Choosing the Nosé-Hoover chain thermostat parameters

The Nosé-Hoover chain thermostat is defined by specifying three parameters: A target kinetic energy, a frequency and a chain length. For the ions, given the target temperature $T_W$, the target kinetic energy is just $gkT_W$, where $g$ is the number of degrees of freedom involved in a common thermostat. For example, if there is one thermostat on the entire ionic system, then $g=3N_{AT}-N_{const}$, where $N_{const}$ is the number of constraints to which the atoms are subject. The frequency for the ionic thermostat should be chosen to be some characteristic frequency of the ionic system for which one wishes to insure equilibration. In water, for example, one could choose the O-H bond vibrational frequency. (Having a precise value for this frequency is not important, as one only wishes to insure that the thermostat will couple to the mode of interest.) The choice of chain length is not terribly important as it only determines how many extra thermostats there will be to absorb energy from the system. Usually a chain length of 4 is sufficient to insure effective equilibration. Longer chains may be used in situations where heating or cooling effects are more dramatic.

For the electrons, the target kinetic energy is not usually known a priori as it is for the ions. However, by performing a short run without thermostats, one can determine a value about which the electron kinetic energy `naturally' fluctuates and take this as the target value. While the precise value is not important, a little experience goes a long way, as a choice that is either too small or too large can cause spurious damping of the ions or departures from the Born-Oppenheimer surface, respectively. A good choice for the frequency of the electron thermostat can be made based on $\Omega_I^{\rm max}$, the maximum frequency in the phonon spectrum. The frequency of the electron thermostat should be at least 2-3 times this value to avoid coupling between the ions and the electron thermostats. As an example, for silicon, the highest frequency in the phonon spectrum is 0.003 a.u., so a good choice for the electron thermostat frequency is 0.01 a.u. The chain length of the electron thermostat can be chosen in the same way as for the ions. 4 is usually sufficient, however longer chains may be used if serious heating is expected. In addition, the electron thermostats have an extra parameter that scales the number of dynamical degrees of freedom for the electrons. ( $1/\beta_e = 2E_e/N_e$, where $E_e$ is the desired electron kinetic energy and $N_e$ is the number of dynamical degrees of freedom for the electrons - see Eq. (3.4) in Ref.[5]). The default value is the true number of dynamical degrees of freedom $N_e = (2*N_{GW}-1)*N_{ST} - N_{ST}^p$, where $p=2$ for orthonormality constraints and $p=1$ for norm constraints. When this number is very large, it may not be possible to integrate the electron chain thermostats stably using a frequency above that top of the phonon spectrum. Should this be the case in your problem, then the number of dynamical degrees of freedom should be scaled to some smaller number such that the system can once again be integrated stably. This parameter has no other effect that to change the relative time scales between the first element of the electron thermostat chain and the other elements of the chain.

In addition to the basic parameters defining the chains themselves, one needs to specify two more parameters related to the integration of the thermostated equations of motion. The first is the order $M_{SUZ}$ of the Suzuki integrator. Experience shows that the choice $M_{SUZ}=3$ is sufficient for most applications. Finally, one must specify the number of times the Suzuki integrator will be applied in a given update. This is the parameter $N_{SUZ}$ which determines the basic Suzuki time step $\delta t$ = $\Delta t/N_{SUZ}$, where $\Delta t$ is the time step being used in the MD run. $N_{SUZ} =2$ or 3 is usually large enough to give stable integration. If more stable integration is required, try $M_{SUZ}=4$ or make $N_{SUZ}$ larger.