Equation-of-motion coupled-cluster method for the study of shape resonance

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.

Observation of Renner-Teller and predissociation coupled vibronic intensity borrowing in dissociative electron attachment to OCS

We use a time-of-flight-based velocity map imaging method to look into the dissociative electron attachment to a linear OCS molecule at electron beam energies ranging from 4.5 to 8.5 eV. The conical time-gated wedge slice imaging method is utilized to extract fragments' slice images, kinetic energy (KE), and angular distributions, which provide a complete kinematic understanding of this experiment on the dissociative electron attachment process. We observe that the formation of S- is relatively higher than the O- product. Three distinct dissociative KE bands of S-/OCS have been observed for the 5.0 and 6.5 eV resonance positions. We notice a prominent rovibrationally coupled bimodality for each KE band in the variation of the most probable KE values. When the electron energy is changed from 5.5 to 6.0 eV, we observed vibronic intensity borrowing in the highest momentum band of S- via the Σ → Π symmetric dipole-forbidden transitions within the 1.5 eV energy gap. Multiple peaks in the angular distributions of S- and their modeling indicate the presence of Renner-Teller vibronic splitting. Using Q-Chem's implemented complex absorbing potential-equation of motion-electron affinity coupled cluster singles and doubles aug-cc-pVDZ+4s3p level of multireference-based electronic structure theory, we confirm the presence of OCS temporary negative ion bending vibrations and Renner-Teller vibronic splittings for the Π symmetric states. Additionally, we notice the presence of a non-radiative predissociation continuum (bringing down the rotational spectrum) and speed-dependent angular anisotropy in the S- fragmentation. Our findings at the resonance of OCS at 6.5 eV closely align with the prediction of vibronic intensity borrowing by Orlandi and Siebrand [J. Chem. Phys. 58, 4513 (1973)].

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.

Shape resonance induced electron attachment to cytosine: The effect of aqueous media

We have investigated the impact of microsolvation on shape-type resonance states of nucleobases, taking cytosine as a case study. To characterize the resonance position and decay width of the metastable states, we employed the newly developed DLPNO-based EA-EOM-CCSD method in conjunction with the resonance via Padé (RVP) method. Our calculations show that the presence of water molecules causes a redshift in the resonance position and an increase in the lifetime for the three lowest-lying resonance states of cytosine. Furthermore, there are some indications that the lowest resonance state in isolated cytosine may get converted to a bound state in the presence of an aqueous environment. The obtained results are extremely sensitive to the basis set used for the calculations.

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.

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.

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.

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.

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.

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.

Uniform vs Partial Scaling within Resonances via Padé Based on the Similarities to Other Non-Hermitian Methods: Illustration for the Beryllium 1s22p3s State

Resonance via Padé (RVP) is an efficient method for calculating autoionization resonance states. It is based on the stabilization technique in which the basis set is scaled. The scaling can be uniform (i.e., all basis functions are scaled) or partial. Herein, we compare the two RVP scaling schemes for calculating an autoionization eigenvalue; moreover, the effect of freezing the core electrons is intertwined within this comparison. In order to study the different behavior of the RVP schemes, we associate each RVP scaling scheme with a complex contour of integration. Similarities between RVP and other non-Hermitian methods emerge from the generated contours, which suggest that RVP introduces similar outgoing boundary conditions as the complex scaling (CS), complex basis function (CBF), and reflection-free complex absorbing potential (RF-CAP) methods. A uniform-RVP contour, unlike a partial one, immediately penetrates the complex plane and influences the interaction region. Hence, uniform scaling within RVP destroys the description of the core electrons, as well as the description of the reference state, and yields less reliable results than partial scaling. The 1s²2p3s ¹P autoionization state of Be, at the equation-of-motion coupled-cluster level, is used as our case study model.

Negative Ion Resonance States: The Fock-Space Coupled-Cluster Way

The negative ion resonance states, which are electron-molecule metastable compound states, play the most important role in free-electron controlled molecular reactions and low-energy free-electron-induced DNA damage. Their electronic structure is often only poorly described but crucial to an understanding of their reaction dynamics. One of the most important challenges to current electronic structure theory is the computation of negative ion resonance states. As a major step forward, coupled-cluster theories, which are well-known for their ability to produce the best approximate bound state electronic eigen solutions, are upgraded to offer the most accurate and effective approximations for negative ion resonance states. The existing Fock-space coupled-cluster (FSCC) and the equation-of-motion coupled-cluster (EOM-CC) approaches that compute bound states are redesigned for the direct and simultaneous determination of both the kinetic energy of the free electron at which the electron-molecule compound states are resonantly formed and the corresponding autodetachment decay rate of the electron from the metastable compound state. This Feature Article reviews the computation of negative ion resonances using the FSCC approach and, in passing, provides the highlights of the equivalent EOM-CC approach.

An Electron Propagator Approach Based on a Multiconfigurational Reference State for the Investigation of Negative-Ion Resonances Using a Complex Absorbing Potential Method

Negative-ion resonances are important metastable states that result from the collision between an electron and a neutral target. The course of many chemical processes in nature is often dictated by how an intermediate resonance state falls apart. This article reports on the development of an electron propagator (EP) based on a Hamiltonian Ĥ perturbed by a complex absorbing potential (CAP) and a multiconfigurational self-consistent field (MCSCF) initial state to study these resonances. Perturbation of Ĥ by a CAP makes the resonances amenable to a bound-state method like MCSCF. Resonances stand out among the non-resonant states as persistent complex eigenvalues of the perturbed Ĥ when the strength (η) of the CAP is varied. The MCSCF method gives a reliable and accurate description of the target states, especially when the non-dynamical correlations are dominant. The resonance energies are obtained from the poles of the EP. We propose three variants of our EP depending on how the effect of the CAP is introduced. We find that the computationally most efficient variant is the one in which the reference state of the EP is an unperturbed MCSCF wavefunction and a non-zero CAP is defined only on the virtual orbital subspace of the reference state. The onset of the CAP is carefully optimized in order to minimize the artifacts due to reflections from the CAP. An extrapolation method (based on a Padé approximant) and a de-perturbation method are adopted in order to account for the limitations of finite basis sets used and determine the resonance energy in the limit of η → 0. ²P Be^-, ²Π_g N₂^-, and ²Π CO^- shape resonances are investigated. The position and width of these resonances computed in this study agree well with those reported earlier in the literature.

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.

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.

Resonance Energies and Lifetimes from the Analytic Continuation of the Coupling Constant Method: Robust Algorithms and a Critical Analysis

The energy of a metastable state can be computed by adding an artificial stabilizing potential to the Hamiltonian, increasing the stabilization until the metastable state is turned into a bound one, and then further increasing the stabilization until enough bound-state data have been collected so that these can be extrapolated back to vanishing stabilization. The lifetime of the metastable state can be obtained from the same data, but only if the extrapolation is performed by analytic continuation. This extrapolation method is called analytic continuation of the coupling constant (ACCC). Here we introduce preconditioning schemes for two of the three established extrapolation algorithms and critically compare results from all three extrapolation schemes in a variety of situations: As examples for resonance states serve the π* temporary anions of ethylene and formaldehyde as well as a model potential, which provides a case where input data with full numeric precision are available. In the data collection step, three different stabilizing potentials are employed, a Coulomb potential, a short-range Coulomb potential, and a soft-box Voronoi potential. Effects of different orders of the extrapolating Padé approximant are investigated, and last, the energy range of input data for the extrapolation is studied. Moreover, all ACCC results are compared to resonance parameters that have been independently obtained with the same theoretical method, but with a different continuum approach-complex scaling for the model and complex absorbing potentials for the temporary anions.

Ab initio calculation of the cross sections for electron impact vibrational excitation of CO via the (2)Π shape resonance

The stabilization method is used to calculate the complex potential energy curve of the (2)Π state of CO(-) as a function of bond length, with the refinement that separate potentials are determined for p-wave and d-wave attachment and detachment of the excess electron. Using the resulting complex potentials, absolute vibrational excitation cross sections are calculated as a function of electron energy and scattering angle. The calculated cross sections agree well with experiment.

A Fresh Look at Resonances and Complex Absorbing Potentials: Density Matrix-Based Approach

A new strategy of using complex absorbing potentials (CAPs) within electronic structure calculations of metastable electronic states, which are ubiquitous in chemistry and physics, is presented. The stumbling block in numerical applications of CAPs is the necessity to optimize the CAP strength for each system, state, and one-electron basis set, while there is no clear metric to assess the quality of the results and no simple algorithm of achieving numerical convergence. By analyzing the behavior of resonance wave functions, we found that robust results can be obtained when considering fully stabilized resonance states characterized by constant density at large η (parameter determining the CAP strength). Then the perturbation due to the finite-strength CAP can be removed by a simple energy correction derived from energy decomposition analysis and response theory. The utility of this approach is illustrated by CAP-augmented calculations of several shape resonances using EOM-EA-CCSD with standard Gaussian basis sets.