Singlet Fission in Lycopene H-Aggregates.

A theory of singlet fission (SF) in carotenoid dimers is applied to explain the SF in lycopene H-aggregates observed after high-energy photoexcitation. The explanation proposed here is that a high energy, delocalized bright 1Bu+ state first relaxes and localizes onto a single lycopene monomer. The high-energy intramonomer state then undergoes internal conversion to the 11Bu- state. Once populated, the 11Bu- state allows exothermic bimolecular singlet fission, while its internal conversion to the 21Ag- state is symmetry forbidden. The simulation of SF predicts that the intramonomer triplet-pair state undergoes almost complete population transfer to the intermonomer singlet-pair state within 100 ps. Simultaneously, ZFS interactions begin to partially populate the intermonomer quintet triplet-pair state up to ca. 2 ns, after which hyperfine interactions thermally equilibrate the triplet-pair states, thus forming free single triplets within 50 ns.

Theory of singlet fission in carotenoid dimers.

We develop a theory of singlet fission in carotenoid dimers. Following photoexcitation of the "bright" state (i.e., a singlet electron-hole pair) in a single carotenoid, the first step in the singlet fission process is ultrafast intramolecular conversion into the highly correlated "dark" (or 2Ag) state. This state has both entangled singlet triplet-pair and charge-transfer character. Our theory is predicated on the assumption that it is the singlet triplet-pair component of the "dark" state that undergoes bimolecular singlet fission. We use valence bond theory to develop a minimal two-chain model of the triplet-pair states. The single and double chain triplet-pair spectra are described, as this helps explain the dynamics and the equilibrated populations. We simulate the dynamics of the initial entangled pair state using the quantum Liouville equation, including both spin-conserving and spin-nonconserving dephasing processes. By computing the intrachain and interchain singlet, triplet, and quintet triplet-pair populations, we show that singlet fission critically depends on the interchain coupling and the driving potential (that determines endothermic vs exothermic fission).

Dynamical Simulations of Carotenoid Photoexcited States Using Density Matrix Renormalization Group Techniques.

We present a dynamical simulation scheme to model the highly correlated excited state dynamics of linear polyenes. We apply it to investigate the internal conversion processes of carotenoids following their photoexcitation. We use the extended Hubbard-Peierls model, H^UVP, to describe the π-electronic system coupled to nuclear degrees of freedom. This is supplemented by a Hamiltonian, H^ϵ, that explicitly breaks both the particle-hole and two-fold rotation symmetries of idealized carotenoid structures. The electronic degrees of freedom are treated quantum mechanically by solving the time-dependent Schrödinger equation using the adaptive time-dependent DMRG (tDMRG) method, while nuclear dynamics are treated via the Ehrenfest equations of motion. By defining adiabatic excited states as the eigenstates of the full Hamiltonian, H^=H^UVP+H^ϵ, and diabatic excited states as eigenstates of H^UVP, we present a computational framework to monitor the internal conversion process from the initial photoexcited 11Bu+ state to the singlet triplet-pair states of carotenoids. We further incorporate Lanczos-DMRG to the tDMRG-Ehrenfest method to calculate transient absorption spectra from the evolving photoexcited state. We describe in detail the accuracy and convergence criteria for DMRG, and show that this method accurately describes the dynamical processes of carotenoid excited states. We also discuss the effect of the symmetry-breaking term, H^ϵ, on the internal conversion process, and show that its effect on the extent of internal conversion can be described by a Landau-Zener-type transition. This methodological paper is a companion to our more explanatory discussion of carotenoid excited state dynamics in Manawadu, D.; Georges, T. N.; Barford, W. Photoexcited State Dynamics and Singlet Fission in Carotenoids. J. Phys. Chem. A 2023, 127, 1342.

Photoexcited State Dynamics and Singlet Fission in Carotenoids.

We describe our simulations of the excited state dynamics of the carotenoid neurosporene, following its photoexcitation into the "bright" (nominally 11Bu+) state. To account for the experimental and theoretical uncertainty in the relative energetic ordering of the nominal 11Bu+ and 21Ag- states at the Franck-Condon point, we consider two parameter sets. In both cases, there is ultrafast internal conversion from the "bright" state to a "dark" singlet triplet-pair state, i.e., to one member of the "2Ag" family of states. For one parameter set, internal conversion from the 11Bu+ to 21Ag- states occurs via the dark, intermediate 11Bu- state. In this case, there is a cross over of the 11Bu+ and 11Bu- diabatic energies within 5 fs and an associated avoided crossing of the S2 and S3 adiabatic energies. After the adiabatic evolution of the S2 state from predominately 11Bu+ character to predominately 11Bu- character, there is a slower nonadiabatic transition from S2 to S1, accompanied by an increase in the population of the 21Ag- state. For the other parameter set, the 21Ag- energy lies higher than the 11Bu+ energy at the Franck-Condon point. In this case, there is cross over of the 21Ag- and 11Bu+ energies and an avoided crossing of the S1 and S2 energies, as the S1 state evolves adiabatically from being of 11Bu+ character to 21Ag- character. We make a direct connection from our predictions to experimental observables by calculating the time-resolved excited state absorption. For the case of direct 11Bu+ to 21Ag- internal conversion, we show that the dominant transition at ca. 2 eV, being close to but lower in energy than the T1 to T1* transition, can be attributed to the 21Ag- component of S1. Moreover, we show that it is the charge-transfer exciton component of the 21Ag- state that is responsible for this transition (to a higher-lying exciton state), and not its triplet-pair component. These simulations are performed using the adaptive tDMRG method on the extended Hubbard model of π-conjugated electrons. The Ehrenfest equations of motion are used to simulate the coupled nuclei dynamics. We next discuss the microscopic mechanism of "bright" to "dark" state internal conversion and emphasize that this occurs via the exciton components of both states. Finally, we describe a mechanism relying on torsional relaxation whereby the strongly bound intrachain triplet-pairs of the "dark" state may undergo interchain exothermic dissociation.

Singlet Triplet-Pair Production and Possible Singlet-Fission in Carotenoids.

Internal conversion from the photoexcited state to a correlated singlet triplet-pair state is believed to be the precursor of singlet fission in carotenoids. We present numerical simulations of this process using a π-electron model that fully accounts for electron-electron interactions and electron-nuclear coupling. The time-evolution of the electrons is determined rigorously using the time-dependent density matrix renormalization group method, while the nuclei are evolved via the Ehrenfest equations of motion. We apply this to zeaxanthin, a carotenoid chain with 18 fully conjugated carbon atoms. We show that the internal conversion of the primary photoexcited state, S2, to the singlet triplet-pair state occurs adiabatically via an avoided crossing within ∼50 fs with a yield of ∼60%. We further discuss whether this singlet triplet-pair state will undergo exothermic versus endothermic intra- or interchain singlet fission.

Ultrafast Fluorescence Depolarization in Conjugated Polymers.

We report on large-scale simulations of intrachain exciton dynamics in poly(para-phenylenevinylene). Our theoretical model describes Frenkel exciton coupling to both fast, quantized C-C bond vibrations and slow, classical torsional modes. We also incorporate system-bath interactions. The dynamics is simulated using the time evolution block decimation method, which avoids the failures of the Ehrenfest approximation to describe decoherence processes and nonadiabatic interstate conversion. System-bath interactions are modeled using quantum trajectories and Lindblad quantum jump operators. We find that following photoexcitation, the quantum mechanical entanglement of the exciton and C-C bond phonons causes exciton-site decoherence. Next, system-bath interactions cause the stochastic collapse of high-energy delocalized excitons into chromophores. Finally, torsional relaxation causes additional exciton-density localization. We relate these dynamical processes to the predicted fluorescence depolarization, extract the time scales corresponding to them, and thus interpret the observed sub-ps fluorescence depolarization.

Using spectroscopy to probe relaxation, decoherence, and localization of photoexcited states in π-conjugated polymers.

We use the coarse-grained Frenkel-Holstein model to simulate the relaxation, decoherence, and localization of photoexcited states in conformationally disordered π-conjugated polymers. The dynamics are computed via wave-packet propagation using matrix product states and the time evolution block decimation method. The ultrafast (i.e., t < 10 fs) coupling of an exciton to C-C bond vibrations creates an exciton-polaron. The relatively short (ca. 10 monomers) exciton-phonon correlation length causes ultrafast exciton-site decoherence, which is observable on conformationally disordered chains as fluorescence depolarization. Dissipative coupling to the environment (modelled via quantum jumps) causes the localization of quasi-extended exciton states (QEESs) onto local exciton ground states (LEGSs, i.e., chromophores). This is observable as lifetime broadening of the 0-0 transition (and vibronic satellites) of the QEES in two-dimensional electronic coherence spectroscopy. However, as this process is incoherent, neither population increases of the LEGSs nor coherences with LEGSs are observable.

Torsionally induced exciton localization and decoherence in π-conjugated polymers.

We develop a model of excitons coupled to the rotational motion of monomers to study the torsionally induced relaxation and decoherence of excitons in π-conjugated polymers. The model assumes that the monomer units are described by elastically uncoupled harmonic oscillators and that there is a linear exciton-roton coupling. Although the rotational degrees of freedom are much slower than the exciton, so that the adiabatic approximation is generally expected to be valid, we also investigate possible quantized roton corrections via coupled time evolving block decimation-Ehrenfest equations of motion. For the relaxation of the lowest-excited exciton, we find that (1) for a polymer chain with a ground state spiral torsional conformation, the equilibrium angular displacement of each monomer is proportional to the difference of the exciton bond-orders on the neighboring bridging bonds. Consequently, this displacement vanishes in the long chain limit and a classical (Landau) exciton-polaron is not formed. (2) For a polymer chain with a ground state staggered torsional conformation, the equilibrium angular displacement of each monomer is proportional to the sum of the exciton bond-orders on the neighboring bridging bonds. Consequently, there is significant angular displacement and local planarization causing exciton density localization. A classical (Landau) exciton-polaron is formed where the staggered angular displacement is proportional to the exciton density. (3) Generally, in the adiabatic limit, the decay of off-diagonal long-range order (i.e., exciton decoherence) mirrors the localization of the exciton density. However, quantum corrections to the rotational motion alter this adiabatic prediction because of correlated exciton-roton dynamics within the first rotational half-period. In particular, exciton-polaron quasiparticle formation causes more rapid and oscillatory exciton decoherence and slower exciton density localization.

Extracting structural information from MEH-PPV optical spectra.

The Frenkel-Holstein model in the Born-Oppenheimer regime is used to interpret temperature-dependent photoluminescence spectra of solutions made with the poly(p-phenylene vinylene) derivative MEH-PPV. Using our recently developed structural optimization method and assuming only intrachain electronic coupling, we predict the structure of emissive MEH-PPV chromophores in terms of a mean torsional angle ϕ0 and its static fluctuations σϕ, assuming no cis-trans defects. This allows us to fully account for the observed changes in spectra, and the chromophore structures obtained are consistent with the known phase transition at 180 K between a "red" and "blue" phase.

Structural Information for Conjugated Polymers from Optical Modeling.

We use a Frenkel-Holstein model of uncoupled chains in the adiabatic limit to simulate the optical spectra of the conjugated polymer ladder-type poly( p-phenylene) derivative (MeLPPP), which is a planar conjugated polymer with especially low interchain interactions. The theoretical calculations correctly reproduce the vibronic spectra and yield reasonable torsion angles between adjacent phenyl rings. The success of this approach indicates that, in contrast to interchain coupling, the strong electronic coupling along a polymer chain is more appropriately described in the adiabatic limit.

Ultra-fast relaxation, decoherence, and localization of photoexcited states in π-conjugated polymers.

The exciton relaxation dynamics of photoexcited electronic states in poly(p-phenylenevinylene) are theoretically investigated within a coarse-grained model, in which both the exciton and nuclear degrees of freedom are treated quantum mechanically. The Frenkel-Holstein Hamiltonian is used to describe the strong exciton-phonon coupling present in the system, while external damping of the internal nuclear degrees of freedom is accounted for by a Lindblad master equation. Numerically, the dynamics are computed using the time evolving block decimation and quantum jump trajectory techniques. The values of the model parameters physically relevant to polymer systems naturally lead to a separation of time scales, with the ultra-fast dynamics corresponding to energy transfer from the exciton to the internal phonon modes (i.e., the C-C bond oscillations), while the longer time dynamics correspond to damping of these phonon modes by the external dissipation. Associated with these time scales, we investigate the following processes that are indicative of the system relaxing onto the emissive chromophores of the polymer: (1) Exciton-polaron formation occurs on an ultra-fast time scale, with the associated exciton-phonon correlations present within half a vibrational time period of the C-C bond oscillations. (2) Exciton decoherence is driven by the decay in the vibrational overlaps associated with exciton-polaron formation, occurring on the same time scale. (3) Exciton density localization is driven by the external dissipation, arising from "wavefunction collapse" occurring as a result of the system-environment interactions. Finally, we show how fluorescence anisotropy measurements can be used to investigate the exciton decoherence process during the relaxation dynamics.

Perspective: Optical spectroscopy in π-conjugated polymers and how it can be used to determine multiscale polymer structures.

Exciton delocalization in conjugated polymer systems is determined by polymer conformations and packing. Since exciton delocalization determines the photoluminescent vibronic progression, optical spectroscopy provides an indirect link to polymer multiscale structures. This perspective describes our current theoretical understanding of how exciton delocalization in π-conjugated polymers determines their optical spectroscopy and further shows how exciton delocalization is related to conformational and environmental disorder. If the multiscale structures in conjugated polymer systems are known, then using first-principles modeling of excitonic processes it is possible to predict a wide-range of spectroscopic observables. We propose a reverse-engineering protocol of using these experimental observables in combination with theoretical and computational modeling to determine the multiscale polymers structures, thus establishing quantitative structure-function predictions.

Theory of optical transitions in curved chromophores.

Using first order perturbation theory in the Born-Oppenheimer regime of the Frenkel-Holstein model, we develop a theory for the optical transitions in curved chromophores of π-conjugated polymers. Our key results are that for absorption, A, and emission, I, polarized parallel to the 0-0 transition, I01/I00 ≃ A01/A00 = S(N), where S(N) = S(1)/IPR is the effective Huang-Rhys parameter for a chromophore of N monomers and IPR is the inverse participation ratio. In contrast, absorption and emission polarized perpendicular to the 0-0 transition acquires vibronic intensity via the Herzberg-Teller effect. This intensity generally increases as the curvature increases and consequently I01/I00 increases (where I01 is the total 0-1 emission intensity). This effect is enhanced for long chromophores and in the anti-adiabatic regime. We show via DMRG calculations that this theory works well in the adiabatic regime relevant to π-conjugated polymers, i.e., h ω/|J| ≲ 0.2.

Theory of optical transitions in π-conjugated macrocycles.

We describe a theoretical and computational investigation of the optical properties of π-conjugated macrocycles. Since the low-energy excitations of these systems are Frenkel excitons that couple to high-frequency dispersionless phonons, we employ the quantized Frenkel-Holstein model and solve it via the density matrix renormalization group (DMRG) method. First we consider optical emission from perfectly circular systems. Owing to optical selection rules, such systems radiate via two mechanisms: (i) within the Condon approximation, by thermally induced emission from the optically allowed j = ± 1 states and (ii) beyond the Condon approximation, by emission from the j = 0 state via coupling with a totally non-symmetric phonon (namely, the Herzberg-Teller effect). Using perturbation theory, we derive an expression for the Herzberg-Teller correction and show via DMRG calculations that this expression soon fails as h ω/J and the size of the macrocycle increase. Next, we consider the role of broken symmetry caused by torsional disorder. In this case the quantum number j no longer labels eigenstates of angular momentum, but instead labels localized local exciton groundstates (LEGSs) or quasi-extended states (QEESs). As for linear polymers, LEGSs define chromophores, with the higher energy QEESs being extended over numerous LEGSs. Within the Condon approximation (i.e., neglecting the Herzberg-Teller correction) we show that increased disorder increases the emissive optical intensity, because all the LEGSs are optically active. We next consider the combined role of broken symmetry and curvature, by explicitly evaluating the Herzberg-Teller correction in disordered systems via the DMRG method. The Herzberg-Teller correction is most evident in the emission intensity ratio, I00/I01. In the Condon approximation I00/I01 is a constant function of curvature, whereas in practice it vanishes for closed rings and only approaches a constant in the limit of vanishing curvature. We calculate the optical spectra of a model system, cyclo-poly(para-phenylene ethynylene), for different amounts of torsional disorder within and beyond the Condon approximation. We show how broken symmetry and the Herzberg-Teller effect explain the spectral features. The Herzberg-Teller correction to the 0-1 emission vibronic peak is always significant. Finally, we note the qualitative similarities between the optical properties of conformationally disordered linear polymers and macrocycles in the limit of sufficiently large disorder, because in both cases they are determined by the optical properties of curved chromophores.

Polarons in π-Conjugated Polymers: Anderson or Landau?

Using both analytical expressions and the density matrix renormalization group method, we study the fully quantized disordered Holstein model to investigate the localization of charges and excitons by vibrational or torsional modes-i.e., the formation of polarons-in conformationally disordered π-conjugated polymers. We identify two distinct mechanisms for polaron formation, namely Anderson localization via disorder (causing the formation of Anderson polarons) and self-localization by self-trapping via normal modes (causing the formation of Landau polarons). We identify the regimes where either description is more valid. The key distinction between Anderson and Landau polarons is that for the latter the particle wave function is a strong function of the normal coordinates, and hence the "vertical" and "relaxed" wave functions are different. This has theoretical and experimental consequences for Landau polarons. Theoretically, it means that the Condon approximation is not valid, and so care needs to be taken when evaluating transition rates. Experimentally, it means that the self-localization of the particle as a consequence of its coupling to the normal coordinates may lead to experimental observables, e.g., ultrafast fluorescence depolarization. We apply these ideas to poly(p-phenylenevinylene). We show that the high frequency C-C bond oscillation only causes Landau polarons for a very narrow parameter regime; generally we expect disorder to dominate and Anderson polarons to be a more applicable description. Similarly, for the low frequency torsional fluctuations we show that Anderson polarons are expected for realistic parameters.

Charge mobility induced by Brownian fluctuations in π-conjugated polymers in solution.

We study the motion of a doped charge in a π-conjugated polymer chain in solution subject to Brownian fluctuations. Specifically, we take poly(para-phenylene) to be our model system where the Brownian fluctuations cause rotational motion of the phenylene rings. The instantaneous torsional fluctuations cause Anderson localization of the charge wavefunction, with the lower-energy spectrum being composed of local ground states and the higher-energy spectrum being composed of quasi-extended states. At low temperatures, additional charge localization occurs via torsional relaxation. The dynamical torsional fluctuations lead to two distinct modes of motion of the charge: adiabatic and non-adiabatic. Adiabatic motion is a 'crawling' motion of the charge along the polymer chain while the charge remains in its local ground state. Non-adiabatic motion is a rapid 'hopping' motion as the charge is excited into higher energy quasi-extended states and travels ballistically along the chain before relaxing into a local ground state. The adiabatic motion dominates at low temperatures, and exhibits a linear temperature dependence and thus a constant zero-field charge mobility. Non-adiabatic motion begins to dominate as the temperature is increased, as the charge is thermally excited into higher energy states. At high temperatures the diffusion constant becomes almost temperature independent, indicating a decrease in the charge mobility with increasing temperature, which we attribute to the charge localization length being a decreasing function of temperature at high temperatures.