# Exciton dynamics

Excitons are electron-hole pairs bound by Coulomb interaction which can be generated in semiconductors or insulators by interaction with light. In the simplest case, they involve excitations of electrons from the (highest) occupied to the (lowest) unoccupied molecular orbital, or from valence to conduction band. Regarded as quasi-particles in solid-state materials, excitons can transport energy without transporting net electric charge. Eventually they release their energy by recombination, coupling to lattice vibrations, or dissociation into separate charges. Efficient excitonic energy transport is of paramount importance in a variety of opto-electronic applications. For example, in photovoltaic solar cells, excitons have to migrate from "antenna" sites of efficient light absorption to active interfaces such as electrodes or embedded catalytic sites in order for charge separation to occur.

## Exciton dynamics in organic semiconductors

Rupert Klein with Burkhard Schmidt

Cooperations with Patrick Gelss, Felix Henneke, Sebastian Matera

Support by ECMath (Einstein Center for Mathematics Berlin) through project SE 20 (2017/18)

Support by MATH+ (Berlin Mathematics Research Center) through projects AA2-2 (2019/20) and AA2-11 (2021/22)

In organic semiconductors such as molecular crystals or conjugated polymer chains, excitons are typically localized (Frenkel excitons), and their transport is normally modeled in terms of excitons diffusively hopping between sites. An improved understanding of excitonic energy transport has to account for the role of electron-phonon coupling (EPC). We limit ourselves to the use of rather simple models of quantum dynamics of excitons, i.e., only two electronic states with nearest-neighbor interactions, only harmonic lattice vibrations, and only linear EPC (known as Frenkel, Holstein, Fröhlich, Davydov, and/or Peierls Hamiltonians).

Despite of these models being under investigation for several decades already, and despite of their apparent simplicity, solving the corresponding quantum-mechanical Schrödinger equation still represents a major challenge. Analytic solutions are elusive, and numerical approaches suffer from the curse of dimensionality, i.e. the exponential growth of computational effort with the number of sites involved. To cope with that problem, we employ a hierarchy of different approaches detailed in the following.

### Fully quantum-mechanical approaches

Our work on a fully quantum-mechanical approach to coupled excitons and phonons focuses on the use of efficient low-rank tensor decomposition techniques to mitigate the curse of dimensionality as much as possible. The limitation to chain structures with nearest neighbor interactions in the Fröhlich-Holstein type Hamiltonians for coupled excitons and phonons suggests the use of tensor train formats, also known as matrix product states, representing a good compromise between storage consumption and computational robustness.

For the time-independent Schrödinger equation [86], we introduced an approach which directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems in the TT format. This has lead to almost linear scaling of storage consumption in the number of sites, along with an only slightly faster increase of the CPU time. This technique has allowed us to directly tackle the phenomenon of mutual self-trapping, see the figure avove. We were able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians.

We also investigated tensor-train approaches to the solution of the time-dependent Schrödinger equation [88]. One class of propagation schemes that we explored builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions. In addition to the first order Lie-Trotter and the second order Strang-Marchuk splitting schemes, we also implemented a 4th order Yoshida-Neri and an 8th order Kahan-Li symplectic composition which are best when very high accuracy is required for shorter chains. Another class of propagators involves explicit, time-symmetrized Euler integrators for which we also implemented methods of 4th and 6th order, the former of which represents a good compromise between accuracy and computational effort for longer chains.

Transfer of our methodological work to applications has been guaranteed by making models, algorithms, and software freely available through the open-source WaveTrain software package for numerical simulations of chain-like quantum systems with nearest-neighbor interactions [89]. This Python package is centered around TT format representations of Hamiltonian operators and state vectors. It builds on the Python tensor train toolbox Scikit_TT, which provides efficient construction methods and storage schemes for the TT format. WaveTrain software is freely available from the GitHub platform where it will also be further developed. Moreover, it is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics.

### Mixed quantum-classical approaches

In cases where the space and time scales governing the dynamics of excitons and phonons are well separated, mixed quantum-classical molecular dynamics (QCMD) provides a suitable approximation for exciton-phonon coupling. There, the electronic degrees of freedom (excitons) are treated quantum-mechanically while the ionic motions (molecular vibrations and/or lattice vibrations, aka phonons) are treated classically. In Ehrenfeld (mean field) approaches, the latter ones are subject to forces averaged over the quantum states of the former ones. An alternative is the widely used concept of surface hopping trajectories (SHT) algorithms, where the ionic positions are modeled by classical trajectories that may stochastically switch between electronic states thus resembling non-adiabatic transitions. In cooperation with L. Cancissu Araujo and C. Lasser at TU Munich, we implemented and evaluated various non-standard SHT variants for the case of Holstein-type Hamiltonians typically used to describe the dynamics of excitons coupled to phonon modes [82], see also our page on quantum-classical dynamics.

### Semi-classical approaches

In another line of activities, we combine advanced semi-classical simulation techniques for the ionic degrees of freedom with multiscale asymptotics to take advantage of a systematic scaling behavior of exciton-nuclear couplings in models of conjugated polymer chains. Our work rests on work by Hagedorn who extended the well-established theory of approximate Gaussian wave packet solutions to the time-dependent Schrödinger equation toward moving and deforming complex Gaussian packets multiplied by Hermite polynomials, yielding semi-classical approximations which are valid on (at least) the Ehrenfest time scale, i.e., the characteristic time scale of the motion of the ions. Lubich and Lasser, see their 2020 review article, developed numerical approximations based on those ideas. Their variational approaches rely on approximations to wave functions by linear combinations of (frozen or thawed) Gauss or Hagedorn functions. In principle, error bounds of any prescribed order in the semi-classical smallness parameter can be obtained, and also estimators for both the temporal and spatial discretization can be obtained efficiently, thus paving the way for fully adaptive propagation.

While the techniques sketched above will serve to overcome (at least the worst of) the curse of dimensionality, we aim at a further reduction of complexity by employing multi-scale analysis. We will utilize the fact that the expected displacements are small and that exciton-phonon coupling is much slower than the exciton transfer rate along the chain, so that we have a fast spreading of excitations that are only weakly coupled to the lattice degrees of freedom. This justifies, on sufficiently long time scales, an asymptotic WKB-like ansatz involving weak variation, or long-wave behavior. We expect to obtain the desired further complexity reduction by focusing on the associated low wave number modes. This will be particularly relevant for a large number of sites, in which case the numerical calculation of will become very expensive otherwise. In summary, building on multiscale analysis and semi-classical asymptotics, the present project aims at providing analytical insight, and hence understanding, at least in some interesting limit regimes of the relevant parameter space.