We validate that off-resonant electron transport across ultra-short oligomer molecular junctions is characterised by a conductance which decays exponentially with length, and we discuss a method to determine the damping factor via the energy spectrum of a periodic structure as a function of complex wavevector. An exact mapping to the complex wavevector is demonstrated by first-principle-based calculations of: (a) the conductance of molecular junctions of phenyl-ethynylene wires covalently bonded to graphitic ribbons as a function of the bridge length, and (b) the complex-band structure of poly-phenyl-ethynylene. (C) 2004 Elsevier B.V. All rights reserved.