Nesta presente tese, abordaremos três problemas sobre processos estocásticos unidimensionais governados pela equação mestra. Através do Ansatz do Produto Matricial (MPA) determinaremos as condições suficientes para garantir a integrabilidade de um novo processo de difusão num meio com impurezas. Investigando o espectro de tal modelo, computaremos o expoente crítico z que determina como os observáveis atingem o estado estacionário. Em seguida, estudaremos o clássico modelo de 6-vértices bidimensional definido na matriz de transferência diagonal-diagonal, como um modelo de trafego unidimensional com dinâmica síncrona e assíncrona. E para concluir nosso trabalho, investigaremos alguns modelos de processos de contato com difusão, utilizando a teoria de Campo Médio em Cluster. / In this thesis, we discuss three problems on dimensional stochastic processes governed by master equation. By Product Matrix Ansatz (MPA) we determine the conditions sufficient to ensure integrability of a new process of diffusion in a medium with impurities. Investigating the spectrum of this model, we compute the critical exponent z that determines how the observable flow to stationary state. In the folowing, we study the classical 6-vertex model defined in two-dimensional diagonal-diagonal matrix transfer as a unidimensional model of traffic with synchronous and asynchronous dinamics. And to finish our work, we study models of diffusion processes of contact, using the theory of Cluster Mean-Field

Nesta presente tese, abordaremos três problemas sobre processos estocásticos unidimensionais governados pela equação mestra. Através do Ansatz do Produto Matricial (MPA) determinaremos as condições suficientes para garantir a integrabilidade de um novo processo de difusão num meio com impurezas. Investigando o espectro de tal modelo, computaremos o expoente crítico z que determina como os observáveis atingem o estado estacionário. Em seguida, estudaremos o clássico modelo de 6-vértices bidimensional definido na matriz de transferência diagonal-diagonal, como um modelo de trafego unidimensional com dinâmica síncrona e assíncrona. E para concluir nosso trabalho, investigaremos alguns modelos de processos de contato com difusão, utilizando a teoria de Campo Médio em Cluster. / In this thesis, we discuss three problems on dimensional stochastic processes governed by master equation. By Product Matrix Ansatz (MPA) we determine the conditions sufficient to ensure integrability of a new process of diffusion in a medium with impurities. Investigating the spectrum of this model, we compute the critical exponent z that determines how the observable flow to stationary state. In the folowing, we study the classical 6-vertex model defined in two-dimensional diagonal-diagonal matrix transfer as a unidimensional model of traffic with synchronous and asynchronous dinamics. And to finish our work, we study models of diffusion processes of contact, using the theory of Cluster Mean-Field

Strongly correlated systems have been a major challenge for a long time in the field of theoretical chemistry. For such systems, the relevant portion of the Hilbert space scales exponentially, preventing efficient simulation on large systems. However, in many cases, the Hilbert space can be partitioned into clusters on the basis of strong and weak interactions. In this work, we mainly focus on an approach where we partition the system into smaller orbital clusters in which we can define many-particle cluster states and use traditional many-body methods to capture the rest of the inter-cluster correlations. This dissertation can be mainly divided into two parts. In the first part of this dissertation, the clustered ansatz, termed as tensor product states (TPS), is used to study large strongly correlated systems. In the second part, we study a particular type of strongly correlated system, correlated triplet pair states that arise in singlet fission. The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. We extend the incremental full configuration interaction originally proposed for a Slater determinant to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean-field reference TPS configuration and includes the correlation contribution of the excited TPSs using a many-body expansion. This method, named cluster many-body expansion (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from an RHF reference. The performance of the cMBE method is also tested on a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 1066 determinants for the exact FCI solution. Selected CI (SCI) using determinants becomes intractable for large systems with strong correlation. We introduce a method for SCI algorithms using tensor product states which exploits local molecular structure to significantly reduce the number of SCI variables. We demonstrate the potential of this method, called tensor product selected configuration interaction (TPSCI), using a few model Hamiltonians and molecular examples. These numerical results show that TPSCI can be used to significantly reduce the number of SCI variables in the variational space, and thus paving a path for extending these deterministic and variational SCI approaches to a wider range of physical systems. The extension of the TPSCI algorithm for excited states is also investigated. TPSCI with perturbative corrections provides accurate excitation energies for low-lying triplet states with respect to extrapolated results. In the case of traditional SCI methods, accurate excitation energies are obtained only after extrapolating calculations with large variational dimensions compared to TPSCI. We provide an intuitive connection between lower triplet energy mani- folds with Hückel molecular orbital theory, providing a many-body version of Hückel theory for excited triplet states. The n-body Tucker ansatz (which is a truncated TPS wavefunction) developed in our group provides a good approximation to the low-lying states of a clusterable spin system. In this approach, a Tucker decomposition is used to obtain local cluster states which can be truncated to prune the full Hilbert space of the system. As a truncated variational approach, it has been observed that the self-consistently optimized n-body Tucker method is not size- extensive, a property important for many-body methods. We explore the use of perturbation theory and linearized coupled-cluster methods to obtain a robust yet efficient approximation. Perturbative corrections to the n-body Tucker method have been implemented for the Heisenberg Hamiltonian and numerical data for various lattices and molecular systems has been presented to show the applicability of the method. In the second part of this dissertation, we focus on studying a particular type of strongly correlated states that occurs in singlet fission material. The correlated triplet pair state 1(TT) is a key intermediate in the singlet fission process, and understanding the mechanism by which it separates into two independent triplet states is critical for leveraging singlet fission for improving solar cell efficiency. This separation mechanism is dominated by two key interactions: (i) the exchange interaction (K) between the triplets which leads to the spin splitting of the biexciton state into 1(TT),3(TT) and 5(TT) states, and (ii) the triplet-triplet energy transfer integral (t) which enables the formation of the spatially separated (but still spin entangled) state 1(T...T). We develop a simple ab initio technique to compute both the triplet-triplet exchange (K) and triplet-triplet energy transfer coupling (t). Our key findings reveal new conditions for successful correlated triplet pair state dissociation. The biexciton exchange interaction needs to be ferromagnetic or negligible compared to the triplet energy transfer for favorable dissociation. We also explore the effect of chromophore packing to reveal geometries where these conditions are achieved for tetracene. We also provide a simple connectivity rule to predict whether the through-bond coupling will be stabilizing or destabilizing for the (TT) state in covalently linked singlet fission chromophores. By drawing an analogy between the chemical system and a simple spin-lattice, one is able to determine the ordering of the multi-exciton spin state via a generalized usage of Ovchinnikov's rule. In the case of meta connectivity, we predict 5(TT) to be formed and this is later confirmed by experimental techniques like time-resolved electron spin resonance (TR-ESR). / Doctor of Philosophy / The study of the correlated motion of electrons in molecules and materials allows scientists to gain useful insights into many physical processes like photosynthesis, enzyme catalysis, superconductivity, chemical reactions and so on. Theoretical quantum chemistry tries to study the electronic properties of chemical species. The exact solution of the electron correlation problem is exponentially complex and can only be computed for small systems. Therefore, approximations are introduced for practical calculations that provide good results for ground state properties like energy, dipole moment, etc. Sometimes, more accurate calculations are required to study the properties of a system, because the system may not adhere to the as- sumptions that are made in the methods used. One such case arises in the study of strongly correlated molecules. In this dissertation, we present methods which can handle strongly correlated cases. We partition the system into smaller parts, then solve the problem in the basis of these smaller parts. We refer to this block-based wavefunction as tensor product states and they provide accurate results while avoiding the exponential scaling of the full solution. We present accurate energies for a wide variety of challenging cases, including bond breaking, excited states and π conjugated molecules. Additionally, we also investigate molecular systems that can be used to increase the efficiency of solar cells. We predict improved solar efficiency for a chromophore dimer, a result which is later experimentally verified.