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Resonances for model potentials in quantum mechanics
Rašek, Petr ; Houfek, Karel (advisor) ; Čížek, Martin (referee)
Resonance phenomena appear in various areas of physics. A case of shape resonance is associated with the scattering of particles on a potential barrier, through which particles pass due to tunneling effect. The energy spectrum is the output of solving the eigenvalue problem with a modified Hamiltonian operator which enables calculation of the reso- nances. The eigenvalues corresponding to resonances are complex and are located in the fourth quadrant of the complex plane. Numerical methods such as Finite Element Me- thod (FEM), Discrete Variable Representation (DVR), Exterior Complex Scaling (ECS), and the QR algorithm are used for their computation. The thesis focuses on numerical simulations of the method properties with an emphasis on the precision of resonances cal- culation and coverage of the complex plane for several one-dimensional model potentials. The utilization of potentials is also illustrated by resonance data for electron scattering on selected diatomic molecules. For these purposes, an open-source program in Python has been created. 1
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Study of conical intersections of metastable states in continuum using the molecular symmetry.
Šmahlík, Václav ; Čížek, Martin (advisor) ; Houfek, Karel (referee)
The bachelor thesis deals with the study of vibrational interaction effects in molecu- les modeled by quantum dots embeded into the electronic continuum, modeled as "s" and "p" partial waves, using non-local theory. We show some methods for constructing molecular vibrational coordinates and their context in group theory. We focus on illustra- ting conical intersections in the well-known JT (E × e) model extended to higher orders and we shortly discuss the nature of some intersections. For the three-point vibrational model with C3v symmetry and the four-point vibrational model with tetrahedral Td sy- mmetry, we derive general form of matrix of adiabatic potential energy surfaces, forms of continuum-coupling matrices and matrix forms of non-local potentials using various tools of group theory. We comment briefly the Berry Geometric Phase effect present in adiabatic walks along hypersurfaces. Drawing inspiration from previous ab initio results, we partially discuss the topology of the three-point PJT (A1 + E) × e complex model up to the third order. Towards the end of the work, we discuss some intersections in the four-point complex model of vibrational interactions transitioning to the continuum and we focus on visualization of these peculiar hypersurfaces for various parameters. 1
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Model of coherent electron dynamics in molecules
Šenk, Jan ; Kolorenč, Přemysl (advisor) ; Houfek, Karel (referee)
An ultrashort laser pulse photoionizing the molecule produces a superposition of a few cationic states. The coupling between the electrons and the nuclei has been predicted to cause fast decoherence. In this thesis, a two-dimensional model of coupled electron- nuclear dynamics in molecules is constructed. It is based on a harmonic potential in the nuclear degree of freedom and a double harmonic potential representing two centers that bind the electrons. The electronic potential's dependence on the nuclear configuration facilitates the coupling. Thanks to the simplicity of the model, it is numerically exactly solvable. We use the basis of its eigenstates to calculate the evolution of any initial state. Several quantities are used to measure decoherence and unravel the underlying mecha- nisms. Especially useful is the Wigner quasiprobability distribution. A few fundamental cases of the model are analyzed, and it is used to approximate the coherence dynamics in the normal modes of the H2O+ cation. 1
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Two-dimensional model of electron collisions with molecules
Ivánek, Richard ; Houfek, Karel (advisor) ; Čížek, Martin (referee)
In this master thesis we study convergence of selected iterative methods applied to linear systems originating from the finite element method, dis- crete variable representation and exterior complex scaling in the context of models of electron collisions with molecules. Benchmarks and comparisons with direct methods are made and convergence of methods is analyzed for a one-dimensional two-channel model and a more realistic two-dimensional model of four different molecular systems. Along the way, a review of the mentioned numerical methods and solvers is presented as well as a discussion of the sparsity patterns and numbers of non-zero elements of the used matrix representations. Finally, possible pathologies of preconditioners are shown.
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Modelling of quantum dynamics in anharmonic potentials using model of interacting harmonic oscillators.
Kubíček, Václav ; Čížek, Martin (advisor) ; Houfek, Karel (referee)
Thematically, the work pursues a means of numerical description of anhar- monic molecular vibrations regarding vibronic coupling. The main concern is with the possibility of replacing motion in adiabatic anharmonic potential with a set of vibroni- cally interacting harmonic potentials. The work initially provides an accurately tested implementation of a numerical method for finding stationary states in anharmonic po- tentials. And mainly, there is included a thorough discussion of how well the spectrum and states of the vibranically coupled model correspond to the adiabatic approximation of the same model. 1
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A model of resonant collisions of electrons with molecules and molecular ions
Váňa, Martin ; Houfek, Karel (advisor) ; Cejnar, Pavel (referee) ; Mašín, Zdeněk (referee)
A two-dimensional model of the resonant electron-molecule collision processes with one nuclear and one electronic degree of freedom introduced by Houfek, Rescigno and McCurdy [Phys. Rev. A 73, 032721 (2006)] and a similar two- dimensional model of the dissociative recombination with potential proposed by Hamilton [Ph.D. thesis, University of Colorado, (2003)] are formulated within the time-dependent framework and solved numerically using the finite-element method with the discrete variable representation basis, the exterior complex scaling method and the generalized Crank-Nicolson method. On the model of electron-molecule collisions we illustrate how the time-dependent calculations can provide a deep insight into the origin of oscillatory structures in the vibrational excitation cross sections if one evaluates the cross sections not only at sufficiently large time to obtain the final cross sections, but rather at several characteristic times which are given by the evolution of the system. With use of the time- dependent calculations we demonstrate the complex nature of the dissociative recombination model dynamics and we propose the interpretation of the recom- bination process mechanism. We also propose few techniques for the explanation of the sharp structures in the dissociative recombination cross sections...
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Účinné průřezy srážek elektronů s atomy vodíku
Benda, Jakub ; Houfek, Karel (advisor) ; Zamastil, Jaroslav (referee)
Although the collisions of electrons and atomic hydrogen has been studied for several decades, there is still neither a complete database of scattering data, nor a universal method that would let generate such data. For astronomical and other purposes the cross sections of electron-hydrogen collisions are necessary, in a broad range of energies - from tenths of electronvolt to millions of electronvolts. In this work the author concentrates on several established approaches to electron-atom scattering and confronts results of his own implementations of these methods against the published data and results of freely available computer codes. A special attention is given to the overlaps of different methods, so that in the end a database with easy user interface can be offered for common practical usage of scattering data in applied fields.
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Time evolution of resonant collisions of electrons with molecules
Mašín, Zdeněk ; Houfek, Karel (advisor) ; Čížek, Martin (referee)
In the present work we study an alternative formulation of the so called nonlocal resonant model, which describes collisions of electrons with molecules. In our approach we solve a system of differential equations describing time-dependent nuclear dynamics of the collision process instead of solving a standard equation with nonlocal potential. The goal of this work was to find a suitable numerical method for solution of those equations, test it on a model problem and on the problem of resonant collision of electron with molecule H2. Obtained results are in very good agreement with reference results from the standard calculations.
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