The Quantum Chemistry of Open-Shell Species

Vibronic Coupling Effects in the Photoelectron Spectrum of Ozone: A Coupled-Cluster Approach

One of the most important areas of application for equation-of-motion coupled-cluster (EOM-CC) theory is the prediction, simulation, and analysis of various types of electronic spectra. In this work, the EOM-CC method for ionized states, known as EOM-IP-CC, is applied to the closely lying and coupled pair of states of the ozone cation─X̃2A1 and Ã2B2─using highly accurate treatments including up to the full single, double, triple, and quadruple excitations (EOM-IP-CCSDTQ). Combined with a venerable and powerful method for calculating vibronic spectra from the Hamiltonian produced by EOM-IP-CC calculations, the simulations yield a spectrum that is in good agreement with the photoelectron spectrum of ozone. Importantly, the calculations suggest that the adiabatic gap separating these two electronic states is somewhat smaller than currently thought; an assignment of the simulated spectrum together with more precise band positions of the experimental measurements suggests T00 = 1,368 ± 65 cm-1.

Photoswitching Molecules Functionalized with Optical Cycling Centers Provide a Novel Platform for Studying Chemical Transformations in Ultracold Molecules

A novel molecular structure that bridges the fields of molecular optical cycling and molecular photoswitching is presented. It is based on a photoswitching molecule azobenzene functionalized with one and two CaO- groups, which can act as optical cycling centers (OCCs). This paper characterizes the electronic structure of the resulting model systems, focusing on three questions: (1) how the electronic states of the photoswitch are impacted by a functionalization with an OCC; (2) how the states of the OCC are impacted by the scaffold of the photoswitch; and (3) whether the OCC can serve as a spectroscopic probe of isomerization. The experimental feasibility of the proposed design and the advantages that organic synthesis can offer in the further functionalization of this molecular scaffold are also discussed. This work brings into the field of molecular optical cycling a new dimension of chemical complexity intrinsic to only polyatomic molecules.

Analytical Gradients for Electron-Attached and Ionized States for the Algebraic-Diagrammatic Construction Scheme for the Electron Propagator up to Third Order

The derivation and implementation of analytical gradients for methods based on the non-Dyson algebraic diagrammatic construction for the electron propagator, IP-ADC and EA-ADC, up to the third order is presented. Using nuclear gradients, ground-state equilibrium structures for small open-shell systems are calculated. In addition, we investigated the performance of IP/EA-ADC methods for the calculation of adiabatic ionization potentials and electron affinities for medium-sized organic molecules.

The Semiexperimental Approach at Work: Equilibrium Structure of Radical Species

The so-called semiexperimental (SE) approach is a powerful technique for obtaining highly accurate equilibrium structures for isolated systems. This Featured Article describes its extension to open-shell species, thus providing the first systematic investigation on radical equilibrium geometries to be used for benchmarking purposes. The small yet significant database obtained demonstrates that there is no reduction in accuracy when moving from closed-shell species to radicals. We also provide an extension of the applicability of the SE approach to medium-/large-sized radicals by exploiting the so-called "Lego-brick" approach, which is based on the assumption that a molecular system can be seen as formed by smaller fragments for which the SE equilibrium structure is available. In this Featured Article we show that this model can be successfully applied also to open-shell species.

Augmentation of NIR Circularly Polarized Luminescence Activity in Shibasaki-Type Lanthanide Complexes Supported by the Spirane Sphenol

We report two new circularly polarized luminescence (CPL)-active lanthanide complexes emissive in the near-infrared (NIR) region; using sphenol as a supporting ligand, we provide the first reported example of an NIR-emissive lanthanide complex supported by a chiral spirane. Inclusion of a quaternary carbon to impart axial chirality results in dramatic augmentation of the CPL strength of the resultant sphenolate complexes (glum ≤ 0.77 for [(sphenol)3ErNa3(thf)6]) compared to that of their contemporary biaryl-based axially chiral analogues (glum ≤ 0.47 for [(binol)3ErNa3(thf)6]). Despite similar structural parameters, the rigid spiro carbon of sphenol enables the strongest dissymmetry factors observed to date from Shibasaki-type complexes for both Yb and Er. We also demonstrate the sensitivity of the reported chiroptical measurements to small variations in instrumental parameters, such as bandpass, and suggest a standardized method or at least that additional detail should be included in future reports to allow for direct comparisons between newly published CPL emitters.

Linear-Scaling Local Natural Orbital CCSD(T) Approach for Open-Shell Systems: Algorithms, Benchmarks, and Large-Scale Applications

The extension of the highly optimized local natural orbital (LNO) coupled cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented for high-spin open-shell molecules based on restricted open-shell references. The techniques enabling the outstanding efficiency of the closed-shell LNO-CCSD(T) variant are adopted, including the iteration- and redundancy-free second-order Møller-Plesset and (T) formulations as well as the integral-direct, memory- and disk use-economic, and OpenMP-parallel algorithms. For large molecules, the efficiency of our open-shell LNO-CCSD(T) method approaches that of its closed-shell parent method due to the application of restricted orbital sets for demanding integral transformations and a novel approximation for higher-order long-range spin-polarization effects. The accuracy of open-shell LNO-CCSD(T) is extensively tested for radicals and reactions thereof, ionization processes, as well as spin-state splittings, and transition-metal compounds. At the size range where the canonical CCSD(T) reference is accessible (up to 20-30 atoms), the average open-shell LNO-CCSD(T) correlation energies are found to be 99.9 to 99.95% accurate, which translates into average absolute deviations of a few tenths of kcal/mol in the investigated energy differences already with the default settings. For more extensive molecules, the local errors may grow, but they can be estimated and decreased via affordable systematic convergence studies. This enables the accurate modeling of large systems with complex electronic structures, as illustrated on open-shell organic radicals and transition-metal complexes of up to 179 atoms as well as on challenging biochemical systems, including up to 601 atoms and 11,000 basis functions. While the protein models involve difficulties for local approximations, such as the spin states of a bounded iron ion or an extremely delocalized singly occupied orbital, the corresponding single-node LNO-CCSD(T) computations were feasible in a matter of days with 10s to 100 GB of memory use. Therefore, the new LNO-CCSD(T) implementation enables highly accurate computations for open-shell systems of unprecedented size and complexity with widely accessible hardware.

State-Interaction Approach for Evaluating g-Tensors within EOM-CC and RAS-CI Frameworks: Theory and Benchmarks

Among various techniques designed for studying open-shell species, electron paramagnetic resonance (EPR) spectroscopy plays an important role. The key quantity measured by EPR is the g-tensor, describing the coupling between an external magnetic field and molecular electronic spin. One theoretical framework for quantum chemistry calculations of g-tensors is based on response theory, which involves substantial developments that are specific to the underlying electronic structure models. A simplified and easier-to-implement approach is based on the state-interaction scheme, in which perturbation is included by considering a small number of states. We describe and benchmark the state-interaction approach using equation-of-motion coupled-cluster and restricted-active-space configuration interaction wave functions. The analysis confirms that this approach can deliver accurate results and highlights caveats of applying it, such as a choice of the reference state, convergence with respect to the number of states used in calculations, etc. The analysis also contributes toward a better understanding of challenges in calculations of higher-order properties using approximate wave functions.

Theoretical Study of the Photochemical Mechanisms of the Electronic Quenching of NO(A2Σ+) with CH4, CH3OH, and CO2

The electronic quenching of NO(A2Σ+) with molecular partners occurs through complex non-adiabatic dynamics that occurs on multiple coupled potential energy surfaces. Moreover, the propensity for NO(A2Σ+) electronic quenching depends heavily on the strength and nature of the intermolecular interactions between NO(A2Σ+) and the molecular partner. In this paper, we explore the electronic quenching mechanisms of three systems: NO(A2Σ+) + CH4, NO(A2Σ+) + CH3OH, and NO(A2Σ+) + CO2. Using EOM-EA-CCSD calculations, we rationalize the very low electronic quenching cross-section of NO(A2Σ+) + CH4 as well as the outcomes observed in previous NO + CH4 photodissociation studies. Our analysis of NO(A2Σ+) + CH3OH suggests that it will undergo facile electronic quenching mediated by reducing the intermolecular distance and significantly stretching the O-H bond of CH3OH. For NO(A2Σ+) + CO2, intermolecular attractions lead to a series of low-energy ON-OCO conformations in which the CO2 is significantly bent. For both the NO(A2Σ+) + CH3OH and NO(A2Σ+) + CO2 systems, we see evidence of the harpoon mechanism and low-energy conical intersections between NO(A2Σ+) + M and NO(X2Π) + M. Overall, this work provides the first detailed theoretical study on the NO(A2Σ+) + M potential energy surface of each of these systems and will inform future velocity map imaging experiments.

F12+EOM Quartic Force Fields for Rovibrational Predictions of Electronically Excited States

Quartic force fields (QFFs) constructed using a sum of ground-state CCSD(T)-F12b energies with EOM-CCSD excitation energies are proposed for computation of spectroscopic properties of electronically excited states. This is dubbed the F12+EOM approach and is shown to provide similar accuracy to previous methodologies at lower computational cost. Using explicitly correlated F12 approaches instead of canonical CCSD(T), as in the corresponding (T)+EOM approach, allows for 70-fold improvement in computational time. The mean percent difference between the two methods for anharmonic vibrational frequencies is only 0.10%. A similar approach is also developed herein which accounts for core correlation and scalar relativistic effects, named F12cCR+EOM. The F12+EOM and F12cCR+EOM approaches both match to within 2.5% mean absolute error of experimental fundamental frequencies. These new methods should help in clarifying astronomical spectra by assigning features to vibronic and vibrational transitions of small astromolecules when such data are not available experimentally.

Theory, implementation, and disappointing results for two-photon absorption cross sections within the doubly electron-attached equation-of-motion coupled-cluster framework

The equation-of-motion coupled-cluster singles and doubles method with double electron attachment (EOM-DEA-CCSD) is capable of computing reliable energies, wave functions, and first-order properties of excited states in diradicals and polyenes that have a significant doubly excited character with respect to the ground state, without the need for including the computationally expensive triple excitations. Here, we extend the capabilities of the EOM-DEA-CCSD method to the calculations of a multiphoton property, two-photon absorption (2PA) cross sections. Closed-form expressions for the 2PA cross sections are derived within the expectation-value approach using response wave functions. We analyze the performance of this new implementation by comparing the EOM-DEA-CCSD energies and 2PA cross sections with those computed using the CC3 quadratic response theory approach. As benchmark systems, we consider transitions to the states with doubly excited character in twisted ethene and in polyenes, for which EOM-EE-CCSD (EOM-CCSD for excitation energies) performs poorly. The EOM-DEA-CCSD 2PA cross sections are comparable with the CC3 results for twisted ethene; however, the discrepancies between the two methods are large for hexatriene. The observed trends are explained by configurational analysis of the 2PA channels.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Molecular-orbital-based machine learning for open-shell and multi-reference systems with kernel addition Gaussian process regression

We introduce a novel machine learning strategy, kernel addition Gaussian process regression (KA-GPR), in molecular-orbital-based machine learning (MOB-ML) to learn the total correlation energies of general electronic structure theories for closed- and open-shell systems by introducing a machine learning strategy. The learning efficiency of MOB-ML(KA-GPR) is the same as the original MOB-ML method for the smallest criegee molecule, which is a closed-shell molecule with multi-reference characters. In addition, the prediction accuracies of different small free radicals could reach the chemical accuracy of 1 kcal/mol by training on one example structure. Accurate potential energy surfaces for the H₁₀ chain (closed-shell) and water OH bond dissociation (open-shell) could also be generated by MOB-ML(KA-GPR). To explore the breadth of chemical systems that KA-GPR can describe, we further apply MOB-ML to accurately predict the large benchmark datasets for closed- (QM9, QM7b-T, and GDB-13-T) and open-shell (QMSpin) molecules.

Potential energy surfaces of a stacked dimer of benzene and its radical cation: what remains and what appears

A stepwise qualitative consideration of the reduction in symmetry for the highly symmetrical "right sandwich" and "twisted sandwich" structures of the stacked benzene dimer caused by the pseudo-Jahn-Teller effect made it possible to construct a scheme of the potential energy surface (PES) of this dimer. Thirty-six equivalent structures of minimum energy are ordered on this extremely flat surface, transforming into each other in an almost barrier-free manner. There are two kinds of these transformations, both of which are pseudorotation. The transformation pathways inherent in this neutral dimer are also characteristic of its radical cation (RC). Structural transformations of the RC are inextricably linked with changes in its electronic state since its stationary structures relate to different electronic states. (C₆H₆)₂⁺˙ exists in the form of two orbital isomers, each of which "pseudorotates" on its own area of the PES. During the pseudorotation, the SOMO distribution on the fragments of (C₆H₆)₂⁺˙ changes, which is identical to what occurs during the pseudorotation of the Jahn-Teller benzene RC. These areas are connected by pairwise interconversions of their minimum energy structures. The interconversion of the orbital isomers is a bypassing of conical intersections between corresponding electronic states, which occurs through a synchronized pseudorotation of the fragments. The conclusions of the qualitative consideration and the results of quantum chemical calculations of various levels performed for (C₆H₆)₂⁺˙ are in full agreement with each other. The revealed features of the structure of the PES of the two reference systems in studying the intermolecular and ion-molecule interactions are the basis for considering their more complex analogs, primarily their less symmetrical ones.

Reactive quenching of NO (A2Σ+) with H2O leads to HONO: a theoretical analysis of the reactive and nonreactive electronic quenching mechanisms

In this study, we develop a mechanistic understanding of the pathways for nonreactive and reactive electronic quenching of NO (A2Σ+) with H2O. In doing so, we identify a photochemical mechanism for HONO production in the upper atmosphere.

Stereodynamic Control of Collision-Induced Nonadiabatic Dynamics of NO (A2Σ+) with H2, N2, and CO: Intermolecular Interactions Drive Collision Outcomes

Intermolecular interactions, stereodynamics, and coupled potential energy surfaces (PESs) all play a significant role in determining the outcomes of molecular collisions. A detailed knowledge of such processes is often essential for a proper interpretation of spectroscopic observations. For example, nitric oxide (NO), an important radical in combustion and atmospheric chemistry, is commonly quantified using laser-induced fluorescence on the A²Σ⁺ ← X²Π transition band. However, the electronic quenching of NO (A²Σ⁺) with other molecular species provides alternative nonradiative pathways that compete with fluorescence. While the cross sections and rate constants of NO (A²Σ⁺) electronic quenching have been experimentally measured for a number of important molecular collision partners, the underlying photochemical mechanisms responsible for the electronic quenching are not well understood. In this paper, we describe the development of high-quality PESs that provide new physical insights into the intermolecular interactions and conical intersections that facilitate the branching between the electronic quenching and scattering of NO (A²Σ⁺) with H₂, N₂, and CO. The PESs are calculated at the EOM-EA-CCSD/d-aug-cc-pVTZ//EOM-EA-CCSD/aug-cc-pVDZ level of theory, an approach that ensures a balanced treatment of the valence and Rydberg electronic states and an accurate description of the open-shell character of NO. Our PESs show that H₂ is incapable of electronically quenching NO (A²Σ⁺) at low collision energies; instead, the two molecules will likely undergo scattering. The PESs of NO (A²Σ⁺) with N₂ and CO are highly anisotropic and demonstrate evidence of electron transfer from NO (A²Σ⁺) into the lowest unoccupied molecular orbital of the collision partner, that is, the harpoon mechanism. In the case of ON + CO, the PES becomes strongly attractive at longer intermolecular distances and funnels population to a conical intersection between NO (A²Σ⁺) + CO and NO (X²Π) + CO. In contrast, for ON + N₂, the conical intersection is preceded by an ∼0.40 eV barrier. Overall, our work shines new light into the impact of coupled PESs on the nonadiabatic dynamics of open-shell systems.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

(T)+EOM Quartic Force Fields for Theoretical Vibrational Spectroscopy of Electronically Excited States

(T)+EOM quartic force fields (QFFs) are proposed for ab initio rovibrational properties of electronically excited states of small molecules. The (T)+EOM method is a simple treatment of the potential surface of the excited state using a composite energy from the CCSD(T) energy for the ground-state configuration and the EOM-CCSD excitation energy for the target state. The method is benchmarked with two open-shell species, HOO and HNF, and two closed-shell species, HNO and HCF. A (T)+EOM QFF with a complete basis set extrapolation (C) and corrections for core correlation (cC) and scalar relativity (R), dubbed (T)+EOM/CcCR, achieves a mean absolute error (MAE) as low as 1.6 cm^-1 for the à ²A' state of HOO versus an established benchmark QFF with CCSD(T)-F12b/cc-pVTZ-F12 (F12-TZ) for this variationally accessible electronically excited state. The MAE for anharmonic frequencies for (T)+EOM/CcCR versus F12-TZ for HNF is 7.5 cm^-1. The closed-shell species are compared directly with the experiment, where a simpler (T)+EOM QFF using the aug-cc-pVTZ basis set compares more favorably than the more costly (T)+EOM/CcCR, suggesting a possible influence of decreasing accuracy with basis set size. Scans along internal coordinates are also provided which show reasonable modeling of the potential surface by (T)+EOM compared to benchmark QFFs computed for variationally accessible electronic states. The agreement between (T)+EOM/CcCR with F12-TZ and CcCR benchmarks is also shown to be quite accurate for rotational constants and geometries, with an MAE of 0.008 MHz for the rotational constants of (T)+EOM/CcCR versus CcCR for à ²A' HOO and agreement within 0.003 Å for bond lengths.

Spin contamination in MP2 and CC2, a surprising issue

When calculating the spin multiplicity at either the second-order Møller-Plesset (MP2) or the iterative second-order approximate coupled-cluster singles and doubles (CC2) levels of theory using the same strategy for the calculation of the expectation value as in regular CC theory together with the usual definitions of the MP2 and CC2 density matrices, artificial spin contamination occurs in closed-shell molecules. Non-intuitively, for open-shell systems, results at the MP2 or CC2 levels of theory based on this procedure even suggest stronger contamination at the correlated level than for the Hartree-Fock reference, although treatment of electron correlation should lower spin contamination. In this Communication, the reasons behind this inconsistency are investigated and a solution is proposed, which removes spin contamination for closed-shell molecules and leads to physically meaningful results for open-shell cases. Additionally, we show that CC2 significantly outperforms MP2 in describing systems with a strongly spin-contaminated reference with a performance similar to that of full coupled-cluster with singles and doubles substitutions (CCSD).

Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.

Equation-of-motion coupled-cluster method with double electron-attaching operators: Theory, implementation, and benchmarks

We report a production-level implementation of the equation-of-motion (EOM) coupled-cluster (CC) method with double electron-attaching (DEA) EOM operators of 2p and 3p1h types, EOM-DEA-CCSD. This ansatz, suitable for treating electronic structure patterns that can be described as two-electrons-in-many orbitals, represents a useful addition to the EOM-CC family of methods. We analyze the performance of EOM-DEA-CCSD for energy differences and molecular properties. By considering reduced quantities, such as state and transition one-particle density matrices, we compare EOM-DEA-CCSD wave functions with wave functions computed by other EOM-CCSD methods. The benchmarks illustrate that EOM-DEA-CCSD is capable of treating diradicals, bond-breaking, and some types of conical intersections.

A Fock space coupled cluster based probing of the single- and double-ionization profiles for the poly-cyclic aromatic hydrocarbons and conjugated polyenes

Sequential formation of a poly-cyclic aromatic hydrocarbon (PAH) dication in the H I regions of the interstellar medium (ISM) is proposed to be a function of internal energy of the doubly ionized PAHs, which, in turn, is dependent on the single- and double-ionization potentials of the system. This sets a limit on the single- and double-ionization energies of the system(s) that can further undergo sequential absorption of two photons, leading to a dication (PAH⁺²). Here, we report the single-ionization (I⁺¹) and double-ionization (I⁺²) energies and the I⁺²/I⁺¹ ratio for some selected PAHs and conjugated polyenes obtained using the Fock space coupled cluster technique, enabling simultaneous consideration of several electronic states of different characters. The I⁺² to I⁺¹ ratio bears a constant ratio, giving allowance to determine I⁺² from the knowledge of single-ionization (I⁺¹) and vice versa. Our observations are in good agreement with the established literature findings, confirming the reliability of our estimates. The measured single- and double-ionization energies further demonstrate that the sequential formation and fragmentation of a PAH dication in the H I regions of the ISM for systems such as benzene and conjugated polyenes such as ethylene and butadiene are quite unlikely because I⁺²-I⁺¹ for such system(s) is higher than the available photon energy in the H I regions of the ISM. Present findings may be useful to understand the formation and underlying decay mechanisms of multiply charged ions from PAHs and related compounds that may accentuate the exploration of the phenomenon of high-temperature superconductivity.

An assessment of different electronic structure approaches for modeling time-resolved x-ray absorption spectroscopy

We assess the performance of different protocols for simulating excited-state x-ray absorption spectra. We consider three different protocols based on equation-of-motion coupled-cluster singles and doubles, two of them combined with the maximum overlap method. The three protocols differ in the choice of a reference configuration used to compute target states. Maximum-overlap-method time-dependent density functional theory is also considered. The performance of the different approaches is illustrated using uracil, thymine, and acetylacetone as benchmark systems. The results provide guidance for selecting an electronic structure method for modeling time-resolved x-ray absorption spectroscopy.

Feshbach-Fano approach for calculation of Auger decay rates using equation-of-motion coupled-cluster wave functions. I. Theory and implementation

X-ray absorption creates electron vacancies in the core shell. These highly excited states often relax by Auger decay-an autoionization process in which one valence electron fills the core hole and another valence electron is ejected into the ionization continuum. Despite the important role of Auger processes in many experimental settings, their first-principles modeling is challenging, even for small systems. The difficulty stems from the need to describe many-electron continuum (unbound) states, which cannot be tackled with standard quantum-chemistry methods. We present a novel approach to calculate Auger decay rates by combining Feshbach-Fano resonance theory with the equation-of-motion coupled-cluster single double (EOM-CCSD) framework. We use the core-valence separation scheme to define projectors into the bound (square-integrable) and unbound (continuum) subspaces of the full function space. The continuum many-body decay states are represented by products of an appropriate EOM-CCSD state and a free-electron state, described by a continuum orbital. The Auger rates are expressed in terms of reduced quantities, two-body Dyson amplitudes (objects analogous to the two-particle transition density matrix), contracted with two-electron bound-continuum integrals. Here, we consider two approximate treatments of the free electron: a plane wave and a Coulomb wave with an effective charge, which allow us to evaluate all requisite integrals analytically; however, the theory can be extended to incorporate a more sophisticated description of the continuum orbital.

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.

Interplay of Open-Shell Spin-Coupling and Jahn-Teller Distortion in Benzene Radical Cation Probed by X-ray Spectroscopy

We report a theoretical investigation and elucidation of the X-ray absorption spectra of neutral benzene and of the benzene cation. The generation of the cation by multiphoton ultraviolet (UV) ionization and the measurement of the carbon K-edge spectra of both species using a table-top high-harmonic generation source are described in the companion experimental paper [Epshtein, M.; et al. J. Phys. Chem. A http://dx.doi.org/10.1021/acs.jpca.0c08736]. We show that the 1s_C → π transition serves as a sensitive signature of the transient cation formation, as it occurs outside of the spectral window of the parent neutral species. Moreover, the presence of the unpaired (spectator) electron in the π-subshell of the cation and the high symmetry of the system result in significant differences relative to neutral benzene in the spectral features associated with the 1s_C → π* transitions. High-level calculations using equation-of-motion coupled-cluster theory provide the interpretation of the experimental spectra and insight into the electronic structure of benzene and its cation. The prominent split structure of the 1s_C → π* band of the cation is attributed to the interplay between the coupling of the core → π* excitation with the unpaired electron in the π-subshell and the Jahn-Teller distortion. The calculations attribute most of the splitting (∼1-1.2 eV) to the spin coupling, which is visible already at the Franck-Condon structure, and we estimate the additional splitting due to structural relaxation to be around ∼0.1-0.2 eV. These results suggest that X-ray absorption with increased resolution might be able to disentangle electronic and structural aspects of the Jahn-Teller effect in the benzene cation.

Crystalline Diradical Dianions of Pyrene-Fused Azaacenes

Although diradicals and azaacenes have been greatly attractive in fundamental chemistry and functional materials, the isolable diradical dianions of azaacenes are still unknown. Herein, we describe the first isolation of pyrene-fused azaacene diradical dianion salts [(18-c-6)K(THF)₂ ]⁺ [(18-c-6)K]⁺ ⋅1^2-.. and [(18-c-6)K(THF)]²⁺ ⋅2^2-.. by reduction of the neutral pyrene-fused azaacene derivatives 1 and 2 with excess potassium graphite in THF in the presence of 18-crown-6. Their electronic structures were investigated by various experiments, in conjunction with theoretical calculations. It was found that both dianions are open-shell singlets in the ground state and their triplet states are thermally readily accessible owing to the small singlet-triplet energy gap. This work provides the first examples of crystalline diradical dianions of azaacenes with considerable diradical character.

Two Cycling Centers in One Molecule: Communication by Through-Bond Interactions and Entanglement of the Unpaired Electrons

Many applications in quantum information science (QIS) rely on the ability to laser-cool molecules. The scope of applications can be expanded if laser-coolable molecules possess two or more cycling centers, i.e., moieties capable of scattering photons via multiple absorption-emission events. Here we employ the equation-of-motion coupled-cluster method for double electron attachment (EOM-DEA-CCSD) to study the electronic structure of hypermetallic molecules with two alkaline-earth metals connected by an acetylene linker. The electronic structure of the molecules is similar to that of two separated alkali metals; however, the interaction between the two electrons is weak and largely dominated by through-bond interactions. The communication between the two cycling centers is quantified by the extent of the entanglement of the two unpaired electrons associated with the two cycling centers. This contribution highlights the rich electronic structure of hypermetallic molecules that may advance various applications in QIS and beyond.

Machine Learning in Nanoscience: Big Data at Small Scales

Recent advances in machine learning (ML) offer new tools to extract new insights from large data sets and to acquire small data sets more effectively. Researchers in nanoscience are experimenting with these tools to tackle challenges in many fields. In addition to ML's advancement of nanoscience, nanoscience provides the foundation for neuromorphic computing hardware to expand the implementation of ML algorithms. In this Mini Review, we highlight some recent efforts to connect the ML and nanoscience communities by focusing on three types of interaction: (1) using ML to analyze and extract new insights from large nanoscience data sets, (2) applying ML to accelerate material discovery, including the use of active learning to guide experimental design, and (3) the nanoscience of memristive devices to realize hardware tailored for ML. We conclude with a discussion of challenges and opportunities for future interactions between nanoscience and ML researchers.

General framework for calculating spin-orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., M_s = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupled-cluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.

Ab Initio Probing of the Ground State of Tetraradicals: Breakdown of Hund's Multiplicity Rule

The electronic structure of organic σ-type polyradical including 2,4,6-tridehydropyridine radical cation (246-TDHP) and three isomers of tetradehydrobenzene (TDHB) have been studied using a computationally robust and cost-effective second-order multireference perturbative model which provides a balanced treatment of nondynamic and dynamic contributions to the electron correlation problem in the ground or excited electronic states which are imperative for predicting structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of polyradicals. Energy gaps are useful to capture insight into the degree of interaction between the radical sites. An important finding of this study is that the tetraradicals considered here possess singlet ground states, contrary to Hund's rule. Present findings are in close agreement with the available high-level ab initio estimates at attainable cost implying that a perturbative description of the systems is adequate. The impact of N⁺ on the nature of ground state for the 246-TDHP have also analyzed. The singlet-triplet energy gaps for 1245- and 1234-TDHB are smaller than for o-benzyne mainly due to the ring strain. 1235-TDHB is 14.42 and 11.05 kcal/mol lower in energy than 1245- and 1234-isomers, respectively. IVO-SSMRPT predicts ¹A₁-³B₂ and ¹A₁-⁵B₂ gaps of 25.84 and 105.15 kcal/mol, respectively for the 246-TDHP cation.

The role of the multiconfigurational character of nitronyl-nitroxide in the singlet–triplet energy gap of its diradicals

CAS(2,2) reference may not be sufficient for the computation of singlet–triplet energy gap by DDCI.