Combining spin-adapted open-shell TD-DFT with spin–orbit coupling

Restricted open-shell time-dependent density functional theory with perturbative spin-orbit coupling

When using quantum chemical methods to study electronically excited states of open-shell molecules, it is often beneficial to start with wave functions that are spin eigenfunctions. For excited states of molecules containing heavy elements, spin-orbit coupling (SOC) is important and needs to be included as well. An efficient approach is to include SOC perturbatively on top of a restricted open-shell Kohn-Sham (ROKS) time-dependent density functional theory, which can be combined with the Tamm-Dancoff approximation (TDA) to suppress numerical instabilities. We implemented and assessed the potential of such a ROKS-TDA-SOC method, also featuring the possibility of calculating transition dipole moments between states to allow for full spectrum simulation. Our study shows that the ROKS-TDA-SOC formalism yields a clear and easy-to-use method to obtain electronically excited states of open-shell molecules that are of moderate size and contain heavy elements.

Brightening triplet excitons enable high-performance white-light emission in organic small molecules via integrating n-π*/π-π* transitions

Luminescent materials that simultaneously embody bright singlet and triplet excitons hold great potential in optoelectronics, signage, and information encryption. However, achieving high-performance white-light emission is severely hampered by their inherent unbalanced contribution of fluorescence and phosphorescence. Herein, we address this challenge by pressure treatment engineering via the hydrogen bonding cooperativity effect to realize the mixture of n-π*/π-π* transitions, where the triplet state emission was boosted from 7% to 40% in isophthalic acid (IPA). A superior white-light emission based on hybrid fluorescence and phosphorescence was harvested in pressure-treated IPA, and the photoluminescence quantum yield was increased to 75% from the initial 19% (blue-light emission). In-situ high-pressure IR spectra, X-ray diffraction, and neutron diffraction reveal continuous strengthening of the hydrogen bonds with the increase of pressure. Furthermore, this enhanced hydrogen bond is retained down to the ambient conditions after pressure treatment, awarding the targeted IPA efficient intersystem crossing for balanced singlet/triplet excitons population and resulting in efficient white-light emission. This work not only proposes a route for brightening triplet states in organic small molecules, but also regulates the ratio of singlet and triplet excitons to construct high-performance white-light emission.

Theoretical study on the photophysical properties of thiophene-fused-type BODIPY series molecules in fluorescence imaging and photodynamic therapy

As a class of photosensitizers (PSs) with dual functions of photodynamic therapy (PDT) and fluorescence imaging, the relationship between the structure and dual-function of thiophene-fused-type BODIPY dyes has not been studied in depth before. We found that the thiophene-fused-type BODIPY triplet photosensitizer is produced according to the energy level matching rule and the introduction of the thiophene ring significantly reduces the energy gap ΔEST between singlet and triplet states, as revealed by our investigation of the excited state structures and energies of thieno-fused BODIPY dyes. At the same time, a tiny ΔEST also results in a greatly enhanced intersystem crossing (ISC) rate, kISC. The kISC value of MeO-BODIPY, having the highest singlet oxygen quantum yield (ΦΔ), is the largest. Substitution with a strong electron donor N,N-dimethylaminophenyl (DMA) leads to the vertical configuration in the T1 state. The small ΔE (0.0029 eV) between the HOMO and HOMO-1 triggers the photo induced electron transfer (PET) of inhibiting ISC and fluorescence. When thieno-fused BODIPYs react with pyrrole, the increase of π-conjugation and smaller ΔEHOMO-LUMO explain the redshift in emission wavelength of thieno-pyrrole-fused BODIPY. The more planar configuration of the S1 state and the stronger oscillator intensity reflect a higher fluorescence quantum yield (ΦF). The extension of π-conjugation can cause molecules to transition to higher-level singlet excited states (Sn states, n ≥ 1) after absorbing energy and reduce the energy level of the excited state, resulting in multiple channels and favoring 1O2 production for thieno-pyrrole-fused BODIPYs with electron-withdrawing groups at the para-position of the phenyl groups. Due to ΔES0-T1 < 0.980 eV, the substitution of electron-donating groups cannot produce 1O2. In this work, we have revealed the mechanism of ISC and the fluorescence emission process in the thiophene-fused-type BODIPY dye, which has provided a theoretical foundation and guidance for the future design of BODIPY-based heavy-atom-free PSs for molecular applications in PDT.

Combining Functional Units to Design Organic Materials with Dynamic Room-Temperature Phosphorescence under Continuous Ultraviolet Irradiation

Developing materials with dynamic room-temperature phosphorescence (RTP) properties is crucial for expanding the applications of organic light-emitting materials. In this study, we designed and synthesized two novel RTP molecules by combining functional units, incorporating the folded unit thianthrene into the classic luminescent cores thioxanthone or anthraquinone to construct TASO and TA2O. In this combination, the TA unit contributes to the enhancement of spin-orbit coupling (SOC), while the luminescent core governs the triplet energy level. After the strategic manipulation of SOC using the thianthrene unit, the target molecules exhibited a remarkable enhancement in RTP performance. This strategy led to the successful development of TASO and TA2O molecules with outstanding dynamic RTP properties when exposed to continuous ultraviolet irradiation, a result that can be ascribed to their efficient RTP, improved absorption ability, and oxygen-sensitive RTP properties. Leveraging the oxygen-mediated ultraviolet-radiation-induced RTP enhancement in TASO-doped polymer films, we developed a novel time-resolved detection technique for identifying phase separation in polymers with varying oxygen permeability. This research offers a promising approach for constructing materials with dynamic RTP properties.

Glutathione-responsive Aggregation-induced Emission Photosensitizers for Enhanced Photodynamic Therapy of Lung Cancer

Lung cancer, a highly prevalent and lethal form of cancer, is often associated with oxidative stress. Photodynamic therapy (PDT) has emerged as a promising alternative therapeutic tool in cancer treatments, but its efficacy is closely correlated to the photosensitizers generating reactive oxygen species (ROS) and the antioxidant capacity of tumor cells. In particular, glutathione (GSH) can reduce the ROS and thus compromise PDT efficacy. In this study, a GSH-responsive near-infrared photosensitizer (TBPPN) based on aggregation-induced emission for real-time monitoring of GSH levels and enhanced PDT for lung cancer treatment is developed. The strategic design of TBPPN, consisting of a donor-acceptor structure and incorporation of dinitrobenzene, enables dual functionality by not only the fluorescence being activated by GSH but also depleting GSH to enhance the cytotoxic effect of PDT. TBPPN demonstrates synergistic PDT efficacy in vitro against A549 lung cancer cells by specifically targeting different cellular compartments and depleting intracellular GSH. In vivo studies further confirm that TBPPN can effectively inhibit tumor growth in a mouse model with lung cancer, highlighting its potential as an integrated agent for the diagnosis and treatment of lung cancer. This approach enhances the effectiveness of PDT for lung cancer and deserves further exploration of its potential for clinical application.

Boosting organic phosphorescence in adaptive host-guest materials by hyperconjugation

Phosphorescence is ubiquitous in heavy atom-containing organic phosphors, which attracts considerable attention in optoelectronics and bioelectronics. However, heavy atom-free organic materials with efficient phosphorescence are rare under ambient conditions. Herein, we report a series of adaptive host-guest materials derived from dibenzo-heterocyclic analogues, showing host-dependent color-tunable phosphorescence with phosphorescence efficiency of up to 98.9%. The adaptive structural deformation of the guests arises from the hyperconjugation, namely the n→π* interaction, enabling them to inhabit the cavity of host crystals in synergy with steric effects. Consequently, a perfect conformation match between host and guest molecules facilitates the suppression of triplet exciton dissipation, thereby boosting the phosphorescence of these adaptive materials. Moreover, we extend this strategy to a ternary host-guest system, yielding both excitation- and time-dependent phosphorescence with a phosphorescence efficiency of 92.0%. This principle provides a concise way for obtaining efficient and color-tunable phosphorescence, making a major step toward potential applications in optoelectronics.

Comparison of Variational and Perturbative Spin-Orbit Coupling within Two-Component CASSCF

The modeling of spin-orbit coupling (SOC) remains a challenge in computational chemistry due to the high computational cost. With the rising popularity of spin-driven processes and f-block metals in chemistry and materials science, it is incumbent on the community to develop accurate multiconfigurational SOC methods that scale to large systems and understand the limits of different treatments of SOC. Herein, we introduce an implementation of perturbative SOC in scalar-relativistic two-component CASSCF (srX2C-CASSCF-SO). Perspectives on the limitations and accuracy of srX2C-CASSCF-SO are presented via benchmark calculations.

Diradical-Based Strategy in Designing Narrowband Thermally Activated Delayed Fluorescence Molecules with Tunable Emission Wavelengths

In the design of thermally activated delayed fluorescence (TADF) materials, narrow-band emission is of particular importance for the development of organic light-emitting diodes (OLEDs). In this work, we proposed a new strategy for designing TADF molecules utilizing degenerate nonbonding (NB) orbitals of diradical parent molecules, and these designed molecules are termed NB-TADF molecules. Based on this strategy, a series of NB-TADF molecules is finely designed and systematically studied by theoretical calculations. Taking advantage of the nonbonding properties, these NB-TADF molecules exhibit desirable narrowband emissions and high quantum yields. More importantly, the emission bands can be easily tuned from blue to near-infrared by changing the conjugate length of the parent group in the NB-TADF molecules. We hope that this new strategy can open a new door for the design of novel TADF materials.

Unified construction of relativistic Hamiltonians

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Toward the Rational Design of Organic Catalysts for Organocatalysed Atom Transfer Radical Polymerisation

Thanks to their diversity, organic photocatalysts (PCs) have been widely used in manufacturing polymeric products with well-defined molecular weights, block sequences, and architectures. Still, however, more universal property-performance relationships are needed to enable the rational design of such PCs. That is, a set of unique descriptors ought to be identified to represent key properties of the PCs relevant for polymerisation. Previously, the redox potentials of excited PCs (PC*) were used as a good descriptor for characterising very structurally similar PCs. However, it fails to elucidate PCs with diverse chromophore cores and ligands, among which those used for polymerisation are a good representative. As showcased by model systems of organocatalysed atom transfer radical polymerisation (O-ATRP), new universal descriptors accounting for additional factors, such as the binding and density overlap between the PC* and initiator, are proposed and proved to be successful in elucidating the experimental performances of PCs in polymerisation. While O-ATRP is exemplified here, the approach adopted is general for studying other photocatalytic systems.

When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin

Open-shell molecules rarely fluoresce, due to their typically faster non-radiative relaxation rates compared to closed-shell ones. Even rarer is the fluorescence from states that have two more unpaired electrons than the open-shell ground state, since they involve excitations from closed-shell orbitals to vacant-shell orbitals, which are typically higher in energy compared to excitations from or out of open-shell orbitals. States that are dominated by the former type of excitations are known as tripdoublet states when they can be described as a triplet excitation antiferromagnetically coupled to a doublet state, and their description by unrestricted single-reference methods (e.g., U-TDDFT) is notoriously inaccurate due to large spin contamination. In this work, we applied our spin-adapted TDDFT method, X-TDDFT, and the efficient and accurate static-dynamic-static second order perturbation theory (SDSPT2), to the study of the excited states as well as their relaxation pathways of copper(II) porphyrin; previous experimental works suggested that the photoluminescence of some substituted copper(II) porphyrins originate from a tripdoublet state, formed by a triplet ligand π → π* excitation antiferromagnetically coupled with the unpaired d electron. Our results demonstrated favorable agreement between the X-TDDFT, SDSPT2 and experimental excitation energies, and revealed noticeable improvements of X-TDDFT compared to U-TDDFT, not only for vertical excitation energies but also for adiabatic energy differences. These suggest that X-TDDFT is a reliable tool for the study of tripdoublet state fluorescence. Intriguingly, we showed that the aforementioned tripdoublet state is only slightly above the lowest doublet excited state and lies only slightly higher than the lowest quartet state, which suggests that the tripdoublet of copper(II) porphyrin is long-lived enough to fluoresce due to a lack of efficient non-radiative relaxation pathways; an explanation for this unusual state ordering is given. Indeed, thermal vibration correlation function (TVCF)-based calculations of internal conversion, intersystem crossing, and radiative transition rates confirm that copper(II) porphyrin emits thermally activated delayed fluorescence (TADF) and a small amount of phosphorescence at low temperature (83 K), in accordance with experiment. The present contribution is concluded by a few possible approaches of designing new molecules that fluoresce from tripdoublet states.

Low-Pressure Sensitive Piezochromic Fluorescence Switching of Tetraphenylethylene-Anthraquinone

Sensing of low-pressure signals is of great importance for cutting-edge technologies. Organic piezochromic molecules offer a promising library of pressure sensitive materials which can be tailor-designed toward specific requirements. However, very few examples of low-pressure sensitive piezochromic fluorescent molecules have been obtained till date, and the underlying mechanisms are still in its infancy. Herein, we report highly sensitive piezochromic fluorescent switching under low-pressure regimes (∼60 kPa) of tetraphenylethylene-anthraquinone (TPE-AQ) based on the controlled molecular design and polymorphic phase strategy. The influence of both intramolecular conformation effect and variations of intermolecular stacking modes on the piezochromic property of TPE-AQ is investigated. The underlying mechanism of the low-pressure sensitive piezochromic fluorescence switching is demonstrated to be closely related to the loosely packed molecular orientation, as confirmed by X-ray diffraction measurements combined with simulations. This work provides a way to design highly efficient pressure sensors based on organic molecular systems.

Smart Reversible Transformations between Chiral Superstructures of Copper Clusters for Optical and Chiroptical Switching

Superstructures made from nanoscale clusters with new collective properties are promising in high-tech applications; however, chiral superstructures remain elusive, and the limited intercluster coupling effect at room temperature hampers the tailoring of collective properties. Here, we show that from chiral monomeric copper clusters to two enantiomeric pairs of supercrystals with distinct phases, the absorption band edge red-shifts by over 1.3 eV, with photoluminescence and circularly polarized phosphorescence from visible (572 nm) to near-infrared (NIR, 858 nm). These supercrystals with high NIR quantum yields of up to 45% at room temperature are prototyped for night-vision imaging. In response to solvent and temperature stimuli, chiral supercrystal-to-supercrystal transformations occurred, concomitant with high-contrast optical/chiroptical switching. In situ single-crystal X-ray diffraction (SCXRD), steady-state and time-resolved optical spectroscopy, and response experiments combined with theoretical calculations demonstrate that distance-sensitive intercluster orbital interactions contribute to the exceptional collective optical responses. Such chiral supercrystals built from subnanoscale metal clusters with novel collective chiroptical responses would be useful in the fields of information storage and NIR optical devices.

State Interaction Linear Response Time-Dependent Density Functional Theory with Perturbative Spin-Orbit Coupling: Benchmark and Perspectives

Spin-orbit coupling (SOC) is an important driving force in photochemistry. In this work, we develop a perturbative spin-orbit coupling method within the linear response time-dependent density function theory framework (TDDFT-SO). A full state interaction scheme, including singlet-triplet and triplet-triplet coupling, is introduced to describe not only the coupling between the ground and excited states, but also between excited states with all couplings between spin microstates. In addition, expressions to compute spectral oscillator strengths are presented. Scalar relativity is included variationally using the second-order Douglas-Kroll-Hess Hamiltonian, and the TDDFT-SO method is validated against variational SOC relativistic methods for atomic, diatomic, and transition metal complexes to determine the range of applicability and potential limitations. To demonstrate the robustness of TDDFT-SO for large-scale chemical systems, the UV-Vis spectrum of Au₂₅(SR)₁₈ ^- is computed and compared to experiment. Perspectives on the limitation, accuracy, and capability of perturbative TDDFT-SO are presented via analyses of benchmark calculations. Additionally, an open-source Python software package (PyTDDFT-SO) is developed and released to interface with the Gaussian 16 quantum chemistry software package to perform this calculation.

Evaluation of molecular photophysical and photochemical properties using linear response time-dependent density functional theory with classical embedding: Successes and challenges

Time-dependent density functional theory (TDDFT) based approaches have been developed in recent years to model the excited-state properties and transition processes of the molecules in the gas-phase and in a condensed medium, such as in a solution and protein microenvironment or near semiconductor and metal surfaces. In the latter case, usually, classical embedding models have been adopted to account for the molecular environmental effects, leading to the multi-scale approaches of TDDFT/polarizable continuum model (PCM) and TDDFT/molecular mechanics (MM), where a molecular system of interest is designated as the quantum mechanical region and treated with TDDFT, while the environment is usually described using either a PCM or (non-polarizable or polarizable) MM force fields. In this Perspective, we briefly review these TDDFT-related multi-scale models with a specific emphasis on the implementation of analytical energy derivatives, such as the energy gradient and Hessian, the nonadiabatic coupling, the spin-orbit coupling, and the transition dipole moment as well as their nuclear derivatives for various radiative and radiativeless transition processes among electronic states. Three variations of the TDDFT method, the Tamm-Dancoff approximation to TDDFT, spin-flip DFT, and spin-adiabatic TDDFT, are discussed. Moreover, using a model system (pyridine-Ag20 complex), we emphasize that caution is needed to properly account for system-environment interactions within the TDDFT/MM models. Specifically, one should appropriately damp the electrostatic embedding potential from MM atoms and carefully tune the van der Waals interaction potential between the system and the environment. We also highlight the lack of proper treatment of charge transfer between the quantum mechanics and MM regions as well as the need for accelerated TDDFT modelings and interpretability, which calls for new method developments.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Folding-Induced Spin-Orbit Coupling Enhancement for Efficient Pure Organic Room-Temperature Phosphorescence

For the direct luminescence of triplet excitons, different mechanisms have been proposed for realizing pure organic room-temperature phosphorescence (RTP). To further verify the mechanism of folding-induced spin-orbit coupling (SOC) enhancement, two analogues of thianthrene (TA) were introduced by gradually replacing the sulfur atom with an oxygen atom for a systematical comparison, corresponding to phenoxathiine (PX) and dioxins (DX) molecules with increasing folding dihedral angles (or decreasing degrees of folding). Photophysical measurements show an obviously enhanced RTP efficiency from DX and PX to TA, which is consistent with their greatly enhanced SOC with a decrease in folding dihedral angle. The folding angle-dependent SOC calculations for each molecule reveal that this enhanced RTP is dominated by folding-induced SOC enhancement, in contrast with the negligible heavy-atom effect from oxygen to sulfur. This work further validates the rationality of the folding-induced SOC enhancement mechanism, which provides an innovative molecular design strategy for developing efficient pure organic RTP materials using folding structures.

Efficient Room-Temperature Phosphorescence from Discrete Molecules Based on Thianthrene Derivatives for Oxygen Sensing and Detection

In this work, two thianthrene (TA) derivatives, 1-phenylthianthrene (TA1P) and 2-phenylthianthrene (TA2P), were synthesized with single-phenyl modification for pure organic discrete-molecule room-temperature phosphorescence (RTP). They both show the dual emission of fluorescence and RTP in amorphous polymer matrix after deoxygenation, as a result of a new mechanism of folding-induced spin-orbit coupling (SOC) enhancement. Compared with TA1P, TA2P exhibits a higher RTP efficiency and a larger spectral separation between fluorescence and RTP, which is ascribed to the substituent effect of TA at the 2-position. With decreasing oxygen concentration from 1.61% to 0%, the discrete-molecule TA2P shows an about 18-fold increase in RTP intensity and an almost constant fluorescence intensity, which can make TA2P as a self-reference ratiometric optical oxygen sensing probe at low oxygen concentrations. The oxygen quenching constant (K_SV) of TA2P is estimated as high as 10.22 KPa⁻¹ for polymethyl methacrylate (PMMA)-doped film, and even reach up to 111.86 KPa⁻¹ for Zeonex^®-doped film, which demonstrates a very high sensitivity in oxygen sensing and detection. This work provides a new idea to design pure organic discrete-molecule RTP materials with high efficiency, and TA derivatives show a potential to be applied in quantitative detection of oxygen as a new-generation optical oxygen-sensing material.

New Light on an Old Story: Breaking Kasha's Rule in Phosphorescence Mechanism of Organic Boron Compounds and Molecule Design

In this work, the phosphorescence mechanism of (E)-3-(((4-nitrophenyl)imino)methyl)-2H-thiochroman-4-olate-BF2 compound (S-BF2) is investigated theoretically. The phosphorescence of S-BF2 has been reassigned to the second triplet state (T2) by the density matrix renormalization group (DMRG) method combined with the multi-configurational pair density functional theory (MCPDFT) to approach the limit of theoretical accuracy. The calculated radiative and non-radiative rate constants support the breakdown of Kasha's rule further. Our conclusion contradicts previous reports that phosphorescence comes from the first triplet state (T1). Based on the revised phosphorescence mechanism, we have purposefully designed some novel compounds in theory to enhance the phosphorescence efficiency from T2 by replacing substitute groups in S-BF2. Overall, both S-BF2 and newly designed high-efficiency molecules exhibit anti-Kasha T2 phosphorescence instead of the conventional T1 emission. This work provides a useful guidance for future design of high-efficiency green-emitting phosphors.

Theoretical Insight Into the Ultralong Room-Temperature Phosphorescence of Nonplanar Aromatic Hydrocarbon

Purely aromatic hydrocarbon materials with ultralong room-temperature phosphorescence (RTP) were reported recently, but which is universally recognized as unobservable. To reveal the inherent luminescent mechanism, two compounds, i.e., PT with a faint RTP and HD with strong RTP featured by nonplanar geometry, were chosen as a prototype to study their excited-state electronic structures by using quantum mechanics/molecular mechanics (QM/MM) model. It is demonstrated that the nonplanar ethylene brides can offer σ-electron to strengthen spin-orbit coupling (SOC) between singlet and triplet excited states, which can not only promote intersystem crossing (ISC) of S₁→T_n to increase the population of triplet excitons, but also accelerate the radiative decay rate of T₁→S₀, and thus improving RTP. Impressively, the nonradiative decay rate only has a small increase, owing to the synergistic effect between the increase of SOC and the reduction of reorganization energy of T₁→S₀ caused by the restricted torsional motions of aromatic rings. Therefore, a bright and long-lived RTP was obtained in aromatic hydrocarbon materials with twisted structure. This work provided a new insight into the ultralong RTP in pure organic materials.

NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

First-order nonadiabatic coupling (NAC) matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically nonadiabatic processes that are ubiquitous in molecular physics and chemistry. Given the large size of systems of chemical interests, time-dependent density functional theory (TDDFT) is usually the first choice of methods. However, the lack of many-electron wave functions in TDDFT renders the formulation of NAC-TDDFT for fo-NACMEs conceptually difficult. Because of this, various variants of NAC-TDDFT have been proposed in the literature from different standing points, including the Hellmann-Feynman-like expression and auxiliary/pseudo-wave function (AWF)-, equation-of-motion (EOM)-, and time-dependent perturbation theory (TDPT)-based formulations. Based on critical analyses, the following conclusions are made here: (1) The Hellmann-Feynman-like expression, which is rooted in exact wave function theory, is hardly useful due to huge demand on basis sets. (2) Although most popular, the AWF variants of NAC-TDDFT are not theoretically founded and become ambiguous particularly for the fo-NACMEs between two excited states, although they do agree with the EOM and TDPT variants under the Tamm-Dancoff approximation. (3) The TDPT variant of NAC-TDDFT is theoretically most rigorous but suffers from numerical instabilities on the one hand and does not differ to a significant extent from the EOM variant on the other hand. (4) As such, the EOM variant of NAC-TDDFT for the fo-NACMEs between the ground and excited states and between two excited states is solely the right choice in practice. These formal analyses are fully supported by numerical experimentations, taking azulene as a showcase. The proper implementation of the EOM variant of NAC-TDDFT is also highlighted, showing that the fo-NACMEs between the ground and excited states and between two excited states are computationally very much the same as the analytic energy gradients of DFT and TDDFT, respectively. Possible future developments of the EOM variant of NAC-TDDFT are also highlighted. Its extensions to spin-adapted open-shell TDDFT and proper treatment of spin-orbit couplings (which are another source of force for electronically nonadiabatic processes) are particularly warranted in the near future.

Dynamic-then-Static Approach for Core Excitations of Open-Shell Molecules

Delta self-consistent-field methods are widely used in studies of electronically excited states. However, the nonaufbau determinants are generally spin-contaminated. Here, we describe a general approach for spin-coupling interactions of open-shell molecules, making use of multistate density functional theory (MSDFT). In particular, the effective exchange integrals that determine spin coupling are obtained by enforcing the multiplet degeneracy of the S+1 state in the M_S = S manifold. Consequently, they are consistent with the energy of the high-spin state that is adequately treated by Kohn-Sham density functional theory (DFT) and, thereby, free of double counting of correlation. The method was applied to core excitations of open-shell molecules and compared with those by spin-adapted time-dependent DFT. An excellent agreement with experiment was found employing the BLYP functional and aug-cc-pCVQZ basis set. Overall, MSDFT provides an effective combination of the strengths of DFT and wave function theory to achieve efficiency and accuracy.

Low-Lying Electronic States of the Nickel Dimer

The generalized Van Vleck second order multireference perturbation theory (GVVPT2) method was used to investigate the low-lying electronic states of Ni₂. Because the nickel atom has an excitation energy of only 0.025 eV to its first excited state (the least in the first row of transition elements), Ni₂ has a particularly large number of low-lying states. Full potential energy curves (PECs) of more than a dozen low-lying electronic states of Ni₂, resulting from the atomic combinations ³F₄ + ³F₄ and ³D₃ + ³D₃, were computed. In agreement with previous theoretical studies, we found the lowest lying states of Ni₂ to correlate with the ³D₃ + ³D₃ dissociation limit, and the holes in the d-subshells were in the subspace of delta orbitals (i.e., the so-dubbed δδ-states). In particular, the ground state was determined as X ¹Γ_g and had spectroscopic constants: bond length (R_e) = 2.26 Å, harmonic frequency (ω_e) = 276.0 cm⁻¹, and binding energy (D_e) = 1.75 eV; whereas the 1 ¹Σ_g⁺ excited state (with spectroscopic constants: R_e = 2.26 Å, ω_e = 276.8 cm⁻¹, and D_e = 1.75) of the ³D₃ + ³D₃ dissociation channel lay at only 16.4 cm⁻¹ (0.002 eV) above the ground state at the equilibrium geometry. Inclusion of scalar relativistic effects through the spin-free exact two component (sf-X2C) method reduced the bond lengths of both of these two states to 2.20 Å, and increased their binding energies to 1.95 eV and harmonic frequencies to 296.0 cm⁻¹ for X ¹Γ_g and 297.0 cm⁻¹ for 1 ¹Σ_g⁺. These values are in good agreement with experimental values of R_e = 2.1545 ± 0.0004 Å, ω_e = 280 ± 20 cm⁻¹, and D₀ = 2.042 ± 0.002 eV for the ground state. All states considered within the ³F₄ + ³F₄ dissociation channel proved to be energetically high-lying and van der Waals-like in nature. In contrast to most previous theoretical studies of Ni₂, full PECs of all considered electronic states of the molecule were produced.

Modeling Spin-Crossover Dynamics

In this article, we review nonadiabatic molecular dynamics (NAMD) methods for modeling spin-crossover transitions. First, we discuss different representations of electronic states employed in the grid-based and direct NAMD simulations. The nature of interstate couplings in different representations is highlighted, with the main focus on nonadiabatic and spin-orbit couplings. Second, we describe three NAMD methods that have been used to simulate spin-crossover dynamics, including trajectory surface hopping, ab initio multiple spawning, and multiconfiguration time-dependent Hartree. Some aspects of employing different electronic structure methods to obtain information about potential energy surfaces and interstate couplings for NAMD simulations are also discussed. Third, representative applications of NAMD to spin crossovers in molecular systems of different sizes and complexities are highlighted. Finally, we pose several fundamental questions related to spin-dependent processes. These questions should be possible to address with future methodological developments in NAMD.

Suppressing triplet decay in quinoidal singlet fission materials: the role of molecular planarity and rigidity

Singlet fission, in which one singlet exciton is split into two triplet excitons, provides the potential to exceed the Shockley-Queisser limit for the power conversion efficiencies of organic solar cells. However, the charge transfer from the triplet state is found to be slow in singlet fission materials, so suppression of the triplet decay is crucial for effective utilization of singlet fission. Here, we first investigated triplet decay for the singlet fission molecular materials of ThBF and TThBF, which are characteristic of twisted and flexible quinoidal backbones. It is found that these compounds show rapid nonradiative decay in the Franck-Condon region and through the T₁/S₀ crossing point. Interestingly, upon locking the backbone twist by methylene, the LThBF and LTThBF compounds exhibit much higher energy barriers from T₁ to the T₁/S₀ crossing point, vanishing spin-orbit couplings, and decreased reorganization energies due to the planar and rigid structures. Consequently, both the triplet decay pathways are effectively suppressed. Our work reveals the importance of molecular planarity and rigidity in suppressing triplet decay and will be very helpful for full utilization of singlet fission in organic photovoltaics.

Investigation of different photochemical reactions of avobenzone derivatives by ultrafast transient absorption spectroscopy

Avobenzone (AB) is one of the most widely used UVA sunscreens, and it is viewed as a model compound for studying the photoisomerization process. In recent years, Miranda and co-workers studied photophysical and photochemical reactions of several AB derivatives. However, there is still a gap in the data of these compounds in the ultrafast time region. To get a better understanding of the photophysical and photochemical reaction mechanisms, selected AB derivatives of AB-Me, AB-Pr, AB-Br and AB-Cl were studied using ultrafast transient absorption spectroscopy and density functional theory calculations in the present study. It is unravelled that alkylated substituted AB compounds of AB-Me and AB-Pr exhibit an efficient intersystem crossing with the generation of the corresponding triplet state species, which further leads to the Norrish type II reaction for AB-Pr. On the other hand, AB-Br and AB-Cl prefer photochemical reactions via the singlet state surface. Based on the DFT calculations, the spin-orbit coupling constant between the singlet and triplet states, the energy difference between the singlet and triplet states and the natural transition orbital separations of the studied AB compounds were found to be the leading reasons accounting for their corresponding photochemical activities via singlet and triplet states.

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

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

Efficient and Long-Lived Room-Temperature Organic Phosphorescence: Theoretical Descriptors for Molecular Designs

Room-temperature phosphorescence (RTP) with long afterglow from pure organic materials has attracted great attention for its potential applications in biological imaging, digital encryption, optoelectronic devices, and so on. Organic materials have been long considered to be nonphosphorescent owing to their weak molecular spin-orbit coupling and high sensitivity to temperature. However, recently, some purely organic compounds have demonstrated highly efficient RTP with long afterglow upon aggregation, while others fail. Namely, it remains a challenge to expound on the underlying mechanisms. In this study, we present the molecular descriptors to characterize the phosphorescence efficiency and lifetime. For a prototypical RTP system consisting of a carbonyl group and π-conjugated segments, the excited states can be regarded as an admixture of n → π* (with portion α) and π → π* (portion β). Starting from the phosphorescent process and El-Sayed rule, we deduced that (i) the intersystem crossing (ISC) rate of S₁ → T _n is mostly governed by the modification of the product of α and β and (ii) the ISC rate of T₁ → S₀ is determined by the β value of T₁. Thus, the descriptors (γ = α × β, β) can be employed to describe the RTP character of organic molecules. From hybrid quantum mechanics and molecular mechanics (QM/MM) calculations, we illustrated the relationships among the descriptors (γ, β), phosphorescence efficiency and lifetime, and spin-orbit coupling constants. We stressed that the large γ and β values are favorable for the strong and long-lived RTP in organic materials. Experiments have reported confirmations of these molecular design rules.

Relativistic time-dependent density functional theories

The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed.

Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems

We present a new wave function ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as [Formula: see text], where [Formula: see text] is the spin projector for total spin S and |Ψ_MPS^(N,M)⟩ is an MPS wave function with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin adaptation (i.e., to create eigenfunctions of Ŝ²) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ_MPS^(N,M)⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for nonsinglet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of |Ψ_MPS^(N,M)⟩ as its underlying state, the SP-MPS can be closely connected to broken-symmetry mean-field states. This allows one to straightforwardly generate the large number of broken-symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector [Formula: see text] naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin-projected theories. Unlike usual spin-projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ_MPS^(N,M)⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.

Combining the spin-separated exact two-component relativistic Hamiltonian with the equation-of-motion coupled-cluster method for the treatment of spin–orbit splittings of light and heavy elements

An accurate and efficient treatment of spin–orbit splittings has been achieved by combining the sf-X2C+soc-DKH1 Hamiltonian with the equation-of-motion coupled-cluster method.