Nose-Hoover

To construct Nose-Hoover constant-temperature molecular dynamics, an additional coordinate, s, and its conjugate momentum p, are introduced. The Hamiltonian of the extended system of the N particles plus extended degrees of freedom can be expressed... 

In Nose-Hoover methods the heat bath is considered an integral part of the system, and enters the simulation on an equal footing with the other variables. 

The constant k is the compressibility of the system. Alternatively the pressure may be maintained by a Nose-Hoover approach. 

Mean square displacements of C100 at 509°K derived from Nose-Hoover-MD... 

By applying the Taylor expansion as we did in Eq. (9.8), it is possible to derive an extension of the Verlet algorithm that allows these equations to be integrated numerically. This approach to controlling the temperature is known as the Nose-Hoover thermostat... 

To conclude our brief overview of ab initio MD, we note that the dynamics defined by Eq. (9.16) define a microcanonical ensemble. That is, trajectories defined by this Lagrangian will conserve the total energy of the system. Similar to the situation for classical MD simulations, it is often more useful to calculate trajectories associated with dynamics at a constant temperature. One common and effective way to do this is to add additional terms to the Lagrangian so that calculations can be done in the canonical ensemble (constant N, V, and T) using the Nose-Hoover thermostat introduced in Section 9.1.2. 

For the finite-temperature simulations, the temperature of the Si ions were controlled with a chain of five, linked Nose-Hoover thermostats." Because the electrons are always quenched back onto the Born-Oppenheimer surface after every timestep, no additional thermostat is needed for the electrons. Details of the configurations were similar to those with the CP scheme, except that the in-plane cells consisted of 16 atoms per layer and the basic timestep of the simulation was 100 a.u. 

Natural imits were used in these simulations, where m = = ku = 1. The sampling of the initial position and momentum centroids were made through the Nose-Hoover chain dynamics (NHC) on the effective potential of Vcm, More details of these calculations can again be foimd in Ref 10. 

An alternative coupling scheme for temperature and pressure, the Nose-Hoover scheme, adds new, independent variables that control these quantities to the simulation (Nose 1984 Hoover 1985). These variables are then propagated along with the position and momentum variables. 

We applied the Langevin temperature control [57] (temperature 300 K, damping coefficient 1/ps) and the Nose-Hoover Langevin piston pressure control [58,59] (target pressure 1 atm, oscillation period 100 fs, and oscillation decay time 50 fs). To ensure stable temperature and pressure, an equilibra-... 

Write and explain the equation of motion by Nose-Hoover. 

The Nose-Hoover thermostat exhibits non-ergodicity problems for some systems, e.g. the classical harmonic oscillator. These problems can be solved by using a chain... 

The Nose-Hoover thermostat, or chain of thermostats, can be used as well to control the wave function temperature, i.e. the fictitious kinetic energy. This prevents drifting of the wave function from the Born-Oppenheimer PES during long simulations. Wave function thermostats are introduced in a similar way to Eqs. 7-9. 

Controlling the temperature implies specifying the parameters characteristic for the thermostat method used in the simulation. In the case of the most popular Nose-Hoover thermostat the basic quantities are the target temperature and the thermostat frequency. 

The time-step of 0.5 fs is used to simulate the dynamic system to 4.0 ps. The temperature of 300 K is used throughout the simulations. The MD simulations are performed using the Nose-Hoover thermostat for temperature control. The Hellmann-Feynman forces acting on the atoms are calculated from the ground-state electronic energies at each time step and are subsequently used in the integration of Newton s equation of motion. 

Liquids and proteins are complex systems for which the smdy of dynamical systems has wide applicability. In the conference, relaxation in liquids (s-entropy by Douglas at the National Institute of Standards and Technology, nonlinear optics by Saito, and energy bottlenecks by Shudo and Saito), energy redistribution in proteins (Leitner and Straub et al.), structural changes in proteins (Kidera at Yokohama City University), and a new formulation of the Nose-Hoover chain (Ezra at Cornell University) were discussed. Kidera s talk discussed time series analyses in molecular dynamics, and it is closely related to the problem of data mining. In the second part of the volume, we collect the contributions by Leitner and by Straub s group, and the one by Shudo and Saito in the third part. 

Constant temperatnre is maintained by Nose-Hoover thermostat and the equations of motion were integrated using the two time scale r-RESPA with a large time step of 2 fs and a small time step of 0.2 fs. Equilibration using these initial configurations was then carried out for at least 2 ns before beginning any produc-... 

Minimum Energy Path (MEP), 344, 390 Nose-Hoover method, tor sunulations, 386 solvahon, 396 ... 

Most of the previous algorithms for generating isothermal and or isobaric ensembles can be shown to be special cases of the Nose -Hoover approach. As noted by Frenkel and Smit [2], problems appear most often when one attempts to simulate fluctuations in the various ensembles, but the averages of quantities such as the average energy and the pressure tensor elements are less sensitive to the choice of simulation algorithm. 

S. Nose, An Extension of the Canonical Ensemble Molecular Dynamics Method, Mol. Phys. 57 (1986) 187-191 S. Nose, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, J. Chem. Phys. 81 (1984) 511-519 D. J. Evans and B. L. Holian, The Nose-Hoover Thermostat, J. Chem. Phys. 83 (1985) 4069-4074 B. L. Holian, A. F. Voter and R. Ravelo, Thermostatted Molecular Dynamics How to avoid the Toda Demon Hidden in Nose-Hoover Dynamics, Phys. Rev. E 52 (1995), 2338-2347 Luis F. Rull, J.J. Morales and F. Cuadros, Isothermal Molecular-Dynamics Calculations, Phys. Rev. B 32 (1985) 6050-6052. 

Two of more sophisticated and commonly used approaches are the Nose-Hoover thermostat [79,80] and Langevin method [81]. In the Langevin method, additional terms are added to the equations of motion corresponding to a fiiction term and a random force. The Langevin equation of motion is given by... 

Standard classical equations of motion that can be thought of as either scaling time or inertia, so that the system spends more time in the regions of phase space where the potential energy is a minimum. The equation of motion in Nose-Hoover form is... 

The Berendsen [82] and Gauss [83] thermostats are also among other methods used. The Berendsen thermostat [82] was developed starting from the Langevin formalism by eliminating the random forces and replacing the friction term with one that depends on the ratio of the desired temperature to current kinetic temperature of the system. The resulting equation of motion takes the same form as the Nose-Hoover equation with... 

The basic mechanism inherent in the Nose-Hoover equations of motion is that the variable p acts as a dynamic friction coefficient that controls the... 

It was noted by Hoover that Eqs. [65] are not guaranteed to yield ergodic trajectories, in which case a dynamical simulation based on these equations of motion would not generate a canonical distribution in H p, q). This was seen most dramatically in the example of a single harmonic oscillator coupled to a Nose-Hoover thermostat, where a distribution radically different from the correct canonical distribution was generated as a result of nonergodicity. Thus far, two different solutions to this problem based on continuous dynamics have been proposed. 

---