| 279 |
|
v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r}, |
| 280 |
|
\label{eq:dampCoulomb} |
| 281 |
|
\end{equation} |
| 282 |
< |
the shifted potential (Eq. \ref{eq:WolfSP}) can be recovered |
| 283 |
< |
\textit{via} equation \ref{eq:shiftingForm}, |
| 282 |
> |
the shifted potential (Eq. (\ref{eq:WolfSP})) can be recovered |
| 283 |
> |
using eq. (\ref{eq:shiftingForm}), |
| 284 |
|
\begin{equation} |
| 285 |
< |
v_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}(\alpha r)}{r}-\frac{\textrm{erfc}(\alpha R_\textrm{c})}{R_\textrm{c}}\right) \quad r\leqslant R_\textrm{c}. |
| 285 |
> |
v_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}(\alpha r)}{r}-\frac{\textrm{erfc}(\alpha R_\textrm{c})}{R_\textrm{c}}\right) \quad r\leqslant R_\textrm{c}, |
| 286 |
|
\label{eq:DSPPot} |
| 287 |
< |
\end{equation}, |
| 287 |
> |
\end{equation} |
| 288 |
|
with associated forces, |
| 289 |
|
\begin{equation} |
| 290 |
|
f_{\textrm{DSP}}(r) = q_iq_j\left(-\frac{\textrm{erfc}(\alpha r)}{r^2}-\frac{2\alpha}{\pi^{1/2}}\frac{\exp{(-\alpha^2r^2)}}{r}\right) \quad r\leqslant R_\textrm{c}. |
| 292 |
|
\end{equation} |
| 293 |
|
Again, this damped shifted potential suffers from a discontinuity and |
| 294 |
|
a lack of the image charges in the forces. To remedy these concerns, |
| 295 |
< |
one may derive a Shifted-Force variant by including the derivative |
| 296 |
< |
term in equation \ref{eq:shiftingForm}, |
| 295 |
> |
one may derive a {\sc sf} variant by including the derivative |
| 296 |
> |
term in eq. (\ref{eq:shiftingForm}), |
| 297 |
|
\begin{equation} |
| 298 |
|
\begin{split} |
| 299 |
|
v_\mathrm{DSF}(r) = q_iq_j\Biggr{[}&\frac{\mathrm{erfc}\left(\alpha r\right)}{r}-\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}} \\ &\left.+\left(\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}}\right)\left(r-R_\mathrm{c}\right)\ \right] \quad r\leqslant R_\textrm{c}. |
| 308 |
|
\end{split} |
| 309 |
|
\end{equation} |
| 310 |
|
|
| 311 |
< |
This new Shifted-Force potential is similar to equation |
| 311 |
> |
This new {\sc sf} potential is similar to equation |
| 312 |
|
\ref{eq:ZahnPot} derived by Zahn \textit{et al.}; however, there are |
| 313 |
|
two important differences.\cite{Zahn02} First, the $v_\textrm{c}$ term |
| 314 |
|
from eq. (\ref{eq:shiftingForm}) is equal to |
| 333 |
|
performed by the Ewald sum. |
| 334 |
|
|
| 335 |
|
\subsection{Other alternatives} |
| 336 |
< |
|
| 337 |
< |
Reaction Field blah |
| 338 |
< |
|
| 339 |
< |
Group-based cutoff blah |
| 336 |
> |
In addition to the methods described above, we will consider some |
| 337 |
> |
other techniques that commonly get used in molecular simulations. The |
| 338 |
> |
simplest of these is group-based cutoffs. Though of little use for |
| 339 |
> |
non-neutral molecules, collecting atoms into neutral groups takes |
| 340 |
> |
advantage of the observation that the electrostatic interactions decay |
| 341 |
> |
faster than those for monopolar pairs.\cite{Steinbach94} When |
| 342 |
> |
considering these molecules as groups, an orientational aspect is |
| 343 |
> |
introduced to the interactions. Consequently, as these molecular |
| 344 |
> |
particles move through $R_\textrm{c}$, the energy will drift upward |
| 345 |
> |
due to the anisotropy of the net molecular dipole |
| 346 |
> |
interactions.\cite{Rahman71} To maintain good energy conservation, |
| 347 |
> |
both the potential and derivative need to be smoothly switched to zero |
| 348 |
> |
at $R_\textrm{c}$.\cite{Adams79} This is accomplished using a |
| 349 |
> |
switching function, |
| 350 |
> |
\begin{equation} |
| 351 |
> |
S(r) = \begin{cases} 1 &\quad r\leqslant r_\textrm{sw} \\ |
| 352 |
> |
\frac{(R_\textrm{c}+2r-3r_\textrm{sw})(R_\textrm{c}-r)^2}{(R_\textrm{c}-r_\textrm{sw})^3} &\quad r_\textrm{sw}<r\leqslant R_\textrm{c} \\ |
| 353 |
> |
0 &\quad r>R_\textrm{c} |
| 354 |
> |
\end{cases}, |
| 355 |
> |
\end{equation} |
| 356 |
> |
where the above form is for a cubic function. If a smooth second |
| 357 |
> |
derivative is desired, a fifth (or higher) order polynomial can be |
| 358 |
> |
used.\cite{Andrea83} |
| 359 |
|
|
| 360 |
+ |
Group-based cutoffs neglect the surroundings beyond $R_\textrm{c}$, |
| 361 |
+ |
and to incorporate their effect, a method like Reaction Field ({\sc |
| 362 |
+ |
rf}) can be used. The original theory for {\sc rf} was originally |
| 363 |
+ |
developed by Onsager,\cite{Onsager36} and it was applied in |
| 364 |
+ |
simulations for the study of water by Barker and Watts.\cite{Barker73} |
| 365 |
+ |
In application, it is simply an extension of the group-based cutoff |
| 366 |
+ |
method where the net dipole within the cutoff sphere polarizes an |
| 367 |
+ |
external dielectric, which reacts back on the central dipole. The |
| 368 |
+ |
same switching function considerations for group-based cutoffs need to |
| 369 |
+ |
made for {\sc rf}, with the additional pre-specification of a |
| 370 |
+ |
dielectric constant. |
| 371 |
|
|
| 372 |
|
\section{Methods} |
| 373 |
|
|
| 551 |
|
|
| 552 |
|
\subsection{Electrostatic Summation Methods}\label{sec:ESMethods} |
| 553 |
|
Electrostatic summation method comparisons were performed using SPME, |
| 554 |
< |
the Shifted-Potential and Shifted-Force methods - both with damping |
| 554 |
> |
the {\sc sp} and {\sc sf} methods - both with damping |
| 555 |
|
parameters ($\alpha$) of 0.0, 0.1, 0.2, and 0.3 \AA$^{-1}$ (no, weak, |
| 556 |
|
moderate, and strong damping respectively), reaction field with an |
| 557 |
|
infinite dielectric constant, and an unmodified cutoff. Group-based |
| 614 |
|
particularly with a cutoff radius greater than 12 \AA . Use of a |
| 615 |
|
larger damping parameter is more helpful for the shortest cutoff |
| 616 |
|
shown, but it has a detrimental effect on simulations with larger |
| 617 |
< |
cutoffs. In the Shifted-Force sets, increasing damping results in |
| 617 |
> |
cutoffs. In the {\sc sf} sets, increasing damping results in |
| 618 |
|
progressively poorer correlation. Overall, the undamped case is the |
| 619 |
|
best performing set, as the correlation and quality of fits are |
| 620 |
|
consistently superior regardless of the cutoff distance. This result |
| 647 |
|
in the previous $\Delta E$ section. The unmodified cutoff results are |
| 648 |
|
poor, but using group based cutoffs and a switching function provides |
| 649 |
|
a improvement much more significant than what was seen with $\Delta |
| 650 |
< |
E$. Looking at the Shifted-Potential sets, the slope and $R^2$ |
| 650 |
> |
E$. Looking at the {\sc sp} sets, the slope and $R^2$ |
| 651 |
|
improve with the use of damping to an optimal result of 0.2 \AA |
| 652 |
|
$^{-1}$ for the 12 and 15 \AA\ cutoffs. Further increases in damping, |
| 653 |
|
while beneficial for simulations with a cutoff radius of 9 \AA\ , is |
| 654 |
|
detrimental to simulations with larger cutoff radii. The undamped |
| 655 |
< |
Shifted-Force method gives forces in line with those obtained using |
| 655 |
> |
{\sc sf} method gives forces in line with those obtained using |
| 656 |
|
SPME, and use of a damping function results in minor improvement. The |
| 657 |
|
reaction field results are surprisingly good, considering the poor |
| 658 |
|
quality of the fits for the $\Delta E$ results. There is still a |
| 675 |
|
torque vector magnitude results in figure \ref{fig:trqMag} are still |
| 676 |
|
similar to those seen for the forces; however, they more clearly show |
| 677 |
|
the improved behavior that comes with increasing the cutoff radius. |
| 678 |
< |
Moderate damping is beneficial to the Shifted-Potential and helpful |
| 679 |
< |
yet possibly unnecessary with the Shifted-Force method, and they also |
| 678 |
> |
Moderate damping is beneficial to the {\sc sp} and helpful |
| 679 |
> |
yet possibly unnecessary with the {\sc sf} method, and they also |
| 680 |
|
show that over-damping adversely effects all cutoff radii rather than |
| 681 |
|
showing an improvement for systems with short cutoffs. The reaction |
| 682 |
|
field method performs well when calculating the torques, better than |
| 705 |
|
show the improvement afforded by choosing a longer simulation cutoff. |
| 706 |
|
Increasing the cutoff from 9 to 12 \AA\ typically results in a halving |
| 707 |
|
of the distribution widths, with a similar improvement going from 12 |
| 708 |
< |
to 15 \AA . The undamped Shifted-Force, Group Based Cutoff, and |
| 708 |
> |
to 15 \AA . The undamped {\sc sf}, Group Based Cutoff, and |
| 709 |
|
Reaction Field methods all do equivalently well at capturing the |
| 710 |
|
direction of both the force and torque vectors. Using damping |
| 711 |
< |
improves the angular behavior significantly for the Shifted-Potential |
| 712 |
< |
and moderately for the Shifted-Force methods. Increasing the damping |
| 711 |
> |
improves the angular behavior significantly for the {\sc sp} |
| 712 |
> |
and moderately for the {\sc sf} methods. Increasing the damping |
| 713 |
|
too far is destructive for both methods, particularly to the torque |
| 714 |
|
vectors. Again it is important to recognize that the force vectors |
| 715 |
|
cover all particles in the systems, while torque vectors are only |
| 751 |
|
\end{table} |
| 752 |
|
|
| 753 |
|
Although not discussed previously, group based cutoffs can be applied |
| 754 |
< |
to both the Shifted-Potential and Shifted-Force methods. Use off a |
| 754 |
> |
to both the {\sc sp} and {\sc sf} methods. Use off a |
| 755 |
|
switching function corrects for the discontinuities that arise when |
| 756 |
|
atoms of a group exit the cutoff before the group's center of mass. |
| 757 |
|
Though there are no significant benefit or drawbacks observed in |
| 760 |
|
\ref{tab:groupAngle} shows the angular variance values obtained using |
| 761 |
|
group based cutoffs and a switching function alongside the standard |
| 762 |
|
results seen in figure \ref{fig:frcTrqAng} for comparison purposes. |
| 763 |
< |
The Shifted-Potential shows much narrower angular distributions for |
| 763 |
> |
The {\sc sp} shows much narrower angular distributions for |
| 764 |
|
both the force and torque vectors when using an $\alpha$ of 0.2 |
| 765 |
< |
\AA$^{-1}$ or less, while Shifted-Force shows improvements in the |
| 765 |
> |
\AA$^{-1}$ or less, while {\sc sf} shows improvements in the |
| 766 |
|
undamped and lightly damped cases. Thus, by calculating the |
| 767 |
|
electrostatic interactions in terms of molecular pairs rather than |
| 768 |
|
atomic pairs, the direction of the force and torque vectors are |
| 769 |
|
determined more accurately. |
| 770 |
|
|
| 771 |
|
One additional trend to recognize in table \ref{tab:groupAngle} is |
| 772 |
< |
that the $\sigma^2$ values for both Shifted-Potential and |
| 773 |
< |
Shifted-Force converge as $\alpha$ increases, something that is easier |
| 772 |
> |
that the $\sigma^2$ values for both {\sc sp} and |
| 773 |
> |
{\sc sf} converge as $\alpha$ increases, something that is easier |
| 774 |
|
to see when using group based cutoffs. Looking back on figures |
| 775 |
|
\ref{fig:delE}, \ref{fig:frcMag}, and \ref{fig:trqMag}, show this |
| 776 |
|
behavior clearly at large $\alpha$ and cutoff values. The reason for |
| 789 |
|
high would introduce error in the molecular torques, particularly for |
| 790 |
|
the shorter cutoffs. Based on the above findings, empirical damping |
| 791 |
|
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 792 |
< |
unnecessary when using the Shifted-Force method. |
| 792 |
> |
unnecessary when using the {\sc sf} method. |
| 793 |
|
|
| 794 |
|
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 795 |
|
|
| 796 |
< |
In the previous studies using a Shifted-Force variant of the damped |
| 796 |
> |
In the previous studies using a {\sc sf} variant of the damped |
| 797 |
|
Wolf coulomb potential, the structure and dynamics of water were |
| 798 |
|
investigated rather extensively.\cite{Zahn02,Kast03} Their results |
| 799 |
< |
indicated that the damped Shifted-Force method results in properties |
| 799 |
> |
indicated that the damped {\sc sf} method results in properties |
| 800 |
|
very similar to those obtained when using the Ewald summation. |
| 801 |
|
Considering the statistical results shown above, the good performance |
| 802 |
|
of this method is not that surprising. Rather than consider the same |
| 807 |
|
\begin{figure} |
| 808 |
|
\centering |
| 809 |
|
\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
| 810 |
< |
\caption{Power spectra obtained from the velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, Shifted-Force ($\alpha$ = 0, 0.1, \& 0.2), and Shifted-Potential ($\alpha$ = 0.2). Apodization of the correlation functions via a cubic switching function between 40 and 50 ps was used to clear up the spectral noise resulting from data truncation, and had no noticeable effect on peak location or magnitude. The inset shows the frequency region below 100 cm$^{-1}$ to highlight where the spectra begin to differ.} |
| 810 |
> |
\caption{Power spectra obtained from the velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, {\sc sf} ($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). Apodization of the correlation functions via a cubic switching function between 40 and 50 ps was used to clear up the spectral noise resulting from data truncation, and had no noticeable effect on peak location or magnitude. The inset shows the frequency region below 100 cm$^{-1}$ to highlight where the spectra begin to differ.} |
| 811 |
|
\label{fig:methodPS} |
| 812 |
|
\end{figure} |
| 813 |
|
|
| 819 |
|
methods differ. Considering the low-frequency inset (expanded in the |
| 820 |
|
upper frame of figure \ref{fig:dampInc}), at frequencies below 100 |
| 821 |
|
cm$^{-1}$, the correlated motions are blue-shifted when using undamped |
| 822 |
< |
or weakly damped Shifted-Force. When using moderate damping ($\alpha |
| 823 |
< |
= 0.2$ \AA$^{-1}$) both the Shifted-Force and Shifted-Potential |
| 822 |
> |
or weakly damped {\sc sf}. When using moderate damping ($\alpha |
| 823 |
> |
= 0.2$ \AA$^{-1}$) both the {\sc sf} and {\sc sp} |
| 824 |
|
methods give near identical correlated motion behavior as the Ewald |
| 825 |
|
method (which has a damping value of 0.3119). The damping acts as a |
| 826 |
|
distance dependent Gaussian screening of the point charges for the |
| 844 |
|
shift to higher frequency in exponential fashion. Though not shown, |
| 845 |
|
the spectrum for the simple undamped electrostatic potential is |
| 846 |
|
blue-shifted such that the lowest frequency peak resides near 325 |
| 847 |
< |
cm$^{-1}$. In light of these results, the undamped Shifted-Force |
| 847 |
> |
cm$^{-1}$. In light of these results, the undamped {\sc sf} |
| 848 |
|
method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is |
| 849 |
|
quite respectable; however, it appears as though moderate damping is |
| 850 |
|
required for accurate reproduction of crystal dynamics. |
| 851 |
|
\begin{figure} |
| 852 |
|
\centering |
| 853 |
|
\includegraphics[width = \linewidth]{./comboSquare.pdf} |
| 854 |
< |
\caption{Upper: Zoomed inset from figure \ref{fig:methodPS}. As the damping value for the Shifted-Force potential increases, the low-frequency peaks red-shift. Lower: Low-frequency correlated motions for NaCl crystals at 1000 K when using SPME and a simple damped Coulombic sum with damping coefficients ($\alpha$) ranging from 0.15 to 0.3 \AA$^{-1}$. As $\alpha$ increases, the peaks are red-shifted toward and eventually beyond the values given by SPME. The larger $\alpha$ values weaken the real-space electrostatics, explaining this shift towards less strongly correlated motions in the crystal.} |
| 854 |
> |
\caption{Upper: Zoomed inset from figure \ref{fig:methodPS}. As the damping value for the {\sc sf} potential increases, the low-frequency peaks red-shift. Lower: Low-frequency correlated motions for NaCl crystals at 1000 K when using SPME and a simple damped Coulombic sum with damping coefficients ($\alpha$) ranging from 0.15 to 0.3 \AA$^{-1}$. As $\alpha$ increases, the peaks are red-shifted toward and eventually beyond the values given by SPME. The larger $\alpha$ values weaken the real-space electrostatics, explaining this shift towards less strongly correlated motions in the crystal.} |
| 855 |
|
\label{fig:dampInc} |
| 856 |
|
\end{figure} |
| 857 |
|
|
| 862 |
|
electrostatic summation techniques than the Ewald summation, chiefly |
| 863 |
|
methods derived from the damped Coulombic sum originally proposed by |
| 864 |
|
Wolf \textit{et al.}\cite{Wolf99,Zahn02} In particular, the |
| 865 |
< |
Shifted-Force method, reformulated above as equation \ref{eq:SFPot}, |
| 865 |
> |
{\sc sf} method, reformulated above as eq. (\ref{eq:DSFPot}), |
| 866 |
|
shows a remarkable ability to reproduce the energetic and dynamic |
| 867 |
|
characteristics exhibited by simulations employing lattice summation |
| 868 |
|
techniques. The cumulative energy difference results showed the |
| 869 |
< |
undamped Shifted-Force and moderately damped Shifted-Potential methods |
| 869 |
> |
undamped {\sc sf} and moderately damped {\sc sp} methods |
| 870 |
|
produced results nearly identical to SPME. Similarly for the dynamic |
| 871 |
< |
features, the undamped or moderately damped Shifted-Force and |
| 872 |
< |
moderately damped Shifted-Potential methods produce force and torque |
| 871 |
> |
features, the undamped or moderately damped {\sc sf} and |
| 872 |
> |
moderately damped {\sc sp} methods produce force and torque |
| 873 |
|
vector magnitude and directions very similar to the expected values. |
| 874 |
|
These results translate into long-time dynamic behavior equivalent to |
| 875 |
|
that produced in simulations using SPME. |
