Typical masses, forces, temperatures, etc, involved in most chemical processes strongly suggest that the field of molecular dynamics is situated on the border of quantum and classical mechanics. Hence, 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, where few (but important!) degrees of freedom are modeled quantum-mechanically while the remaining ones are treated within the classical approximation, often provide a promising approach on the way to high dimensionality.
Surface hopping trajectories
Burkhard Schmidt with Rupert Klein
Cooperation with Leonardo Cancissu Araujo and Caroline Lasser (TU München)
The existence of two space/time scales governing the dynamics of ions and electrons found in most molecular systems suggests the use of hybrid quantum-classical molecular dynamics (QCMD) models. The most widely used numerical schemes for QCMD are based on the concept of surface hopping trajectories (SHT), i. e., the ionic positions are modeled by bundles of trajectories that may stochastically switch between electronic states thus representing non-adiabatic transitions. This overcomes the main limitation of the mean field (Ehrenfest) approach where the trajectories are governed by averaged potential energy surfaces .
Jointly with the group at TU Munich, we found that "single switch surface hopping" (SSSH) offers substantial advantages over the traditional "fewest switches surface hopping" (FSSH). In the latter type of simulations, one needs the first order non-adiabatic coupling vectors for integrating the Schrödinger equations attached to each of the trajectories. In marked contrast, this is not required in SSSH simulations which are based on Landau-Zener (LZ) formulae for the evaluation of non-adiabatic transition probabilities. Especially the variants involving adiabatic energy gaps only lead to a substantial reduction of computational effort while reproducing the long-time population transfer with similar precision as FSSH, also for the case of densities passing a conical intersections multiple times, as is typically found in Jahn-Teller dynamics of coupled electronic and vibrational degrees of freedom .
Surface hopping Gaussians
Burkhard Schmidt with Illia Horenko and Christof Schütte
Support by Deutsche Forschungsgemeinschaft through SFB 450
A mathematically rigorous approach to mixed (or hybrid) quantum-classical mechanics 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 . In our surface hopping Gaussian (SHG) algorithm  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  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  .
Niels Bohr in a letter to Wolfgang Pauli (Dec 11, 1924)
... 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 ...
... but even though from a logical point of view it might be a crime, I have to confess that I am nevertheless convinced that the swindle of intermingling classical and quantum theory will prove fruitful in many way in tracking down the secrets of nature ...