Thermostat Nose-Hoover

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 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-... 

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... 

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. 

The first of these, proposed by Martyna, Tuckerman, and Klein (MTK), was based on the notion that the variable py, itself, has a canonical (Gaussian) distribution exp(- 3p /Q). However, there is nothing in the equations of motion to control its fluctuations. MTK proposed that the Nose-Hoover thermostat should, itself, be connected to a thermostat, and that this thermostat should also be connected to a thermostat. The result is that a chain of thermostats is introduced whereby each element of the chain controls the fluctuations of the element just preceding it. The equations of motion for such a thermostat chain are ... 

For simplicity, Eqs. [76] show the coupling of the system to an ordinary Nose-Hoover thermostat. However, in practice, any desired thermostating scheme can be used. In addition, the system variables (p, q] as well as the volume are shown coupled to the same thermostat. In practice, it is useful to couple the system and the volume to separate thermostats, because the time scales of their fluctuations are usually considerably different. 

Once again, Eqs. [84] utilize only a single Nose-Hoover thermostat, although in practice, one should couple separate thermostats to the cell matrix and to the system variables. The ensemble average of p is, from Eqs. [84],... 

Here, we have coupled the GSLLOD equations to a Nose-Hoover thermostat. It is simple to show that the computation of q, in the unthermostated form of the GSLLOD equations will yield Eq. [133]. 

From the earlier section on Molecular Dynamics and Equilibrium Statistical Mechanics, we hope that we have made clear that a conserved quantity is the starting point for phase space analysis and the derivation of a probability distribution function. Following the same analysis that led to the distribution functions for NVE, NVT, and NPT dynamics, the new distribution /(q, p,, I) for GSLLOD coupled to a Nose-Hoover thermostat is given by... 

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - " are ... 

The original scheme utilized Gaussian isokinetic thermostats, whereas in Eqs. [217] we have replaced it with a Nose-Hoover thermostat. In this equation, the true local streaming velocity is given by iyy, + Usi(q,). In principle, there are no restrictions on u i, so the steady state velocity can be of any form hence, Evans and Morriss refer to Eqs. [217] as profile-unbiased thermostats (PUT). The PUT scheme requires only a reasonable prescription for determining the true local streaming velocity. 

For SLLOD dynamics coupled to a Nose-Hoover thermostat, the metric determinant is given by The first and third terms in Eq. [229] become Nfe f feqPr /Q with Opposite signs and thus cancel. We are left with the expression... 

In derivation of the steady state relations, artificial, reversible thermostats such as the Gaussian thermostat or Nose-Hoover thermostat have been used. However, Williams et al.22S have recently used MD simulations to verify assumptions made in their derivation that shows that the FR is insensitive to the details of the thermostatting mechanism. 

Fig. 1. The momentum distribution function for a one-dimension free particle coupled to a Nose-Hoover thermostat solid line) [rn = 1, kT = 1, p(0) = 1, q 0) = 0, Q = 1, Pr 0) = 1, time step At = 0.05] compared to the analytical result dashed line) and the canonical ensemble distribution dot dashed line)...

Similarly, if thermostating (e.g. using a Nose-Hoover thermostat [169,170]) is performed while integrating the equations of motions for the nuclei, this will generate the canonical distribution... 

The motion equations have been solved by the Verlet Leap-frog algorithm subject to periodic boundary conditions in a cubic simulation cell and a time step of 2 fs. The simulations have been performed in the NVT ensemble with the Nose-Hoover thermostat [62]. The SHAKE constraints scheme [65] was used. The spherical cutoff radius comprises 1.2 nm. The Ewald sum method was used to treat long-range electrostatic interactions. 

If these equations are equipped with an appropriate thermostat, for example, a stochastic Andersen thermostat [53] or a deterministic Nose-Hoover thermostat [37], the resulting distribution in path space is consistent with exp( = ab[x( ]- Alternatively, a stochastic Lan-... 

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