An algorithm for automated extraction of resonance parameters from the stabilization method

The application of the stabilization method (Hazi and Taylor, 1970 [1]) to extract accurate energy and lifetimes of resonance states is challenging: The process requires labor-intensive numerical manipulation of a large number of eigenvalues of a parameter-dependent Hamiltonian matrix, followed by a fitting procedure. In this article, we present ReSMax, an efficient algorithm implemented as an open-access Python code, which offers full automation of the stabilization diagram analysis in a user-friendly environment while maintaining high numerical precision of the computed resonance characteristics. As a test case, we use ReSMax to analyze the natural parity doubly-excited resonance states (${}^1\text{S}^{\text{e}}$, ${}^3\text{S}^{\text{e}}$, ${}^1\text{P}^{\text{o}}$, and ${}^3\text{P}^{\text{o}}$) of helium, demonstrating the accuracy and efficiency of the developed methodology. The presented algorithm is applicable to a wide range of resonances in atomic, molecular, and nuclear systems.

Program summary

Program Title: ReSMax

CPC Library link to program files: https://doi.org/10.17632/8yny7jycgz.1

Developer's repository link: https://github.com/giogina/ReSMax

Licensing provisions: MIT

Programming language: Python

Nature of problem: The stabilization method is a widely used indirect approach for identifying resonance states (RSs) in atomic and molecular systems. It analyzes the behavior of energy eigenvalues of the system's Hamiltonian as a function of a basis set parameter, as visualized in stabilization diagrams (SDs) [1]. Resonance states manifest as plateaus in these SDs and are characterized by the position and width of the associated Lorentzian peaks in the density of states (DOS) [2]. However, applying this method in practice remains labor-intensive and error-prone: analyzing large eigenvalue datasets, identifying plateau regions, and fitting DOS peaks manually requires significant effort and expert judgment. These difficulties limit the method's scalability and reproducibility, especially for systems with many closely spaced resonances or high angular momentum states. There is currently no widely available open-source tool that automates the entire workflow in a robust, accurate, and user-friendly way.

Solution method: ReSMax is an open-source Python program that automates the extraction of resonance parameters from stabilization diagrams. Given an input file containing the eigenvalue spectrum of a Hamiltonian across a range of basis set parameter values, it computes the density of states (DOS) for each root, identifies local maxima corresponding to potential resonances, and fits them to Lorentzian functions. Peaks are grouped into resonance candidates based on energy proximity and root uniqueness, and the best-fitting peak is selected for each resonance. A combination of numerical filtering, symmetry checks, and fit quality metrics ensures robust peak detection. Automatic resonance detection completes in a few seconds. An interactive interface allows optional refinement of resonance assignments before final export of the results.

Additional comments including restrictions and unusual features: Resonances indicated by descending plateaus in the SD — which can appear due to insufficient basis set size for fluorescence-active resonances as well as directly below ionization thresholds — are assigned approximate energies and flagged for manual inspection. While ReSMax was developed for helium-like ions, the method is general and capable of detecting resonance states of a wide range of systems.

References

• [1]
  A.U. Hazi, H.S. Taylor, Phys. Rev. A 1 (1970) 1109.
• [2]
  V.A. Mandelshtam, T.R. Ravuri, H.S. Taylor, Phys. Rev. Lett. 70 (1993) 1932.