The NPT ensemble

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]. 

Second virial coefficients represent the first approximation to the system equation of state. Yethiraj and Hall [148] obtained the compressibility factor, i.e., pV/kgTn, for small stars. They found no significant differences with respect to the linear chains in the pressure vs volume behavior. Escobedo and de Pablo [149] performed simulations in the NPT ensemble (constant pressure) with an extended continuum configurational bias algorithm to determine volumetric properties of small branched chains with a squared-well attractive potential... 

Allen and Bevan (80) have applied the SMD technique to the study of reversible inhibitors of monoamine oxidase B, and this paper will be used as an example for discussion of the constant velocity SMD pulling method. They used the Gromacs suite of biomolecular simulation programs (18) with the united-atom Gromos 43al force field to parameterize the lipid bilayer, protein, and small-molecule inhibitors. The protein was inserted into their mixed bilayer composed of phosphatidyl choline (POPC) and phosphatidyl ethanolamine (POPE) lipids in a ratio known to be consistent for a mitochondrial membrane. Each inhibitor-bound system studied was preequilibrated in a periodic box of SPC water (20) with the simulations run using the NPT ensemble at 300 K and 1 atm pressure for 20 ns. Full atomic coordinates and velocities were saved in 200-ps increments giving five replicates for each inhibitor-bound system. A dummy atom was attached to an atom (the SMD atom shown in Fig. 7) of the inhibitor nearest to the... 

The ab initio molecular simulation was carried out in the NPT ensemble with version 3.9.1 of CPMD [29]. In the calculation, 20 water and one acetic acid molecules were simulated in a periodic cubic cell under an ambient condition of a constant pressure 1 bar, a constant temperature 300 K, and density of 1.0 gem-3. The starting structure was constructed in a simple flexible cubic cell with a length of 8.8794 A, by randomly adding 11 water molecules to an optimized hydration compound composed of a single acetic acid molecule and 9 water molecules. 

Hill discusses how to make reasonable choices for the discrete volumes in the sum [85 ] and more recent authors have developed a complete theory of the NPT ensemble where continuous volume integrals are made unitless by the proper normalization [37, 38,115]. 

The coefficient of thermal expansion was calculated by two different procedures. The first involves the statistical mechanical fluctuation formula for the NPT ensemble [39],... 

Tave and Pave are the average temperature and pressure during each simulation, and V are the total energy and volume, with standard deviations cte and calculated using the NPT ensemble fluctuation formula [39],... 

In the work of Edwards et al [36], which included interactions only to the third neighbor shell (approximately 4.4 A), the problem of energy instability for a 1000 K simulation did not arise the timestep was lO s at all temperatures. Since no analysis of the effects of timestep and cutoff was presented, it is difficult to determine the reasons for the differences in the behavior observed by Edwards et al and in the present study. Some possibilities are (a) They may have employed a minimum image approach for the periodic boundary conditions, whereas we have calculated both primaryprimary and primary-image interactions for each atom, (b) Their system was much larger than ours (2048 vs 256 atoms), (c) Edwards et al used the NVE ensemble for the production simulations, in contrast to our choice of the NPT ensemble. 

Three trajectories were calculated as follows the system was first allowed to vary its volume by performing a NPT (Number of molecules. Pressure and Temperature constant) run at 298 K and 1.0 atm for a duration of 100 ps the cell edge at this point came down to about 22 A. A 100 ps NVT (Number of molecules. Volume and Temperature constant) run at 700 K was then performed and the system was finally equilibrated for 50 ps at 298 K in the NPT ensemble. All these points were then discarded and a final run of 200 ps was performed recording the trajectory. The runs were stopped at 200 ps as no significant variation was observed in the average total energy over the last 100 ps. 

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... 

In the earlier subsection on the Dynamical Generation of the NPT Ensemble, we introduced equations of motion to perform equilibrium MD under constant temperature and pressure conditions. These equations of motion can be augmented with terms involving the shear rate from the SLLOD equations and can be written as follows ... 

At equilibrium, the box matrix h evolves to respond to the internal stresses with respect to an externally set pressure in the NPT ensemble (Eqs. [84]) in both the isotropic and flexible cell cases. The application of the external field contributes an additional term to its evolution. It is important to apply appropriate boundary conditions that are consistent with the nature of the dynamics. The contribution of the field to the time evolution of the simulation box can be represented as follows ... 

This exercise, which validatts Eq. [192], can also be performed to see the validity of the evolution of h in the NPT ensemble, (Eqs. [84]). To simulate flows in bulk systems, Eq. [192] must be used in conjunction with SLLOD or GSLLOD equations. We call Eq. [192] the box dynamics method. 

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