  By performing a critical analysis of the fundamental equations of
linear-response (LR) formalism in molecules, we explore the interplay between
locality of the response density operator and numerical convergence of
LR-related quantities. We show that for frequencies below the first ionization
potential (IP) of the system, it is possible to express the response density by
employing localized states only. Above this threshold energy, such a locality
property cannot be achieved. Such considerations may be transposed in terms of
the molecule's excited states. We show that not all the system's excitations
can be considered on equal footing. There is a discrete sector of excitations
-- which may also extend above IP -- that can be parametrized by observable,
localized states, which can be computationally expressed with high precision,
provided an adequate level of completeness. We present indicators that can help
to quantify such potential observable properties of an excitation, that can be
evaluated in any discretization scheme. The remaining excitation modes belong
to a continuum spectrum that, on the contrary, is not directly associated to
observable properties and can only be effectively represented in a given
computational setup. Such considerations are important not only for
reproducibility of the results among different computer codes employing diverse
formalisms, but also in view of providing a deeper understanding on the impact
of models' approximations on the scientific outcomes of the simulation.
