| 77 |
|
leading to an effect excluded from the pair interactions within a unit |
| 78 |
|
box. In large systems, excessively large cutoffs need to be used to |
| 79 |
|
accurately incorporate their effect, and since the computational cost |
| 80 |
< |
increases proportionally with the cutoff sphere, it quickly becomes an |
| 81 |
< |
impractical task to perform these calculations. |
| 80 |
> |
increases proportionally with the cutoff sphere, it quickly becomes |
| 81 |
> |
very time-consuming to perform these calculations. |
| 82 |
|
|
| 83 |
+ |
There have been many efforts to address this issue of both proper and |
| 84 |
+ |
practical handling of electrostatic interactions, and these have |
| 85 |
+ |
resulted in the availability of a variety of |
| 86 |
+ |
techniques.\cite{Roux99,Sagui99,Tobias01} These are typically |
| 87 |
+ |
classified as implicit methods (i.e., continuum dielectrics, static |
| 88 |
+ |
dipolar fields),\cite{Born20,Grossfield00} explicit methods (i.e., |
| 89 |
+ |
Ewald summations, interaction shifting or |
| 90 |
+ |
trucation),\cite{Ewald21,Steinbach94} or a mixture of the two (i.e., |
| 91 |
+ |
reaction field type methods, fast multipole |
| 92 |
+ |
methods).\cite{Onsager36,Rokhlin85} The explicit or mixed methods are |
| 93 |
+ |
often preferred because they incorporate dynamic solvent molecules in |
| 94 |
+ |
the system of interest, but these methods are sometimes difficult to |
| 95 |
+ |
utilize because of their high computational cost.\cite{Roux99} In |
| 96 |
+ |
addition to this cost, there has been some question of the inherent |
| 97 |
+ |
periodicity of the explicit Ewald summation artificially influencing |
| 98 |
+ |
systems dynamics.\cite{Tobias01} |
| 99 |
+ |
|
| 100 |
+ |
In this paper, we focus on the common mixed and explicit methods of |
| 101 |
+ |
reaction filed and smooth particle mesh |
| 102 |
+ |
Ewald\cite{Onsager36,Essmann99} and a new set of shifted methods |
| 103 |
+ |
devised by Wolf {\it et al.} which we further extend.\cite{Wolf99} |
| 104 |
+ |
These new methods for handling electrostatics are quite |
| 105 |
+ |
computationally efficient, since they involve only a simple |
| 106 |
+ |
modification to the direct pairwise sum, and they lack the added |
| 107 |
+ |
periodicity of the Ewald sum. Below, these methods are evaluated using |
| 108 |
+ |
a variety of model systems and comparison methodologies to establish |
| 109 |
+ |
their useability in molecular simulations. |
| 110 |
+ |
|
| 111 |
|
\subsection{The Ewald Sum} |
| 112 |
< |
The complete accumulation electrostatic interactions in a system with periodic boundary conditions (PBC) requires the consideration of the effect of all charges within a simulation box, as well as those in the periodic replicas, |
| 112 |
> |
The complete accumulation electrostatic interactions in a system with |
| 113 |
> |
periodic boundary conditions (PBC) requires the consideration of the |
| 114 |
> |
effect of all charges within a simulation box, as well as those in the |
| 115 |
> |
periodic replicas, |
| 116 |
|
\begin{equation} |
| 117 |
|
V_\textrm{elec} = \frac{1}{2} {\sum_{\mathbf{n}}}^\prime \left[ \sum_{i=1}^N\sum_{j=1}^N \phi\left( \mathbf{r}_{ij} + L\mathbf{n},\bm{\Omega}_i,\bm{\Omega}_j\right) \right], |
| 118 |
|
\label{eq:PBCSum} |
| 119 |
|
\end{equation} |
| 120 |
< |
where the sum over $\mathbf{n}$ is a sum over all periodic box replicas |
| 121 |
< |
with integer coordinates $\mathbf{n} = (l,m,n)$, and the prime indicates |
| 122 |
< |
$i = j$ are neglected for $\mathbf{n} = 0$.\cite{deLeeuw80} Within the |
| 123 |
< |
sum, $N$ is the number of electrostatic particles, $\mathbf{r}_{ij}$ is |
| 124 |
< |
$\mathbf{r}_j - \mathbf{r}_i$, $L$ is the cell length, $\bm{\Omega}_{i,j}$ are |
| 125 |
< |
the Euler angles for $i$ and $j$, and $\phi$ is Poisson's equation |
| 126 |
< |
($\phi(\mathbf{r}_{ij}) = q_i q_j |\mathbf{r}_{ij}|^{-1}$ for charge-charge |
| 127 |
< |
interactions). In the case of monopole electrostatics, |
| 128 |
< |
eq. (\ref{eq:PBCSum}) is conditionally convergent and is discontiuous |
| 129 |
< |
for non-neutral systems. |
| 120 |
> |
where the sum over $\mathbf{n}$ is a sum over all periodic box |
| 121 |
> |
replicas with integer coordinates $\mathbf{n} = (l,m,n)$, and the |
| 122 |
> |
prime indicates $i = j$ are neglected for $\mathbf{n} = |
| 123 |
> |
0$.\cite{deLeeuw80} Within the sum, $N$ is the number of electrostatic |
| 124 |
> |
particles, $\mathbf{r}_{ij}$ is $\mathbf{r}_j - \mathbf{r}_i$, $L$ is |
| 125 |
> |
the cell length, $\bm{\Omega}_{i,j}$ are the Euler angles for $i$ and |
| 126 |
> |
$j$, and $\phi$ is Poisson's equation ($\phi(\mathbf{r}_{ij}) = q_i |
| 127 |
> |
q_j |\mathbf{r}_{ij}|^{-1}$ for charge-charge interactions). In the |
| 128 |
> |
case of monopole electrostatics, eq. (\ref{eq:PBCSum}) is |
| 129 |
> |
conditionally convergent and is discontiuous for non-neutral systems. |
| 130 |
|
|
| 131 |
|
This electrostatic summation problem was originally studied by Ewald |
| 132 |
|
for the case of an infinite crystal.\cite{Ewald21}. The approach he |
| 573 |
|
investigated through measurement of the angle ($\theta$) formed |
| 574 |
|
between those computed from the particular method and those from SPME, |
| 575 |
|
\begin{equation} |
| 576 |
< |
\theta_f = \cos^{-1} \hat{f}_\textrm{SPME} \cdot \hat{f}_\textrm{Method}, |
| 576 |
> |
\theta_f = \cos^{-1} \left(\hat{f}_\textrm{SPME} \cdot \hat{f}_\textrm{Method}\right), |
| 577 |
|
\end{equation} |
| 578 |
|
where $\hat{f}_\textrm{M}$ is the unit vector pointing along the |
| 579 |
|
force vector computed using method $M$. |
| 603 |
|
when using the reference method (SPME). |
| 604 |
|
|
| 605 |
|
\subsection{Short-time Dynamics} |
| 606 |
< |
|
| 607 |
< |
\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods} |
| 577 |
< |
Evaluation of the long-time dynamics of charged systems was performed |
| 578 |
< |
by considering the NaCl crystal system while using a subset of the |
| 606 |
> |
Evaluation of the short-time dynamics of charged systems was performed |
| 607 |
> |
by considering the 1000 K NaCl crystal system while using a subset of the |
| 608 |
|
best performing pairwise methods. The NaCl crystal was chosen to |
| 609 |
|
avoid possible complications involving the propagation techniques of |
| 610 |
< |
orientational motion in molecular systems. To enhance the atomic |
| 611 |
< |
motion, these crystals were equilibrated at 1000 K, near the |
| 612 |
< |
experimental $T_m$ for NaCl. Simulations were performed under the |
| 613 |
< |
microcanonical ensemble, and velocity autocorrelation functions |
| 614 |
< |
(Eq. \ref{eq:vCorr}) were computed for each of the trajectories, |
| 610 |
> |
orientational motion in molecular systems. All systems were started |
| 611 |
> |
with the same initial positions and velocities. Simulations were |
| 612 |
> |
performed under the microcanonical ensemble, and velocity |
| 613 |
> |
autocorrelation functions (Eq. \ref{eq:vCorr}) were computed for each |
| 614 |
> |
of the trajectories, |
| 615 |
|
\begin{equation} |
| 616 |
< |
C_v(t) = \langle v_i(0)\cdot v_i(t)\rangle. |
| 616 |
> |
C_v(t) = \frac{\langle v_i(0)\cdot v_i(t)\rangle}{\langle v_i(0)\cdot v_i(0)\rangle}. |
| 617 |
|
\label{eq:vCorr} |
| 618 |
|
\end{equation} |
| 619 |
< |
Velocity autocorrelation functions require detailed short time data |
| 620 |
< |
and long trajectories for good statistics, thus velocity information |
| 621 |
< |
was saved every 5 fs over 100 ps trajectories. The power spectrum |
| 622 |
< |
($I(\omega)$) is obtained via Fourier transform of the autocorrelation |
| 623 |
< |
function |
| 619 |
> |
Velocity autocorrelation functions require detailed short time data, |
| 620 |
> |
thus velocity information was saved every 2 fs over 10 ps |
| 621 |
> |
trajectories. Because the NaCl crystal is composed of two different |
| 622 |
> |
atom types, the average of the two resulting velocity autocorrelation |
| 623 |
> |
functions was used for comparisons. |
| 624 |
> |
|
| 625 |
> |
\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods} |
| 626 |
> |
Evaluation of the long-time dynamics of charged systems was performed |
| 627 |
> |
by considering the NaCl crystal system, again while using a subset of |
| 628 |
> |
the best performing pairwise methods. To enhance the atomic motion, |
| 629 |
> |
these crystals were equilibrated at 1000 K, near the experimental |
| 630 |
> |
$T_m$ for NaCl. Simulations were performed under the microcanonical |
| 631 |
> |
ensemble, and velocity information was saved every 5 fs over 100 ps |
| 632 |
> |
trajectories. The power spectrum ($I(\omega)$) was obtained via |
| 633 |
> |
Fourier transform of the velocity autocorrelation function |
| 634 |
|
\begin{equation} |
| 635 |
|
I(\omega) = \frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt, |
| 636 |
|
\label{eq:powerSpec} |
| 637 |
|
\end{equation} |
| 638 |
< |
where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. |
| 638 |
> |
where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. Again, because the |
| 639 |
> |
NaCl crystal is composed of two different atom types, the average of |
| 640 |
> |
the two resulting power spectra was used for comparisons. |
| 641 |
|
|
| 642 |
|
\subsection{Representative Simulations}\label{sec:RepSims} |
| 643 |
|
A variety of common and representative simulations were analyzed to |
| 923 |
|
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 924 |
|
unnecessary when using the {\sc sf} method. |
| 925 |
|
|
| 926 |
< |
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 926 |
> |
\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals} |
| 927 |
|
|
| 928 |
|
In the previous studies using a {\sc sf} variant of the damped |
| 929 |
|
Wolf coulomb potential, the structure and dynamics of water were |
| 938 |
|
|
| 939 |
|
\begin{figure} |
| 940 |
|
\centering |
| 941 |
+ |
\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
| 942 |
+ |
\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.} |
| 943 |
+ |
\label{fig:vCorrPlot} |
| 944 |
+ |
\end{figure} |
| 945 |
+ |
|
| 946 |
+ |
The short-time decays through the first collision are nearly identical |
| 947 |
+ |
in figure \ref{fig:vCorrPlot}, but the peaks and troughs of the |
| 948 |
+ |
functions show how the methods differ. The undamped {\sc sf} method |
| 949 |
+ |
has deeper troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher |
| 950 |
+ |
peaks than any of the other methods. As the damping function is |
| 951 |
+ |
increased, these peaks are smoothed out, and approach the SPME |
| 952 |
+ |
curve. The damping acts as a distance dependent Gaussian screening of |
| 953 |
+ |
the point charges for the pairwise summation methods; thus, the |
| 954 |
+ |
collisions are more elastic in the undamped {\sc sf} potental, and the |
| 955 |
+ |
stiffness of the potential is diminished as the electrostatic |
| 956 |
+ |
interactions are softened by the damping function. With $\alpha$ |
| 957 |
+ |
values of 0.2 \AA$^{-1}$, the {\sc sf} and {\sc sp} functions are |
| 958 |
+ |
nearly identical and track the SPME features quite well. This is not |
| 959 |
+ |
too surprising in that the differences between the {\sc sf} and {\sc |
| 960 |
+ |
sp} potentials are mitigated with increased damping. However, this |
| 961 |
+ |
appears to indicate that once damping is utilized, the form of the |
| 962 |
+ |
potential seems to play a lesser role in the crystal dynamics. |
| 963 |
+ |
|
| 964 |
+ |
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 965 |
+ |
|
| 966 |
+ |
The short time dynamics were extended to evaluate how the differences |
| 967 |
+ |
between the methods affect the collective long-time motion. The same |
| 968 |
+ |
electrostatic summation methods were used as in the short time |
| 969 |
+ |
velocity autocorrelation function evaluation, but the trajectories |
| 970 |
+ |
were sampled over a much longer time. The power spectra of the |
| 971 |
+ |
resulting velocity autocorrelation functions were calculated and are |
| 972 |
+ |
displayed in figure \ref{fig:methodPS}. |
| 973 |
+ |
|
| 974 |
+ |
\begin{figure} |
| 975 |
+ |
\centering |
| 976 |
|
\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
| 977 |
|
\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.} |
| 978 |
|
\label{fig:methodPS} |
| 979 |
|
\end{figure} |
| 980 |
|
|
| 981 |
< |
Figure \ref{fig:methodPS} shows the power spectra for the NaCl |
| 982 |
< |
crystals (from averaged Na and Cl ion velocity autocorrelation |
| 983 |
< |
functions) using the stated electrostatic summation methods. While |
| 984 |
< |
high frequency peaks of all the spectra overlap, showing the same |
| 985 |
< |
general features, the low frequency region shows how the summation |
| 986 |
< |
methods differ. Considering the low-frequency inset (expanded in the |
| 987 |
< |
upper frame of figure \ref{fig:dampInc}), at frequencies below 100 |
| 988 |
< |
cm$^{-1}$, the correlated motions are blue-shifted when using undamped |
| 989 |
< |
or weakly damped {\sc sf}. When using moderate damping ($\alpha |
| 990 |
< |
= 0.2$ \AA$^{-1}$) both the {\sc sf} and {\sc sp} |
| 991 |
< |
methods give near identical correlated motion behavior as the Ewald |
| 992 |
< |
method (which has a damping value of 0.3119). The damping acts as a |
| 993 |
< |
distance dependent Gaussian screening of the point charges for the |
| 994 |
< |
pairwise summation methods. This weakening of the electrostatic |
| 995 |
< |
interaction with distance explains why the long-ranged correlated |
| 920 |
< |
motions are at lower frequencies for the moderately damped methods |
| 921 |
< |
than for undamped or weakly damped methods. To see this effect more |
| 922 |
< |
clearly, we show how damping strength affects a simple real-space |
| 923 |
< |
electrostatic potential, |
| 981 |
> |
While high frequency peaks of the spectra in this figure overlap, |
| 982 |
> |
showing the same general features, the low frequency region shows how |
| 983 |
> |
the summation methods differ. Considering the low-frequency inset |
| 984 |
> |
(expanded in the upper frame of figure \ref{fig:dampInc}), at |
| 985 |
> |
frequencies below 100 cm$^{-1}$, the correlated motions are |
| 986 |
> |
blue-shifted when using undamped or weakly damped {\sc sf}. When |
| 987 |
> |
using moderate damping ($\alpha = 0.2$ \AA$^{-1}$) both the {\sc sf} |
| 988 |
> |
and {\sc sp} methods give near identical correlated motion behavior as |
| 989 |
> |
the Ewald method (which has a damping value of 0.3119). This |
| 990 |
> |
weakening of the electrostatic interaction with increased damping |
| 991 |
> |
explains why the long-ranged correlated motions are at lower |
| 992 |
> |
frequencies for the moderately damped methods than for undamped or |
| 993 |
> |
weakly damped methods. To see this effect more clearly, we show how |
| 994 |
> |
damping strength alone affects a simple real-space electrostatic |
| 995 |
> |
potential, |
| 996 |
|
\begin{equation} |
| 997 |
|
V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r})}{r}\right]S(r), |
| 998 |
|
\end{equation} |
| 1007 |
|
shift to higher frequency in exponential fashion. Though not shown, |
| 1008 |
|
the spectrum for the simple undamped electrostatic potential is |
| 1009 |
|
blue-shifted such that the lowest frequency peak resides near 325 |
| 1010 |
< |
cm$^{-1}$. In light of these results, the undamped {\sc sf} |
| 1011 |
< |
method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is |
| 1012 |
< |
quite respectable; however, it appears as though moderate damping is |
| 1013 |
< |
required for accurate reproduction of crystal dynamics. |
| 1010 |
> |
cm$^{-1}$. In light of these results, the undamped {\sc sf} method |
| 1011 |
> |
producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is quite |
| 1012 |
> |
respectable and shows that the shifted force procedure accounts for |
| 1013 |
> |
most of the effect afforded through use of the Ewald summation. |
| 1014 |
> |
However, it appears as though moderate damping is required for |
| 1015 |
> |
accurate reproduction of crystal dynamics. |
| 1016 |
|
\begin{figure} |
| 1017 |
|
\centering |
| 1018 |
|
\includegraphics[width = \linewidth]{./comboSquare.pdf} |
| 1056 |
|
standard by which these simple pairwise sums are judged. However, |
| 1057 |
|
these results do suggest that in the typical simulations performed |
| 1058 |
|
today, the Ewald summation may no longer be required to obtain the |
| 1059 |
< |
level of accuracy most researcher have come to expect |
| 1059 |
> |
level of accuracy most researchers have come to expect |
| 1060 |
|
|
| 1061 |
|
\section{Acknowledgments} |
| 1062 |
|
\newpage |
