| 2 |
|
%\documentclass[aps,prb,preprint]{revtex4} |
| 3 |
|
\documentclass[11pt]{article} |
| 4 |
|
\usepackage{endfloat} |
| 5 |
< |
\usepackage{amsmath} |
| 5 |
> |
\usepackage{amsmath,bm} |
| 6 |
|
\usepackage{amssymb} |
| 7 |
|
\usepackage{epsf} |
| 8 |
|
\usepackage{times} |
| 81 |
|
impractical task to perform these calculations. |
| 82 |
|
|
| 83 |
|
\subsection{The Ewald Sum} |
| 84 |
< |
blah blah blah Ewald Sum Important blah blah blah |
| 84 |
> |
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, |
| 85 |
> |
\begin{equation} |
| 86 |
> |
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], |
| 87 |
> |
\label{eq:PBCSum} |
| 88 |
> |
\end{equation} |
| 89 |
> |
where the sum over $\mathbf{n}$ is a sum over all periodic box replicas |
| 90 |
> |
with integer coordinates $\mathbf{n} = (l,m,n)$, and the prime indicates |
| 91 |
> |
$i = j$ are neglected for $\mathbf{n} = 0$.\cite{deLeeuw80} Within the |
| 92 |
> |
sum, $N$ is the number of electrostatic particles, $\mathbf{r}_{ij}$ is |
| 93 |
> |
$\mathbf{r}_j - \mathbf{r}_i$, $L$ is the cell length, $\bm{\Omega}_{i,j}$ are |
| 94 |
> |
the Euler angles for $i$ and $j$, and $\phi$ is Poisson's equation |
| 95 |
> |
($\phi(\mathbf{r}_{ij}) = q_i q_j |\mathbf{r}_{ij}|^{-1}$ for charge-charge |
| 96 |
> |
interactions). In the case of monopole electrostatics, |
| 97 |
> |
eq. (\ref{eq:PBCSum}) is conditionally convergent and is discontiuous |
| 98 |
> |
for non-neutral systems. |
| 99 |
|
|
| 100 |
+ |
This electrostatic summation problem was originally studied by Ewald |
| 101 |
+ |
for the case of an infinite crystal.\cite{Ewald21}. The approach he |
| 102 |
+ |
took was to convert this conditionally convergent sum into two |
| 103 |
+ |
absolutely convergent summations: a short-ranged real-space summation |
| 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}{(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$^{-1}$, $\mathbf{k}$ 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,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 |
| 127 |
+ |
spherical real-space cutoff with the reciprocal sum and consider the |
| 128 |
+ |
surrounding dielectric. |
| 129 |
|
\begin{figure} |
| 130 |
|
\centering |
| 131 |
|
\includegraphics[width = \linewidth]{./ewaldProgression.pdf} |
| 139 |
|
\label{fig:ewaldTime} |
| 140 |
|
\end{figure} |
| 141 |
|
|
| 142 |
+ |
The Ewald summation in the straight-forward form is an |
| 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 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 |
+ |
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 an 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 |
| 189 |
> |
for the accurate accumulation of electrostatic interactions in an |
| 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\}. |
| 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} |
| 204 |
|
\end{equation} |
| 205 |
|
Eq. (\ref{eq:WolfPot}) is essentially the common form of a shifted |
| 206 |
|
potential. However, neutralizing the charge contained within each |
| 207 |
|
cutoff sphere requires the placement of a self-image charge on the |
| 208 |
|
surface of the cutoff sphere. This additional self-term in the total |
| 209 |
< |
potential enables Wolf {\it et al.} to obtain excellent estimates of |
| 209 |
> |
potential enabled Wolf {\it et al.} to obtain excellent estimates of |
| 210 |
|
Madelung energies for many crystals. |
| 211 |
|
|
| 212 |
|
In order to use their charge-neutralized potential in molecular |
| 214 |
|
derivative of this potential prior to evaluation of the limit. This |
| 215 |
|
procedure gives an expression for the forces, |
| 216 |
|
\begin{equation} |
| 217 |
< |
F^{\textrm{Wolf}}(r_{ij}) = q_i q_j\left\{-\left[\frac{\textrm{erfc}(\alpha r_{ij})}{r^2_{ij}}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{(-\alpha^2r_{ij}^2)}}{r_{ij}}\right]+\left[\frac{\textrm{erfc}\left(\alpha R_{\textrm{c}}\right)}{R_{\textrm{c}}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}\right]\right\}, |
| 217 |
> |
F_{\textrm{Wolf}}(r_{ij}) = q_i q_j\left\{\left[\frac{\textrm{erfc}\left(\alpha r_{ij}\right)}{r^2_{ij}}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r_{ij}^2\right)}}{r_{ij}}\right]-\left[\frac{\textrm{erfc}\left(\alpha R_{\textrm{c}}\right)}{R_{\textrm{c}}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}\right]\right\}, |
| 218 |
|
\label{eq:WolfForces} |
| 219 |
|
\end{equation} |
| 220 |
|
that incorporates both image charges and damping of the electrostatic |
| 222 |
|
|
| 223 |
|
More recently, Zahn \textit{et al.} investigated these potential and |
| 224 |
|
force expressions for use in simulations involving water.\cite{Zahn02} |
| 225 |
< |
In their work, they pointed out that the method that the forces and |
| 226 |
< |
derivative of the potential are not commensurate. Attempts to use |
| 227 |
< |
both Eqs. (\ref{eq:WolfPot}) and (\ref{eq:WolfForces}) together will |
| 228 |
< |
lead to poor energy conservation. They correctly observed that taking |
| 229 |
< |
the limit shown in equation (\ref{eq:WolfPot}) {\it after} calculating |
| 230 |
< |
the derivatives is mathematically invalid. |
| 225 |
> |
In their work, they pointed out that the forces and derivative of |
| 226 |
> |
the potential are not commensurate. Attempts to use both |
| 227 |
> |
Eqs. (\ref{eq:WolfPot}) and (\ref{eq:WolfForces}) together will lead |
| 228 |
> |
to poor energy conservation. They correctly observed that taking the |
| 229 |
> |
limit shown in equation (\ref{eq:WolfPot}) {\it after} calculating the |
| 230 |
> |
derivatives gives forces for a different potential energy function |
| 231 |
> |
than the one shown in Eq. (\ref{eq:WolfPot}). |
| 232 |
|
|
| 233 |
|
Zahn \textit{et al.} proposed a modified form of this ``Wolf summation |
| 234 |
|
method'' as a way to use this technique in Molecular Dynamics |
| 236 |
|
\ref{eq:WolfForces}, they proposed a new damped Coulomb |
| 237 |
|
potential, |
| 238 |
|
\begin{equation} |
| 239 |
< |
V^{\textrm{Zahn}}(r_{ij}) = q_iq_j\left\{\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}}-\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_{ij}-R_\mathrm{c}\right)\right\}. |
| 239 |
> |
V_{\textrm{Zahn}}(r_{ij}) = q_iq_j\left\{\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}}-\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_{ij}-R_\mathrm{c}\right)\right\}. |
| 240 |
|
\label{eq:ZahnPot} |
| 241 |
|
\end{equation} |
| 242 |
|
They showed that this potential does fairly well at capturing the |
| 247 |
|
|
| 248 |
|
The potentials proposed by Wolf \textit{et al.} and Zahn \textit{et |
| 249 |
|
al.} are constructed using two different (and separable) computational |
| 250 |
< |
tricks: \begin{itemize} |
| 250 |
> |
tricks: \begin{enumerate} |
| 251 |
|
\item shifting through the use of image charges, and |
| 252 |
|
\item damping the electrostatic interaction. |
| 253 |
< |
\end{itemize} Wolf \textit{et al.} treated the |
| 253 |
> |
\end{enumerate} Wolf \textit{et al.} treated the |
| 254 |
|
development of their summation method as a progressive application of |
| 255 |
|
these techniques,\cite{Wolf99} while Zahn \textit{et al.} founded |
| 256 |
|
their damped Coulomb modification (Eq. (\ref{eq:ZahnPot})) on the |
| 270 |
|
\textit{et al.} and Zahn \textit{et al.} by considering the standard |
| 271 |
|
shifted potential, |
| 272 |
|
\begin{equation} |
| 273 |
< |
v^\textrm{SP}(r) = \begin{cases} |
| 273 |
> |
v_\textrm{SP}(r) = \begin{cases} |
| 274 |
|
v(r)-v_\textrm{c} &\quad r\leqslant R_\textrm{c} \\ 0 &\quad r > |
| 275 |
|
R_\textrm{c} |
| 276 |
|
\end{cases}, |
| 278 |
|
\end{equation} |
| 279 |
|
and shifted force, |
| 280 |
|
\begin{equation} |
| 281 |
< |
v^\textrm{SF}(r) = \begin{cases} |
| 282 |
< |
v(r)-v_\textrm{c}-\left(\frac{\textrm{d}v(r)}{\textrm{d}r}\right)_{r=R_\textrm{c}}(r-R_\textrm{c}) |
| 281 |
> |
v_\textrm{SF}(r) = \begin{cases} |
| 282 |
> |
v(r)-v_\textrm{c}-\left(\frac{d v(r)}{d r}\right)_{r=R_\textrm{c}}(r-R_\textrm{c}) |
| 283 |
|
&\quad r\leqslant R_\textrm{c} \\ 0 &\quad r > R_\textrm{c} |
| 284 |
|
\end{cases}, |
| 285 |
|
\label{eq:shiftingForm} |
| 291 |
|
potential is smooth at the cutoff radius |
| 292 |
|
($R_\textrm{c}$).\cite{Allen87} |
| 293 |
|
|
| 294 |
< |
|
| 295 |
< |
|
| 207 |
< |
|
| 208 |
< |
If the derivative term is taken to be zero, we are left with the shifted Coulomb potential devised by Wolf \textit{et al.},\cite{Wolf99} |
| 294 |
> |
The forces associated with the shifted potential are simply the forces |
| 295 |
> |
of the unshifted potential itself (when inside the cutoff sphere), |
| 296 |
|
\begin{equation} |
| 297 |
< |
V^\textrm{SP}(r_{ij}) = q_iq_j\left(\frac{1}{r_{ij}}-\frac{1}{R_\textrm{c}}\right) \quad r_{ij}\leqslant R_\textrm{c}. \label{eq:WolfSP} |
| 297 |
> |
f_{\textrm{SP}} = -\left( \frac{d v(r)}{dr} \right), |
| 298 |
|
\end{equation} |
| 299 |
< |
The forces associated with this potential are obtained by taking the derivative, resulting in the following, |
| 299 |
> |
and are zero outside. Inside the cutoff sphere, the forces associated |
| 300 |
> |
with the shifted force form can be written, |
| 301 |
|
\begin{equation} |
| 302 |
< |
F^\textrm{SP}(r_{ij}) = q_iq_j\left(-\frac{1}{r_{ij}^2}\right) \quad r_{ij}\leqslant R_\textrm{c}. |
| 303 |
< |
\label{eq:FWolfSP} |
| 302 |
> |
f_{\textrm{SF}} = -\left( \frac{d v(r)}{dr} \right) + \left(\frac{d |
| 303 |
> |
v(r)}{dr} \right)_{r=R_\textrm{c}}. |
| 304 |
|
\end{equation} |
| 305 |
< |
These forces are identical to the forces of the standard electrostatic interaction, and this was addressed by Wolf \textit{et al.} as undesirable. They pointed out that the effect of the image charges is neglected in the forces when they would expect there to be some pressure exerted due to their presence.\cite{Wolf99} As a consequence the forces, though mathematically valid, may not be physically correct due to this missing component. Additionally, there is a discontinuity in the forces. This can be remedied with the use of a switching function to zero the potential and forces smoothly as particles near $R_\textrm{c}$. |
| 305 |
> |
|
| 306 |
> |
If the potential ($v(r)$) is taken to be the normal Coulomb potential, |
| 307 |
> |
\begin{equation} |
| 308 |
> |
v(r) = \frac{q_i q_j}{r}, |
| 309 |
> |
\label{eq:Coulomb} |
| 310 |
> |
\end{equation} |
| 311 |
> |
then the Shifted Potential ({\sc sp}) forms will give Wolf {\it et |
| 312 |
> |
al.}'s undamped prescription: |
| 313 |
> |
\begin{equation} |
| 314 |
> |
v_\textrm{SP}(r) = |
| 315 |
> |
q_iq_j\left(\frac{1}{r}-\frac{1}{R_\textrm{c}}\right) \quad |
| 316 |
> |
r\leqslant R_\textrm{c}, |
| 317 |
> |
\label{eq:SPPot} |
| 318 |
> |
\end{equation} |
| 319 |
> |
with associated forces, |
| 320 |
> |
\begin{equation} |
| 321 |
> |
f_\textrm{SP}(r) = q_iq_j\left(\frac{1}{r^2}\right) \quad r\leqslant R_\textrm{c}. |
| 322 |
> |
\label{eq:SPForces} |
| 323 |
> |
\end{equation} |
| 324 |
> |
These forces are identical to the forces of the standard Coulomb |
| 325 |
> |
interaction, and cutting these off at $R_c$ was addressed by Wolf |
| 326 |
> |
\textit{et al.} as undesirable. They pointed out that the effect of |
| 327 |
> |
the image charges is neglected in the forces when this form is |
| 328 |
> |
used,\cite{Wolf99} thereby eliminating any benefit from the method in |
| 329 |
> |
molecular dynamics. Additionally, there is a discontinuity in the |
| 330 |
> |
forces at the cutoff radius which results in energy drift during MD |
| 331 |
> |
simulations. |
| 332 |
|
|
| 333 |
< |
If the derivative term in equation \ref{eq:shiftingForm} is evaluated, we obtain an hitherto undiscussed shifted force Coulomb potential, |
| 333 |
> |
The shifted force ({\sc sf}) form using the normal Coulomb potential |
| 334 |
> |
will give, |
| 335 |
|
\begin{equation} |
| 336 |
< |
V^\textrm{SF}(r_{ij}) = q_iq_j\left[\frac{1}{r_{ij}}-\frac{1}{R_\textrm{c}}+\left(\frac{1}{R_\textrm{c}^2}\right)(r_{ij}-R_\textrm{c})\right] \quad r_{ij}\leqslant R_\textrm{c}. |
| 336 |
> |
v_\textrm{SF}(r) = q_iq_j\left[\frac{1}{r}-\frac{1}{R_\textrm{c}}+\left(\frac{1}{R_\textrm{c}^2}\right)(r-R_\textrm{c})\right] \quad r\leqslant R_\textrm{c}. |
| 337 |
|
\label{eq:SFPot} |
| 338 |
|
\end{equation} |
| 339 |
< |
Taking the derivative of this shifted force potential gives the |
| 225 |
< |
following forces, |
| 339 |
> |
with associated forces, |
| 340 |
|
\begin{equation} |
| 341 |
< |
F^\textrm{SF}(r_{ij} = q_iq_j\left(-\frac{1}{r_{ij}^2}+\frac{1}{R_\textrm{c}^2}\right) \quad r_{ij}\leqslant R_\textrm{c}. |
| 341 |
> |
f_\textrm{SF}(r) = q_iq_j\left(\frac{1}{r^2}-\frac{1}{R_\textrm{c}^2}\right) \quad r\leqslant R_\textrm{c}. |
| 342 |
|
\label{eq:SFForces} |
| 343 |
|
\end{equation} |
| 344 |
< |
Using this formulation rather than the simple shifted potential |
| 345 |
< |
(Eq. \ref{eq:WolfSP}) means that there are no discontinuities in the |
| 346 |
< |
forces in addition to the potential. This form also has the benefit |
| 347 |
< |
that the image charges have a force presence, addressing the concerns |
| 348 |
< |
about a missing physical component. One side effect of this treatment |
| 349 |
< |
is a slight alteration in the shape of the potential that comes about |
| 350 |
< |
from the derivative term. Thus, a degree of clarity about the |
| 351 |
< |
original formulation of the potential is lost in order to gain |
| 352 |
< |
functionality in dynamics simulations. |
| 344 |
> |
This formulation has the benefits that there are no discontinuities at |
| 345 |
> |
the cutoff distance, while the neutralizing image charges are present |
| 346 |
> |
in both the energy and force expressions. It would be simple to add |
| 347 |
> |
the self-neutralizing term back when computing the total energy of the |
| 348 |
> |
system, thereby maintaining the agreement with the Madelung energies. |
| 349 |
> |
A side effect of this treatment is the alteration in the shape of the |
| 350 |
> |
potential that comes from the derivative term. Thus, a degree of |
| 351 |
> |
clarity about agreement with the empirical potential is lost in order |
| 352 |
> |
to gain functionality in dynamics simulations. |
| 353 |
|
|
| 354 |
|
Wolf \textit{et al.} originally discussed the energetics of the |
| 355 |
< |
shifted Coulomb potential (Eq. \ref{eq:WolfSP}), and they found that |
| 356 |
< |
it was still insufficient for accurate determination of the energy. |
| 357 |
< |
The energy would fluctuate around the expected value with increasing |
| 358 |
< |
cutoff radius, but the oscillations appeared to be converging toward |
| 359 |
< |
the correct value.\cite{Wolf99} A damping function was incorporated to |
| 360 |
< |
accelerate convergence; and though alternative functional forms could |
| 361 |
< |
be used,\cite{Jones56,Heyes81} the complimentary error function was |
| 362 |
< |
chosen to draw parallels to the Ewald summation. Incorporating |
| 363 |
< |
damping into the simple Coulomb potential, |
| 355 |
> |
shifted Coulomb potential (Eq. \ref{eq:SPPot}), and they found that |
| 356 |
> |
it was still insufficient for accurate determination of the energy |
| 357 |
> |
with reasonable cutoff distances. The calculated Madelung energies |
| 358 |
> |
fluctuate around the expected value with increasing cutoff radius, but |
| 359 |
> |
the oscillations converge toward the correct value.\cite{Wolf99} A |
| 360 |
> |
damping function was incorporated to accelerate the convergence; and |
| 361 |
> |
though alternative functional forms could be |
| 362 |
> |
used,\cite{Jones56,Heyes81} the complimentary error function was |
| 363 |
> |
chosen to mirror the effective screening used in the Ewald summation. |
| 364 |
> |
Incorporating this error function damping into the simple Coulomb |
| 365 |
> |
potential, |
| 366 |
|
\begin{equation} |
| 367 |
< |
v(r_{ij}) = \frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}}, |
| 367 |
> |
v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r}, |
| 368 |
|
\label{eq:dampCoulomb} |
| 369 |
|
\end{equation} |
| 370 |
< |
the shifted potential (Eq. \ref{eq:WolfSP}) can be rederived |
| 371 |
< |
\textit{via} equation \ref{eq:shiftingForm}, |
| 370 |
> |
the shifted potential (Eq. (\ref{eq:SPPot})) can be reacquired using |
| 371 |
> |
eq. (\ref{eq:shiftingForm}), |
| 372 |
|
\begin{equation} |
| 373 |
< |
V^{\textrm{DSP}}(r_{ij}) = q_iq_j\left(\frac{\textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\frac{\textrm{erfc}(\alpha R_\textrm{c})}{R_\textrm{c}}\right) \quad r_{ij}\leqslant R_\textrm{c}. |
| 373 |
> |
v_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r}-\frac{\textrm{erfc}\left(\alpha R_\textrm{c}\right)}{R_\textrm{c}}\right) \quad r\leqslant R_\textrm{c}, |
| 374 |
|
\label{eq:DSPPot} |
| 375 |
|
\end{equation} |
| 376 |
< |
The derivative of this Shifted-Potential can be taken to obtain forces |
| 261 |
< |
for use in MD, |
| 376 |
> |
with associated forces, |
| 377 |
|
\begin{equation} |
| 378 |
< |
F^{\textrm{DSP}}(r_{ij}) = q_iq_j\left(-\frac{\textrm{erfc}(\alpha r_{ij})}{r^2_{ij}}-\frac{2\alpha}{\pi^{1/2}}\frac{\exp{(-\alpha^2r_{ij}^2)}}{r_{ij}}\right) \quad r_{ij}\leqslant R_\textrm{c}. |
| 378 |
> |
f_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r^2\right)}}{r}\right) \quad r\leqslant R_\textrm{c}. |
| 379 |
|
\label{eq:DSPForces} |
| 380 |
|
\end{equation} |
| 381 |
< |
Again, this Shifted-Potential suffers from a discontinuity in the |
| 382 |
< |
forces, and a lack of an image-charge component in the forces. To |
| 383 |
< |
remedy these concerns, a Shifted-Force variant is obtained by |
| 384 |
< |
inclusion of the derivative term in equation \ref{eq:shiftingForm} to |
| 270 |
< |
give, |
| 381 |
> |
Again, this damped shifted potential suffers from a discontinuity and |
| 382 |
> |
a lack of the image charges in the forces. To remedy these concerns, |
| 383 |
> |
one may derive a {\sc sf} variant by including the derivative |
| 384 |
> |
term in eq. (\ref{eq:shiftingForm}), |
| 385 |
|
\begin{equation} |
| 386 |
|
\begin{split} |
| 387 |
< |
V^\mathrm{DSF}(r_{ij}) = q_iq_j\Biggr{[}&\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}}-\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_{ij}-R_\mathrm{c}\right)\ \right] \quad r_{ij}\leqslant R_\textrm{c}. |
| 387 |
> |
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}. |
| 388 |
|
\label{eq:DSFPot} |
| 389 |
|
\end{split} |
| 390 |
|
\end{equation} |
| 391 |
< |
The derivative of the above potential gives the following forces, |
| 391 |
> |
The derivative of the above potential will lead to the following forces, |
| 392 |
|
\begin{equation} |
| 393 |
|
\begin{split} |
| 394 |
< |
F^\mathrm{DSF}(r_{ij}) = q_iq_j\Biggr{[}&-\left(\frac{\textrm{erfc}(\alpha r_{ij})}{r^2_{ij}}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{(-\alpha^2r_{ij}^2)}}{r_{ij}}\right) \\ &\left.+\left(\frac{\textrm{erfc}(\alpha R_{\textrm{c}})}{R_{\textrm{c}}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}\right)\right] \quad r_{ij}\leqslant R_\textrm{c}. |
| 394 |
> |
f_\mathrm{DSF}(r) = q_iq_j\Biggr{[}&\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r^2\right)}}{r}\right) \\ &\left.-\left(\frac{\textrm{erfc}\left(\alpha R_{\textrm{c}}\right)}{R_{\textrm{c}}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}\right)\right] \quad r\leqslant R_\textrm{c}. |
| 395 |
|
\label{eq:DSFForces} |
| 396 |
|
\end{split} |
| 397 |
|
\end{equation} |
| 398 |
+ |
If the damping parameter $(\alpha)$ is chosen to be zero, the undamped |
| 399 |
+ |
case, eqs. (\ref{eq:SPPot}-\ref{eq:SFForces}) are correctly recovered |
| 400 |
+ |
from eqs. (\ref{eq:DSPPot}-\ref{eq:DSFForces}). |
| 401 |
|
|
| 402 |
< |
This new Shifted-Force potential is similar to equation |
| 403 |
< |
\ref{eq:ZahnPot} derived by Zahn \textit{et al.}; however, there are |
| 404 |
< |
two important differences.\cite{Zahn02} First, the $v_\textrm{c}$ term |
| 405 |
< |
from equation \ref{eq:shiftingForm} is equal to equation |
| 406 |
< |
\ref{eq:dampCoulomb} with $R_\textrm{c}$ supplied for $r_{ij}$. This |
| 407 |
< |
term is not present in the Zahn potential, resulting in a |
| 408 |
< |
discontinuity as particles cross $R_\textrm{c}$. Second, the sign of |
| 409 |
< |
the derivative portion is different. The constant $v_\textrm{c}$ term |
| 410 |
< |
is not going to have a presence in the forces after performing the |
| 411 |
< |
derivative, but the negative sign does effect the derivative. In |
| 412 |
< |
fact, it introduces a discontinuity in the forces at the cutoff, |
| 413 |
< |
because the force function is shifted in the wrong direction and |
| 414 |
< |
doesn't cross zero at $R_\textrm{c}$. Thus, these alterations make |
| 415 |
< |
for an electrostatic summation method that is continuous in both the |
| 299 |
< |
potential and forces and incorporates the pairwise sum considerations |
| 300 |
< |
stressed by Wolf \textit{et al.}\cite{Wolf99} |
| 402 |
> |
This new {\sc sf} potential is similar to equation \ref{eq:ZahnPot} |
| 403 |
> |
derived by Zahn \textit{et al.}; however, there are two important |
| 404 |
> |
differences.\cite{Zahn02} First, the $v_\textrm{c}$ term from |
| 405 |
> |
eq. (\ref{eq:shiftingForm}) is equal to eq. (\ref{eq:dampCoulomb}) |
| 406 |
> |
with $r$ replaced by $R_\textrm{c}$. This term is {\it not} present |
| 407 |
> |
in the Zahn potential, resulting in a potential discontinuity as |
| 408 |
> |
particles cross $R_\textrm{c}$. Second, the sign of the derivative |
| 409 |
> |
portion is different. The missing $v_\textrm{c}$ term would not |
| 410 |
> |
affect molecular dynamics simulations (although the computed energy |
| 411 |
> |
would be expected to have sudden jumps as particle distances crossed |
| 412 |
> |
$R_c$). The sign problem would be a potential source of errors, |
| 413 |
> |
however. In fact, it introduces a discontinuity in the forces at the |
| 414 |
> |
cutoff, because the force function is shifted in the wrong direction |
| 415 |
> |
and doesn't cross zero at $R_\textrm{c}$. |
| 416 |
|
|
| 417 |
+ |
Eqs. (\ref{eq:DSFPot}) and (\ref{eq:DSFForces}) result in an |
| 418 |
+ |
electrostatic summation method that is continuous in both the |
| 419 |
+ |
potential and forces and which incorporates the damping function |
| 420 |
+ |
proposed by Wolf \textit{et al.}\cite{Wolf99} In the rest of this |
| 421 |
+ |
paper, we will evaluate exactly how good these methods ({\sc sp}, {\sc |
| 422 |
+ |
sf}, damping) are at reproducing the correct electrostatic summation |
| 423 |
+ |
performed by the Ewald sum. |
| 424 |
+ |
|
| 425 |
+ |
\subsection{Other alternatives} |
| 426 |
+ |
In addition to the methods described above, we will consider some |
| 427 |
+ |
other techniques that commonly get used in molecular simulations. The |
| 428 |
+ |
simplest of these is group-based cutoffs. Though of little use for |
| 429 |
+ |
non-neutral molecules, collecting atoms into neutral groups takes |
| 430 |
+ |
advantage of the observation that the electrostatic interactions decay |
| 431 |
+ |
faster than those for monopolar pairs.\cite{Steinbach94} When |
| 432 |
+ |
considering these molecules as groups, an orientational aspect is |
| 433 |
+ |
introduced to the interactions. Consequently, as these molecular |
| 434 |
+ |
particles move through $R_\textrm{c}$, the energy will drift upward |
| 435 |
+ |
due to the anisotropy of the net molecular dipole |
| 436 |
+ |
interactions.\cite{Rahman71} To maintain good energy conservation, |
| 437 |
+ |
both the potential and derivative need to be smoothly switched to zero |
| 438 |
+ |
at $R_\textrm{c}$.\cite{Adams79} This is accomplished using a |
| 439 |
+ |
switching function, |
| 440 |
+ |
\begin{equation} |
| 441 |
+ |
S(r) = \begin{cases} 1 &\quad r\leqslant r_\textrm{sw} \\ |
| 442 |
+ |
\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} \\ |
| 443 |
+ |
0 &\quad r>R_\textrm{c} |
| 444 |
+ |
\end{cases}, |
| 445 |
+ |
\end{equation} |
| 446 |
+ |
where the above form is for a cubic function. If a smooth second |
| 447 |
+ |
derivative is desired, a fifth (or higher) order polynomial can be |
| 448 |
+ |
used.\cite{Andrea83} |
| 449 |
+ |
|
| 450 |
+ |
Group-based cutoffs neglect the surroundings beyond $R_\textrm{c}$, |
| 451 |
+ |
and to incorporate their effect, a method like Reaction Field ({\sc |
| 452 |
+ |
rf}) can be used. The original theory for {\sc rf} was originally |
| 453 |
+ |
developed by Onsager,\cite{Onsager36} and it was applied in |
| 454 |
+ |
simulations for the study of water by Barker and Watts.\cite{Barker73} |
| 455 |
+ |
In application, it is simply an extension of the group-based cutoff |
| 456 |
+ |
method where the net dipole within the cutoff sphere polarizes an |
| 457 |
+ |
external dielectric, which reacts back on the central dipole. The |
| 458 |
+ |
same switching function considerations for group-based cutoffs need to |
| 459 |
+ |
made for {\sc rf}, with the additional pre-specification of a |
| 460 |
+ |
dielectric constant. |
| 461 |
+ |
|
| 462 |
|
\section{Methods} |
| 463 |
|
|
| 304 |
– |
\subsection{What Qualities are Important?}\label{sec:Qualities} |
| 464 |
|
In classical molecular mechanics simulations, there are two primary |
| 465 |
|
techniques utilized to obtain information about the system of |
| 466 |
|
interest: Monte Carlo (MC) and Molecular Dynamics (MD). Both of these |
| 469 |
|
|
| 470 |
|
In MC, the potential energy difference between two subsequent |
| 471 |
|
configurations dictates the progression of MC sampling. Going back to |
| 472 |
< |
the origins of this method, the Canonical ensemble acceptance criteria |
| 473 |
< |
laid out by Metropolis \textit{et al.} states that a subsequent |
| 474 |
< |
configuration is accepted if $\Delta E < 0$ or if $\xi < \exp(-\Delta |
| 475 |
< |
E/kT)$, where $\xi$ is a random number between 0 and |
| 476 |
< |
1.\cite{Metropolis53} Maintaining a consistent $\Delta E$ when using |
| 477 |
< |
an alternate method for handling the long-range electrostatics ensures |
| 478 |
< |
proper sampling within the ensemble. |
| 472 |
> |
the origins of this method, the acceptance criterion for the canonical |
| 473 |
> |
ensemble laid out by Metropolis \textit{et al.} states that a |
| 474 |
> |
subsequent configuration is accepted if $\Delta E < 0$ or if $\xi < |
| 475 |
> |
\exp(-\Delta E/kT)$, where $\xi$ is a random number between 0 and |
| 476 |
> |
1.\cite{Metropolis53} Maintaining the correct $\Delta E$ when using an |
| 477 |
> |
alternate method for handling the long-range electrostatics will |
| 478 |
> |
ensure proper sampling from the ensemble. |
| 479 |
|
|
| 480 |
< |
In MD, the derivative of the potential directs how the system will |
| 480 |
> |
In MD, the derivative of the potential governs how the system will |
| 481 |
|
progress in time. Consequently, the force and torque vectors on each |
| 482 |
< |
body in the system dictate how it develops as a whole. If the |
| 483 |
< |
magnitude and direction of these vectors are similar when using |
| 484 |
< |
alternate electrostatic summation techniques, the dynamics in the near |
| 485 |
< |
term will be indistinguishable. Because error in MD calculations is |
| 486 |
< |
cumulative, one should expect greater deviation in the long term |
| 487 |
< |
trajectories with greater differences in these vectors between |
| 488 |
< |
configurations using different long-range electrostatics. |
| 482 |
> |
body in the system dictate how the system evolves. If the magnitude |
| 483 |
> |
and direction of these vectors are similar when using alternate |
| 484 |
> |
electrostatic summation techniques, the dynamics in the short term |
| 485 |
> |
will be indistinguishable. Because error in MD calculations is |
| 486 |
> |
cumulative, one should expect greater deviation at longer times, |
| 487 |
> |
although methods which have large differences in the force and torque |
| 488 |
> |
vectors will diverge from each other more rapidly. |
| 489 |
|
|
| 490 |
|
\subsection{Monte Carlo and the Energy Gap}\label{sec:MCMethods} |
| 491 |
< |
Evaluation of the pairwise summation techniques (outlined in section |
| 492 |
< |
\ref{sec:ESMethods}) for use in MC simulations was performed through |
| 493 |
< |
study of the energy differences between conformations. Considering |
| 494 |
< |
the SPME results to be the correct or desired behavior, ideal |
| 495 |
< |
performance of a tested method was taken to be agreement between the |
| 496 |
< |
energy differences calculated. Linear least squares regression of the |
| 497 |
< |
$\Delta E$ values between configurations using SPME against $\Delta E$ |
| 498 |
< |
values using tested methods provides a quantitative comparison of this |
| 499 |
< |
agreement. Unitary results for both the correlation and correlation |
| 500 |
< |
coefficient for these regressions indicate equivalent energetic |
| 501 |
< |
results between the methods. The correlation is the slope of the |
| 502 |
< |
plotted data while the correlation coefficient ($R^2$) is a measure of |
| 503 |
< |
the of the data scatter around the fitted line and tells about the |
| 504 |
< |
quality of the fit (Fig. \ref{fig:linearFit}). |
| 491 |
> |
The pairwise summation techniques (outlined in section |
| 492 |
> |
\ref{sec:ESMethods}) were evaluated for use in MC simulations by |
| 493 |
> |
studying the energy differences between conformations. We took the |
| 494 |
> |
SPME-computed energy difference between two conformations to be the |
| 495 |
> |
correct behavior. An ideal performance by an alternative method would |
| 496 |
> |
reproduce these energy differences exactly. Since none of the methods |
| 497 |
> |
provide exact energy differences, we used linear least squares |
| 498 |
> |
regressions of the $\Delta E$ values between configurations using SPME |
| 499 |
> |
against $\Delta E$ values using tested methods provides a quantitative |
| 500 |
> |
comparison of this agreement. Unitary results for both the |
| 501 |
> |
correlation and correlation coefficient for these regressions indicate |
| 502 |
> |
equivalent energetic results between the method under consideration |
| 503 |
> |
and electrostatics handled using SPME. Sample correlation plots for |
| 504 |
> |
two alternate methods are shown in Fig. \ref{fig:linearFit}. |
| 505 |
|
|
| 506 |
|
\begin{figure} |
| 507 |
|
\centering |
| 510 |
|
\label{fig:linearFit} |
| 511 |
|
\end{figure} |
| 512 |
|
|
| 513 |
< |
Each system type (detailed in section \ref{sec:RepSims}) studied |
| 514 |
< |
consisted of 500 independent configurations, each equilibrated from |
| 515 |
< |
higher temperature trajectories. Thus, 124,750 $\Delta E$ data points |
| 516 |
< |
are used in a regression of a single system type. Results and |
| 517 |
< |
discussion for the individual analysis of each of the system types |
| 518 |
< |
appear in the supporting information, while the cumulative results |
| 519 |
< |
over all the investigated systems appears below in section |
| 520 |
< |
\ref{sec:EnergyResults}. |
| 513 |
> |
Each system type (detailed in section \ref{sec:RepSims}) was |
| 514 |
> |
represented using 500 independent configurations. Additionally, we |
| 515 |
> |
used seven different system types, so each of the alternate |
| 516 |
> |
(non-Ewald) electrostatic summation methods was evaluated using |
| 517 |
> |
873,250 configurational energy differences. |
| 518 |
> |
|
| 519 |
> |
Results and discussion for the individual analysis of each of the |
| 520 |
> |
system types appear in the supporting information, while the |
| 521 |
> |
cumulative results over all the investigated systems appears below in |
| 522 |
> |
section \ref{sec:EnergyResults}. |
| 523 |
|
|
| 524 |
|
\subsection{Molecular Dynamics and the Force and Torque Vectors}\label{sec:MDMethods} |
| 525 |
< |
Evaluation of the pairwise methods (outlined in section |
| 526 |
< |
\ref{sec:ESMethods}) for use in MD simulations was performed through |
| 527 |
< |
comparison of the force and torque vectors obtained with those from |
| 528 |
< |
SPME. Both the magnitude and the direction of these vectors on each |
| 529 |
< |
of the bodies in the system were analyzed. For the magnitude of these |
| 530 |
< |
vectors, linear least squares regression analysis can be performed as |
| 531 |
< |
described previously for comparing $\Delta E$ values. Instead of a |
| 532 |
< |
single value between two system configurations, there is a value for |
| 533 |
< |
each particle in each configuration. For a system of 1000 water |
| 534 |
< |
molecules and 40 ions, there are 1040 force vectors and 1000 torque |
| 535 |
< |
vectors. With 500 configurations, this results in 520,000 force and |
| 536 |
< |
500,000 torque vector comparisons samples for each system type. |
| 525 |
> |
We evaluated the pairwise methods (outlined in section |
| 526 |
> |
\ref{sec:ESMethods}) for use in MD simulations by |
| 527 |
> |
comparing the force and torque vectors with those obtained using the |
| 528 |
> |
reference Ewald summation (SPME). Both the magnitude and the |
| 529 |
> |
direction of these vectors on each of the bodies in the system were |
| 530 |
> |
analyzed. For the magnitude of these vectors, linear least squares |
| 531 |
> |
regression analyses were performed as described previously for |
| 532 |
> |
comparing $\Delta E$ values. Instead of a single energy difference |
| 533 |
> |
between two system configurations, we compared the magnitudes of the |
| 534 |
> |
forces (and torques) on each molecule in each configuration. For a |
| 535 |
> |
system of 1000 water molecules and 40 ions, there are 1040 force |
| 536 |
> |
vectors and 1000 torque vectors. With 500 configurations, this |
| 537 |
> |
results in 520,000 force and 500,000 torque vector comparisons. |
| 538 |
> |
Additionally, data from seven different system types was aggregated |
| 539 |
> |
before the comparison was made. |
| 540 |
|
|
| 541 |
< |
The force and torque vector directions were investigated through |
| 542 |
< |
measurement of the angle ($\theta$) formed between those from the |
| 543 |
< |
particular method and those from SPME |
| 541 |
> |
The {\it directionality} of the force and torque vectors was |
| 542 |
> |
investigated through measurement of the angle ($\theta$) formed |
| 543 |
> |
between those computed from the particular method and those from SPME, |
| 544 |
|
\begin{equation} |
| 545 |
< |
\theta_F = \frac{\vec{F}_\textrm{SPME}}{|\vec{F}_\textrm{SPME}|}\cdot\frac{\vec{F}_\textrm{Method}}{|\vec{F}_\textrm{Method}|}. |
| 545 |
> |
\theta_f = \cos^{-1} \hat{f}_\textrm{SPME} \cdot \hat{f}_\textrm{Method}, |
| 546 |
|
\end{equation} |
| 547 |
+ |
where $\hat{f}_\textrm{M}$ is the unit vector pointing along the |
| 548 |
+ |
force vector computed using method $M$. |
| 549 |
+ |
|
| 550 |
|
Each of these $\theta$ values was accumulated in a distribution |
| 551 |
< |
function, weighted by the area on the unit sphere. Non-linear fits |
| 552 |
< |
were used to measure the shape of the resulting distributions. |
| 551 |
> |
function, weighted by the area on the unit sphere. Non-linear |
| 552 |
> |
Gaussian fits were used to measure the width of the resulting |
| 553 |
> |
distributions. |
| 554 |
|
|
| 555 |
|
\begin{figure} |
| 556 |
|
\centering |
| 563 |
|
non-linear fits. The solid line is a Gaussian profile, while the |
| 564 |
|
dotted line is a Voigt profile, a convolution of a Gaussian and a |
| 565 |
|
Lorentzian. Since this distribution is a measure of angular error |
| 566 |
< |
between two different electrostatic summation methods, there is |
| 567 |
< |
particular reason for the profile to adhere to a specific shape. |
| 568 |
< |
Because of this and the Gaussian profile's more statistically |
| 569 |
< |
meaningful properties, Gaussian fits was used to compare all the |
| 570 |
< |
tested methods. The variance ($\sigma^2$) was extracted from each of |
| 571 |
< |
these fits and was used to compare distribution widths. Values of |
| 572 |
< |
$\sigma^2$ near zero indicate vector directions indistinguishable from |
| 405 |
< |
those calculated when using SPME. |
| 566 |
> |
between two different electrostatic summation methods, there is no |
| 567 |
> |
{\it a priori} reason for the profile to adhere to any specific shape. |
| 568 |
> |
Gaussian fits was used to compare all the tested methods. The |
| 569 |
> |
variance ($\sigma^2$) was extracted from each of these fits and was |
| 570 |
> |
used to compare distribution widths. Values of $\sigma^2$ near zero |
| 571 |
> |
indicate vector directions indistinguishable from those calculated |
| 572 |
> |
when using the reference method (SPME). |
| 573 |
|
|
| 574 |
+ |
\subsection{Short-time Dynamics} |
| 575 |
+ |
|
| 576 |
|
\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 |
| 641 |
|
|
| 642 |
|
\subsection{Electrostatic Summation Methods}\label{sec:ESMethods} |
| 643 |
|
Electrostatic summation method comparisons were performed using SPME, |
| 644 |
< |
the Shifted-Potential and Shifted-Force methods - both with damping |
| 644 |
> |
the {\sc sp} and {\sc sf} methods - both with damping |
| 645 |
|
parameters ($\alpha$) of 0.0, 0.1, 0.2, and 0.3 \AA$^{-1}$ (no, weak, |
| 646 |
|
moderate, and strong damping respectively), reaction field with an |
| 647 |
|
infinite dielectric constant, and an unmodified cutoff. Group-based |
| 660 |
|
the energies and forces calculated. Typical molecular mechanics |
| 661 |
|
packages default this to a value dependent on the cutoff radius and a |
| 662 |
|
tolerance (typically less than $1 \times 10^{-4}$ kcal/mol). Smaller |
| 663 |
< |
tolerances are typically associated with increased accuracy in the |
| 664 |
< |
real-space portion of the summation.\cite{Essmann95} The default |
| 665 |
< |
TINKER tolerance of $1 \times 10^{-8}$ kcal/mol was used in all SPME |
| 663 |
> |
tolerances are typically associated with increased accuracy, but this |
| 664 |
> |
usually means more time spent calculating the reciprocal-space portion |
| 665 |
> |
of the summation.\cite{Perram88,Essmann95} The default TINKER |
| 666 |
> |
tolerance of $1 \times 10^{-8}$ kcal/mol was used in all SPME |
| 667 |
|
calculations, resulting in Ewald Coefficients of 0.4200, 0.3119, and |
| 668 |
|
0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ respectively. |
| 669 |
|
|
| 705 |
|
particularly with a cutoff radius greater than 12 \AA . Use of a |
| 706 |
|
larger damping parameter is more helpful for the shortest cutoff |
| 707 |
|
shown, but it has a detrimental effect on simulations with larger |
| 708 |
< |
cutoffs. In the Shifted-Force sets, increasing damping results in |
| 708 |
> |
cutoffs. In the {\sc sf} sets, increasing damping results in |
| 709 |
|
progressively poorer correlation. Overall, the undamped case is the |
| 710 |
|
best performing set, as the correlation and quality of fits are |
| 711 |
|
consistently superior regardless of the cutoff distance. This result |
| 738 |
|
in the previous $\Delta E$ section. The unmodified cutoff results are |
| 739 |
|
poor, but using group based cutoffs and a switching function provides |
| 740 |
|
a improvement much more significant than what was seen with $\Delta |
| 741 |
< |
E$. Looking at the Shifted-Potential sets, the slope and $R^2$ |
| 741 |
> |
E$. Looking at the {\sc sp} sets, the slope and $R^2$ |
| 742 |
|
improve with the use of damping to an optimal result of 0.2 \AA |
| 743 |
|
$^{-1}$ for the 12 and 15 \AA\ cutoffs. Further increases in damping, |
| 744 |
|
while beneficial for simulations with a cutoff radius of 9 \AA\ , is |
| 745 |
|
detrimental to simulations with larger cutoff radii. The undamped |
| 746 |
< |
Shifted-Force method gives forces in line with those obtained using |
| 746 |
> |
{\sc sf} method gives forces in line with those obtained using |
| 747 |
|
SPME, and use of a damping function results in minor improvement. The |
| 748 |
|
reaction field results are surprisingly good, considering the poor |
| 749 |
|
quality of the fits for the $\Delta E$ results. There is still a |
| 766 |
|
torque vector magnitude results in figure \ref{fig:trqMag} are still |
| 767 |
|
similar to those seen for the forces; however, they more clearly show |
| 768 |
|
the improved behavior that comes with increasing the cutoff radius. |
| 769 |
< |
Moderate damping is beneficial to the Shifted-Potential and helpful |
| 770 |
< |
yet possibly unnecessary with the Shifted-Force method, and they also |
| 769 |
> |
Moderate damping is beneficial to the {\sc sp} and helpful |
| 770 |
> |
yet possibly unnecessary with the {\sc sf} method, and they also |
| 771 |
|
show that over-damping adversely effects all cutoff radii rather than |
| 772 |
|
showing an improvement for systems with short cutoffs. The reaction |
| 773 |
|
field method performs well when calculating the torques, better than |
| 796 |
|
show the improvement afforded by choosing a longer simulation cutoff. |
| 797 |
|
Increasing the cutoff from 9 to 12 \AA\ typically results in a halving |
| 798 |
|
of the distribution widths, with a similar improvement going from 12 |
| 799 |
< |
to 15 \AA . The undamped Shifted-Force, Group Based Cutoff, and |
| 799 |
> |
to 15 \AA . The undamped {\sc sf}, Group Based Cutoff, and |
| 800 |
|
Reaction Field methods all do equivalently well at capturing the |
| 801 |
|
direction of both the force and torque vectors. Using damping |
| 802 |
< |
improves the angular behavior significantly for the Shifted-Potential |
| 803 |
< |
and moderately for the Shifted-Force methods. Increasing the damping |
| 802 |
> |
improves the angular behavior significantly for the {\sc sp} |
| 803 |
> |
and moderately for the {\sc sf} methods. Increasing the damping |
| 804 |
|
too far is destructive for both methods, particularly to the torque |
| 805 |
|
vectors. Again it is important to recognize that the force vectors |
| 806 |
|
cover all particles in the systems, while torque vectors are only |
| 842 |
|
\end{table} |
| 843 |
|
|
| 844 |
|
Although not discussed previously, group based cutoffs can be applied |
| 845 |
< |
to both the Shifted-Potential and Shifted-Force methods. Use off a |
| 845 |
> |
to both the {\sc sp} and {\sc sf} methods. Use off a |
| 846 |
|
switching function corrects for the discontinuities that arise when |
| 847 |
|
atoms of a group exit the cutoff before the group's center of mass. |
| 848 |
|
Though there are no significant benefit or drawbacks observed in |
| 851 |
|
\ref{tab:groupAngle} shows the angular variance values obtained using |
| 852 |
|
group based cutoffs and a switching function alongside the standard |
| 853 |
|
results seen in figure \ref{fig:frcTrqAng} for comparison purposes. |
| 854 |
< |
The Shifted-Potential shows much narrower angular distributions for |
| 854 |
> |
The {\sc sp} shows much narrower angular distributions for |
| 855 |
|
both the force and torque vectors when using an $\alpha$ of 0.2 |
| 856 |
< |
\AA$^{-1}$ or less, while Shifted-Force shows improvements in the |
| 856 |
> |
\AA$^{-1}$ or less, while {\sc sf} shows improvements in the |
| 857 |
|
undamped and lightly damped cases. Thus, by calculating the |
| 858 |
|
electrostatic interactions in terms of molecular pairs rather than |
| 859 |
|
atomic pairs, the direction of the force and torque vectors are |
| 860 |
|
determined more accurately. |
| 861 |
|
|
| 862 |
|
One additional trend to recognize in table \ref{tab:groupAngle} is |
| 863 |
< |
that the $\sigma^2$ values for both Shifted-Potential and |
| 864 |
< |
Shifted-Force converge as $\alpha$ increases, something that is easier |
| 863 |
> |
that the $\sigma^2$ values for both {\sc sp} and |
| 864 |
> |
{\sc sf} converge as $\alpha$ increases, something that is easier |
| 865 |
|
to see when using group based cutoffs. Looking back on figures |
| 866 |
|
\ref{fig:delE}, \ref{fig:frcMag}, and \ref{fig:trqMag}, show this |
| 867 |
|
behavior clearly at large $\alpha$ and cutoff values. The reason for |
| 880 |
|
high would introduce error in the molecular torques, particularly for |
| 881 |
|
the shorter cutoffs. Based on the above findings, empirical damping |
| 882 |
|
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 883 |
< |
unnecessary when using the Shifted-Force method. |
| 883 |
> |
unnecessary when using the {\sc sf} method. |
| 884 |
|
|
| 885 |
|
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 886 |
|
|
| 887 |
< |
In the previous studies using a Shifted-Force variant of the damped |
| 887 |
> |
In the previous studies using a {\sc sf} variant of the damped |
| 888 |
|
Wolf coulomb potential, the structure and dynamics of water were |
| 889 |
|
investigated rather extensively.\cite{Zahn02,Kast03} Their results |
| 890 |
< |
indicated that the damped Shifted-Force method results in properties |
| 890 |
> |
indicated that the damped {\sc sf} method results in properties |
| 891 |
|
very similar to those obtained when using the Ewald summation. |
| 892 |
|
Considering the statistical results shown above, the good performance |
| 893 |
|
of this method is not that surprising. Rather than consider the same |
| 898 |
|
\begin{figure} |
| 899 |
|
\centering |
| 900 |
|
\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
| 901 |
< |
\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.} |
| 901 |
> |
\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.} |
| 902 |
|
\label{fig:methodPS} |
| 903 |
|
\end{figure} |
| 904 |
|
|
| 910 |
|
methods differ. Considering the low-frequency inset (expanded in the |
| 911 |
|
upper frame of figure \ref{fig:dampInc}), at frequencies below 100 |
| 912 |
|
cm$^{-1}$, the correlated motions are blue-shifted when using undamped |
| 913 |
< |
or weakly damped Shifted-Force. When using moderate damping ($\alpha |
| 914 |
< |
= 0.2$ \AA$^{-1}$) both the Shifted-Force and Shifted-Potential |
| 913 |
> |
or weakly damped {\sc sf}. When using moderate damping ($\alpha |
| 914 |
> |
= 0.2$ \AA$^{-1}$) both the {\sc sf} and {\sc sp} |
| 915 |
|
methods give near identical correlated motion behavior as the Ewald |
| 916 |
|
method (which has a damping value of 0.3119). The damping acts as a |
| 917 |
|
distance dependent Gaussian screening of the point charges for the |
| 922 |
|
clearly, we show how damping strength affects a simple real-space |
| 923 |
|
electrostatic potential, |
| 924 |
|
\begin{equation} |
| 925 |
< |
V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r_{ij}})}{r_{ij}}\right]S(r), |
| 925 |
> |
V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r})}{r}\right]S(r), |
| 926 |
|
\end{equation} |
| 927 |
|
where $S(r)$ is a switching function that smoothly zeroes the |
| 928 |
|
potential at the cutoff radius. Figure \ref{fig:dampInc} shows how |
| 935 |
|
shift to higher frequency in exponential fashion. Though not shown, |
| 936 |
|
the spectrum for the simple undamped electrostatic potential is |
| 937 |
|
blue-shifted such that the lowest frequency peak resides near 325 |
| 938 |
< |
cm$^{-1}$. In light of these results, the undamped Shifted-Force |
| 938 |
> |
cm$^{-1}$. In light of these results, the undamped {\sc sf} |
| 939 |
|
method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is |
| 940 |
|
quite respectable; however, it appears as though moderate damping is |
| 941 |
|
required for accurate reproduction of crystal dynamics. |
| 942 |
|
\begin{figure} |
| 943 |
|
\centering |
| 944 |
|
\includegraphics[width = \linewidth]{./comboSquare.pdf} |
| 945 |
< |
\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.} |
| 945 |
> |
\caption{Regions of spectra showing the low-frequency correlated motions for NaCl crystals at 1000 K using various electrostatic summation methods. The upper plot is a zoomed inset from figure \ref{fig:methodPS}. As the damping value for the {\sc sf} potential increases, the low-frequency peaks red-shift. The lower plot is of spectra 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.} |
| 946 |
|
\label{fig:dampInc} |
| 947 |
|
\end{figure} |
| 948 |
|
|
| 953 |
|
electrostatic summation techniques than the Ewald summation, chiefly |
| 954 |
|
methods derived from the damped Coulombic sum originally proposed by |
| 955 |
|
Wolf \textit{et al.}\cite{Wolf99,Zahn02} In particular, the |
| 956 |
< |
Shifted-Force method, reformulated above as equation \ref{eq:SFPot}, |
| 956 |
> |
{\sc sf} method, reformulated above as eq. (\ref{eq:DSFPot}), |
| 957 |
|
shows a remarkable ability to reproduce the energetic and dynamic |
| 958 |
|
characteristics exhibited by simulations employing lattice summation |
| 959 |
|
techniques. The cumulative energy difference results showed the |
| 960 |
< |
undamped Shifted-Force and moderately damped Shifted-Potential methods |
| 960 |
> |
undamped {\sc sf} and moderately damped {\sc sp} methods |
| 961 |
|
produced results nearly identical to SPME. Similarly for the dynamic |
| 962 |
< |
features, the undamped or moderately damped Shifted-Force and |
| 963 |
< |
moderately damped Shifted-Potential methods produce force and torque |
| 962 |
> |
features, the undamped or moderately damped {\sc sf} and |
| 963 |
> |
moderately damped {\sc sp} methods produce force and torque |
| 964 |
|
vector magnitude and directions very similar to the expected values. |
| 965 |
|
These results translate into long-time dynamic behavior equivalent to |
| 966 |
|
that produced in simulations using SPME. |
