Application of the Friedel sum rule to symmetric molecular conductors

Recent developments in nanotechnology have made it possible to measure the resistance of individual molecular wires self-assembled between two metallic pads. In this paper we present a simple expression for the resistance of a symmetric molecule in terms of a set of M phase shifts, M being the number of conducting channels connecting the molecule to the metallic pads. These phase-shifts are related by a sum rule to ${\mathrm{N}}_{\mathrm{d}}$=${\mathrm{N}}_{\mathrm{even}}$-${\mathrm{N}}_{\mathrm{odd}}$, where ${\mathrm{N}}_{\mathrm{even}}$ and ${\mathrm{N}}_{\mathrm{odd}}$ are the number of electrons occupying the even and odd eigenstates respectively. For many molecular wires of interest all the phase shifts are nearly zero except for one (per spin), allowing us to write a particularly simple relation for the resistance: ${\mathrm{R}}^{\mathrm{-}1}$=(${2\mathrm{e}}^2$/h)sin ${}_{}{}^2{}_{}{}^{}$(π${\mathrm{N}}_{\mathrm{d}}$/2). The relations presented here are based on the Friedel sum rule and should be valid even in the presence of interactions.