Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.

The present thesis deals with a theoretical study of electronic structures in -conjugated molecular materials with focus on their application in organic elec-tronics. We also discuss a modified and efficient symmetrized DMRG algorithm for studying excited states in these systems. In recent times, organic conjugated systems have emerged as potential candidates in a wide range of fascinating fields by virtue of their tunable electronic properties, easy processability and low cost. Tunability in the electronic and optical properties primarily are centered on the or-dering and nature of the low-lying excited states. Probing these important excited states also demands development of efficient and adaptable techniques. Chapter 1 provides a basic overview of conjugated organic polymers which have been utilized over decades in diverse fields as in organic light emitting diodes (OLED), organic solar cells (OSC) and non-linear optical (NLO) devices. These systems also contribute significantly to theoretical understanding as they pro vide important insights of one and quasi-one dimensional systems. In this chapter, we have given basic description of the electronic processes in OLED and OSC along with a brief theoretical description of -conjugated organic systems. Chapter 2 gives an account of the numerical techniques which are necessary for the study of low-dimensional strongly correlated systems like -conjugated sys-tems. For this purpose, effective low-energy model Hamiltonians viz. Huckel,¨ Hubbard and Pariser-Parr-Pople Hamiltonians are discussed. Exact diagonalization technique within the diagrammatic valence bond (DVB) basis and density matrix renormalization group (DMRG) technique are discussed in details. We have also given brief accounts of the methods employed to study real-time dynamics. A short description of different computational techniques for the study of NLO properties in -conjugated systems is also provided. Engineering the position of the lowest triplet state (T1) relative to the first excited singlet state (S1) is of great importance in improving the efficiencies of organic light emitting diodes and organic photovoltaic cells. In chapter 3, we have carried out model exact calculations of substituted polyene chains to understand the fac-tors that affect the energy gap between S1 and T1. The factors studied are backbone dimerization, different donor-acceptor substitutions and twisted backbone geome-try. The largest system studied is an eighteen carbon polyene which spans a Hilbert space of about 991 million in the triplet subspace. We show that for reverse inter-system crossing (RISC) process, the best choice involves substituting all carbon sites on one half of the polyene with donors and the other half with acceptors. Singlet fission (SF) is a potential pathway for significant enhancement of efficiency in OSC. In chapter 4, we study singlet fission in a pair of polyene molecules in two different stacking arrangements employing exact many-body wave packet dy-namics. In the non-interacting model, SF is absent. The individual molecules are treated within Hubbard and Pariser-Parr-Pople (PPP) models and the interac-tion between them involves transfer terms, intersite electron repulsions and site-charge—bond-charge repulsion terms. Initial wave packet is construc ted from ex-cited singlet state of one molecule and ground state of the other. Time develop-ment of this wave packet under the influence of intermolecular interactions is fol-lowed within the Schrodinger¨ picture by an efficient predictor-corrector scheme. In unsubstituted Hubbard and PPP chains, 21A state leads to significant SF yield while the 11B state gives negligible fission yield. On substitution by donor-acceptor groups of moderate strength, the lowest excited state will have sufficient 2 1A char-acter and hence gives significant SF yield. Because of rapid internal c onversion, the nature of the lowest excited singlet will determine the SF contribution to OSC effi - ciency. Furthermore, we find the fission yield depends considerably on th e stacking arrangement of the polyene molecules. In chapter 5, we have given an account of a new modified algorithm for symmetry adaptation within symmetrized density matrix renormalization group (SDMRG) technique. SDMRG technique has been an efficient method for studying low-lying eigenstates in one and quasi-one dimensional electronic systems. However, SDMRG method until now, had bottlenecks involving construction of linearly in-dependent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. Our modified algorithm overcomes this bottleneck. T he new method incorporates end-to-end interchange symmetry (C2), electron-hole symmetry (J) and parity or spin-flip symmetry (P) in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new ba-sis with maximum sparseness, just one non-zero matrix element per row. Using methods similar to those employed in exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the eighties, it is possible to construct or-thogonal SDMRG basis states while bypassing the slow step of Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited states of 1,12-benzoperylene. In chapter 6, we have studied the correlated excited states of coronene and ova-lene within Pariser-Parr-Pople (PPP) model employing symmetry adapted density matrix renormalization group technique. These polynuclear aromatic hydrocar-bons can be considered as graphene nanoflakes and study of their ele ctronic struc-tures will shed light on the electron correlation effects in these finite-size gr aphene analogues. The electron correlation effect usually diminishes on going from one-dimensional to higher-dimensional systems, yet, it is significant within these fin ite-size graphene derivatives where it depends on the molecular topology. We have characterized these low-lying energy states by calculating bond orders, spin den-sities in the lowest triplet state and two-photon absorption cross-sections for low-lying two-photon states. vi

This thesis deals with out-of-equilibrium transport phenomena in strongly correlated low-dimensional fermionic systems, with special emphasis on π-conjugated molecular materials. The focus of this work is to study real-time dynamics of spin and charge transport in these systems in order to investigate non-equilibrium transport in single-molecule electronic and spintronic devices. Chapter 1 describes the electronic structure and dynamics of strongly correlated fermionic systems in general, and in one-dimension, in particular. For this purpose, effective low-energy model Hamiltonians (used in this work) are discussed. Whenever applicable, approximate analytical and numerical methods commonly used in the literature to deal with these model Hamiltonians, are outlined. In the context of one-dimensional strongly correlated fermionic systems, analytical techniques like the Bethe ansatz and bosonization, and numerical procedures like exact diagonalization and DMRG, used for solving finite systems, are discussed in detail. Chapter 2 provides an overview of the different zero-temperature (T = 0) time-dependent DMRG algorithms, which have been used to study out-of-equilibrium time-dependent phenomena in low-dimensional strongly correlated systems. In Chapter 3 we employ the time-dependent DMRG algorithm proposed by Luo, Xiang and Wang [Phys. Rev. Lett. 91, 049701 (2003)], to study the role of dimerization and electronic correlations on the dynamics of spin-charge separation. We employ the H¨uckel and Hubbard models for our studies. We have modified the algorithm proposed by Luo et. al to overcome some of its limitations. Chapter 4 presents a generalized adaptive time-dependent density matrix renormalization group (DMRG) scheme developed by us, called the Double Time Window Targeting (DTWT) technique, which is capable of giving accurate results with lesser computational resources than required by the existing methods. This procedure originates from the amalgamation of the features of pace keeping DMRG algorithm, first proposed by Luo et. al, [Phys.Rev. Lett. 91, 049701 (2003)], and the time-step targeting (TST) algorithm by Feiguin and White [Phys. Rev. B 72, 020404 (2005)]. In chapter 5 we apply the Double Time Window Targeting (DTWT) technique, which was discussed in the previous chapter, for studying real-time quantum dynamics of spin-charge separation in π-conjugated polymers. We employ the Pariser-Parr-Pople (PPP) model which has long-range electron-electron interactions. For investigating real-time dynamics of spin and charge transport, we inject a hole at one end of polyene chains of different lengths and study the temporal evolution of its spin and charge degrees of freedom, using the DTWT td-DMRG algorithm. Chapter 6 we investigate the effect of terminal substituents on the dynamics of spin and charge transport in donor-acceptor substituted polyenes (D- (CH)x- A) chains, also known as push-pull polyenes. We employ long-range correlated model Hamiltonian for the D- (CH)x- A system and, real-time DMRG dynamics for time propagating the wave packet obtained by injecting a hole at a terminal site in the ground state of the system. Our studies reveal that the end groups do not affect the spin and charge velocities in any significant way, but change the amount of charge transported. We have compared these with the polymethineimine (CN)x system in which besides electron affinities, the nature of pz orbitals in conjugation also alternate from site to site. Chapter 7 presents our investigation on the effect of static electron-phonon coupling (dimerization) on the dynamics of spin-charge separation in particular, and transport in general, in π-conjugated polyene chains. The polyenes are modeled by the Pariser-Parr-Pople Hamiltonian, having long-range electron-electron correlations. Our studies reveal that spin and charge velocities depend both on the chain length and dimerization. The spin and charge velocities increase as dimerization increases, but the amount of charge and spin transported along the chain decrease with enhancement in dimerization. Furthermore, in the range 0.3≤ δ≤0.5, it is observed that the dynamics of spin-charge separation becomes complicated, and the charge degree of freedom is affected more by electron-phonon coupling compared to the spin degree of freedom.

We study unconventional superconductivity in two-dimensional systems. Unbiased numerical calculations within two-dimensional Hubbard models have found no evidence for long-range superconducting order. Most of the two-dimensional theories suggest that the superconducting state can be obtained by destabilizing an antiferromagnetic or spin-liquid insulating state. An antiferromagnet is a half-filled system because each site has one electron or hole. However, in anisotropic triangular lattices, numerical calculation finds pairing enhancement at quarter-filling but no long-range superconducting order. Many organic superconductors are dimerized in nature. Such a dimer lattice is effectively half-filled because each dimer has one electron or hole. Some theories suggest that magnetic fluctuation in such a system can give superconductivity. However, at zero temperature, we performed density matrix renormalization group (DMRG) calculations in such a system, and we find no superconducting long-range order. We also find that the antiferromagnetic order is not necessary to get a superconducting state. Failure in explaining superconductivity in two-dimensional systems suggests that only repulsive interactions between electrons are not sufficient, and other interactions are required. The most likely candidate is the electron-phonon interaction. However, existing theories of superconductivity emphasize either electron-electron or electron-phonon interactions, each of which tends to cancel the effect of the other. We present direct evidence from quantum Monte Carlo calculations of cooperative, as opposed to competing, effects of electron-electron and electron-phonon interactions within the frustrated Hubbard Hamiltonian, uniquely at the band-filling of one-quarter. Bond-coupled phonons and the onsite Hubbard U cooperatively reinforce d-wave superconducting pair-pair correlations at this filling while competing with one another at all other densities. Our work further gives new insight into how intertwined charge-order and superconductivity appear in real materials.