# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 7 Overcoming the Leakage via the Adaptive Quantum Thermal Bath (adQTB)
### A Cure through Strict Enforcement of the First Kind FDT  
To design a cure for ZPEL, an adaptive procedure was proposed in order to compensate the energy unbalance measured via the first-kind FDT [17]. To that end, the second-kind FDT is broken and two different frequency-dependent coupling coefficients are introduced: \(\gamma_{r}(\omega)\), that replaces \(\gamma\) in the random force power spectrum, Equation (14), and \(\gamma_{f}(\omega)\) that characterizes the memory kernel of the friction force that is rendered non-Markovian (it is related to the memory kernel in Equation (6), by \(\tilde{K}(\omega)=2m\text{Re}[\gamma_{f}(\omega)]\)). The first-kind FDT is then written as [17]:  
\[\text{Re}[C_{\varphi R}(\omega)]=m\gamma_{r}(\omega)C_{\varphi R}(\omega) \tag{34}\]  
Figure 11: Effective temperature \(T_{\text{eff}}\) versus frequency as defined by Equation (33) in liquid water at 300 K described with the q-TIP4P/F model and simulated with the QTB method using \(\gamma=20\) ps\({}^{-1}\). The blue dashed line corresponds to oxygen (it is average over all O atoms and over all three directions \(x\), \(y\) and \(z\)), the dotted orange line to hydrogen, and the black continuous line is the reference \(k_{B}^{-1}\theta(\omega,T)\) that corresponds to the first-kind FDT. The simulations were performed using Tinker-HP software [81] and parameters similar to that in Ref. [18]. The inset represents a zoom on the low-frequency region and the vibrational power spectrum \(mC_{\varphi\varphi}(\omega)\) is shown as a dashed-dotted grey line.  
Figure 12: Radial distribution functions (RDF) in liquid water at 300 K described with the q-TIP4P/F model and simulated with different molecular dynamics methods (left, oxygen–oxygen; middle, oxygen–hydrogen). The friction coefficient used for QTB and adQTB methods is \(\gamma=20\) ps\({}^{-1}\), the adapted random force coefficients \(\gamma_{f}(\omega)\) are plotted on the right panel for oxygen (dashed blue) and hydrogen (dotted orange) atoms. The simulations were performed using Tinker-HP software [81] and parameters similar to that in Ref. [18].  
In the adaptive QTB (adQTB) method, the frequency-dependent coefficients \(\gamma_{r}\) and \(\gamma_{f}\) are adjusted on the fly during the simulation through a relaxation process that nullifies the difference between the two sides of Equation (34). This means that the random and the friction forces adapt by modifying the energy balance for each frequency to compensate for ZPEL and enforce the proper quantum energy distribution. In practice, only one of the two coefficients \(\gamma_{f}\) or \(\gamma_{r}\) has to be adapted. Both possibilities were tested in [17] and provide similar accuracy in compensating the ZPEL, although the variant where \(\gamma_{r}\) is adapted while \(\gamma_{f}\) is kept fixed is much simpler to apply, as it does not require the implementation of a non-Markovian friction force.  
The adaptive procedure was successfully checked on several model systems and on Ne\({}_{13}\) clusters where it was able to stop the spurious melting of the solid clusters that ZPEL causes in QTB simulations [17,49]. Plots of \(\gamma_{r}(\omega)\) on Figure 13 clearly show how the bath adapts, reducing the energy input from the bath at low frequencies, while increasing it at high frequencies.