Extended system method

The Car-Parrinello method is similar in spirit to the extended system methods [37] for constant temperature [38, 39] or constant pressure dynamics [40], Extensions of the original scheme to the canonical NVT-ensemble, the NPT-ensemble, or to variable cell constant-pressure dynamics [41] are hence in principle straightforward [42, 43]. The treatment of quantum effects on the ionic motion is also easily included in the framework of a path-integral formalism [44-47]. 

Finally, with the advent of high speed computers, parallel architecture, and the popularity of extended system methods, NEMD has seen yet another resurgence. Researchers are beginning to access the resources that allow one to compute the physical properties of real lubricants. The aim of this chapter is to provide the theoretical and technical foundations to enable simulators to model realistic systems under nonequilibrium conditions. 

The original work of Andersen and of Parrinello and Rahman on the generation of the NPT or isothermal-isobaric ensemble using an extended phase space predates Nose s work on the NVT ensemble, as noted above. Applying the extended system method to generate the NPT ensemble involves the inclusion of the volume into the phase space as a dynamical variable along... 

Constant pressure MD can be implemented either by extended system methods that couple the dynamical system to an external variable V which is the volume of the simulation box (Andersen, 1980) or by constraint methods that use a Lagrange multiplier determined from Gauss principle (Evans and Morriss,... 

It is possible to devise extended-system methods [79, ] and constrained-system methods [M] to simulate the constant-APT ensemble using MD. The general methodology is similar to that employed for constant-... 

The extended system method was developed by Nose l and subsequently by Hoover who considered the thermal reservoir to be an integral part of the system. The inclusion of the reservoir requires an additional degree of freedom, defined as s, be added to the system. The potential energy for this additional degree of freedom is calculated as... 

While the Berendsen thermostat is efficient for achieving a target temperature within your system, the use of a thermostat that represents a canonical ensemble once the system has reached a thermal equilibrium. The extended system method, which was originally introduced by Nosd [1984a, b] and then further developed by Hoover (1985), introduces additional degrees of freedom into the Hamiltonian that describes the system, from which equations of motion can be determined. 

The extended system method considers the external heat bath as an integral part of the system by including an additional degree of freedom in the Hamiltonian of the system that is represented by the variable s. As a result, the potential energy of the reservoir is... 

Similar to the Nosd-Hoover thermostat, the extended system method has been applied to create a barostat (Hoover 1986) that is coupled with a Nose-Hoover thermostat. In this case, the extra... 

The Berendsen thermostat is quite efficient for relaxing a system to the target temperature. However, once yom system has reached equihbrium, it is important to probe a correct canonical ensemble. The Nose-Hoover thermostat is an extended-system method for controlling the temperatare of simulated system (Huenenberger 2005 Thijssen 1999). It allows temperatures to fluctuate about an average value, and uses a friction factor to control particle velocities. This particular thermostat can oscillate when a system is not in equilibrium. 

Nos developed a deterministic approach to constant temperature simulations based on an extended Lagrangian which does not disturb the dynamic properties of the system. Since then, modifications of this approach have been developed to generate constant pressure and temperature simulations. These extended system methods have been known to suffer from stability problems, as well as occasional failures in ergodicity. Klein and co-workers introduced the concept of Nos6-Hoover chains to overcome these problems, and used a Liouville operator formalism to obtain reversible integrators generating these chains. 

Due to problems with early versions of the above extended system methods, and the need to control temperature drift in systems employing cutoff of electrostatic interactions, Berendsen et al. introduced a method of weak coupling to... 

The idea of a thermostat based on an extended-system method is due to Nose [61]. A simpler formulation of the equations of motion was later proposed simultaneously ... 

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... 

The Extended Iliickel method also allows the inclusion ofd orbitals for third row elements (specifically, Si. P, Sand CD in the basis set. Since there arc more atomic orbitals, choosing this option resn Its in a Ion ger calc ii 1 at ion. Th e m ajor reason to in cin de d orbitals is to improve the description of the molecular system. 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

To include the volume as a dynamic variable, the equations of motion are determined in the analysis of a system in which the positions and momenta of all particles are scaled by a factor proportional to the cube root of the volume of the system. Andersen [23] originally proposed a method for constant-pressure MD that involves coupling the system to an external variable, V, the volume of the simulation box. This coupling mimics the action of a piston on a real system. The piston has a mass [which has units of (mass)(length) ]. From the Fagrangian for this extended system, the equations of motion for the particles and the volume of the cube are... 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

In the foregoing treatments of pressure feedback, the simulation volume retains its cubic form, so changes consist of uniform contractions and expansions. The method is readily extended to the case of a simulation region in which the lengths and directions of the edges are allowed to vary independently. Parrinello and Rahman [31] and Nose and Klein [32] extended the Andersen method to the case of noncubic simulation cells and derived a new Lagrangian for the extended system. Though their equations of motion are... 

In comparison with the more standard Configuration Interaction (Cl) method, the one-particle Green s function approach offers the essential advantages, in the outlook of numerical applications on extended systems, of a stronger and systematic compactness (30) of the configuration spaces in high order approximations and of energy separability (5,31) in the dissociation limit (size-consistency). The latter is a necessary prerequisite ( ) for a correct (i.e. size-... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

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