| 105 |
|
\label{eq:DSFPot} |
| 106 |
|
\end{split} |
| 107 |
|
\end{equation} |
| 108 |
< |
(In these potentials and in all electrostatic quantities that follow, |
| 109 |
< |
the standard $4 \pi \epsilon_{0}$ has been omitted for clarity.) |
| 108 |
> |
In these potentials and in all electrostatic quantities that follow, |
| 109 |
> |
the standard $4 \pi \epsilon_{0}$ has been omitted for clarity. |
| 110 |
|
|
| 111 |
|
The damped SF method is an improvement over the SP method because |
| 112 |
|
there is no discontinuity in the forces as particles move out of the |
| 236 |
|
functions for higher multipole potentials and forces. Each subsequent |
| 237 |
|
damping function includes one additional term, and we can simplify the |
| 238 |
|
procedure for obtaining these terms by writing out the following |
| 239 |
< |
generating function, |
| 239 |
> |
recurrence relation, |
| 240 |
|
\begin{equation} |
| 241 |
|
c_n(r_{ij}) = \frac{2^n(\alpha r_{ij})^{2n-1}e^{-\alpha^2r^2_{ij}}} |
| 242 |
|
{(2n-1)!!\sqrt{\pi}} + c_{n-1}(r_{ij}), |
| 324 |
|
correction.\cite{Stillinger74,Jorgensen83,Berendsen81,Berendsen87} |
| 325 |
|
Without this correction, the pressure term on the central particle |
| 326 |
|
from the surroundings is missing. When this correction is included, |
| 327 |
< |
systems of these particles expand to compensate for this added |
| 328 |
< |
pressure term and under-predict the density of water under standard |
| 329 |
< |
conditions. In developing TIP5P-E, Rick preserved the geometry and |
| 330 |
< |
point charge magnitudes in TIP5P and focused on altering the |
| 331 |
< |
Lennard-Jones parameters to correct the density at 298~K. With the |
| 332 |
< |
density corrected, he compared common water properties for TIP5P-E |
| 333 |
< |
using the Ewald sum with TIP5P using a 9~\AA\ cutoff. |
| 327 |
> |
the system expands to compensate for this added pressure and therefore |
| 328 |
> |
under-predicts the density of water under standard conditions. In |
| 329 |
> |
developing TIP5P-E, Rick preserved the geometry and point charge |
| 330 |
> |
magnitudes in TIP5P and focused on altering the Lennard-Jones |
| 331 |
> |
parameters to correct the density at 298~K. With the density |
| 332 |
> |
corrected, he compared common water properties for TIP5P-E using the |
| 333 |
> |
Ewald sum with TIP5P using a 9~\AA\ cutoff. |
| 334 |
|
|
| 335 |
|
In the following sections, we compare these same properties calculated |
| 336 |
|
from TIP5P-E using the Ewald sum with TIP5P-E using the damped SF |
| 365 |
|
simulations.\cite{Rick04} In order to improve statistics around the |
| 366 |
|
density maximum, 3~ns trajectories were accumulated at 0, 12.5, and |
| 367 |
|
25$^\circ$C, while 2~ns trajectories were obtained at all other |
| 368 |
< |
temperatures. The average densities were calculated from the later |
| 368 |
> |
temperatures. The average densities were calculated from the latter |
| 369 |
|
three-fourths of each trajectory. Similar to Mahoney and Jorgensen's |
| 370 |
|
method for accumulating statistics, these sequences were spliced into |
| 371 |
|
200 segments, each providing an average density. These 200 density |
| 381 |
|
reciprocal-space portion of the Ewald summation, leading to slightly |
| 382 |
|
lower densities. This effect is more visible with the 9~\AA\ cutoff, |
| 383 |
|
where the image charges exert a greater force on the central |
| 384 |
< |
particle. The error bars for the SF methods show the average one-sigma |
| 385 |
< |
uncertainty of the density measurement, and this uncertainty is the |
| 386 |
< |
same for all the SF curves.} |
| 384 |
> |
particle. The representative error bar for the SF methods shows the |
| 385 |
> |
average one-sigma uncertainty of the density measurement, and this |
| 386 |
> |
uncertainty is the same for all the SF curves.} |
| 387 |
|
\label{fig:t5peDensities} |
| 388 |
|
\end{figure} |
| 389 |
|
Figure \ref{fig:t5peDensities} shows the densities calculated for |
| 437 |
|
\begin{figure} |
| 438 |
|
\includegraphics[width=\linewidth]{./figures/tip5peGofR.pdf} |
| 439 |
|
\caption{The oxygen-oxygen pair correlation functions calculated for |
| 440 |
< |
TIP5P-E at 298~K and 1atm while using the Ewald summation |
| 440 |
> |
TIP5P-E at 298~K and 1~atm while using the Ewald summation |
| 441 |
|
(Ref. \citen{Rick04}) and the SF technique with varying |
| 442 |
|
parameters. Even with the lower densities obtained using the SF |
| 443 |
|
technique, the correlation functions are essentially identical.} |
| 636 |
|
Ewald sum, this screening parameter was tethered to the simulation box |
| 637 |
|
size (as was the $R_\textrm{c}$).\cite{Rick04} In general, systems |
| 638 |
|
with larger screening parameters reported larger dielectric constant |
| 639 |
< |
values, the same behavior we see here with {\sc sf}; however, the |
| 639 |
> |
values, the same behavior we see here with SF; however, the |
| 640 |
|
choice of cutoff radius also plays an important role. |
| 641 |
|
|
| 642 |
|
\subsection{Optimal Damping Coefficients for Damped |
| 659 |
|
cutoff radius for TIP5P-E and for a point-dipolar water model |
| 660 |
|
(SSD/RF). To calculate the static dielectric constant, we performed |
| 661 |
|
5~ns $NPT$ calculations on systems of 512 water molecules and averaged |
| 662 |
< |
over the converged region (typically the later 2.5~ns) of equation |
| 662 |
> |
over the converged region (typically the latter 2.5~ns) of equation |
| 663 |
|
(\ref{eq:staticDielectric}). The selected cutoff radii ranged from 9, |
| 664 |
|
10, 11, and 12~\AA , and the damping parameter values ranged from 0.1 |
| 665 |
|
to 0.35~\AA$^{-1}$. |
