of spontaneous emission noise acting on each mode.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Characterization of phosphate structures in biochar from swine bones

The objective of this work was to develop an alternative methodology to study and characterize the phosphate crystalline properties, directly associated with solubility and plant availability, in biochar from swine bones. Some phosphate symmetry properties of pyrolyzed swine bones were established, using solid state nuclear magnetic resonance spectroscopy, principal component analysis, and multivariate curve resolution analysis, on four pyrolyzed samples at different carbonization intensities. Increasing carbonization parameters (temperature or residence time) generates diverse phosphate structures, increasing their symmetry and decreasing the crossed polarizability of the pair ¹H-31P, producing phosphates with, probably, lower solubility than the ones produced at lower carbonization intensity. Additionally, a new methodology is being developed to study and characterize phosphate crystalline properties directly associated with phosphate solubility and availability to plants.

Femtosecond real-time probing of reactions. 12. Vectorial dynamics of transition states

Femtosecond time-resolved techniques with KETOF (kinetic energy time-of-flight) detection in a molecular beam are developed for studies of the vectorial dynamics of transition states. Application to the dissociation reaction of IHgI is presented. For this system, the complex [I---Hg---I](++)* is unstable and, through the symmetric and asymmetric stretch motions, yields different product fragments: [I---Hg---I](++)* -> HgI(X^2/sigma^+) + I(^2P_3/2) [or I*(^2P_l/2)] (1a); [I---Hg---I](++)* -> Hg(^1S_0) + I(^2P_3/2) + I(^2P_3/2) [or I* (^2P_1/2)] (1 b). These two channels, (1a) and (1b), lead to different kinetic energy distributions in the products. It is shown that the motion of the wave packet in the transition-state region can be observed by MPI mass detection; the transient time ranges from 120 to 300 fs depending on the available energy. With polarized pulses, the vectorial properties (transition moments alignment relative to recoil direction) are studied for fragment separations on the femtosecond time scale. The results indicate the nature of the structure (symmetry properties) and the correlation to final products. For 311-nm excitation, no evidence of crossing between the I and I* potentials is found at the internuclear separations studied. (Results for 287-nm excitation are also presented.) Molecular dynamics simulations and studies by laser-induced fluorescence support these findings.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.