Figure 1

Mean total ($⟨E⟩$), potential ($⟨V⟩$), and kinetic ($⟨K⟩$) energy for a proton in the external potential of Eq. (4) as a function of $\omega$, for $k=1{\AA }^{-1}$ and $T=100\mathrm{K}$. In the inset, the Fourier transform of the momentum distribution is reported for the fully quantum, classical, and colored-noise thermostat simulations at $\omega =200{\mathrm{cm}}^{-1}$. Both in the main figure and in the inset, dashed black lines correspond to the quantum expectation values, while the dark (red) series are results from present work, using a set of parameters fitted over a frequency range from 2 to $2000{\mathrm{cm}}^{-1}$. Light (blue) lines correspond to the classical expectation values.Reuse & Permissions

Figure 2

Lattice parameter of diamond as a function of temperature, computed by isothermal-isobaric runs at atmospheric pressure [23]. Black dashed lines correspond to PIMC results from Ref. 19, and red circles to averages computed from runs using our properly fitted colored-noise thermostat. In inset (a), the average kinetic temperature $T^{\star }$, projected on a few selected normal modes, is plotted for three target temperatures (from top to bottom, 1000, 500, and 200 K) as a function of the mode frequency. The continuous lines are the expected quantum $T^{\star }(\omega )$ curves. In the inset (b), we compare the total internal energy (kinetic plus potential) as computed with PIMC (black dashed line, Ref. 19) and with colored-noise thermostat fitted to the quantum $E(\omega )$, with (red circles) and without (gray lozenges) ZPE. The latter (gray) points have been aligned to the PIMC results at $T\to 0$.Reuse & Permissions

Figure 3

Radial distribution functions for ice at $T=220\mathrm{K}$. Black, dashed line corresponds to the PIMD results of Hernández de la Peña et al. [21], dark (red) and light (blue) series to colored-noise and purely classical runs, respectively. Runs have been performed in isothermal ensemble for a box of 360 water molecules at experimental density [24], and are 125 ps-long. In inset (a), we show the experimental proton-momentum distribution for ice at 269 K (black dashed line) [25], and corresponding distribution as computed from a classical (blue line) and colored-noise (red line) simulation using a flexible TIP4P-like potential [22] at the rescaled temperature (247 K, i.e., 4 K below the quantum melting point of the potential).Reuse & Permissions