# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 8 Quantumness from Baths: State of the Art and Open Issues
Moreover, this solution to the ZPE leakage made it possible to distinguish the intrinsic inaccuracies of the QTB method from those that are induced by the leakage, thus opening the way to other possible improvements of quantum baths. We note that, although the adaptive procedure has so far only been explored in the context of the QTB, in principle it could be extended to the QT formalism as well. We stress that, at variance with their classical counterpart, the equilibrium distributions obtained from quantum baths cannot be derived analytically apart from a few cases, such as the harmonic oscillators. We illustrated through a few examples the performances of quantum baths (focusing more particularly on the QTB) both on equilibrium, time-independent, averages, and time-dependent correlation functions. Quite a wide experience has thus been gained on various systems, differing by the kind of interatomic bonding and the thermodynamic conditions. Rather generally, the QTB yields the maxima of the particle distributions at the minima of the potential energy surface, as do the classical ones. This is a minor shortcoming for mildly anharmonic cases, but might be a serious limitation when dealing with strongly anharmonic systems, characterized by double wells or when shallow potential energy surfaces are combined with large ZPE contributions; in such cases, a comparison with exact probability distributions as obtained via path-integral-based simulations could be helpful. However, in many realistic problems, and even close to phase transitions (for which the potential typically presents double well features), quantum baths much improve the classical description, capturing the correct trends and yielding quantitatively reliable results for a number of observables. It turns out the quantum baths provide also encouraging results when dynamical processes are under scrutiny. They combine approximate sampling of the quantum thermal distribution with an essentially classical time propagation. The vibrational spectra, which can be easily obtained and analyzed as in the purely classical dynamics, can provide a relevant description of vibrational properties, including anharmonic properties, overtones and resonances, although a built-in criterion for attaining the convergence with respect to fully quantum simulations has not emerged yet. Many questions remain open in that respect, and quantum transport phenomena represent an interesting challenge at the time of writing. Another recent use of quantum baths relies on their coupling to other methods. In this respect, the combination of quantum baths with the path-integral formalism often provides a good compromise: while path-integral-based methods converge towards exact equilibrium states when the number of beads is increased, they are costly. When the beads are thermalized or coupled to quantum baths, their number can be significantly reduced. This enables the simulations of molecules and condensed matter systems at very low temperatures, which are otherwise out of reach within pure path-integral molecular dynamics, by keeping the number of beads relatively small. To conclude, quantum baths have known an increasing number of applications for the simulation of material properties since their conception. In this review, we analyzed the difficulties in their use as well as their shortcomings; at the same time, we prospected new solutions for making them more reliable and physically sound. We anticipate that other improvements will be built, so as to make generalized (quantum) baths an increasingly reliable simulation tool for nuclear quantum effects in material science. Writing--original draft, review & editing, S.H., T.P., S.B., P.D. and F.F. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The published data are available on request. The authors declare no conflict of interest.