01.01.2021 − 31.12.2022
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 utilized here can be tested against fully quantum-mechanical approaches of MATH+ project AA2-2. There, a representation of Hamiltonians and quantum states in terms of tensor trains (aka matrix product states) has been explored. For the time-independent Schrödinger equation, this has lead to almost linear scaling of storage consumption and computational effort in the number of sites, thus enabling us precise calculations of self-trapping beyond the Davydov theory.
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)
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.