| 104 |
|
and a long-ranged reciprocal-space summation, |
| 105 |
|
\begin{equation} |
| 106 |
|
\begin{split} |
| 107 |
< |
V_\textrm{elec} = \frac{1}{2}& \sum_{i=1}^N\sum_{j=1}^N \Biggr[ q_i q_j\Biggr( {\sum_{\mathbf{n}}}^\prime \frac{\textrm{erfc}\left( \alpha|\mathbf{r}_{ij}+\mathbf{n}|\right)}{|\mathbf{r}_{ij}+\mathbf{n}|} \\ &+ \frac{1}{\pi L^3}\sum_{\mathbf{k}\ne 0}\frac{4\pi^2}{|\mathbf{k}|^2} \exp{\left(-\frac{\pi^2|\mathbf{k}|^2}{\alpha^2}\right) \cos\left(\mathbf{k}\cdot\mathbf{r}_{ij}\right)}\Biggr)\Biggr] \\ &- \frac{\alpha}{\pi^{1/2}}\sum_{i=1}^N q_i^2 + \frac{2\pi}{3L^3}\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2, |
| 107 |
> |
V_\textrm{elec} = \frac{1}{2}& \sum_{i=1}^N\sum_{j=1}^N \Biggr[ q_i q_j\Biggr( {\sum_{\mathbf{n}}}^\prime \frac{\textrm{erfc}\left( \alpha|\mathbf{r}_{ij}+\mathbf{n}|\right)}{|\mathbf{r}_{ij}+\mathbf{n}|} \\ &+ \frac{1}{\pi L^3}\sum_{\mathbf{k}\ne 0}\frac{4\pi^2}{|\mathbf{k}|^2} \exp{\left(-\frac{\pi^2|\mathbf{k}|^2}{\alpha^2}\right) \cos\left(\mathbf{k}\cdot\mathbf{r}_{ij}\right)}\Biggr)\Biggr] \\ &- \frac{\alpha}{\pi^{1/2}}\sum_{i=1}^N q_i^2 + \frac{2\pi}{(2\epsilon_\textrm{S}+1)L^3}\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2, |
| 108 |
|
\end{split} |
| 109 |
|
\label{eq:EwaldSum} |
| 110 |
|
\end{equation} |
| 111 |
|
where $\alpha$ is a damping parameter, or separation constant, with |
| 112 |
< |
units of \AA$^{-1}$, and $\mathbf{k}$ are the reciprocal vectors and |
| 113 |
< |
equal $2\pi\mathbf{n}/L^2$. The final two terms of |
| 112 |
> |
units of \AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and equal |
| 113 |
> |
$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric |
| 114 |
> |
constant of the encompassing medium. The final two terms of |
| 115 |
|
eq. (\ref{eq:EwaldSum}) are a particle-self term and a dipolar term |
| 116 |
|
for interacting with a surrounding dielectric.\cite{Allen87} This |
| 117 |
|
dipolar term was neglected in early applications in molecular |
| 118 |
|
simulations,\cite{Brush66,Woodcock71} until it was introduced by de |
| 119 |
|
Leeuw {\it et al.} to address situations where the unit cell has a |
| 120 |
|
dipole moment and this dipole moment gets magnified through |
| 121 |
< |
replication of the periodic images.\cite{deLeeuw80} This term is zero |
| 122 |
< |
for systems where $\epsilon_{\rm S} = \infty$. Figure |
| 121 |
> |
replication of the periodic images.\cite{deLeeuw80,Smith81} If this |
| 122 |
> |
term is taken to be zero, the system is using conducting boundary |
| 123 |
> |
conditions, $\epsilon_{\rm S} = \infty$. Figure |
| 124 |
|
\ref{fig:ewaldTime} shows how the Ewald sum has been applied over |
| 125 |
|
time. Initially, due to the small size of systems, the entire |
| 126 |
|
simulation box was replicated to convergence. Currently, we balance a |
| 143 |
|
$\mathscr{O}(N^2)$ algorithm. The separation constant $(\alpha)$ |
| 144 |
|
plays an important role in the computational cost balance between the |
| 145 |
|
direct and reciprocal-space portions of the summation. The choice of |
| 146 |
< |
the magnitude of this value allows one to whether the real-space or |
| 147 |
< |
reciprocal space portion of the summation is an $\mathscr{O}(N^2)$ |
| 148 |
< |
calcualtion, with the other being $\mathscr{O}(N)$.\cite{Sagui99} With |
| 149 |
< |
appropriate choice of $\alpha$ and thoughtful algorithm development, |
| 150 |
< |
this cost can be brought down to |
| 146 |
> |
the magnitude of this value allows one to select whether the |
| 147 |
> |
real-space or reciprocal space portion of the summation is an |
| 148 |
> |
$\mathscr{O}(N^2)$ calcualtion (with the other being |
| 149 |
> |
$\mathscr{O}(N)$).\cite{Sagui99} With appropriate choice of $\alpha$ |
| 150 |
> |
and thoughtful algorithm development, this cost can be brought down to |
| 151 |
|
$\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route taken to |
| 152 |
< |
accelerate the Ewald summation is to se |
| 152 |
> |
reduce the cost of the Ewald summation further is to set $\alpha$ such |
| 153 |
> |
that the real-space interactions decay rapidly, allowing for a short |
| 154 |
> |
spherical cutoff, and then optimize the reciprocal space summation. |
| 155 |
> |
These optimizations usually involve the utilization of the fast |
| 156 |
> |
Fourier transform (FFT),\cite{Hockney81} leading to the |
| 157 |
> |
particle-particle particle-mesh (P3M) and particle mesh Ewald (PME) |
| 158 |
> |
methods.\cite{Shimada93,Luty94,Luty95,Darden93,Essmann95} In these |
| 159 |
> |
methods, the cost of the reciprocal-space portion of the Ewald |
| 160 |
> |
summation is from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N \log N)$. |
| 161 |
> |
|
| 162 |
> |
These developments and optimizations have led the use of the Ewald |
| 163 |
> |
summation to become routine in simulations with periodic boundary |
| 164 |
> |
conditions. However, in certain systems the intrinsic three |
| 165 |
> |
dimensional periodicity can prove to be problematic, such as two |
| 166 |
> |
dimensional surfaces and membranes. The Ewald sum has been |
| 167 |
> |
reformulated to handle 2D |
| 168 |
> |
systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89}, but the new |
| 169 |
> |
methods have been found to be computationally |
| 170 |
> |
expensive.\cite{Spohr97,Yeh99} Inclusion of a correction term in the |
| 171 |
> |
full Ewald summation is a possible direction for enabling the handling |
| 172 |
> |
of 2D systems and the inclusion of the optimizations described |
| 173 |
> |
previously.\cite{Yeh99} |
| 174 |
> |
|
| 175 |
> |
Several studies have recognized that the inherent periodicity in the |
| 176 |
> |
Ewald sum can also have an effect on systems that have the same |
| 177 |
> |
dimensionality.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00} |
| 178 |
> |
Good examples are solvated proteins kept at high relative |
| 179 |
> |
concentration due to the periodicity of the electrostatics. In these |
| 180 |
> |
systems, the more compact folded states of a protein can be |
| 181 |
> |
artificially stabilized by the periodic replicas introduced by the |
| 182 |
> |
Ewald summation.\cite{Weber00} Thus, care ought to be taken when |
| 183 |
> |
considering the use of the Ewald summation where the intrinsic |
| 184 |
> |
perodicity may negatively affect the system dynamics. |
| 185 |
|
|
| 186 |
+ |
|
| 187 |
|
\subsection{The Wolf and Zahn Methods} |
| 188 |
|
In a recent paper by Wolf \textit{et al.}, a procedure was outlined |
| 189 |
|
for the accurate accumulation of electrostatic interactions in an |
| 190 |
< |
efficient pairwise fashion.\cite{Wolf99} Wolf \textit{et al.} observed |
| 191 |
< |
that the electrostatic interaction is effectively short-ranged in |
| 192 |
< |
condensed phase systems and that neutralization of the charge |
| 193 |
< |
contained within the cutoff radius is crucial for potential |
| 194 |
< |
stability. They devised a pairwise summation method that ensures |
| 195 |
< |
charge neutrality and gives results similar to those obtained with |
| 196 |
< |
the Ewald summation. The resulting shifted Coulomb potential |
| 197 |
< |
(Eq. \ref{eq:WolfPot}) includes image-charges subtracted out through |
| 198 |
< |
placement on the cutoff sphere and a distance-dependent damping |
| 199 |
< |
function (identical to that seen in the real-space portion of the |
| 200 |
< |
Ewald sum) to aid convergence |
| 190 |
> |
efficient pairwise fashion and lacks the inherent periodicity of the |
| 191 |
> |
Ewald summation.\cite{Wolf99} Wolf \textit{et al.} observed that the |
| 192 |
> |
electrostatic interaction is effectively short-ranged in condensed |
| 193 |
> |
phase systems and that neutralization of the charge contained within |
| 194 |
> |
the cutoff radius is crucial for potential stability. They devised a |
| 195 |
> |
pairwise summation method that ensures charge neutrality and gives |
| 196 |
> |
results similar to those obtained with the Ewald summation. The |
| 197 |
> |
resulting shifted Coulomb potential (Eq. \ref{eq:WolfPot}) includes |
| 198 |
> |
image-charges subtracted out through placement on the cutoff sphere |
| 199 |
> |
and a distance-dependent damping function (identical to that seen in |
| 200 |
> |
the real-space portion of the Ewald sum) to aid convergence |
| 201 |
|
\begin{equation} |
| 202 |
|
V_{\textrm{Wolf}}(r_{ij})= \frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}. |
| 203 |
|
\label{eq:WolfPot} |
| 572 |
|
when using the reference method (SPME). |
| 573 |
|
|
| 574 |
|
\subsection{Short-time Dynamics} |
| 575 |
< |
|
| 576 |
< |
\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods} |
| 542 |
< |
Evaluation of the long-time dynamics of charged systems was performed |
| 543 |
< |
by considering the NaCl crystal system while using a subset of the |
| 575 |
> |
Evaluation of the short-time dynamics of charged systems was performed |
| 576 |
> |
by considering the 1000 K NaCl crystal system while using a subset of the |
| 577 |
|
best performing pairwise methods. The NaCl crystal was chosen to |
| 578 |
|
avoid possible complications involving the propagation techniques of |
| 579 |
< |
orientational motion in molecular systems. To enhance the atomic |
| 580 |
< |
motion, these crystals were equilibrated at 1000 K, near the |
| 581 |
< |
experimental $T_m$ for NaCl. Simulations were performed under the |
| 582 |
< |
microcanonical ensemble, and velocity autocorrelation functions |
| 583 |
< |
(Eq. \ref{eq:vCorr}) were computed for each of the trajectories, |
| 579 |
> |
orientational motion in molecular systems. All systems were started |
| 580 |
> |
with the same initial positions and velocities. Simulations were |
| 581 |
> |
performed under the microcanonical ensemble, and velocity |
| 582 |
> |
autocorrelation functions (Eq. \ref{eq:vCorr}) were computed for each |
| 583 |
> |
of the trajectories, |
| 584 |
|
\begin{equation} |
| 585 |
< |
C_v(t) = \langle v_i(0)\cdot v_i(t)\rangle. |
| 585 |
> |
C_v(t) = \frac{\langle v_i(0)\cdot v_i(t)\rangle}{\langle v_i(0)\cdot v_i(0)\rangle}. |
| 586 |
|
\label{eq:vCorr} |
| 587 |
|
\end{equation} |
| 588 |
< |
Velocity autocorrelation functions require detailed short time data |
| 589 |
< |
and long trajectories for good statistics, thus velocity information |
| 590 |
< |
was saved every 5 fs over 100 ps trajectories. The power spectrum |
| 591 |
< |
($I(\omega)$) is obtained via Fourier transform of the autocorrelation |
| 592 |
< |
function |
| 588 |
> |
Velocity autocorrelation functions require detailed short time data, |
| 589 |
> |
thus velocity information was saved every 2 fs over 10 ps |
| 590 |
> |
trajectories. Because the NaCl crystal is composed of two different |
| 591 |
> |
atom types, the average of the two resulting velocity autocorrelation |
| 592 |
> |
functions was used for comparisons. |
| 593 |
> |
|
| 594 |
> |
\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods} |
| 595 |
> |
Evaluation of the long-time dynamics of charged systems was performed |
| 596 |
> |
by considering the NaCl crystal system, again while using a subset of |
| 597 |
> |
the best performing pairwise methods. To enhance the atomic motion, |
| 598 |
> |
these crystals were equilibrated at 1000 K, near the experimental |
| 599 |
> |
$T_m$ for NaCl. Simulations were performed under the microcanonical |
| 600 |
> |
ensemble, and velocity information was saved every 5 fs over 100 ps |
| 601 |
> |
trajectories. The power spectrum ($I(\omega)$) was obtained via |
| 602 |
> |
Fourier transform of the velocity autocorrelation function |
| 603 |
|
\begin{equation} |
| 604 |
|
I(\omega) = \frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt, |
| 605 |
|
\label{eq:powerSpec} |
| 606 |
|
\end{equation} |
| 607 |
< |
where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. |
| 607 |
> |
where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. Again, because the |
| 608 |
> |
NaCl crystal is composed of two different atom types, the average of |
| 609 |
> |
the two resulting power spectra was used for comparisons. |
| 610 |
|
|
| 611 |
|
\subsection{Representative Simulations}\label{sec:RepSims} |
| 612 |
|
A variety of common and representative simulations were analyzed to |
| 892 |
|
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 893 |
|
unnecessary when using the {\sc sf} method. |
| 894 |
|
|
| 895 |
< |
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 895 |
> |
\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals} |
| 896 |
|
|
| 897 |
|
In the previous studies using a {\sc sf} variant of the damped |
| 898 |
|
Wolf coulomb potential, the structure and dynamics of water were |
| 907 |
|
|
| 908 |
|
\begin{figure} |
| 909 |
|
\centering |
| 910 |
+ |
\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
| 911 |
+ |
\caption{Velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset is a magnification of the first trough. The times to first collision are nearly identical, but the differences can be seen in the peaks and troughs, where the undamped to weakly damped methods are stiffer than the moderately damped and SPME methods.} |
| 912 |
+ |
\label{fig:vCorrPlot} |
| 913 |
+ |
\end{figure} |
| 914 |
+ |
|
| 915 |
+ |
The short-time decays through the first collision are nearly identical |
| 916 |
+ |
in figure \ref{fig:vCorrPlot}, but the peaks and troughs of the |
| 917 |
+ |
functions show how the methods differ. The undamped {\sc sf} method |
| 918 |
+ |
has deeper troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher |
| 919 |
+ |
peaks than any of the other methods. As the damping function is |
| 920 |
+ |
increased, these peaks are smoothed out, and approach the SPME |
| 921 |
+ |
curve. The damping acts as a distance dependent Gaussian screening of |
| 922 |
+ |
the point charges for the pairwise summation methods; thus, the |
| 923 |
+ |
collisions are more elastic in the undamped {\sc sf} potental, and the |
| 924 |
+ |
stiffness of the potential is diminished as the electrostatic |
| 925 |
+ |
interactions are softened by the damping function. With $\alpha$ |
| 926 |
+ |
values of 0.2 \AA$^{-1}$, the {\sc sf} and {\sc sp} functions are |
| 927 |
+ |
nearly identical and track the SPME features quite well. This is not |
| 928 |
+ |
too surprising in that the differences between the {\sc sf} and {\sc |
| 929 |
+ |
sp} potentials are mitigated with increased damping. However, this |
| 930 |
+ |
appears to indicate that once damping is utilized, the form of the |
| 931 |
+ |
potential seems to play a lesser role in the crystal dynamics. |
| 932 |
+ |
|
| 933 |
+ |
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 934 |
+ |
|
| 935 |
+ |
The short time dynamics were extended to evaluate how the differences |
| 936 |
+ |
between the methods affect the collective long-time motion. The same |
| 937 |
+ |
electrostatic summation methods were used as in the short time |
| 938 |
+ |
velocity autocorrelation function evaluation, but the trajectories |
| 939 |
+ |
were sampled over a much longer time. The power spectra of the |
| 940 |
+ |
resulting velocity autocorrelation functions were calculated and are |
| 941 |
+ |
displayed in figure \ref{fig:methodPS}. |
| 942 |
+ |
|
| 943 |
+ |
\begin{figure} |
| 944 |
+ |
\centering |
| 945 |
|
\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
| 946 |
|
\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.} |
| 947 |
|
\label{fig:methodPS} |
| 948 |
|
\end{figure} |
| 949 |
|
|
| 950 |
< |
Figure \ref{fig:methodPS} shows the power spectra for the NaCl |
| 951 |
< |
crystals (from averaged Na and Cl ion velocity autocorrelation |
| 952 |
< |
functions) using the stated electrostatic summation methods. While |
| 953 |
< |
high frequency peaks of all the spectra overlap, showing the same |
| 954 |
< |
general features, the low frequency region shows how the summation |
| 955 |
< |
methods differ. Considering the low-frequency inset (expanded in the |
| 956 |
< |
upper frame of figure \ref{fig:dampInc}), at frequencies below 100 |
| 957 |
< |
cm$^{-1}$, the correlated motions are blue-shifted when using undamped |
| 958 |
< |
or weakly damped {\sc sf}. When using moderate damping ($\alpha |
| 959 |
< |
= 0.2$ \AA$^{-1}$) both the {\sc sf} and {\sc sp} |
| 960 |
< |
methods give near identical correlated motion behavior as the Ewald |
| 961 |
< |
method (which has a damping value of 0.3119). The damping acts as a |
| 962 |
< |
distance dependent Gaussian screening of the point charges for the |
| 963 |
< |
pairwise summation methods. This weakening of the electrostatic |
| 964 |
< |
interaction with distance explains why the long-ranged correlated |
| 885 |
< |
motions are at lower frequencies for the moderately damped methods |
| 886 |
< |
than for undamped or weakly damped methods. To see this effect more |
| 887 |
< |
clearly, we show how damping strength affects a simple real-space |
| 888 |
< |
electrostatic potential, |
| 950 |
> |
While high frequency peaks of the spectra in this figure overlap, |
| 951 |
> |
showing the same general features, the low frequency region shows how |
| 952 |
> |
the summation methods differ. Considering the low-frequency inset |
| 953 |
> |
(expanded in the upper frame of figure \ref{fig:dampInc}), at |
| 954 |
> |
frequencies below 100 cm$^{-1}$, the correlated motions are |
| 955 |
> |
blue-shifted when using undamped or weakly damped {\sc sf}. When |
| 956 |
> |
using moderate damping ($\alpha = 0.2$ \AA$^{-1}$) both the {\sc sf} |
| 957 |
> |
and {\sc sp} methods give near identical correlated motion behavior as |
| 958 |
> |
the Ewald method (which has a damping value of 0.3119). This |
| 959 |
> |
weakening of the electrostatic interaction with increased damping |
| 960 |
> |
explains why the long-ranged correlated motions are at lower |
| 961 |
> |
frequencies for the moderately damped methods than for undamped or |
| 962 |
> |
weakly damped methods. To see this effect more clearly, we show how |
| 963 |
> |
damping strength alone affects a simple real-space electrostatic |
| 964 |
> |
potential, |
| 965 |
|
\begin{equation} |
| 966 |
|
V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r})}{r}\right]S(r), |
| 967 |
|
\end{equation} |
| 976 |
|
shift to higher frequency in exponential fashion. Though not shown, |
| 977 |
|
the spectrum for the simple undamped electrostatic potential is |
| 978 |
|
blue-shifted such that the lowest frequency peak resides near 325 |
| 979 |
< |
cm$^{-1}$. In light of these results, the undamped {\sc sf} |
| 980 |
< |
method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is |
| 981 |
< |
quite respectable; however, it appears as though moderate damping is |
| 982 |
< |
required for accurate reproduction of crystal dynamics. |
| 979 |
> |
cm$^{-1}$. In light of these results, the undamped {\sc sf} method |
| 980 |
> |
producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is quite |
| 981 |
> |
respectable and shows that the shifted force procedure accounts for |
| 982 |
> |
most of the effect afforded through use of the Ewald summation. |
| 983 |
> |
However, it appears as though moderate damping is required for |
| 984 |
> |
accurate reproduction of crystal dynamics. |
| 985 |
|
\begin{figure} |
| 986 |
|
\centering |
| 987 |
|
\includegraphics[width = \linewidth]{./comboSquare.pdf} |
| 1025 |
|
standard by which these simple pairwise sums are judged. However, |
| 1026 |
|
these results do suggest that in the typical simulations performed |
| 1027 |
|
today, the Ewald summation may no longer be required to obtain the |
| 1028 |
< |
level of accuracy most researcher have come to expect |
| 1028 |
> |
level of accuracy most researchers have come to expect |
| 1029 |
|
|
| 1030 |
|
\section{Acknowledgments} |
| 1031 |
|
\newpage |
