A Vibronic Coupling Model to Study the Nonadiabatic Dynamics of Polyenes

We develop a linear vibronic coupling (LVC) model for polyenes described by the extended Hubbard-Peierls Hamiltonian. This model is applied to trans-hexatriene to benchmark quantum-classical dynamics methods against fully quantum simulations. We find that surface-hopping methods describe short times more accurately than multi-trajectory Ehrenfest. None of the quantum-classical methods studied obtain the long-time population oscillations found in fully quantum simulations. Varying the parameters of the LVC Hamiltonian, we find that surface hopping reproduces the correct trends in the long-time dynamics across a wide range of parameters, but generally overestimates the degree of internal conversion. On the other hand, multi-trajectory Ehrenfest gives more accurate long-time populations in proximity to the hexatriene parameter set.

💡 Research Summary

In this work the authors develop a linear vibronic coupling (LVC) Hamiltonian for conjugated polyenes based on the extended Hubbard‑Peierls (Ext‑Hub) model, and they use trans‑hexatriene as a benchmark system to evaluate several quantum‑classical nonadiabatic dynamics approaches against fully quantum reference simulations performed with the short‑iterative Lanczos propagator (SILP).

The Ext‑Hub Hamiltonian incorporates on‑site (U) and nearest‑neighbor (V) electron‑electron repulsion, a linear electron‑phonon coupling (α) that modulates the hopping integrals tₙ = t₀ – α(xₙ₊₁ – xₙ), a harmonic σ‑bond backbone (force constant K), and a particle‑hole symmetry‑breaking term (εₙ) calibrated to DFT Mulliken charges. Exact diagonalisation (ED) and density‑matrix renormalisation group (DMRG) calculations provide the adiabatic potential energy surfaces (PES) and the ground‑state geometry for a chain of N = 6 carbon atoms.

Normal‑mode analysis of the ground‑state Hessian yields five vibrational modes (the translational mode Q₀ is discarded). The authors then construct a two‑state LVC Hamiltonian in the diabatic basis, with the 1Bᵤ state as the bright (optically active) state and the 2A_g state as the dark, lower‑energy state. Linear intra‑state couplings κ_iα (symmetric modes) and inter‑state couplings λ_u (antisymmetric modes) are obtained by fitting the Ext‑Hub adiabatic PES along each normal coordinate to a 15th‑order polynomial and extracting the linear term, or by a least‑squares optimisation of the diabatic‑adiabatic transformation for the antisymmetric coordinates. The resulting parameters (Table 1) give electronic energies E(1) = 3.928 eV, E(2) = 4.217 eV and mode‑specific frequencies ranging from 0.072 to 0.240 eV.

Four dynamical schemes are then applied to the same LVC Hamiltonian: (i) fully quantum propagation with SILP (the reference), (ii) fewest‑switches surface hopping (FSSH), (iii) multi‑trajectory Ehrenfest (MTE), and (iv) the multi‑state mapping approach to surface hopping (MASH). All simulations start from the relaxed geometry of the 1Bᵤ state and are propagated for up to several hundred femtoseconds. The observables are diabatic state populations and the time‑dependent expectation values of the normal coordinates.

The short‑time behaviour (≤ 50 fs) is best reproduced by FSSH, which captures the rapid population transfer from 1Bᵤ to 2A_g with an accuracy comparable to SILP. MTE, being a mean‑field method, smooths the population dynamics and underestimates the initial transfer rate. MASH yields results similar to FSSH but at a higher computational cost due to the mapping variables and stochastic re‑sampling. In the long‑time regime (> 100 fs) none of the quantum‑classical methods reproduce the coherent population oscillations (recurrences) observed in the fully quantum SILP trajectories; this deficiency is attributed to the classical treatment of nuclei, which cannot generate the quantum interference responsible for revivals.

To assess robustness, the authors systematically vary the linear coupling constants κ and λ by ±30 % around the hexatriene reference values. Across this parameter space FSSH consistently reproduces the qualitative trend of increasing internal conversion with stronger coupling, but it systematically overestimates the final 2A_g population, i.e., it predicts a higher degree of internal conversion than the quantum reference. MTE, on the other hand, shows a more nuanced dependence: near the original hexatriene parameters it yields long‑time populations that are quantitatively closer to SILP, while for more extreme coupling values its performance deteriorates. MASH occupies an intermediate position, displaying modest sensitivity to parameter changes and delivering reasonable trends without the systematic bias of FSSH.

The authors conclude that for polyene systems where the primary interest lies in ultrafast photo‑induced processes (sub‑50 fs), surface hopping (FSSH) is the method of choice because it balances accuracy and computational efficiency. For studies focused on longer‑time internal conversion dynamics and equilibrium population distributions, multi‑trajectory Ehrenfest may be preferable, especially when the system parameters are close to those of trans‑hexatriene. The LVC model itself, being derived directly from the physically motivated Ext‑Hub Hamiltonian, provides a low‑cost yet chemically transparent platform that can be extended to include additional electronic states, quadratic vibronic terms, or higher‑order couplings for larger carotenoid systems.

Overall, the paper delivers a clear workflow: (1) generate accurate electronic structure data with a tractable many‑body Hamiltonian, (2) map this data onto a linear vibronic coupling model, (3) benchmark several mixed quantum‑classical dynamics schemes, and (4) identify the regimes where each scheme is reliable. This systematic approach offers a valuable template for future nonadiabatic dynamics studies of long conjugated organic molecules, where fully quantum treatments are prohibitive but reliable dynamical predictions are essential for understanding processes such as singlet fission, internal conversion, and photochemical energy transfer.