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Vacancies in the L1 shell of atoms and molecules can decay non-radiatively via Coster-Kronig decay whereby the vacancy is filled by an electron from the L2,3 shell while a second electron is emitted into the ionization continuum. This process is akin to Auger decay, but in contrast to Auger electrons, Coster-Kronig electrons have rather low kinetic energies of less than 50 eV. In the present work, we extend recently introduced methods for the construction of molecular Auger spectra that are based on complex-scaled equation-of-motion coupled-cluster theory to Coster-Kronig decay. We compute ionization energies as well as total and partial decay widths for the 2s-1 states of argon and hydrogen sulfide and construct the L1L2,3M Coster-Kronig and L1MM Auger spectra of these species. Whereas our final spectra are in good agreement with the available experimental and theoretical data, substantial disagreements are found for various branching ratios suggesting that spin-orbit coupling makes a major impact on Coster-Kronig decay already in the third period of the periodic table., Comment: 10 pages, 5 figures, 83 references

The transfer of charges, including electrons and holes, is a key step in heterogeneous catalysis, taking part in the reduction and oxidation of adsorbate species on catalyst surfaces. In plasmonic catalysis, electrons can transfer from photo-excited metal nanoparticles to molecular adsorbates, forming transient negative ions that can easily undergo reactions such as dissociation, desorption, or other chemical transformations. However, ab initio characterization of these anionic states has proven challenging, and little is known about the topology of their potential energy surfaces. In this work, we investigate the dissociative adsorption of HCl on Au(111) as a representative catalytic process with relatively low reaction probabilities, which could potentially be enhanced by electron transfer from photo-excited gold nanoparticles to HCl. We employ projection-based density embedding that combines the equation-of-motion electron-attachment coupled-cluster singles and doubles (EOM-EA-CCSD) method with density functional theory (DFT), and build dissociation curves of HCl$^-$ on Au(111) along the H-Cl bond distance. The HCl anion in the gas phase is unbound at equilibrium distances and only becomes bound as the bond stretches. However, our results show that, upon adsorption on Au(111), HCl$^-$ remains a stable, bound anion at all bond lengths due to charge delocalization to the metal. Forming bound anions is easier, and dissociation of HCl$^-$ on Au(111) is further facilitated, with its dissociation energy reduced by 0.61 eV compared to its neutral counterpart on Au(111), and by 1.16 eV relative to HCl. These results underscore the efficacy of embedded EOM-CCSD methods in addressing surface science challenges and highlight the potential of plasmonic catalysis proceeding via bound, rather than transient, anionic states.

We report the first coupled-cluster study of Auger decay in heavy metals. The zinc atom is used as a case study due to its relevance to the Auger emission properties of the $^{67}$Ga radionuclide. Coupled-cluster theory combined with complex basis functions is used to describe the transient nature of the core-ionized zinc atom. We also introduce second-order M{\o}ller-Plesset perturbation theory as an alternative method for computing partial Auger decay widths. Scalar-relativistic effects are included in our approach for computing Auger electron energies by means of the spin-free exact two-component one-electron Hamiltonian, while spin-orbit coupling is treated by means of perturbation theory. We center our attention on the K-edge Auger decay of zinc dividing the spectrum into three parts (K-LL, K-LM, and K-MM) according to the shells involved in the decay. The computed Auger spectra are in good agreement with experimental results. The most intense peak is found at an Auger electron energy of 7432 eV, which corresponds to a $^1$D$_2$ final state arising from K-L$_2$L$_3$ transitions. Our results highlight the importance of relativistic effects for describing Auger decay in heavier nuclei. Furthermore, the effect of a first solvation shell is studied by modeling of Auger decay in the hexaaqua-zinc (II) complex. We find that K-edge Auger decay is slightly enhanced by the presence of the water molecules as compared to the bare atom.

We perform ab initio simulations of the total and partial Auger decay widths of 1s^-1, 2s^-1, and 2p^-1 ionized hydrogen sulfide and 2s^-1 ionized argon with non-Hermitian quantum chemistry. We use coupled cluster theory with single and double substitutions (CCSD) and equation of motion CCSD (EOM-CCSD) and discuss the novel application of (equation of motion-) second order M{\o}ller-Plesset perturbation theory (MP2). We find good agreement between the methods for the 1s^-1 hole of H2S, whereas for the other holes we can only use the EOM methods. We obtain very large decay widths of the 2s^-1-vacant states due to intense Coster-Kronig transitions with excellent agreement to experiments. The three 2p^-1 holes show completely different spectra because a decay channel is only significant when one of the final holes is spatially aligned with the initial hole. Lastly, we observe that triplet channels are much more important for the 2s^-1 and 2p^-1 holes than for the 1s^-1 hole, for which it is well known that triplet channels only contribute weakly to the total Auger intensity.

We compute EOM-EA-CCSD and EOM-EA-CCSDT potential energy curves and one-electron properties of several anions at bond lengths close to where these states become unbound. In the potential energy curves of the totally symmetric anions of HCl and pyrrole, which are associated with s-wave scattering states at the equilibrium bond lengths of the parent neutral molecules, we observe on inclusion of diffuse basis functions a pronounced bending effect near the crossing points with the potential energy curves of the neutral molecules. Additionally, we observe that the Dyson orbital and second moment of the electron density become extremely large in this region. In particular, the second moment of the HCl anion becomes 5 orders of magnitude times larger over a range of 5 pm. This behaviour is very different to the well-characterised non-totally symmetric anions of dinitrogen and dihydrogen that correspond to electronic resonances at the equilibrium bond lengths of their parent neutral molecules. Our work thus shows that bound state electronic-structure methods can distinguish between anions that turn into electronic resonances and those associated with s-wave scattering states., Comment: 17 pages, 6 figures

Complex absorbing potentials (CAPs) are artificial potentials added to electronic Hamiltonians to make the wave function of metastable electronic states square-integrable. This makes electronic-structure theory of resonances comparable to that of bound states, thus reducing the complexity of the problem. However, the most often used box and Voronoi CAPs depend on several parameters that have a substantial impact on the numerical results. Among these parameters are the CAP strength and a set of spatial parameters that define the onset of the CAP. It has been a common practice to minimize the perturbation of the resonance states due to the CAP by optimizing the strength parameter while fixing the onset parameters although the performance of this approach strongly depends on the chosen onset. Here we introduce a more general approach that allows one to optimize not only the CAP strength but also the spatial parameters. We show that fixing the CAP strength and optimizing the spatial parameters is a reliable way for minimizing CAP perturbations. We illustrate the performance of this new approach by computing resonance energies and widths of the temporary anions of dinitrogen, ethylene, and formic acid. This is done at the Hartree-Fock and equation-of-motion coupled-cluster singles and doubles levels of theory, using full and projected box and Voronoi CAPs., Comment: 35 pages

We compute autoionization widths of various Rydberg states of neon and dinitrogen by equation-of-motion coupled-cluster theory combined with complex scaling and complex basis functions. This represents the first time that complex-variable methods are applied to Rydberg states represented in Gaussian basis sets. A new computational protocol based on Kaufmann basis functions is designed to make these methods applicable to atomic and molecular Rydberg states. As a first step, we apply our protocol to the neon atom and computed widths of the $3s$, $3p$, $4p$ and $3d$ Rydberg states. We then proceed to compute the widths of the $3s\sigma_g$, $3d\sigma_g$, and $3d\pi_g$ Rydberg states of dinitrogen, which belong to the Hopfield series. Our results demonstrate a decrease in the decay width for increasing angular momentum and principal quantum number within both Rydberg series.

We present the theory for the evaluation of non-adiabatic couplings (NACs) involving resonance states within the complex absorbing potential equation-of-motion coupled-cluster (CAP-EOM-CC) framework implemented within the singles and doubles approximation. Resonance states are embedded in the continuum and undergo rapid decay through autodetachment. In addition, nuclear motions can facilitate transitions between different resonances and between resonances and bound states. These non-adiabatic transitions affect the chemical fate of resonances and have distinct spectroscopic signatures. The NAC vector is a central quantity needed to model such effects. In the CAP-EOM-CC framework, resonance states are treated on the same footing as bound states. Using the example of fumaronitrile, which supports a bound radical anion and several anionic resonances, we analyze the non-adiabatic coupling between bound states and pseudocontinuum states, between bound states and resonances and between two resonances. We find that the NAC between a bound state and a resonance is nearly independent of the CAP strength and thus straightforward to evaluate whereas the NAC between two resonance states or between a bound state and a pseudocontinuum state is more difficult to evaluate.

When a vacancy is created in an inner-valence orbital of a dimer of atoms or molecules, the resulting species can undergo interatomic/intermolecular Coulombic decay (ICD): the hole is filled through a relaxation process that leads to a doubly ionized cluster with two positively charged atoms or molecules. Since they are subject to electronic decay, inner-valence ionized states are not bound states but electronic resonances whose transient nature can only be described with special quantum-chemical methods. In this work, we explore the capacity of equation-of-motion coupled-cluster theory combined with two techniques from non-Hermitian quantum mechanics, complex basis functions and Feshbach-Fano projection, to describe ICD. To this end, we compute decay rates of several dimers: Ne_2, NeAr, NeMg, and (HF)_2, among which the energy of the outgoing electron varies between 0.3 eV and 16 eV. We observe that both methods deliver better results when the outgoing electron is fast, but the characteristic R^{-6} distance dependence of the ICD width is captured much better with complex basis functions.

Complex absorbing potentials (CAPs) are artificial potentials added to electronic Hamiltonians to make the wavefunction of metastable electronic states square-integrable. This makes the electronic structure problem of electronic resonances comparable to that of electronic bound states, thus reducing the complexity of the problem. CAPs depend on two types of parameters: the coupling parameter $\eta$ and a set of spatial parameters which define the onset of the CAP. It has been a common practice over the years to minimize the CAP perturbation on the physical electronic Hamiltonian by running an $\eta-$trajectory, whereby one fixes the spatial parameters and varies $\eta$. The optimal $\eta$ is chosen according to the minimum log-velocity criterion. But the effectiveness of an $\eta-$trajectory strongly depends on the values of the fixed spatial parameters. In this work, we propose a more general criterion, called the $\xi-$criterion, which allows one to minimize any CAP parameter, including the CAP spatial parameters. Indeed, we show that fixing $\eta$ and varying the spatial parameters according to a scheme (i.e., running a spatial trajectory) is a more efficient and reliable way of minimizing the CAP perturbations (which is assessed using the $\xi-$criterion). We illustrate the method by determining the resonance energy and width of the temporary anion of dinitrogen, at the Hartree-Fock and EOM-EA-CCSD levels, using two different types of CAPs: the box- and the smooth Voronoi-CAPs., Comment: 13 pages, 8 figures

Dissociative electron attachment, that is, the cleavage of chemical bonds induced by low-energy electrons, is difficult to model with standard quantum-chemical methods because the involved anions are not bound but subject to autodetachment. We present here a new computational development for simulating the dynamics of temporary anions on complex-valued potential energy surfaces. The imaginary part of these surfaces describes electron loss, whereas the gradient of the real part represents the force on the nuclei. In our method, the forces are computed analytically based on Hartree-Fock theory with a complex absorbing potential. $Ab\ initio$ molecular dynamics simulations for the temporary anions of dinitrogen, ethylene, chloroethane, and the five mono- to tetrachlorinated ethylenes show qualitative agreement with experiments and offer mechanistic insights into dissociative electron attachments. The results also demonstrate how our method evenhandedly deals with molecules that may undergo dissociation upon electron attachment and those which only undergo autodetachment., Comment: Manuscript: 10 pages, 4 figures. Supplementary Material: 41 pages, 43 figures

Electronic resonances are states that are unstable towards loss of electrons. They play critical roles in high-energy environments across chemistry, physics, and biology but are also relevant to processes under ambient conditions that involve unbound electrons. This feature article focuses on complex-variable techniques such as complex scaling and complex absorbing potentials that afford a treatment of electronic resonances in terms of discrete square-integrable eigenstates of non-Hermitian Hamiltonians with complex energy. Fundamental aspects of these techniques as well their integration into molecular electronic-structure theory are discussed and an overview of some recent developments is given: analytic gradient theory for electronic resonances, the application of rank-reduction techniques and quantum embedding to them, as well as approaches for evaluating partial decay widths., Comment: 19 pages, 10 figures, 269 references

The emission of an Auger electron is the predominant relaxation mechanism of core-vacant states in molecules composed of light nuclei. In this non-radiative decay process, one valence electron fills the core vacancy while a second valence electron is emitted into the ionization continuum. Because of this coupling to the continuum, core-vacant states represent electronic resonances that can be tackled with standard quantum-chemical methods only if they are approximated as bound states, meaning that Auger decay is neglected. Here, we present an approach to compute Auger decay rates of core-vacant states from coupled-cluster and equation-of-motion coupled-cluster wave functions combined with complex scaling of the Hamiltonian or, alternatively, complex-scaled basis functions. Through energy decomposition analysis, we illustrate how complex-scaled methods are capable of describing the coupling to the ionization continuum without the need to model the wave function of the Auger electron explicitly. In addition, we introduce in this work several approaches for the determination of partial decay widths and Auger branching ratios from complex-scaled coupled-cluster wave functions. We demonstrate the capabilities of our new approach by computations on core-ionized states of neon, water, dinitrogen, and benzene. Coupled-cluster and equation-of-motion coupled-cluster theory in the singles and doubles approximation both deliver excellent results for total decay widths, whereas we find partial widths more straightforward to evaluate with the former method. We also observe that the requirements towards the basis set are less arduous for Auger decay than for other types of resonances so that extensions to larger molecules are readily possible., Comment: 15 pages, 6 figures, 9 tables

The projection-based quantum embedding method is applied to electronically excited states of valence, Rydberg, and charge-transfer character, valence- and core-ionized states, as well as bound and temporary radical anions. We embed different variants of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory in density functional theory and investigate the performance of the resulting methods using small organic molecules microsolvated by a varying number of water molecules as test cases. States that are unstable towards electron loss are treated by means of a complex-absorbing potential. Besides transition energies, we also present Dyson orbitals and natural transition orbitals for embedded EOM-CCSD. Our results illustrate that embedded EOM-CCSD describes ionization and valence excitation very well and that these transitions are quite insensitive towards technical details of the embedding procedure. On the contrary, more care is required when dealing with Rydberg excitations or electron attachment. For the latter type of transition in particular, the use of long-range corrected density functionals is mandatory and truncation of the virtual orbital space -- which is indispensable for the application of projection-based embedding to large systems -- proves to be difficult., Comment: 21 pages, 3 figures, 9 tables. Final version before typesetting

Ionization rates and Stark shifts of H$_2$, CO, O$_2$, H$_2$O, and CH$_4$ in static electric fields have been computed with coupled-cluster methods in a basis set of atom-centered Gaussian functions with complex-scaled exponent. Consideration of electron correlation is found to be of great importance even for a qualitatively correct description of the dependence of ionization rates and Stark shifts on the strength and orientation of the external field. The analysis of the second moments of the molecular charge distribution suggests a simple criterion for distinguishing tunnel and barrier suppression ionization in polyatomic molecules., Comment: 23 pages in preprint format including 6 figures

The impact of residual electron correlation beyond the equation-of-motion coupled-cluster singles and doubles approximation (EOM-CCSD) on positions and widths of electronic resonances is investigated. To establish a method that accomplishes this task in an economical manner, several approaches proposed for the approximate treatment of triples excitations are reviewed with respect to their performance in the electron attachment (EA) variant of EOM-CC theory. The recently introduced EOM-CCSD(T)(a)* method, which includes non-iterative corrections to the reference and the target states, reliably reproduces vertical attachment energies from EOM-EA-CC calculations with singles, doubles, and full triples excitations in contrast to schemes in which non-iterative corrections are applied only to the target states. Applications of EOM-EA-CCSD(T)(a)* augmented by a complex absorbing potential (CAP) to several temporary anions illustrate that shape resonances are well described by EOM-EA-CCSD, but that residual electron correlation often makes a non-negligible impact on their positions and widths. The positions of Feshbach resonances, on the other hand, are significantly improved when going from CAP-EOM-EA-CCSD to CAP-EOM-EA-CCSD(T)(a)*, but the correct energetic order of the relevant electronic states is still not achieved., Comment: 23 pages in preprint format including 3 figures, 2 tables, and 76 references