Nosé-Hoover chain thermostat
The standard Nose Hoover suffers from well known issues, such as the ergodicity violation in the case of simple harmonic oscillator[1]. As proposed by Martyna and Klein[1], these problems can be solved by using multiple Nose Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:
- [math]\displaystyle{ \mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) + \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + (3N-N_c)k_{B} T \eta_1 + k_{B} T \sum\limits_{j=2}^{M} \eta_j }[/math]
where [math]\displaystyle{ \mathcal{H}(\bold{r},\bold{p}) }[/math] is the Hamiltonian of the physical system, [math]\displaystyle{ M }[/math], [math]\displaystyle{ N }[/math] and [math]\displaystyle{ N_c }[/math] are the numbers of thermostats, atoms in the cell, and geometric constraints, respectively, and [math]\displaystyle{ \eta_{j} }[/math], [math]\displaystyle{ p_{\eta_j} }[/math], and [math]\displaystyle{ Q_{j} }[/math] are the position, momentum, and mass-like parameter associated with the thermostat [math]\displaystyle{ j }[/math]. Just like the total energy in NVE ensemble,[math]\displaystyle{ \mathcal{H'} }[/math] is valuable for diagnostics purposes. Indeed, a significant drift in [math]\displaystyle{ \mathcal{H'} }[/math] indicate that the corresponding computational setting is suboptimal. Typical reasons for this behavior involve noisy forces (e.g., because of a poor SCF convergence) and/or a too large integration step (defined via POTIM).
The number of thermostats is controlled by the flag NHC_NCHAINS. Typically, this flag is set to a value between 1 and 5, the maximal allowed value is 20. In the special case of NHC_NCHAINS=0, the thermostat is switched off, leading to a MD in microcanonical ensemble. Another special case of NHC_NCHAINS=1 corresponds to the standard Nose-Hoover thermostat.
The only thermostat parameter is NHC_PERIOD, corresponding to a characteristic time scale ([math]\displaystyle{ \tau }[/math]) of the system expressed in time steps. This variable is used to setup the mass-like variables via the relations:
- [math]\displaystyle{ Q_1 = 3 (N -N_c)k_{B} T \tau^2 }[/math]
- [math]\displaystyle{ Q_j = k_{B} T \tau^2; \; \; \; j=2,\dots,M }[/math]