2. Quantum Transport Simulations Based on Time Dependent Density Functional Theory

First principles simulations of electronic quantum transport through nanostructured materials have become an area of intense research over the past years. Energy based approaches in the spirit of Landauer theory are well established in this field, but recently also methods that aim at the solution of the time dependent many electron problem become increasingly popular and highlight conduction as a dynamical process. In the first part of this chapter, we review the corresponding literature with a focus on time dependent density functional theory (TDDFT) as electronic structure method. The covered material is categorized according to the way the open boundary conditions are implemented. This division is not a mere technical point but also helps to elucidate conceptual and fundamental differences between the methods. In the second part a more detailed overview is given over one of the possible approaches: the Liouville-von Neumann scheme in TDDFT. We discuss the foundations of the method in terms of the holographic electron density theorem for open systems and present the relevant equations of motion as well as appropriate approximations. The chapter closes with a sample application of this method.