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].