Constant pressure dynamics

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]. 

Qin Yueping, et al. 2012. Constant Pressure Dynamic Law of Gas Desorption Experiments. Journal of Liaoning Technical University (Natural Science), 31(5) 581-586. 

Andersen H C 1980 Molecular dynamics simulations at constant pressure and/or temperature J. Chem. 

A typical molecular dynamics simulation comprises an equflibration and a production phase. The former is necessary, as the name imphes, to ensure that the system is in equilibrium before data acquisition starts. It is useful to check the time evolution of several simulation parameters such as temperature (which is directly connected to the kinetic energy), potential energy, total energy, density (when periodic boundary conditions with constant pressure are apphed), and their root-mean-square deviations. Having these and other variables constant at the end of the equilibration phase is the prerequisite for the statistically meaningful sampling of data in the following production phase. 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. 

In a normal molecular dynamics simulation with repeating boundary conditions (i.e., periodic boundary condition), the volume is held fixed, whereas at constant pressure the volume of the system must fluemate. In some simulation cases, such as simulations dealing with membranes, it is more advantageous to use the constant-pressure MD than the regular MD. Various schemes for prescribing the pressure of a molecular dynamics simulation have also been proposed and applied [23,24,28,29]. In all of these approaches it is inevitable that the system box must change its volume. 

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. 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

Compressor performance can be represented in various ways. The commonly accepted practice is to plot the speed lines as a function of the pressure delivered and the flow. Figure 3-9 is a performance map for a centrifugal compressor. The constant speed lines shown in Figure 3-9 are constant aero-dynamic speed lines, not constant mechanical speed lines. 

An interesting extension of the original methodology was proposed by Lopes and Tildesley to allow the study of more than two phases at equilibrium [21], The extension is based on setting up a simulation with as many boxes as the maximum number of phases expected to be present. Kristof and Liszi [22, 23] have proposed an implementation of the Gibbs ensemble in which the total enthalpy, pressure and number of particles in the total system are kept constant. Molecular dynamics versions of the Gibbs ensemble algorithm are also available [24-26]. 

Tu, K. Klein, M. Tobias, D. J., Constant-pressure molecular dynamics investigations of cholesterol effects in a dipalmitoylphosphatidylchohne bilayer, Biophys. J. 1998, 75, 2147-2156. 

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

Similar schemes to the above can be used in molecular dynamics simulations in other ensembles such as those at constant temperature or constant pressure (see Frenkel and Smit, and Allen and Tildesley (Further reading)). A molecular dynamics simulation is computationally much more intensive than an energy minimization. Typically with modern computers the real time sampled in a simulation run for large cells is of the order of nanoseconds (106 time steps). Dynamical processes operating on longer time-scales will thus not be revealed. 

Venable, R. M., Brooks, B. R. and Pastor, R. W. (2000). Molecular dynamics simulations of gel phase lipid bilayers in constant pressure and constant surface area ensembles, J. Chem. Phys., 112, 4822-4832. 

Berger, O., Edholm, O. and Jahnig F. (1997). Molecular dynamics simulations of a fluid bilayer of dipalmitoylphosphatidylcholine at full hydration, constant pressure and constant temperature, Biophys. J., 72, 2002-2013. 

Chem. Solids, 56, 501 (1995). First-Principle-Constant Pressure Molecular Dynamics. 

Figure 5 Relationship among loci of structural, dynamic, and thermodynamic anomalies in SPC/E water. The structurally anomalous region is bounded by the loci of q maxima (upward-pointing triangles) and t minima (downward-pointing triangles). Inside of this region, water becomes more disordered when compressed. The loci of diffusivity minima (circles) and maxima (diamonds) define the region of dynamic anomalies, where self-diffusivity increases with density. Inside of the thermodynamically anomalous region (squares), the density increases when water is heated at constant pressure. Reprinted with permission from Ref. 29.

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