  The interplay of electronic and nuclear degrees of freedom presents an
outstanding problem in condensed matter physics and chemistry. Computational
challenges arise especially for large systems, long time scales, in
nonequilibrium, or in systems with strong correlations. In this work, we show
how downfolding approaches facilitate complexity reduction on the electronic
side and thereby boost the simulation of electronic properties and nuclear
motion - in particular molecular dynamics (MD) simulations. Three different
downfolding strategies based on constraining, unscreening, and combinations
thereof are benchmarked against full density functional calculations for
selected charge density wave (CDW) systems, namely 1H-TaS$_2$, 1T-TiSe$_2$,
1H-NbS$_2$, and a one-dimensional carbon chain. We find that the downfolded
models can reproduce potential energy surfaces on supercells accurately and
facilitate computational speedup in MD simulations by about five orders of
magnitude in comparison to purely ab initio calculations. For monolayer
1H-TaS$_2$ we report classical replica exchange and quantum path integral MD
simulations, revealing the impact of thermal and quantum fluctuations on the
CDW transition.
