**Project Heads**

*Rupert Klein*

**Project Members**

Burkhard Schmidt (FU)

**Project Duration**

01.01.2019 – 31.12.2020

**Located at**

FU Berlin

This project aims at modeling and simulation of π-conjugated polymer chains for use in photovoltaics. The excitonic energy transport results from a complex interplay of three scales in time and space, the efficient numerical treatment of which necessitates dedicated multiscale approaches. Thus, the chain stretching modes are in the classical regime, while electrons and torsional/bending motions call for quantum dynamics: the high dimensionality is tackled by exploiting analogies of layered multiconfigurational approaches with hierarchical tensor tree formats.

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 numeric approaches suffer from the curse of dimensionality, i.e. the exponential growth of computational effort with the number of sites involved.

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 beat the curse of dimensionality. The limitation to chain structures with nearest neighbor interactions in the electron-phonon Hamiltonians mentioned above suggests the use of tensor train formats, also known as matrix product states, representing a good compromise between storage consumption and computational robustness. The time-independent Schrödinger equation is solved using an alternating linear scheme (ALS), and higher quantum states are obtained by an approach that directly incorporates the Wielandt deflation technique into the ALS for the solution of eigenproblems. In test calculations for homogeneous systems, we find that the tensor-train ranks of the state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the CPU time increases slightly faster with the chain length because the ALS requires more iterations to achieve convergence for longer chains and a given rank.

As a first test, tensor train approaches based on a SLIM decomposition of the Hamiltonian have been used to investigate the phenomenon of self-trapping, i.e., the formation of localized excitons “dressed” with deformations of the ionic scaffold. Within a certain range of the parameters involved, our calculations exactly reproduce the predictions by Davydov’s soliton theory of excitonic energy transport, but we are also able to explore cases where the rigorous assumptions of that approximate analytic theory do not apply.

**Project Webpages**

**Selected Publications
**

*B. Schmidt,**R. Klein,**L. Cancissu Araujo:*WavePacket: A Matlab package for numerical quantum dynamics. III: Quantum-classical simulations and surface hopping trajectories.

Journal of Computational Chemistry**40**(30), 2677-2688 (2019)- F. Henneke, L. Lin, C. Vorwerk, C. Draxl, R. Klein, C. Yang:

Fast optical absorption spectra calculations for periodic solid state systems.

Communications in Applied Mathematics and Computational Science,**15**(1), 89-113 (2020) - P. Gelß, R. Klein, S. Matera, B. Schmidt:

Solving the time-independent Schrödinger equation for chains of

coupled excitons and phonons using tensor trains.

Journal of Chemical Physics**156**(2), 024109 (2022)

**Selected Pictures
**

Self-trapping of excitons and phonons obtained from tensor train calculations. Note that the excitonic probability distribution exactly matches the analytic form predicted by the Davydov theory for excitonic energy transfer.

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