Electronic polarization in quasilinear chains

Starting with a finite $k$-mesh version of a well-known equation of Blount, we show how various definitions proposed for the polarization of a long chain are related. Expressions used for infinite periodic chains in the `modern theory of polarization' are thereby obtained along with a new single-particle formulation. Separate intracellular and intercellular contributions to the polarization are identified and in application to infinite chains, the traditional sawtooth definition is found to be missing the latter. For a finite open chain the dipole moment depends upon how the chain is terminated, but the intracellular and intercellular polarization do not. All these results are illustrated through calculations with a simple H\"uckel-like model.