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1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2210 Femtosecond torsional relaxation Theoretical methodology: J. Clark, S. Tretiak, T. Nelson, G. Cirmi & G. Lanzani To model non-adiabatic excited state dynamics in photoexcited polyfluorene pentamer and rationalize experimentally observed relaxation tamescales and concomitant conformational motions, we use recently developed NA-ESMD code. The NA-ESMD approach combines the Tully s Fewest Switches Surface Hopping (FSSH) algorithm, as it is used in the Molecular Dynamics with Quantum Transitions (MDQT) method [1], with the ''on the fly" calculation of the electronic energies, gradients, and nonadiabatic coupling vectors for the excited states. The configuration interaction singles (CIS) or time-dependent Hartree-Fock (TDHF) formalism combined with the Austin Model 1 (AM1) semiempirical model Hamiltonian serves as a numerically efficient technique for computing excited states in large systems. A detailed description of the NA-ESMD implementation can be found elsewhere [2,3]. MDQT treats the electronic degrees of freedom quantum mechanically, while the motion of the nuclei is treated classically. The nuclei of each trajectory are evolved on a single adiabatic potential energy surface (PES) following the classical Langevin equation of motion [4]. The probability for a quantum transition from the current excited state to any other excited state depends on the strength of the nonadiabatic couplings between excited states calculated at each integration step along the trajectory. An appropriate statistical sampling is required to compute detailed photoinduced wavepacket evolution (such as branching and relaxation timescales). Consequently, a swarm (typically few hundreds) of classical trajectories with initial conditions selected should be propagated along different excited state PESs. Such treatment of nonadiabatic effects allows to preserve the detailed balance conditions and to account for trajectory branching. Consequently, surface hopping methods have been extensively explored worldwide by providing a reasonable compromise between an accuracy and computational cost [5,6]. Theoretical results: Figure 1A shows the chemical structure of polyfluorene pentamer. The AM1 ground state optimal geometry has ~42 0 torsional angle between neighboring fluorine units, a value consistent with previous theoretical results [7]. The structure of significant electronic levels is schematically shown in Fig. 1B. Transitions S 0 -S 1, S 0 -S m, and S 1 -S n are optically allowed, permitting to interrogate the excited state dynamics pertinent to S 1, S n and S m states via ultrafast pump-probes. All these three states are excitonic transition with various degree of delocalization. Previously develop two-dimensional analysis of the transition density matrices [7] allows to quantify the spatial extent of the excitation corresponding to the separation between an electron and a hole (excitonic size). Figure 1C shows these transition density matrices plots for S 1, S n and NATURE PHYSICS 1

2 S m states. Each plot depicts probabilities of an electron moving from one molecular position (horizontal axis) to another (vertical axis). S1 state is a strongly bound exciton corresponding to the lowest band-gap transition. The exciton size (maximal distance between electron and hole) is less than 2 fluorene units (largest off-diagonal extent of the nonzero matrix area). A S n state is a very delocalized transition with weak electron-hole interaction due to their large spatial separation of about 3 repeat units. The energy difference between S 1 and S n state has been previously used to evaluate exciton binding energies in a number of conjugated polymers. Finally, S m state is delocalized benzene-like transition where both an electron and a hole are confined to a single phenyl ring. Ground and excited state linear absorption spectra calculated at AM1-CIS level are shown in Fig. 2. These spectra have been calculated at the ground state optimal geometry using Gaussian lineshape with empirical 0.1eV broadening parameter. In spite of many excited state present (Fig. 2A), most of these transitions are forbidden by selection rules. Consequently, only two states appear in the ground state absorption (S 0 -S 1 and S 0 -S m transitions), and one major peak shows up in the excited state absorption (S1-Sn transition). Overall, calculated spectra qualitatively agree with experimental results. We note that the computed energy of the S n state is blue-shifted compared to the experimental value, which is a known fault of single reference methods due to lack of higher-level electronic correlations. To investigate coupling of these states to the torsional degrees of freedom, we next calculate the potential energy slices scanned along the torsional coordinate. Figure 3 displays the resulting curves and emphasizes that the state delocalization is directly relevant to the steepness of the torsional potential. S m state is very localized, so it has torsional potential similar to the ground state S 0. The S 1 state is delocalized to some extent, consequently a flatter excited state geometry is favored with torsion of ~30 0. Finally, the S n state, being the most delocalized, is very sensitive to the torsions having the steepest torsional potential out of 4 considered states. Finally, to check if the semiempirical modeling reproduces torsional potentials calculated with higher accuracy methodology, we use Time-dependent Density Functional Theory technique as implemented in the Gaussian 09 package. Figure 4 shows the resulting scans calculated for S 0 and S 1 states at B3LYP/6-31G* and CAM-B3LYP/6-31G* levels. B3LYP is a very popular hybrid functional with 20% of orbital exchange component. However, in the context of conjugated polymers, it tends to overestimate the excitonic delocalization. CAM-B3LYP is its asymptotically corrected variation where the fraction of the electronic orbital exchange varies from 20% to 65% depending on the distances between the centers. Comparison of Fig. 3 and Fig. 4 reveal torsion potentials for S 0 and S 1 states show the same qualitative trends for all methods used, which suggests that AM1-CIS approach being numerically inexpensive but adequate technique. Our next step is modeling of the photoinduced dynamics of PFO-pentamer at room temperature (300 K). We start with the AM1 ground state molecular dynamics simulations of 400 ps long Born-Oppenheimer (BO) trajectories at 300 K with the time step Δt = 0.5 fs. The system was heated and allowed to equilibrate to a final temperature of 300 K during the first 20 ps. The 2

3 Langevin thermostat was used to keep the temperature constant with a friction coefficient ζ = 2.0 ps -1. The rest of the trajectory was used to collect a set of initial positions and momenta for the subsequent simulations of the excited states. Configurations were sampled for a total of 500 configurations. All of the subsequent data should be interpreted as the average value over the swarm of 500 trajectories. Excited-state dynamics trajectories were started from these initial configurations after photoexcitation. We considered three distinct cases. Case I: Dynamics of the ground state S 0 in which no photoexcitation was produced. Here the nuclei are propagated on a single BO PES. These simulations serve as a reference point. Case II: Dynamics after a vertical excitation to the lowest excited S 1 state. Here the nuclei are also propagated on a single BO PES of the S 1 state. Finally, cases III and IV correspond to the photoexcited dynamics after populating the highly excited S n and S m states, respectively. Here we target photoinduced dynamics of the interband relaxation via multiple BO surfaces as the system passes nonadiabatic regions and state switches occur. Fifty lowest excited states have been calculated for every snapshot to determine Energetic positions of S n and S m states. We further use NA-ESMD code to propagate all trajectories for 500fs at 300 K. A classical step Δt = 0.1 fs and Nq = 3 quantum steps were used in all simulations, resulting in about a week of computational time for trajectories involving nonadiabatic relaxation from S n and S m states using 50 excited states to allow upward energy transitions. Overall, only a few hops are observed to the higher excited-state energies. The top panel in Fig. 5 shows variation of the potential energies averaged over 500 trajectories during the S 0, S 1, and S m and S n simulations. As expected, the potential energy remains constant during simulation of the ground state evidencing that the simulated wavepaket represents wellequilibrated conformational space of the molecule. After the S 1 photoexcitation, the system is drawn away from the equilibrium, and subsequent during electron-vibrational dynamics the excess of vibrational energy is dissipated to phonons. A few wiggles in the beginning of the trajectory (0-30 fs) correspond to a coherent (in phase) C-C vibrational excitation across the ensemble of all snapshots (i.e., coherent phonons). Following vibrational relaxation, S 1 remains at a constant average energy. Molecular dynamics of the pentamer are shown in movie form in the accompanying supplementary gif files. Both Born-Oppenheimer and nonadiabatic dynamics occur simultaneously in the S m and S n simulations. The molecule undergoes vibrational relaxation on a single PES and can also hop to a different electronic state where the system will undergo vibrational relaxation on the new PES. Subsequently, the population of the S 1 state shown in Fig. 5 (bottom) steadily rises reflecting the wavepacket relaxation timescale. Note, the ground state remains unpopulated during our nonadiabatic dynamics simulation since the nonradiative relaxation over the large gap from excited states to the ground state occurs on a much slower (nanosecond) timescales. 3

4 The results of our simulations unambiguously show anomalous ultrafast relaxation S n -S 1 rate, which is several times faster, compared to S m -S 1 relaxation timescale, even though the S n -S 1 and S m -S 1 transition energies are comparable. Thus nearly 2eV of electronic energy is getting damped into the phonon degrees of freedom within ~100fs during S n -S 1 relaxation. These findings compare well and confirm experimental data discussed in this article. Such ultrafast relaxation from the Sn state occurs as a combination of several reasons. First we note that that the underlying non-adiabatic dynamics samples multiple excited states (see Fig. 1A) lying in between S n (S m ) and S 1. Most of these states are delocalized excitons []. Generally, the nonadiabatic coupling vector between states I and J ( d IJ = ψ I (r;r ) R ψ J (r ;R ) ) is large when both states are delocalized and are coupled to the same vibrational modes (here C-C stretches and torsions). In contrast to Sn, non-adiabatic couplings between localized states S m to the majority of delocalized states below it are limited due to small wavefunction overlaps and coupling to different vibrational modes (e.g., S m is decoupled from torsional motion). Consequently, S n undergoes efficient non-adiabatic relaxation via dense manifold of delocalized excited states, whereas S m cannot utilize this pathway efficiently. Moreover, PESs of delocalized excitons (particularly S n ) are relatively steep along not only torsional degree of freedom, but also along the high-frequency C-C stretching motion. Consequently, the wavepacket gains substantial velocity along these two strongly coupled vibrational coordinates, transferring an excess of the electronic energy into vibrations. Such significant vibrational couplings are also getting reflected on the larger magnitude of the non-adiabatic coupling vector. Subsequently, if a hop is realized, following Tully s prescription, the excess of electronic energy is distributes into the vibtational kinetic energy in the direction of the non-adiabatic coupling vector, thus, providing an additional efficient channel for electronic energy dumping into phonon modes. Altogether, these processes continue fuelling specific vibrational motions ( sinks of electronic energy) during the entire non-adiabatic relaxation. Figure 6 demonstrates that the molecule efficiently planarizes locally during S n relaxation, being a very special case among all dynamics considered. The top panel in Fig. 6 shows that the average torsion angle does not change significantly in the respective dynamics of S 0, S 1, S n and S m states: the S 0 torsion barely varies around its equilibrium value, whereas S 1, S n, and S m torsions weakly drift to the lower values (Fig. 6, top). A different picture emerges if we will follow the evolution of the smallest torsional angle out of four along the chain. Here S 0 and S m do not change as a reflection of the flat potential energy surface with respect to the torsional coordinates (see Fig. 3). S 1 torsion reduces by ~5 degree over 400fs showing Born-Oppenheimer relaxation along the torsional PES. Finally, non-adiabatic dynamics starting from the S n state quickly flattens molecule locally (typically 2 out of 5 repeat units are getting flat), taking an advantage of constant infusion of electronic energy during non-adiabatic relaxation. Notably, that the resulting near-zero average torsional angle appear as arithmetic averaging of minor portion of trajectories that overshoot by flipping the neighboring fluorine units to the negative value of the respective dihedral angle with other 4

5 trajectories. Finally, we would note that due to significant anharmonicities, torsional motions can efficiently re-distribute an excess of the vibrational energy into the other vibrations and bath degrees of freedom, playing the role of a strongly overdamped oscillator in this relaxation process. In our simulations, however, dissipation of the energy into the bath degrees of freedom is captured only qualitatively due to an empirical friction coefficient ζ = 2.0 ps -1 used in the numerical simulations. 5

6 Figure 1. A) Calculated ground state optimized geometry of PFO pentamer. B) Scheme of photoinduced pathways after excitation of S m and S n states relaxing to the bend-gap excitation S 1. Yellow lines show multiple excited states in between S 1 and S n /S m states. C) Two-dimensional plots of the transition density matrices of excited states in question, showing relatively delocalized S 1 transition (strongly bound exciton), very delocalized S n transition (nearly unbound exciton), and very localized (on single phenyl ring) S m transition. 6

7 Figure 2. A) Calculated density of excited states of PFO pentamer. B) Calculated ground state absorption spectrum showing dipolar allowed peaks due to S 0 -S 1 and S 0 -S m transitions. C) Calculated excited (S 1 ) state absorption spectrum showing dipolar allowed peak due to S 1 -S n transition, distinctly separation so called ma g and ka g state contributions. 7

8 Figure 3. Slices of the molecular multidimensional PESs along the torsion coordinate for S 0, S 1, S n, and S m states calculated at AM-1-CIS level. The torsion angle was simultaneously changed between units as shown by red arrows. For each torsion angle, a few dynamics steps are made to allow relaxation along the fast degrees of freedom. 8

9 Figure 4. Slices of the molecular multidimensional PESs along the torsion coordinate for S 0 and S 1 calculated using time-dependent Density Functional Theory (TDDFT) approach at B3LYP/6-31G* and CAM-B3LYP/6-31G* levels. The torsion angle was simultaneously changed between units as shown by the red arrows starting from the ground state optimal geometry. 9

10 Figure 5. The results of photoinduced dynamics simulations for S 0, S 1, S n and S m states, averaged over 500 trajectories to obtain statistical averages. Top: variation of the average potential energy. The curve for S 0 is flat, showing no time-dependence; S 1 state shows weak vibrational relaxation over the first 10fs; potential energy of S n quickly drops to the S 1 value within ~100fs, showing ultrafast non-adiabatic relaxation; potential energy of S m state decreases on much slower timescale (~400fs). Bottom: An increase of population of S 1 state during non-adiabatic dynamics starting from S m (slow) and S n (fast) states. 10

11 Figure 6. Variation of torsion angles during Born-Oppenheimer dynamics of S 0 and S 1 states, and non-adiabatic dynamics of S n and S m states. Top: the average values over 4 dihedral angles on each molecule were then averaged over 500 molecules (snapshots) for a given moment of time along the trajectory. Bottom: the value of the minimum dihedral angle on each molecule was then averaged over 500 molecules (snapshots) for a given moment of time along the trajectory. 11

12 References: [1] Tully, J. J. Chem. Phys. 93, (1990). [2] T. Nelson, S. F. Alberti, V. Chernyak, A. Roitberg, S. Tretiak, J. Phys. Chem. B 115, 5402 (2011). [3] S. F. Alberti, V. Kleiman, S. Tretiak, A. Roitberg J. Phys. Chem. Lett., (2010). [4] Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: USA, 1990 [5] Craig, C. F.; Duncan, W. R.; Prezhdo, O. V. Phys. Rev. Lett. 95, (2005). [6] Duncan, W. R.; Prezhdo, O. V. Annu. Rev. Phys. Chem. 58, (2007). [7] Chen et al., J. Phys. Chem. B, 113 (25), (2009). [8] Tretiak, S.; Mukamel, S. Chem. Rev. 102, (2002). 12