Nose thermostat

To achieve an evenly distributed thermal excitation, the nuclei were brought to a temperature of 300 K by applying a sequence of 30 sinusoidal pulses, each of which was chosen to raise the temperature by 10 K. Each of the excitation vectors was chosen to be orthogonal to the already excited modes. The warmed-up systems were equilibrated for the 10 000 timesteps. The time step of 7 au. was used. Constraints were maintained by SHAKE algorithm.36 A temperature of 300 K was controlled by a Nose thermostat.23,24 The fictitious kinetic energy of the electrons was controlled in a similar fashion by a Nose thermostat.52... 

Despite the reservations set down above, to carry out a molecular dynamics study of the diffusion process itself one resorts to a computational cell of the type described earlier. The temperature is assigned and maintained via some scheme such as the Nose thermostat (Frenkel and Smit, 1996), and the atomic-level trajectories are obtained via a direct integration of the equations of motion. In fig. 3.22, we showed the type of resuiting trajectories in the case of surface... 

Lynch and Pettitt (36) employed a pure Nose thermostat and RATTLE-like algorithm for bond constraint, as well as a single fraction particle. Our approach (40) used quaternions (41) to allow separate Nose-Hoover thermostats for translational and rotational modes. We found that the system s tendency to freeze in metastable states could be overcome by introduction of multiple fractional particles (four were used). Curve (b) of Fig. 3 is an extreme example of runaway insertions at high chemical potential as compared to curve (a) where the actual chemical potential of water resulted in a stable density virtually undistin-guishable from the experimental one. 

The proof that the Nose thermostat samples a canonical ensemble of noicrostates, provided that g = Ndf + l (virtual-time sampling) ox g = Ndf (real-time sampling), is as follows [53]. The partition function of the microcanonical ensemble generated for the extended system using virtual-time sampling (i.e., using the natural time evolution of the extended system) reads... 

In practice modifications are made to incorporate thermostats or barostats that may destroy the time-reversible and symplectic properties. While extended-system algorithms such as Nose dynamics [41] can be designed on the principles of the reversible operators, methods that use proportional velocity or coordinate scaling [42] cannot. Such methods arc very... 

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. 

The Langrage function of the system L can be generalized to describe the effective interaction of the system with a thermostat or the isobaric behavior of the system. For example, for the system connected with a thermostat, one can use the Langrage function by Nose [59] ... 

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. 

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-... 

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