  The localization spread gives a criterion to decide between metallic versus
insulating behaviour of a material. It is defined as the second moment cumulant
of the many-body position operator, divided by the number of electrons.
Different operators are used for systems treated with Open or Periodic Boundary
Conditions. In particular, in the case of periodic systems, we use the
complex-position definition, that was already used in similar contexts for the
treatment of both classical and quantum situations. In this study, we show that
the localization spread evaluated on a finite ring system of radius $R$ with
Open Boundary Conditions leads, in the large $R$ limit, to the same formula
derived by Resta et al. for 1D systems with periodic Born-von K\'arm\'an
boundary conditions. A second formula, alternative to the Resta's one, is also
given, based on the sum-over-state formalism, allowing for an interesting
generalization to polarizability and other similar quantities.
