Figure 1

The autocorrelation time of the total energy for harmonic oscillators of frequency $\omega$ [cf. Equation (5)] is plotted for different values of the thermostat parameters. Dark curves correspond to high friction (${\gamma }^{-1}=20\mathrm{fs}$) whereas light ones correspond to a more gentle thermostat (${\gamma }^{-1}=1\mathrm{ps}$). Dotted lines correspond to white noise (${\tau }_F=0$) and full ones to colored noise with ${\tau }_F=2\mathrm{fs}$. The curves are superimposed on the vibrational density of states (DoS) for a polarizable force-field simulation of crystalline calcite, which was obtained from the Fourier transform of the velocity-velocity autocorrelation function. For reference, we report the shell vibrational modes as obtained from a run where we artificially heated the shells to 300 K.Reuse & Permissions

Figure 2

Shell temperature ($T_S$) for calcite as a function of the thermostat parameters. The points are joined by continuous lines, for clarity sake. In both panels, we distinguish the strength of the ion thermostat by the line color. Lighter or darker (blue or red in the online version) curves correspond, respectively, to a strong (${\gamma }^{-1}=20\mathrm{fs}$) or mild (${\gamma }^{-1}=1\mathrm{ps}$) friction. In panel (a), we plot $T_S$ against ${\tau }_F$, and we choose two extreme values of the shell friction, ${\gamma }_S^{-1}=1\mathrm{ps}$ and ${\gamma }_S^{-1}=50\mathrm{fs}$, which are represented, respectively, with full and dashed lines. In panel (b), we plot the dependence of $T_S$ versus ${\gamma }_S$. Here, full and dashed lines correspond, respectively, to a physically meaningful filter (${\tau }_F=2\mathrm{fs}$) and to white noise (${\tau }_F=0$).Reuse & Permissions

Figure 3

Autocorrelation functions for the squares of (a) the symmetric stretching and (b) the bending modes of a heavy water molecule in vacuum, performed in the $NVT$ ensemble at $T=300\mathrm{K}$. We use a fictitious mass $\mu =200$ a.u., and a time step of 4 a.u., in order to minimize the errors on the forces [24]. The Nosé-Hoover thermostat with chain length 4 has been used and its mass chosen so as to maximize the coupling to the stretching mode. The NH correlation functions (lighter lines, blue in the online version) are highly oscillating and decay very slowly. The shading highlights the curve’s envelope. In contrast, using the new thermostat (darker lines, red in the online version), we find a much sharper decay, which in the case of the stretching requires an enlarged scale to be appreciated [inset of panel (a)]. In the inset of panel (b) we show the relation between ${\tau }_H$ and $\omega$ for our thermostat. The parameters have been optimized to obtain a sharp decay of the response for frequencies above the stretching mode.Reuse & Permissions