Theory

Two types of ab initio computations were carried out in order to provide an interpretation of the experimental features observed upon doubly photoionizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {O_3}$$\end{document}. We computed the adiabatic (ADIE) and the vertical (VDIE) double ionization energies of O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}_3$$\end{document} and mapped its electronic states lying in the 0–8 eV energy range with respect to the electronic ground dicationic state. We used the approaches described in Ref.25 as implemented in the MOLPRO package (version 2024)26 where the atoms were described using the basis sets27–29 of Dunning and co-workers together with their associated density fitting sets30 while using the F12 approaches. In the following we will briefly describe these computations.

The potentials of O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}_3^{2+}$$\end{document} were mapped using the standard and explicitly correlated versions of the internally contracted multi-reference configuration interaction (MRCI31–33 and MRCI-F1234–36) methods on top of state-averaging complete active-space self-consistent field (SA-CASSCF) computations32,37. In CASSCF, the active space is composed by considering the orbitals from HOMO-9 up to HOMO+4 (i.e. all the 9 valence orbitals and the 4 next virtual orbitals) as active. All valence electrons were correlated. All electronic states having the same spin-multiplicity were averaged together using the state averaging procedure of MOLPRO. At the MRCI or MRCI-F12 level, the active space was constructed by considering all single and double excitations from the CASSCF wavefunctions, retaining only those having CI coefficients \ge 0.05. For each spin multiplicity, we asked for 4 states for each C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}_s$$\end{document} symmetry component.

To determine ADIEs and VDIEs of the lowest singlet and triplet states of O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}_3^{2+}$$\end{document}, we used the (R)CCSD(T)-F12b/aug-cc-pVTZ (opt)//RCCSD(T)/aug-cc-pVTZ + _Δ_CV + _Δ_SR + _Δ_ZPVE (SP) composite scheme, where “opt” and “SP” denote full geometrical optimisations without constraints (in the C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {_s}$$\end{document} point group) and single point computations, respectively. (R)CCSD(T)38–41 and (R)CCSD(T)-F1242–45 are the standard and the explicitly correlated versions of the coupled clusters method with single, double, and perturbative triple excitations employed. _Δ_CV, _Δ_SR, _Δ_ZPVE correspond to the core-valence, scalar relativistic and zero-point vibrational energy corrections. _Δ_ZPVE was evaluated after a PBE0/aug-cc-pVTZ geometry optimisation followed by anharmonic frequency calculations. ADIEs of the lowest singlet and triplet states of O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}3^{2+}$$\end{document} were evaluated as the energy difference between the energies of the respective dicationic electronic state and the one of the neutral ground state, taken at their respective equilibrium geometries, whereas their VDIEs were determined as the energy difference between the energies of the dicationic electronic state and the one of the neutral ground state at the equilibrium geometry of the neutral species. The expected accuracy of the computed ADIEs is 0.02 eV. Then, the upper dication electronic states were located using their relative energies with respect the lowest O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}3^{2+}$$\end{document} singlet as computed at the MRCI-F12/CASSCF/aug-cc-pVTZ level. These computations were done in the C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}{2v}$$\end{document} point group where we requested 4 states per C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vphantom{0}{2v}$$\end{document} symmetry for a given spin multiplicity. The expected accuracy is 0.2 eV or better.