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In this work we will analyse the capabilities of several numerical techniques for the description of different physical systems. Thereby, the considered systems range from quantum over semiclassical to classical and from few- to many-particle systems. For each case we address an interesting, partly unsolved question. Despite the different topics we address in the individual chapters, the problems under study are somehow related because we focus on the time evolution of the system. In chapter 1 we investigate the behaviour of a single quantum particle in the presence of an external disordered background (static potentials). Starting from the quantum percolation problem, we address the fundamental question of a disorder induced (Anderson-) transition from extended to localised single-particle eigenstates. Distinguishing isolating from conducting states by applying a local distribution approach for the local density of states (LDOS), we detect the quantum percolation threshold in two- and three-dimensions. Extending the quantum percolation model to a quantum random resistor model, we comment on the possible relevance of our results to the influence of disorder on the conductivity in graphene sheets. Furthermore, we confirm the localisation properties of the 2D percolation model by calculating the full quantum time evolution of a given initial state. For the calculation of the LDOS as well as for the Chebyshev expansion of the time evolution operator, the kernel polynomial method (KPM) is the key numerical technique. In chapter 2 we examine how a single quantum particle is influenced by retarded bosonic fields that are inherent to the system. Within the Holstein model, these bosonic degrees of freedom (phonons) give rise to an infinite dimensional Hilbert space, posing a true many-particle problem. Constituting a minimal model for polaron formation, the Holstein model allows us to study the optical absorption and activated transport in polaronic systems. Using a two-dimensional variant of the KPM, we calculate for the first time quasi-exactly the optical absorption and dc-conductivity as a function of temperature. Concerning the numerical technique, the close relation to the time evolution in the other chapters get clear if we identify temperature with an imaginary time. In chapter 3 we come back to the time evolution of a quantum particle in an external, static potential and investigate the capability of semiclassical approximations to it. Considering various one-dimensional geometries, we address basic quantum effects as tunneling, interference and anharmonicity. The question is, to which extend and at which numerical costs, several semiclassical methods can reproduce the exact result for the quantum dynamics, calculated by Chebyshev expansion. To this end we consider the linearised semiclassical propagator method, the Wigner-Moyal approach and the recently proposed quantum tomography. A conceptually very interesting aspect of the compared semiclassical methods is their relation to different representations of quantum mechanics (wave function/density matrix, Wigner function, quantum tomogram). Finally, in chapter 4 we calculate the dynamics of a classical many-particle system under the influence of external fields. Considering a low-temperature rf-plasma, we investigate the interplay of the plasma dynamics and the motion of dust particles, immersed into the plasma for diagnostic reasons. In addition to the huge number of involved particles, the numerical description of this systems faces the challenge of a large range of involved time and length scales. Exploiting the mass differences of plasma constituents and dust particles allows for separating the PIC description of the plasma from the MD simulation of the dust particles in the effective surrounding plasma.
Computational chemical physics can give important input to astrophysical modelling and other fields of physics, where molecular properties are of importance. Understanding of spectroscopic and reactive behaviour is crucial for many systems of astrophysical interests like stars, interstellar medium and comets. Especially stellar atmospheres are of interest, because the complex physics of stars are not yet completely understood. Stars are in an unstable balance of gravitation and radiation pressure and the atmospheric dynamics have been subject of extensive modelling. Complete and accurate spectroscopic information of the atoms and molecules in these atmospheres is necessary for this attempt. In addition, the only information we have about astrophysical systems is light which is emitted or absorbed by particles in these media. This is not only true for astrophysics. In plasma physics sometimes the usage of invasive diagnostics, like Langmuir probes, is not wanted because they disturb the system. In these cases some information of the system can be regained by passively measuring infrared spectra of the plasma or by active induction of electronic transition like the laser-induced fluorescence method. Another remote sensing application is the measurement of the atmospheric composition on earth. Here, larger particles in the atmosphere as well as greenhouse gases are of current interest. Unfortunately, the experimental spectroscopic data, which is needed for the understanding and interpretation of the measured spectra, is often incomplete. This gap can be, to some extend, filled by computational chemical physics. The aim of this work was to investigate the capabilities and limitations of ab initio based potential energy surfaces for spectroscopic and reactive studies and to apply these methods to problems of rovibrational and rovibronic spectroscopy and reaction dynamics. The choice of ab initio methods and the potential fitting methods is critical for the computational chemical physics, as all further quantities directly depend on their quality. In this work modified versions of the Braams polynomial potential energy surface were used. A high level coupled cluster ab initio method was used to build potentials for a series of small hydrocarbons. Hydrocarbons can be found almost everywhere on earth and in the universe. They exist in laboratory plasmas, stellar and planetary atmospheres and interstellar gases. In all these cases, light emitted or absorbed by the molecules is an important diagnostics of the system. The potential constructed in this work partly included a cluster expansion, which adds reactant configuration spaces to the fits. This could not be done for CH_3 and higher hydrocarbons, because of the limitations of the Coupled Cluster ab initio method, which is well suited for the potential wells, but not for the dissociation regions. The examples of methyl and methane show how the potentials can be used for rovibrational spectroscopy. Results of radiation transport simulations illustrate the importance of as complete-as-possible line lists for radiation transport calculations.\\ The rovibronic spectroscopy of diatomic molecules is another important aspect for the stellar atmospheric modelling. Metal hydrides and oxides add opacity to the atmosphere in the visible light and ultraviolet frequency regions, as well as do the hydrocarbons in the infrared one. In addition the spectra of metal hydrides/oxides can be used to gather information about metal and their isotope abundances. They are used as markers for the conditions in the atmospheres of stars. In this work a new code was developed, that efficiently calculates bound-bound transitions between electronic states and bound-continuum cross sections for diatomic molecules. It also offers an adequate treatment of quasi-bound rovibrational states. One important representative of the diatoms is magnesium hydride, MgH. Before this work, line lists and photodissociation cross section were available involving the three lowest doublet states of MgH. In this work new potential energy curves were calculated and adapted to updated experimental data. This causes changes in the relative energies between the electronic states and therefore shifts in the line lists. These are important, because accurate line positions are needed for the identification of spectral lines. In addition two further electronic states were included in the calculations. This expands the spectral range of MgH into the near ultraviolet region. Radiation transport models showed significant absorption by MgH from the newly added electronic states. A second usage of the diatomic potential energy curves are photodissociation cross sections. As interstellar environments are chemically active, such data is necessary for a complete picture of the ongoing processes. The photodissociation cross sections of MgH reveal a stronger dependence of the underlying potential than the bound-bound lines. In the case of MgH the cross sections are rather weak, besides occasional resonance lines which can be several orders of magnitude stronger. As mentioned, not only spectroscopic, but also reactive behaviour of molecules is important in astrophysics. A current problem connected with this is the abundance of CH^+ in interstellar clouds. Its measured abundances do not fit the predictions from theoretical models. In addition Gerlich and co-workers recently measured low temperature H + CH^+ -> C^+ + H_2 reaction rates, which diverge from the theoretical picture and which could not be explained. In this work a reactive potential energy surface was built for the CH_2^+ system, which was then used to perform extensive calculations with quasi-classical trajectory and quantum scattering methods. It was found out, that the potentials used in previous works are not accurate enough to allow low temperature calculations. Results from these potentials must be taken with care. Furthermore, the results from the new potential energy surface indicate significantly reduced reaction rates compared to previous numerical studies. This is in agreement with the new results of Gerlich and co-workers. Nevertheless, the large error bars in the low temperature range for experimental as well as numerical results strongly suggest refined methods to be developed for both, before a final conclusion can be made. This work demonstrated the possibility of modern computational chemical physics to supply consistent data for spectroscopy and reaction dynamics. These are necessary and important inputs for fields like astrophysics, plasma physics and chemistry.