View source for Nosé-Hoover chain thermostat

The [[Nosé-Hoover thermostat|standard Nosé-Hoover thermostat]] suffers from well-known issues, such as the ergodicity violation in the case of simple harmonic oscillator{{cite|martyna:jcp:92}}. As proposed by Martyna and Klein{{cite|martyna:jcp:92}}, these problems can be solved by using multiple Nosé-Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term: 

::<math>
\mathcal{H'} = \mathcal{H}(\mathbf{r},\mathbf{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>\mathcal{H}(\mathbf{r},\mathbf{p})</math> is the Hamiltonian of the physical system, <math>M</math>, <math>N</math> and <math>N_c</math> are the numbers of thermostats, atoms in the cell, and geometric constraints, respectively, and <math>\eta_{j}</math>, <math>p_{\eta_j}</math>, and <math>Q_{j}</math> are the position, momentum, and mass-like parameter associated with the thermostat <math>j</math>. Just like the total energy in the NVE ensemble,<math>\mathcal{H'}</math> is valuable for diagnostics purposes. Indeed, a significant drift in <math>\mathcal{H'}</math> indicates 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 {{TAG|POTIM}}).   

The number of thermostats is controlled by the flag  {{TAG|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 {{TAG|NHC_NCHAINS}}=0, the thermostat is switched off, leading to a MD in the microcanonical ensemble. Another special case of {{TAG|NHC_NCHAINS}}=1 corresponds to the  standard [[Nosé-Hoover thermostat]]. 

The only parameter of this thermostat is the characteristic time scale (<math>\tau</math>), defined via flag {{TAG|NHC_PERIOD}}. This parameter is used to setup the mass-like variables via the relations: 

::<math>
Q_1 = 3 (N -N_c)k_{B} T \tau^2
</math>
::<math>
Q_j = k_{B} T \tau^2;   \; \; \;   j=2,\dots,M
</math>

Furthermore, due to rapidly varying forces in thermostat variables propagators, the standard velocity Verlet algorithm with fixed integration step might be insufficiently accurate. As proposed by Tuckerman<ref>M. E. Tuckerman, Statistical mechanics: theory and molecular simulation, Oxford University Press Inc., New York, 2010; pp 194-199.</ref>, the RESPA<ref>[https://pubs.aip.org/aip/jcp/article/97/3/1990/221848/Reversible-multiple-time-scale-molecular M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, 1900 (1992)]</ref> methodology can be used to overcome this problem, in which the integration step used in thermostat variables propagation is split into {{TAG|NHC_NRESPA}} equal parts, each of which may be further divided into {{TAG|NHC_NS}} smaller parts treated by Suzuki-Yoshida scheme of fourth or sixth order. 

==Related tags and articles==
[[Molecular dynamics calculations|Molecular-dynamics calculations]], [[Andersen thermostat]], [[Nosé-Hoover thermostat]], [[Langevin thermostat]], [[CSVR thermostat]], {{TAG|ISIF}}, {{TAG|MDALGO}},{{TAG|NHC_NCHAINS}},{{TAG|NHC_PERIOD}},{{TAG|NHC_NRESPA}},{{TAG|NHC_NS}}

==References==
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