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Typical masses, forces, temperatures, etc, involved in many physical/chemical processes strongly suggest that the field of molecular dynamics is situated on the border of quantum and classical mechanics. Mathematical models developed to describe molecular processes should (at least partly!) account for quantum effects. Due to the exceedingly high effort of fully quantum dynamical simulations of complex molecular systems, mixed quantum-classical simulations often provide a promising approach on the way to high dimensionality.

Exciton dynamics in organic semiconductors

Rupert Klein with Burkhard Schmidt
Cooperation with Felix Henneke

Support by ECMath (Einstein Center for Mathematics Berlin) through project SE 20 (2017/18)
Support by Math+ (Berlin Mathematics Research Center) through project AA2-2 (2019/20)

The performance of optoelectronic devices, such as photovoltaic cells, is critically influenced by the transport of excitonic energy because the majority of photo-excitations occur in the bulk of the crystal from where the energy has to be transported to the interfaces with the electrodes, where charge generation often only occurs. In organic semiconductors, e.g. molecular crystals, polymer chains or dendrimers, the excitons are strongly localized, and the energy transport is normally modeled in terms of excitons diffusively hopping between sites. The present proposal aims at an improved understanding of the excitonic energy transport in organic semiconductors, which is relevant for the characterization of organic solar cells, on a microscopic basis, with emphasis on the role of the electron-phonon coupling. Using mixed quantum-classical dynamics schemes, the electronic degrees of freedom (excitons) are to be treated quantum-mechanically while the nuclear motions (phonons, molecular vibrations) are treated classically. To this end, stochastic surface-hopping algorithms shall be applied and further developed.

Most of the quantum and quantum-classical simulations have been carried out with our WavePacket software package (freely available via the SourceForge platform), which has been developed in parallel with these projects. For an extensive documentation, see also [75] [78] [82].

Figure adapted from DOI:10.1039/C5EE00925A

Mixed Quantum-Classical Dynamics

Burkhard Schmidt (previously with Illia Horenko)
Cooperation with Leonardo Cancissu Araujo and Caroline Lasser (TU München)

In mixed (or hybrid) quantum-classical mechanics few (but important!) degrees of freedom are modeled quantum-mechanically while the remaining ones are treated within the classical approximation. 
 A mathematically rigorous approach to this is based on the technique of the partial Wigner transform of the quantum Liouville-von Neumann equation for systems with two-components of disparate masses. As a first order approximation, the quantum-classical Liouville equation (QCLE) describes consistently the evolution of densities and coherences in phase space thereby overcoming the main limitation of the mean field (Ehrenfest) approach to mixed quantum-classical dynamics where all densities are subject to the same (mean field) potential.

Our approaches to efficient numerical propagation schemes for the QCLE are based on a representation of densities as well as coherences in terms of Gaussian packets in phase space [38]. In our surface hopping Gaussian (SHG) algorithm [39] these Gaussian packets evolve independently; non-adiabatic transitions at (avoided or genuine) crossings or at conical intersections are modeled by stochastic hopping of these Gaussians, thus representing a major step beyond the stochastic surface hopping trajectory approch (SHT). In other work, deterministic approaches have been cultivated: Based on Rothe methods, the trapezoidal rule adaptive integrator for Liouville dynamics (TRAIL) offers the advantage of full adaptivity [43] where the quality of the spatial approximation can be controlled by dynamic creation and/or annihilation of Gaussians. We also developed theoretical QCLE-based models for quantum dynamics driven by external fields based on Floquet representations [34].

Figure adapted from  Basile  F. E. Curchod

... aber wenn es auch von einem logischen Standpunkt aus gesehen vielleicht ein Verbrechen ist, muss ich gestehen, dass ich nichtsdestoweniger davon überzeugt bin, dass der Schwindel des Vermischens der klassischen Theorie und der Quantentheorie sich noch auf viele Weisen beim Aufspüren der Geheimnisse der Natur als fruchtbar erweisen wird ...

N. Bohr an W. Pauli, 11. Dez. 1924
Burkhard Schmidt,
Jun 9, 2017, 11:37 AM
Burkhard Schmidt,
Jun 9, 2017, 11:37 AM