Constant Temperature and Pressure Molecular Dynamics

The isobaric, isothermal or NPT ensemble models a system in contact with a heat bath (temperature T) and a pressure reservoir maintained at pressure P. The pressure is defined by the formula 

An isothermal-isobaric system exchanges energy with the bath in such a way as to maintain the constant temperature at the same time the volume fluctuates to control the pressure. The state of the system can be expressed in terms of particle positions and momenta as well as the volume V, i.e. we think of the probabiUty 

E be the number of atoms, volume and energy, respectively, of the system of interest, and N2, V2, E2 be the corresponding quantities describing the bath system we have N + N2 = N, V + V2 — V, E+E2 = E all constant. We view the entropy as a function of the other variables and expand in a Taylor series in both volume and energy  

Bear in mind that here E and V are small relative perturbations of the total energy 

The method of Andersen [11] ( Andersen s piston ) is to control the volume by reparamaterizing the coordinates to lie in a cubic box with unit side. Assuming the system occupies a cubic domain, say, [—Z,/2, L/2] x [-L/2, L/2] x [—L/2, L/2], whereL = we may write, for each three-dimensional atomic coordinate vector 9i 

---

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). 

Two unusual features can be observed in these plots (and, at least for the self-diffusion coefficient, this behaviour is common to all hydrogen-bonded liquids). This ratio is a function of temperature. At constant temperature and pressure, rotation and translation reveal the same isotope effect. From simple sphere dynamics one would expect the rotation to scale as the square root of the ratio of the moments of inertia (=1.38) while for translational mobility the square root of the ratio of the molecular masses ( = 1.05) should be found. This is clearly not the case, indicating that the dynamics of liquid water are really the dynamics of the hydrogen-bond network. The hydrogen bonds in D2O are stronger than those in H2O and thus the mobility in the D2O network decreases more rapidly as the temperature decreases. 

S. Nose, in Computer Simulation in Materials Science, M. Meyer and V. Pontikis, Eds., NATO ASl Series, Kluwer Academic Publishers, Dordrecht, 1991, Vol. 205, pp. 21-41. Molecular Dynamics Simulations at Constant Temperature and Pressure. 

Methods and algorithms for molecular dynamics simulations of molecular fluids are described. Those techniques are emphasized that in the experience of the authors are reliable and easy to implement. This implies the use of cartesian coordinates throughout, the use of the SHAKE procedure to satisfy constraints and the incorporation of an adjustable coupling to an external bath with constant temperature and pressure. 

In molecular dynamics (MD) simulation atoms are moved in space along their lines of force (which are determined from the first derivative of the potential energy function) using finite difference methods [27, 28]. At each time step the evolution of the energy and forces allow the accelerations on each atom to be determined, in turn allowing the atom changes in velocities and positions to be evaluated and hence allows the system clock to move forward, typically in time steps of the order of a few fs. Bulk system properties such as temperature and pressure are easily determined from the atom positions and velocities. As a result simulations can be readily performed at constant temperature and volume (NVT ensemble) or constant temperature and pressure (NpT ensemble). The constant temperature and pressure constraints can be imposed using thermostats and barostat [29-31] in which additional variables are coupled to the system which act to modify the equations of motion. 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

Twenty years ago Car and Parrinello introduced an efficient method to perform Molecular Dynamics simulation for classical nuclei with forces computed on the fly by a Density Functional Theory (DFT) based electronic calculation [1], Because the method allowed study of the statistical mechanics of classical nuclei with many-body electronic interactions, it opened the way for the use of simulation methods for realistic systems with an accuracy well beyond the limits of available effective force fields. In the last twenty years, the number of applications of the Car-Parrinello ab-initio molecular d3mam-ics has ranged from simple covalent bonded solids, to high pressure physics, material science and biological systems. There have also been extensions of the original algorithm to simulate systems at constant temperature and constant pressure [2], finite temperature effects for the electrons [3], and quantum nuclei [4]. 

The principle of Le Chatelier-Braun states that any reaction or phase transition, molecular transformation or chemical reaction that is accompanied by a volume decrease of the medium will be favored by HP, while reactions that involve an increase in volume will be inhibited. Qn the other hand, the State Transition Theory points out that the rate constant of a reaction in a liquid phase is proportional to the quasi-equilibrium constant for the formation of active reactants (Mozhaev et al., 1994 Bordarias, 1995 Lopez-Malo et al., 2000). To fully imderstand the dynamic behavior of biomolecules, the study of the combined effect of temperature and pressure is necessary. The Le Chatelier-Braim Principle states that changes in pressure and temperature cause volume and energy changes dependent on the magnitude of pressure and temperature levels and on the physicochemical properties of the system such as compressibility. "If y is a quantity characteristic of equilibrium or rate process, then the influence of temperature (7 and pressure (P) can be written as ... 

Molecular dynamics is traditionally performed in the constant NVE (or NVEP) ensemble. Although thermodynamic results can be transformed between ensembles, this is strictly only possible in the limit of infirate system size ( the thermodjmamic limit ). It may therefore be desired to perform the simulation in a different ensemble. The two most common alternative ensembles are the constant NVT and the constant NPT ensembles. In this section we will therefore consider how molecular dynamics simulations can be performed under conditions of constant temperature and/ or constant pressure. 

In principle, the modeller has the choice of using either the Monte Carlo or molecular d)mamics technique for a given simulation. In practice one technique must be chosen over the other. Sometimes the decision is a trivial one, for example because a suitable program is readily available. In other cases there are clear reasons for choosing one method instead of the other. For example, molecular dynamics is required if one wishes to calculate time-dependent quantities such as transport coefficients. Conversely, Monte Carlo is often the most appropriate method to investigate systems in certain ensembles for example, it is much easier to perform simulations at exact temperatures and pressures with the Monte Carlo method than using the sometimes awkward and ill-defined constant temperature and constant pressure molecular d)mamics simulation methods. The Monte Carlo method is also well suited to certain types of models such as the lattice models. 

---