In contrast to bound states, electronically metastable states or resonances still represent a major challenge for quantum chemistry and molecular physics. The reason lies in the embedding continuum: Bound states represent a many-body problem, while resonances represent a simultaneous scattering and many-body problem. Here we focus on so-called L 2 -methods, which treat the continuum only implicitly, but rather take the 'decaying state' perspective and emphasize electron correlation in the decaying state. These methods represent a natural extension of quantum chemistry into the metastable domain and are suitable for, say, modeling electron-induced reactions or resonant photo detachment. The three workhorse L 2methods are complex absorbing potentials, the stabilization method, and regularized analytic continuation. However, even for these three methods, making comparisons is less than straightforward as each method works best with a unique blend of electronic structure methods and basis sets. Here we address this issue by considering a model potential. For a model, we can establish a reliable reference resonance energy by using the complex scaling method and a discrete variable representation. Then, we can study the performance of the three workhorse methods as well as effects of more approximate Gaussian basis sets.

The alkaline earth metal trimer cluster dianions Be 3 2− and Mg 3 2− lie energetically above their respective monoanions and can therefore decay by electron autodetachment. Consequently, these dianions possess only short-lived resonance states, and here we study these states using regularized analytic continuation as well as complex absorbing potentials combined with a wide a variety of quantum chemistry methods including CCSD(T), SACCI, EOM-CCSD, CASPT2, and NEVPT2. For both Be 3 2− and Mg 3 2− , four low-energy resonance states corresponding to different occupation patterns of the two excess electrons in the two lowest p−σ and p−π orbitals are identified: Two states are dominated by doubly occupied configurations and can be characterized as showing σ and π aromatic character. The other two states correspond to the open-shell singlet/triplet pair. All dianion states are found to be highly unstable and to possess short lifetimes: They show resonance positions in the energy range 2.3−4.3 eV above the ground states of their respective monoanions and broad widths between 1 and 1.5 eV translating into femtosecond lifetimes. For both Be 3 2− and Mg 3 2− , the differences between the four states are small, but the triplet states tend to be slightly more stable than the three singlet states. Thus, in the case of the multicharged ion aromatic character of the excess electrons takes second stage while Coulomb repulsion takes front and center. In addition to the two isolated cluster dianions, model stabilization by small water clusters is explored. Our results show a dramatic drop in resonance position and width corresponding to a lifetime increase by 2 orders of magnitude. However, the "solvated" clusters are still resonances, and a more pronounced perturbation by, for example, yet larger water clusters or a ligand environment providing larger bond dipoles will be needed to fully stabilize two excess electrons localized on a small system such as an alkaline metal trimer.