| 29 |
|
tabular listing of the results for the $\Delta E$, force and torque |
| 30 |
|
vector magnitude, and force and torque vector direction comparisons. |
| 31 |
|
|
| 32 |
< |
\section{\label{app-water}Liquid Water} |
| 32 |
> |
\section{\label{app:water}Liquid Water} |
| 33 |
|
|
| 34 |
|
500 liquid state configurations were generated as described in the |
| 35 |
|
Methods section using the SPC/E model of water.\cite{Berendsen87} The |
| 66 |
|
& 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ |
| 67 |
|
& 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ |
| 68 |
|
GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ |
| 69 |
< |
RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
| 70 |
< |
|
| 69 |
> |
RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
| 70 |
|
\midrule |
| 72 |
– |
|
| 71 |
|
PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ |
| 72 |
|
SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ |
| 73 |
|
& 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ |
| 79 |
|
& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
| 80 |
|
GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 81 |
|
RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 84 |
– |
|
| 82 |
|
\midrule |
| 86 |
– |
|
| 83 |
|
PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ |
| 84 |
|
SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ |
| 85 |
|
& 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ |
| 179 |
|
nothing for cases were error is introduced by overdamping the |
| 180 |
|
potentials. |
| 181 |
|
|
| 182 |
< |
\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
| 182 |
> |
\section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$} |
| 183 |
|
|
| 184 |
|
In addition to the disordered molecular system above, the ordered |
| 185 |
|
molecular system of ice I$_\textrm{c}$ was also considered. The |
| 302 |
|
by actively enforcing a zeroed dipole moment within each cutoff |
| 303 |
|
sphere. |
| 304 |
|
|
| 305 |
< |
\section{\label{app-melt}NaCl Melt} |
| 305 |
> |
\section{\label{app:melt}NaCl Melt} |
| 306 |
|
|
| 307 |
|
A high temperature NaCl melt was tested to gauge the accuracy of the |
| 308 |
|
pairwise summation methods in a highly charge disordered system. The |
| 374 |
|
\label{tab:meltAng} |
| 375 |
|
\end{table} |
| 376 |
|
|
| 377 |
< |
The molten NaCl system shows the a |
| 377 |
> |
The molten NaCl system shows more sensitivity to the electrostatic |
| 378 |
> |
damping than the water systems. The most noticeable point is that the |
| 379 |
> |
undamped {\sc sf} method does very well at replicating the {\sc spme} |
| 380 |
> |
configurational energy differences and forces. Light damping appears |
| 381 |
> |
to minimally improve the dynamics, but this comes with a deterioration |
| 382 |
> |
of the energy gap results. In contrast, this light damping improves |
| 383 |
> |
the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic |
| 384 |
> |
damping reduce the agreement with {\sc spme} for both methods. From |
| 385 |
> |
these observations, the undamped {\sc sf} method is the best choice |
| 386 |
> |
for disordered systems of charges. |
| 387 |
|
|
| 388 |
< |
\section{\label{app-salt}NaCl Crystal} |
| 388 |
> |
\section{\label{app:salt}NaCl Crystal} |
| 389 |
|
|
| 390 |
|
A 1000K NaCl crystal was used to investigate the accuracy of the |
| 391 |
|
pairwise summation methods in an ordered system of charged |
| 464 |
|
\label{tab:saltAng} |
| 465 |
|
\end{table} |
| 466 |
|
|
| 467 |
< |
\section{\label{app-sol1}Weak NaCl Solution} |
| 467 |
> |
The crystalline NaCl system is the most challenging test case for the |
| 468 |
> |
pairwise summation methods, as evidenced by the results in tables |
| 469 |
> |
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
| 470 |
> |
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
| 471 |
> |
choices. These methods match well with {\sc spme} across the energy |
| 472 |
> |
gap, force magnitude, and force directionality tests. The {\sc sp} |
| 473 |
> |
method struggles in all cases with the exception of good dynamics |
| 474 |
> |
reproduction when using weak electrostatic damping with a large cutoff |
| 475 |
> |
radius. |
| 476 |
|
|
| 477 |
+ |
The moderate electrostatic damping case is not as good as we would |
| 478 |
+ |
expect given the good long-time dynamics results observed for this |
| 479 |
+ |
system. Since these results are a test of instantaneous dynamics, this |
| 480 |
+ |
indicates that good long-time dynamics comes in part at the expense of |
| 481 |
+ |
short-time dynamics. Further indication of this comes from the full |
| 482 |
+ |
power spectra shown in the main text. It appears as though a |
| 483 |
+ |
distortion is introduced between 200 to 300 cm$^{-1}$ with increased |
| 484 |
+ |
$\alpha$. |
| 485 |
+ |
|
| 486 |
+ |
\section{\label{app:solnWeak}Weak NaCl Solution} |
| 487 |
+ |
|
| 488 |
|
In an effort to bridge the charged atomic and neutral molecular |
| 489 |
|
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
| 490 |
|
the liquid water system. This low ionic strength system consists of 4 |
| 595 |
|
\label{tab:solnWeakAng} |
| 596 |
|
\end{table} |
| 597 |
|
|
| 598 |
< |
\section{\label{app-sol10}Strong NaCl Solution} |
| 598 |
> |
This weak ionic strength system can be considered as a perturbation of |
| 599 |
> |
the pure liquid water system. The {\sc sp} and {\sc sf} methods are |
| 600 |
> |
not significantly affected by the inclusion of a few ions. The aspect |
| 601 |
> |
of cutoff sphere neutralization aids in the smooth incorporation of |
| 602 |
> |
these ions; thus, all of the observations regarding these methods |
| 603 |
> |
carry over from section \ref{app:water}. The differences between these |
| 604 |
> |
systems are visible for the {\sc rf} method. Though good force |
| 605 |
> |
reproduction is still maintained, the energy gaps show a significant |
| 606 |
> |
increase in the data scatter. This foreshadows the breakdown of the |
| 607 |
> |
method as we introduce system inhomogeneities. |
| 608 |
|
|
| 609 |
+ |
\section{\label{app:solnStr}Strong NaCl Solution} |
| 610 |
+ |
|
| 611 |
|
The bridging of the charged atomic and neutral molecular systems was |
| 612 |
< |
furthered by considering a high ionic strength system consisting of 40 |
| 613 |
< |
ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for |
| 614 |
< |
the energy gap comparisons and the force and torque vector magnitude |
| 615 |
< |
comparisons are shown in table \ref{tab:solnWeak}. The force and |
| 616 |
< |
torque vector directionality results are displayed separately in table |
| 617 |
< |
\ref{tab:solnWeakAng}, where the effect of group-based cutoffs and |
| 618 |
< |
switching functions on the {\sc sp} and {\sc sf} potentials are |
| 619 |
< |
investigated. |
| 612 |
> |
further developed by considering a high ionic strength system |
| 613 |
> |
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
| 614 |
> |
M). The results for the energy gap comparisons and the force and |
| 615 |
> |
torque vector magnitude comparisons are shown in table |
| 616 |
> |
\ref{tab:solnWeak}. The force and torque vector directionality |
| 617 |
> |
results are displayed separately in table\ref{tab:solnWeakAng}, where |
| 618 |
> |
the effect of group-based cutoffs and switching functions on the {\sc |
| 619 |
> |
sp} and {\sc sf} potentials are investigated. |
| 620 |
|
|
| 621 |
|
\begin{table}[htbp] |
| 622 |
|
\centering |
| 711 |
|
\label{tab:solnStrAng} |
| 712 |
|
\end{table} |
| 713 |
|
|
| 714 |
< |
\section{\label{app-argon}Argon Sphere in Water} |
| 714 |
> |
The {\sc rf} method struggles with the jump in ionic strength. The |
| 715 |
> |
configuration energy difference degrade to unuseable levels while the |
| 716 |
> |
forces and torques degrade in a more modest fashion. The {\sc rf} |
| 717 |
> |
method was designed for homogeneous systems, and this restriction is |
| 718 |
> |
apparent in these results. |
| 719 |
|
|
| 720 |
+ |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
| 721 |
+ |
their agreement with {\sc spme}. With these results, we still |
| 722 |
+ |
recommend no to moderate damping for the {\sc sf} method and moderate |
| 723 |
+ |
damping for the {\sc sp} method, both with cutoffs greater than 12 |
| 724 |
+ |
\AA. |
| 725 |
+ |
|
| 726 |
+ |
\section{\label{app:argon}Argon Sphere in Water} |
| 727 |
+ |
|
| 728 |
|
The final model system studied was 6 \AA\ sphere of Argon solvated by |
| 729 |
|
SPC/E water. The results for the energy gap comparisons and the force |
| 730 |
|
and torque vector magnitude comparisons are shown in table |
| 831 |
|
\label{tab:argonAng} |
| 832 |
|
\end{table} |
| 833 |
|
|
| 834 |
+ |
This system appears not to show in any significant deviation in the previously observed results. The {\sc sp} and {\sc sf} methods give result qualities similar to those observed in section \ref{app:water}. The only significant difference is the improvement for the configuration energy differences for the {\sc rf} method. This is surprising in that we are introducing an inhomogeneity to the system; however, this inhomogeneity is charge-neutral and does not result in charged cutoff spheres. The charge-neutrality, which the {\sc sp} and {\sc sf} methods explicity enforce, seems to play a greater role in the stability of the {\sc rf} method than the necessity of a homogeneous environment. |
| 835 |
+ |
|
| 836 |
|
\newpage |
| 837 |
|
|
| 838 |
|
\bibliographystyle{jcp2} |
