Green's function methods for single molecule junctions

We present a brief pedagogical review of theoretical Green's function methods applicable to open quantum systems out of equilibrium, in general, and single molecule junctions, in particular. We briefly describe experimental advances in molecular electronics and then discuss different theoretical approaches. We then focus on Green's function methods. Two characteristic energy scales governing the physics are many-body interactions within the junctions and molecule-contact coupling. We, therefore, discuss weak interactions and weak coupling as two limits that can be conveniently treated within, respectively, the standard nonequilibrium Green's function (NEGF) method and its many-body flavors (pseudoparticle and Hubbard NEGF). We argue that the intermediate regime, where the two energy scales are comparable, can in many cases be efficiently treated within the recently introduced superperturbation dual fermion approach. Finally, we review approaches for going beyond these analytically accessible limits, as embodied by recent developments in numerically exact methods based on Green's functions.