Spin-flip Time-dependent Density Functional Theory Research Articles

OpenQP: A Quantum Chemical Platform Featuring MRSF-TDDFT with an Emphasis on Open-Source Ecosystem.

The OpenQP (Open Quantum Platform) is a new open-source quantum chemistry library developed to tackle sustainability and interoperability challenges in the field of computational chemistry. OpenQP provides various popular quantum chemical theories as autonomous modules such as energy and gradient calculations of HF, DFT, TDDFT, SF-TDDFT, and MRSF-TDDFT, thereby allowing easy interconnection with third-party software. A scientifically notable feature is the innovative mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) and its customized exchange-correlation functionals such as the DTCAM series of VAEE, XI, XIV, AEE, and VEE, which significantly expand the applicability scope of DFT and TDDFT. OpenQP also supports parallel execution and is optimized with BLAS and LAPACK for high performance. Future enhancements such as extended Koopman's theorem (EKT)-MRSF-TDDFT and spin-orbit coupling (SOC)-MRSF-TDDFT will further expand OpenQP's capabilities. Additionally, a Python wrapper PyOQP is provided that performs tasks such as geometry optimization, conical intersection searches, and nonadiabatic coupling calculations, among others, by prototyping the modules of the OpenQP library in combination with third-party libraries. Overall, OpenQP aligns with modern trends in high-performance scientific software development by offering flexible prototyping and operation while retaining the performance benefits of compiled languages like Fortran and C. They enhance the sustainability and interoperability of quantum chemical software, making OpenQP a crucial platform for accelerating the development of advanced quantum theories like MRSF-TDDFT.

UMRSF-TDDFT: Unrestricted Mixed-Reference Spin-Flip-TDDFT.

An unrestricted version of Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (UMRSF-TDDFT) was developed based on unrestricted Kohn-Sham orbitals (UKS) with a new molecular orbital (MO) reordering scheme. Additionally, a simple yet accurate method for estimating ⟨S2⟩ expectation values was devised. UMRSF-TDDFT was benchmarked against cases where DFT, TDDFT, and SF-TDDFT traditionally fail to provide accurate descriptions. In an application to the ground and excited states of a Be atom, UMRSF-TDDFT successfully recovers the degenerate states, with its energies slightly reduced compared to its RO counterpart, due to the additional variational flexibility of UKS. A clear difference between UMRSF and U-SF-TDDFT is evident in the bond breaking of the hydrogen fluoride system, as the latter misses an important configuration. In the case of the Jahn-Teller distortion of trimethylenemethane (TMM), the relative singlet energy compared to the triplet is lower by 0.1 and 0.2 eV for UMRSF and U-SF-TDDFT, respectively, than that of MRSF-TDDFT. The reduction in UMRSF energy is attributed to spatial orbital relaxations, whereas the reduction in U-SF-TDDFT energy results from spin contamination. Overall, the additional orbital relaxations afforded by unrestricted Kohn-Sham (UKS) orbitals in UMRSF-TDDFT lead to lower total system energies compared to their restricted open-shell counterparts. This enhancement adds a practical and accurate quantum chemical theory to the existing RO variant for addressing challenging systems where traditional quantum theories suffer.

Towards the accurate simulation of multi-resonance emitters using mixed-reference spin-flip time-dependent density functional theory

Multi-resonant Thermally Activated Delayed Fluorescent (MR-TADF) materials have received significant research interest owing to their potential use as emitters in high-performance Organic Light Emitting Diodes (OLEDs). Despite their advantages, including narrow emission spectra leading to high colour purity, several challenges remain in optimising the performance of these materials. One key issue is the typically long delayed fluorescence lifetime which arises from a large gap and weak coupling between the lowest lying singlet and triplet states. To develop high-performing materials, in silico design is an important step and consequently it is crucial to develop and deploy computational methods that accurately model their excited state properties. Previous studies have highlighted the importance of double excitations, which are not accounted for within the framework of Linear Response Time-Dependent Density Functional Theory (LR-TDDFT), contributing to the poor performance of this method for these materials. Consequently, in this work, we employ Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) to calculate the properties of MR-TADF materials. Our findings indicate that this approach accurately predicts the excited state properties including the crucial ΔEST, the energy difference between the lowest singlet (S1) and triplet (T1) excited states. We further use this method to explore the excited state properties of systems designed to enhance the coupling between singlet and triplet states by increasing the density of states and enhancing spin–orbit coupling through metal perturbation. The results in this work sets the foundation for computationally efficient in silico development high-performing MR-TADF materials within the framework of MRSF-TDDFT.

Chemoexcited Formation and Radiationless Decay Dynamics of Firefly Chromophore.

Multistate nonadiabatic dynamics combined with Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) were performed to investigate the chemoexcitation dynamics of firefly dioxetanone (FDO- in S0) to oxyluciferin (OxyLH- in S1) and its subsequent decay dynamics. The formation of oxyluciferin occurs within approximately 100 fs and is primarily controlled by oscillatory CO2 decarboxylation. Unexpected radiationless decay from oxyluciferin was also observed, facilitated by intramolecular rotation. Simulations under three thermal conditions reveal that higher initial thermal energy not only enhances the formation of oxyluciferin but also increases radiationless decay by surpassing barriers to the ground state. Conversely, lower thermal energy conditions reduce oxyluciferin formation but suppress radiationless decay. These findings suggest that optimal conditions for higher chemiluminescence quantum yield involve initial high thermal energy to accelerate CO2 decarboxylation and gradual thermal dissipation to prevent intramolecular rotation of oxyluciferin. This approach could enhance the chemiluminescence quantum yield beyond the current limit of 40%, offering significant potential for applications in biological imaging and analytical chemistry.

The photochemical trans → cis and thermal cis → trans isomerization pathways of azobenzo-13-crown ether: A computational study on a strained cyclic azobenzene system.

The isomerization of azobenzo-13-crown ether can be expected to be hindered due to the polyoxyethylene linkage connecting the 2,2'-positions of azobenzene. The mixed reference spin-flip time-dependent density functional theory results reveal that the planar and rotational minima of the first photo-excited singlet state (S1) of the trans-isomer pass through a barrier (2.5-5.0 kcal/mol) as it goes toward the torsional conical intersection (S0/S1) geometry (<CNNC ≈ 98°), which is responsible for the cis isomer formation. The second excited singlet state (S2) of the trans form has a nearly planar minimum along the N-N stretching mode, which approaches a sloped S2/S1 intersection geometry. This excited state has a rotational minimum (<CNNC ≈ 99°) as well. Both these minima have a characteristic S2-S1 energy gap of 9 kcal/mol and may undergo internal conversion. A comparison of this system with an analogous 2,2'-dimethoxy-substituted azobenzene system reveals less strain in the rotational path of the latter on the S1 surface, indicating the possibility of its better trans → cis yield than the azocrown. The completely planar S2 geometry of the dimethoxy system has easy access to the linear concerted inversion path, which seems to be the reason behind its reported slightly lower π-π*(S2) yield than n-π*(S1). The thermal cis → trans isomerization path of the azobenzo-13-crown passes through a transition state (frequency 453i cm-1), which corresponds to Gibbs free energy of activation value of 26 kcal/mol in the gas-phase and isooctane. Our study also confirms that its trans isomer strongly binds Li+ among the alkali metal ions, and this observation may open up possibilities for practical applications of this azobenzo-crown.

Ultrafast Photoinduced Dynamics in 1,3-Cyclohexadiene: A Comparison of Trajectory Surface Hopping Schemes†.

Photoinduced nonadiabatic processes play a crucial role in a wide range of disciplines, from fundamental steps in biology to modern applications in advanced materials science. A theoretical understanding of these processes is highly desirable, and trajectory surface hopping (TSH) has proven to be a well-suited framework for a wide range of systems. In this work, we present a comprehensive comparison between two TSH algorithms, the conventional Tully's fewest switches surface hopping (FSSH) scheme and the Landau-Zener surface hopping (LZSH), to study the photoinduced ring-opening of 1,3-cyclohexadiene (CHD) to 1,3,5-hexatriene at the spin-flip time-dependent density functional theory (SF-TDDFT) level of theory. Additionally, we compare our results with a literature study at the extended multistate complete active space second-order perturbation theory method (XMS-CASPT2) level of theory. Our results show that the average population and lifetimes estimated with LZSH using SF-TDDFT are closer to the literature (using multireference methods) than those estimated with FSSH using SF-TDDFT. The latter speaks in favor of applying LZSH in combination with the SF-TDDFT method to study larger and more complex systems such as molecular photoswitches where the CHD molecule acts as a backbone. In addition, we present an implementation of Tully's FSSH algorithm as an extension to the PySurf software package.

Singlet Fission as the Gateway to Triplet Generation in Heavy Atom-Free Organic Molecules.

Triplet generations in heavy atom-free organic molecules are primarily revealed to proceed through singlet fissions (SFs) by investigating the contributions of SFs and intersystem crossings to the generation rates. The spin-flip long-range corrected time-dependent density functional theory calculations on 11 organic molecules known for triplet generation under photoirradiation are performed. The correlation between the descriptors for SF and the experimental singlet-to-triplet conversion rates strongly supports the predominance of SF progressions in all these molecules, corroborated by experimental observations of their triplet-triplet annihilations. Based on these findings, we propose updated conditions for SF progression: There is a high-absorption singlet state just above the triplet-triplet excitation of the chromophore dimer, or the singlet (triplet-triplet) excitation itself is responsible for photoabsorption. To the best of our knowledge, all organic molecules known for rapid triplet state generation fulfill these conditions.

Quantum-Chemistry Study of the Photophysical Properties of 4-Thiouracil and Comparisons with 2-Thiouracil.

DNA in living beings is constantly damaged by exogenous and endogenous agents. However, in some cases, DNA photodamage can have interesting applications, as it happens in photodynamic therapy. In this work, the current knowledge on the photophysics of 4-thiouracil has been extended by further quantum-chemistry studies to improve the agreement between theory and experiments, to better understand the differences with 2-thiouracil, and, last but not least, to verify its usefulness as a photosensitizer for photodynamic therapy. This study has been carried out by determining the most favorable deactivation paths of UV-vis photoexcited 4-thiouracil by means of the photochemical reaction path approach and an efficient combination of the complete-active-space second-order perturbation theory//complete-active-space self-consistent field (CASPT2//CASSCF), (CASPT2//CASPT2), time-dependent density functional theory (TDDFT), and spin-flip TDDFT (SF-TDDFT) methodologies. By comparing the data computed herein for both 4-thiouracil and 2-thiouracil, a rationale is provided on the relatively higher yields of intersystem crossing, triplet lifetime and singlet oxygen production of 4-thiouracil, and the relatively higher yield of phosphorescence of 2-thiouracil.

Survey of T1 and T2 Energies of Intramolecular Singlet Fission Chromophores

Singlet fission is a desired process in photovoltaics since it enhances photoelectric conversion efficiency. Intramolecular singlet fission is of special interest as the fission efficiency can be improved through tuning configurations between chromophore units that are covalently connected. However, intramolecular singlet fission chromophores feature a large tetraradical character, and may tend to dissatisfy the ET2&gt;2ET1 criterion for all singlet fission chromophores, intramolecular or not. We performed spin-flip time-dependent density functional theory calculations for a collection of representative intramolecular singlet fission chromophores to show that this is indeed the case.

Singlet fission initiating organic photosensitizations

The feasibility of singlet fission (SF) in organic photosensitizers is investigated through spin-flip long-range corrected time-dependent density functional theory. This study focuses on four major organic photosensitizer molecules: benzophenone, boron-dipyrromethene, methylene blue, and rose bengal. Calculations demonstrate that all these molecules possess moderate pi-stacking energies and closely-lying singlet (S) and quintet (triplet–triplet, TT) excitations, satisfying the essential conditions for SF: (1) Near-degenerate low-lying S and (TT) excitations with a significant S–T energy gap, and (2) Moderate pi-stacking energy of chromophores, slightly higher than solvation energy, enabling dissociation for triplet-state chromophore generation. Moreover, based on the El-Sayed rule, intersystem crossing is found to simultaneously proceed at very slow rates in all these photosensitizers. This is attributed to the fact that the lowest singlet excitation of the monomers partly involves npi ^* transitions alongside the main pi pi ^* transitions. The proposed mechanisms are strongly substantiated by comparisons with experimental studies.

Roles of Singlet Fission in the Photosensitization of Silicon Phthalocyanine.

The roles of singlet fission in the triplet generation of silicon phthalocyanine (SiPc), a compound analogous to the IRDye700DX photosensitizer used in near-infrared photoimmunotherapy, are investigated by considering the energetical relation between the excitations of this compound. These excitations are obtained through spin-flip long-range corrected time-dependent density functional theory calculations. To initiate singlet fission, chromophores must meet two conditions: (1) near-degenerate low-lying singlet and quintet (triplet-triplet) excitations with a considerable energy gap of the lowest singlet and triplet excited states and (2) moderate π-stacking energy of chromophores, which is higher than but not far from the solvation energy, to facilitate the dissociation and generation of triplet-state chromophores. The present calculations demonstrate that SiPc satisfies both of these conditions after the formation of π-stacking irrespective of the presence of an axial ligand(s), suggesting that singlet fission plays a crucial role in the triplet generation process, although intersystem crossing occurs simultaneously at a very slow rate.

Excited State Properties of Point Defects in Semiconductors and Insulators Investigated with Time-Dependent Density Functional Theory.

We present a formulation of spin-conserving and spin-flip hybrid time-dependent density functional theory (TDDFT), including the calculation of analytical forces, which allows for efficient calculations of excited state properties of solid-state systems with hundreds to thousands of atoms. We discuss an implementation on both GPU- and CPU-based architectures along with several acceleration techniques. We then apply our formulation to the study of several point defects in semiconductors and insulators, specifically the negatively charged nitrogen-vacancy and neutral silicon-vacancy centers in diamond, the neutral divacancy center in 4H silicon carbide, and the neutral oxygen-vacancy center in magnesium oxide. Our results highlight the importance of taking into account structural relaxations in excited states in order to interpret and predict optical absorption and emission mechanisms in spin defects.

Solvent Effects and pH Dependence of the X-ray Absorption Spectra of Proline from Electrostatic Embedding Quantum Mechanics/Molecular Mechanics and Mixed-Reference Spin-Flip Time-dependent Density-Functional Theory.

The accurate description of solvent effects on X-ray absorption spectra (XAS) is fundamental for comparing the simulated spectra with experiments in solution. Currently, few protocols exist that can efficiently reproduce the effects of the solute/solvent interactions on XAS. Here, we develop an efficient and accurate theoretical protocol for simulating the solvent effects on XAS. The protocol combines electrostatic embedding QM/MM based on electrostatic potential fitted operators for describing the solute/solvent interactions and mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) for simulating accurate XAS spectra. To demonstrate the capabilities of our protocol, we compute the X-ray absorption of neutral proline in the gas phase and ionic proline in water in all relevant K-edges, showing excellent agreement with experiments. We show that states represented by core to π* transitions are almost unaffected by the interaction with water, whereas the core to σ* transitions are more impacted by the fluctuation of proline structure and the electrostatic interaction with the solvent. Finally, we reconstruct the pH-dependent XAS of proline in solution, determining that the N K-edge can be used to distinguish its three protonation states.

Doubly Tuned Exchange-Correlation Functionals for Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory.

It is demonstrated that significant accuracy improvements in MRSF-TDDFT can be achieved by introducing two different exchange-correlation (XC) functionals for the reference Kohn-Sham DFT and the response part of the calculations, respectively. Accordingly, two new XC functionals of doubly tuned Coulomb attenuated method-vertical excitation energy (DTCAM-VEE) and DTCAM-AEE were developed on the basis of the "adaptive exact exchange (AEE)" concept in the framework of the Coulomb-attenuating XC functionals. The values by DTCAM-VEE are in excellent agreement with those of Thiel's set [mean absolute errors (MAEs) and the interquartile range (IQR) values of 0.218 and 0.327 eV, respectively]. On the other hand, DTCAM-AEE faithfully reproduced the qualitative aspects of conical intersections (CIs) of trans-butadiene and thymine and the nonadiabatic molecular dynamics (NAMD) simulations on thymine. The latter functional also remarkably exhibited the exact 1/R asymptotic behavior of the charge-transfer state of an ethylene-tetrafluoroethylene dimer and the accurate potential energy surfaces (PESs) along the two torsional angles of retinal protonated Schiff base model with six double bonds (rPSB6). Overall, DTCAM-AEE generally performs well, as its MAE (0.237) and IQR (0.41 eV) are much improved as compared to BH&HLYP. The current idea can also be applied to other XC functionals as well as other variants of linear response theories, opening a new way of developing XC functionals.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory: Multireference Advantages with the Practicality of Linear Response Theory.

The density functional theory (DFT) and linear response (LR) time-dependent (TD)-DFT are of the utmost importance for routine computations. However, the single reference formulation of DFT suffers in the description of open-shell singlet systems such as diradicals and bond-breaking. LR-TDDFT, on the other hand, finds difficulties in the modeling of conical intersections, doubly excited states, and core-level excitations. In this Perspective, we demonstrate that many of these limitations can be overcome by recently developed mixed-reference (MR) spin-flip (SF)-TDDFT, providing an alternative yet accurate route for such challenging situations. Empowered by the practicality of the LR formalism, it is anticipated that MRSF-TDDFT can become one of the major workhorses for general routine tasks.

Ultrafast Photocontrolled Rotation in a Molecular Motor Investigated by Machine Learning-Based Nonadiabatic Dynamics Simulations.

The thermal helix inversion (THI) of the overcrowded alkene-based molecular motors determines the speed of the unidirectional rotation due to the high reaction barrier in the ground state, in comparison with the ultrafast photoreaction process. Recently, a phosphine-based motor has achieved all-photochemical rotation experimentally, promising to be controlled without a thermal step. However, the mechanism of this photochemical reaction has not yet been fully revealed. The comprehensive computational studies on photoisomerization still resort to nonadiabatic molecular dynamics (NAMD) simulations based on electronic structure calculations, which remains a high computational cost for large systems such as molecular motors. Machine learning (ML) has become an accelerating tool in NAMD simulations recently, where excited-state potential energy surfaces (PESs) are constructed analytically with high accuracy, providing an efficient approach for simulations in photochemistry. Herein the reaction pathway is explored by a spin-flip time-dependent density functional theory (SF-TDDFT) approach in combination with ML-based NAMD simulations. According to our computational simulations, we notice that one of the key factors of fulfilling all-photochemical rotation in the phosphine-based motor is that the excitation energies of four isomers are similar. Additionally, a shortcut photoinduced transformation between unstable isomers replaces the THI step, which shares the conical intersection (CI) with photoisomerization. In this study, we provide a practical approach to speed up the NAMD simulations in photochemical reactions for a large system that could be extended to other complex systems.

High-efficient luminescence induced by the restriction of benzothiazole group torsion for the HBT-H-H molecule in the aggregate state

The aggregation-induced emission (AIE) effect has been demonstrated to have great potential application in different areas, from organic electronics to biomedical research and physical process monitoring. In general, molecules with AIE characteristic exhibit fluorescence enhancement in the aggregated state by restricting intramolecular motion consumption. The combination of AIE and excited-state intramolecular proton transfer (ESIPT) is meaningful for promoting luminescence. Recently, HBT-H-H molecule, as a derivative of 2-(2-Hydroxyphenyl)benzothiazole (HBT), has drawn extensive attention from researchers. The molecule possesses the intramolecular hydrogen bonding structure which has the potential for ESIPT. Moreover, the fluorescence quantum yield of HBT-H-H in the aggregation state is 35 times higher than that in Toluene. However, the interplay between excited state dynamics and the AIE effect for this molecule is not clear. Especially, how does AIE effect beat non-radiative transition channel by affecting motions of molecular structure. Herein, we investigated the excited state dynamics of HBT-H-H molecule by the spin-flip time-dependent density functional theory and QM/MM method. We found that the molecule relaxes to the conical intersection region through the twisting motion of the benzothiazole group in Toluene solvent. While the AIE effect effectively inhibits this process by preventing the torsion of benzothiazole group, which induces the emission enhancement. The interplay between the excited-state dynamics and AIE effect for the HBT-H-H molecule delineated in this work not only benefits the deep understanding of molecular behavior to the aggregate level, but also provides a guide for the synthesis of AIE materials with favorable performance.

Turn on the luminescence channel by the suppression of conical intersection process for the sulfur-containing hydrogen-bonded molecules

The sulfur-containing hydrogen bonds exhibit notable potential in gene and protein engineering. As an important research direction of sulfur-containing hydrogen bonding dynamics, the excited-state intramolecular proton transfer (ESIPT) reaction of the thiol proton has garnered significant interest. Recently, Chou et al. synthesized a series of sulfur-containing H-bonded molecules, inclusive of 3-thiolflavone (3 TF), its trifluoromethyl derivative (3FTF), and its diethylamino derivative (3NTF), thereby inaugurating a new chapter in the ESIPT reaction of the thiol proton. Nevertheless, many open issues cannot be answered solely by experimental means and call for a detailed theoretical investigation, especially the relationship between the molecular excited-state properties and different luminescence properties of the three molecules. Herein, we validate that the molecules undergo the ESIPT reaction of the thiol proton and identify the time scale through the time-dependent density functional theory (TDDFT) calculation. Moreover, employing the spin-flip TDDFT approach, the analysis of the non-radiative transition process reveals that difference in Gibbs free energy of activation (ΔG⧧) for the transition from Frank-Condon point to minimum energy conical intersection (MECI) point primarily accounts for the different luminescence properties of the three molecules. Besides, we also reveal the reason for the abnormal blue-shifted fluorescence of the 3NTF in the aggregated state by the ONIOM (TDDFT: UFF) strategy. These findings are favorable to understanding the luminescence mechanism of sulfur-containing H-bonded molecules and provide a reference for the synthesis and design of the luminescent materials based on the molecules with thiol proton transfer properties.

Unveiling controlling factors of the S0/S1 minimum-energy conical intersection (3): Frozen orbital analysis based on the spin-flip theory.

Conical intersections (CIs), which indicate the crossing of two or more adiabatic electronic states, are crucial in the mechanisms of photophysical, photochemical, and photobiological processes. Although various geometries and energy levels have been reported using quantum chemical calculations, the systematic interpretation of the minimum energy CI (MECI) geometries is unclear. A previous study [Nakai et al., J. Phys. Chem. A 122, 8905 (2018)] performed frozen orbital analysis (FZOA) based on time-dependent density functional theory (TDDFT) at the MECI formed between the ground and first electronic excited states (S0/S1 MECI), thereby inductively clarifying two controlling factors. However, one of the factors that the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) energy gap became close to the HOMO-LUMO Coulomb integral was not valid in the case of spin-flip TDDFT (SF-TDDFT), which is frequently used as a means of the geometry optimization of MECI [Inamori et al., J. Chem. Phys. 152, 144108 (2020)]. This study revisited the controlling factors using FZOA for the SF-TDDFT method. Based on spin-adopted configurations within a minimum active space, the S0-S1 excitation energy is approximately represented by the HOMO and LUMO energy gap ΔεHL, a contribution from Coulomb integrals JHL″ and that from the HOMO-LUMO exchange integral KHL″. Furthermore, numerical applications of the revised formula at the SF-TDDFT method confirmed the control factors of S0/S1 MECI.

Carbon Condensation via [4 + 2] Cycloaddition of Highly Unsaturated Carbon Chains.

We present computational studies of reaction pathways for alkyne/polyyne dimerization that represent plausible early steps in mechanisms for carbon condensation. A previous computational study of the ring coalescence and annealing model of C60 formation revealed that a 1,4-didehydrobenzocyclobutadiene intermediate (p-benzyne derivative) has little to no barrier to undergoing an unproductive retro-Bergman cyclization, which brings into question the relevance of that reaction pathway. The current study investigates an alternative model, which proceeds through an initial [4 + 2] cycloaddition instead of a [2 + 2] cycloaddition. In this pathway, the problematic intermediate is avoided, with the reaction proceeding via a (potentially) more kinetically stable tetradehydronaphthalene derivative. The computational studies of the [2 + 2] and [4 + 2] model systems, with increasing alkyne substitutions, reveal that the para-benzyne diradical of the [4 + 2] pathway has a significantly greater barrier to ring opening than the analogous intermediates of the [2 + 2] pathway and that alkyne substitution has little effect on this important barrier. These studies employ spin-flip, time-dependent density functional theory (SF-TDDFT) to provide suitable treatment of open-shell diradical intermediates.