# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 4 Critical Analysis of Quantum Baths: Model Systems
### Morse Potential in One Dimension
#### 4.2.1 A 'Mildly' Anharmonic Example  
Simulations were carried out with the following parameters: \(V_{0}=3.0\) eV, \(x_{0}=1.0\) A, \(d=0.4\) A, \(T=300\) K, \(\gamma=0.1\) THz and cutoff frequency \(f_{\rm max}=500\) THz. The mass of the particle is that of a proton. These parameters yield a harmonic frequency of 3185 cm\({}^{-1}\), of the same order of magnitude as that of OH stretching modes in various systems. The zero-point energy over \(V_{0}\) ratio is 0.065, so this example can be considered as mildly anharmonic, given that, at this low temperature, only the ground state is significantly populated.  
The quantum analysis and the QTB both yield a mean position \(\left\langle x\right\rangle=1.02\) A, while the classical simulation yields 1.00 A. The quantum total energy (actually the zero-point energy at this temperature) turns out to be 194 meV while the QTB gives 197 meV (for the small friction coefficient \(\gamma\) used in this case, the kinetic energy overestimation effect mentioned in Section 4.1 is minor and the dependence on the frequency cutoff is negligible) and the classical only 26 meV which, of course, does not take into account the large ZPE. The quantum distribution is shown in green in Figure 3 while the classical distribution (yellow) in much sharper. The distribution generated by the QTB (blue) is adequately broadened, however, its shape is slightly modified: the maximum is at the the classical position, while its tail reaches out further.  
Figure 4 shows the corresponding spectra. The QTB spectrum is broadened and red-shifted with respect to the classical one as the anharmonic part of the potential is explored further, but red-shift of the actual quantum transition (between the ground state and first excited states) is even larger; therefore, the QTB captures the correct trend while slightly underestimating the quantum effect. The inset shows the overtone resonance at approximately twice the main peak frequency. Similar shifts are observed between the classical, QTB and exact quantum results as for the main resonance. The detailed perturbation analysis performed in [16] shows that an analogy can be drawn between the QTB and linearized semiclassical initial value representation (LSC-IVR) methods, that combine quantum phase space sampling with classical molecular dynamics for the evaluation of time correlation functions [41; 42]. Indeed, in QTB simulations, the short-term dynamics is essentially classical, whereas the coupling with the bath introduces quantum statistics in the phase space sampling. It can be shown that the overtone intensity is underestimated by approximately a factor of two due to the absence of position-momentum correlation in the QTB phase space sampling [43; 44]. This error on the overtone intensity can be detected  
Figure 3: A ‘mildly’ anharmonic Morse potential (magenta, left-hand scale), with the position distributions (right-hand scale) for a classical simulation (yellow), a quantum solution through Schrödinger’s equation (green) and the Quantum Thermal Bath (light blue).  
(and potentially corrected) using the first-kind fluctuation-dissipation theorem criterion defined in Section 7, and it remains limited compared to that of path integral methods (see Section 6), which yield a much lower overtone intensity, similar to that of classical simulations [16].