Trajectory Surface-Hopping Dynamics Including Intersystem Crossing

Letter pubs.acs.org/JPCL

Trajectory Surface-Hopping Dynamics Including Intersystem Crossing in [Ru(bpy)3]2+ Andrew J. Atkins* and Leticia González* Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, A-1090 Vienna, Austria S Supporting Information *

ABSTRACT: Surface-hopping dynamics coupled to linear response TDDFT and explicit nonadiabatic and spin−orbit couplings have been used to model the ultrafast intersystem crossing (ISC) dynamics in [Ru(bpy)3]2+. Simulations using an ensemble of trajectories starting from the singlet metal-to-ligand charge transfer (1MLCT) band show that the manifold of 3MLCT triplet states is first populated from high-lying singlet states within 26 ± 3 fs. ISC competes with an intricate internal conversion relaxation process within the singlet manifold to the lowest singlet state. Normal-mode analysis and principal component analysis, combined with further dynamical simulations where the nuclei are frozen, unequivocally demonstrate that it is not only the high density of states and the large spin−orbit couplings of the system that promote ISC. Instead, geometrical relaxation involving the nitrogen atoms is required to allow for state mixing and efficient triplet population transfer.

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(bpy)3]2+ or any transition metal complex. In 2011, Tavernelli and co-workers used surface-hopping to study the photoexcitation of [Ru(bpy)3]2+ in solution using energies and forces on-the-fly for singlet states but estimating a posteriori the probability for ISC.29 This approach coupled to linear response time-dependent density functional theory30−32 (LR-TDDFT) and two trajectories provided an ISC time to the 3MLCT state of ∼50 fs; the solvent-mediated electron localization and dynamics in the long-lived 3MLCT state were then studied for several picoseconds by hybrid DFT/classical molecular dynamics simulations.33 A similar approach has been used to investigate the ultrafast relaxation dynamics of photoexcited [Cu(dmp)2]+ employing nine trajectories, where only nonadiabatic transitions between singlet states were allowed and the ISC probabilities where obtained from Landau−Zener probabilities calculated a posteriori at the crossing points between the singlet active state and the energetically close triplet states.34 A related approach to manually estimate ISC probabilities based on Landau−Zener probabilities was employed to simulate the photoinduced dynamics of a Ru nitrosyl complex.35 ISC rate constants can also be directly calculated from the Golden Rule,36 as, for example, done in ref 37 for the fac-tris(2-phenylpyridine)iridium ( fac-Ir(ppy)3) complex. The benefit of the latter method is that it can predict much longer ISC time scales than direct propagation methods; however, it misses the electronic and geometrical changes underlying the ISC process. Moreover, the Golden Rule estimates of the ISC rate are valid only for long times, when stationary wave functions have been formed. In ultrafast ISC

etal-polypyridine complexes are commonly used as photosensitizers in photovoltaics, photocatalysis, and photodynamical therapy1−4 due to their remarkable photophysical properties.5 After photoexcitation, these complexes typically undergo charge transfer from the d orbital centered on the metal to the π orbital of the ligand, that is, they are promoted to a so-called metal-to-ligand-charge transfer (MLCT) state. One feature exploited in many of these applications is their efficient intersystem crossing (ISC), that is, nonradiative decay from the singlet 1MLCT to a triplet 3 MLCT state, as well as the long stability of this triplet state.6 Among such complexes, ruthenium trisbipyridine ([Ru(bpy)3]2+) is considered a prototype for this family of compounds. A number of studies using time-resolved techniques have evidenced that ISC in [Ru(bpy)3]2+ occurs in less than 30 fs7,8 with almost unity yield.9,10 However, the intramolecular redistribution of vibrational energy of the molecule during and after this process has been largely discussed without a clear-cut mechanism underlying the ultrafast ISC.11−15 Because ISC involves changes in the electronic structure that should be accompanied by corresponding geometrical changes, it can be best monitored by ultrafast optical methods16 and by nonadiabatic dynamical simulations.17,18 Regarding the latter, a number of approaches are currently available to perform fulldimensional dynamical simulations including ISC, one being SHARC (Surface-Hopping including ARbitrary Couplings).18,19 Following SHARC and the seminal work of Persico,20 other strategies have followed that can deal with spin−orbit coupling and nonadiabatic dynamical effects on-the-fly, including that of Thiel21 and Tavernelli.22 While for small organic molecules these methods have demonstrated their wide applicability,23−28 none of these approaches have yet been applied to [Ru© XXXX American Chemical Society

Received: June 11, 2017 Accepted: August 2, 2017 Published: August 2, 2017 3840

DOI: 10.1021/acs.jpclett.7b01479 J. Phys. Chem. Lett. 2017, 8, 3840−3845

Letter

The Journal of Physical Chemistry Letters

eV to take into account the possible variance of the incident wavelength and inaccuracy of the calculated energies. The first interesting fact to realize is that even if the excitation energy range is small (0.5 eV) there is congestion of electronic states lying very close energetically. Therefore, the trajectories cannot be simply started from the bright S8 and S9 states, but a large number of trajectories is necessary to adequately represent the initial excitation as occurring experimentally. Accordingly, from the pool of 1500 initial conditions, 101 random permissible trajectories within the chosen excitation range were initially distributed following the ratio of absorbing states within the chosen energy range: 1 in S6, 6 in S7, 18 in S8, 23 in S9, 14 in S10, 10 in S11, 7 in S12, 12 in S13, and 10 in S14. The trajectories were then propagated for 30 fs with a nuclear time step of 0.5 fs and an electronic time step of 0.02 fs, within 15 singlet and 15 triplet states, that is, 60 states, taking into account the multiplet components of the triplet.18 The spin−orbit couplings are collected in Table S2 and depicted schematically in Figure S2. Figure 2A shows the time evolution of the participating singlet and triplet states normalized for all of the trajectories. Already after a few fs, very efficient ISC to the triplet states can be observed. The rapid increase in triplet population, tailing off to ∼65% total triplet population after 30 fs, concomitant with the immediate decrease of singlet population, is more evident summing up all singlet and triplet states (Figure 2B). A global fitting of the population decay and the bootstrapping method45 provides a time constant of 26 ± 3 fs, in very good agreement with the experimental time constant of 15 ± 10 fs provided for the 1MLCT state decay.8 Using this time constant, a triplet yield of 99% requires ∼120 fs, which is also in very good agreement with the experimental observation that the 3MLCT state is populated within 110 fs.8 One should note here that it is not a single state that is depopulated or populated but that it is a thick manifold of near-degenerate states responsible for the singlet−triplet population transfer and associated time scales. A closer look at the individual state populations (Figures 2A and S3) reveals that the lowest-lying singlet states take some time to populate nonadiabatically from the higher ones, while the lowest triplet states have a higher population in the same time period. As an example, the S1 state only starts populating after ∼25 fs and reaches 1% after 30 fs. In comparison, the T1 is populated after 20 fs and after 30 fs reaches 3% population (see also Table S3). Before the S1 and T1 are populated, both internal conversion from the brighter S9 and S8 states and ISC from high-lying singlet states to high-lying triplet states compete from the onset of the simulation; see the inset in Figure S3. Clearly, as there is only a minor increase of population of the lower singlet states upon a decrease in the higher singlet states, ISC does not occur following Kasha’s rule,46 according to which, first, “vertical” nonadiabatic deactivation from the highly excited singlet excited states all the way down to the S1 state takes place and then ISC to the triplet manifold occurs. Instead, in [Ru(bpy)3]2+, ISC is a “horizontal” process between high-lying singlet and triplet states, as suggested by Cannizzo and co-workers8 and proposed in other Ru polypyridyl complexes47 and Ir complexes,48 followed by further relaxation within the triplet manifold. As shown in Figure 2A, it takes about 25 fs for the S1 to populate, from which fluorescence can take place.8 Such a “horizontal” ISC process could hint that it is essentially the high number of pseudodegenerate excited states and the substantial spin−orbit coupling between them

cases, the notion of a stationary wave function fails and the Golden Rule cannot be applied. In this Letter, we report the first trajectory surface-hopping dynamics study on [Ru(bpy)3]2+ with spin−orbit couplings onthe-fly and, therefore, able to capture ISC as it takes place. This means that hops between the active singlet and triplet states are not estimated a posteriori but occur stochastically based on the rate of change of the actual electronic populations, allowing one to investigate the precise molecular deformations and vibrational energy redistributions that accompany the population transfer from the singlet to the triplet states in [Ru(bpy)3]2+. To this aim, we use gas-phase LR-TDDFT-based nonadiabatic dynamics, which has been implemented in a local development version of the SHARC suite38 in conjunction with the ADF program package.39 Nonadiabatic couplings are calculated from the electronic wave function overlaps.40 In passing, we note that the present implementation is based on atom-centered basis set LR-TDDFT dynamics, while that employed in refs 22 and 34 relies on a plane wave basis. Energies, gradients, spin−orbit couplings, and wave function overlaps are calculated on-the-fly using the PBE functional41 with the TZP basis set on the Ru atom and DZP on all other atoms, hereafter denoted as TZP(Ru)-DZP. The PBE functional has been chosen as it is most suitable to describe the energy differences between the electronic states.42 Further computational details can be found in the Supporting Information (SI). Figure 1 shows the calculated absorption spectrum generated from a Wigner distribution including 1500 initial geometries,

Figure 1. UV−vis absorption spectrum of [Ru(bpy)3]2+ calculated from a Wigner distribution of 1500 geometries and a fwhm of 0.1 eV per excitation per geometry. The contributing states (S1−S14) are color labeled, as indicated. The gray area denotes the excitation energy range from which trajectories will be launched.

with TD-PBE/TZP(Ru)-DZP excitation energies for 14 electronic excited singlet states. The absorption spectrum compares well to experimental data recorded in solution16 (see Figure S1 in the SI) with a slight red shift (ca. 0.3 eV), which is not uncommon for GGA functionals.43 The gray area indicates the energetic window (2.75 ± 0.25 eV) that will be employed for excitation. In the experiment,8 the system is excited at 400 nm (3.1 eV), that is, 0.15 eV higher in energy than the bright state located at 2.95 eV.44 This maximum can be assigned to the S7 and S8 singlet excited states at the Franck−Condon (FC) or equilibrium geometry (see Table S1) and to the S8 and S9 within the Wigner distribution (see Figure 1). Because these states are predicted theoretically at 2.6 eV, a similar situation as that in the experiment corresponds to excite 0.15 eV higher in energy, that is, at 2.75 eV. We then consider a window of ±0.25 3841


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Figure 2. (A) Time-resolved normalized singlet (bluish) and triplet (brownish) populations over 30 fs. For the triplet states, the different ms states are summed as one. Highlighted states are the S1 in blue, the S8 in green, the S9 in light blue, and the T1 in red. (B) Normalized total population of all of the singlets (pale blue) and all of the triplets (red) summed over all of the trajectories. (C) Normalized total population of the singlets and triplets summed over all trajectories when the initial geometries are frozen (case I). (D) Normalized total population of the singlets and triplets summed over all trajectories run from the FC geometry allowing geometrical changes (case II). (E) Normalized total population of the singlets and triplets summed over all trajectories run from the frozen FC geometry (case III).

Figure 3. (A) Total standard deviation of each normal mode for [Ru(bpy)3]2+ for all trajectories at different time intervals. Modes are only shown if the difference in the total standard deviation between any two intervals is at least 0.09. The values shown are independent of whether the motions are coherent or random and are just an indication of how each mode is activated over all trajectories. (B) Total standard deviation of each normal mode for an averaged trajectory in [Ru(bpy)3]2+ at different time intervals. Modes are only shown if the difference in the standard deviation between any two intervals is at least 0.025. Note the different axis for the average trajectory standard deviation.

(maximum of 350 cm−1) that promotes a spin−flip already at the FC geometry. In order to test this hypothesis, we have performed three additional sets of molecular dynamics simulations. In case I, the initial geometries from the Wigner distribution were frozen during the dynamics (Figure 2C). In cases II and III, only the FC geometry was allowed to evolve from the nine different initial excited states, with (case II) or without (case III) allowing for geometrical relaxation (Figure 2D,E, respectively; see also Figure S4). One should note that by freezing the molecule, internal conversion is shut off because nonadiabatic couplings perturb the dynamics of the system from adiabatic evolution in proportion to the nuclear velocities,

and the values of the spin−orbit couplings are fixed as they depend on the molecular geometry. Therefore, any population transfer within each manifold must occur by successive spin− orbit-mediated transitions between the corresponding manifold. Interestingly, the frozen dynamics (Figure 2C) shows a significant decrease in the population transfer to the triplet states from the singlets (15% in triplets after 30 fs) compared to the “normal” dynamics (Figure 2B). This difference indicates that dynamical molecular relaxation does play a role within the process of ISC in [Ru(bpy)3]2+ and ISC is not only due to the density of states and large spin−orbit coupling. This initial conclusion is further supported by comparing the differences 3842


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normal modes. The resulting modes that change the most are shown in Figure 3B. As can be seen, modes 33, 36, and 60 show the most significant coherence between the different trajectories, and these modes correspond to the motions of the hydrogen, nitrogen, and ruthenium atoms; see, for example, modes 33 and 60 in Figure 4. A principal component analysis or essential dynamics analysis is complementary to the previous normal-mode analysis as it allows one to identify the primary motions being followed within a trajectory. The most relevant motions for each analyzed time interval are shown in Figure S8. The major motions also involve hydrogen bond bending, in agreement with the normal-mode analysis. Three example trajectories starting from the bright S9 state are shown in Figure 5, where ISC occurs immediately at t = 0.5 (A), at 18.0 fs (B), and where it does not occur (C). The first conspicuous fact is that all of the states remain nearly degenerate in time regardless of ISC. The energies of the states are colored according to the spin expectation value, that is, 0 (blue) for singlet and 2 (red) for triplet. Accordingly, it is relevant to note that many states couple strongly during the dynamics, represented by an intermediate (green) color, while others retained most of their singlet (red) or triplet (blue) character. In conclusion, a newly implemented combination of LRTDDFT with SHARC for nonadiabatic ab initio molecular dynamics has been employed to model for the first time the relaxation dynamics of the model complex [Ru(bpy)3]2+ after light irradiation with explicit nonadiabatic and spin−orbit couplings. The propagations show that the decay from the 1 MLCT states to the 3MLCT states occurs with a time constant of 26 ± 3 fs, in very good agreement with the experiment.8 Internal conversion within the singlet manifold is directly competing with ISC. The ISC involves “horizontal” population transfer to high-lying triplet states from which nonadiabatic relaxation via internal conversion to the lowest triplet state also takes place. Accordingly, within the time scale probed here (30 fs), the population remains in the higher-energy singlet and triplet states, while a small population of the T1 and S1 states is present. Such behavior is different from that found in organic molecules, which typically follow Kasha’s rule. Importantly, it is shown that the high density of states and large spin−orbit coupling are not enough to fully promote ISC. Instead, dynamical relaxation is clearly manifested as an important ingredient to enhance ISC, as was also found in a Rhenium complex by Cannizzo and co-workers.51 A normal-mode analysis and principal component analysis identified that the motion on the hydrogens, the motion of the bonded nitrogen atoms, and the motion of the ruthenium are activated significantly. The motions in the nitrogen atoms and the ruthenium atom are expected to be the most important for promoting ISC through excited-state mixing. As the nitrogen vibrations should not be largely affected by water molecules, the present simulations in the gas phase should be a good approximation of the early dynamics of [Ru(bpy)3]2+ after excitation. The effect of the solvent is primarily important on longer time scales than those studied here, as demonstrated in refs 29 and 33. Another relevant conclusion is that, due to the high number of near-degenerate states, numerical simulations in [Ru(bpy)3]2+ require one to sample a large number of states to adequately represent the initial excitation.

between the dynamics of panels D and E in Figure 2; even if in both cases only a single geometry is considered, molecular relaxation enhances ISC as there is more triplet population transfer in case II than in III. The comparison of population between cases I and II (C vs D in Figure 2) also indicates that the inclusion of the zero-point energy and, thus, the initial kinetic energy distributed within the system (see also Table S4) influence and increase substantially the amount of ISC. The natural question now is to identify the internal molecular motions that promote ISC. For this purpose, we performed a time-resolved normal-mode analysis and a principal component analysis of the trajectories;49,50 see the SI for details. The normal-mode analysis allows one to decompose the motions of the molecule into its normal modes taking the D3 FC geometry as a reference, while the principal component analysis or essential dynamics tells us which motion(s) occurs primarily within the dynamics, regardless of whether they can be mapped into a normal mode or not. We analyze first the evolution of the normal modes during the dynamics. From the 177 vibrations (see Table S5) available in [Ru(bpy)3]2+, Figure 3A shows the 12 modes that show significant changes in time over all trajectories. These are expected to influence ISC the most, in particular, those involving the metal center where spin−orbit coupling is dominant. In general, the lower-frequency normal modes tend to have the largest deviations. Most of the represented modes are primarily motions of the hydrogen atoms, and in some cases (e.g., mode 60), additional motion in the Ru−N bond is present. Selected modes and corresponding frequencies are displayed in Figure 4. In order to provide additional information on any concerted motions between the individual trajectories, we now average all of the trajectories into a single one and then analyze again the

Figure 4. Normal modes changing most during the dynamics, selected from both the total standard deviation of all trajectories (Figure 3A) and the averaged trajectory (Figure 3B). 3843


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Figure 5. Example trajectories starting from the S9 state, where ISC occurs at t = 0.5 (A), and 18.0 fs (B) and where it does not occur (C). The dots indicate the active state. The states are labeled by the spin expectation value. The inset magnifies the states around the time that ISC takes place. The arrows indicate the time at which the trajectory is in the triplet state.

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(3) Hagfeldt, A.; Boschloo, G.; Sun, L.; Kloo, L.; Pettersson, H. DyeSensitized Solar Cells. Chem. Rev. 2010, 110, 6595−6663. (4) Berardi, S.; Drouet, S.; Francàs, L.; Gimbert-Suriñach, C.; Guttentag, M.; Richmond, C.; Stoll, T.; Llobet, A. Molecular Artificial Photosynthesis. Chem. Soc. Rev. 2014, 43, 7501−7519. (5) Juris, A.; Balzani, V.; Barigelletti, F.; Campagna, S.; Belser, P.; von Zelewsky, A. Ru(II) polypyridine complexes: photophysics, photochemistry, eletrochemistry, and chemiluminescence. Coord. Chem. Rev. 1988, 84, 85−277. (6) Müller, P.; Brettel, K. [Ru(bpy)3]2+ as a reference in transient absorption spectroscopy: differential absorption coefficients for formation of the long-lived 3MLCT excited state. Photochem. Photobiol. Sci. 2012, 11, 632−636. (7) Bräm, O.; Messina, F.; El-Zohry, A. M.; Cannizzo, A.; Chergui, M. Polychromatic femtosecond fluorescence studies of metalpolypyridine complexes in solution. Chem. Phys. 2012, 393, 51−57. (8) Cannizzo, A.; van Mourik, F.; Gawelda, W.; Zgrablic, G.; Bressler, C.; Chergui, M. Broadband Femtosecond Fluorescence Spectroscopy of [Ru(bpy)3]2+. Angew. Chem. 2006, 118, 3246−3248. (9) Damrauer, N. H.; Cerullo, G.; Yeh, A.; Boussie, T. R.; Shank, C. V.; McCusker, J. K. Femtosecond Dynamics of Excited-State Evolution in [Ru(bpy)3]2+. Science 1997, 275, 54−57. (10) Yeh, A. T.; Shank, C. V.; McCusker, J. K. Ultrafast electron localization dynamics following photo-induced charge transfer. Science 2000, 289, 935−938. (11) Hartmann, P.; Leiner, M. J. P.; Draxler, S.; Lippitsch, M. E. Specific solvent effects of linear alcohols on the emission spectrum and the excited state decay of tris(2,2′-bipyridyl) ruthenium(I). Chem. Phys. 1996, 207, 137−146. (12) Yersin, H.; Humbs, W.; Strasser, J. Low-lying electronic states of [Rh(bpy)3]3+, [Pt(bpy)2]2+, and [Ru(bpy)3]2+. A comparative study based on highly resolved and time-resolved spectra. Coord. Chem. Rev. 1997, 159, 325−358. (13) Bhasikuttan, A. C.; Suzuki, M.; Nakashima, S.; Okada, T. Ultrafast Fluorescence Detection in Tris(2,2‘-bipyridine)ruthenium(II) Complex in Solution: Relaxation Dynamics Involving Higher Excited States. J. Am. Chem. Soc. 2002, 124, 8398−8405.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01479. Additional computational details and results, including a full vibrational mode analysis and principal component analysis (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.J.A.). *E-mail: [email protected] (L.G.). ORCID

Leticia González: 0000-0001-5112-794X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The Austrian Science Fund (FWF), Project M 1815-N28 is acknowledged for funding. The authors thank Sebastian Mai for his input on the SHARC-ADF implementation and Felix Plasser as well as Sebastian Fernandez-Alberti for useful discussion about the normal-mode analysis. The computational results presented have been partially achieved using the Vienna Scientific Cluster (VSC).

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Trajectory Surface-Hopping Dynamics Including Intersystem Crossing

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