Letter pubs.acs.org/JPCL
Chaotic Soliton Dynamics in Photoexcited trans-Polyacetylene Leonardo Bernasconi* Scientific Computing Department, STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, United Kingdom ABSTRACT: We study the photogeneration of topological solitons in trans-polyacetylene and their time evolution using ab initio excited-state dynamics. The system is excited to the optically allowed 11Bu state, and the atoms are then propagated classically using quantum mechanical forces computed using hybrid time-dependent density functional theory (TD-DFT). A soliton/antisoliton pair nucleates spontaneously and creates two independent solitons moving at constant velocity, similar to simulations based on uncorrelated lattice models like the Su−Schrieffer−Heeger (SSH) Hamiltonian [Su, W. P.; Schrieffer, J. R.; Heeger, A. J. Phys. Rev. Lett. 1979, 42, 1698]. At T = 0, the solitons coalesce into bound pairs with a two-soliton functional form, whereas chaotic dynamics, in the form of 2-bounce resonances, is observed at soliton/antisoliton collisions at T ≠ 0. This behavior is related to the onset of a strong correlation regime at short intersoliton distance, which is not accounted for by SSH simulations.
ground-state value u = u0 ≠ 0. Because E0(u0) = E0(−u0), the ground state is doubly degenerate, with ±u0 corresponding to the two possible C−C bond alternation patterns, and is described by a broken-symmetry state in the form of a bond order wave directed along the chain length. The SSH model also supports nonlinear excitations in the form of moving domain walls separating ±u0 phases.1,4,5,7,8 These distortions can be described using the staggered displacement field (SDF)
C
onjugated polymers exhibit a variety of intriguing electronic and optical properties and form a class of very attractive materials for technological applications in photovoltaics, optoelectronics, nonlinear optics, and electromagnetic shielding.2,3 Strong electron−electron and electron− phonon coupling can lead to the appearance of exotic features in the electronic and vibrational response of even the simplest polymers, including topological and nontopological localized lattice distortions.4−6 Trans-polyacetylene (PA), (C2H2)n, is structurally the simplest conjugated polymer. The ground state of PA has been extensively studied using a modified Hückel model, the Su−Schrieffer−Heeger (SSH) lattice Hamiltonian.1 The SSH model accounts for the coupling of the π electrons with the PA backbone distortion via an electron−lattice (e−l) coupling term. The mean amplitude of the distortion u is related to the dimerization coordinate un, specifying the displacement of the nth CH site along the chain direction, by u = N−1 ∑Nn (−1)nun (Figure 1), where N is the number of CH sites. The total adiabatic ground-state energy E0, given by the sum of the electronic and elastic energies, reaches a minimum at the
φn = ( −1)n un
which for long enough chains approaches ±u0 for n → ±∞ and shows deviations from ±u0 around its centroid n0. The spread of the lattice distortion along the chain is measured by a width parameter ξ, and in the continuum limit, the adiabatic energy of the SSH Hamiltonian is minimized by φ ≃ u0 tanh[(n − n0)a/ ξ], with a as the separation between successive C atoms in the chain direction and ξ ≃ 7a for typical SSH parameters. This functional form is characteristic of topological kink solitons, which are well-known solutions of dispersive nonlinear partial differential equations, like the sine-Gordon equation in the large-amplitude wave limit.9 The SSH model provides an accurate description of the e−l interactions in the ground state, but an explicit treatment of electron−electron (e−e) correlation coupling is required in order to obtain a consistent model of the chain dimerization and of the optical response of PA.3,10−12 Correlation effects have been shown to substantially modify the structures and energy ordering of the lowest excited states (e.g., of the lowest optically allowed “ionic” 11B−u state relative to the forbidden “covalent” 21A+g state) of long PA chains relative to the SSH limit.13 This has important implications for the electroluminescence properties of PA.
Figure 1. Dimerization coordinate un in PA. The displacement of the CH sites (black dots) relative to the undimerized structure (gray) by (−1) nu (short arrows) leads to the formation of alternating double (d) and single (s) bonds. © 2015 American Chemical Society
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Received: January 23, 2015 Accepted: February 23, 2015 Published: February 23, 2015 908
DOI: 10.1021/acs.jpclett.5b00159 J. Phys. Chem. Lett. 2015, 6, 908−912
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Figure 2. The 0.025 Å−3 isosurfaces of the HO (a) and LU (b) KS orbital wave functions corresponding to the hole and particle, respectively, in the independent-particle KS approximation. (c) The 0.025 Å−3 LDR isosurface, representing the flow of electron density for the HO → LU excitation at ω = Eo. Different colors indicate the positive and negative phases in all plots. The triangles mark the positions of the double CC bonds. The PA geometry used in this correspond to an instantaneous configuration (t = 1.5 ps) from the excited-state dynamics shown in Figure Figure 3, which is slightly distorted relative to the optimized PA structure owing to thermal relaxation.
response (LDR), is essential to obtain quantitative estimates of excitonic absorption energies and binding energies.20−24 In the approach used in this work, electronic excitation energies ωi are estimated in the linear response and adiabatic approximations from the solution of the generalized eigenvalue equation
Owing to the importance of the electronic correlation, ab initio one-particle theories, for example, Hartree−Fock (HF) and Kohn−Sham (KS) density functional theory (DFT), may not provide a reliable description of the optical response of PA10 (see, however, ref 14). Many-body electronic effects act in two ways, first by screening the HF or KS electron and hole and second by correlating the motion of the screened electron and hole.15 The first contribution typically results in a simple shift in the energies of the KS bands (quasi-particle shift). The screened electron−hole interaction leads to far more substantial deviations from the independent particle limit and may create electronic transitions to gap states with bound excitonic character. These effects have been studied in PA at the twoparticle Green’s function level (LDA+GW+BSE).16 The lowest optically allowed excitation has been identified with a transition to an excitonic state at E0 = 1.7 eV, with a binding energy Eb = Eg − Eo = 0.4 eV (E0 and Eg indicate here the optical and band gaps, respectively), which has been correlated to the intense absorption band experimentally observed at 1.7 eV.17 In this work, we use time-dependent DFT (TD-DFT)18,19 to study (a) the (linear) optical response of an isolated, infinite PA chain and (b) the adiabatic relaxation of the lowest optically allowed excited electronic state and its dynamics at room temperature over time scales of 1−5 ps. As a formally exact theory of the response of a many-electron system to an external time-dependent perturbing potential,15 TD-DFT is in principle able to account for all contributions determining the optical behavior of a finite or infinite system. In practice, however, the accuracy of the TD-DFT optical spectrum depends crucially on the quality of the ground-state exchange−correlation potential and of its response. In particular, spatial nonlocality in the exchange−correlation response kernel, defined as the functional derivative of the KS potential with respect to the linear density
(A − B)1/2 (A + B)(A − B)1/2 Ti = ωi2Ti
(2)
where the RPA-like matrices A and B couple independentparticle transitions corresponding to singly excited configurations (see, e.g., refs 23 and 25 for further details). We solve eq 2 in the Tamm−Dancoff approximation26 (TDA) with the PBE0 exchange−correlation functional27 and the corresponding nonlocal response kernel.28 The TDA has been shown to provide considerably more accurate estimates of excitation energies in polyene chains than TD-DFT itself when used in conjunction with hybrid functionals.29 Transition probabilities are computed from the eigenvectors Ti using the generalized Berry phase method of ref 30. We first consider a 1D periodic supercell structure containing 32 CH units at the experimental value u0 = 0.0245 Å, for which we estimate a KS band gap Eg = 1.993 ± 0.01 eV (where the uncertainty derives from the finite size of the supercell). This value is close to the LDA+GW estimate of ref 16, Eg = 2.1 eV. The good agreement between the DFT/PBE0 band gap and the GW quasi-particle gap of PA is in line with recent findings on the nature of the electronic transport mechanism in conjugated polymers.31 The TD-DFT optical gap amounts to Eo = 1.753 eV, in excellent agreement with the experimental estimate of ref 17. The calculated exciton binding energy is Eb = 0.24 eV. This value is suggestive of a weakly bound Wannier− Mott-like excitonic state, originating from a single-particle excitation of odd symmetry (11B−u ) from the highest occupied 909
DOI: 10.1021/acs.jpclett.5b00159 J. Phys. Chem. Lett. 2015, 6, 908−912
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The Journal of Physical Chemistry Letters (HO) to the lowest unoccupied (LU) KS π orbital (Figure 2a and b). We notice that, consistent with the results of ref 29, our calculations indicate that a transition to the optically allowed 11B−u state, rather than to the dark 21A+g state observed experimentally, is the lowest excitation in PA. The electron− hole attraction leads to a LDR n(1)(r; ω) =
∫ dr′ χ(r, r′; ω)Vext(r′; ω)
(3)
which localizes in real space with a fwhm of about eight CH units at ω = Eo (Figure 2c). Here, χ(r,r′;ω) is the TD-DFT many-body density−density response function, and Vext(r′;ω) is the external perturbing potential oscillating at an energy ω = Eo. The LDR represents the displacement of the electronic density induced by the transition to the 11B−u state and leads to the creation of a localized 11B−u exciton wave function from a KS electron/hole pair spread over the infinite chain. After photoexcitation, we optimize the supercell parameter for the 11B−u state. This calculation is carried out on a smaller supercell (20 CH units), for which the finite size error in Eg is estimated to be below 0.05 eV. The optimized supercell parameter is found to increase from its ground-state value of 24.6 to 26.0 Å. This expansion is consistent with the π → π* character of the electronic transition to the 11B−u state. The atomic positions are relaxed using excited-state ab initio molecular dynamics32 (MD), starting from a perfectly dimerized PA chain at the optimized supercell parameter lsc = 26 Å. The force acting on atom I when the system is in an atomic configuration 00 is excited to the 11B−u state, and FI is given by FI[00] = −∇R I Etot[00] − ∇R I ωi[00]
Figure 3. (a) Time evolution of the SDF relative at lsc = 26 Å during adiabatic dynamics of the 11B−u state. The colors indicate the phase of SDF, φn ≥ 0 (A) and φn < 0 (B), where n is the C atom index. (b−d) Instantaneous SDFs (dots) at t ≃ (b) 0, (c) 25, and (d) 90 fs. The red curves represent the (continuous) theoretical SDF distribution from the SSH model for (b) a perfectly dimerized phase A chain, (c) a wellseparated soliton/antisoliton pair at n = 6 and 14, and (d) a colliding soliton/antisoliton pair at n = 9 and 11.
fast generation of two domain walls, separated by a distance of eight CH units. This corresponds to the nucleation of a soliton S and an antisoliton S− (Figure 3c). The topological nature of these defects is clear from the alternation of A and B phases once the pair is formed at ∼20 fs. S/S− is therefore analogous to a kink/antikink soliton pair. The S/S− nucleation is estimated to complete within tc ≃ 2 × 10−14 s. This prediction is in good agreement with estimates from MD simulations based on the SSH model, tc ≃ 1.4 × 10−14 s.4,7 After nucleation, S and S− move with similar constant velocities. The widths of S and S− are on the order of 2−4 CH units, which compare well with typical spread values from calculations based on the SSH model. From the slope of the S/ S− center displacements with time, the velocity v of the soliton and antisoliton can be estimated to be on the order of 1.6 × 103 m s−1, roughly 1 order of magnitude smaller than the speed of sound ca. This value is again in good agreement with typical SSH estimates for small-width kinks,4 for example, v = 0.6ca for ξ = 2.75a, to be compared to more spread out defects, v = 2.6ca for ξ = 7a. Owing to the ring topology of the calculation setup, S and S− collide recurrently at regular intervals Tc = 10a/v ≃ 60 fs. Unlike nontopological solitons, kinks moving in opposite directions cannot pass through each other unchanged; after each collision, they invert their directions of propagation and recover their uniform motion. At, or in close proximity to, a collision, a complex oscillatory time-symmetric pattern is observed in the SDF (Figure 4). This pattern is associated with a short period of time (∼50 fs) in which S and S− collide twice before recovering their translation at uniform velocity in opposite directions. This behavior has not been reported before in simulations based on either the SSH model or on correlated approaches (see, e.g., ref 36) but has been observed in accurate
(4)
where Etot[00] is the total ground-state energy for the configuration 00 , ωi[00] are excitation energies obtained from the solution of eq 2, and ∇RI indicates the derivative with respect to a nuclear displacement. The first term on the righthand side of eq 4 is computed using standard derivative techniques from ab initio MD.33 The second term, corresponding the first-order derivative of an excitation energy with respect to a nuclear displacement, is computed in TD-DFT by solving a set of coupled-perturbed equations (Handy−Schaefer Z-vector equation33,34), similar to the standard coupled-perturbed equations of, for example, ref 23 in the limit of ω = 0. From the solution of eq 4, the nuclei are then propagated as classical particles35 to a new configuration 01, using a time step δt = 0.5 fs. Each MD step thus involves one ground-state DFT calculation, the solution of eq 2, and the calculation of the excited-state gradients from eq 4. In practice, for each MD step, we compute the excitation energy to the lowest optically allowed state from eq 2, ω̃ i, and then, we solve eq 4 for the corresponding nuclear gradients. For 00 , ω̃ i corresponds to the transition energy to the 11B−u state, but at later times, different low-lying excited states can also contribute to the nuclear dynamics. The simulation is carried out at a fixed supercell parameter, and the nuclear temperature is controlled using a Nosé thermostat with a thermostat frequency of 1100 cm−1. The time evolution of the SDF relative to a reference dimerized structure at lsc = 26 Å and T = 298 K, [φn(t) − u0]/ u0, is plotted in Figure 3. At t = 0, the system is in the dimerized phase conventionally indicated as A. The creation of the (bound) particle−hole pair by photoexcitation occurs instantaneously within the BO approximation7 and is followed by the 910
DOI: 10.1021/acs.jpclett.5b00159 J. Phys. Chem. Lett. 2015, 6, 908−912
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resonance window.37 In PA, the formation of a bound, neutral soliton/antisoliton pair indicates the dominance of attractive e−e correlation effects over repulsive e−l interactions.40 According to our TD-DFT simulations, this attraction is sufficient to hold the pair together at T = 0, whereas at T ≃ 298 K and for sufficiently large distances, S and S− behave as independent particles. The 2-bounce collision pattern in the TD-DFT simulations is therefore likely to originate from a competition between opposing e−l and e−e interactions when S and S− are at sufficiently short distance. In general, the spontaneous generation of solitons by optical excitation occurs only if the formation energy of the soliton Es is lower than the chemical potential of the system, μ = Eg/2, which is the energy required to create a particle and a hole. For a typical SSH parameter set, it can be estimated that the energy to create a soliton at rest is 0.61Eg/2,4 which, for Eg = 2.1 eV, corresponds to Es = 0.64 eV. From the energy of the relaxed 11B−u geometry at T = 0 relative to the ground state, we estimate a formation energy Es = 0.68 eV at T = 0, which is in good agreement with the SSH model and confirms that the optical generation of topological solitons is preferred over the (bound) electron−hole pair creation. At T = 298 K, a thermally averaged soliton pair formation energy can be estimated of 0.85 eV. This value is again well below Eg/2, indicating that mobile solitons remain stable also at room temperature. In summary, we have examined the energy and structure of the lowest optically allowed singlet excited state of PA (11B−u ) using hybrid (PBE0) TD-DFT and excited-state BO dynamics. We have shown that this approach accounts quantitatively for the electron−hole attraction responsible for the formation of a localized excitonic state even in the perfectly dimerized PA structure. Upon lattice relaxation, topological S/S− pairs form spontaneously in the 11B−u state and propagate as independent particles at room temperature, consistent with results from simulations based on the uncorrelated SSH model. At T = 0, bound S/S− pairs are observed, with a SDF described by the two-soliton form predicted by PPPP theory. S/S− collisions exhibit a pattern (2-bounce resonance) characteristic of nonintegrable dynamics that is inconsistent with the SSH prediction of a purely deterministic sine-Gordon equation of motion. We attribute the difference to the approximate inclusion of short-ranged (dynamical) correlation between the excited electron and hole in TD-DFT. This effect is sufficient to completely screen the repulsive S−S− interaction originating from e−l coupling at T = 0, which leads to the formation of bound S/S− pairs. At room temperature, the screening only suffices to perturb the dynamics of the solitons when these are close to each other, resulting in a deviation from the free particle motion predicted by the SSH model.
Figure 4. SDF near a 2-bounce resonance. S and S− collide first at C1 (t ≃ 75 fs), move apart to the points S1 and S2 (t ≃ 90 fs), and then collide again at C2 (t ≃ 100 fs).
numerical simulations of the time-propagation of the kink solutions of the ϕ4 equation,37,38 a strongly nonlinear equation with a double-well potential. Unlike the related sine-Gordon equation, the ϕ4 equation is not completely integrable. At high velocities, colliding ϕ4 kink/antikinks are immediately reflected. For discrete sets of so-called resonance window velocities, 2bounce resonances are observed, in which the kinks first coalesce, start to move apart, and then turn around to collide a second time before finally escaping each other’s influence. This behavior is consistent with the pattern shown in Figure 4. It is also interesting to study the behavior of the solitons in PA when the nuclear kinetic energy is gradually decreased, to reach T = 0. In this case S and S− are found to coalesce into a single nontopological defect (Figure 5), similar to a polaron
Figure 5. SDF at zero temperature (dots). The periodic supercell size is indicated by the arrow. Interpolations with the two-soliton function, eq 5, are also plotted, with φ0 = 0.011 Å, n0 = 5, and the values of ξ indicated in the caption.
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Corresponding Author
with ξ ≃ 5a. The formation of a strongly bound polaronic 11B−u state with the two-soliton functional form39 ⎡ ⎤ ⎛ 2n a ⎞ φn = φ0⎢1 + tanh⎜ 0 ⎟{tanh A− − tanh A+}⎥ ⎝ ξ ⎠ ⎣ ⎦
AUTHOR INFORMATION
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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±
where A = (n ± n0)a/ξ, is predicted by calculations based on the correlated Pariser−Parr−Pople−Peierls (PPPP) model for any nonzero value of the Coulomb interaction.3 Bound kink/ antikink pairs are also observed in simulations of the ϕ4 equation for collisions occurring at velocities below the
ACKNOWLEDGMENTS
I thank W. Barford, N. Gidopoulos, E. K. U. Gross, and A. Rubio for useful discussion. This work is supported by EPSRC through a Service Level Agreement with STFC Scientific Computing Department. Support from the UK Materials 911
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(25) Dreuw, A.; Head-Gordon, M. Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules. Chem. Rev. 2005, 105, 4009−4037. (26) Hirata, S.; Head-Gordon, M. Time-Dependent Density Functional Theory within the Tamm−Dancoff Approximation. Chem. Phys. Lett. 1999, 314, 291−299. (27) Enzerhof, M.; Scuseria, G. E. Assessment of the Perdew− Burke−Ernzerhof Exchange−Correlation Functional. J. Chem. Phys. 1999, 110, 5029−5036. (28) Plane-wave pseudopotential calculations were performed using a locally modified version of CPMD 3.9 (http://www.cpmd.org/), with standard norm-conserving pseudopotentials and kinetic energy cutoffs varying between 80 and 100 Ry depending on the supercell size. Reference all-electron calculations to test the reliability of the planewave results were carried out using CRYSTAL09 (http://www.crystal. unito.it/), with 6-21G* and 5-11G* basis sets for C and H, respectively. (29) Hsu, C.-P.; Hirata, S.; Head-Gordon, M. Excitation Energies from Time-Dependent Density Functional Theory for Linear Polyene Oligomers: Butadiene to Decapentacene. J. Phys. Chem. A 2001, 105, 451−458. (30) Bernasconi, L.; Sprik, M.; Hutter, J. Time Dependent Density Functional Theory Study of Charge-Transfer and Intramolecular Electronic Excitations in Acetone−Water Systems. J. Chem. Phys. 2003, 119, 12417−12431. (31) Ferretti, A.; Mallia, G.; Martin-Samos, L.; Bussi, G.; Ruini, A.; Montanari, B.; Harrison, N. M. Ab Initio Complex Band Structure of Conjugated Polymers. Phys. Rev. B 2012, 85, 235105. (32) Hutter, J. Excited State Nuclear Forces from the Tamm− Dancoff Approximation to Time-Dependent Density Functional Theory within the Plane Wave Basis Set Framework. J. Chem. Phys. 2003, 118, 3928−3934. (33) Marx, D.; Hutter, J. In Modern Methods and Algorithms of Quantum Chemistry; Grotendorst, J., Ed.; John von Neumann Institute for Computing: Julich, Germany, 2000. (34) Handy, N. C.; Schaefer, H. F. On the Evaluation of Analytic Energy Derivatives for Correlated Wave Functions. J. Chem. Phys. 1984, 81, 5031−5033. (35) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, U.K., 1987. (36) Ma, H.; Schollwöck, U. Effect of Electron−Electron Interactions on the Charge Carrier Transitions in trans-Polyacetylene. J. Phys. Chem. A 2010, 114, 5439−5444. (37) Goodman, R. H.; Haberman, R. Kink−Antikink Collisions in the ϕ4 Equation. SIAM J. Appl. Dyn. Syst. 2005, 4, 1195−1227. (38) Goodman, R. H.; Haberman, R. Chaotic Scattering and the nBounce Resonance in Solitary-Wave Interactions. Phys. Rev. Lett. 2007, 98, 104103. (39) Grabowski, M.; Hone, D.; Schrieffer, J. R. Photogenerated Solitonic States in trans-Polyacetylene. Phys. Rev. B 1985, 31, 7850− 7854. (40) Orenstein, J.; Vardeny, Z.; Baker, G. L.; Eagle, G.; Eternad, S. Mechanism for Photogeneration of Charge Carriers in Polyacetylene. Phys. Rev. B 1984, 30, 786−794.
Chemistry Consortium (Grant EP/L000202) and from STFC Hartree Centre is gratefully acknowledged.
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