The Parrinello-Rahman-Nosé Extended Lagrangian

The starting point of our derivation of the multilevel integrator for the NPT ensemble is the Parrinello-Rahman-Nosé Lagrangian for a molecular system with $N$ molecules or groups ^3.2 each containing $n_{i}$ atoms and subject to a potential $V$. In order to construct the Lagrangian we define a coordinate scaling and a velocity scaling, i.e.

$$r_{ik\alpha} = R_{i \alpha}+ l_{ik\alpha}= \sum_{\beta} h_{\alpha \beta}S_{i \beta}+ l_{ik\alpha}$$ (3.1)

$$\dot R_{i \alpha}' = \dot R_{i \alpha}s$$ (3.2)

$$\dot l_{ik\alpha}' = \dot l_{ik\alpha}s$$

Here, the indices $i$ and $k$ refer to molecules and atoms, respectively, while Greek letters are used to label the Cartesian components. $r_{ik\alpha}$ is the $\alpha$ component of the coordinates of the $k$-th atom belonging to the $i$-th molecule; $R_{i \alpha}$ is the center of mass coordinates; $S_{i\beta}$ is the scaled coordinate of the $i$-th molecular center of mass. $l_{ik\alpha}$ is the coordinate of the $k$-th atom belonging to the $i$-th molecule expressed in a frame parallel at any instant to the fixed laboratory frame, but with origin on the instantaneous molecular center of mass. The set of $l_{ik\alpha}$ coordinates satisfies $3N$ constraints of the type $\sum_{k=1}^{n_{i}} l_{ik\alpha}= 0$.

The matrix ${\bf h}$ and the variable $s$ control the pressure an temperature of the extended system, respectively. The columns of the matrix ${\bf h}$ are the Cartesian components of the cell edges with respect to a fixed frame. The elements of this matrix allow the simulation cell to change shape and size and are sometimes called the ``barostat'' coordinates. The volume of the MD cell is related to ${\bf h}$ through the relation

$$\Omega = \det({\bf h}).$$ (3.3)

$s$ is the coordinates of the so-called ``Nosé thermostat'' and is coupled to the intramolecular and center of mass velocities,

We define the ``potentials'' depending on the thermodynamic variables $P$ and $T$

$$V_{P} = P \det({\bf h})$$

$$V_{T} = {g \over \beta} \ln s.$$ (3.4)

Where $P$ is the external pressure of the system, $\beta = k_B T$, and $g$ is a constant related to total the number of degrees of freedom in the system. This constant is chosen to correctly sample the $N{\bf P}T$ distribution function.

The extended $N{\bf P}T$ Lagrangian is then defined as

$${\cal L} = {1 \over 2} \sum_{i}^{N} M_{i} s^{2} {\bf\dot S}_{i}^{t}{\bf h^{t... ... l^{t}}_{ik} \dot {\bf l}_{ik} + {1 \over 2} W s^{2} tr({\bf\dot h^{t} \dot h})$$ (3.5)

$$+ {1 \over 2 } Q \dot s^{2} - V - P_{ext} \Omega - {g\over \beta} \ln s$$ (3.6)

The arbitrary parameters $W$ and $Q$ are the ``masses'' of the barostat and of the thermostats, respectively^3.3. They do not affect the sampled distribution function but only the sampling efficiency [26,94,95]. For a detailed discussion of the sampling properties of this Lagrangian the reader is referred to Refs. [91,27].