The behavior of the Position-Spread Tensor, Λ, in a series of light diatomic molecules (either neutral or negative ions) is investigated at a Full Configuration Interaction level. This tensor, which is the second moment cumulant of the total position operator, is invariant with respect to molecular translations, while its trace is also rotationally invariant. Moreover, the tensor is additive in the case of non-interacting subsystems, and can be seen as an intrinsic property of a molecule. In the present work, it is shown that the longitudinal component of the tensor. Λ||, which is small for internuclear distances close to the equilibrium, tends to grow if the bond is stretched. A maximum is reached in the region of the bond breaking, then Λ|| decreases and converges towards the isolated-atom value. The degenerate transversal components, Λ, on the other hand, have usually a monotonic growth towards the atomic value. The Position Spread is extremely sensitive to reorganization of the molecular wavefunction, and it becomes larger in the case of an increase of the electron mobility, as illustrated by the neutral-ionic avoided crossing in LiF. For these reasons, the Position Spread can be an extremely useful property that characterizes the nature of the wavefunction in a molecular system.