Switching to Other Ensembles

The $N{\bf P}T$ extended system is the most general among all possible extended Lagrangians. All other ensemble can be in fact obtained within the same computational framework. We must stress [27] that the computational overhead of the extended system formulation, due to the introduction and handling of the extra degrees of freedom of the barostat and thermostat variables, is rather modest and is negligible with respect to a NVE simulation for large samples ( $N_{f} > 2000$) [26,25,27]. Therefore, a practical, albeit inelegant way of switching among ensembles is simply to set the inertia of the barostat and/or thermostat to a very large number. This must be of course equivalent to decouple the barostat and/or the thermostat from the true degrees of freedom. In fact, by setting $W$ to infinity^3.11 in Eqs. (3.23-3.27) we recover the $NVT$ canonical ensemble equations of motion. Putting instead $Q$ to infinity the $NPH$ equations of motion are obtained. Finally, setting both $W$ and $Q$ to infinity the $NVE$ equations of motion are recovered.

Switching to the $NPT$ isotropic stress ensemble is less obvious. One may define the kinetic term associated to barostat in the extended Lagrangian as

$$K = {1 \over 2} \sum_{\alpha\beta} W_{\alpha\beta} s^{2} \dot h_{\alpha\beta}^{2}$$ (3.78)

such that a different inertia may in principle be assigned to each of 9 extra degrees of freedom of the barostat. Setting for example

$$W_{\alpha\beta} = W ~~~~ {\rm for}~ \alpha \le \beta$$ (3.79)

$$W_{\alpha\beta} = \infty ~~~~ {\rm for}~ \alpha > \beta$$ (3.80)

one inhibits cell rotations [27].

This trick does not work, unfortunately, to change to isotropic stress tensor. In this case, there is only one independent barostat degrees of freedom, namely the volume of the system. In order to simulate isotropic cell fluctuations a set of five constraints on the ${\bf h}$ matrix are introduced which correspond to the conditions:

$$\frac{h_{\alpha\beta}}{h_{11} } - \frac{h^{0}_{\alpha\beta}}{h^{0}_{11} } =$$ 0

$$\Dot{h}_{\alpha\beta} - \frac{h^{0}_{\alpha\beta}}{h^{0}_{11} } \Dot{h}_{11} = 0 ~~~~~~~~~\rm { for~~ }\alpha \leq \beta$$ (3.81)

with ${\bf h}^{0}$ being some reference ${\bf h}$ matrix. These constraints are implemented naturally in the framework of the multi time step velocity Verlet using the RATTLE algorithm which evaluates iteratively the constraints force to satisfy the constraints on both coordinates ${\bf h}$ and velocities $\dot {\bf h}$ [27]. In Ref. [27] it is proved that the phase space sampled by the $N{\bf P}T$ equations with the addition of the constraints Eq. (3.83) correspond to that given by $NPT$ distribution function.