Energy Transfer Mechanisms in Large Low-Bandgap Polymers from Time-Resolved Experiments and Nonadiabatic Molecular Dynamics Calculations

Abstract

Conjugated polymers offer unprecedented chemical tunability for modulating energy transfer in a multitude of infrared light applications. In this work, we use a combination of time-resolved spectroscopic experiments and nonadiabatic molecular dynamics calculations to probe the photochemistry and nonradiative transitions in a recently synthesized narrow bandgap donor–acceptor conjugated polymer based on alternating cyclopentadithiophene and electronegative benzothiadiazole heterocycles. Using large-scale semi-empirical nonadiabatic molecular dynamics, which can treat a large 260-atom hexamer, we calculate an S₅ → S₁ lifetime of 34.75 fs, which is consistent with our time-resolved spectroscopic data. Our simulations suggest that vibronic motions of the central carbons in the cyclopentadithiophene functional groups are predominantly involved in the nonradiative transitions, and the excitation becomes more localized on a monomer fragment over time. The combined use of time-resolved experiments and nonadiabatic molecular dynamics calculations in this work provides mechanistic insight into chemical functionalities that can be tuned to enhance energy transfer in other prospective low-bandgap polymer materials.

Introduction

The absorption of infrared (IR) light is integral to a growing range of applications, including solar energy harvesting, , surveillance and tracking industrial inspection , and biomedical sensing. However, the widespread adoption of existing IR-active inorganic semiconductors, such as HgCdTe and InGaAs, is limited by their costly epitaxial fabrication, active cooling requirements, and lattice mismatch with existing Si electronics. , These shortcomings have driven research into solution processable alternatives, including colloidal quantum dots, 2D materials, perovskites, and organic semiconductors (OSCs). Among the broad OSC landscape, conjugated polymers (CPs) offer a flexible platform for tuning their electronic structure via chemical synthesis and morphological control. CPs with extended conjugation, quinoidal bonding patterns, and alternating electron donor and acceptor units have narrow energy gaps required of IR absorbers. ,

The dynamics of the photogenerated exciton have an enormous impact on device performance. Following the Energy Gap Law, nonradiative recombination in molecular systems increases exponentially as the bandgap narrows due to increasing vibronic overlap between the ground and excited states. , The increased nonradiative recombination rate shortens exciton lifetimes and directly competes with charge carrier generation in optoelectronic devices. Several mechanisms that combat the Energy Gap Law have emerged, including ultrafast charge generation in molecular aggregates or via transient delocalization and decoupling of the exciton from molecular vibrations. However, molecular design guidelines to effectively use these features to produce high-performance, low-bandgap CPs are still emerging. Thus, it is crucial to study how exciton-vibrational coupling in these systems impacts the nonradiative energy-transfer pathways.

In this work, we examine the excited-state dynamics of a recently-synthesized polymer (P1, Figure a) whose low bandgap arises from a pro-quinoidal cyclopentadithiophene donor and a highly electronegative benzothiadiazole acceptor that promotes a low bond length alternation and a highly planar backbone. Using a combination of time-resolved spectroscopic experiments and nonadiabatic molecular dynamics calculations, we characterize the mechanisms underlying the nonradiative pathways (including energy transfer) in large hexamers of P1, denoted as P1 ₆, which contains 260 atoms (see Figure b). It is worth mentioning that conventional excited-state molecular dynamics (ESMD) simulations cannot be used for these studies since they are unable to capture the intricate nonradiative pathways that occur in these low-bandgap materials. For this reason, we utilized nonadiabatic molecular dynamics (NAMD) to capture the strongly coupled electronic and nuclear degrees of freedom that occur in these nonradiative mechanisms. We provide a detailed comparison between our NAMD simulations and time-resolved experiments to shed mechanistic insight into these energy-transfer mechanisms and conclude with a perspective of approaches that can be used for other large, low-bandgap polymer systems.

1.

(a) Skeletal structure showing the monomer subunit of P1. (b) 3D view showing the initial molecular configurations of P1 ₆ for the NAMD simulations.

Computational Methodology

Ground State Molecular Dynamics Sampling

It is computationally costly to use conventional quantum mechanical methods to simulate a sufficiently long molecular dynamics (MD) trajectory (100 s of ps) due to the large size of P1 ₆ (260 atoms). Therefore, we utilized classical MD to sample the conformational space of the ground-state potential energy surface (PES) efficiently. The geometry of P1 ₆ was optimized, and the atomic charges were calculated in the gas phase using the RM1 semiempirical method. We generated the parameters for the generalized AMBER force field (GAFF) and topology files using the atomic charges and bond charge correction methodology via the AMBER antechamber program. The structure was then minimized and heated from 0 to 300 K with a timestep of 0.5 fs over a 10 ps duration using a Langevin thermostat (collision frequency 20 ps^–1) and GAFF. We then used classical MD to simulate hundreds of picoseconds and sampled 101 geometries every one ps for use in 101 semiempirical ground state MD trajectories. Using the sampled geometries, we performed semiempirical (RM1) ground-state MD with implicit solvation using a conductor-like screening model (COSMO) with dielectric constant (ϵ) = 5.62 (it is worth noting that the ground-state optimization and MD calculations used an equilibrium solvation model, whereas the AMD calculations used a linear-response solvation approach , ). We used a 0.5 fs timestep for 500 fs at a temperature of 300 K using a Langevin thermostat (collision frequency = 20 ps^–1). The resulting 101 geometries and velocities were then used as initial conditions for the 101 NAMD trajectories (see Figure ). The absorption spectrum was simulated using the 101 sampled geometries with a broadening width of k _B ·300 K. We chose the starting electronic state for the NAMD simulations by taking the difference in energy between the pump laser and the experimental peak (≈0.17 eV) to identify the energetically nearest state (S₅) in the simulated spectrum. While it would be more physically correct to simulate an ensemble of molecules in different excited states since the S₁ through S₇ peaks overlap, this complicates the analysis. Furthermore, since the decay from S₅ to S₁ occurs on an ultrafast time scale, the observables from the NAMD simulation are likely not sensitive to the initial electronic state.

Nonadiabatic Molecular Dynamics Trajectories

One of the most widely used variants of NAMD is fewest switches surface hopping (FSSH), where transitions between adiabatic electronic surfaces are accounted for in the form of “hops.” The ability to simulate strong electronic-nuclear coupling is also crucial for describing dynamics involving nonradiative decay and energy transfer, making NAMD well-suited for gaining mechanistic insight into the P1 polymer system. NAMD simulations of molecular systems comprised of hundreds of atoms using first-principles electronic structure methods (such as time-dependent density functional theory) are prohibitively expensive. Multiple approaches to address this computational limitation have been developed, including the neglect-of-back-reaction approximation, approximating the nonadiabatic couplings between states, or using a computationally cheaper electronic structure method. For our study, we chose to employ semiempirical quantum mechanics (RM1) with FSSH via the NEXMD software package, , which computes NAMD couplings − to enable simulations of large systems with reasonable accuracy. The NAMD trajectories for a single polymer chain of P1 ₆ were started in the S₅ state using the RM1 semiempirical method, CIS for excited states, Newtonian dynamics (0.5 fs timestep), Tully’s fewest switches surface hopping (FSSH), and the conductor-like screening model (COSMO) for implicit solvation (ϵ = 5.62).

Excited State Molecular Dynamics Trajectories

After 500 fs, the trajectories remain largely on the S₁ PES, and we used adiabatic excited state molecular dynamics (ESMD) to extend the simulation duration. Since CIS with the RM1 method has no S₁–S₀ coupling and transitions to higher electronic excited states are rare after 500 fs, the NAMD and ESMD results will be qualitatively identical and quantitatively very similar. We performed S₁ ESMD trajectories using the RM1 semiempirical method, CIS for excited states, Newtonian dynamics, and COSMO for implicit solvation (ϵ = 5.62).

Bootstrapping Methodology and Spectra

Following the method described in ref. we carried out a bootstrap sampling of 1000 samples from the 101 trajectories.

We used the oscillator strengths from the NAMD and ESMD simulations to compute fluorescence spectra using the Einstein A coefficient given by

$$A=\frac{2c}{a_0}{\left(\frac{\mathrm{\Delta }E}{m_e{c}^2}\right)}^{2}f$$ 1

where ΔE is excitation energy, f is oscillator strength, c is the speed of light, m _e is the electron mass, and a ₀ is the Bohr radius. Both the absorption and fluorescence spectra were normalized.

Experimental Methodology

Our time-resolved experiments show that exciting polymer chains with a large amount of excess energy results in an enhancement of the exciton lifetime over the first few picoseconds. The exciton lifetime in our experiments was measured using transient optical spectroscopy with a NIR probe that spans the spectral range of 850–1500 nm. This enables us to measure the dynamics of the primary ground state bleach (GSB) feature near 900 nm and a broad excited-state absorption feature extending into the infrared. We have previously characterized and associated these features with an excitonic state in this polymer and a series of related structures. Our data were taken on dilute and filtered solutions to ensure that the measured dynamics correspond to individual polymer chains (additional experimental spectra and details are provided in Figure S6). The wavelength corresponding to the minima of the GSB in the transient optical spectra of the P1 polymer is constant as a function of time, which is consistent with an unaggregated structure with nearly identical exciton sites. The photon energy of the pump pulse is tuned to be either resonant with the S₁ state (E _pump = 1.35 eV) or to impart excess energy $\mathrm{\Delta }E=E_{\mathrm{pump}}-E_{{\mathrm{S}}_1}$ into the system to populate a set of S _n states.

Results and Discussion

To benchmark our computational approaches, we compared simulated absorption with available experimental data. To this end, we used geometries sampled from ground-state molecular dynamics (MD) to simulate the absorption spectrum. In Figure a, the simulated absorption peak is higher in energy than the experimental absorption peak by ≈0.41 eV, which is in good agreement considering the semi-empirical methods used in this work; most importantly, the overall lineshapes agree. It should be noted that long-chain polymers with several repeat units are used in the time-resolved experiments, whereas our simulations used a finite-sized hexamer due to computational constraints from the NAMD calculations. To confirm that a hexamer was an appropriate choice for our simulations, we carried out a series of calculations to obtain an extrapolated S₁ excitation energy of the infinite-length P1 polymer (see Figure S1). From this extrapolation, we find that the infinite-length P1 polymer would have an S1 excitation of 1.77 eV, which is only 0.06 eV lower than the S₁ excitation for a hexamer (1.83 eV). Because these excitation energies are in close agreement and since NAMD calculations with periodic boundary conditions are not currently possible with RM1, our hexamer calculations provide relatively accurate predictions for the polymer system, allowing us to proceed with the NAMD calculations described below.

2.

(a) Experimental and simulated absorption spectra with excited-state contributions in the latter. The prominent experimental absorption peak was obtained from ref. (b) Simulated time-resolved and time-integrated fluorescence spectrum broadened by k _b ·300 K. The time-integrated experimental fluorescence spectrum from ref. (dashed) is also shown for comparison.

The oscillator and excitation energies from our NAMD simulations were used to compute the fluorescence spectra. In Figure b, the time-resolved fluorescence starts at a higher energy partly due to the contributions from S₅–S₂ but stabilizes around 1.7 eV once the ensemble has relaxed on S₁. The integrated (over time) simulated fluorescence peak shows the weighted average of the time-resolved spectra and indicates that RM1 consistently overestimates the energy by ≈0.51 eV, although the lineshapes are similar. While RM1 does overestimate the excitation energy, it is worth re-emphasizing that finite-size effects in a hexamer are expected to give different results than a periodic polymer, and more π conjugation in a larger oligomer (if computationally feasible) would decrease the excitation energy.

To understand the dynamics in the P1 ₆ system, we first analyzed the orbitals associated with the electronic transitions. Natural transition orbitals (NTOs), which are obtained from a singular value decomposition of the transition density matrix, provide a visualization of an electronic excited state. The NTOs for an electronic excited state depict the orbital of the excited electron and the corresponding hole. The first five singlet excited-state NTOs are shown in Figure and can be characterized as ππ* excitonic transitions (the electron and hole are spatially close). S₇ can also be characterized as a ππ* excitonic transition. It is worth noting that the higher-energy states are more spatially localized (in accordance with quantum confinement effects) which can be clearly seen in Figure , where the S₁, S₂, S₃, S₄, and S₅ excitations roughly span 4, 3, 2.5, 1.5, and 1 monomer units, respectively.

3.

NTOs displayed at a contour value of ±0.01 (left column: hole, right column: excited electron) for (a) S₁, (b) S₂, (c) S₃, (d) S₄, and (e) S₅ at the S₀ RM1-optimized geometry.

Longer exciton lifetimes are essential for enabling charge generation in materials, sensors, and photovoltaic devices. As such, understanding the electronic transitions between excited states is crucial to rationally designing systems with longer lifetimes. Within our simulations, the nonradiative lifetime can be obtained from the nuclear population, which is the fraction of trajectories in the ensemble on a given PES. In Figure , an exponential fit of the nuclear population on S₁ is used to estimate an S₅ to S₁ lifetime $({\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1})$ of 34.75 fs. Moreover, Figure shows the minor accumulation of the population of each state. For example, the inset of Figure shows the sequential accumulation of S₄, followed by S₃, S₂, and S₁. Nonradiative lifetimes are an excellent estimation of excited-state lifetimes in this system because the nonradiative pathways kinetically out-compete the radiative pathways.

4.

(a) Electronic and (b) nuclear populations of the ensemble of trajectories. The solid lines indicate the mean (μ) of the respective electronic state, and the shaded regions indicate a bootstrap confidence interval (1000 samples) of two standard deviations (μ ± 2σ). Fitting the nuclear S₁ population data to $f(t)=101(1-e^{-\frac{t}{\tau }})$ yields a ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}=34.75$ fs.

A direct experimental measurement of ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}$ is not feasible because the experimental data includes the time to decay to S₀ whereas the simulations do not capture the nonradiative transitions from S₁ to S₀ (this occurs because there is no S₁–S₀ coupling in CIS, and the S₁ to S₀ energy gap is overestimated by RM1). However, an indirect comparison of the S₅ to S₁ lifetime against experiment estimates the upper limit of ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}$ to be 50 fs, which is consistent with the calculated 34.75 fs. The effect of cooling on the overall exciton lifetime (τ _t ) can be directly inferred by comparing the initial exciton decay for S₁ and S _n pumping with the expression ${\tau }_t={\tau }_{{\mathrm{S}}_n\to {\mathrm{S}}_1}+{\tau }_{{\mathrm{S}}_1\to {\mathrm{S}}_0}$ , where ${\tau }_{{\mathrm{S}}_n\to {\mathrm{S}}_1}$ is the hot carrier lifetime. The intrinsic exciton lifetime, ${\tau }_{{\mathrm{S}}_1\to {\mathrm{S}}_0}$ , is directly measured through resonant pumping of the S₁ state (shown in Figure ) and determined to be ≈200 fs. This anomalous short lifetime has been previously identified as an issue with infrared organic systems, indicating significant mixing of electronic excited states and high-lying vibrational levels of the ground state. A longer lifetime component that was previously assigned to an autoionization process in residual aggregates is ignored in this current analysis as it appears to be relatively independent of excitation energy. As shown in Table , the experimental ΔE of 0.37 eV corresponds well to the simulated ΔE of 0.34 corresponding to the S₅ excitation. The overall exciton lifetime appears to be nearly constant until the system is pumped with a photon with an excess energy of more than 0.6 eV, for which τ _t = 330 fs. Since τ _t at ΔE = 0 is experimentally indistinguishable from τ _t at ΔE = 0.37 and assuming that the excess energy should increase τ _t , we can infer that ${\tau }_{{\mathrm{S}}_n\to {\mathrm{S}}_1}<{\sigma }_{\mathrm{IRF}}=100$ fs. This agrees extremely well with the computed ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}=34.75$ fs value and further supports that the simulated S₅ → S₁ dynamics are reasonable and consistent with experiment. This experimental comparison suggests that while NAMD simulations cannot capture S₁–S₀ transitions, mechanistic insight into the transitions and dynamics from S₅ to S₁ may still be gained.

5.

Experimental exciton relaxation dynamics (solid lines) and fits (bold solid lines) measured with infrared transient optical spectroscopy. The lifetimes for resonant S₁ pumping at 1.35 eV and pumping with a moderate amount of excess energy (1.72 eV) are nearly identical; however, a significant change in the exciton lifetime is observed as the excess energy is increased (1.97 eV).

1. Comparison Between Calculated Energies and Experiment .

Calculated			Experiment
State	E (eV)	ΔE (eV)	Epump (eV)	ΔE (eV)	σIRF (fs)	τ t (fs)
S1	1.83	0	1.35	0	50	210
S5	2.17	0.34	1.72	0.37	45	190
S7	2.46	0.63	1.97	0.62	100	330

^aSummary of the photon pump energy (E _pump), excess energy above the S₁ state (ΔE), and measured overall exciton lifetime (τ _t ).

^bThe fits and measured lifetimes account for the finite instrument response function, which can be approximated by a Gaussian with standard deviation of σ_IRF.

Nonradiative Mechanisms

For mechanistic insight, we analyze the nonradiative transitions (“surface hops”), nonadiabatic derivative coupling vectors (NAC vectors), and the transition density from our calculations. Figure shows that the nonradiative transitions from each electronic excited state mostly occur from consecutive electronic states, which is generally expected since the strength of the nonadiabatic coupling depends inversely on the energy gap between states. Furthermore, Figure also illustrates that S₁ → S₂ transitions are common, whereas other transitions to higher excited states are uncommon. Since the consecutive hops are the main net contributors to the nonradiative transitions, we primarily focus on the mechanisms of these transitions and their relation to the chemical structure of the polymers.

6.

Matrix plot showing the cumulative “hops”/nonradiative transitions between electronic states of the ensemble of NAMD trajectories at 0.5 ps.

The magnitude of the nonadiabatic derivative coupling (NAC) vectors − ∥d _i,i‑1∥, at each nuclei are shown visually via the colors in Figure (visualization of the NAC vectors are given in the SI). Generally, the NAC vectors are largest on the backbone of the polymer, particularly on the carbons in the five-membered rings in each monomer unit. The S₅ → S₄ transitions have the most contributions from the five-member ring carbons on monomers 1 and 6 (numbered left to right). The S₄ → S₃ and S₃ → S₂ transitions have contributions similar to S₅ → S₄ but with some more contribution coming from monomer unit 5. The S₂ → S₁ transitions are more delocalized among all monomer units, but the largest contributions are from monomer units 1, 2, 5, and 6. These results suggest that the vibronic motions responsible for nonradiative transitions between electronic states primarily involve the backbone nuclei. As such, introducing chemical functional groups that could enhance these couplings could be a promising route to modulating energy transfer in other similar polymer systems.

7.

Schematic of polymer P1 ₆, which is color-coded by the relative magnitude of the NAC vectors (∥d _i , _{i –1}∥) for (a) S₅ → S₄, (b) S₄ → S₃, (c) S₃ → S₂, and (d) S₂ → S₁.

The spatial location of the transition density on each monomer can be computed and tracked throughout the simulation − (a visualization of the NAC vectors is given in the Supporting Information), which gives a physical sense of the spatial movement of the exciton as the molecule transitions from S₅ to S₁. As seen in Figure , the transition density is initially delocalized and spread among all monomer fragments. However, as time progresses, fragment 3 (green) increases its contribution, and the molecule transitions to S₁, which is more localized on fragment 3. However, as time progresses and the molecule transitions to S₁, the contribution of fragment 3 (green) increases, signifying that the excitation becomes more localized on fragment 3.

8.

Ensemble fraction of transition density on each monomer fragment as a function of time. The solid lines indicate the mean (μ) of the respective electronic state, and the shaded regions indicate a bootstrap confidence interval (1000 samples) of two standard deviations (μ ± 2σ).

Conclusions

In conclusion, we have performed a series of time-resolved spectroscopic experiments and nonadiabatic molecular dynamics calculations to characterize nonradiative pathways in large oligomers of a low-bandgap P1 polymer. Our NAMD simulations showed no transitions to the ground state within 1.5 ps, resulting in a ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}$ lifetime of 34.75 fs. The experimental ${\tau }_{{\mathrm{S}}_5\to {\mathrm{S}}_1}$ (deduced from differences in measurements) is within a measurement resolution of 50 fs, which supports our S₅ → S₁ NAMD results. Our calculations show that most of the nonradiative transitions arise from adjacent electronic states and that the movements of the carbon atoms in the five-member rings are predominantly responsible for coupling the electronic states. Furthermore, our calculations show that the exciton is initially delocalized along the backbone of the monomer subunits and becomes more localized as energy transfer occurs. The analyses enabled by our large-scale NAMD calculations provide detailed mechanistic insight into the electron-nuclear processes that mediate photo-induced dynamics in these complex polymer systems. Most importantly, the large-scale NAMD calculations and analyses used in this work demonstrate that these techniques can be used to understand energy-transfer mechanisms in large systems, which we are currently applying to other low-bandgap polymer materials.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.chemmater.5c00311.

• Linear fit of S₁ excitation energies as a function of 1/N, visualization of the nonadiabatic derivative coupling vectors and transition densities at geometries where nonradiative transitions occur, and additional experimental spectra and details (PDF)

The authors declare no competing financial interest.