Development and applicarions of time-dependent density matrix functional theory

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor in Physics of Nanostructures and Advanced Materials.In thesis we analyse and implement first-principle reduced density matrix (RDM) theories as one promising framework for ab-inito calculations, both for equilibrium and nonequilibrium situations. We spend the first part of the thesis concentrating on the basic properties of density matrix theories at equilibrium. We will exploit simple systems made of two particles which are harmonically confined, as a ground to illustrate usefulness of density matrices. For such systems, we use the one-body RDM to calculate some quantities of interest, such as momentum density, from the one-body RDM. Next, we obtain natural orbitals and their occupation numbers for some of the models, and for one model we express exactly the two-body RDM in terms of the one-body RDM. Moreover, we utilise the differential virial theorem to establish a unified description of the entire family of these model systems in terms of density. In the second part we will deal with nonequilibrium cases where we need to calculate the time evolution of RDMs. We analyse the Born, Bogoliubov, Green, Kirkwood and Yvon (BBGKY) hierarchy of equations for describing the full time-evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground state energy, when applied to describe driven electron dynamics exhibit undesirable and unphysical behaviour, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyse these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations, and positive semidefiniteness of reduced density matrices in instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive-definitiveness is violated. We propose some partial solutions of such problem and point the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.The PhD research described in this thesis was carried out in the Nanobio Spectroscopy Group at the University of the Basque Country, Centro de Fisica de Materiales (CFM), CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, E-20018, Donostia-San Sebastian. The research was financially supported largely by Nanoquanta Network of Excellence (the European Commission’s Sixth Framework Programme) and the Spanish Council for Scientific Research (CSIC) through their “Junta para la Ampliacion de Estudios” fellowship (JAE-Predoc-2008). Also, some financial support was provided by the Donostia International Physics Center (DIPC).Peer reviewe