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7.2 NPH Molecular Dynamics

Andersen (1980) introduced an additional (``synthetic'') energy that was coupled to changes of the system volume. Putting $\Delta E_{pot}=P_{0} V$ and $\Delta E_{kin}=Q \dot{V}^{2}/2$ (with a generalized mass $Q$ which may be visualized as the mass of some piston) he wrote the Hamiltonian as
H=E_{pot}+\Delta E_{pot}+E_{kin}+\Delta E_{kin}
\end{displaymath} (7.10)

Introducing scaled position vectors $\vec{s}_{i} \equiv \vec{r}_{i}  V^{-1/3}$ he derived the equations of motion
$\displaystyle \ddot{\vec{s}_{i}}$ $\textstyle =$ $\displaystyle \frac{\vec{K}_{i}}{m V^{1/3}}
- \frac{2}{3}  \frac{\dot{V}}{V} \dot{\vec{s}_{i}}$ (7.11)
$\displaystyle \ddot{V}$ $\textstyle =$ $\displaystyle \left( P-P_{0}\right)/Q$ (7.12)

These equations of motion conserve the enthalpy, as appropriate in an isobaric ensemble.

Parrinello and Rahman (1980) extended the NPH method of Andersen to allow for non-isotropic stretching and shrinking of box sides. Important applications are structural phase transitions in solids.

Scaling of the position vectors now follows the equation $\vec{r}_{i}= \mbox{${\bf H}$} \cdot \vec{s}_{i}$ instead of $=V^{1/3} \vec{s}_{i}$.

The matrix $\mbox{${\bf H}$}\equiv \left\{ \vec{h}_{1},\vec{h}_{2},\vec{h}_{3}\right\}$ describes the anisotropic transformation of the basic cell. The cell volume is
V \equiv \left\vert \mbox{${\bf H}$}\right\vert
= \vec{h}_{1} \cdot \left[\vec{h}_{2} \times \vec{h}_{3}\right]
\end{displaymath} (7.13)

The additional terms in the Hamiltonian are
\Delta E_{pot} = P_{0}  V\;\;\;\;{\rm and} \;\;\;\;
\Delta ...
\end{displaymath} (7.14)

and the equations of motion are
$\displaystyle \ddot{s}_{i}$ $\textstyle =$ $\displaystyle \frac{1}{m}\mbox{${\bf H}$}^{-1} \cdot \vec{K}_{i}
-\mbox{${\bf G}$}^{-1}\cdot \dot{\mbox{${\bf G}$}} \cdot \dot{\vec{s}}$ (7.15)
$\displaystyle \ddot{\mbox{${\bf H}$}}$ $\textstyle =$ $\displaystyle \frac{1}{Q}   \left[ \mbox{${\bf P}$}-\mbox{${\bf I}$}P_{0}\right]
  V   (\mbox{${\bf H}$}^{-1})^{T}$ (7.16)

where $\mbox{${\bf G}$}=\mbox{${\bf H}$}^{T} \cdot \mbox{${\bf H}$}$ is a metric tensor, $Q$ is a virtual mass (of dimension mass), and the pressure/stress tensor is defined as
(\mbox{${\bf P}$})_{\alpha \beta} = \frac{1}{V}
\left[ \sum_...
  \left( \vec{K}_{ij}\right)_{\beta}
\end{displaymath} (7.17)

Morriss and Evans (1983) devised another type of NPH dynamics. They suggested to constrain the pressure not by an inert piston but by a generalized constraint force in the spirit of Gaussian dynamics.

The same idea may be carried over to constrain $E_{kin}$ in addition to the pressure. In this way one arrives at a $NPT$ molecular dynamics procedure. (See Allen-Tildesley, Ch. 7.)
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F. J. Vesely / University of Vienna