Locality and computational reliability of linear response calculations for molecular systems

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.