**Project Heads**

Rupert Klein

**Project Members**

Burkhard Schmidt

**Project Duration**

01.01.2021 − 31.12.2022

**Located at**

FU Berlin

In this project 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 numeric calculations 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.

At least for a small to medium number of sites, the semi-classical and multiscale methods can be tested against fully quantum-mechanical approaches to coupled excitons and phonons modeled by Fröhlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. In order to mitigate the curse of dimensionality as much as possible, representations of Hamiltonians and quantum states in terms of tensor trains (TTs, aka matrix product states) have been explored.

For the time-independent Schrödinger equation [1], 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 below. 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 [2]. 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 [3]. 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.

**External Website**

**Related Publications
**

- 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) - P. Gelß, R. Klein, S. Matera, B. Schmidt:

Quantum Dynamics of Coupled Excitons and Phonons in Chain-Like Systems: Tensor Train Approaches and Higher-Order Propagators.

arXiv:2302.03568 (2023) - J. Riedel, P. Gelß, R. Klein, B. Schmidt:

WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-Like Systems Based on Tensor Trains.

Journal of Chemical Physics**158**(16), 164801 (2023)

**Selected Pictures
**

Effect of self-trapping of excitons and phonons as a function of the coupling strength σ and for various values of the chain length N. Top: Energetic stabilization, with the straight line representing the power law as predicted by the Davydov theory. Bottom: Total degree of excitation of the lattice vibrations.