# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 7 Overcoming the Leakage via the Adaptive Quantum Thermal Bath (adQTB)
### Quantitative Assessment of the ZPEL  
As mentioned in Sections 4.3 and 5, ZPEL and its consequences were observed in several studies with either the QTB or the QT methods. The ZPEL can be mitigated by increasing the damping coefficient \(\gamma\), or equivalently in the QT formalism, by designing more strongly coupled GLE thermostats [34]. However, strong system-bath couplings also affect the accuracy of the GLE methods (even for harmonic systems, the QTB and the QT are only rigorously exact in the weak coupling limit), and distort the vibrational spectra. Furthermore, even for large \(\gamma\) values, the ZPEL is not completely suppressed andit can still have non-negligible effects. Therefore, a first step to design an effective cure is to provide a reliable diagnostic tool. Such a diagnosis is derived from the first kind fluctuation-dissipation theorem (see Section 2.1) that, in the context of the QTB, can be written as:  
\[2\theta(\omega,T)\ \text{Re}[\chi(\omega)]=C_{vv}(\omega) \tag{30}\]  
where \(\theta(\omega,T)\) is the quantum energy distribution, \(\chi(\omega)\) is the complex admittance or susceptibility that characterizes the response of the system to a perturbation, and \(C_{vv}(\omega)\) is the Fourier transform of the velocity-velocity correlation function. By using the quantum first-kind FDT (Equation (3)), the symmetrized autocorrelation spectrum \(C_{vv}^{s}(\omega)\) (replaced here by \(C_{vv}(\omega)\)), can be evaluated from the Fourier transform of the QTB trajectories as it would be in a classical framework, consistently with the quasiclassical approximation at the root of the method. As introduced in Section 2.1, the first-kind FDT is a very general relation that holds for a quantum system at thermal equilibrium and indicates that the vibration power density is distributed according to the quantum energy distribution \(\theta(\omega,T)\). The first-kind FDT is more fundamental than the second-kind FDT as used in the original formulation of the QTB, which relates the strength of the friction and random forces in the generalized Langevin equation. Indeed, the QTB would thermalize the vibrational modes with the appropriate average energy if they were non interacting, but in general anharmonic systems, the modes are coupled. This generates an energy flow from high to low frequencies, even though the system-bath coupling obeys the second-kind FDT. This flow creates an energy unbalance that can be detected through the violation of the first-kind FDT (30). To exploit this relation, it was shown that, in a Langevin dynamics, the linear susceptibility can be estimated from the velocity-random-force correlation [17]:  
\[\text{Re}[\chi(\omega)]=\frac{\text{Re}[C_{vR}(\omega)]}{C_{RR}(\omega)}=\frac {\text{Re}[C_{vR}(\omega)]}{2m\gamma\theta(\omega,T)} \tag{31}\]  
The first-kind FDT can then be rewritten as (here we use one-dimensional notations; generally, the FDT should be satisfied for each degree of freedom and for each frequency independently):  
\[\text{Re}[C_{vR}(\omega)]=m\gamma C_{vv}(\omega) \tag{32}\]  
where \(C_{vR}(\omega)\) is the Fourier transform of the velocity-random-force correlation function. All terms in Equation (32) are now easily computed during a simulation run, and deviations from it are a clear indication that ZPEL occurs. Indeed, the ratio between the two sides of the equation can be used to define an effective frequency-dependent temperature \(T_{\text{eff}}(\omega)\), that, if the first kind FDT holds, should follow the quantum distribution of energy:  
\[T_{\text{eff}}(\omega)=\frac{\text{Re}[C_{vR}(\omega)]}{m\gamma C_{vv}(\omega )}k_{B}^{-1}\theta(\omega,T)=k_{B}^{-1}\theta(\omega,T)\quad\text{ iff FDT holds} \tag{33}\]  
The effect of ZPEL on this effective temperature is represented in Figure 11 in the case of liquid water. Even with a relatively large value of \(\gamma\), the effective temperature deviates from the first kind FDT reference (shown as a black line), which is a clear evidence of ZPEL.