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

Reference CC3 Excitation Energies for Organic Chromophores: Benchmarking TD-DFT, BSE/GW, and Wave Function Methods

To expand the QUEST database of highly accurate vertical transition energies, we consider a series of large organic chromogens ubiquitous in dye chemistry, such as anthraquinone, azobenzene, BODIPY, and naphthalimide. We compute, at the CC3 level of theory, the singlet and triplet vertical transition energies associated with the low-lying excited states. This leads to a collection of more than 120 new highly accurate excitation energies. For several singlet transitions, we have been able to determine CCSDT transition energies with a compact basis set, finding minimal deviations from the CC3 values for most states. Subsequently, we employ these reference values to benchmark a series of lower-order wave function approaches, including the popular ADC(2) and CC2 schemes, as well as time-dependent density-functional theory (TD-DFT), both with and without applying the Tamm-Dancoff approximation (TDA). At the TD-DFT level, we evaluate a large panel of global, range-separated, local, and double hybrid functionals. Additionally, we assess the performance of the Bethe-Salpeter equation (BSE) formalism relying on both G0W0 and evGW quasiparticle energies evaluated from various starting points. It turns out that CC2 and ADC(2.5) are the most accurate models among those with respective O ( N 5 ) and O ( N 6 ) scalings with system size. In contrast, CCSD does not outperform CC2. The best performing exchange-correlation functionals include BMK, M06-2X, M06-SX, CAM-B3LYP, ωB97X-D, and LH20t, with average deviations of approximately 0.20 eV or slightly below. Errors on vertical excitation energies can be further reduced by considering double hybrids. Both SOS-ωB88PP86 and SOS-ωPBEPP86 exhibit particularly attractive performances with overall quality on par with CC2, whereas PBE0-DH and PBE-QIDH are only slightly less efficient. BSE/evGW calculations based on Kohn-Sham starting points have been found to be particularly effective for singlet transitions, but much less for their triplet counterparts.

A mountaineering strategy to excited states: Accurate vertical transition energies and benchmarks for substituted benzenes

In an effort to expand the existing QUEST database of accurate vertical transition energies [Véril et al. WIREs Comput. Mol. Sci. 2021, 11, e1517], we have modeled more than 100 electronic excited states of different natures (local, charge-transfer, Rydberg, singlet, and triplet) in a dozen of mono- and di-substituted benzenes, including aniline, benzonitrile, chlorobenzene, fluorobenzene, nitrobenzene, among others. To establish theoretical best estimates for these vertical excitation energies, we have employed advanced coupled-cluster methods including iterative triples (CC3 and CCSDT) and, when technically possible, iterative quadruples (CC4). These high-level computational approaches provide a robust foundation for benchmarking a series of popular wave function methods. The evaluated methods all include contributions from double excitations (ADC(2), CC2, CCSD, CIS(D), EOM-MP2, STEOM-CCSD), along with schemes that also incorporate perturbative or iterative triples (ADC(3), CCSDR(3), CCSD(T)(a)  ⋆ , and CCSDT-3). This systematic exploration not only broadens the scope of the QUEST database but also facilitates a rigorous assessment of different theoretical approaches in the framework of a homologous chemical series, offering valuable insights into the accuracy and reliability of these methods in such cases. We found that both ADC(2.5) and CCSDT-3 can provide very consistent estimates, whereas among less expensive methods SCS-CC2 is likely the most effective approach. Importantly, we show that some lower order methods may offer reasonable trends in the homologous series while providing quite large average errors, and vice versa. Consequently, benchmarking the accuracy of a model based solely on absolute transition energies may not be meaningful for applications involving a series of similar compounds.

Photophysical Outcomes of Water-Solvated Heterocycles: Single-Conformation Ultraviolet and Infrared Spectroscopy of Microsolvated 2-Phenylpyrrole

The molecular chromophores within brown carbon (BrC) aerosols absorb solar radiation at visible and near-ultraviolet wavelengths. This contributes to the overall warming of the troposphere and the photochemical aging of aerosols. In this investigation, we combine a suite of experimental and theoretical methods to reveal the conformation-specific ultraviolet and infrared spectroscopy of 2-phenylpyrrole (2PhPy)─an extended π-conjugated pyrrole derivative and a model BrC chromophore─along with its water microsolvated molecular complexes (2PhPy:nH2O, n = 1-3). Using resonant two-photon ionization and double-resonance holeburning techniques alongside MP3 (ground state) and ADC(3) (excited state) torsional potential energy surfaces and discrete variable representation simulations, we characterized the ultraviolet spectra of 2PhPy and 2PhPy:1H2O. This analysis revealed evidence for Herzberg-Teller vibronic coupling along the CH wagging and NH stretching coordinates of the aromatic rings. Conformation-specific infrared spectroscopy revealed extended hydrogen-bonding networks of the 2PhPy:nH2O complexes. Upon stepwise addition of H2O solvation, the nearest H2O acceptor forms a strong, noncovalent interaction with the pyrrole NH donor, while the second and third H2O partners interface with the phenyl and pyrrole aromatic rings through growing van der Waals π/H atom stabilization. A local-mode Hamiltonian approach was employed for comparison with the experimental spectra, thus identifying the vibrational spectral signatures to specific 2PhPy:nH2O oscillators.

Performance Tests of the Second-Order Approximate Internally Contracted Multireference Coupled-Cluster Singles and Doubles Method icMRCC2

Benchmark results are presented for the second-order approximation of the internally contracted multireference coupled-cluster method with single and double excitations, icMRCC2 [Köhn, Bargholz, J. Chem. Phys. 2019, 151, 041106], which was designed as a multireference analogue of the single-reference second-order approximate coupled-cluster method CC2 [Christiansen, Koch, Jørgensen, Chem. Phys. Lett. 1995, 243, 409-418]. Vertical excitation energies of various small to medium-sized organic molecules are investigated based on established test sets from the literature. Additionally, the spectroscopic constants of ground and excited states of diatomics and the geometric parameters of excited triatomic molecules were determined and compared to the experimental data. The results show that the method clearly extends the applicability of single-reference CC2, including doubly excited states, and also artifacts of CC2 like too low Rydberg excitations and too weak multiple bonds are eliminated. The method is computationally more demanding than standard multireference second-order perturbation theories but improves significantly in accuracy, as shown by the benchmark results. In addition, it is demonstrated that small active spaces are often sufficient to obtain accurate energies with icMRCC2. Example applications like the automerization of cyclobutadiene, the deactivation pathway of ethylene, and the excited states of an iron complex with a noninnocent nitrosyl ligand demonstrate the potential of icMRCC2 in cases with strong multireference character.

Algebraic Diagrammatic Construction Schemes Employing the Intermediate State Formalism: Theory, Capabilities, and Interpretation

Algebraic diagrammatic construction (ADC) schemes represent a family of ab initio methods for the calculation of excited electronic states and electron-detached and -attached states. All ADC methods have been demonstrated to possess great potential for molecular applications, e.g., for the calculation of absorption or photoelectron spectra or electron attachment processes. ADC originates from Green's function or propagator theory; however, most recent ADC developments heavily rely on the intermediate state representation or effective Liouvillian formalisms, which comprise new ADC methods and computational schemes for high-order properties. The different approaches for the calculation of excitation energies, ionization potentials, and electron affinities are intimately related, and they provide a coherent description of these quantities at equivalent levels of theory and with comparable errors. Most quantum chemical program packages contain ADC methods; however, the most complete ADC suite of methods can be found in the recent release of Q-Chem.

Excitation and fragmentation of the dielectric gas C4F7N: Electrons vs photons

C₄F₇N is a promising candidate for the replacement of sulfur hexafluoride as an insulating medium, and it is important to understand the chemical changes initiated in the molecule by collision with free electrons, specifically the formation of neutral fragments. The first step of neutral fragmentation is electronic excitation, yet neither the absorption spectrum in the vacuum ultraviolet (VUV) region nor the electron energy loss spectrum have previously been reported. Here, we experimentally probed the excited states by VUV photoabsorption spectroscopy and electron energy loss spectroscopy (EELS). We found that the distribution of states populated upon electron impact with low-energy electrons is significantly different from that following photoabsorption. This difference was confirmed and interpreted with ab initio modeling of both VUV and EELS spectra. We propose here a new computational protocol for the simulation of EELS spectra combining the Born approximation with approximate forms of correlated wave functions, which allows us to calculate the (usually very expensive) scattering cross sections at a cost similar to the calculation of oscillator strengths. Finally, we perform semi-classical non-adiabatic dynamics simulations to investigate the possible neutral fragments of the molecule formed through electron-induced neutral dissociation. We show that the product distribution is highly non-statistical.

Large-Scale Benchmarking of Multireference Vertical-Excitation Calculations via Automated Active-Space Selection

We have calculated state-averaged complete-active-space self-consistent-field (SA-CASSCF), multiconfiguration pair-density functional theory (MC-PDFT), hybrid MC-PDFT (HMC-PDFT), and n-electron valence state second-order perturbation theory (NEVPT2) excitation energies with the approximate pair coefficient (APC) automated active-space selection scheme for the QUESTDB benchmark database of 542 vertical excitation energies. We eliminated poor active spaces (20-40% of calculations) by applying a threshold to the SA-CASSCF absolute error. With the remaining calculations, we find that NEVPT2 performance is significantly impacted by the size of the basis set the wave functions are converged in, regardless of the quality of their description, which is a problem absent in MC-PDFT. Additionally, we find that HMC-PDFT is a significant improvement over MC-PDFT with the translated PBE (tPBE) density functional and that it performs about as well as NEVPT2 and second-order coupled cluster on a set of 373 excitations in the QUESTDB database. We optimized the percentage of SA-CASSCF energy to include in HMC-PDFT when using the tPBE on-top functional, and we find the 25% value used in tPBE0 to be optimal. This work is by far the largest benchmarking of MC-PDFT and HMC-PDFT to date, and the data produced in this work are useful as a validation of HMC-PDFT and of the APC active-space selection scheme. We have made all the wave functions produced in this work (orbitals and CI vectors) available to the public and encourage the community to utilize this data as a tool in the development of further multireference model chemistries.

Reference Energies for Cyclobutadiene: Automerization and Excited States

Cyclobutadiene is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. Indeed, due to its high spatial symmetry, especially at the D_4h square geometry but also in the D_2h rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multiconfigurational characters and single-reference methods, such as standard adiabatic time-dependent density-functional theory (TD-DFT) or standard equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies at D_2h and D_4h equilibrium structures. In particular, selected configuration interaction (SCI), multireference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. The spin-flip formalism, which is known to provide a qualitatively correct description of these diradical states, is also tested within TD-DFT (combined with numerous exchange-correlation functionals) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.

A Mountaineering Strategy to Excited States: Revising Reference Values with EOM-CC4

In the framework of the computational determination of highly accurate vertical excitation energies in small organic compounds, we explore the possibilities offered by the equation-of-motion formalism relying on the approximate fourth-order coupled-cluster (CC) method, CC4. We demonstrate, using an extended set of more than 200 reference values based on CC including up to quadruples excitations (CCSDTQ), that CC4 is an excellent approximation to CCSDTQ for excited states with a dominant contribution from single excitations with an average deviation as small as 0.003 eV. We next assess the accuracy of several additive basis set correction schemes, in which vertical excitation energies obtained with a compact basis set and a high-order CC method are corrected with lower-order CC calculations performed in a larger basis set. Such strategies are found to be overall very beneficial, though their accuracy depends significantly on the actual scheme. Finally, CC4 is employed to improve several theoretical best estimates of the QUEST database for molecules containing between four and six (nonhydrogen) atoms, for which previous estimates were computed at the CCSDT level.

Benchmarking CASPT3 Vertical Excitation Energies

Based on 280 reference vertical transition energies of various natures (singlet, triplet, valence, Rydberg, n → π∗, π → π∗, and double excitations) extracted from the QUEST database, we assess the accuracy of third-order multireference perturbation theory, CASPT3, in the context of molecular excited states. When one applies the disputable ionization- potential-electron-affinity (IPEA) shift, we show that CASPT3 provides a similar accuracy as its second-order counterpart, CASPT2, with the same mean absolute error of 0.11 eV. However, as already reported, we also observe that the accuracy of CASPT3 is almost insensitive to the IPEA shift, irrespective of the transition type and system size, with a small reduction of the mean absolute error to 0.09 eV when the IPEA shift is switched off.

Revisiting the Performance of Time-Dependent Density Functional Theory for Electronic Excitations: Assessment of 43 Popular and Recently Developed Functionals from Rungs One to Four

In this paper, the performance of more than 40 popular or recently developed density functionals is assessed for the calculation of 463 vertical excitation energies against the large and accurate QuestDB benchmark set. For this purpose, the Tamm-Dancoff approximation offers a good balance between computational efficiency and accuracy. The functionals ωB97X-D and BMK are found to offer the best performance overall with a root-mean square error (RMSE) of around 0.27 eV, better than the computationally more demanding CIS(D) wave function method with a RMSE of 0.36 eV. The results also suggest that Jacob's ladder still holds for time-dependent density functional theory excitation energies, though hybrid meta generalized-gradient approximations (meta-GGAs) are not generally better than hybrid GGAs. Effects of basis set convergence, gauge invariance correction to meta-GGAs, and nonlocal correlation (VV10) are also studied, and practical basis set recommendations are provided.

Exploring the accuracy and usefulness of semi-empirically scaled ADC schemes by blending second and third order terms

Different approaches to mixed-order algebraic-diagrammatic construction (ADC) schemes are investigated. The performance of two different strategies for scaling third-order contributions to the ADC secular matrix is evaluated. Both considered schemes employ a single tuning parameter and conserve general properties inherent to all ADC methods such as hermiticity and size-consistency.The first approach, scaled-matrix ADC[(2)+x(3)], scales all contributions first occurring in ADC(3) equally and leads to an improvement of the accuracy of excitation energies compared to ADC(3) for x=0.4-0.5. However, with respect to excited state dipole moments this method provides lower accuracy than ADC(3). The second scaling approach, MP[(1)+x(2)]-ISR(3), scales the second order contributions of the ground-state wavefunction and derives a rigorous ADC scheme via the intermediate state representation (ISR) formalism. Although the error in excitation energies is not improved, this method provides insight into the relevance of the individual terms of the ADC(3) matrix and indicates, that the MP(2) wavefunction is indeed the optimal reference wavefunction for deriving a third-order single-reference ADC scheme.

Assessing the Performances of CASPT2 and NEVPT2 for Vertical Excitation Energies

Methods able to simultaneously account for both static and dynamic electron correlations have often been employed, not only to model photochemical events but also to provide reference values for vertical transition energies, hence allowing benchmarking of lower-order models. In this category, both the complete-active-space second-order perturbation theory (CASPT2) and the N-electron valence state second-order perturbation theory (NEVPT2) are certainly popular, the latter presenting the advantage of not requiring the application of the empirical ionization-potential-electron-affinity (IPEA) and level shifts. However, the actual accuracy of these multiconfigurational approaches is not settled yet. In this context, to assess the performances of these approaches, the present work relies on highly accurate (±0.03 eV) aug-cc-pVTZ vertical transition energies for 284 excited states of diverse character (174 singlet, 110 triplet, 206 valence, 78 Rydberg, 78 n → π*, 119 π → π*, and 9 double excitations) determined in 35 small- to medium-sized organic molecules containing from three to six non-hydrogen atoms. The CASPT2 calculations are performed with and without IPEA shift and compared to the partially contracted (PC) and strongly contracted (SC) variants of NEVPT2. We find that both CASPT2 with IPEA shift and PC-NEVPT2 provide fairly reliable vertical transition energy estimates, with slight overestimations and mean absolute errors of 0.11 and 0.13 eV, respectively. These values are found to be rather uniform for the various subgroups of transitions. The present work completes our previous benchmarks focused on single-reference wave function methods ( J. Chem. Theory Comput. 2018, 14, 4360; J. Chem. Theory Comput. 2020, 16, 1711), hence allowing for a fair comparison between various families of electronic structure methods. In particular, we show that ADC(2), CCSD, and CASPT2 deliver similar accuracies for excited states with a dominant single-excitation character.

A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Bicyclic Systems

Pursuing our efforts to define highly accurate estimates of the relative energies of excited states in organic molecules, we investigate, with coupled-cluster methods including iterative triples (CC3 and CCSDT), the vertical excitation energies of 10 bicyclic molecules (azulene, benzoxadiazole, benzothiadiazole, diketopyrrolopyrrole, furofuran, phthalazine, pyrrolopyrrole, quinoxaline, tetrathiafulvalene, and thienothiophene). In total, we provide aug-cc-pVTZ reference vertical excitation energies for 91 excited states of these relatively large systems. We use these reference values to benchmark various wave function methods, i.e., CIS(D), EOM-MP2, CC2, CCSD, STEOM-CCSD, CCSD(T)(a)*, CCSDR(3), CCSDT-3, ADC(2), ADC(2.5), and ADC(3), as well as some spin-scaled variants of both CC2 and ADC(2). These results are compared to those obtained previously on smaller molecules. It turns out that while the accuracy of some methods is almost unaffected by system size, e.g., CIS(D) and CC3, the performance of others can significantly deteriorate as the systems grow, e.g., EOM-MP2 and CCSD, whereas others, e.g., ADC(2) and CC2, become more accurate for larger derivatives.

Triangular boron carbon nitrides: an unexplored family of chromophores with unique properties for photocatalysis and optoelectronics

It has recently been shown that cycl[3.3.3]azine and heptazine (1,3,4,6,7,9,9b-heptaazaphenalene) as well as related azaphenalenes exhibit inverted singlet and triplet states, that is, the energy of the lowest singlet excited state (S1) is below the energy of the lowest triplet excited state (T1). This feature is unique among all known aromatic chromophores and is of outstanding relevance for applications in photocatalysis and organic optoelectronics. Heptazine is the building block of the polymeric material graphitic carbon nitride which is an extensively explored photocatalyst in hydrogen evolution photocatalysis. Derivatives of heptazine have also been identified as efficient emitters in organic light emitting diodes (OLEDs). In both areas, the inverted singlet-triplet gap of heptazine is a highly beneficial feature. In photocatalysis, the absence of a long-lived triplet state eliminates the activation of atmospheric oxygen, which is favourable for long-term operational stability. In optoelectronics, singlet-triplet inversion implies the possibility of 100% fluorescence efficiency of electron-hole recombination. However, the absorption and luminescence wavelengths of heptazine and the S1-S0 transition dipole moment are difficult to tune for optimal functionality. In this work, we employed high-level ab initio electronic structure theory to devise and characterize a large family of novel heteroaromatic chromophores, the triangular boron carbon nitrides. These novel heterocycles inherit essential spectroscopic features from heptazine, in particular the inverted singlet-triplet gap, while their absorption and luminescence spectra and transition dipole moments are widely tuneable. For applications in photocatalysis, the wavelength of the absorption maximum can be tuned to improve the overlap with the solar spectrum at the surface of earth. For applications in OLEDs, the colour of emission can be adjusted and the fluorescence yield can be enhanced.

High-resolution vacuum ultraviolet absorption spectra of 2,3- and 2,5-dihydrofuran

Using a synchrotron-based Fourier-transform spectrometer, the high-resolution absorption spectra of the C₁-symmetric 2,3-dihydrofuran (23DHF) and C_2v-symmetric 2,5-dihydrofuran (25DHF) have been measured from 5.5 eV to 9.4 eV with an absolute absorption cross section scale. Oscillator strengths and vertical excitation energies of the lowest 18 states have been computed using the average of the second- and third-order algebraic diagrammatic construction polarization propagator method and the equation-of-motion coupled-cluster method at the level of singles and doubles model. These show that the bright valence transitions of ππ^*-character are embedded into Rydberg transitions, whose oscillator strengths are at least one order of magnitude lower. To account for intensity borrowing, the first broad valence transition between 5.5 eV and 6.8 eV was simulated using a nuclear ensemble, and the agreement between experiment and theory is excellent. Whereas 23DHF only exhibits one broad valence transition followed by d/f Rydberg series converging to the ionization energy, the absorption spectrum of 25DHF has four bands, attributed to a valence nπσ → π^*-transition, nπσ → 3p_x,z/3d_xz transitions, a second valence nπ → π^*-transition followed by d/f Rydberg series converging to the ionization energy, respectively. All Rydberg series converging to the ionization energy have been characterized in terms of their quantum defects.

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.

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.

VUV excited-state dynamics of cyclic ethers as a function of ring size

The ultrafast vacuum ultraviolet (VUV) Rydberg state dynamics of cyclic ethers reflects ring strain.

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.