Overcoming Low Orbital Overlap and Triplet Instability Problems in TDDFT

Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex

The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.

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)⋆ $$ {}^{\star } $$ , 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.

The Role of Aqueous Solvation on the Intersystem Crossing of Nitrophenols

The photochemistry of nitrophenols is a source of smog as nitrous acid is formed from their photolysis. Nevertheless, computational studies of the photochemistry of these widespread toxic molecules are scarce. In this work, the initial photodeactivation of ortho-nitrophenol and para-nitrophenol is modeled, both in gas phase and in aqueous solution to simulate atmospheric and aerosol environments. A large number of excited states, six for ortho-nitrophenol and 11 for para-nitrophenol, have been included and were all populated during the decay. Moreover, periodic time-dependent density functional theory (TDDFT) is used for both the explicitly included solvent and the solute. A comparison to periodic QM/MM (TDDFT/MM), with electrostatic embedding, is made, showing notable differences between the decays of solvated nitrophenols simulated with QM/MM and full (TD)DFT. A reduced intersystem crossing in aqueous solution could be observed thanks to the surface hopping approach using explicit, periodic TDDFT solvation including spin-orbit couplings.

Chiroptical activity of benzannulated N-heterocyclic carbene rhenium(I) tricarbonyl halide complexes: towards efficient circularly polarized luminescence emitters

The design of enantiomerically pure circularly polarized luminescent (CPL) emitters would enormously benefit from the accurate and in-depth interpretation of the chiroptical properties by means of jointly (chiroptical) photophysical measurements and state-of-the-art theoretical investigation. Herein, computed and experimental (chiro-)optical properties of a series of eight enantiopure phosphorescent rhenium(I) tricarbonyl complexes are systematically compared in terms of electronic circular dichroism (ECD) and CPL. The compounds have general formula fac-[ReX(CO)3(N^CNHC)], where N^CNHC is a pyridyl benzannulated N-heterocyclic carbene deriving from a (substituted) 2-(pyridin-2-yl)imidazo[1,5-a]pyridin-2-ium proligand and X = Cl, Br and I, and display structured red phosphorescence with long-lived (τ = 7.0-19.1 μs) excited-state lifetime and dissymmetry factors |gLum| up to 4 × 10-3. The mixing of the character of the lowest-lying emitting triplet excited state is finely modulated between ligand centred (3LC), metal-to-ligand charge transfer (3MLCT) and halogen-to-ligand charge transfer (3XLCT) by the nature of the ancillary halogen and the chromophoric N^CNHC ligand. The study unravels the effect exerted by the nature of the excited state onto the ECD and CPL activity and will help to pave the way to construct efficient CPL emitters by chemical design.

Visualizing and characterizing excited states from time-dependent density functional theory

Time-dependent density functional theory (TD-DFT) is the most widely-used electronic structure method for excited states, due to a favorable combination of low cost and semi-quantitative accuracy in many contexts, even if there are well recognized limitations. This Perspective describes various ways in which excited states from TD-DFT calculations can be visualized and analyzed, both qualitatively and quantitatively. This includes not just orbitals and densities but also well-defined statistical measures of electron-hole separation and of Frenkel-type exciton delocalization. Emphasis is placed on mathematical connections between methods that have often been discussed separately. Particular attention is paid to charge-transfer diagnostics, which provide indicators of when TD-DFT may not be trustworthy due to its categorical failure to describe long-range electron transfer. Measures of exciton size and charge separation that are directly connected to the underlying transition density are recommended over more ad hoc metrics for quantifying charge-transfer character.

Introducing an orthogonally polarized electron-rich alkene: synthesis of a zwitterionic boron-containing π-conjugated system

The synthesis of an alkene is reported which is concurrently twisted (twist angle = 86.6(8)°), push-pull (dipole moment = 7.48 D), and electron-rich (E1/2 = -1.45 V and -0.52 V vs. Fc/Fc+) in nature, comprising a unique trinity combination for the alkene class of compounds. Subsequently, this newly synthesized alkene-motif was used as a donor for the synthesis of a zwitterionic boron-containing π-conjugated compound (dipole moment = 12.17 D) through an intramolecular charge transfer process exploiting the π-conjugated donor-acceptor system.

Computational Approaches for Organic Semiconductors: From Chemical and Physical Understanding to Predicting New Materials

While a complete understanding of organic semiconductor (OSC) design principles remains elusive, computational methods─ranging from techniques based in classical and quantum mechanics to more recent data-enabled models─can complement experimental observations and provide deep physicochemical insights into OSC structure-processing-property relationships, offering new capabilities for in silico OSC discovery and design. In this Review, we trace the evolution of these computational methods and their application to OSCs, beginning with early quantum-chemical methods to investigate resonance in benzene and building to recent machine-learning (ML) techniques and their application to ever more sophisticated OSC scientific and engineering challenges. Along the way, we highlight the limitations of the methods and how sophisticated physical and mathematical frameworks have been created to overcome those limitations. We illustrate applications of these methods to a range of specific challenges in OSCs derived from π-conjugated polymers and molecules, including predicting charge-carrier transport, modeling chain conformations and bulk morphology, estimating thermomechanical properties, and describing phonons and thermal transport, to name a few. Through these examples, we demonstrate how advances in computational methods accelerate the deployment of OSCsin wide-ranging technologies, such as organic photovoltaics (OPVs), organic light-emitting diodes (OLEDs), organic thermoelectrics, organic batteries, and organic (bio)sensors. We conclude by providing an outlook for the future development of computational techniques to discover and assess the properties of high-performing OSCs with greater accuracy.

Excited state potential energy surfaces of N-phenylpyrrole upon twisting: reference values and comparison between BSE/GW and TD-DFT

The puzzling case of the mixing between the charge transfer (CT) and local excited (LE) characters upon twisting of the geometry of N-phenylpyrrole (N-PP) is investigated considering the six low-lying singlet excited states (ES). The theoretical calculations of the potential energy surfaces (PES) have been performed for these states using a Coupled Cluster method accounting for the impact of the contributions from the triples, many-body Green's function GW and Bethe-Salpeter equation (BSE) formalisms, as well as Time-Dependent Density Functional Theory (TD-DFT) using various exchange-correlation functionals. Our findings confirm that the BSE formalism is more reliable than TD-DFT for close-lying ES with mixed CT/LE nature. More specifically, BSE/GW yields a more accurate evolution of the excited state PES than TD-DFT when compared to the reference coupled cluster values. BSE/GW PES curves also show negligible exchange-correlation functional starting point dependency in sharp contrast with their TD-DFT counterparts.

Efficient Narrowband Circularly Polarized Light Emitters Based on 1,4-B,N-embedded Rigid Donor-Acceptor Helicenes

Narrow-band emission is essential for applicable circularly polarized luminescence (CPL) active materials in ultrahigh-definition CP-OLEDs. One of the most promising classes of CPL active molecules, helicenes, however, typically exhibit broad emission with a large Stokes shift. We present, herein, a design strategy capitalizing on intramolecular donor-acceptor interactions between nitrogen and boron atoms to address this issue. 1,4-B,N-embedded configurationally stable single- and double helicenes were synthesized straightforwardly. Both helicenes show unprecedentedly narrow fluorescence and CPL bands (full width at half maximum between 17-28 nm, 0.07-0.13 eV) along with high fluorescence quantum yields (72-85 %). Quantum chemical calculations revealed that the relative localization of the natural transition orbitals, mainly on the rigid core of the molecule, and small values of root-mean-square displacements between S₀ and S₁ state geometries, contribute to the narrower emission.

Design and Mechanism of Rare-Earth Singlet Oxygen Sensing: An Experimental and Quantum Chemical Approach

The sensitive detection of singlet oxygen (¹O₂) is one key issue in various photochemical analyses, reactions, and processes; it is indispensable for designing catalysts for photodynamic therapies. Corresponding fluorescence-based organic ¹O₂ monitor luminophores may be equipped with rare-earth complexes with several intrinsic advantages. The design of the necessary ligands being a tedious, time-consuming effort, often involving empirical guesswork, we decided to support our experimental work with quantum chemical calculations. Hence, next to the experimental core, this paper suggests the additional use of time-dependent density functional theory (TDDFT) on suitable, free β-diketonate ligands to devise corresponding Eu³⁺ complexes as ¹O₂ probes eventually; the free ligand calculations obviously allow profoundly reduced computational efforts. Novel β-diketonate-substituted dimethyl anthracene complexes of Eu³⁺, Tb³⁺, and Gd³⁺ and their endoperoxidized descendants were thus synthesized, compared to known related complexes and analyzed with regard to their electronic characteristics; in addition, spectroscopy of a Eu³⁺ complex with ancillary epoxiphenanthroline for subsequent attachment to biological substrates featuring -NH₂ or -SH groups was included. The spectroscopic determination of the decisive lowest triplet (T₁) states of the Gd complexes could be matched by the Tamm-Dancoff approximation (TDA)/TDDFT calculations on the free ligands satisfactorily if suitable functionals were applied. Most significantly, the results suffice to describe the luminescence "switch-on" mechanism of this complex in the presence of ¹O₂.

Efficient implementation of time-dependent auxiliary density functional theory

The random phase approximation of time-dependent auxiliary density functional theory (TDADFT) is rederived from auxiliary density perturbation theory. Our exhaustive validation of TDADFT reveals an upshift of the excitation energies by ∼0.1 eV with respect to standard time-dependent density functional theory. For the computationally efficient implementation of TDADFT, floating point operation optimized three-center electron repulsion integral recurrence relations and their double asymptotic expansions are implemented into the Davidson solver. The computational efficiency of TDADFT is benchmarked with four sets of molecules comprising alkanes, fullerenes, DNA fragments, and zeolites. The results show that TDADFT has a computational scaling between 1.3 and 1.9 with respect to the number of basis functions, which is lower than the scaling of standard time-dependent density functional theory. Due to its computational simplifications, TDADFT is particularly well suited for Born-Oppenheimer molecular dynamics simulations. As illustrative examples, we present the temperature effects on the gas-phase absorption spectra of benzene, naphthalene, and anthracene.

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S₁ → S₀ transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 10⁶-10¹⁶ s^-1 compared with an experimental rate of the order 10⁷ s^-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

TDDFT versus GW/BSE Methods for Prediction of Light Absorption and Emission in a TADF Emitter

Design concepts for organic light emitting diode (OLED) emitters, which exhibit thermally activated delayed fluorescence (TADF) and thereby achieve quantum yields exceeding 25%, depend on singlet-triplet splitting energies of order kT to allow reverse intersystem crossing at ambient temperatures. Simulation methods for these systems must be able to treat relatively large organic molecules, as well as predict their excited state energies, transition energies, singlet-triplet splittings, and absorption and emission cross sections with reasonable accuracy, in order to prove useful in the design process. Here we compare predictions of TDDFT with M06-2X and ωB97X-D exchange-correlation functionals and a G_oW_o@HF/BSE method for these quantities in the well-studied DPTZ-DBTO2 TADF emitter molecule. Geometry optimization is performed for ground state (GS) and lowest donor-acceptor charge transfer (CT) state for each functional. Optical absorption and emission cross sections and energies are calculated at these geometries. Relaxation energies are on the order of 0.5 eV, and the importance of obtaining excited state equilibrium geometries in predicting delayed fluorescence is demonstrated. There are clear trends in predictions of G_oW_o@HF/BSE, and TDDFT/ωB97X-D and M06-2X methods in which the former method favors local exciton (LE) states while the latter favors DA CT states and ωB97X-D makes intermediate predictions. G_oW_o@HF/BSE suffers from triplet instability for LE states but not CT states relevant for TADF. Shifts in HOMO and LUMO levels on adding a conductor-like polarizable continuum model dielectric background are used to estimate changes in excitation energies on going from the gas phase to a solvated molecule.

Either Accurate Singlet-Triplet Gaps or Excited-State Structures: Testing and Understanding the Performance of TD-DFT for TADF Emitters

The energy gap between the lowest singlet and triplet excited states (ΔE_ST) is a key property of thermally activated delayed fluorescence (TADF) emitters, where these states are dominated by charge-transfer (CT) character. Despite its well-known shortcomings concerning CT states, time-dependent density functional theory (TD-DFT) is widely used to predict this gap and study TADF. Moreover, polar CT states exhibit a strong interaction with their molecular environment, which further complicates their computational description. Addressing these two major challenges, this work studies the performance of Tamm-Dancoff-approximated TD-DFT (TDA-DFT) on the recent STGABS27 benchmark set,¹ exploring different strategies to include orbital and structural relaxation, as well as dielectric embedding. The results show that the best-performing strategy is to calculate ΔE_ST at the ground-state structure using functionals with a surprisingly small amount of Fock exchange of ≈10% and without a (complete) solvent model. However, as this approach heavily relies on error cancellation to mimic dielectric relaxation, it is not robust and exhibits large systematic deviations in excited state energies, state characters, and structures. More rigorous approaches, including state-specific solvation, do not share these systematic deviations, but their predicted ΔE_ST values exhibit larger statistical errors. We thus conclude that for the description of CT states in dielectric environments, none of the tested TDA-DFT methods is competitive with the recently presented ROKS/PCM approach regarding robustness, accuracy, and computational efficiency.

Photoprocesses in Derivatives of 1,4- and 1,3-Diazadistyryldibenzenes

Photoprocesses in 1,4-diazadistyrylbenzene (1) and 1,3-diazadistyrylbenzene derivative (2) diperchlorates in MeCN were studied by absorption, luminescence, and kinetic laser spectroscopies. For compound 1, trans-cis-photoisomerization and intersystem crossing to a triplet state are observed. For compound 2, photoelectrocyclization is suggested. Quantum chemical calculations of diazadistyrylbenzene structures in the ground and excited states were carried out. The schemes for photoreactions were proposed.

Fluorescence vs. Phosphorescence: Which Scenario Is Preferable in Au(I) Complexes with Benzothiadiazoles?

The photoluminescence of Au(I) complexes is generally characterized by long radiative lifetimes owing to the large spin-orbital coupling constant of the Au(I) ion. Herein, we report three brightly emissive Au(I) coordination compounds, 1, 2a, and 2b, that reveal unexpectedly short emission lifetimes of 10-20 ns. Polymorphs 2a and 2b exclusively exhibit fluorescence, which is quite rare for Au(I) compounds, while compound 1 reveals fluorescence as the major radiative pathway, and a minor contribution of a microsecond-scale component. The fluorescent behaviour for 1-2 is rationalized by means of quantum chemical (TD)-DFT calculations, which reveal the following: (1) S₀-S₁ and S₀-T₁ transitions mainly exhibit an intraligand nature. (2) The calculated spin-orbital coupling (SOC) between the states is small, which is a consequence of overall small metal contribution to the frontier orbitals. (3) The T₁ state features much lower energy than the S₁ state (by ca. 7000 cm^-1), which hinders the SOC between the states. Thus, the S₁ state decays in the form of fluorescence, rather than couples with T₁. In the specific case of complex 1, the potential energy surfaces for the S₁ and T₂ states intersect, while the vibrationally resolved S₁-S₀ and T₂-S₀ calculated radiative transitions show substantial overlap. Thus, the microsecond-scale component for complex 1 can stem from the coupling between the S₁ and T₂ states.

An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe–Salpeter equation calculations of molecules

The ab initio GW plus Bethe–Salpeter equation (GW-BSE, where G is the one particle Green's function and W is the screened Coulomb interaction) approach has emerged as a leading method for predicting excitations in both solids and molecules with a predictive power contingent upon several factors. Among these factors are the (1) generalized Kohn–Sham eigensystem used to construct the GW self-energy and to solve the BSE and (2) the efficacy and suitability of the Tamm–Dancoff approximation. Here, we present a detailed benchmark study of low-lying singlet excitations from a generalized Kohn–Sham (gKS) starting point based on an optimally tuned range-separated hybrid (OTRSH) functional. We show that the use of this gKS starting point with one-shot G0W0 and G0W0-BSE leads to the lowest mean absolute errors (MAEs) and mean signed errors (MSEs), with respect to high-accuracy reference values, demonstrated in the literature thus far for the ionization potentials of the GW100 benchmark set and for low-lying neutral excitations of Thiel’s set molecules in the gas phase, without the need for self-consistency. The MSEs and MAEs of one-shot G0W0-BSE@OTRSH excitation energies are comparable to or lower than those obtained with other functional starting points after self-consistency. Additionally, we compare these results with linear-response time-dependent density functional theory (TDDFT) calculations and find GW-BSE to be superior to TDDFT when calculations are based on the same exchange-correlation functional. This work demonstrates tuned range-separated hybrids used in combination with GW and GW-BSE can greatly suppress starting point dependence for molecules, leading to accuracy similar to that for higher-order wavefunction-based theories for molecules without the need for costlier iterations to self-consistency.

Towards Optimized Photoluminescent Copper(I) Phenanthroline-Functionalized Complexes: Control of the Photophysics by Symmetry-Breaking and Spin–Orbit Coupling

The electronic and structural alterations induced by the functionalization of the 1,10-phenanthroline (phen) ligand in [Cu(I) (phen-R₂)₂]⁺ complexes (R=H, CH₃, tertio-butyl, alkyl-linkers) and their consequences on the luminescence properties and thermally activated delay fluorescence (TADF) activity are investigated using the density functional theory (DFT) and its time-dependent (TD) extension. It is shown that highly symmetric molecules with several potentially emissive nearly-degenerate conformers are not promising because of low S₁/S₀ oscillator strengths together with limited or no S₁/T₁ spin–orbit coupling (SOC). Furthermore, steric hindrance, which prevents the flattening of the complex upon irradiation, is a factor of instability. Alternatively, linking the phenanthroline ligands offers the possibility to block the flattening while maintaining remarkable photophysical properties. We propose here two promising complexes, with appropriate symmetry and enough rigidity to warrant stability in standard solvents. This original study paves the way for the supramolecular design of new emissive devices.

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.

Can domain-based local pair natural orbitals approaches accurately predict phosphorescence energies?

Since the discovery of the peculiar conducting and optical properties of aromatics, many efforts have been made to characterize and predict their phosphorescence. This physical process is exploited in modern Organic Emitting Light Diodes (OLEDs), and it is also one of the processes decreasing the efficiency of Dye-sensitized solar cells (DSSCs). Herein, we propose a computational strategy for the accurate calculation of singlet-triplet gaps of aromatic compounds, which provides results that are in excellent agreement with available experimental data. Our approach relies on the domain-based local pair natural orbital (DLPNO) variant of the "gold standard" CCSD(T) method. The convergence of our results with respect to the key technical parameters of the calculation, such as the basis set used, the approximations employed in the perturbative triples correction, and the dimension of the PNOs space, was thoroughly discussed.

Assessing Recent Time-Dependent Double-Hybrid Density Functionals on Doublet-Doublet Excitations

This work is the first thorough investigation of time-dependent double-hybrid density functionals (DHDFs) for the calculation of doublet-doublet excitation energies. It sheds light on the current state-of-the-art techniques in the field and clarifies if there is still room for future improvements. Overall, 29 hybrid functionals and DHDFs are investigated. We separately analyze the individual impacts of the Tamm-Dancoff approximation (TDA), range separation, and spin-component/opposite scaling (SCS/SOS) on 45 doublet-doublet excitations in 23 radicals before concluding with an overarching analysis that includes and excludes challenging excitations with double-excitation or multireference character. Our results show again that so-called "nonempirical" DHDFs are outperformed by semiempirical ones. While the best assessed functionals are DHDFs, some of the worst are also DHDFs and outperformed by all assessed hybrids. SCS/SOS is particularly beneficial for range-separated DHDFs. Spin-scaled, range-separated DHDFs paired with the TDA belong to the best tested methods here, and we particularly highlight SCS-ωB2GP-PLYP, SOS-ωB2PLYP, SOS-ωB2GP-PLYP, SOS-ωB88PP86, SOS-RSX-QIDH, and SOS-ωPBEPP86. When comparing our functional rankings with previous studies on singlet-singlet and singlet-triplet excitations, we recommend TDA-SOS-ωB88PP86 and TDA-SOS-ωPBEPP86 as robust methods for excitation energies in general until further improvements have been achieved that surpass the chemical accuracy threshold for challenging open-shell excitations without increasing the computational effort.

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.

Combining Localized Orbital Scaling Correction and Bethe-Salpeter Equation for Accurate Excitation Energies

We applied localized orbital scaling correction (LOSC) in Bethe-Salpeter equation (BSE) to predict accurate excitation energies for molecules. LOSC systematically eliminates the delocalization error in the density functional approximation and is capable of approximating quasiparticle (QP) energies with accuracy similar or better than the GW Green's function approach and with much less computational cost. The QP energies from LOSC instead of commonly used G0 W0 and ev GW are directly used in BSE. We show that the BSE/LOSC approach greatly outperforms the commonly used BSE/ G0W0 approach for predicting excitations with different characters. For the calculations for Truhlar-Gagliardi test set containing valence, charge transfer (CT) and Rydberg excitations, BSE/LOSC with the Tamm-Dancoff approximation provides a comparable accuracy to time-dependent density functional theory (TDDFT) and BSE/ev GW. For the calculations of Stein CT test set and Rydberg excitations of atoms, BSE/LOSC considerably outperforms both BSE/ G0W0 and TDDFT approaches with a reduced starting point dependence. BSE/LOSC is thus a promising and efficient approach to calculate excitation energies for molecular systems.

Charge-Transfer Excitations within Density Functional Theory: How Accurate Are the Most Recommended Approaches?

The performance of the most recent density functionals is assessed for charge-transfer (CT) excitations using comprehensive intra- and intermolecular CT benchmark sets with high-quality reference values. For this comparison, the state-of-the-art range-separated (RS) and long-range-corrected (LC) double hybrid (DH) approaches are selected, and global DH and LC hybrid functionals are also inspected. The correct long-range behavior of the exchange–correlation (XC) energy is extensively studied, and various CT descriptors are compared as well. Our results show that the most robust performance is attained by RS-PBE-P86/SOS-ADC(2), as it is suitable to describe both types of CT excitations with outstanding accuracy. Furthermore, concerning the intramolecular transitions, unexpectedly excellent results are obtained for most of the global DHs, but their limitations are also demonstrated for bimolecular complexes. Despite the outstanding performance of the LC-DH methods for common intramolecular excitations, serious deficiencies are pointed out for intermolecular CT transitions, and the wrong long-range behavior of the XC energy is revealed. The application of LC hybrids to such transitions is not recommended in any respect.

Electronic excitations through the prism of mean-field decomposition techniques

The potential of mean-field decomposition techniques in interpreting electronic transitions in molecules is explored, in particular, the usefulness of these for offering computational signatures of different classes of such excitations. When viewed as a conceptual lens for this purpose, decomposed results are presented for ground- and excited-state energies and dipole moments of selected prototypical organic dyes, and the discrete nature of these properties as well as how they change upon transitioning from one state to another is analyzed without recourse to a discussion based on the involved molecular orbitals. On the basis of results obtained both with and without an account of continuum solvation, our work is further intended to shed new light on practical and pathological differences in between various functional approximations in orbital-optimized Kohn-Sham density functional theory for excited states, equipping practitioners and developers in the field with new probes and possible validation tools.

Synthesis and photophysical properties of linear gold(I) complexes based on a CCC carbene

The reaction between allenylpyridine (L1) and (Me₂S)AuCl resulted in the quantitative formation of the (Indolizy)gold chloride complex 1 (Indolizy = indolizin-2-ylidene). The reaction of 1 with carbazole in the presence of KO^tBu affords the corresponding (Indolizy)Au(Cz) complex 2. Both compounds show high air- and temperature stability. The crystal structure of 2 confirmed the linear co-planar geometry. Complex 1 shows an intense low energy absorption of mixed character in the UV-vis spectrum, ascribed to intraligand and (M + Hal)L charge transfer processes, and exhibits bright yellow phosphorescence with an excited state lifetime of 62.8 μs in the crystal and a luminescence quantum yield up to 65%. On the other hand, the carbazolate complex 2 in a polystyrene matrix shows bright red delayed fluorescence at 617 nm with a sub-microsecond excited state lifetime and a quantum yield of 21.6%.

Assessment of hybrid functionals for singlet and triplet excitations: Why do some local hybrid functionals perform so well for triplet excitation energies?

The performance of various hybrid density functionals is assessed for 105 singlet and 105 corresponding triplet vertical excitation energies from the QUEST database. The overall lowest mean absolute error is obtained with the local hybrid (LH) functional LH12ct-SsirPW92 with individual errors of 0.11 eV (0.11 eV) for singlet (triplet) n → π* excitations and 0.29 eV (0.17 eV) for π → π* excitations. This is slightly better than with the overall best performing global hybrid M06-2X [n → π*: 0.13 eV (0.17 eV), π → π*: 0.30 eV (0.20 eV)], while most other global and range-separated hybrids and some LHs suffer from the "triplet problem" of time-dependent density functional theory. This is exemplified by correlating the errors for singlet and triplet excitations on a state-by-state basis. The excellent performance of LHs based on a common local mixing function, i.e., an LMF constructed from the spin-summed rather than the spin-resolved semilocal quantities, is systematically investigated by the introduction of a spin-channel interpolation scheme that allows us to continuously modulate the fraction of opposite-spin terms used in the LMF. The correlation of triplet and singlet errors is systematically improved for the n → π* excitations when larger fractions of the opposite-spin-channel are used in the LMF, whereas this effect is limited for the π → π* excitations. This strongly supports a previously made hypothesis that attributes the excellent performance of LHs based on a common LMF to cross-spin-channel nondynamical correlation terms.

Are all charge-transfer parameters created equally? A study of functional dependence and excited-state charge-transfer quantification across two dye families

Small molecule organic dyes have many potential uses in medicine, textiles, forensics, and light-harvesting technology. Being able to computationally predict the spectroscopic properties of these dyes could greatly expedite screening efforts, saving time and materials. Time-dependent density functional theory (TD-DFT) has been shown to be a good tool for this in many instances, but characterizing electronic excitations with charge-transfer (CT) character has historically been challenging and can be highly sensitive to the chosen exchange-correlation functional. Here we present a combined experimental and computational study of the excited-state electronic structure of twenty organic dyes obtained from the Max Weaver Dye Library at NCSU. Results of UV-vis spectra calculations on these dyes with six different exchange-correlation functionals, BP86, B3LYP, PBE0, M06, BH and HLYP, and CAM-B3LYP, were compared against their measured UV-vis spectra. It was found that hybrid functionals with modest amounts (20-30%) of included Hartree-Fock exchange are the most effective at matching the experimentally determined λ_max. The interplay between the observed error, the functional chosen, and the degree of CT was analyzed by quantifying the CT character of λ_max using four orbital and density-based metrics, Λ, Δr, S_C and D_CT, as well as the change in the dipole moment, Δμ. The results showed that the relationship between CT character and the functional dependence of error is not straightforward, with the observed behavior being dependent both on how CT was quantified and the functional groups present in the molecules themselves. It is concluded that this may be a result of the examined excitations having intermediate CT character. Ultimately it was found that the nature of the molecular "family" influenced how a given functional behaved as a function of CT character, with only two of the examined CT quantification methods, Δr and D_CT, showing consistent behavior between the different molecular families. This suggests that further work needs to be done to ensure that currently used CT quantification methods show the same general trends across large sets of multiple dye families.

Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation Energies

Following the work on spin-component and spin-opposite scaled (SCS/SOS) global double hybrids for singlet-singlet excitations by Schwabe and Goerigk [ J. Chem. Theory Comput. 2017, 13, 4307-4323] and our own works on new long-range corrected (LC) double hybrids for singlet-singlet and singlet-triplet excitations [ J. Chem. Theory Comput. 2019, 15, 4735-4744 and J. Chem. Phys. 2020, 153, 064106], we present new LC double hybrids with SCS/SOS that demonstrate further improvement over previously published results and methods. We introduce new unscaled and scaled versions of different global and LC double hybrids based on Becke88 or PBE exchange combined with LYP, PBE, or P86 correlation. For singlet-singlet excitations, we cross-validate them on six benchmark sets that cover small to medium-sized chromophores with different excitation types (local-valence, Rydberg, and charge transfer). For singlet-triplet excitations, we perform the cross-validation on three different benchmark sets following the same analysis as in our previous work in 2020. In total, 203 excitations are analyzed. Our results confirm and extend those of Schwabe and Goerigk regarding the superior performance of SCS and SOS variants compared to their unscaled parents by decreasing mean absolute deviations, root-mean-square deviations, or error spans by more than half and bringing absolute mean deviations closer to zero. Our SCS/SOS variants are shown to be highly efficient and robust for the computation of vertical excitation energies, which even outperform specialized double hybrids that also contain an LC in their perturbative part. In particular, our new SCS/SOS-ωPBEPP86 and SCS/SOS-ωB88PP86 functionals are four of the most accurate and robust methods tested in this work, and we fully recommend them for future applications. However, if the relevant SCS and SOS algorithms are not available to the user, we suggest ωPBEPP86 as the best unscaled method in this work.

Reliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2'-bipyridine)-Metal Complexes

An efficient computational protocol for the prediction of vibrationally resolved phosphorescence spectra is developed and validated for five tris(2,2'-bipyridine)-metal complexes ([M(bpy)₃]ⁿ⁺, where M = Zn, Ru, Rh, Os, Ir). The outstanding feature of this protocol is the use of full linear-response time-dependent density functional theory (TDDFT) for the excited-state triplet calculation, i.e., the commonly seen strategies employing the Tamm-Dancoff approximation (TDA) or unrestricted density functional theory (DFT) calculations for the T₁ state are not needed. This is achieved by the use of a local hybrid functional (LH12ct-SsirPW92) that features a real-space dependent admixture of exact exchange governed by a local mixing function. The excellent performance of this LH for triplet excitation energies known from previous studies transfers to a remarkable mean absolute error of 0.06 eV for the phosphorescence 0-0 energies investigated herein, while the popular B3PW91 functional gives an error of 0.27 eV in TDDFT and 0.09 eV in unrestricted DFT calculations, respectively. The advantages of the local hybrid are particularly apparent for excited states with a mixed-valence character. The influence of spin-orbit coupling was found to be significant for [Os(bpy)₃]²⁺ red-shifting the 0-0 energy for phosphorescence by 0.17 eV, while the effect is negligible for the other complexes (<0.03 eV). The influence of the basis-set and integration-grid sizes is evaluated, and a computationally lighter protocol is validated that leads to drastic savings in computation time with negligible loss in accuracy.

Excited-state dynamics of [Mn(im)(CO)3(phen)]+: PhotoCORM, catalyst, luminescent probe?

Mn(I) α-diimine carbonyl complexes have shown promise in the development of luminescent CO release materials (photoCORMs) for diagnostic and medical applications due to their ability to balance the energy of the low-lying metal-to-ligand charge transfer (MLCT) and metal-centered (MC) states. In this work, the excited state dynamics of [Mn(im)(CO)₃(phen)]⁺ (im = imidazole; phen = 1,10-phenanthroline) is investigated by means of wavepacket propagation on the potential energy surfaces associated with the 11 low-lying S_n singlet excited states within a vibronic coupling model in a (quasi)-diabatic representation including 16 nuclear degrees of freedom. The results show that the early time photophysics (<400 fs) is controlled by the interaction between two MC dissociative states, namely, S₅ and S₁₁, with the lowest S₁-S₃ MLCT bound states. In particular, the presence of S₁/S₅ and S₂/S₁₁ crossings within the diabatic picture along the Mn-CO_axial dissociative coordinate (q_Mn-COaxial) favors a two-stepwise population of the dissociative states, at about 60-70 fs (S₁₁) and 160-180 fs (S₅), which reaches about 10% within 200 fs. The one-dimensional reduced densities associated with the dissociative states along q_Mn-COaxial as a function of time clearly point to concurrent primary processes, namely, CO release vs entrapping into the S₁ and S₂ potential wells of the lowest luminescent MLCT states within 400 fs, characteristics of luminescent photoCORM.

TD-DFT and Experimental Methods for Unraveling the Energy Distribution of Charge-Transfer Triplet/Singlet States of a TADF Molecule in a Frozen Matrix

Reverse intersystem crossing (RISC) rate of a thermally activated delayed fluorescence (TADF) molecule is sensitive to the energy alignment of the singlet charge-transfer state (¹CT), triplet charge-transfer state (³CT), and locally excited triplet state (³LE). However, the energy distribution of the charge-transfer states originating from the conformational distribution of TADF molecules in a solid matrix inevitably generated during the preparation of a solid sample due to the rotatable donor-acceptor linkage is rarely considered. Moreover, the investigation of the energy distribution of the ³CT state is both theoretically and experimentally difficult due to the triplet instabilities of time-dependent density functional (TD-DFT) calculations and difficulties in phosphorescence measurements, respectively. As a result, the relationships between conformational distribution, configurations of excited state transition orbitals, and excited state energies/dynamics have not been clearly explained. In this work, we determined the energy distribution of CT states of the TADF emitter TPSA in frozen toluene at 77 K by the measurement of time-resolved spectra in the full time range (1 ns to 30 s) of emission including prompt fluorescence, TADF, ³CT phosphorescence, and ³LE phosphorescence. We obtained the energy band of CT states where ¹CT and ³CT states are distributed in the range of 2.85-3.00 and 2.64-2.96 eV, respectively. We tested various global hybrid and long-range corrected functionals for the TD-DFT calculation of ³CT energy of TPSA and found that only the M11 functional shows consistent results without triplet instability. We performed TD-DFT with the M11* functional optimized for a robust dihedral angle scan of ³CT states without triplet instability and reproduced the energy band structure obtained from the experiment. Through TD-DFT and experimental investigations, it is estimated that the dihedral angles of donor-acceptor (θ_D-A) and acceptor-linker (θ_A) of TPSA in frozen toluene lie within the range 70° ≤ θ_D-A ≤ 90° and 0° ≤ θ_A ≤ 30° respectively. Our results show that the dihedral angle distribution must be considered for further investigation of the photophysics of TADF molecules and the development of stable and efficient TADF emitters.

Vibrational Radiationless Transition from Triplet States of Chromophores at Room Temperature

The radiationless transition rate based on intramolecular vibrations from the lowest excited triplet state (T₁) at room temperature [k_nr(RT)] is crucial for triplet energy harvesting in optoelectronics and photonics applications. Although a decrease of k_nr(RT) of chromophores with strong intermolecular interactions is often proposed, scientific evidence for this has not been reported. Here we report a method to predict k_nr(RT). We optically estimated k_nr(RT) of various molecularly dispersed chromophores with a variety of transition characteristics from T₁ to the ground state (S₀) under appropriate inert liquid or solid host conditions. Spin-orbit coupling (SOC) without considering molecular vibrations was not correlated with the estimated k_nr(RT). However, the estimated k_nr(RT) was strongly correlated with a multiplication of SOC considering vibrations freely allowed at room temperature and the Franck-Condon factor. This correlation revealed that k_nr(RT) of many heavy-atom-free chromophores with a visible T₁-S₀ transition energy and local excited T₁-S₀ transition characteristics is intrinsically less than 10⁰ s^-1 even when vibrations freely occur. This information will assist researchers to appropriately design materials without limitations regarding intermolecular interactions to control T₁ lifetime at room temperature and facilitate triplet energy harvesting.

Effect of Perturbative Vibronic Correction for Weak Fluorescence in Thermally Activated Delayed Fluorescence Systems

Minimizing the energy difference between the lowest singlet (S₁) and the lowest triplet states, ΔE_ST, is the main strategy to design thermally activated delayed fluorescence (TADF) molecules, and spatially separating the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is the general method in the design. However, such a separation also tends to reduce the oscillator strength of the S₁ state. In real systems, vibrations change the S₁ oscillator strength, and thus one needs to consider the vibronic coupling toward searching for TADF candidate molecules. Here, we evaluate the importance of vibronic coupling by including the first-order perturbative correction to the transition dipole moments of carbazolyl-phthalonitrile derivatives. Indeed, some molecules display large enhancements in their oscillator strengths, with their fluorescence lifetimes reduced by 2 orders of magnitude. The twisting mode between the carbazole groups and phthalonitrile is the most important mode in inducing the perturbations. Thus, performing the perturbative correction is crucial in attaining more reliable predictions on their fluorescence propensities. We also observe that some other molecules, whose zeroth-order predicted fluorescence rates are much slower than the actual experimental data, are affected little by the same first-order correction. For these molecules, we deduce that the geometry-dependent excited-state switching kicks in. Our results demonstrate the significance of vibronic coupling in TADF molecules and the importance of adopting correction schemes as the guidelines for screening of useful TADF molecules.

Organic room-temperature phosphorescence from halogen-bonded organic frameworks: hidden electronic effects in rigidified chromophores

Development of purely organic materials displaying room-temperature phosphorescence (RTP) will expand the toolbox of inorganic phosphors for imaging, sensing or display applications. While molecular solids were found to suppress non-radiative energy dissipation and make the RTP process kinetically favourable, such an effect should be enhanced by the presence of multivalent directional non-covalent interactions. Here we report phosphorescence of a series of fast triplet-forming tetraethyl naphthalene-1,4,5,8-tetracarboxylates. Various numbers of bromo substituents were introduced to modulate intermolecular halogen-bonding interactions. Bright RTP with quantum yields up to 20% was observed when the molecule is surrounded by a Br⋯O halogen-bonded network. Spectroscopic and computational analyses revealed that judicious heavy-atom positioning suppresses non-radiative relaxation and enhances intersystem crossing at the same time. The latter effect was found to be facilitated by the orbital angular momentum change, in addition to the conventional heavy-atom effect. Our results suggest the potential of multivalent non-covalent interactions for excited-state conformation and electronic control.

The number and position of halogen substituents in purely organic π–π* chromophores critically affect the efficiency of phosphorescence.

Energy transfer mechanism in luminescence Eu(III) and Tb(III) complexes of coumarin-3-carboxylic acid: A theoretical study

Excited state energy level diagrams of coumarin-3-carboxylic acid (HCCA) chromophore, Eu(CCA)Cl₂(H₂O)₂ (1), Eu(CCA)₂Cl(H₂O)₂ (2)_, Eu(CCA)₃(H₂O)₃ (3), Tb(CCA)₂Cl(H₂O) (4) and Tb(CCA)₂(NO₃)(H₂O) (5) in gas phase and polar solution have been calculated by means of DFT/TDDFT/ωB97XD methods. Based on these results, the ability of CCA to sensitize Eu(III) and Tb(III) luminescence has been examined. The competitive excited state processes in the complexes - fluorescence, intersystem crossing (ISC) and phosphorescence, were analyzed depending on the environment, number of the ligands, Ln(III) ion type (Eu and Tb) and counteranion (Cl^- and NO₃^-). It has been found that the environment altered the S₁ state energy, oscillator strength, fluorescence lifetime as well as the S₁ character - polar solution stabilized the S₁(ππ*) state, whereas non-polar solution (gas phase, solid state) stabilized the S₁(nπ*) state. The S₁(nπ*) state was decisive for the efficient energy transfer as it suppressed the S₁ emission of CCA and favored ISC or direct transfer to the emitting levels of Eu(III). The HCCA triplet (T₁) state minimum energy (~2.7, ~2.6^ZPE eV) and (ππ*) character were retained in Eu/Tb-CCA complexes regardless of the environment. The energy gap between the higher energy T₁ donor state and the acceptor levels ⁵D₁ of Eu(III) (~0.5 eV) and ⁵D₄ of Tb(III) (~0.1 eV) provided optimal resonance conditions for effective energy transfer for Eu(III), but less probability for Tb(III). The nonradiative energy (CCA → Eu(III)) transfer rates and quantum luminescence yield for 2 and 3 were calculated by a strategy combining DFT geometries, INDO/S excitation energies and calculated Judd-Ofelt parameters. The excitation channel T₁ → ⁵D₀ through an exchange mechanism was predicted as the most probable one to populate the main emissive Eu-centered state in complexes 2 and 3. The more efficient luminescence of 3 than that of 2 was discussed and explained.

In-Depth Ab Initio Study of Thermally Activated Delayed Fluorescence in 4,5-Di(9H-carbazol-9-yl)-phthalonitrile

Molecules capable of thermally activated delayed fluorescence (TADF) are promising as emitters in organic light-emitting devices. Processes leading to and competing with TADF in 4,5-di(9H-carbazol-9-yl)-phthalonitrile are analyzed in detail. It is demonstrated that the key features of an efficient TADF emitter include the presence of two triplet states of different natures with potential energy surfaces crossing between the T₁ and S₁ minima and a noticeable dependence of the S₁ → S₀ oscillator strength on molecular deformations from low-frequency antisymmetric vibrational modes. These conclusions can be useful in the targeted design of efficient TADF emitters.

Electrostatic Field-Induced Oscillator Strength Focusing in Molecules

The development of light-harvesting devices based on molecular materials depends critically on the ability to focus the electronic oscillator strength of molecules into the UV-vis spectral window. Typical molecular chromophores have only about 1% of their total electronic oscillator strength in this spectral region and thus perform at a small fraction of their possible effectiveness. This theoretical study finds that the electronic oscillator strength of polyenes in the UV-vis region may be enhanced by 1 order of magnitude using electrostatic fields, motivating specific experimental studies of oscillator strength focusing. We find scaling relationships between the polyene length, the intensity of the applied field, and the field-induced increase in oscillator strength that are useful for the implementation of light-harvesting strategies based on polyenes.