# 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
#### 4.2.2 A 'Strongly' Anharmonic Example  
Now the following parameters are used: \(V_{0}=1.7\) eV, \(x_{0}=1.0\) A, \(d=0.2\) A and \(T=300\) K. This yields a harmonic frequency at 3185 cm\({}^{-1}\) and ground state energy over depth ratio of 0.75, that is significantly more anharmonic. Another qualitative criterion for anharmonicity can be found in how the exact quantum distribution obtained from solving [PERSON]'s equation extends beyond the potential inflexion point: beyond this point, the restoring force decreases instead of increasing, and therefore contributes strongly to anharmonic behavior. In this example, the exact distribution (green line in Figure 5) does significantly extend beyond that point (\(x\approx 1.14\) A), while in the previous (Figure 3) it does not.  
Figure 5 shows the results. The features already noticed for the mildly anharmonic case are enhanced: the maximum still remains at the classical position, while the tail extends even further towards larger distances, requiring the introduction of \(x_{\rm max}=2.6\) A to prevent the particle from escaping the well: in this strongly anharmonic situation, the calculated QTB mean position therefore will depend on how \(x_{\rm max}\) is chosen and ends up being unreliable (\(\langle x\rangle_{\rm QTB}=1.27\) A, versus the exact 1.04 A). The total energy is also over-estimated (exact quantum: 184.4 meV, QTB: 250.4 meV, classical: 26.3 meV), mainly because the increased probability at large distances \(x\) leads to an over-estimation of the average potential energy. The spectra show qualitatively the same features as above, but are much broader as expected from the increased anharmonicity.  
Figure 4: \(C_{vv}(\omega)\) spectra for the ‘mildly’ anharmonic Morse potential, for the classical MD simulation (magenta), the QTB (blue) and the first quantum transition between the ground state and first excited state (green). The inset shows show the overtones (the classical spectrum is multiplied by a factor of 10).  
In these simple 1D examples, the QTB can be considered to a certain extent as a classical dynamics simulation at a higher effective temperature: \(T_{\text{eff}}\simeq 1800\) K in the above example. Indeed, Figure 6 shows that the classical distribution, obtained at \(T=T_{\text{eff}}\), is quite similar to that of the QTB at \(T=300\) K. This should not mean that it suffices to estimate the effective temperature and carry out a classical simulation to describe NQEs correctly! In realistic systems with many degrees of freedom, each vibration mode is characterized by a different frequency and therefore a different effective temperature, so that performing classical simulations at an averaged \(T_{\text{eff}}\) would yield erroneous results [45]. The main advantage of the QTB and QT approaches is that the effective temperature imposed to each mode naturally corresponds to their frequency by construction of the generalized bath (including approximate anharmonic shifts, as shown above), without requiring prior knowledge on the system nor harmonic frequency calculations.  
Figure 5: Probability distributions (**top**) and \(C_{\text{ev}}(\omega)\) spectra (**bottom**) for the strongly anharmonic Morse potential, for the classical MD simulation (magenta), the QTB (blue) and the first quantum transition between the ground state and first excited state (green). The inset shows show the overtones. Note that the classical spectrum is multiplied by a factor of 10.