Theory

The development of pressure controls is similar to those of the thermostats. However the pressure controls do not only alter the motion of the particles by modifying either the velocities or the accelerations but also vary the box size.

The first approach [2] used volume scaling exclusively. A method analogous to the Berendsen thermostat has been devised where the system is coupled to a pressure bath [3]. The instantaneous pressure $P$ has to obey the equation [1]

$$\dot P = (P_{ext} - P)/t_P$$

while the volume is scaled by a factor $\chi$ given by

$$\chi = 1 - \frac{\delta t}{t_P'} (P_{ext} - P).$$

$P_{ext}$ is the desired pressure, $t_P$, $t'_P$ are time constants and $\delta t$ is the time step. As in the case of the temperature control this does not sample the isobaric-isenthalpic ensemble.

Nosé accomplished this task again by introducing an additional degree of freedom. In the general case not only the volume is allowed to change but also the shape of the cell [14,15]. One can also include the rotational degree of freedom for molecules but here we are interested in monatomic systems only. Additionally we restrict ourselves to cubic cells. The real coordinates $\mathbf{r}$ are scaled with respect to the volume $V$, i.e. $\mathbf{r}_i=V^{1/3}\mathbf{x}_i$ The corresponding Hamiltonian becomes

$$\mathcal{H} = \sum_i \frac{\mathbf{p}_i^2}{2 m_i s^2 V^{2/3}} + \phi(V^{1/3} \mathbf{x}) + \frac{W \dot V^2}{2} - P_{ext}V$$ (2)

from which the equations of motion can be obtained in the usual way. The coupling between the system and the pressure control is adjusted by the parameter $W$.

As it has been pointed out [12] one is free to choose a pair of variables from the set $V$-$\dot V$- $\dot\varepsilon$- $\ddot\varepsilon$ where $V$ denotes the volume $\dot\varepsilon$ the strain rate and $\dot V$, $\ddot\varepsilon$ the corresponding time derivatives. Specifically to describe the evolution of volume with time either $\dot V$ (Nosé and Andersen) or $\dot\varepsilon$ can be used. Hoover decided to take the latter one. Again using the transformations to eliminate the scaling parameter described above $s$ he ended up with the following set of equations:

$$\left. \begin{array}{rcl} \mathbf{\ddot{x}} &=& \frac{\mathbf{F}_... ...ot\varepsilon\\ \ddot\varepsilon &=& (P-P_{ext})V/kT\tau^2 \end{array}\right\}$$ (3)

$\tau$ determines the coupling of the system to the pressure control. It can be interpreted as the relaxation time of the hypothetical 'piston' exerting the external pressure. $\mathbf{x}_i$ are the scaled coordinates introduced earlier.

Now we are in the position to combine the Nosé-Hoover thermostat and pressure control to end up with a formulation that allows simulations in the isobaric-isothermal ensemble.