NPT ensemble

Most simulations have been performed in the mieroeanonieal, eanonieal, or NPT ensemble with a fixed number of moleeules. These systems typieally require an iterative adjustment proeess until one part of the system exhibits the required properties, like, eg., the bulk density of water under ambient eonditions. Systems whieh are equilibrated earefully in sueh a fashion yield valuable insight into the physieal and, in some eases, ehemieal properties of the materials under study. However, the speeifieation of volume or pressure is at varianee with the usual experimental eonditions where eontrol over the eomposition of the interfaeial region is usually exerted through the ehemieal potential, i.e., the interfaeial system is in thermodynamie and ehemieal equilibrium with an extended bulk phase. Sueh systems are best simulated in the grand eanonieal ensemble where partiele numbers are allowed to fluetuate. Only a few simulations of aqueous interfaees have been performed to date in this ensemble, but this teehnique will undoubtedly beeome more important in the future. Partieularly the amount of solvent and/or solute in random disordered or in ordered porous media ean hardly be estimated by a judieious equilibration proeedure. Chemieal potential eontrol is mandatory for the simulation of these systems. We will eertainly see many applieations in the near future. 

The reduction in the number of degrees of freedom can lead to an incorrect pressure in the simulation of the coarse-grained systems in NVT ensembles or to an incorrect density in NPT ensembles [24], The pressure depends linearly on the pair-forces in the system, hence the effect of the reduced number of degrees of freedom can be accounted for during the force matching procedure [24], If T is the temperature, V the volume, N the number of degrees of freedom of the system, and kb the Boltzmann constant then the pressure P of a system is given by... 

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. 

The Kohn-Sham orbitals were expanded in a plane wave basis set up to an energy cutoff of 70 Ry. A factious mass of 1,100 a.u. was used in the thermostating equation of motion. A time step of 7a.u. (0.169 fs) was used in the simulation. The trajectory data were collected every 10 steps during in the 16.9 ps production runs after its 1.69 ps initialization with an NPT ensemble. 

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]. 

NPT ensemble anti used the shell-model to describe polarizability. All simulation runs were performed at atmospheric pressure and in the temperature range 10 - 1100 K. For all three surfaces at both 300 and 1100 K it was found that the surface mean square displacements are generally larger for the oxide ions than for the cations and that the out-of-plane surface motion is usually larger than the in-plane surface motion. At room temperature, the oxygen mean square displacements at the (111) surface arc a factor 1.2 larger than in the bulk, a factor 1.6 for the (Oil) surface and approximately five limes larger at the metastable (001) surface compared to the bulk. The effect of the presence of a surface on the ion dynamics (and on the structure for (011)) persists all the way to the slab centers, even for these rather thick slabs. 

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. 

A similar approach exists for performing constant temperature MD calculations. The Lagrangian (Eq. [48]) is extended by terms for additional (fictitious) degrees of freedom that couple the system to an external heat reser-voir. A combination of the Parrinello-Raman and Nose schemes is possible to sample an isobaric-isothermic (NPT) ensemble. These methods as well as other techniques for performing calculations in different statistical ensembles are reviewed by Allen and Tildesley in detail. 

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