# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 4 Critical Analysis of Quantum Baths: Model Systems
### Several Degrees of Freedom and Zero-Point Energy Leakage
#### 4.3.2 Zero-Point Energy Leakage  
Table 1 shows different energy contributions calculated exactly, with the QTB and with classical MD. We easily notice that the QTB provides a massive energy correction to the classical MD and that the total energy is in good agreement with the exact one. The total kinetic energy in QTB is slightly overestimated due to spectral broadening effects for finite \(\gamma\) coefficients (as presented in Section 3.1). The most notable energy errors are found in the stretching and bending contributions to the potential energy: the former is underestimated while the latter is overestimated. This discrepancy is typical of the zero-point energy leakage (ZPEL). Note that the impact of the ZPEL is reduced when the coupling coefficient \(\gamma\) is increased. Zero-point energy leakage is a well-known issue, initially described in the context of LSC-IVR simulations, in which the system is first prepared with the appropriate quantum energy distribution, after which short classical simulations are carried out to assess the dynamics. As the dynamics are indeed classical, the equipartition theorem will take over and eventually destroy the quantum distribution [46; 47; 48]. This is clearly connected with \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\langle\mathbf{E_{\text{kin}}}\rangle\) & \(\langle\mathbf{E_{\text{stretch}}}\rangle\) & \(\langle\mathbf{E_{\text{bend}}}\rangle\) & \(\langle\mathbf{E_{\text{tot}}}\rangle\) \\ \hline Classical & 26 & 13 & 13 & 52 \\ QTB (\(\gamma=1\) THz) & 143 & 87 & 64 & 294 \\ QTB (\(\gamma=10\) THz) & 146 & 90 & 60 & 296 \\ QTB (\(\gamma=20\) THz) & 149 & 93 & 56 & 298 \\ Quantum & 143 & 99 & 49 & 291 \\ \hline \end{tabular}
\end{table} 
Table 1: Average energies for the 2D model of O–H bond at 300 K (energy values in meV). Quantum results are obtained from a PIMD simulation with 96 beads. Figure 7: Quantum probability distribution (solid black), classical distribution (dotted blue) and QTB probability distribution (red), at \(T=300\) K for the 2D O–H potential. Contour levels are 1.0 Å\({}^{-2}\), 5.0 Å\({}^{-2}\), 9.0 Å\({}^{-2}\), 13.0 Å\({}^{-2}\), 17.0 Å\({}^{-2}\). anharmonicity that allows vibrational modes to exchange energy, resulting, in practice, in a flow of energy from high-frequency modes to low-frequency modes. As the effective temperature can be very high, the outcome might be dramatic, as the melting down of the system in solid-state simulations. In the case of LSC-IVR simulations, the problem can be addressed by carrying out short enough classical dynamics, so that the leakage does not significantly alter the energy distribution and thus the dynamical results. It is not so with the quantum baths methods [34; 49] as the quantum energy distribution is generated on the fly through the colored Langevin thermostat. The result is therefore a compromise between how much energy is being pumped into the system by the colored noise and how fast this energy is flowing to low frequency vibrational modes finally to be dissipated by the friction forces. A systematic endeavour [50] to measure ZPEL in QTB simulations as a function of anharmonicity and the coupling coefficient \(\gamma\) in simple model systems such as non-linearly coupled harmonic oscillators showed that ZPEL indeed increases with anharmonicity and that increasing the coupling coefficient \(\gamma\) significantly reduces the effects of ZPEL. Similar observations were made regarding the QT in the early papers on this method [14; 34] and strongly coupled generalized baths were designed in order to mitigate the effect of ZPEL, an approach comparable to the increase of the coupling coefficient \(\gamma\) of the QTB. However, the spectral broadening induced by large coupling coefficients can hinder the use of quantum baths for the computation of vibration spectra. Even more importantly, large system-bath couplings do not completely suppress ZPEL, which can remain significant in strongly anharmonic systems such as liquid water where it has massive consequences, as examined in Section 7.3.