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Desenvolupament computacional de la semblança molecular quàntica

La present Tesi Doctoral, titulada desenvolupament computacional de la semblança molecular quàntica, tracta, fonamentalment, els aspectes de càlcul de mesures de semblança basades en la comparació de funcions de densitat electrònica.El primer capítol, Semblança quàntica, és introductori. S'hi descriuen les funcions de densitat de probabilitat electrònica i llur significança en el marc de la mecànica quàntica. Se n'expliciten els aspectes essencials i les condicions matemàtiques a satisfer, cara a una millor comprensió dels models de densitat electrònica que es proposen. Hom presenta les densitats electròniques, mencionant els teoremes de Hohenberg i Kohn i esquematitzant la teoria de Bader, com magnituds fonamentals en la descripció de les molècules i en la comprensió de llurs propietats.En el capítol Models de densitats electròniques moleculars es presenten procediments computacionals originals per l'ajust de funcions densitat a models expandits en termes de gaussianes 1s centrades en els nuclis. Les restriccions físico-matemàtiques associades a les distribucions de probabilitat s'introdueixen de manera rigorosa, en el procediment anomenat Atomic Shell Approximation (ASA). Aquest procediment, implementat en el programa ASAC, parteix d'un espai funcional quasi complert, d'on se seleccionen variacionalment les funcions o capes de l'expansió, d'acord als requisits de no negativitat. La qualitat d'aquestes densitats i de les mesures de semblança derivades es verifica abastament. Aquest model ASA s'estén a representacions dinàmiques, físicament més acurades, en quant que afectades per les vibracions nuclears, cara a una exploració de l'efecte de l'esmorteïment dels pics nuclears en les mesures de semblança molecular. La comparació de les densitats dinàmiques respecte les estàtiques evidencia un reordenament en les densitats dinàmiques, d'acord al que constituiria una manifestació del Principi quàntic de Le Chatelier. El procediment ASA, explícitament consistent amb les condicions de N-representabilitat, s'aplica també a la determinació directe de densitats electròniques hidrogenoides, en un context de teoria del funcional de la densitat.El capítol Maximització global de la funció de semblança presenta algorismes originals per la determinació de la màxima sobreposició de les densitats electròniques moleculars. Les mesures de semblança molecular quàntica s'identifiquen amb el màxim solapament, de manera es mesuri la distància entre les molècules, independentment dels sistemes de referència on es defineixen les densitats electròniques. Partint de la solució global en el límit de densitats infinitament compactades en els nuclis, es proposen tres nivells de aproximació per l'exploració sistemàtica, no estocàstica, de la funció de semblança, possibilitant la identificació eficient del màxim global, així com també dels diferents màxims locals. Es proposa també una parametrització original de les integrals de recobriment a través d'ajustos a funcions lorentzianes, en quant que tècnica d'acceleració computacional. En la pràctica de les relacions estructura-activitat, aquests avenços possibiliten la implementació eficient de mesures de semblança quantitatives, i, paral·lelament, proporcionen una metodologia totalment automàtica d'alineació molecular. El capítol Semblances d'àtoms en molècules descriu un algorisme de comparació dels àtoms de Bader, o regions tridimensionals delimitades per superfícies de flux zero de la funció de densitat electrònica. El caràcter quantitatiu d'aquestes semblances possibilita la mesura rigorosa de la noció química de transferibilitat d'àtoms i grups funcionals. Les superfícies de flux zero i els algorismes d'integració usats han estat publicats recentment i constitueixen l'aproximació més acurada pel càlcul de les propietats atòmiques. Finalment, en el capítol Semblances en estructures cristal·lines hom proposa una definició original de semblança, específica per la comparació dels conceptes de suavitat o softness en la distribució de fonons associats a l'estructura cristal·lina. Aquests conceptes apareixen en estudis de superconductivitat a causa de la influència de les interaccions electró-fonó en les temperatures de transició a l'estat superconductor. En aplicar-se aquesta metodologia a l'anàlisi de sals de BEDT-TTF, s'evidencien correlacions estructurals entre sals superconductores i no superconductores, en consonància amb les hipòtesis apuntades a la literatura sobre la rellevància de determinades interaccions.Conclouen aquesta tesi un apèndix que conté el programa ASAC, implementació de l'algorisme ASA, i un capítol final amb referències bibliogràfiques.

Carbon suboxide: the vibrational dependence of the v7 bending potential function

Data on the vibrational energy levels and rotational constants of carbon suboxide for the low-wavenumber bending mode ν7 are reviewed, in the ground-state manifold, and in the ν2-, ν3-, ν4-, and ν2 + ν4-state manifolds. Following the procedure developed by Duckett, Mills, and Robiette [J. Mol. Spectrosc. 63, 249 (1976)] the data have been inverted to give the effective bending potential in ν7 for each of these five states. Values are obtained for various other parameters in the effective vibration-rotation Hamiltonian. The potential and rotational constants in ν2 + ν4 are given to a close approximation by linear extrapolation from the ground state through the ν2 and ν4 states.

Mapping and controlling molecular processes: The application of control theory and system identification algorithms to ASOPS and other fast pulse spectroscopies

Asynchronous Optical Sampling (ASOPS) [1,2] and frequency comb spectrometry [3] based on dual Ti:saphire resonators operated in a master/slave mode have the potential to improve signal to noise ratio in THz transient and IR sperctrometry. The multimode Brownian oscillator time-domain response function described by state-space models is a mathematically robust framework that can be used to describe the dispersive phenomena governed by Lorentzian, Debye and Drude responses. In addition, the optical properties of an arbitrary medium can be expressed as a linear combination of simple multimode Brownian oscillator functions. The suitability of a range of signal processing schemes adopted from the Systems Identification and Control Theory community for further processing the recorded THz transients in the time or frequency domain will be outlined [4,5]. Since a femtosecond duration pulse is capable of persistent excitation of the medium within which it propagates, such approach is perfectly justifiable. Several de-noising routines based on system identification will be shown. Furthermore, specifically developed apodization structures will be discussed. These are necessary because due to dispersion issues, the time-domain background and sample interferograms are non-symmetrical [6-8]. These procedures can lead to a more precise estimation of the complex insertion loss function. The algorithms are applicable to femtosecond spectroscopies across the EM spectrum. Finally, a methodology for femtosecond pulse shaping using genetic algorithms aiming to map and control molecular relaxation processes will be mentioned.

The geometry of optimal control solutions on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E³, the spheres S³ and the hyperboloids H³ with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions is illustrated.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Studying reactive processes with classical dynamics: rebinding dynamics in MbNO

A new surface-crossing algorithm suitable for describing bond-breaking and bond-forming processes in molecular dynamics simulations is presented. The method is formulated for two intersecting potential energy manifolds which dissociate to different adiabatic states. During simulations, crossings are detected by monitoring an energy criterion. If fulfilled, the two manifolds are mixed over a finite number of time steps, after which the system is propagated on the second adiabat and the crossing is carried out with probability one. The algorithm is extensively tested (almost 0.5 mu s of total simulation time) for the rebinding of NO to myoglobin. The unbound surface ((FeNO)-N-...) is represented using a standard force field, whereas the bound surface (Fe-NO) is described by an ab initio potential energy surface. The rebinding is found to be nonexponential in time, in agreement with experimental studies, and can be described using two time constants. Depending on the asymptotic energy separation between the manifolds, the short rebinding timescale is between 1 and 9 ps, whereas the longer timescale is about an order of magnitude larger. NO molecules which do not rebind within 1 ns are typically found in the Xenon-4 pocket, indicating the high affinity of NO to this region in the protein.

Temperature dependent SANS from ferritin and apoferritin

SANS from deuterated ferritin and apoferritin solutions over the temperature range 5 to 300 K is presented. Above the freezing point the SANS is well described by Percus-Yevick hard sphere packing. On freezing, highly correlated, partially crystallised, clusters of the proteins form and grow with decreasing temperature. The resulting scattering, characterised by a squared Lorentzian structure factor, indicates a spatial extent of 1000 8, for the protein clusters.

Optimal steady motions for oriented vehicles

Motivated by the motion planning problem for oriented vehicles travelling in a 3-Dimensional space; Euclidean space E3, the sphere S3 and Hyperboloid H3. For such problems the orientation of the vehicle is naturally represented by an orthonormal frame over a point in the underlying manifold. The orthonormal frame bundles of the space forms R3,S3 and H3 correspond with their isometry groups and are the Euclidean group of motion SE(3), the rotation group SO(4) and the Lorentzian group SO(1; 3) respectively. Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. For constant twist motions or helical motions, the corresponding curves g(t) 2 SE(3) are given in closed form by using the well known Rodrigues’ formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw/helical motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Wavelet based detection of changes in the composition of RLC networks

The current work discusses the compositional analysis of spectra that may be related to amorphous materials that lack discernible Lorentzian, Debye or Drude responses. We propose to model such response using a 3-dimensional random RLC network using a descriptor formulation which is converted into an input-output transfer function representation. A wavelet identification study of these networks is performed to infer the composition of the networks. It was concluded that wavelet filter banks enable a parsimonious representation of the dynamics in excited randomly connected RLC networks. Furthermore, chemometric classification using the proposed technique enables the discrimination of dielectric samples with different composition. The methodology is promising for the classification of amorphous dielectrics.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

ON INTEGRABLE CODIMENSION ONE ANOSOV ACTIONS OF R(k)

In this paper, we consider codimension one Anosov actions of R(k), k >= 1, on closed connected orientable manifolds of dimension n vertical bar k with n >= 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of R(k) admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k = 1, we show that, under some assumptions about the smoothness of the sub-bundle E(ss) circle plus E(uu), and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of Z(k) on T(n).