Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix

Toward Consistent Predictions of Core/Valence Ionization Potentials and Valence Excitation Energies by MRSF-TDDFT

Optimizing exchange-correlation functionals for both core/valence ionization potentials (cIPs/vIPs) and valence excitation energies (VEEs) at the same time in the framework of MRSF-TDDFT is self-contradictory. To overcome the challenge, within the previous "adaptive exact exchange" or double-tuning strategy on Coulomb-attenuating XC functionals (CAM), a new XC functional specifically for cIPs and vIPs was first developed by enhancing exact exchange to both short- and long-range regions. The resulting DTCAM-XI functional achieved remarkably high accuracy in its predictions with errors of less than half eV. An additional concept of "valence attenuation", where the amount of exact exchange for the frontier orbital regions is selectively suppressed, was introduced to consistently predict both VEEs and IPs at the same time. The second functional, DTCAM-XIV, exhibits consistent overall prediction accuracy at ∼0.64 eV. By preferentially optimizing VEEs within the same "valence attenuation" concept, a third functional, DTCAM-VAEE, was obtained, which exhibits improved performance as compared to that of the previous DTCAM-VEE and DTCAM-AEE in the prediction of VEEs, making it an attractive alternative to BH&HLYP. As the combination of "adaptive exchange" and "valence attenuation" is operative, it would be exciting to explore its potential with a more tunable framework in the future.

Enhancing the Potential of Fused Heterocycle-Based Triarylhydrazone Photoswitches

Triarylhydrazones represent an attractive class of photochromic compounds offering many interesting features including high molar absorptivity, good addressability, and extraordinary thermal stability. In addition, unlike most other hydrazone-based photoswitches, they effectively absorb light above 365 nm. However, previously prepared triaryhydrazones suffer from low quantum yields of the Z→E photoisomerization. Here, we have designed a new subclass of naphthoyl-benzothiazole hydrazones that balance the most beneficial features of previously reported naphthoyl-quinoline and benzoyl-pyridine triarylhydrazones. These preserve the attractive absorption characteristics, exhibit higher thermal stability of the metastable form than the former and enhance the rate of the Z→E photoisomerization compared to the later, as a result of the weakening of the intramolecular hydrogen bonding between the hydrazone hydrogen and the benzothiazole moiety. Introducing the benzothiazole motif extends the tunability of the photochromic behaviour of hydrazone-based switches.

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.

High-performance strategies for the recent MRSF-TDDFT in GAMESS

Multiple ERI (Electron Repulsion Integral) tensor contractions (METC) with several matrices are ubiquitous in quantum chemistry. In response theories, the contraction operation, rather than ERI computations, can be the major bottleneck, as its computational demands are proportional to the multiplicatively combined contributions of the number of excited states and the kernel pre-factors. This paper presents several high-performance strategies for METC. Optimal approaches involve either the data layout reformations of interim density and Fock matrices, the introduction of intermediate ERI quartet buffer, and loop-reordering optimization for a higher cache hit rate. The combined strategies remarkably improve the performance of the MRSF (mixed reference spin flip)-TDDFT (time-dependent density functional theory) by nearly 300%. The results of this study are not limited to the MRSF-TDDFT method and can be applied to other METC scenarios.

Accurate Spin-Orbit Coupling by Relativistic Mixed-Reference Spin-Flip-TDDFT

Relativistic mixed-reference spin-flip (MRSF)-TDDFT is developed considering the spin-orbit coupling (SOC) within the mean-field approximation. The resulting SOC-MRSF faithfully reproduces the experiments with very high accuracy, which is also consistent with the values by four-component (4c) relativistic CASSCF and 4c-CASPT2 in the spin-orbit-energy splitting calculations of the C, Si, and Ge atoms. Even for the fifth-row element Sn, the SOC-MRSF yielded accurate splittings (∼ 3 % error). In the SOC calculations of the molecular 4-thiothymine with a third-row element, SOC-MRSF values are in excellent agreement with those of the SO-GMC-QDPT2 level, regardless of geometries and exchange-correlation functionals. The same SOC-MRSF predicted the anticipated chance of S₁ (nπ*) → T₁ (ππ*) intersystem crossing, even in thymine with only second-row elements. With its accuracy and practicality, thus, SOC-MRSF is a promising electronic structure protocol in challenging situations such as nonadiabatic molecular dynamics (NAMD) incorporating both internal conversions and intersystem crossings in large systems.

Prototropically Controlled Dynamics of Cytosine Photodecay

The effect of the existence of several prototropic tautomers of cytosine on its UV/vis spectra and the excited state decay dynamics is studied by spectral and nonadiabatic molecular dynamics (NAMD) simulations in connection with the mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) method. Simulated UV/vis spectra provide a strong indication that the H3N keto-amino cytosine tautomer (the least anticipated species) may be present under experimental conditions. The NAMD simulations yield a wide range of excited state decay constants for various tautomers of cytosine, ranging from ∼1.3 ps for the biologically relevant H1N keto-amino tautomer to ∼0.1 ps for the keto-imino tautomer. The slowness of the H1N decay dynamics follows from the presence of a barrier on the excited state energy surface separating the Franck-Condon structure from the major decay funnel, the conical intersection seam. It is suggested that the experimentally observed photodecay dynamics may result from a combination of the decay processes of various tautomers (H3N in particular) present simultaneously under the experimental conditions.

Ultrafast Excited State Aromatization in Dihydroazulene

Excited-state aromatization dynamics in the photochemical ring opening of dihydroazulene (DHA) is investigated by nonadiabatic molecular dynamics simulations in connection with the mixed-reference spin-flip (MRSF)-TDDFT method. It is found that, in the main reaction channel, the ring opening occurs in the excited state in a sequence of steps with increasing aromaticity. The first stage lasting ca. 200 fs produces an 8π semiaromatic S₁ minimum (S_1, min) through an ultrafast damped bond length alternation (BLA) movement synchronized with a partial planarization of the cycloheptatriene ring. An additional ca. 200 fs are required to gain the vibrational energy needed to overcome a ring-opening transition state characterized by an enhanced Baird aromaticity. Unlike other BLA motions of ππ* state, it was shown that their damping is a characteristic feature of aromatic bond-equalization process. In addition, some minor channels of the reaction have also been discovered, where noticeably higher barriers of the S₁ non/antiaromatic transition structures must be surmounted. These anti-Baird channels led to reformation of DHA or other closed-ring products. The observed competition between the Baird and anti-Baird channels suggests that the quantum yield of photochemical products can be controllable by tipping their balance. Hence, here we suggest including the concept of anti-Baird, which would expand the applicability of Baird rule to much broader situations.

Photochemistry of Thymine in Solution and DNA Revealed by an Electrostatic Embedding QM/MM Combined with Mixed-Reference Spin-Flip TDDFT

The photochemistry of nucleobases, important for their role as building blocks of DNA, is largely affected by the electrostatic environment in which they are soaked. For example, despite the numerous studies of thymine in solution and DNA, there is still a debate on the photochemical deactivation pathways after UV absorption. Many theoretical models are oversimplified due to the lack of computationally accurate and efficient electronic structure methodologies that capture excited state electron correlation effects when nucleobases are embedded in large electrostatic media. Here, we combine mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) with electrostatic embedding QM/MM using electrostatic potential fittingfitted (ESPF) atomic charges, as a strategy to accurately and efficiently describe the electronic structure of chromophores polarized by an electrostatic medium. In particular, we develop analytic expressions for the energy and gradient of MRSF/MM based on the ESPF coupling using atom-centered grids and total charge conservation. We apply this methodology to the study of solvation effects on thymine photochemistry in water and thymine dimers in DNA. In the former, the combination of trajectory surface hopping (TSH) nonadiabatic molecular dynamics (NAMD) with MRSF/MM remarkably revealed accelerated deactivation decay pathways, which is consistent with the experimental decay time of ∼400 fs. The enhanced hopping rate can be explained by the preferential stabilization of corresponding conical interactions due to their increased dipole moments. Structurally, it is a consequence of characteristic methyl puckered geometries near the conical intersection region. For the thymine dimer in B-DNA, we found new photochemical pathways through conical intersections that could explain the formation of cyclobutadiene dimers and 6-4 photoproducts.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory for Accurate X-ray Absorption Spectroscopy

It is demonstrated that the challenging core-hole particle (CHP) orbital relaxation for core electron spectra can be readily achieved by the mixed-reference spin-flip (MRSF)-time-dependent density functional theory (TDDFT). With the additional scalar relativistic effects on K-edge excitation energies of 24 second- and 17 third-row molecules, the particular ΔCHP-MRSF(R) exhibited near perfect predictions with RMSE ∼0.5 eV, featuring a median value of 0.3 and an interquartile range of 0.4. Overall, the CHP effect is 2-4 times stronger than relativistic ones, contributing more than 20 eV in the cases of sulfur and chlorine third-row atoms. Such high precision allows to explain the splitting and spectral shapes of O, N, and C atom K-edges in the ground state of thymine with atom as well as orbital specific accuracy. The same protocol with a double hole particle relaxation also produced remarkably accurate K-edge spectra of core to valence hole excitation energies from the first (n_O8π*) and second (ππ*) excited states of thymine, confirming the assignment of 1s → n excitation for the experimentally observed 526.4 eV peak. Regarding both accuracy and practicality, therefore, MRSF-TDDFT provides a promising protocol for core electron spectra of both ground and excited electronic states alike.

Using diketopyrrolopyrroles to stabilize double excitation and control internal conversion

Diketopyrrolopyrrole (DPP) is a pivotal functional group to tune the physicochemical properties of novel organic photoelectronic materials. Among multiple uses, DPP-thiophene derivatives forming a dimer through a vinyl linker were recently shown to quench the fluorescence observed in their isolated monomers. Here, we explain this fluorescence quenching using computational chemistry. The DPP-thiophene dimer has a low-lying doubly excited state that is not energetically accessible for the monomer. This state delays the fluorescence allowing internal conversion to occur first. We characterize the doubly excited state wavefunction by systematically changing the derivatives to tune the π-scaffold size and the acceptor and donor characters. The origin of this state's stabilization is related to the increase in the π-system and not to the charge-transfer features. This analysis delivers core conceptual information on the electronic properties of organic chromophores arranged symmetrically around a vinyl linker, opening new ways to control the balance between luminescence and internal conversion.

A Plausible Mechanism of Uracil Photohydration Involves an Unusual Intermediate

It is well-known that photolysis of pyrimidine nucleobases, such as uracil, in an aqueous environment results in the formation of hydrate as one of the main products. Although several hypotheses regarding photohydration have been proposed in the past, e.g., the zwitterionic and "hot" ground-state mechanisms, its detailed mechanism remains elusive. Here, theoretical nonadiabatic simulations of the uracil photodynamics reveal the formation of a highly energetic but kinetically stable intermediate that features a half-chair puckered pyrimidine ring and a strongly twisted intracyclic double bond. The existence and the kinetic stability of the intermediate are confirmed by a variety of computational chemistry methods. According to the simulations, the unusual intermediate is mainly formed almost immediately (∼50-200 fs) upon photoabsorption and survives long enough to engage in a hydration reaction with a neighboring water. A plausible mechanism of uracil photohydration is proposed on the basis of the modeling of nucleophilic insertion of water into the twisted double bond of the intermediate.

Reference Energies for Cyclobutadiene: Automerization and Excited States

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

Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential

Light-induced chemical processes are ubiquitous in nature and have widespread technological applications. For example, photoisomerization can allow a drug with a photo-switchable scaffold such as azobenzene to be activated with light. In principle, photoswitches with desired photophysical properties like high isomerization quantum yields can be identified through virtual screening with reactive simulations. In practice, these simulations are rarely used for screening, since they require hundreds of trajectories and expensive quantum chemical methods to account for non-adiabatic excited state effects. Here we introduce a diabatic artificial neural network (DANN), based on diabatic states, to accelerate such simulations for azobenzene derivatives. The network is six orders of magnitude faster than the quantum chemistry method used for training. DANN is transferable to azobenzene molecules outside the training set, predicting quantum yields for unseen species that are correlated with experiment. We use the model to virtually screen 3100 hypothetical molecules, and identify novel species with high predicted quantum yields. The model predictions are confirmed using high-accuracy non-adiabatic dynamics. Our results pave the way for fast and accurate virtual screening of photoactive compounds.

Difficulty of the evaluation of the barrier height of an open-shell transition state between closed shell minima: The case of small C4n rings

C_4n cyclacenes exhibit strong bond-alternation in their equilibrium geometry. In the two equivalent geometries, the system keeps an essentially closed-shell character. The two energy minima are separated by a transition state suppressing the bond-alternation, where the wave function is strongly diradical. This paper discusses the physical factors involved in this energy difference and possible evaluations of the barrier height. The barrier given as the energy difference between the restricted density functional theory (DFT)/B3LYP for the equilibrium and the broken symmetry DFT/B3LYP of the transition state is either negative or small, in contradiction with the most reliable Wave Function Theory calculations. The minimal (two electrons in two molecular orbitals) Complete Active Space self-consistent field (CASSCF) overestimates the barrier, and the subsequent second-order perturbation cancels it. Due to the collective character of the spin-polarization effect, it is necessary to perform a full π CASSCF + second-order perturbation to reach a reasonable value of the barrier, but this type of treatment cannot be applied to large molecules. DFT procedures treating on an equal foot the closed-shell and open-shell geometries have been explored, such as Mixed-Reference Spin-Flip Time-dependent-DFT and a new spin-decontamination proposal, namely, DFT-dressed configuration interaction, but the results still depend on the density functional. M06-2X without or with spin-decontamination gives the best agreement with the accurate wave function results.

Generalized Formulation of the Density Functional Tight Binding-Based Restricted Ensemble Kohn-Sham Method with Onsite Correction to Long-Range Correction

We present a generalized formulation for the combination of the density functional tight binding (DFTB) approach and the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method by considering onsite correction (OC) as well as the long-range corrected (LC) functional. The OC contribution provides more accurate energies and analytic gradients for individual microstates, while the multireference character of the SSR provides the correct description for conical intersections. We benchmark the LC-OC-DFTB/SSR method against various DFTB calculation methods for excitation energies and conical intersection structures with π/π* or n/π* characters. Furthermore, we perform excited-state molecular dynamics simulations with a molecular rotary motor with variations of LC-OC-DFTB/SSR approaches. We show that the OC contribution to the LC functional is crucial to obtain the correct geometry of conical intersections.

Singlet Oxygen Generation from Polyaminoglycerol by Spin-Flip-Based Electron Transfer

Reactive oxygen species have drawn attention owing to their strong oxidation ability. In particular, the singlet oxygen (¹O₂) produced by energy transfer is the predominant species for controlling oxidation reactions efficiently. However, conventional ¹O₂ generators, which rely on enhanced energy transfer, frequently suffer from poor solubility, low stability, and low biocompatibility. Herein, we introduce a hyperbranched aliphatic polyaminoglycerol (hPAG) as a ¹O₂ generator, which relies on spin-flip-based electron transfer. The coexistence of a lone pair electron on the nitrogen atom and a hydrogen-bonding donor (the protonated form of nitrogen and hydroxyl group) affords proximity between hPAG and O₂. Subsequent direct electron transfer after photo-irradiation induces hPAG^•+-O₂ ^•- formation, and the following spin-flip process generates ¹O₂. The spin-flip-based electron transfer pathway is analyzed by a series of photophysical, electrochemical, and computational studies. The ¹O₂ generator, hPAG, is successfully employed in photodynamic therapy and as an antimicrobial reagent.

Dicarbonyl anthracenes and phenanthrenes as singlet fission chromophores

Singlet fission is a highly desired process in photovoltaic devices as it can significantly enhance photoelectric conversion efficiency. Exploitation of this process in photovoltaics is hindered by the lack of appropriate chromophores. We used mixed-reference spin-flipping time-dependent density functional theory to investigate five di-carbonyl anthracenes and phenanthrenes, with the purpose to design singlet fission chromophores. Two molecules were found to be promising candidates. For all the dicarbonyl molecules, the oxygen lone pair orbitals were found to be involved in the excited states that are relevant to singlet fission.

Exploring Dyson's Orbitals and Their Electron Binding Energies for Conceptualizing Excited States from Response Methodology

The molecular orbital (MO) concept is a useful tool, which relates the molecular ground-state energy with the energies (and occupations) of the individual orbitals. However, analysis of the excited states from linear response computations is performed in terms of the initial state MOs or some other forms of orbitals, e.g., natural or natural transition orbitals. Because these orbitals lack the respective energies, they do not allow developing a consistent orbital picture of the excited states. Herein, we argue that Dyson's orbitals enable description of the response states compatible with the concepts of molecular orbital theory. The Dyson orbitals and their energies obtained by mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) for the response ground state are remarkably similar to the canonical MOs obtained by the usual DFT calculation. For excited states, the Dyson orbitals provide a chemically sensible picture of the electronic transitions, thus bridging the chasm between orbital theory and response computations.

Internal Conversion between Bright (11Bu+) and Dark (21Ag-) States in s-trans-Butadiene and s-trans-Hexatriene

Internal conversion (IC) between the two lowest singlet excited states, 1¹B_u⁺ and 2¹A_g^-, of s-trans-butadiene and s-trans-hexatriene is investigated using a series of single- and multi- reference wave function and density functional theory (DFT) methodologies. Three independent types of the equation-of-motion coupled-cluster (EOMCC) theory capable of providing an accurate and balanced description of one- as well as two-electron transitions, abbreviated as δ-CR-EOMCC(2,3), DIP-EOMCC(4h2p){N_o}, and DEA-EOMCC(4p2h){N_u} or DEA-EOMCC(3p1h,4p2h){N_u}, consistently predict that the 1¹B_u⁺/2¹A_g^- crossing in both molecules occurs along the bond length alternation coordinate. However, the analogous 1¹B_u⁺ and 2¹A_g^- potentials obtained with some multireference approaches, such as CASSCF and MRCIS(D), as well as with the linear-response formulation of time-dependent DFT (TDDFT), do not cross. Hence, caution needs to be exercised when studying the low-lying singlet excited states of polyenes with conventional multiconfigurational methods and TDDFT. The multistate many-body perturbation theory methods, such as XMCQDPT2, do correctly reproduce the curve crossing. Among the simplest and least expensive computational methodologies, the DFT approaches that incorporate the contributions of doubly excited configurations, abbreviated as MRSF (mixed reference spin-flip) TDDFT and SSR(4,4), accurately reproduce our best EOMCC results. This is highly promising for nonadiabatic molecular dynamics simulations in larger systems.

Nonadiabatic dynamics with spin-flip vs linear-response time-dependent density functional theory: A case study for the protonated Schiff base C5H6NH2. J Chem Phys 2021;155:124111. [PMID: 34598550 DOI: 10.1063/5.0062757]

Nonadiabatic trajectory surface hopping simulations are reported for trans-C₅H₆NH₂ ⁺, a model of the rhodopsin chromophore, using the augmented fewest-switches algorithm. Electronic structure calculations were performed using time-dependent density functional theory (TDDFT) in both its conventional linear-response (LR) and its spin-flip (SF) formulations. In the SF-TDDFT case, spin contamination in the low-lying singlet states is removed by projecting out the lowest triplet component during iterative solution of the TDDFT eigenvalue problem. The results show that SF-TDDFT qualitatively describes the photoisomerization of trans-C₅H₆NH₂ ⁺, with favorable comparison to previous studies using multireference electronic structure methods. In contrast, conventional LR-TDDFT affords qualitatively different photodynamics due to an incorrect excited-state potential surface near the Franck-Condon region. In addition, the photochemistry (involving pre-twisting of the central double bond) appears to be different for SF- and LR-TDDFT, which may be a consequence of different conical intersection topographies afforded by these two methods. The present results contrast with previous surface-hopping studies suggesting that the LR-TDDFT method's incorrect topology around S₁/S₀ conical intersections is immaterial to the photodynamics.

Relief of excited-state antiaromaticity enables the smallest red emitter

It is commonly accepted that a large π-conjugated system is necessary to realize low-energy electronic transitions. Contrary to this prevailing notion, we present a new class of light-emitters utilizing a simple benzene core. Among different isomeric forms of diacetylphenylenediamine (DAPA), o- and p-DAPA are fluorescent, whereas m-DAPA is not. Remarkably, p-DAPA is the lightest (FW = 192) molecule displaying red emission. A systematic modification of the DAPA system allows the construction of a library of emitters covering the entire visible color spectrum. Theoretical analysis shows that their large Stokes shifts originate from the relief of excited-state antiaromaticity, rather than the typically assumed intramolecular charge transfer or proton transfer. A delicate interplay of the excited-state antiaromaticity and hydrogen bonding defines the photophysics of this new class of single benzene fluorophores. The formulated molecular design rules suggest that an extended π-conjugation is no longer a prerequisite for a long-wavelength light emission.

Electronic Structure Methods for the Description of Nonadiabatic Effects and Conical Intersections

Nonadiabatic effects are ubiquitous in photophysics and photochemistry, and therefore, many theoretical developments have been made to properly describe them. Conical intersections are central in nonadiabatic processes, as they promote efficient and ultrafast nonadiabatic transitions between electronic states. A proper theoretical description requires developments in electronic structure and specifically in methods that describe conical intersections between states and nonadiabatic coupling terms. This review focuses on the electronic structure aspects of nonadiabatic processes. We discuss the requirements of electronic structure methods to describe conical intersections and nonadiabatic couplings, how the most common excited state methods perform in describing these effects, and what the recent developments are in expanding the methodology and implementing nonadiabatic couplings.

Variations of the Hartree-Fock fractional-spin error for one electron

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H⁺ ₂ cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the Møller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.

Description of Sudden Polarization in the Excited Electronic States with an Ensemble Density Functional Theory Method

Sudden polarization (SP) is one of the manifestations of electron transfer in the electronically excited states of molecules. Proposed initially to explain the unusual reactivity of photoexcited olefins, SP often occurs in the excited states of molecules possessing strongly correlated diradical ground state. Theoretical description of SP involves mixing between the singly excited and the doubly excited zwitterionic states, which makes it inaccessible with the use of the popular linear-response time-dependent density functional theory methods. In this work, an extended variant of the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS, or SSR) method is applied to study SP in a number of organic diradical systems. To this end, the analytical derivative formalism is derived and implemented for the SSR(3,2) method (see the main text for explanation of the acronym), which enables the automatic geometry optimization and obtains the relaxed density matrices as well as the electron binding energies and respective Dyson's orbitals. Application of the new method to SP in the lowest singlet excited state of ethylene agrees with the results obtained previously with the use of multireference methods of wavefunction theory. A number of interesting manifestations of SP are observed, such as the charge transfer in photoexcited tetramethyleneethene (TME) diradical mediated by the vibrational motion and conductivity switching in the excited state of a donor-acceptor dyad placed in an external electric field.

Fast and Accurate Computation of Nonadiabatic Coupling Matrix Elements Using the Truncated Leibniz Formula and Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory

We present a fast and accurate numerical algorithm for computing the first-order nonadiabatic coupling matrix element (NACME). The algorithm employs the truncated Leibniz formula (TLF) approximation within the finite-difference method, which makes it easily applicable in connection with any wave function-based methodology. In this work, we used the algorithm in connection with the recently developed mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT, MRSF for brevity). The accuracy is assessed for NACME between the singlet electronic states of a dissociating hydrogen molecule. It is demonstrated that an intermediate approximation, TLF(1), affords a negligible numeric error on the order of ∼10^-10 a.u. while enabling a fast computation of NACME. As the MRSF method yields the correct description of the dissociation curves of H₂ for all the electronic states involved, the numeric TLF(1)/MRSF NACME values are in excellent agreement with the reference analytical values obtained by the full configuration interaction. For polyatomic molecules, the MRSF NAC vectors agree very closely with the MRCISD NAC vectors. Hence, the proposed protocol is a promising tool for the evaluation of NACMEs.

Impact of the Dynamic Electron Correlation on the Unusually Long Excited-State Lifetime of Thymine

Non-radiative relaxation of the photoexcited thymine in the gas phase shows an unusually long excited-state lifetime, and, over the years, a number of models, i.e., S₁-trapping, S₂-trapping, and S₁&S₂-trapping, have been put forward to explain its mechanism. Here, we investigate this mechanism using non-adiabatic molecular dynamics (NAMD) simulations in connection with the recently developed mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) method. We show that the previously predicted S₂-trapping model was due to an artifact caused by an insufficient account of the dynamic electron correlation. The current work supports the S₁-trapping mechanism with two lifetimes, τ₁ = 30 ± 1 fs and τ₂ = 6.1 ± 0.035 ps, quantitatively consistent with the recent time-resolved experiments. Upon excitation to the S₂ (ππ*) state, thymine undergoes an ultrafast (ca. 30 fs) S₂→S₁ internal conversion and resides around the minimum on the S₁ (n_Oπ*) surface, slowly decaying to the ground state (ca. 6.1 ps). While the S₂→S₁ internal conversion is mediated by fast bond length alternation distortion, the subsequent S₁→S₀ occurs through several conical intersections, involving a slow puckering motion.

Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties

We review recent developments in the framework of simplified quantum chemistry for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to molecular response calculations of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent density functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solutions to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can extract chemical structure-property relationships from excited state or response calculations are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.

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

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

Optimization of Three State Conical Intersections by Adaptive Penalty Function Algorithm in Connection with the Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory Method (MRSF-TDDFT)

A new adaptive algorithm for penalty function optimization for minimum-energy three-states conical intersections (ME3CI) is suggested. The new algorithm differs from the original penalty function algorithm by (a) removing the redundancy in the target function, (b) using an adaptive increment for the penalty function weighting factor, and (c) using tighter convergence criteria for the energy gap. The latter was introduced to guarantee convergence to a true conical intersection rather than to a narrowly avoided crossing geometry. The new algorithm was tested in the optimization of the ME3CI geometries in butadiene and malonaldehyde, where all of the previously found true ME3CI geometries were recovered. The previously found butadiene's CI_3/2/1 turned out to be a narrowly avoided crossing. For butadiene, seven new ME3CI geometries have been located. Because of the removal of the redundancy and the use of the adaptive weighting factor, the convergence rate of the new algorithm is noticeably improved as compared to that of the previously proposed penalty function algorithm. The application to malonaldehyde and butadiene demonstrates that the three-state conical intersections may be more abundant and hence more involved in the photochemistry than previously thought. The recently developed mixed-reference spin flip (MRSF)-TDDFT method yields ME3CI geometries and relative energies quantitatively consistent with the previously reported calculations at a much reduced computational cost.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) as a Simple yet Accurate Method for Diradicals and Diradicaloids

Due to their multiconfigurational nature featuring strong electron correlation, accurate description of diradicals and diradicaloids is a challenge for quantum chemical methods. The recently developed mixed-reference spin-flip (MRSF)-TDDFT method is capable of describing the multiconfigurational electronic states of these systems while avoiding the spin-contamination pitfalls of SF-TDDFT. Here, we apply MRSF-TDDFT to study the adiabatic singlet-triplet (ST) gaps in a series of well-known diradicals and diradicaloids. On average, MRSF displays a very high prediction accuracy of the adiabatic ST gaps with the mean absolute error (MAE) amounting to 0.14 eV. In addition, MRSF is capable of accurately describing the effect of the Jahn-Teller distortion occurring in the trimethylenemethane diradical, the violation of the Hund rule in a series of the didehydrotoluene diradicals, and the potential energy surfaces of the didehydrobenzene (benzyne) diradicals. A convenient criterion for distinguishing diradicals and diradicaloids is suggested on the basis of the easily obtainable quantities. In all of these cases, which are difficult for the conventional methods of density functional theory (DFT), MRSF shows results consistent with the experiment and the high-level ab initio computations. Hence, the present study documents the reliability and accuracy of MRSF and lays out the guidelines for its application to strongly correlated molecular systems.

Diboron- and Diaza-Doped Anthracenes and Phenanthrenes: Their Electronic Structures for Being Singlet Fission Chromophores

We used quantum chemistry methods at the levels of mixed-reference spin-flip time-dependent density functional theory and multireference perturbation theory to study diboron- and diaza-doped anthracenes and phenanthrenes. This class of structures recently surged as potential singlet fission chromophores. We studied electronic structures of their excited states and clarified the reasons why they satisfy or fail to satisfy the energy criteria for singlet fission chromophores. Many studied structures have their S₁ states not dominated by HOMO → LUMO excitation, so they cannot be described using the conventional two site model. This is attributed to frontier orbital energy shifts induced by the doping and different charge-transfer energies in different one-electron singlet excitations or, in other words, different polarizations of hole and/or particle orbitals in their S₁ and T₁ states. There is a mirror relation between the orbital energy shifts induced by diboron- and diaza-dopings, which together with alternant hydrocarbon pairings of occupied and unoccupied orbitals, leads to more mirror relations between the excited states of the two types of doped structures.

Computation of Molecular Ionization Energies Using an Ensemble Density Functional Theory Method

Computation of the ionization energies and of the respective Dyson orbitals based on the use of the extended Koopmans theorem (EKT) is implemented in connection with an ensemble density functional theory (eDFT) method, the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method. The new methodology enables fast computation of the ionization energies and evaluation of the respective Dyson orbitals, the square norms of which are related with the ionization probabilities, in the ground and excited electronic states of molecules. As the application of EKT recycles the intermediate quantities from the SSR analytical energy gradient, evaluation of the ionization energies and probabilities can be carried out on-the-fly during the nonadiabatic molecular dynamics simulations. This opens up a perspective for fast theoretical simulation of the time-resolved photoelectron spectroscopy observations. In the present work, the new methodology is tested in the computation of the ionization energies and Dyson orbitals of several molecules in the ground and excited electronic states, including strongly correlated species, such as the ozone molecule, dissociating chemical bonds, and conical intersections.

Photoinduced Double Proton Transfer in the Glyoxal-Methanol Complex Revisited: The Role of the Excited States

Under irradiation in the visible range, the glyoxal-methanol complex in a cryogenic argon matrix undergoes a double proton transfer (DPT) reaction through which the glyoxal molecule isomerizes into hydroxyketene. In this work, we employ electronic structure simulations in order to shed more light on the underlying mechanism. Rewardingly, we find that the lowest singlet excited state (S₁) of the complex acts as a gateway to two previously unknown isomerization pathways, of which one takes place entirely in the singlet manifold and the other also involves the lowest triplet state (T₁). Both of these pathways are fully compatible with the available experimental data, implying that either or both are operative under experimental conditions. In either pathway, the methanol molecule acts as a proton shuttle between the proton-donating and proton-accepting sites of glyoxal, resulting in a dramatic lowering of the potential energy barrier to isomerization with respect to the case of isolated glyoxal. The occurrence of DPT in the singlet manifold is demonstrated directly with the use of nonadiabatic molecular dynamics simulations at the spin-flip time-dependent density functional theory level.

Spin-flip methods in quantum chemistry

This Perspective discusses salient features of the spin-flip approach to strong correlation and describes different methods that sprung from this idea. The spin-flip treatment exploits the different physics of low-spin and high-spin states and is based on the observation that correlation is small for same-spin electrons. By using a well-behaved high-spin state as a reference, one can access problematic low-spin states by deploying the same formal tools as in the excited-state treatments (i.e., linear response, propagator, or equation-of-motion theories). The Perspective reviews applications of this strategy within wave function and density functional theory frameworks as well as the extensions for molecular properties and spectroscopy. The utility of spin-flip methods is illustrated by examples. Limitations and proposed future directions are also discussed.

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

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

Conical Intersections in Organic Molecules: Benchmarking Mixed-Reference Spin-Flip Time-Dependent DFT (MRSF-TD-DFT) vs Spin-Flip TD-DFT

The mixed-reference spin-flip time-dependent density functional theory (MRSF-TD-DFT) method eliminates the erroneous spin contamination of the SF-TD-DFT methodology, while retaining the conceptual and practical simplicity of the latter. The availability of the analytic gradient of the energy of the MRSF-TD-DFT response states enables automatic geometry optimization of the targeted states. Here, we apply the new method to optimize the geometry of several S₁/S₀ conical intersections occurring in typical organic molecules. We demonstrate that MRSF-TD-DFT is capable of producing the correct double-cone topology of the intersections and describing the geometry of the lowest-energy conical intersections and their relative energies with accuracy matching that of the best multireference wavefunction ab initio methods. In this regard, MRSF-TD-DFT differs from many popular single-reference methods, such as, e.g., the linear response TD-DFT method, which fail to produce the correct topology of the intersections. As the new methodology completely eliminates the ambiguity with the identification of the response states as proper singlets or triplets, which is plaguing the SF-TD-DFT calculations, it can be used for automatic geometry optimization and molecular dynamic simulations not requiring constant human intervention.

Design of singlet fission chromophores with cyclic (alkyl)(amino) carbene building blocks

We use MRSF-TDDFT and NEVPT2 methods to design singlet fission chromophores with the building blocks of cyclic (alkyl)(amino)carbenes (CAACs). CAAC dimers with C₂, C₄, and p-phenylene spacers are considered. The substitutions with trifluoromethyls and fluorine atoms at the α C position are investigated. The electronegative substituents enhance the π accepting capability of the α C while maintaining it as a quaternary C atom. The phenylene-connected dimers with the two substitutions are identified as promising candidates for singlet fission chromophores. The cylindrically symmetric C₂ and C₄ spacers allow for substantial structural reorganizations in the S₀-to-S₁ and S₀-to-T₁ excitations. Although the two substituted dimers with the C₄ spacer satisfy (or very close to satisfy) the primary thermodynamics criterion for singlet fission, the significant structural reorganizations result in high barriers so that the fission is kinetically unfavorable.