Spin-adapted open-shell time-dependent density functional theory: towards a simple and accurate method for spin-flip-down excitations

Spin-flip methods provide access to certain electronic states having multireference character while retaining single-reference cost. However, conventional spin-flip time-dependent density functional theory (SF-TDDFT) often suffers from severe spin contamination that may cause inaccurate state ordering or engender ambiguous state character. For singlet excited states, this is largely rectified by a "mixed-reference" formulation (MRSF-TDDFT), while a spin-adapted formalism (SA-SF-TDDFT) addresses spin contamination in a general way for arbitrary multiplicities. Here, we revisit SA-SF-TDDFT and demonstrate that it significantly improves the agreement with reference data compared to other variants and also relative to conventional (spin-conserving) linear response TDDFT. Overall, SA-SF-TDDFT proves to be the most accurate among these methods, for excitation energies of both closed-shell molecules and doublet radicals as well as for singlet-triplet gaps. However, SF methods exhibit a notable limitation in the case of linear and quasi-linear doublet radicals, due to degeneracies in the high-spin quartet reference state.

Computational investigation of the longest wavelength excitations in a series of cyanines and linear n-acenes is undertaken with the use of standard spin-conserving linear response time-dependent density functional theory (TD-DFT) as well as its spin-flip variant and a ΔSCF method based on the ensemble DFT. The spin-conserving linear response TD-DFT fails to accurately reproduce the lowest excitation energy in these π-conjugated systems by strongly overestimating the excitation energies of cyanines and underestimating the excitation energies of n-acenes. The spin-flip TD-DFT is capable of correcting the underestimation of excitation energies of n-acenes by bringing in the non-dynamic electron correlation into the ground state; however, it does not fully correct for the overestimation of the excitation energies of cyanines, for which the non-dynamic correlation does not seem to play a role. The ensemble DFT method employed in this work is capable of correcting for the effect of missing non-dynamic correlation in the ground state of n-acenes and for the deficient description of differential correlation effects between the ground and excited states of cyanines and yields the excitation energies of both types of extended π-conjugated systems with the accuracy matching high-level ab initio multireference calculations.

Accurately calculating the diradical character (y0) of molecular systems remains a significant challenge due to the scarcity of experimental data and the inherent multireference nature of the electronic structure. In this study, various quantum mechanical approaches, including broken symmetry density functional theory (BS-DFT), spin-flip time-dependent density functional theory (SF-TDDFT), mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT), complete active space self-consistent field (CASSCF), complete active space second-order perturbation theory (CASPT2), and multiconfigurational pair-density functional theory (MCPDFT), are employed to compute the singlet-triplet energy gaps (EST) and y0 values in Thiele, Chichibabin, and Müller analogous diradicals. By systematically comparing the results from these computational methods, we identify optimally tuned long-range corrected functional CAM-B3LYP in the BS-DFT framework as a most efficient method for accurately and affordably predicting both EST and y0 values. Additionally, our results demonstrate that (i) MRSF-TDDFT performs much better than SF-TDDFT; (ii) the MCPDFT method is robust in determining EST with minimal dependence on the choice of active space. These findings provide insight into the electronic structure and diradical character of the investigated molecules and highlight effective computational strategies for future studies in this domain. Thus, this work not only advances our understanding of diradical systems but also offers practical guidelines for their computational investigation.

Analytic energy gradients of individual singlet and triplet states with respect to nuclear coordinates are derived and implemented for the collinear mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT), which eliminates the problematic spin-contamination of SF-TDDFT. Dimensional-transformation matrices for the singlet and triplet response spaces are introduced, simplifying the subsequent derivations. These matrices enable the general forms of MRSF-TDDFT equations to be similar to those of SF-TDDFT, suggesting that the computational overhead of singlet or triplet states for MRSF-TDDFT is nearly identical to that of SF-TDDFT. In test calculations, the new MRSF-TDDFT yields quite different optimized structures and energies as compared to SF-TDDFT. These differences turned out to mainly come from the spin-contamination of SF-TDDFT, which are largely cured by MRSF-TDDFT. In addition, it was demonstrated that the clear separation of singlet states from triplets dramatically simplifies the location of minimum energy conical intersection. As a result, it is clear that the MRSF-TDDFT has advantages over SF-TDDFT in terms of both accuracy and practicality. Therefore, it can be a preferred method, which is readily applied to other "black-box" type applications, such as the minimum-energy optimization, reaction path following, and molecular dynamics simulations.

This paper presents the nonadiabatic molecular dynamics simulation in the solution phase using the spin-flip time-dependent density functional theory (SF-TDDFT). Despite the single-reference level of theory, the SF-TDDFT method can generate the correct topology of S0/S1 crossing points, thus providing a natural S1 → S0 nonadiabatic transition. We extend the gas-phase trajectory surface hopping simulation with the SF-TDDFT [N. Minezawa and T. Nakajima, J. Chem. Phys. 150, 204120 (2019)] to the hybrid quantum mechanical/molecular mechanics (QM/MM) scheme. To this end, we modify the code to evaluate the electrostatic interaction between the QM and MM atoms and to extract the classical MM energy and forces from the MM program package. We apply the proposed method to the photoisomerization reaction of aqueous E-azomethane and anionic green fluorescent protein chromophore in water and compare the results with those of the previous simulation studies based on the multireference methods.

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.

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.

In subsystem time-dependent density functional theory (TDDFT) [J. Neugebauer, J. Chem. Phys. 126, 134116 (2007)] localized excitations are used to calculate delocalized excitations in large chromophore aggregates. We have extended this formalism to allow for the Tamm-Dancoff approximation (TDA). The resulting response equations have a form similar to a perturbative configuration interaction singles (CIS) approach. Thus, the inter-subsystem matrix elements in subsystem TDA can, in contrast to the full subsystem-TDDFT case, directly be interpreted as exciton coupling matrix elements. Here, we present the underlying theory of subsystem TDDFT within the TDA as well as first applications. Since for some classes of pigments, such as linear polyenes and carotenoids, TDA has been reported to perform better than full TDDFT, we also report applications of this formalism to exciton couplings in dimers of such pigments and in mixed bacteriochlorophyll-carotenoid systems. The improved description of the exciton couplings can be traced back to a more balanced description of the involved local excitations.

On the calculation of [formula omitted] for electronic excitations in time-dependent density-functional theory

We present benchmark calculations on excitation energies based on time-dependent density functional theory (TDDFT) as well as orbital relaxed self-consistent and constricted variational DFT (RSCF-CV-DFT) with and without use of the Tamm-Dancoff approximation. The compilation contains results for the 3d complexes MnO₄⁻, CrO₄²⁻, and VO₄³⁻, as well as the 4d congeners RuO₄, TcO₄⁻, and MoO₄²⁻, and 5d homologues OsO₄, ReO₄⁻, and WO₄²⁻. Considerations have been given to the local density approximation (LDA) and the functionals BP86 and PBE based on the generalized gradient approximation (GGA), as well as the hybrids B3LYP, BHLYP, and PBE0 and the length corrected functional LCBP86. We find for the 3d complexes that RSCF-CV-DFT fares better than TDDFT. Thus, in the case of RSCF-CV-DFT, the average root-mean-square deviations (RMSDs) are 0.25-0.3 eV for GGAs, 0.1 eV for B3LYP, and 0.45 eV for BHLYP. TDDFT affords RMSDs that on average range from 0.3 eV for local functionals to 0.7 eV for BHLYP with the largest fraction of Hartree-Fock (HF) exchange. TDDFT is seen to fare better among the heavier tetraoxo systems. For the 4d and 5d systems, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange, and LCBP86 have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86* have a somewhat larger RMSD of 0.3 eV. Nearly the same performance is observed for RSCF-CV-DFT with respect to the different functionals in the case of the 4d and 5d systems. Thus, for the heavier tetraoxo systems, the two DFT schemes are comparable in accuracy.

Organic light emitting diodes (OLEDs) based on thermally activated delayed fluorescence (TADF) emitters are an attractive category of materials that have witnessed a booming development in recent years. In the present contribution, we scrutinize the accountability of parameterized and parameter-free single-hybrid (SH) and double-hybrid (DH) functionals through the two formalisms, full time-dependent density functional theory (TD-DFT) and Tamm-Dancoff approximation (TDA), for the estimation of photophysical properties like absorption energy, emission energy, zero-zero transition energy, and singlet-triplet energy splitting of TADF molecules. According to our detailed analyses on the performance of SHs based on TD-DFT and TDA, the TDA-based parameter-free SH functionals, PBE0 and TPSS0, with one-third of exact-like exchange turned out to be the best performers in comparison to other functionals from various rungs to reproduce the experimental data of the benchmarked set. Such affordable SH approximations can thus be employed to predict and design the TADF molecules with low singlet-triplet energy gaps for OLED applications. From another perspective, considering this point that both the nonlocal exchange and correlation are essential for a more reliable description of large charge-transfer excited states, applicability of the functionals incorporating these terms, namely, parameterized and parameter-free DHs, has also been evaluated. Perusing the role of exact-like exchange, perturbative-like correlation, solvent effects, and other related factors, we find that the parameterized functionals B2π-PLYP and B2GP-PLYP and the parameter-free models PBE-CIDH and PBE-QIDH have respectable performance with respect to others. Lastly, besides the recommendation of reliable computational protocols for the purpose, hopefully this study can pave the way toward further developments of other SHs and DHs for theoretical explorations in the field of OLEDs technology.

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.

On-the-fly molecular dynamics (MD) is a classical trajectory method where atomic positions and velocities are propagated by solving Newton’s equations of motion using energy gradients from quantum chemical computations. This chapter introduces spin-flip time-dependent density functional theory (SF-TDDFT) approach, and then reviews some applications of on-the-fly excited-state MD simulations based on SF-TDDFT calculations for photoisomerization of π π*- excited cis-stilbene (cis-SB), 1,1’-dimethyl-cis-stilbene (cis-dmSB), and α-methyl-cis-stilbene (cis-mSB). Stilbene is one of the basic typical molecules in photo-chemistry, showing cis-trans photo-isomerization. The calculated results are compared with available experimental data, and the reaction pathways and reaction dynamics are discussed for the respective photo-isomerization.

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 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.