The electrocatalytic water decomposition has been investigated in this thesis by means of its two half standard reactions: the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER). These reactions occur in different locations in a typical electrochemical cell: the anode and the cathode, respectively. Motivated by the lack of understanding about the reaction mechanisms occurring at the anodes and cathodes, we have proposed first: novel representations of typical TiO2 surfaces, based on small cluster systems, which can be used for a quick and more detailed assessment of the OER activities at modified TiO2 surfaces, and secondly we investigated the HER in two sets of model surfaces which represent recently synthesized materials, based on Mo2C and graphene with promising activities toward the HER. We have employed Density Functional Theory (DFT) based methods within both localized and extended basis sets, as implemented in GAMESS and VASP packages, respectively, to examine the structural, electronic and vibrational properties of the proposed models. We propose new reaction mechanisms for the OER on a number of molecular representations of TiO2 electrodes. For each reaction pathway, the free energy profile is computed, at different biases, from the DFT energies, the entropic and the zero-point energy contributions. The mechanisms explored in this thesis are found to be energetically more feasible than alternative reaction pathways considered in previous theoretical works based on molecular representations of the TiO2 surfaces. The representation of the surface of specific, commonly occurring, titanium dioxide crystals (e.g., rutile and anatase) within the small cluster approximation is able to reproduce qualitatively the rutile (110) outperforming of the anatase (001) surface. We subsequently investigate the influence of doping TiO2 surfaces with transition metals (TMs) on the performance of TiO2 -based electrodes for the water splitting electrochemical reaction. Two cluster models of the TM-doped active sites which resemble both the TiO2 anatase (001) and rutile (110) surfaces, respectively, are considered for the evaluation of the water decomposition reaction when a Ti is replaced by a TM atom. A set of TMs spanning from Vanadium to Nickel is considered. The late TMs explored here: Fe, Co and Ni are found to reproduce the observed experimental trends for the overpotentials in TiO2-doped electrodes. In the case of Cr and Mn, the present study predicts an enhancement of the OER activity for the anatase-like clusters while a reduction of this activity is found for the rutile-like ones. The vanadium-doped structures do not show relevant influence in the OER activity compared to pure TiO2-based cluster models. The last part of this work is devoted to the theoretical study of the HER on recently found materials based on the synergistic combination of molybdenum carbide and graphene layers. We propose two major structural models to describe the HER mechanism within the framework of DFT: Mo2C-based clusters adsorbed on carbon nanosheets and the Mo2C (001) surface covered by pure and nitrogen-doped graphene layers. The former system evaluates the influence of Mo2C nanoparticles adsorbed on carbon nanosheets towards the HER. The second one is employed to gain insight about the high HER activity observed in molybdenum carbide anchored on nitrogen-doped porous carbon nanosheets (Mo2C@2D-NPC), recently synthesized. The H-adsorption free energy has been used as a principal descriptor to asses the HER activity at the proposed model active sites. It resembles the value for the best state of the art catalyst for the HER (i.e., platinum at carbon substrate Pt@C) in some of the proposed structural models. Furthermore, a pH-correction is added within a simplified model, to the H-adsorption free energy barrier in every proposed structure. The pH dependence of the H-adsorption free energy barriers allows the assessment of the HER at acidic and alkaline conditions simultaneously. An overall agreement with experimental results is found and further predictions, promoting the development of better HER catalysts, have been done.

During the last 30 years considerable attention was paid to open systems far from thermal equilibrium. Under certain conditions these dissipative systems show a qualitatively new behavior on macroscopic length scales, which are known as spatiotemporal structures. These new structures arise as a feature of collective behavior of a many-body systems. One particular example of dissipative systems considered in the present Thesis is the systems with reactant birth and death. Such systems arise, e.g., in description of the population growth or the kinetics of chemical reactions. To describe the systems with a large number of particles, one has to impose some restrictions. So, it is assumed that individual properties of particles are not important, only their interaction and interaction result (reaction) are taken into account. A number of rules, which describe the behavior of particles on the microscopic level, are known as a mathematical model. There exist two methods to analyze properties of a mathematical model. The first is analysis based on the master equation. In general, this method fails to describe the properties of spatiotemporal structures. There are no analytical approximations taking into account the effect of long-range particle correlation, which is important for description of the changes on a macroscopic range. The second approach are Monte Carlo (MC) computer simulations, which actually is alternative to experiments. The MC method takes into account long-range reactant correlations. They arise as a result of microscopical model. MC has disadvantages typical for all numerical methods, e.g., a large simulation time. In the present Thesis the Lotka-type and the A+B->0 models are considered in detail. These reactions are commonly found as one of a component in many chemical reactions. The emphasis is made on understanding the basic properties of these models. Further, several physically important modifications of the Lotka-type and the A+B->0 models are made. Firstly, in Chapter 1. the Lotka-type model is extended to investigate the resonance properties. Secondly, the effect of reactant diffusion and interaction is incorporated into Lotka-type model in Chapter 2. Thirdly, the standard A+B->0 reaction is extended to the case of surface reconstruction in Chapter 3. General conclusion is presented at the end of the Thesis, which is ended by four Appendices.

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