The theory of time-dependent currents through small molecules, with attached leads, is examined. The wide-band and narrow-band approximations are used to describe the leads and the time-dependent Schrodinger is applied directly. A variety of possible boundary conditions are considered and, within the approximations of the model, exact solutions are obtained and their form discussed. Examples using two-atom chain molecules are analyzed in detail and series solutions are obtained for longer chains. (C) 2003 American Institute of Physics.