Spin-flip Time-dependent Density Functional Theory Research Articles

Performance Analysis and Optimization of Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) for Vertical Excitation Energies and Singlet-Triplet Energy Gaps.

The mixed-reference spin-flip (MRSF) time-dependent density functional theory (TDDFT) method eliminates the notorious spin contamination of SF-TDDFT, thus enabling identification of states of proper spin-symmetry for automatic geometry optimization and molecular dynamics simulations. Here, we analyze and optimize the MRSF-TDDFT in the calculations of the vertical excitation energies (VEEs) and the singlet-triplet (ST) gaps. The dependence of the obtained VEEs and ST gaps on the intrinsic parameters of the MRSF-TDDFT method is investigated, and prescriptions for the proper use of the method are formulated. For VEEs, MRSF-TDDFT displays similar or better accuracy than SF-TDDFT (ca. 0.5 eV), while considerably outperforming the LR-TDDFT for the ST gaps. As a result, a new functional of STG1X (dubbed here), especially for ST gaps is suggested on the basis of splitting between the components of the atomic multiplets.

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.

Using Density Functional Theory Based Methods to Investigate the Photophysics of Polycyclic Aromatic Hydrocarbon Radical Cations: A Benchmark Study on Naphthalene, Pyrene and Perylene Cations

AbstractUnrestricted DFT (UDFT), time‐dependent DFT (TDDFT) and spin‐flip TDDFT (SF‐TDDFT) were used to investigate the potential energy surfaces of the ground and first two electronic excited states of the naphthalene, pyrene and perylene radical cations. In particular, conical intersections (which play a central role in the photophysics of these cations) were located with these DFT‐based methods. The results are consistent with accurate multiconfigurational wavefunction‐based ab initio methods. These show that naphthalene and pyrene cations can quickly relax nonradiatively from their excited states back down to the original ground state species through easily accessible conical intersections, but the perylene cation cannot do so, due to the absence of any accessible funnels between the lowest excited state and the ground state, leaving radiative decay as the most probable photophysical pathway. This study paves the way for using computationally efficient density functional theory (DFT)‐based methods in future investigations of the photophysics of much larger polycyclic aromatic hydrocarbons, for which multiconfigurational wavefunction‐based methods become prohibitively expensive.

A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals.

Experimentalists working with diradicals are often facing the question of what kind of species among singlet or triplet diradicals or closed-shell molecules are observed. To treat large diradicals with a high density of electronic states, we propose a simplified version of the spin-flip time-dependent density functional theory (SF-TD-DFT) method for a fast computation of their state energies and absorption spectra with an accuracy similar to the nonsimplified scheme. An ultrafast tight-binding variant called SF-sTD-DFT-xTB is also developed to treat even larger systems. For a benchmark set of nine diradicals, good agreement between simplified and conventional SF excitation energies for standard functionals is found. This shows that the proposed parameterization is robust for a wide range of Fock exchange mixing values. With the asymptotically correct response integrals used in SF-sTD-DFT and a correction factor of for the transition moments, the SF-sTD-DFT/B5050LYP/cc-pVDZ method even outperforms the nonsimplified scheme at drastically reduced computational effort when comparing to the experimental absorption spectra for this set of diradicals. To showcase the actual performance of the method, absorption spectra of two μ-hydroxo-bridged dimers of corrole tape Ga(III) complex derivatives were computed and compared to the experiment, providing good qualitative agreement. Finally, a comparison with the high-spin triplet spectrum of a perylene bisimide biradical and the one determined at the SF-sTD-DFT level showed that at room temperature, mostly triplet diradicals are present in solution.

Trajectory surface hopping molecular dynamics simulation by spin-flip time-dependent density functional theory.

This paper presents the nonadiabatic molecular dynamics simulation combined with the spin-flip time-dependent density functional theory (SF-TDDFT). In contrast to the conventional single-reference electronic structure methods, which have difficulty in describing the S0/S1 conical intersections, the SF-TDDFT can yield the correct topology of crossing points. Thus, one expects that the method can take naturally into account the S1 → S0 nonadiabatic transitions. We adopt Tully's fewest switch surface hopping algorithm by introducing the analytic SF-TDDFT nonadiabatic coupling vector. We apply the proposed method to the photoisomerization reactions of E-azomethane, methanimine, and ethene molecules and reproduce the results of previous studies based on the multireference methods. The proposed approach overcomes the ad hoc treatment of S1 → S0 transition at the single-reference calculation level and affords both the dynamics on the S1 state and the recovery of the S0 state with modest computational costs.

Efficient implementations of analytic energy gradient for mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT).

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.

Fast Overlap Evaluations for Nonadiabatic Molecular Dynamics Simulations: Applications to SF-TDDFT and TDDFT.

Fast overlap integral algorithms for the spin-flip time-dependent density functional theory (SF-TDDFT) and the linear response (LR)-TDDFT were proposed on the basis of determinant factorization (DF) and the truncated Leibnitz formula (TLF). These in turn allow efficient computation of nonadiabatic coupling terms (NACTs) in nonadiabatic molecular dynamics simulations. The TLF(0), TLF(1), and TLF(2) were proposed according to the truncation order. The DF and TLF(1) or TLF(2) provide a four order combined performance improvement to the conventional method without introducing additional errors in the finite difference approximation. On the other hand, the DF and TLF(0) provide a five orders performance improvement making it the most efficient algorithm for NACT calculations so far with errors slightly larger than those of the finite difference approximation. The same techniques can be also applicable to other determinantal wave functions.

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.

The use of the mixed reference (MR) reduced density matrix, which combines reduced density matrices of the MS = +1 and -1 triplet-ground states, is proposed in the context of the collinear spin-flip-time-dependent density functional theory (SF-TDDFT) methodology. The time-dependent Kohn-Sham equation with the mixed state is solved by the use of spinor-like open-shell orbitals within the linear response formalism, which enables to generate additional configurations in the realm of TD-DFT. The resulting MR-SF-TDDFT computational scheme has several advantages before the conventional collinear SF-TDDFT. The spin-contamination of the response states of SF-TDDFT is nearly removed. This considerably simplifies the identification of the excited states, especially in the "black-box" type applications, such as the automatic geometry optimization, reaction path following, or molecular dynamics simulations. With the new methodology, the accuracy of the description of the excited states is improved as compared to the collinear SF-TDDFT. Several test examples, which include systems typified by strong non-dynamic correlation, orbital (near) degeneracy, and conical intersections, are given to illustrate the performance of the new method.

On-the-fly molecular dynamics study of the excited-state branching reaction of α-methyl-cis-stilbene

The branching reaction of α-methyl-cis-stilbene (cis-mSB) into its trans-mSB and 4a,4b-dihydrophenanthrene (DHP) forms upon ππ∗ excitation was examined theoretically by exploring the excited-state potential energy surface and using on-the-fly molecular dynamics simulations at the spin-flip time-dependent density functional theory (SF-TDDFT) level of theory. The branching ratio of trajectories was calculated as DHP:twist = 11:29, where twist denotes a mid-region between the cis-form and trans-form, indicating that the trans-mSB is a dominant product. The branching mechanism was analyzed by comparison with the corresponding theoretical studies on stilbene (SB) and 1,1′-dimethyl-stilbene (dmSB). The present computations elucidate the origin of variations in the branching ratio in the photoreactions of cis-SB, cis-mSB, and cis-dmSB. We also found that, because of loss of the slow component of the decay to the ground state, cis-mSB shows a faster decay rate to the ground state than cis-SB and cis-dmSB.

Multireference Density Functional Theory with Generalized Auxiliary Systems for Ground and Excited States.

To describe static correlation, we develop a new approach to density functional theory (DFT), which uses a generalized auxiliary system that is of a different symmetry, such as particle number or spin, from that of the physical system. The total energy of the physical system consists of two parts: the energy of the auxiliary system, which is determined with a chosen density functional approximation (DFA), and the excitation energy from an approximate linear response theory that restores the symmetry to that of the physical system, thus rigorously leading to a multideterminant description of the physical system. The electron density of the physical system is different from that of the auxiliary system and is uniquely determined from the functional derivative of the total energy with respect to the external potential. Our energy functional is thus an implicit functional of the physical system density, but an explicit functional of the auxiliary system density. We show that the total energy minimum and stationary states, describing the ground and excited states of the physical system, can be obtained by a self-consistent optimization with respect to the explicit variable, the generalized Kohn-Sham noninteracting density matrix. We have developed the generalized optimized effective potential method for the self-consistent optimization. Among options of the auxiliary system and the associated linear response theory, reformulated versions of the particle-particle random phase approximation (pp-RPA) and the spin-flip time-dependent density functional theory (SF-TDDFT) are selected for illustration of principle. Numerical results show that our multireference DFT successfully describes static correlation in bond dissociation and double bond rotation.

On low-lying excited states of extended nanographenes.

Low-lying excited states of planarly extended nanographenes are investigated using the long-range corrected (LC) density functional theory (DFT) and the spin-flip (SF) time-dependent density functional theory (TDDFT) by exploring the long-range exchange and double-excitation correlation effects on the excitation energies, band gaps, and exciton binding energies. Optimizing the geometries of the nanographenes indicates that the long-range exchange interaction significantly improves the CC bond lengths and amplify their bond length alternations with overall shortening the bond lengths. The calculated TDDFT excitation energies show that long-range exchange interaction is crucial to provide accurate excitation energies of small nanographenes and dominate the exciton binding energies in the excited states of nanographenes. It is, however, also found that the present long-range correction may cause the overestimation of the excitation energy for the infinitely wide graphene due to the discrepancy between the calculated band gaps and vertical ionization potential (IP) minus electron affinity (EA) values. Contrasting to the long-range exchange effects, the SF-TDDFT calculations show that the double-excitation correlation effects are negligible in the low-lying excitations of nanographenes, although this effect is large in the lowest excitation of benzene molecule. It is, therefore, concluded that long-range exchange interactions should be incorporated in TDDFT calculations to quantitatively investigate the excited states of graphenes, although TDDFT using a present LC functional may provide a considerable excitation energy for the infinitely wide graphene mainly due to the discrepancy between the calculated band gaps and IP-EA values. © 2017 Wiley Periodicals, Inc.

Different conical intersections control nonadiabatic photochemistry of fluorene light-driven molecular rotary motor: A CASSCF and spin-flip DFT study

We report the light-driven isomerization mechanism of a fluorene-based light-driven rotary motor (corresponding to Feringa's 2nd generation rotary motor, [M. M. Pollard et al., Org. Biomol. Chem. 6, 507-512 (2008)]) at the complete active space self-consistent field (CASSCF) and spin-flip time-dependent density functional theory (TDDFT) (SFDFT) levels, combined with the complete active space second-order perturbation theory (CASPT2) single-point energy corrections. The good consistence between the SFDFT and CASSCF results confirms the capability of SFDFT in investigating the photoisomerization step of the light-driven molecular rotary motor, and proposes the CASPT2//SFDFT as a promising and effective approach in exploring photochemical processes. At the mechanistic aspect, for the fluorene-based motor, the S1/S0 minimum-energy conical intersection (MECIs) caused by pyramidalization of a fluorene carbon have relatively low energies and are easily accessible by the reactive molecule evolution along the rotary reaction path; therefore, the fluorene-type MECIs play the dominant role in nonadiabatic decay, as supported by previous experimental and theoretical works. Comparably, the other type of MECIs that results from pyramidalization of an indene carbon, which has been acting as the dominant nonadiabatic decay channel in the stilbene motor, is energetically inaccessible, thus the indene-type MECIs are "missing" in previous mechanistic studies including molecular dynamic simulations. A correlation between the geometric and electronic factors of MECIs and that of the S1 energy profile along the C═C rotary coordinate was found. The findings in current study are expected to deepen the understanding of nonadiabatic transition in the light-driven molecular rotary motor and provide insights into mechanistic tuning of their performance.

Ab Initio Molecular Dynamics Study of the Photoreaction of 1,1′-Dimethylstilbene upon S0 → S1 Excitation

Ab initio molecular dynamics (AIMD) simulations were carried out for ππ*-excited 1,1'-dimethylstilbene (dmSB) at the spin-flip time-dependent density functional theory (SF-TDDFT) level with the TSF-index technique, to get insights into the substitution effects on the photoisomerization dynamics of stilbene (SB). It is found that the reaction path from the Franck-Condon structure of cis-dmSB is oriented toward the 4,4-dihydrophenanthrene (DHP) side from the beginning, which is in contrast to the case of SB where the pathway is oriented toward the twist side in the initial stage. The optimized geometries of minima and the minimum-energy conical intersection (MECI) suggested that molecules in the DHP region could easily decay to the ground state. On the other hand, S1/S0-MECI and S1-minimum in the twist region have a relatively different geometry from each other, which is consistent with the experimental observation of the long lifetime of the perpendicular structure. AIMD simulations showed that more trajectories enter the well of the DHP side than the well of the twist side and that all of the trajectories going to the DHP side reached the S1/S0-CI region within ∼0.2 ps on average, while very few trajectories reached S1/S0-CI even after 1 ps in the twist region. Decrease in the S1 population in the cis and twist regions qualitatively reproduced the temporal profiles of the transient absorption bands of dmSB observed in the visible and ultraviolet regions, respectively.

Spin-flip, tensor equation-of-motion configuration interaction with a density-functional correction: A spin-complete method for exploring excited-state potential energy surfaces.

We revisit the formalism of the spin-adapted, spin-flip (SA-SF) configuration-interaction singles (CIS) method based on a tensor equation-of-motion formalism that affords proper spin eigenstates without sacrificing single-reference simplicity. Matrix elements for SA-SF-CIS are then modified in a manner similar to collinear spin-flip time-dependent density functional theory (SF-TDDFT), to include a DFT exchange-correlation correction. The performance of this method, which we call SA-SF-DFT, is evaluated numerically and we find that it systematically improves the energies of electronic states that exhibit significant spin contamination within the conventional SF-TDDFT approach. The new method cures the state assignment problem that plagues geometry optimizations and ab initio molecular dynamics simulations using traditional SF-TDDFT, without sacrificing computational efficiency, and furthermore provides correct topology at conical intersections, including those that involve the ground state, unlike conventional TDDFT. As such, SA-SF-DFT appears to be a promising method for generating excited-state potential energy surfaces at DFT cost.

Exploration of Quenching Pathways of Multiluminescent Acenes Using the GRRM Method with the SF-TDDFT Method.

The quenching pathways were investigated for three types of multiluminescent acene derivatives, which show environment-dependent fluorescence. Spin-flip time dependent density functional theory (SF-TDDFT) combined with the Global Reaction Route mapping (GRRM) strategy is employed to locate minimum-energy conical intersections (MECIs). The energies and geometries of the MECIs relative to the Franck-Condon (FC) state control the difference in fluorescence behavior among the three derivatives. For the molecule with a phenyamide moiety, a MECI with energy lower than the FC state with large geometrical change from V-type to flat structure provides an efficient internal conversion (quenching) pathway in solution. For the same molecule, in a solid, this large geometrical change is inhibited, and the second MECI, with an energy lower than FC but higher than the first MECI requiring only a small geometry change of CH out-of-plane bending, contributes to the quenching. The molecule with the napthaleneimide moiety has only one low-energy MECI that requires large geometrical change from the V-type to flat structure. Although this MECI provides the quenching pathway in solution, in the solid, this large motion is inhibited, and the molecule will stay in the excited state and emit. The molecule with an anthraceneimide moiety has no conical intersection lower than the FC state, and no quenching pathway is available in solution or solid. In addition, in this molecule, at the local minimum of the excited state, the dipole transition to the ground state is allowed, and this molecule prefers emission rather than internal conversion.

Quantum Chemical Investigation of Light-Activated Spin State Change in Pyrene Coupled to Oxoverdazyl Radical Center.

Low-spin ground states and low-lying excited states of higher spin were investigated for four pyrene oxoverdazyl monoradicals 1-4 and eight pyrene dioxoverdazyl diradicals 5-12. The ground states for quartet and quintet spin symmetries that are in reality excited states were found in the region of 565-775 nm above the respective electronic ground states. We calculated the "adiabatic" magnetic exchange coupling constant in the electronic ground state of each isolated biradical (5-12) by unrestricted density functional theory. A number of hybrid functionals such as B3LYP, PBE0, M06, and M06-2X were used. We also used range-separated functionals such as LC-ωPBE and ωB97XD to compare their effects on the coupling constant and the relative energy of the high-spin state. Molecular geometries were optimized for the doublet and quartet spin states of every monoradical (1-4), and the broken symmetry and triplet solutions were optimized for every biradical (5-12), by systematically using 6-311G, 6-311G(d,p), and 6-311++G(d,p) basis sets with each functional. The geometry of each quintet diradical (5-12) was optimized using 6-311G basis set. B3LYP produced the best spin values. The excited state (quartet or quintet)-ground state energy difference (ΔE) increases in the presence of para-phenylene connectors. These energy differences were predicted here. The nature of spin coupling and consequently the ground state spin agree with spin alternation rule and the calculated atomic spin population. The adiabatic coupling constants were predicted for the biradicals (5-12) in their electronic ground states. Electron paramagnetic resonance parameters were determined at 6-311++G** level for the ground state and the quartet state of 1 and compared with the available experimental data. Low-lying excited states were found for the radical center (oxoverdazyl), pyrene, molecule 1, and diradical 5 by time-dependent density functional theory (TDDFT) method using B3LYP hybrid, 6-311++G(d,p) basis set, and the molecular geometry in the electronic ground state. Data from these calculations were used to discuss possible mechanisms for the achievement of the high-spin (excited) states in 1 and 5 and to predict a similar outcome for radicals 2-4 and 6-12 upon excitation. A comprehensive mechanism for the first excitation is proposed here. In particular, we show that the initial excitation of 1 involves large contributions from mixed transitions between pyrene and oxoverdazyl moieties, whereas the initial excitation of 5 is basically that of only the pyrene fragment. Subsequent internal conversion and intersystem crossing are likely to lead to the high-spin states of lower energy. Sample spin-flip TDDFT calculations were also done to confirm the energetic location and composition of the quartet state of 1 and the quintet state of 5.

Singlet-triplet energy gaps for diradicals from particle-particle random phase approximation.

The particle-particle random phase approximation (pp-RPA) for calculating excitation energies has been applied to diradical systems. With pp-RPA, the two nonbonding electrons are treated in a subspace configuration interaction fashion while the remaining part is described by density functional theory (DFT). The vertical or adiabatic singlet-triplet energy gaps for a variety of categories of diradicals, including diatomic diradicals, carbene-like diradicals, disjoint diradicals, four-π-electron diradicals, and benzynes are calculated. Except for some excitations in four-π-electron diradicals, where four-electron correlation may play an important role, the singlet-triplet gaps are generally well predicted by pp-RPA. With a relatively low O(r(4)) scaling, the pp-RPA with DFT references outperforms spin-flip configuration interaction singles. It is similar to or better than the (variational) fractional-spin method. For small diradicals such as diatomic and carbene-like ones, the error of pp-RPA is slightly larger than noncollinear spin-flip time-dependent density functional theory (NC-SF-TDDFT) with LDA or PBE functional. However, for disjoint diradicals and benzynes, the pp-RPA performs much better and is comparable to NC-SF-TDDFT with long-range corrected ωPBEh functional and spin-flip configuration interaction singles with perturbative doubles (SF-CIS(D)). In particular, with a correct asymptotic behavior and being almost free from static correlation error, the pp-RPA with DFT references can well describe the challenging ground state and charge transfer excitations of disjoint diradicals in which almost all other DFT-based methods fail. Therefore, the pp-RPA could be a promising theoretical method for general diradical problems.

Photochemical ring opening and closing of three isomers of diarylethene: spin-flip time-dependent density functional study.

The reaction mechanism of photochemical ring opening and closing transformation was investigated for diarylethene (DAE), which works as a molecular switch and photodevice. Spin-flip time-dependent density functional theory is employed to map the potential energy surfaces and to elucidate the photochemical mechanism of three isomers (normal, inverse, and mixed types) of 1,2-dithienylethene, a model DAE. The potential energy characteristics including the minimum-energy conical intersection reveals the origin of different product preferences of the three isomers. For the normal type, the excited state from either closed or open form reaches the same conical intersection that gives preferentially the closed product. The inverse type preferentially gives the closed product. The mixed type has two pathways that are easily convertible, and both open and closed reactants give both open and closed products.

Analytic derivative couplings in time-dependent density functional theory: Quadratic response theory versus pseudo-wavefunction approach.

We revisit the formalism for analytic derivative couplings between excited states in time-dependent density functional theory (TDDFT). We derive and implement these couplings using quadratic response theory, then numerically compare this response-theory formulation to couplings implemented previously based on a pseudo-wavefunction formalism and direct differentiation of the Kohn-Sham determinant. Numerical results, including comparison to full configuration interaction calculations, suggest that the two approaches perform equally well for many molecular systems, provided that the underlying DFT method affords accurate potential energy surfaces. The response contributions are found to be important for certain systems with high symmetry, but can be calculated with only a moderate increase in computational cost beyond what is required for the pseudo-wavefunction approach. In the case of spin-flip TDDFT, we provide a formal proof that the derivative couplings obtained using response theory are identical to those obtained from the pseudo-wavefunction formulation, which validates our previous implementation based on the latter formalism.

Vertical excitation energies of linear cyanine dyes by spin-flip time-dependent density functional theory

Vertical excitation energies of linear cyanine dyes are examined using the spin-flip time-dependent density functional theory. The Hartree–Fock exchange (HFX) plays an essential role in predicting the absorption spectra, and the best values are obtained by the combination of collinear approximation and hybrid functionals with ∼50% HFX. The non-collinear approach with pure density functionals underestimates the excitation energy severely. The significant error is due to low excitation energy from the reference triplet to first excited singlet state. The excitation energy decomposition gives small orbital energy difference term and large negative non-collinear kernel.