An Algorithm for Automated Extraction of Resonance Parameters from the Stabilization Method

TL;DR

This work tackles the laborious task of extracting resonance energies $E_r$ and widths $\Gamma$ from stabilization diagrams by introducing ReSMax, an open-source Python tool that automates DOS computation, peak fitting to Lorentzians, and resonance grouping. The method builds on the stabilization approach and uses an explicitly correlated Hylleraas basis to generate energy eigenroots via a generalized eigenvalue problem $\mathbb{H} c = E\mathbb{S} c$, with a tunable basis-set parameter $\gamma$ to produce plateaus corresponding to resonances. ReSMax streamlines analysis with automated DOS peak detection and Lorentzian fitting, while offering an interactive mode for manual refinement; it is tested on the natural-parity doubly-excited resonances of ${}^{\infty}\text{He}$ and, for the first time, reports finite-nuclear-mass resonance data. The tool achieves rapid analysis (on the order of seconds for automatic processing) and improves reproducibility, providing a practical pathway to characterize resonances across atomic, molecular, and nuclear systems by yielding precise $E_r$ and $\Gamma$, with $\Gamma$ inversely related to the lifetime $\tau$ of the state. The results validate known benchmarks and extend resonance data to previously unexplored finite-mass helium cases, highlighting the method's broad applicability and impact on resonance spectroscopy.

Abstract

The application of the stabilization method [A.~U.\ Hazi and H.~S.\ Taylor, Phys.~Rev.~A {\bf 1}, 1109 (1970)]) 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 \dosmax, an efficient algorithm implemented as an open-access \texttt{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 \dosmax to analyze the natural parity doubly-excited resonance states (${}^{1}\textnormal{S}^{\textnormal{e}}$, ${}^{3}\textnormal{S}^{\textnormal{e}}$, ${}^{1}\textnormal{P}^{\textnormal{o}}$, and ${}^{3}\textnormal{P}^{\textnormal{o}}$) 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.

Figures (8)

• Figure 1: ReSMax-generated stabilization diagram (SD) displaying the first $500$${}^{1}\text{P}^{\text{o}}$ energy roots of ${}^{\infty\!}\text{He}$, computed using the basis set given in Eq. \ref{['rwf']} with $n=9$, $\texttt{w}=2$, and $\lambda_{0}=4.0$ a.u., as a function of the basis set parameter $\gamma$. Bound states appear as almost constant lines below $E=-2.0$ a.u.; resonances manifest as plateaus.
• Figure 2: Energy curve $(\gamma_i, E^{(19)}_i)$ (left) and density of states (DOS) curve $(E^{(19)}_i, \rho_i^{(19)})$ (right, rotated by 90$^{\circ}$ to share the energy axis) for root 19 of ${}^{1}\text{P}^{\text{o}}$${}^{\infty\!}\text{He}$ (compare Fig. \ref{['fig:po_he_infm']}) over the energy range from $-0.72$ to $-0.55\text{ a.u.}$ The energy curve exhibits three plateaus at $E_r = -0.693$, $-0.597$ and $-0.564$ a.u. Each plateau indicates a resonance state, and corresponds to a peak in the DOS curve centered at the same energy. Flatter plateaus produce narrower and higher DOS peaks, indicating longer lifetimes. (The DOS peak at $-0.597$ exceeds the plotted $\rho^{(19)}$ range by a factor of 47.)
• Figure 3: ReSMax-generated resonance overview plot corresponding to the three lowest ${}^{1}\text{P}^{\text{o}}$ resonance states of ${}^{\infty\!}\text{He}$, consisting of a stabilization diagram on the left with highlighted plateau regions and detected resonance states (RS), and a scatter plot on the right with the DOS distribution for each roots, confirming the resonance positions.
• Figure 4: Subset of the ReSMax-generated grid plot displaying the DOS points (colored dots) and corresponding Lorentzian fits (black lines) of roots 15, 17, and 18 for the second RS of ${}^1\text{P}^{\text{o}}$${}^{\infty}\text{He}$. Each peak fit plot is annotated with the root number, peak position $E \equiv E_r$ and width $G \equiv \Gamma$, as well as the relative SSR per point $\text{Err} \equiv \chi_{r}$. The top-left subplot displays the DOS data points of all contributing roots together, providing an overview of their relative positions and heights. It demonstrates that the DOS peaks 15 and 16 (fit not shown here) do not have fully Lorentzian character, in contrast to the remaining peaks 17 and 18. The DOS peak of root 17 has been selected as the best fit, as indicated by a label <Selected> and the smallest fit error.
• Figure 5: ReSMax-generated resonance overview plot corresponding to ${}^{1}\text{P}^{\text{o}}$${}^{\infty\!}\text{He}$ in the energy range from $-0.55$ to $-0.5$ a.u., consisting of a stabilization diagram with highlighted plateaus and detected RSs (on the left), and a logarithmic scatter plot of the DOS distributions, indicating resonances as clear peaks (on the right). For each resonance, the representative plateau is indicated in a darker shade, and labeled in bold. Descending plateaus are labeled in red, as are resonances that have a descending plateau as their representative.