The Clusterization Technique: A Systematic Search for the Resonance Energies Obtained via Padé

Atomic and molecular resonances play an important role in many physical processes, hence developing theoretical tools to properly calculate these states is required. Recently, we introduced a method for calculating the electronic resonance complex energies from stabilization graphs via analytical continuation, specifically, using the Padé approximant. This method was shown to be efficient, for example, in interpreting the results of cold molecular collisions. However, we observed that the complex energies obtained by Padé depend on the selected set of input points from the stabilization graph. In addition, unphysical solutions (noise) may appear and need to be eliminated. Therefore, applying the method to systems in which the resonance values are unavailable is difficult. The excited Li-He* Feshbach states, for which autoionization was recently observed, present such a challenge. Herein, we introduce a statistical approach to single out the resonance energy from the false solutions by identifying it as a cluster of Padé solutions. This clusterization technique was applied to study several electronic resonance states, for which we obtained excellent agreement with available data (exact or other theoretical solutions and an experimental result). Following this, the technique successfully identified the most likely Li-He* Feshbach resonance energy. Moreover, we concluded that large input sets generate much noise while restricting the number of points facilitates clusterization, which makes this approach more attractive, since input points are obtained from computationally demanding electronic-structure calculations. Overall, the use of analytical continuation via Padé, along with the statistical technique presented herein, offers an efficient approach to calculate resonances.