Complex Absorbing Potential Equation-of-Motion Coupled-Cluster Method Yields Smooth and Internally Consistent Potential Energy Surfaces and Lifetimes for Molecular Resonances

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.

Complex potential energy surfaces with projected CAP technique: Vibrational excitation of N2

The projected complex absorbing potential (CAP) technique is one of the methods that allow one to extend the bound state methods for computing resonances' energies and widths. Here, we explore the accuracy of the potential energy curves generated with different electronic structure theory methods in combination with the projected CAP technique by considering resonant vibrational excitation (RVE) of N2 by electron impact as a model process. We report RVE cross sections computed using the boomerang model with potential energy curves obtained with CAP-based extended multistate complete active space perturbation theory (XMS-CASPT2) and equation of motion coupled-cluster method for electron attachment with single and double substitution (EOM-EA-CCSD) methods. We also compare potential energy curves computed with several electronic structure methods, including XMS-CASPT2, EOM-EA-CCSD, multireference configuration interaction with singles (MR-CIS) and singles and doubles (MR-CISD). A good agreement is observed between the experiment and simulated RVE cross sections obtained with the potential energy curves generated with XMS-CASPT2 and EOM-EA-CCSD methods, thus highlighting the potential of the projected CAP technique combined with accurate electronic structure methods for dynamical simulations of the processes that proceed through metastable electronic states.

Taming Negative Ion Resonances Using Nonlocal Exchange-Correlation Functionals

The characterization of negative ion resonances poses a fundamental challenge to density functional methods due to the unbound nature of resonances. To overcome this challenge, we propose one-particle nonlocal exchange-correlation (xc) potentials combining the exact-exchange (EXX) and the random phase approximation (RPA) correlation potentials. The negative ion resonances are identified by perturbing the real Hermitian nonlocal xc potentials using complex absorbing local potentials. Our studies show that the nonlocal EXX+RPA potential significantly enhances the description of positions and widths of negative ion resonance states compared to potentials that exclude dynamic polarization in RPA or include only EXX. The use of low-scaling algorithms simplifies the computation of the RPA potential, thereby providing a practical solution for resonance-state characterization within the density functional framework. A theoretical framework and the underlying assumptions required for combining real Hermitian nonlocal xc potentials with complex local potentials are discussed.

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.

Recent Advances in the Study of Negative-Ion Resonances Using Multiconfigurational Propagator and a Complex Absorbing Potential

The transient resonances are a challenge to bound state quantum mechanics. These states lie in the continuum part of the spectrum of the Hamiltonian. For this, one has to treat a continuum problem due to electron-molecule scattering and the many-electron correlation problem simultaneously. Moreover, the description of a resonance requires a wavefunction that bridges the part that resembles a bound state with another that resembles a continuum state such that the continuity of the wavefunction and its first derivative with respect to the distance between the incoming projectile and the target is maintained. A review of the recent advances in the theoretical investigation of the negative-ion resonances (NIR) is presented. The NIRs are ubiquitous in nature. They result from the scattering of electrons off of an atomic or molecular target. They are important for numerous chemical processes in upper atmosphere, space and even biological systems. A contextual background of the existing theoretical methods as well as the newly-developed multiconfigurational propagator tools based on a complex absorbing potential are discussed.

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.

Investigation of negative-ion resonances using a subspace-projected multiconfigurational electron propagator perturbed with a complex absorbing potential

The transient negative-ion resonances found in scattering experiments are important intermediates in many chemical processes. These metastable states correspond to the continuum part of the Hamiltonian of the projectile-target composite system. Usual bound-state electronic structure methods are not applicable for these. In this work, we develop a subspace-projection method in connection with an electron propagator (EP) defined in terms of a complete-active-space self-consistent-field initial state. The target Hamiltonian (Ĥ) is perturbed by a complex absorbing potential (CAP) for the analytical continuation of the spectrum of Ĥ to complex eigenvalues associated with the continuum states. The resonance is identified as a pole of the EP, which is stable with respect to variations in the strength of the CAP. The projection into a small subspace reduces the size of the complex matrices to be diagonalized, minimizes the computational cost, and affords some insight into the orbitals that are likely to play some role in the capture of the projectile. Two molecular (Πg2N₂ ^- and ²Π CO^-) and an atomic shaperesonance (²P Be^-) are investigated using this method. The position and width of the resonances are in good agreement with the previously reported values.

Projected Complex Absorbing Potential Multireference Configuration Interaction Approach for Shape and Feshbach Resonances

Anion resonances are formed as metastable intermediates in low-energy electron-induced reactions. Due to the finite lifetimes of resonances, applying standard Hermitian formalism for their characterization presents a vexing problem for computational chemists. Numerous modifications to conventional quantum chemical methods have enabled satisfactory characterization of resonances, but specific issues remain, especially in describing two-particle one-hole (2p-1h) resonances. An accurate description of these resonances and their coupling with single-particle resonances requires a multireference approach. We propose a projected complex absorbing potential (CAP) implementation within the multireference configuration interaction (MRCI) framework to characterize single-particle and 2p-1h resonances. As a first application, we use the projected-CAP-MRCI approach to characterize and benchmark the ²Π_g shape resonance in N₂^-. We test its performance as a function of the size of the subspace and other parameters, and we compute the complex potential energy surface of the ²Π_g shape resonance to show that a smooth curve is obtained. One key benefit of MRCI is that it can describe Feshbach resonances (most common examples of 2p-1h resonances) at the same footing as shape resonances. Therefore, it is uniquely positioned to describe mixing between the different channels. To test these additional capabilities, we compute Feshbach resonances in H₂O^- and anions of dicyanoethylene isomers. We find that CAP-MRCI can efficiently capture the mixing between the Feshbach and shape resonances in dicyanoethylene isomers, which has significant consequences for their lifetimes.

Theory of electronic resonances: fundamental aspects and recent advances

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 as 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.

Decay Processes in Cationic Alkali Metals in Microsolvated Clusters: A Complex Absorbing Potential Based Equation-of-Motion Coupled Cluster Investigation

We have employed the highly accurate complex absorbing potential based ionization potential equation-of-motion coupled cluster singles and doubles (CAP-IP-EOM-CCSD) method to study the various intermolecular decay processes in ionized metals (Li⁺, Na⁺, K⁺) microsolvated by water molecules. For the Li atom, the electron is ionized from the 1s subshell. However, for Na and K atoms, the electron is ionized from 2s and both 2s and 2p subshells, respectively. We have investigated decay processes for the Li⁺-(H₂O)_n (n = 1-3) systems, as well as Na⁺-(H₂O)_n (n = 1, 2), and K⁺-H₂O. The lithium cation in water can decay only via electron transfer mediated decay (ETMD) as there are no valence electrons in lithium. We have investigated how the various decay processes change in the presence of different alkali metal atoms and how the increasing number of water molecules play a significant role in the decay of microsolvated systems. To see the effect of the environment, we have studied Li⁺-NH₃ in comparison to Li⁺-H₂O. In the case of Na⁺-H₂O, we have studied the impact of bond distance on the decay width. The effect of polarization on decay width was checked for the X⁺-H₂O (X = Li, Na) systems. We used the PCM model to study the polarization effect. We have compared our results with existing theoretical and experimental results wherever available in the literature.

Calculation of the Lowest Resonant States of H− and Li by the Complex Absorbing Potential Method

The analysis of the features of the method of complex absorbing potential (CAP) is carried out for a single-channel problem with an explicit parameterization of the scattering matrix. It is shown that there can be several types of CAP trajectories depending on the choice of the initial conditions. In any case, the estimation of the resonance parameters from the position of the optimal trajectory point can lead to a systematic error or an ambiguous result. In special cases, the search for the optimal point can be replaced by the averaging over a closed section of the trajectory. The CAP trajectories constructed in the H− and Li resonance calculations correlate well with the model trajectories, which have a curl around the resonance. The averaging over a closed area of the trajectory leads to better estimates of the energy and width of the resonance in comparison with the technique of searching for the optimal point.

Variational Solutions for Resonances by a Finite-Difference Grid Method

We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as resonances and provide the energies and inverse lifetimes of the system under study (e.g., molecules) in metastable states. This approach allows incorporating finite grid methods in the study of resonance phenomena in chemistry. Possible applications include the calculation of electronic autoionization resonances which occur when ionization takes place as the bond lengths of the molecule are varied. Alternatively, the method can be applied to calculate nuclear predissociation resonances which are associated with activated complexes with finite lifetimes.

CAP/EA-ADC method for metastable anions: Computational aspects and application to π* resonances of norbornadiene and 1,4-cyclohexadiene

The second- and third-order algebraic-diagrammatic construction schemes for the electron propagator for studies of electron attachment processes [EA-ADC(2) and EA-ADC(3)] have been extended to include the complex absorbing potential (CAP) method for the treatment of electronic resonances. Theoretical and conceptual aspects of the new CAP/EA-ADC methodology are studied in detail at the example of the well-known ²Π_g resonance of the nitrogen anion N₂ ^-. The methodology is further applied to π* shape resonances, for which ethylene is considered as a prototype. Furthermore, the first many-body treatment of the π₊ ^* and π_- ^* resonances of norbornadiene and 1,4-cyclohexadiene is provided, which have served as model systems for the concept of through-space and through-bond interactions for a long time.

Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry

We propose a reformulation of the mathematical formalism of many-electron quantum theory that rests entirely on the physical properties of the electronic system under investigation, rather than conventional mathematical assumption of Hermitian operators in Hilbert space. The formalism is based on a modified dot-product that replaces the familiar complex-conjugation in Hilbert space ℌ (fixed for all physical systems) by time-conjugation in T-space (as generated by the specific spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of interest), yielding different spatial structure for different systems. The usual Hermitian requirement for physical operators is thereby generalized to a self-t-adjoint (“t-reversible”) character, leading to correspondingly generalized theorems of virial and hypervirial type. The T-space reformulation preserves the real values of measurable properties and the Born-probabilistic interpretations of state functions that underlie the present quantum theory of measurement, while also properly distinguishing “temporal” behavior of internal decay (tunneling-type) phenomena from that of applied fields with parametric t-dependence on an external clock. The t-product represents a further generalization of the “c-product” that was previously found useful in complex coordinate-rotation studies of autoionizing resonances.

The quest to uncover the nature of benzonitrile anion

Anionic states of benzonitrile are investigated by high-level electronic structure methods.

Quantum Effects in Cold Molecular Collisions from Spatial Polarization of Electronic Wave Function

The quantum phenomena of electronic and nuclear resonances are associated with structures in measured cross sections. Such structures were recently reported in a cold chemistry experiment of ground-state hydrogen isotopologues (H₂/HD) colliding with helium atoms in the excited triplet P-state (He(2³P)) [Shagam et al. Nature Chem. 2015, 7, 921], but a theoretical explanation of their appearance was not given. This work presents a quantum explanation and simulation of this experiment, which are strictly based on ab initio calculations. We incorporate complex potential energy surfaces into adiabatic variational theory, thereby reducing the multidimensional scattering process to a series of uncoupled 1D scattering "gedanken experiments". Our theoretical result, which is in remarkable agreement with the experimental data, manifests that the structures in the observed reaction rate coefficient are due to the spatial arrangement of the excited He p-orbitals with respect to the interaction axis, consequently changing the system from a normal two-rotor model to a three-rotor one. This theoretical scheme can be applied to explain and predict cross sections or reaction rate coefficients for any resonance-related phenomenon.

Spectroscopy of temporary anion states: Renner–Teller coupling and electronic autodetachment in copper difluoride anion

Internal level structures of temporary anion states (resonances) are probed using action spectroscopy obtained from photoelectron imaging of CuF₂⁻.

EOM-CC guide to Fock-space travel: the C2 edition

Electronic structure calculations for C₂, C₂⁻, and C₂²⁻ using the CC/EOM-CC family of methods. Results illustrate that EOM-CCSD provides an attractive alternative to MR approaches.

Bound and continuum-embedded states of cyanopolyyne anions

Equation-of-motion coupled-cluster calculations reveal systematic trends across bound and continuum-embedded excited states in cyanopolyyne anions.

Complete-active-space second-order perturbation theory (CASPT2//CASSCF) study of the dissociative electron attachment in canonical DNA nucleobases caused by low-energy electrons (0-3 eV)

Low-energy (0-3 eV) ballistic electrons originated during the irradiation of biological material can interact with DNA/RNA nucleobases yielding transient-anion species which undergo decompositions. Since the discovery that these reactions can eventually lead to strand breaking of the DNA chains, great efforts have been dedicated to their study. The main fragmentation at the 0-3 eV energy range is the ejection of a hydrogen atom from the specific nitrogen positions. In the present study, the methodological approach introduced in a previous work on uracil [I. González-Ramírez et al., J. Chem. Theory Comput. 8, 2769-2776 (2012)] is employed to study the DNA canonical nucleobases fragmentations of N-H bonds induced by low-energy electrons. The approach is based on minimum energy path and linear interpolation of internal coordinates computations along the N-H dissociation channels carried out at the complete-active-space self-consistent field//complete-active-space second-order perturbation theory level. On the basis of the calculated theoretical quantities, new assignations for the adenine and cytosine anion yield curves are provided. In addition, the π1 (-) and π2 (-) states of the pyrimidine nucleobases are expected to produce the temporary anions at electron energies close to 1 and 2 eV, respectively. Finally, the present theoretical results do not allow to discard neither the dipole-bound nor the valence-bound mechanisms in the range of energies explored, suggesting that both possibilities may coexist in the experiments carried out with the isolated nucleobases.

Same but Different: Dipole-Stabilized Shape Resonances in CuF(-) and AgF(.)

Electron attachment to closed-shell molecules is a gateway to various important processes in the gas and condensed phases. The properties of an electron-attached state, such as its energy and lifetime as well as the character of the molecular orbital to which the electron is attached, determine the fate of the anion. In this experimental and theoretical study of copper and silver fluoride anions, we introduce a new type of metastable anionic state. Abrupt changes in photoelectron angular distributions point to the existence of autodetaching states. Equation-of-motion coupled-cluster singles and doubles calculations augmented by a complex absorbing potential identify some of these states as Σ and Π dipole-stabilized resonances, a new type of shape resonance. In addition, these molecules support valence and dipole-bound states and a Σ resonance of charge-transfer character. By featuring five different types of anionic states, they provide a vehicle for studying fundamental properties of anions and for validating new theoretical approaches for metastable states.