Reference Energies for Double Excitations

High-level coupled-cluster energetics by merging moment expansions with selected configuration interaction

Inspired by our earlier semi-stochastic work aimed at converging high-level coupled-cluster (CC) energetics [J. E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017) and J. E. Deustua, J. Shen, and P. Piecuch, J. Chem. Phys. 154, 124103 (2021)], we propose a novel form of the CC(P; Q) theory in which the stochastic Quantum Monte Carlo propagations, used to identify dominant higher-than-doubly excited determinants, are replaced by the selected configuration interaction (CI) approach using the perturbative selection made iteratively (CIPSI) algorithm. The advantages of the resulting CIPSI-driven CC(P; Q) methodology are illustrated by a few molecular examples, including the dissociation of F₂ and the automerization of cyclobutadiene, where we recover the electronic energies corresponding to the CC calculations with a full treatment of singles, doubles, and triples based on the information extracted from compact CI wave functions originating from relatively inexpensive Hamiltonian diagonalizations.

Accurate full configuration interaction correlation energy estimates for five- and six-membered rings

Following our recent work on the benzene molecule [P.-F. Loos, Y. Damour, and A. Scemama, J. Chem. Phys. 153, 176101 (2020)], motivated by the blind challenge of Eriksen et al. [J. Phys. Chem. Lett. 11, 8922 (2020)] on the same system, we report accurate full configuration interaction (FCI) frozen-core correlation energy estimates for 12 five- and six-membered ring molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, s-tetrazine, and s-triazine) in the standard correlation-consistent double-ζ Dunning basis set (cc-pVDZ). Our FCI correlation energy estimates, with an estimated error smaller than 1 millihartree, are based on energetically optimized-orbital selected configuration interaction calculations performed with the configuration interaction using a perturbative selection made iteratively algorithm. Having at our disposal these accurate reference energies, the respective performance and convergence properties of several popular and widely used families of single-reference quantum chemistry methods are investigated. In particular, we study the convergence properties of (i) the Møller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), (ii) the iterative approximate coupled-cluster series CC2, CC3, and CC4, and (iii) the coupled-cluster series CCSD, CCSDT, and CCSDTQ. The performance of the ground-state gold standard CCSD(T) as well as the completely renormalized CC model, CR-CC(2,3), is also investigated. We show that MP4 provides an interesting accuracy/cost ratio, while MP5 systematically worsens the correlation energy estimates. In addition, CC3 outperforms CCSD(T) and CR-CC(2,3), as well as its more expensive parent CCSDT. A similar trend is observed for the methods including quadruple excitations, where the CC4 model is shown to be slightly more accurate than CCSDTQ, both methods providing correlation energies within 2 millihartree of the FCI limit.

Electronic Structure of Liquid Alkanes: A Representative Case of Liquid Hexanes and Cyclohexane Studied Using Polarization-Dependent Two-Photon Absorption Spectroscopy

Two-photon absorption (2PA) spectra of liquid cyclohexane and hexanes are reported for the energy range 6.4-8.5 eV (177-145 nm), providing detailed information about their electronic structures in bulk liquid. Using a broadband pump-probe fashion, we measured the continuous 2PA spectra by simultaneous absorption of a 266 nm (4.6 eV) pump photon and one UV-vis probe photon from the white-light continuum (1.8-3.9 eV). Theoretical one-photon absorption (1PA) and 2PA cross sections of isolated gas phase molecules are computed by the equation of motion coupled-cluster method with single and double substitutions (EOM-CCSD) to substantiate the assignment of the experimental spectra, and the natural transition orbital (NTO) analysis provides visualization of the participating orbitals in a transition. Our analysis suggests that upon solvation transitions at the lowest excitation energy involving promotion of electron to the 3s Rydberg orbitals are blue-shifted (∼0.55 eV for cyclohexane and ∼0.18 eV for hexanes) to a greater extent as compared to those involving other Rydberg orbitals, which is similar to the behavior observed for water and alcohols. All other transitions experience negligible (cyclohexane) or minor red-shift by ∼0.15-0.2 eV (hexane) upon solvation. In both alkanes, the spectra are entirely dominated by Rydberg transitions: the most intense bands in 1PA and 2PA spectra are due to the excitation of electrons to the Rydberg "p" and "d" type orbitals, respectively, although one transition terminating in the 3s Rydberg has significant 2PA strength. This work demonstrates that the gas phase electronic transition properties in alkanes are not significantly altered upon solvation. In addition, electronic structure calculations using an isolated-molecule framework appear to provide a reasonable starting point for a semiquantitative picture for spectral assignment and also to analyze the solvatochromic shifts for liquid phase absorption spectra.

Variational coupled cluster for ground and excited states

In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H₄ molecule and the continuous deformation of the square H₄ molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree-Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.

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.

The Quest to Simulate Excited-State Dynamics of Transition Metal Complexes

This Perspective describes current computational efforts in the field of simulating photodynamics of transition metal complexes. We present the typical workflows and feature the strengths and limitations of the different contemporary approaches. From electronic structure methods suitable to describe transition metal complexes to approaches able to simulate their nuclear dynamics under the effect of light, we give particular attention to build a bridge between theory and experiment by critically discussing the different models commonly adopted in the interpretation of spectroscopic experiments and the simulation of particular observables. Thereby, we review all the studies of excited-state dynamics on transition metal complexes, both in gas phase and in solution from reduced to full dimensionality.

Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster

The pair coupled cluster doubles (pCCD) method (where the excitation manifold is restricted to electron pairs) has a series of interesting features. Among others, it provides ground-state energies very close to what is obtained with doubly occupied configuration interaction (DOCI), but with a polynomial cost (compared with the exponential cost of the latter). Here, we address whether this similarity holds for excited states by exploring the symmetric dissociation of the linear H4 molecule. When ground-state Hartree-Fock (HF) orbitals are employed, pCCD and DOCI excited-state energies do not match, a feature that is assigned to the poor HF reference. In contrast, by optimizing the orbitals at the pCCD level (oo-pCCD) specifically for each excited state, the discrepancies between pCCD and DOCI decrease by 1 or 2 orders of magnitude. Therefore, the pCCD and DOCI methodologies still provide comparable energies for excited states, but only if suitable, state-specific orbitals are adopted. We also assessed whether a pCCD approach could be used to directly target doubly excited states, without having to resort to the equation-of-motion (EOM) formalism. In our Δoo-pCCD model, excitation energies are extracted from the energy difference between separate oo-pCCD calculations for the ground state and the targeted excited state. For a set comprising the doubly excited states of CH+, BH, nitroxyl, nitrosomethane, and formaldehyde, we found that Δoo-pCCD provides quite accurate excitation energies, with root-mean-square deviations (with respect to full configuration interaction results) lower than those of CC3 and comparable to those of EOM-CCSDT, two methods with a much higher computational cost.

Is Externally Corrected Coupled Cluster Always Better Than the Underlying Truncated Configuration Interaction?

The short answer to the question in the title is "no". We identify classes of truncated configuration interaction (CI) wave functions for which the externally corrected coupled-cluster (ec-CC) approach using the three-body (T₃) and four-body (T₄) components of the cluster operator extracted from CI does not improve the results of the underlying CI calculations. Implications of our analysis, illustrated by numerical examples, for the ec-CC computations using truncated and selected CI methods are discussed. We also introduce a novel ec-CC approach using the T₃ and T₄ amplitudes obtained with the selected CI scheme abbreviated as CIPSI, correcting the resulting energies for the missing T₃ correlations not captured by CIPSI with the help of moment expansions similar to those employed in the completely renormalized CC methods.

How accurate are EOM-CC4 vertical excitation energies?

We report the first investigation of the performance of EOM-CC4-an approximate equation-of-motion coupled-cluster model, which includes iterative quadruple excitations-for vertical excitation energies in molecular systems. By considering a set of 28 excited states in 10 small molecules for which we have computed CC with singles, doubles, triples, quadruples, and pentuples and full configuration interaction reference energies, we show that, in the case of excited states with a dominant contribution from the single excitations, CC4 yields excitation energies with sub-kJ mol^-1 accuracy (i.e., error below 0.01 eV), in very close agreement with its more expensive CC with singles, doubles, triples, and quadruples parent. Therefore, if one aims at high accuracy, CC4 stands as a highly competitive approximate method to model molecular excited states, with a significant improvement over both CC3 and CC with singles, doubles, and triples. Our results also evidence that, although the same qualitative conclusions hold, one cannot reach the same level of accuracy for transitions with a dominant contribution from the double excitations.

Orbital Optimized Density Functional Theory for Electronic Excited States

Density functional theory (DFT) based modeling of electronic excited states is of importance for investigation of the photophysical/photochemical properties and spectroscopic characterization of large systems. The widely used linear response time-dependent DFT (TDDFT) approach is, however, not effective at modeling many types of excited states, including (but not limited to) charge-transfer states, doubly excited states, and core-level excitations. In this perspective, we discuss state-specific orbital optimized (OO) DFT approaches as an alterative to TDDFT for electronic excited states. We motivate the use of OO-DFT methods and discuss reasons behind their relatively restricted historical usage (vs TDDFT). We subsequently highlight modern developments that address these factors and allow efficient and reliable OO-DFT computations. Several successful applications of OO-DFT for challenging electronic excitations are also presented, indicating their practical efficacy. OO-DFT approaches are thus increasingly becoming a useful route for computing excited states of large chemical systems. We conclude by discussing the limitations and challenges still facing OO-DFT methods, as well as some potential avenues for addressing them.

Spin-Conserved and Spin-Flip Optical Excitations from the Bethe-Salpeter Equation Formalism

Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe–Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is “blind” to double (and higher) excitations, which are ubiquitous, for example, in conjugated molecules like polyenes. Here, we apply the spin-flip ansatz (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access, in particular, double excitations. The present scheme is based on a spin-unrestricted version of the GW approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations. Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment. The performance of the present spin-flip BSE formalism is illustrated by computing excited-state energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

Estimates of electron correlation based on density expansions

Two methods for estimating the correlation energy of molecules and other electronic systems are discussed based on the assumption that the correlation energy can be partitioned between atomic regions. In the first method, the electron density is expanded in terms of atomic contributions using rigorous electron repulsion bounds, and in the second method, correlation contributions are associated with basis function pairs. These methods do not consider the detailed nature of localized excitations but instead define a correlation energy per electron factor that is unique to a specific atom. The correlation factors are basis function dependent and are determined by configuration interaction (CI) calculations on diatomic and hydride molecules. The correlation energy estimates are compared with the results of high-level CI calculations for a test set of 27 molecules representing a wide range of bonding environments (average error of 2.6%). An extension based on truncated CI calculations in which d-type and hydrogen p-type functions are eliminated from the virtual space combined with estimates of dynamical correlation contributions using atomic correlation factors is discussed and applied to the dissociation of several molecules.

Generalization of Block-Localized Wave Function for Constrained Optimization of Excited Determinants

The block-localized wave function method is useful to provide insights on chemical bonding and intermolecular interactions through energy decomposition analysis. The method relies on block localization of molecular orbitals (MOs) by constraining the orbitals to basis functions within given blocks. Here, a generalized block-localized orbital (GBLO) method is described to allow both physically localized and delocalized MOs to be constrained in orbital-block definitions. Consequently, GBLO optimization can be conveniently tailored by imposing specific constraints. The GBLO method is illustrated by three examples: (1) constrained polarization response orbitals through dipole and quadrupole perturbation in a water dimer complex, (2) the ground and first excited-state potential energy curves of ethene about its C-C bond rotation, and (3) excitation energies of double electron excited states. Multistate density functional theory is used to determine the energies of the adiabatic ground and excited states using a minimal active space (MAS) comprising specifically charge-constrained and excited determinant configurations that are variationally optimized by the GBLO method. We find that the GBLO expansion that includes delocalized MOs in configurational blocks significantly reduces computational errors in comparison with physical block localization, and the computed ground- and excited-state energies are in good accordance with experiments and results obtained from multireference configuration interaction calculations.

A hybrid approach to excited-state-specific variational Monte Carlo and doubly excited states

We extend our hybrid linear-method/accelerated-descent variational Monte Carlo optimization approach to excited states and investigate its efficacy in double excitations. In addition to showing a superior statistical efficiency when compared to the linear method, our tests on small molecules show good energetic agreement with benchmark methods. We also demonstrate the ability to treat double excitations in systems that are too large for a full treatment by using selected configuration interaction methods via an application to 4-aminobenzonitrile. Finally, we investigate the stability of state-specific variance optimization against collapse to other states' variance minima and find that symmetry, Ansatz quality, and sample size all have roles to play in achieving stability.

Dynamical kernels for optical excitations

We discuss the physical properties and accuracy of three distinct dynamical (i.e., frequency-dependent) kernels for the computation of optical excitations within linear response theory: (i) an a priori built kernel inspired by the dressed time-dependent density-functional theory kernel proposed by Maitra et al. [J. Chem. Phys. 120, 5932 (2004)], (ii) the dynamical kernel stemming from the Bethe-Salpeter equation (BSE) formalism derived originally by Strinati [Riv. Nuovo Cimento 11, 1-86 (1988)], and (iii) the second-order BSE kernel derived by Zhang et al. [J. Chem. Phys. 139, 154109 (2013)]. The principal take-home message of the present paper is that dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated with higher-order excitations (such as the infamous double excitations), an unappreciated feature of dynamical quantities. We also analyze, for each kernel, the appearance of spurious excitations originating from the approximate nature of the kernels, as first evidenced by Romaniello et al. [J. Chem. Phys. 130, 044108 (2009)]. Using a simple two-level model, prototypical examples of valence, charge-transfer, and Rydberg excited states are considered.

Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation

The Bethe-Salpeter equation (BSE) formalism is a computationally affordable method for the calculation of accurate optical excitation energies in molecular systems. Similar to the ubiquitous adiabatic approximation of time-dependent density-functional theory, the static approximation, which substitutes a dynamical (i.e., frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the BSE formalism. Here, going beyond the static approximation, we compute the dynamical correction of the electron-hole screening for molecular excitation energies, thanks to a renormalized first-order perturbative correction to the static BSE excitation energies. The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation. Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing one to assess the clear improvement brought by the dynamical correction for both singlet and triplet optical transitions.

Frequency and Time Domain Nuclear-Electronic Orbital Equation-of-Motion Coupled Cluster Methods: Combination Bands and Electronic-Protonic Double Excitations

The accurate description of excited vibronic states is important for modeling a wide range of photoinduced processes. The nuclear-electronic orbital (NEO) approach, which treats specified protons on the same level as the electrons, can describe excited electronic-protonic states. Herein the multicomponent equation-of-motion coupled cluster with singles and doubles (NEO-EOM-CCSD) method and its time-domain counterpart, TD-NEO-EOM-CCSD, are developed and implemented. The application of these methods to the HCN molecule highlights their capabilities. These methods predict qualitatively reasonable energies and intensities for a combination band corresponding to simultaneous excitation of two vibrational modes, as well as an overtone. These methods also describe states with double excitation character, such as excited electronic-protonic states corresponding to the simultaneous excitation of an electron and a proton. The ability of the NEO-EOM-CCSD method and its time-dependent counterpart to describe combination bands, overtones, and double excitations will enable a wide range of photochemical applications.

Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies

The combination of the many-body Green's function GW approximation and the Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems. The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). Here, we study the topology of the ground-state potential energy surfaces (PESs) of several diatomic molecules near their equilibrium bond length. Using comparisons with state-of-art computational approaches (CC3), we show that ACFDT@BSE is surprisingly accurate and can even compete with lower-order coupled cluster methods (CC2 and CCSD) in terms of total energies and equilibrium bond distances for the considered systems. However, we sometimes observe unphysical irregularities on the ground-state PES in relation with difficulties in the identification of a few GW quasiparticle energies.

The Quest for Highly Accurate Excitation Energies: A Computational Perspective

We provide an overview of the successive steps that made it possible to obtain increasingly accurate excitation energies with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and medium-size molecules. First, we describe the evolution of ab initio methods employed to define benchmark values, with the original Roos CASPT2 method, then the CC3 method as in the renowned Thiel set, and more recently the resurgence of selected configuration interaction methods. The latter method has been able to deliver consistently, for both single and double excitations, highly accurate excitation energies for small molecules, as well as medium-size molecules with compact basis sets. Second, we describe how these high-level methods and the creation of representative benchmark sets of excitation energies have allowed the fair and accurate assessment of the performance of computationally lighter methods. We conclude by discussing possible future theoretical and technological developments in the field.

Electronic Excited States and UV-Vis Absorption Spectra of the Dihydropyrene/Cyclophanediene Photochromic Couple: a Theoretical Investigation

Dihydropyrene (DHP)/cyclophanediene (CPD) is a fascinating photoswitchable organic system displaying negative photochromism. Upon irradiation in the visible region, the colored DHP can be converted to its open-ring CPD colorless isomer, which can be converted back to DHP by UV light. DHP and CPD thus possess very different absorption spectra whose absorption bands have never been assigned in detail so far. In this work, we characterize the vertical electronic transitions of the first six and seven excited states of DHP and CPD, respectively, aiming for a realistic comparison with experiment. We used state-of-the-art electronic structure methods [e.g., complete active space second-order perturbation theory (CASPT2), n-electron valence-state perturbation theory (NEVPT2), extended multiconfigurational quasi-degenerate perturbation theory (XMCQDPT2), and third-order algebraic diagrammatic construction ADC(3)] capable of describing differential electron correlation. Vertical transition energies were also computed with time-dependent density functional theory (TD-DFT) and compared to these accurate methods. After the reliability of TD-DFT was validated for the main optical transitions, this efficient method was used to simulate the absorption spectra of DHP and CPD in the framework of the Franck-Condon Herzberg-Teller approximation and also using the nuclear ensemble approach. Overall, for both methods, the simulated absorption spectra reproduce nicely the main spectral features of the DHP and CPD isomers, that is, the main four absorption bands of increasing intensity of DHP and the absorption rise below 300 nm for CPD.

Computational Studies of Molecular Materials for Unconventional Energy Conversion: The Challenge of Light Emission by Thermally Activated Delayed Fluorescence

In this paper we describe the mechanism of light emission through thermally activated delayed fluorescence (TADF)—a process able to ideally achieve 100% quantum efficiencies upon fully harvesting the energy of triplet excitons, and thus minimizing the energy loss of common (i.e., fluorescence and phosphorescence) luminescence processes. If successful, this technology could be exploited for the manufacture of more efficient organic light-emitting diodes (OLEDs) made of only light elements for multiple daily applications, thus contributing to the rise of a sustainable electronic industry and energy savings worldwide. Computational and theoretical studies have fostered the design of these all-organic molecular emitters by disclosing helpful structure–property relationships and/or analyzing the physical origin of this mechanism. However, as the field advances further, some limitations have also appeared, particularly affecting TD-DFT calculations, which have prompted the use of a variety of methods at the molecular scale in recent years. Herein we try to provide a guide for beginners, after summarizing the current state-of-the-art of the most employed theoretical methods focusing on the singlet–triplet energy difference, with the additional aim of motivating complementary studies revealing the stronger and weaker aspects of computational modelling for this cutting-edge technology.

Excited State Orbital Optimization via Minimizing the Square of the Gradient: General Approach and Application to Singly and Doubly Excited States via Density Functional Theory

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting square gradient minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant ΔSCF and spin-purified restricted open-shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the maximum overlap method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that ΔSCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer, or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems.

A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules

Following our previous work focusing on compounds containing up to 3 non-hydrogen atoms [J. Chem. Theory Comput. 2018, 14, 4360-4379], we present here highly accurate vertical transition energies obtained for 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione, diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine. To obtain these energies, we use equation-of-motion/linear-response coupled cluster theory up to the highest technically possible excitation order for these systems (CC3, EOM-CCSDT, and EOM-CCSDTQ) and selected configuration interaction (SCI) calculations (with tens of millions of determinants in the reference space), as well as the multiconfigurational n-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least CC3/aug-cc-pVQZ vertical excitation energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of ±0.04 eV, except for transitions with a dominant double excitation character where the error is much larger. The present contribution gathers a large, diverse, and accurate set of more than 200 highly accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, n → π*, π → π*, ...). We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, CC3, and NEVPT2. The results of these benchmarks are compared to the available literature data.

Highly Accurate Prediction of Core Spectra of Molecules at Density Functional Theory Cost: Attaining Sub-electronvolt Error from a Restricted Open-Shell Kohn-Sham Approach

We present the use of the recently developed square gradient minimization (SGM) algorithm for excited-state orbital optimization to obtain spin-pure restricted open-shell Kohn-Sham (ROKS) energies for core excited states of molecules. The SGM algorithm is robust against variational collapse and offers a reliable route to converging orbitals for target excited states at only 2-3 times the cost of ground-state orbital optimization (per iteration). ROKS/SGM with the modern SCAN/ωB97X-V functionals is found to predict the K-edge of C, N, O, and F to a root mean squared error of ∼0.3 eV. ROKS/SGM is equally effective at predicting L-edge spectra of third period elements, provided a perturbative spin-orbit correction is employed. This high accuracy can be contrasted with traditional time-dependent density functional theory (TDDFT), which typically has greater than 10 eV error and requires translation of computed spectra to align with experiment. ROKS is computationally affordable (having the same scaling as ground-state DFT and a slightly larger prefactor) and can be applied to geometry optimizations/ab initio molecular dynamics of core excited states, as well as condensed phase simulations. ROKS can also model doubly excited/ionized states with one broken electron pair, which are beyond the ability of linear response based methods.

Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?

The search for new models rapidly delivering accurate excited-state energies and properties is one of the most active research lines of theoretical chemistry. Along with these developments, the performance of known methods is constantly reassessed on the basis of new benchmark values. In this Letter, we show that the third-order algebraic diagrammatic construction, ADC(3), does not yield transition energies of the same quality as the third-order coupled cluster method, CC3. This is demonstrated by extensive comparisons with several hundred high-quality vertical transition energies obtained with FCI, CCSDTQ, and CCSDT. Direct comparisons with experimental 0-0 energies of small- and medium-size molecules support the same conclusion, which holds for both valence and Rydberg transitions. Considering these results, we introduce a composite approach, ADC(2.5), which consists of averaging the ADC(2) and ADC(3) excitation energies. Although ADC(2.5) does not match the CC3 accuracy, it significantly improves the ADC(3) results, especially for vertical energies.

Weight dependence of local exchange–correlation functionals in ensemble density-functional theory: double excitations in two-electron systems

We discuss the construction of first-rung weight-dependent exchange–correlation density-functional approximations for He and H₂ specifically designed for the computation of double excitations within Gross–Oliveira–Kohn-DFT.