Theory of electronic resonances: fundamental aspects and recent advances

Exploring electronic resonances in pyridine: Insights from orbital stabilization techniques

Electron attachment to pyridine results in electronic resonances, metastable states that can decay through electronic or nuclear degrees of freedom. This study uses orbital stabilization techniques combined with bound electronic structure methods, based on equation of motion coupled cluster or multi-reference methods, to calculate positions and widths of electronic resonances in pyridine that exist below 10 eV. We report four 2B1 and four 2A2 resonances, including one 2B1 not previously reported experimentally and two 2A2 resonances not reported at all in the literature. The two lower energy resonances are one-particle shape resonances, while the remaining are mixed or primarily core-excited resonances. Multi-reference perturbation theory provides the best description of these resonances, especially when their character is mixed. We describe the character of these resonances qualitatively and calculate Dyson orbitals, which provide information about their decay channels.

Ab initio treatment of molecular Coster-Kronig decay using complex-scaled equation-of-motion coupled-cluster theory

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.

Selected Configuration Interaction for Resonances

Electronic resonances are metastable states that can decay by electron loss. They are ubiquitous across various fields of science, such as chemistry, physics, and biology. However, current theoretical and computational models for resonances cannot yet rival the level of accuracy achieved by bound-state methodologies. Here, we generalize selected configuration interaction (SCI) to treat resonances by using the complex absorbing potential (CAP) technique. By modifying the selection procedure and the extrapolation protocol of standard SCI, the resulting CAP-SCI method yields resonance positions and widths of full configuration interaction quality. Initial results for the shape resonances of N2- and CO- reveal the important effect of high-order correlation, which shifts the values obtained with CAP-augmented equation-of-motion coupled-cluster with singles and doubles by more than 0.1 eV. The present CAP-SCI approach represents a cornerstone in the development of highly accurate methodologies for resonances.

Modeling Intermolecular Coulombic Decay with Non-Hermitian Real-Time Time-Dependent Density Functional Theory

In this work, we investigate the capability of using real-time time-dependent density functional theory (RT-TDDFT) in conjunction with a complex absorbing potential (CAP) to simulate the intermolecular Coulombic decay (ICD) processes following the ionization of an inner-valence electron. We examine the ICD dynamics in a series of noncovalent bonded dimer systems, including hydrogen-bonded and purely van der Waals (VdW)-bonded systems. In comparison to previous work, we show that RT-TDDFT simulations with a CAP correctly capture the ICD phenomenon in systems exhibiting a stronger binding energy. The calculated time scales for ICD of the studied systems are in the range of 5-50 fs, in agreement with previous studies. However, there is a breakdown in the accuracy of the methodology for the pure VdW-bonded systems. Overall, the presented RT-TDDFT/CAP methodology provides a powerful tool for differentiating between competing electronic relaxation pathways following inner-valence or core ionization without necessitating any a priori assumptions.

Toward a numerically efficient description of bulk-solvated anionic states

We investigate the vertical electron attachment energy (VAE) of 1-methyl-4-nitroimidazole, a model radiosensitizer, employing quantum mechanics/molecular mechanics (QM/MM) and QM/polarized continuum (QM/PCM) solvation models. We considered the solvent-excluded surface (QM/PCM-SES) and Van der Waals (QM/PCM-VDW) cavities within the PCM framework, the electrostatic embedding QM/MM (EE-QM/MM) model, and the self-consistent sequential QM/MM polarizable electrostatic embedding (scPEE-S-QM/MM) model. Due to slow VAE convergence concerning the number of QM solvent molecules, full QM calculations prove inefficient. Ensemble averages in these calculations do not align with VAEs computed for the representative solute-solvent configuration. QM/MM and QM/PCM calculations show agreement with each other for sufficiently large QM regions, although the QM/PCM-VDW model exhibits artifacts linked to the cavity. QM/MM models demonstrate good agreement between ensemble averages and VAEs calculated with the representative configuration. Notably, the VAE computed with the scPEE-S-QM/MM model achieves faster convergence concerning the number of QM water molecules compared to the EE-QM/MM model, attributed to enhanced efficiency from MM charge polarization in the scPEE-S-QM/MM approach. This emphasizes the importance of QM/classical models with accurate solute-solvent and solvent-solvent mutual polarization for obtaining converged VAEs at a reasonable computational cost. The full-QM approach is very inefficient, while the microsolvation model is inaccurate. Computational savings in QM/MM models result from electrostatic embedding and the representative configuration, with the scPEE-S-QM/MM approach emerging as an efficient tool for describing bulk-solvated anions within the QM/MM framework. Its potential extends to improving transient anion state descriptions in biomolecules and radiosensitizers, especially given the frequent employment of microsolvation models.

Effect of N Atom Substitution on Electronic Resonances: A 2D Photoelectron Spectroscopic and Computational Study of Anthracene, Acridine, and Phenazine Anions

The accommodation of an excess electron by polycyclic aromatic hydrocarbons (PAHs) has important chemical and technological implications ranging from molecular electronics to charge balance in interstellar molecular clouds. Here, we use two-dimensional photoelectron spectroscopy and equation-of-motion coupled-cluster calculations of the radical anions of acridine (C13H9N-) and phenazine (C12H8N2-) and compare our results for these species to those for the anthracene anion (C14H10-). The calculations predict the observed resonances and additionally find low-energy two-particle-one-hole states, which are not immediately apparent in the spectra, and offer a slightly revised interpretation of the resonances in anthracene. The study of acridine and phenazine allows us to understand how N atom substitution affects electron accommodation. While the electron affinity associated with the ground state anion undergoes a sizable increase with the successive substitution of N atoms, the two lowest energy excited anion states are not affected significantly by the substitution. The net result is that there is an increase in the energy gap between the two lowest energy resonances and the bound ground electronic state of the radical anion from anthracene to acridine to phenazine. Based on an energy gap law for the rate of internal conversion, this increased gap makes ground state formation progressively less likely, as evidenced by the photoelectron spectra.

Dynamics of Anions: From Bound to Unbound States and Everything In Between

Gas-phase anions present an ideal playground for the exploration of excited-state dynamics. They offer control in terms of the mass, extent of solvation, internal temperature, and conformation. The application of a range of ion sources has opened the field to a vast array of anionic systems whose dynamics are important in areas ranging from biology to star formation. Here, we review recent experimental developments in the field of anion photodynamics, demonstrating the detailed insight into photodynamical and electron-capture processes that can be uncovered. We consider the electronic and nuclear ultrafast dynamics of electronically bound excited states along entire reaction coordinates; electronically unbound states showing that photochemical concepts, such as chromophores and Kasha's rule, are transferable to electron-driven chemistry; and nonvalence states that straddle the interface between bound and unbound states. Finally, we consider likely developments that are sure to keep the field of anion dynamics buoyant and impactful.

Ab Initio Computation of Auger Decay in Heavy Metals: Zinc about It

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 67Ga 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 Møller-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 1D2 final state arising from K-L2L3 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 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.

Photoelectron spectroscopy of the deprotonated tryptophan anion: the contribution of deprotomers to its photodetachment channels

Photoelectron spectroscopy and electronic structure calculations are used to investigate the electronic structure of the deprotonated anionic form of the aromatic amino acid tryptophan, and its chromophore, indole. The photoelectron spectra of tryptophan, recorded at different wavelengths across the UV, consist of two direct detachment channels and thermionic emission, whereas the hν = 4.66 eV spectrum of indole consists of two direct detachment features. Electronic structure calculations indicate that two deprotomers of tryptophan are present in the ion beam; deprotonation of the carboxylic acid group (Trp(I)-) or the N atom on the indole ring (Trp(II)-). Strong similarities are observed between the direct detachment channels in the photoelectron spectra of tryptophan and indole, which in conjunction with electronic structure calculations, indicate that electron loss from Trp(II)- dominates this portion of the spectra. However, there is some evidence that direct detachment of Trp(I)- is also observed. Thermionic emission is determined to predominantly arise from the decarboxylation of Trp(I)-, mediated by the ππ* excited state near λ = 300 nm, which results in an anionic fragment with a negative electron affinity that readily autodetaches.

Calculation of DC Stark Resonances for the Ammonia Molecule

A model potential previously developed for the ammonia molecule is treated in a single-center partial-wave approximation in analogy with a self-consistent field method developed by Moccia. The latter was used in a number of collision studies. The model potential is used to calculate DC Stark resonance parameters, i.e., resonance positions and shifts using the exterior complex scaling method for the radial coordinate. Three molecular valence orbitals are investigated for fields along the three Cartesian coordinates, i.e., along the molecular axis and in two perpendicular directions. The work extends previous work on the planar-geometry water molecule for which non-monotonic shifts were observed. We find such non-monotonic shifts for fields along the molecular axis. For perpendicular fields, we report the splitting of the 1e orbitals into a fast- and a slow-ionizing orbital.

Signatures of s-wave scattering in bound electronic states

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. We compare the anions of HCl and pyrrole, which are associated with s-wave scattering, with N2 and H2, which correspond to resonances. For HCl and pyrrole, we observe, on inclusion of diffuse basis functions, a pronounced bending effect in the anionic potential energy curves near the crossing points with their corresponding neutral molecules. Additionally, we observe that the Dyson orbital and second moment of the electron density become extremely large in this region; for HCl, the size of the latter becomes 5 orders of magnitude larger over a range of 5 pm. This behaviour is not observed in H2 or N2. 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.

A New Strategy to Optimize Complex Absorbing Potentials for the Computation of Resonance Energies and Widths

Complex absorbing potentials (CAPs) are artificial potentials added to electronic Hamiltonians to make the wave function of metastable electronic states square-integrable. This makes the 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 to minimize 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.

Computing Decay Widths of Autoionizing Rydberg States with Complex-Variable Coupled-Cluster Theory

We compute autoionization widths of various Rydberg states of neon and N2 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 compute widths of the 3s, 3p, 4p and 3d Rydberg states. We then proceed to compute the widths of the 3sσg, 3dσg, and 3dπg Rydberg states of N2, 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.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Analytic Evaluation of Nonadiabatic Couplings within the Complex Absorbing Potential Equation-of-Motion Coupled-Cluster Method

We present the theory for the evaluation of nonadiabatic 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 motion can facilitate transitions between different resonances and between resonances and bound states. These nonadiabatic 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 NAC 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.

Ab Initio Investigation of the Auger Spectra of Methane, Ethane, Ethylene, and Acetylene

We present an ab initio computational study of the Auger spectra of methane, ethane, ethylene, and acetylene. Auger spectroscopy is an established technique to probe the electronic structure of molecules and exploits the Auger-Meitner effect that core-ionized states undergo. We compute partial decay widths using coupled-cluster theory with single and double substitutions (CCSD) and equation-of-motion CCSD theory combined with complex-scaled basis functions and Feshbach-Fano projection. We generate Auger spectra from these partial widths and draw conclusions about the strength of particular decay channels and trends among the four molecules. A connection to experimental results about fragmentation pathways of the electronic states produced by Auger decay is also made.

Can anions possess bound doubly-excited electronic states?

Anions play an important role in many fields of chemistry. Many molecules possess stable anions, but these anions often do not have stable electronic excited states and the anion loses its excess electron once excited. All the known stable valence excited states of anions are singly-excited states, i.e., valence doubly-excited states have not been reported. As excited states are relevant for numerous applications, and constitute basic properties, we searched for valence doubly-excited states which are stable, i.e., exhibit energies below that of the ground state of the respective neutral molecule. We concentrated on two promising prototype candidates, the anions of the smallest endocircular carbon ring Li@C₁₂ and of the smallest endohedral fullerene Li@C₂₀. By employing accurate state-of-the-art many-electron quantum chemistry methods, we investigated the low-lying excited states of these anions and found that they possess several low-lying stable singly-excited states and, in particular, a stable doubly-excited state each. It is noteworthy that the found doubly-excited state of Li@C₁₂^- possesses a cumulenic carbon ring in sharp contrast to the ground and singly-excited states. The findings shed light on how to design anions with stable valence singly- and doubly-excited states. Possible applications are mentioned.

Molecular Anions Perspective

This Perspective attempts to shed light on developments in the theoretical and experimental study of molecular anions highlighting more recent workers in the field. The species I discuss include (i) valence-bound (singly and multiply charged) anions including atmospheric, catalytic, superhalogen, interfacial, and more; (ii) dipole- and correlation-bound anions including their role as doorways to other states and their appearance "in space", and (iii) metastable anions focusing on tools needed for their theoretical treatment. I also briefly discuss angular distributions of photodetached electrons and their growing utilization in experiments and theory. A recurring theme is the dependence of electron binding energies (EBEs) on the surrounding environment. Some anions that are nonexistent as isolated species evolve to be stable but with small EBEs when weakly solvated (e.g., as in a cluster or at an air-solvent interface). Others existing in isolation only as metastable species become stable when the underlying molecular framework contains one or more positively charged group (e.g., protonated side chains in a peptide) that generates a stabilizing Coulomb potential. On the other hand, a destabilizing Coulomb potential between/among negative sites in a multiply charged anion decreases the EBEs of each such site and generates a repulsive Coulomb barrier that can affect stability.

The Auger spectrum of benzene

We present an ab initio computational study of the Auger electron spectrum of benzene. Auger electron spectroscopy exploits the Auger-Meitner effect, and although it is established as an analytic technique, the theoretical modeling of molecular Auger spectra from first principles remains challenging. Here, we use coupled-cluster theory and equation-of-motion coupled-cluster theory combined with two approaches to describe the decaying nature of core-ionized states: (i) Feshbach-Fano resonance theory and (ii) the method of complex basis functions. The spectra computed with these two approaches are in excellent agreement with each other and also agree well with experimental Auger spectra of benzene. The Auger spectrum of benzene features two well-resolved peaks at Auger electron energies above 260 eV, which correspond to final states with two electrons removed from the 1e_1g and 3e_2g highest occupied molecular orbitals. At lower Auger electron energies, the spectrum is less well resolved, and the peaks comprise multiple final states of the benzene dication. In line with theoretical considerations, singlet decay channels contribute more to the total Auger intensity than the corresponding triplet decay channels.

Ab Initio Molecular Dynamics of Temporary Anions Using Complex Absorbing Potentials

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.

Cherry-Picking Resolvents: Recovering the Valence Contribution in X-ray Two-Photon Absorption within the Core-Valence-Separated Equation-of-Motion Coupled-Cluster Response Theory

Calculations of first-order response wave functions in the X-ray regime often diverge within correlated frameworks such as equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), a consequence of the coupling with the valence ionization continuum. Here, we extend our strategy of introducing a hierarchy of approximations to the EOM-EE-CCSD resolvent (or, inversely, the model Hamiltonian) involved in the response equations for the calculation of X-ray two-photon absorption (X2PA) cross sections. We exploit the frozen-core core-valence separation (fc-CVS) scheme to first decouple the core and valence Fock spaces, followed by a separate approximate treatment of the valence resolvent. We demonstrate the robust convergence of X-ray response calculations within this framework and compare X2PA spectra of small benchmark molecules with the previously reported density functional theory results.

Application of Box and Voronoi CAPs for Metastable Electronic States in Molecular Clusters

The complex absorbing potential (CAP) approach offers a practical tool for characterization of energies and lifetimes of metastable electronic states, such as temporary anions and core ionized states. Here, we present an implementation of the smooth Voronoi CAP combined with the equation-of-motion coupled cluster with single and double substitutions method for metastable states. The performances of the smooth Voronoi CAP and box CAP are compared for different classes of systems: resonances in isolated molecules and localized and delocalized resonances in molecular clusters. The benchmark calculations show that the Voronoi CAP is generally more robust when applied to molecular clusters, but box CAPs are equally reliable for localized resonances or in the cases when the resonance does not exhibit significant electron density delocalization into the intramolecular region. As such, the choice of the CAP shape and onset should be guided by the character of the metastable states.

Finding Valence Antibonding Levels while Avoiding Rydberg, Pseudo-continuum, and Dipole-Bound Orbitals

Electronic structure methods are now widely used to assist in the interpretation of many varieties of experimental data. The energies and physical characteristics (e.g., sizes, shapes, and spatial localization) of valence antibonding π* and σ* orbitals play key roles in a variety of chemical processes including photochemical reactions and electron attachment reductions and are used in Woodward-Hoffmann-type analyses to probe reaction energy barriers and energy surface intersections leading to internal conversion or intersystem crossings. One's ability to properly populate such valence antibonding orbitals within electronic structure calculations is often hindered by the presence of other molecular orbitals having similar energies. These intruding orbitals can be of Rydberg, pseudo-continuum, or dipole-bound characteristic. This article shows how, within the most widely available electronic structure codes, one can avoid the pitfalls presented by these intruding orbitals to properly populate a valence π* or σ* orbital and how to subsequently use that orbital in a calculation that includes electron correlation effects and thereby offers the possibility of chemically useful precision. Special emphasis is given to cases in which the electronic state is metastable.