The accurate simulation of timedependent manybody systems is among the most challenging topics in modern theoretical physics. For manybody systems the number of degrees of freedom is large and conservation laws are rare making the numerical solution very demanding. Quantum mechanics adds further complications. It requires the collective description of the system in terms of the manybody wavefunction whose time evolution is governed by the Schrödinger equation. The collective nature of the description forbids to decompose the system into individual entities and makes the calculation of a quantum manybody system particularly complicated. In fact, the direct solution of the Schrödinger equation is not a viable option for atomic and molecular systems consisting of more than two electrons. To overcome this limitation a variety of approaches have been developed for the calculation of ground state properties of multielectron systems relevant for the kinetics of chemical reactions. These methods of quantum chemistry can be roughly divided into two groups: methods that are based on the wavefunction as the fundamental object, such as multiconfigurational HartreeFock, and methods based on reduced quantities, such as density functional theory. While wavefunctionbased methods are very accurate their applicability is limited to small systems due to the exponential scaling with particle number. Density functional theory, on the other hand, can treat large and extended systems, however, at the price of introducing the exchangecorrelation functional whose exact form is unknown and whose approximations are hard to improve systematically. Timedependent formulations have been achieved for both of the aforementioned approaches. The multiconfigurational timedependent HartreeFock method is among the most accurate approaches to simulate dynamical manybody systems. Timedependent density functional theory is regularly employed to simulate timedependent largescale systems. In this thesis, we aim for a method that combines the best of both worlds, the accuracy of wavefunctionbased approaches with the efficiency of polynomial scaling as in density functional theory. To this end we propagate the timedependent twoparticle reduced density matrix. As a hybrid between the electron density and the manybody wavefunction the twoparticle reduced density matrix fully includes twoparticle correlations which is a prerequisite to accurately capture effects that arise from electronelectron interactions. Further observables such as kinetic energy spectra, ionization probabilities or the total energy can be expressed directly without invoking approximate readout functionals as required in density functional theory. We develop a closed equation of motion for the twoparticle reduced density matrix by constructing a novel reconstruction functional for the threeparticle reduced density matrix that preserves norm, energy, and spin symmetries during time propagation. Further, we show how to avoid instabilities associated with the violation of Nrepresentability that have been a considerable limitation in previous approaches. We have implemented the timedependent twoparticle reduced density matrix method to describe highharmonic generation from fully threedimensional multielectron atoms as well as from onedimensional molecules. We benchmark the performance of the timedependent twoparticle reduced density matrix method by comparing it to a state of the art multiconfigurational timedependent HartreeFock calculation as well as to timedependent density functional theory. We find very good agreement between the timedependent twoparticle reduced density matrix method and the multiconfigurational timedependent HartreeFock method while timedependent density functional theory within the local density approximation shows clear deviations indicating that the correct treatment of twoparticle correlations is essential to obtain accurate highharmonic spectra.
