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Revision 2653 by gezelter, Tue Mar 21 20:46:55 2006 UTC

+%\documentclass[aps,prb,preprint]{revtex4}
+\documentclass[11pt]{article}
+\usepackage{endfloat}
-<
+\usepackage{amsmath}
->
+\usepackage{amsmath,bm}
+\usepackage{amssymb}
+\usepackage{epsf}
+\usepackage{times}
+\section{Introduction}
+In molecular simulations, proper accumulation of the electrostatic
-<
+interactions is considered one of the most essential and
-<
+computationally demanding tasks.  The common molecular mechanics force
-<
+fields are founded on representation of the atomic sites centered on
-<
+full or partial charges shielded by Lennard-Jones type interactions.
-<
+This means that nearly every pair interaction involves an
-<
+charge-charge calculation.  Coupled with $r^{-1}$ decay, the monopole
-<
+interactions quickly become a burden for molecular systems of all
-<
+sizes.  For example, in small systems, the electrostatic pair
-<
+interaction may not have decayed appreciably within the box length
-<
+leading to an effect excluded from the pair interactions within a unit
-<
+box.  In large systems, excessively large cutoffs need to be used to
-<
+accurately incorporate their effect, and since the computational cost
-<
+increases proportionally with the cutoff sphere, it quickly becomes an
-<
+impractical task to perform these calculations.
->
+interactions is essential and is one of the most
->
+computationally-demanding tasks.  The common molecular mechanics force
->
+fields represent atomic sites with full or partial charges protected
->
+by Lennard-Jones (short range) interactions.  This means that nearly
->
+every pair interaction involves a calculation of charge-charge forces.
->
+Coupled with relatively long-ranged $r^{-1}$ decay, the monopole
->
+interactions quickly become the most expensive part of molecular
->
+simulations.  Historically, the electrostatic pair interaction would
->
+not have decayed appreciably within the typical box lengths that could
->
+be feasibly simulated.  In the larger systems that are more typical of
->
+modern simulations, large cutoffs should be used to incorporate
->
+electrostatics correctly.
-+
+There have been many efforts to address the proper and practical
-+
+handling of electrostatic interactions, and these have resulted in a
-+
+variety of techniques.\cite{Roux99,Sagui99,Tobias01} These are
-+
+typically classified as implicit methods (i.e., continuum dielectrics,
-+
+static dipolar fields),\cite{Born20,Grossfield00} explicit methods
-+
+(i.e., Ewald summations, interaction shifting or
-+
+truncation),\cite{Ewald21,Steinbach94} or a mixture of the two (i.e.,
-+
+reaction field type methods, fast multipole
-+
+methods).\cite{Onsager36,Rokhlin85} The explicit or mixed methods are
-+
+often preferred because they physically incorporate solvent molecules
-+
+in the system of interest, but these methods are sometimes difficult
-+
+to utilize because of their high computational cost.\cite{Roux99} In
-+
+addition to the computational cost, there have been some questions
-+
+regarding possible artifacts caused by the inherent periodicity of the
-+
+explicit Ewald summation.\cite{Tobias01}
-+
-+
+In this paper, we focus on a new set of shifted methods devised by
-+
+Wolf {\it et al.},\cite{Wolf99} which we further extend.  These
-+
+methods along with a few other mixed methods (i.e. reaction field) are
-+
+compared with the smooth particle mesh Ewald
-+
+sum,\cite{Onsager36,Essmann99} which is our reference method for
-+
+handling long-range electrostatic interactions. The new methods for
-+
+handling electrostatics have the potential to scale linearly with
-+
+increasing system size since they involve only a simple modification
-+
+to the direct pairwise sum.  They also lack the added periodicity of
-+
+the Ewald sum, so they can be used for systems which are non-periodic
-+
+or which have one- or two-dimensional periodicity.  Below, these
-+
+methods are evaluated using a variety of model systems to establish
-+
+their usability in molecular simulations.
-+
+\subsection{The Ewald Sum}
-<
+blah blah blah Ewald Sum Important blah blah blah
->
+The complete accumulation electrostatic interactions in a system with
->
+periodic boundary conditions (PBC) requires the consideration of the
->
+effect of all charges within a (cubic) simulation box as well as those
->
+in the periodic replicas,
->
+\begin{equation}
->
+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],
->
+\label{eq:PBCSum}
->
+\end{equation}
->
+where the sum over $\mathbf{n}$ is a sum over all periodic box
->
+replicas with integer coordinates $\mathbf{n} = (l,m,n)$, and the
->
+prime indicates $i = j$ are neglected for $\mathbf{n} =
->
+$.\cite{deLeeuw80} Within the sum, $N$ is the number of electrostatic
->
+particles, $\mathbf{r}_{ij}$ is $\mathbf{r}_j - \mathbf{r}_i$, $L$ is
->
+the cell length, $\bm{\Omega}_{i,j}$ are the Euler angles for $i$ and
->
+$j$, and $\phi$ is the solution to Poisson's equation
->
+($\phi(\mathbf{r}_{ij}) = q_i q_j |\mathbf{r}_{ij}|^{-1}$ for
->
+charge-charge interactions). In the case of monopole electrostatics,
->
+eq. (\ref{eq:PBCSum}) is conditionally convergent and is divergent for
->
+non-neutral systems.
-+
+The electrostatic summation problem was originally studied by Ewald
-+
+for the case of an infinite crystal.\cite{Ewald21}. The approach he
-+
+took was to convert this conditionally convergent sum into two
-+
+absolutely convergent summations: a short-ranged real-space summation
-+
+and a long-ranged reciprocal-space summation,
-+
+\begin{equation}
-+
+\begin{split}
-+
+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,
-+
+\end{split}
-+
+\label{eq:EwaldSum}
-+
+\end{equation}
-+
+where $\alpha$ is the damping or convergence parameter with units of
-+
+\AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and are equal to
-+
+$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric
-+
+constant of the surrounding medium. The final two terms of
-+
+eq. (\ref{eq:EwaldSum}) are a particle-self term and a dipolar term
-+
+for interacting with a surrounding dielectric.\cite{Allen87} This
-+
+dipolar term was neglected in early applications in molecular
-+
+simulations,\cite{Brush66,Woodcock71} until it was introduced by de
-+
+Leeuw {\it et al.} to address situations where the unit cell has a
-+
+dipole moment which is magnified through replication of the periodic
-+
+images.\cite{deLeeuw80,Smith81} If this term is taken to be zero, the
-+
+system is said to be using conducting (or ``tin-foil'') boundary
-+
+conditions, $\epsilon_{\rm S} = \infty$. Figure
-+
+\ref{fig:ewaldTime} shows how the Ewald sum has been applied over
-+
+time.  Initially, due to the small system sizes that could be
-+
+simulated feasibly, the entire simulation box was replicated to
-+
+convergence.  In more modern simulations, the systems have grown large
-+
+enough that a real-space cutoff could potentially give convergent
-+
+behavior.  Indeed, it has been observed that with the choice of a
-+
+small $\alpha$, the reciprocal-space portion of the Ewald sum can be
-+
+rapidly convergent and small relative to the real-space
-+
+portion.\cite{Karasawa89,Kolafa92}
-+
+\begin{figure}
+\centering
-<
+\includegraphics[width = \linewidth]{./ewaldProgression.pdf}
-<
+\caption{How the application of the Ewald summation has changed with
-<
+the increase in computer power.  Initially, only small numbers of
-<
+particles could be studied, and the Ewald sum acted to replicate the
-<
+unit cell charge distribution out to convergence.  Now, much larger
-<
+systems of charges are investigated with fixed distance cutoffs.  The
-<
+calculated structure factor is used to sum out to great distance, and
-<
+a surrounding dielectric term is included.}
->
+\includegraphics[width = \linewidth]{./ewaldProgression2.pdf}
->
+\caption{The change in the application of the Ewald sum with
->
+increasing computational power.  Initially, only small systems could
->
+be studied, and the Ewald sum replicated the simulation box to
->
+convergence.  Now, much larger systems of charges are investigated
->
+with fixed-distance cutoffs.}
+\label{fig:ewaldTime}
+\end{figure}
-+
+The original Ewald summation is an $\mathscr{O}(N^2)$ algorithm.  The
-+
+convergence parameter $(\alpha)$ plays an important role in balancing
-+
+the computational cost between the direct and reciprocal-space
-+
+portions of the summation.  The choice of this value allows one to
-+
+select whether the real-space or reciprocal space portion of the
-+
+summation is an $\mathscr{O}(N^2)$ calculation (with the other being
-+
+$\mathscr{O}(N)$).\cite{Sagui99} With the appropriate choice of
-+
+$\alpha$ and thoughtful algorithm development, this cost can be
-+
+reduced to $\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route
-+
+taken to reduce the cost of the Ewald summation even further is to set
-+
+$\alpha$ such that the real-space interactions decay rapidly, allowing
-+
+for a short spherical cutoff. Then the reciprocal space summation is
-+
+optimized.  These optimizations usually involve utilization of the
-+
+fast Fourier transform (FFT),\cite{Hockney81} leading to the
-+
+particle-particle particle-mesh (P3M) and particle mesh Ewald (PME)
-+
+methods.\cite{Shimada93,Luty94,Luty95,Darden93,Essmann95} In these
-+
+methods, the cost of the reciprocal-space portion of the Ewald
-+
+summation is reduced from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N
-+
+\log N)$.
-+
-+
+These developments and optimizations have made the use of the Ewald
-+
+summation routine in simulations with periodic boundary
-+
+conditions. However, in certain systems, such as vapor-liquid
-+
+interfaces and membranes, the intrinsic three-dimensional periodicity
-+
+can prove problematic.  The Ewald sum has been reformulated to handle
-+
+D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89}, but the
-+
+new methods are computationally expensive.\cite{Spohr97,Yeh99}
-+
+Inclusion of a correction term in the Ewald summation is a possible
-+
+direction for handling 2D systems while still enabling the use of the
-+
+modern optimizations.\cite{Yeh99}
-+
-+
+Several studies have recognized that the inherent periodicity in the
-+
+Ewald sum can also have an effect on three-dimensional
-+
+systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00}
-+
+Solvated proteins are essentially kept at high concentration due to
-+
+the periodicity of the electrostatic summation method.  In these
-+
+systems, the more compact folded states of a protein can be
-+
+artificially stabilized by the periodic replicas introduced by the
-+
+Ewald summation.\cite{Weber00} Thus, care must be taken when
-+
+considering the use of the Ewald summation where the assumed
-+
+periodicity would introduce spurious effects in the system dynamics.
-+
+\subsection{The Wolf and Zahn Methods}
+In a recent paper by Wolf \textit{et al.}, a procedure was outlined
-<
+for an accurate accumulation of electrostatic interactions in an
-<
+efficient pairwise fashion.\cite{Wolf99} Wolf \textit{et al.} observed
-<
+that the electrostatic interaction is effectively short-ranged in
-<
+condensed phase systems and that neutralization of the charge
-<
+contained within the cutoff radius is crucial for potential
->
+for the accurate accumulation of electrostatic interactions in an
->
+efficient pairwise fashion.  This procedure lacks the inherent
->
+periodicity of the Ewald summation.\cite{Wolf99} Wolf \textit{et al.}
->
+observed that the electrostatic interaction is effectively
->
+short-ranged in condensed phase systems and that neutralization of the
->
+charge contained within the cutoff radius is crucial for potential
+stability. They devised a pairwise summation method that ensures
-<
+charge neutrality and gives results similar to those obtained with
-<
+the Ewald summation.  The resulting shifted Coulomb potential
->
+charge neutrality and gives results similar to those obtained with the
->
+Ewald summation.  The resulting shifted Coulomb potential
+(Eq. \ref{eq:WolfPot}) includes image-charges subtracted out through
+placement on the cutoff sphere and a distance-dependent damping
+function (identical to that seen in the real-space portion of the
+Ewald sum) to aid convergence
+\begin{equation}
-<
+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\}.
->
+V_{\textrm{Wolf}}(r_{ij})= \frac{q_i q_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\}.
+\label{eq:WolfPot}
+\end{equation}
+Eq. (\ref{eq:WolfPot}) is essentially the common form of a shifted
+potential.  However, neutralizing the charge contained within each
+cutoff sphere requires the placement of a self-image charge on the
+surface of the cutoff sphere.  This additional self-term in the total
-<
+potential enables Wolf {\it et al.}  to obtain excellent estimates of
->
+potential enabled Wolf {\it et al.}  to obtain excellent estimates of
+Madelung energies for many crystals.
+In order to use their charge-neutralized potential in molecular
+derivative of this potential prior to evaluation of the limit.  This
+procedure gives an expression for the forces,
+\begin{equation}
-<
+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\},
->
+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\},
+\label{eq:WolfForces}
+\end{equation}
+that incorporates both image charges and damping of the electrostatic
+More recently, Zahn \textit{et al.} investigated these potential and
+force expressions for use in simulations involving water.\cite{Zahn02}
-<
+In their work, they pointed out that the method that the forces and
-<
+derivative of the potential are not commensurate.  Attempts to use
-<
+both Eqs. (\ref{eq:WolfPot}) and (\ref{eq:WolfForces}) together will
-<
+lead to poor energy conservation.  They correctly observed that taking
-<
+the limit shown in equation (\ref{eq:WolfPot}) {\it after} calculating
-<
+the derivatives is mathematically invalid.
->
+In their work, they pointed out that the forces and derivative of
->
+the potential are not commensurate.  Attempts to use both
->
+eqs. (\ref{eq:WolfPot}) and (\ref{eq:WolfForces}) together will lead
->
+to poor energy conservation.  They correctly observed that taking the
->
+limit shown in equation (\ref{eq:WolfPot}) {\it after} calculating the
->
+derivatives gives forces for a different potential energy function
->
+than the one shown in eq. (\ref{eq:WolfPot}).
-<
+Zahn \textit{et al.} proposed a modified form of this ``Wolf summation
-<
+method'' as a way to use this technique in Molecular Dynamics
-<
+simulations.  Taking the integral of the forces shown in equation
-<
+\ref{eq:WolfForces}, they proposed a new damped Coulomb
-<
+potential,
->
+Zahn \textit{et al.} introduced a modified form of this summation
->
+method as a way to use the technique in Molecular Dynamics
->
+simulations.  They proposed a new damped Coulomb potential,
+\begin{equation}
-<
+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\}.
->
+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\},
+\label{eq:ZahnPot}
+\end{equation}
-<
+They showed that this potential does fairly well at capturing the
->
+and showed that this potential does fairly well at capturing the
+structural and dynamic properties of water compared the same
+properties obtained using the Ewald sum.
+The potentials proposed by Wolf \textit{et al.} and Zahn \textit{et
+al.} are constructed using two different (and separable) computational
-<
+tricks: \begin{itemize}
->
+tricks: \begin{enumerate}
+\item shifting through the use of image charges, and
+\item damping the electrostatic interaction.
-<
+\end{itemize}  Wolf \textit{et al.} treated the
->
+\end{enumerate}  Wolf \textit{et al.} treated the
+development of their summation method as a progressive application of
+these techniques,\cite{Wolf99} while Zahn \textit{et al.} founded
+their damped Coulomb modification (Eq. (\ref{eq:ZahnPot})) on the
+\textit{et al.}  and Zahn \textit{et al.} by considering the standard
+shifted potential,
+\begin{equation}
-<
+v^\textrm{SP}(r) =      \begin{cases}
->
+V_\textrm{SP}(r) =      \begin{cases}
+v(r)-v_\textrm{c} &\quad r\leqslant R_\textrm{c} \\ 0 &\quad r >
+R_\textrm{c}
+\end{cases},
+\end{equation}
+and shifted force,
+\begin{equation}
-<
+v^\textrm{SF}(r) =      \begin{cases}
-<
+v(r)-v_\textrm{c}-\left(\frac{\textrm{d}v(r)}{\textrm{d}r}\right)_{r=R_\textrm{c}}(r-R_\textrm{c})
->
+V_\textrm{SF}(r) =      \begin{cases}
->
+v(r)-v_\textrm{c}-\left(\frac{d v(r)}{d r}\right)_{r=R_\textrm{c}}(r-R_\textrm{c})
+&\quad r\leqslant R_\textrm{c} \\ 0 &\quad r > R_\textrm{c}
+                                                \end{cases},
+\label{eq:shiftingForm}
+potential is smooth at the cutoff radius
+($R_\textrm{c}$).\cite{Allen87}
-<
-<
-<
-<
+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}
->
+The forces associated with the shifted potential are simply the forces
->
+of the unshifted potential itself (when inside the cutoff sphere),
+\begin{equation}
-<
+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}
->
+F_{\textrm{SP}} = -\left( \frac{d v(r)}{dr} \right),
+\end{equation}
-<
+The forces associated with this potential are obtained by taking the derivative, resulting in the following,
->
+and are zero outside.  Inside the cutoff sphere, the forces associated
->
+with the shifted force form can be written,
+\begin{equation}
-<
+F^\textrm{SP}(r_{ij}) = q_iq_j\left(-\frac{1}{r_{ij}^2}\right) \quad r_{ij}\leqslant R_\textrm{c}.
-<
+\label{eq:FWolfSP}
->
+F_{\textrm{SF}} = -\left( \frac{d v(r)}{dr} \right) + \left(\frac{d
->
+v(r)}{dr} \right)_{r=R_\textrm{c}}.
+\end{equation}
-<
+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 $R_\textrm{c}$.
->
->
+If the potential, $v(r)$, is taken to be the normal Coulomb potential,
->
+\begin{equation}
->
+v(r) = \frac{q_i q_j}{r},
->
+\label{eq:Coulomb}
->
+\end{equation}
->
+then the Shifted Potential ({\sc sp}) forms will give Wolf {\it et
->
+al.}'s undamped prescription:
->
+\begin{equation}
->
+V_\textrm{SP}(r) =
->
+q_iq_j\left(\frac{1}{r}-\frac{1}{R_\textrm{c}}\right) \quad
->
+r\leqslant R_\textrm{c},
->
+\label{eq:SPPot}
->
+\end{equation}
->
+with associated forces,
->
+\begin{equation}
->
+F_\textrm{SP}(r) = q_iq_j\left(\frac{1}{r^2}\right) \quad r\leqslant R_\textrm{c}.
->
+\label{eq:SPForces}
->
+\end{equation}
->
+These forces are identical to the forces of the standard Coulomb
->
+interaction, and cutting these off at $R_c$ 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 this form is
->
+used,\cite{Wolf99} thereby eliminating any benefit from the method in
->
+molecular dynamics.  Additionally, there is a discontinuity in the
->
+forces at the cutoff radius which results in energy drift during MD
->
+simulations.
-<
+If the derivative term in equation \ref{eq:shiftingForm} is evaluated, we obtain an hitherto undiscussed shifted force Coulomb potential,
->
+The shifted force ({\sc sf}) form using the normal Coulomb potential
->
+will give,
+\begin{equation}
-<
+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}.
->
+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}.
+\label{eq:SFPot}
+\end{equation}
-<
+Taking the derivative of this shifted force potential gives the
-<
+following forces,
->
+with associated forces,
+\begin{equation}
-<
+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}.
->
+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}.
+\label{eq:SFForces}
+\end{equation}
-<
+Using this formulation rather than the simple shifted potential
-<
+(Eq. \ref{eq:WolfSP}) means that there are no discontinuities in the
-<
+forces in addition to the potential.  This form also has the benefit
-<
+that the image charges have a force presence, addressing the concerns
-<
+about a missing physical component.  One side effect of this treatment
-<
+is a slight alteration in the shape of the potential that comes about
-<
+from the derivative term.  Thus, a degree of clarity about the
-<
+original formulation of the potential is lost in order to gain
-<
+functionality in dynamics simulations.
->
+This formulation has the benefits that there are no discontinuities at
->
+the cutoff radius, while the neutralizing image charges are present in
->
+both the energy and force expressions.  It would be simple to add the
->
+self-neutralizing term back when computing the total energy of the
->
+system, thereby maintaining the agreement with the Madelung energies.
->
+A side effect of this treatment is the alteration in the shape of the
->
+potential that comes from the derivative term.  Thus, a degree of
->
+clarity about agreement with the empirical potential is lost in order
->
+to gain functionality in dynamics simulations.
+Wolf \textit{et al.} originally discussed the energetics of the
-<
+shifted Coulomb potential (Eq. \ref{eq:WolfSP}), and they found that
-<
+it was still insufficient for accurate determination of the energy.
-<
+The energy would fluctuate around the expected value with increasing
-<
+cutoff radius, but the oscillations appeared to be converging toward
-<
+the correct value.\cite{Wolf99} A damping function was incorporated to
-<
+accelerate convergence; and though alternative functional forms could
-<
+be used,\cite{Jones56,Heyes81} the complimentary error function was
-<
+chosen to draw parallels to the Ewald summation.  Incorporating
-<
+damping into the simple Coulomb potential,
->
+shifted Coulomb potential (Eq. \ref{eq:SPPot}) and found that it was
->
+insufficient for accurate determination of the energy with reasonable
->
+cutoff distances.  The calculated Madelung energies fluctuated around
->
+the expected value as the cutoff radius was increased, but the
->
+oscillations converged toward the correct value.\cite{Wolf99} A
->
+damping function was incorporated to accelerate the convergence; and
->
+though alternative forms for the damping function could be
->
+used,\cite{Jones56,Heyes81} the complimentary error function was
->
+chosen to mirror the effective screening used in the Ewald summation.
->
+Incorporating this error function damping into the simple Coulomb
->
+potential,
+\begin{equation}
-<
+v(r_{ij}) = \frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}},
->
+v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r},
+\label{eq:dampCoulomb}
+\end{equation}
-<
+the shifted potential (Eq. \ref{eq:WolfSP}) can be rederived
-<
+\textit{via} equation \ref{eq:shiftingForm},
->
+the shifted potential (eq. (\ref{eq:SPPot})) becomes
+\begin{equation}
-<
+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}.
->
+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},
+\label{eq:DSPPot}
+\end{equation}
-<
+The derivative of this Shifted-Potential can be taken to obtain forces
-<
+for use in MD,
->
+with associated forces,
+\begin{equation}
-<
+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}.
->
+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}.
+\label{eq:DSPForces}
+\end{equation}
-<
+Again, this Shifted-Potential suffers from a discontinuity in the
-<
+forces, and a lack of an image-charge component in the forces.  To
-<
+remedy these concerns, a Shifted-Force variant is obtained by
-<
+inclusion of the derivative term in equation \ref{eq:shiftingForm} to
-<
+give,
->
+Again, this damped shifted potential suffers from a
->
+force-discontinuity at the cutoff radius, and the image charges play
->
+no role in the forces.  To remedy these concerns, one may derive a
->
+{\sc sf} variant by including the derivative term in
->
+eq. (\ref{eq:shiftingForm}),
+\begin{equation}
+\begin{split}
-<
+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}.
->
+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}.
+\label{eq:DSFPot}
+\end{split}
+\end{equation}
-<
+The derivative of the above potential gives the following forces,
->
+The derivative of the above potential will lead to the following forces,
+\begin{equation}
+\begin{split}
-<
+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}.
->
+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}.
+\label{eq:DSFForces}
+\end{split}
+\end{equation}
-+
+If the damping parameter $(\alpha)$ is set to zero, the undamped case,
-+
+eqs. (\ref{eq:SPPot} through \ref{eq:SFForces}) are correctly
-+
+recovered from eqs. (\ref{eq:DSPPot} through \ref{eq:DSFForces}).
-<
+This new Shifted-Force potential is similar to equation
-<
+\ref{eq:ZahnPot} derived by Zahn \textit{et al.}; however, there are
-<
+two important differences.\cite{Zahn02} First, the $v_\textrm{c}$ term
-<
+from equation \ref{eq:shiftingForm} is equal to equation
-<
+\ref{eq:dampCoulomb} with $R_\textrm{c}$ supplied for $r_{ij}$.  This
-<
+term is not present in the Zahn potential, resulting in a
-<
+discontinuity as particles cross $R_\textrm{c}$.  Second, the sign of
-<
+the derivative portion is different.  The constant $v_\textrm{c}$ term
-<
+is not going to have a presence in the forces after performing the
-<
+derivative, but the negative sign does effect the derivative.  In
-<
+fact, it introduces a discontinuity in the forces at the cutoff,
->
+This new {\sc sf} potential is similar to equation \ref{eq:ZahnPot}
->
+derived by Zahn \textit{et al.}; however, there are two important
->
+differences.\cite{Zahn02} First, the $v_\textrm{c}$ term from
->
+eq. (\ref{eq:shiftingForm}) is equal to eq. (\ref{eq:dampCoulomb})
->
+with $r$ replaced by $R_\textrm{c}$.  This term is {\it not} present
->
+in the Zahn potential, resulting in a potential discontinuity as
->
+particles cross $R_\textrm{c}$.  Second, the sign of the derivative
->
+portion is different.  The missing $v_\textrm{c}$ term would not
->
+affect molecular dynamics simulations (although the computed energy
->
+would be expected to have sudden jumps as particle distances crossed
->
+$R_c$).  The sign problem is a potential source of errors, however.
->
+In fact, it introduces a discontinuity in the forces at the cutoff,
+because the force function is shifted in the wrong direction and
-<
+doesn't cross zero at $R_\textrm{c}$.  Thus, these alterations make
-<
+for an electrostatic summation method that is continuous in both the
-<
+potential and forces and incorporates the pairwise sum considerations
-<
+stressed by Wolf \textit{et al.}\cite{Wolf99}
->
+doesn't cross zero at $R_\textrm{c}$.
-+
+Eqs. (\ref{eq:DSFPot}) and (\ref{eq:DSFForces}) result in an
-+
+electrostatic summation method in which the potential and forces are
-+
+continuous at the cutoff radius and which incorporates the damping
-+
+function proposed by Wolf \textit{et al.}\cite{Wolf99} In the rest of
-+
+this paper, we will evaluate exactly how good these methods ({\sc sp},
-+
+{\sc sf}, damping) are at reproducing the correct electrostatic
-+
+summation performed by the Ewald sum.
-+
-+
+\subsection{Other alternatives}
-+
+In addition to the methods described above, we considered some other
-+
+techniques that are commonly used in molecular simulations.  The
-+
+simplest of these is group-based cutoffs.  Though of little use for
-+
+charged molecules, collecting atoms into neutral groups takes
-+
+advantage of the observation that the electrostatic interactions decay
-+
+faster than those for monopolar pairs.\cite{Steinbach94} When
-+
+considering these molecules as neutral groups, the relative
-+
+orientations of the molecules control the strength of the interactions
-+
+at the cutoff radius.  Consequently, as these molecular particles move
-+
+through $R_\textrm{c}$, the energy will drift upward due to the
-+
+anisotropy of the net molecular dipole interactions.\cite{Rahman71} To
-+
+maintain good energy conservation, both the potential and derivative
-+
+need to be smoothly switched to zero at $R_\textrm{c}$.\cite{Adams79}
-+
+This is accomplished using a standard switching function.  If a smooth
-+
+second derivative is desired, a fifth (or higher) order polynomial can
-+
+be used.\cite{Andrea83}
-+
-+
+Group-based cutoffs neglect the surroundings beyond $R_\textrm{c}$,
-+
+and to incorporate the effects of the surroundings, a method like
-+
+Reaction Field ({\sc rf}) can be used.  The original theory for {\sc
-+
+rf} was originally developed by Onsager,\cite{Onsager36} and it was
-+
+applied in simulations for the study of water by Barker and
-+
+Watts.\cite{Barker73} In modern simulation codes, {\sc rf} is simply
-+
+an extension of the group-based cutoff method where the net dipole
-+
+within the cutoff sphere polarizes an external dielectric, which
-+
+reacts back on the central dipole.  The same switching function
-+
+considerations for group-based cutoffs need to made for {\sc rf}, with
-+
+the additional pre-specification of a dielectric constant.
-+
+\section{Methods}
-–
+\subsection{What Qualities are Important?}\label{sec:Qualities}
+In classical molecular mechanics simulations, there are two primary
+techniques utilized to obtain information about the system of
+interest: Monte Carlo (MC) and Molecular Dynamics (MD).  Both of these
+techniques utilize pairwise summations of interactions between
+particle sites, but they use these summations in different ways.
-<
+In MC, the potential energy difference between two subsequent
-<
+configurations dictates the progression of MC sampling.  Going back to
-<
+the origins of this method, the Canonical ensemble acceptance criteria
-<
+laid out by Metropolis \textit{et al.} states that a subsequent
-<
+configuration is accepted if $\Delta E < 0$ or if $\xi < \exp(-\Delta
-<
+E/kT)$, where $\xi$ is a random number between 0 and
-<
+.\cite{Metropolis53} Maintaining a consistent $\Delta E$ when using
-<
+an alternate method for handling the long-range electrostatics ensures
-<
+proper sampling within the ensemble.
->
+In MC, the potential energy difference between configurations dictates
->
+the progression of MC sampling.  Going back to the origins of this
->
+method, the acceptance criterion for the canonical ensemble laid out
->
+by Metropolis \textit{et al.} states that a subsequent configuration
->
+is accepted if $\Delta E < 0$ or if $\xi < \exp(-\Delta E/kT)$, where
->
+$\xi$ is a random number between 0 and 1.\cite{Metropolis53}
->
+Maintaining the correct $\Delta E$ when using an alternate method for
->
+handling the long-range electrostatics will ensure proper sampling
->
+from the ensemble.
-<
+In MD, the derivative of the potential directs how the system will
->
+In MD, the derivative of the potential governs how the system will
+progress in time.  Consequently, the force and torque vectors on each
-<
+body in the system dictate how it develops as a whole.  If the
-<
+magnitude and direction of these vectors are similar when using
-<
+alternate electrostatic summation techniques, the dynamics in the near
-<
+term will be indistinguishable.  Because error in MD calculations is
-<
+cumulative, one should expect greater deviation in the long term
-<
+trajectories with greater differences in these vectors between
-<
+configurations using different long-range electrostatics.
->
+body in the system dictate how the system evolves.  If the magnitude
->
+and direction of these vectors are similar when using alternate
->
+electrostatic summation techniques, the dynamics in the short term
->
+will be indistinguishable.  Because error in MD calculations is
->
+cumulative, one should expect greater deviation at longer times,
->
+although methods which have large differences in the force and torque
->
+vectors will diverge from each other more rapidly.
+\subsection{Monte Carlo and the Energy Gap}\label{sec:MCMethods}
-–
+Evaluation of the pairwise summation techniques (outlined in section
-–
+\ref{sec:ESMethods}) for use in MC simulations was performed through
-–
+study of the energy differences between conformations.  Considering
-–
+the SPME results to be the correct or desired behavior, ideal
-–
+performance of a tested method was taken to be agreement between the
-–
+energy differences calculated.  Linear least squares regression of the
-–
+$\Delta E$ values between configurations using SPME against $\Delta E$
-–
+values using tested methods provides a quantitative comparison of this
-–
+agreement.  Unitary results for both the correlation and correlation
-–
+coefficient for these regressions indicate equivalent energetic
-–
+results between the methods.  The correlation is the slope of the
-–
+plotted data while the correlation coefficient ($R^2$) is a measure of
-–
+the of the data scatter around the fitted line and tells about the
-–
+quality of the fit (Fig. \ref{fig:linearFit}).
-+
+The pairwise summation techniques (outlined in section
-+
+\ref{sec:ESMethods}) were evaluated for use in MC simulations by
-+
+studying the energy differences between conformations.  We took the
-+
+SPME-computed energy difference between two conformations to be the
-+
+correct behavior. An ideal performance by an alternative method would
-+
+reproduce these energy differences exactly (even if the absolute
-+
+energies calculated by the methods are different).  Since none of the
-+
+methods provide exact energy differences, we used linear least squares
-+
+regressions of energy gap data to evaluate how closely the methods
-+
+mimicked the Ewald energy gaps.  Unitary results for both the
-+
+correlation (slope) and correlation coefficient for these regressions
-+
+indicate perfect agreement between the alternative method and SPME.
-+
+Sample correlation plots for two alternate methods are shown in
-+
+Fig. \ref{fig:linearFit}.
-+
+\begin{figure}
+\centering
+\includegraphics[width = \linewidth]{./dualLinear.pdf}
-<
+\caption{Example least squares regressions of the configuration energy differences for SPC/E water systems. The upper plot shows a data set with a poor correlation coefficient ($R^2$), while the lower plot shows a data set with a good correlation coefficient.}
-<
+\label{fig:linearFit}
->
+\caption{Example least squares regressions of the configuration energy
->
+differences for SPC/E water systems. The upper plot shows a data set
->
+with a poor correlation coefficient ($R^2$), while the lower plot
->
+shows a data set with a good correlation coefficient.}
->
+\label{fig:linearFit}
+\end{figure}
-<
+Each system type (detailed in section \ref{sec:RepSims}) studied
-<
+consisted of 500 independent configurations, each equilibrated from
-<
+higher temperature trajectories. Thus, 124,750 $\Delta E$ data points
-<
+are used in a regression of a single system type.  Results and
-<
+discussion for the individual analysis of each of the system types
-<
+appear in the supporting information, while the cumulative results
-<
+over all the investigated systems appears below in section
-<
+\ref{sec:EnergyResults}.
->
+Each system type (detailed in section \ref{sec:RepSims}) was
->
+represented using 500 independent configurations.  Additionally, we
->
+used seven different system types, so each of the alternative
->
+(non-Ewald) electrostatic summation methods was evaluated using
->
+,250 configurational energy differences.
-+
+Results and discussion for the individual analysis of each of the
-+
+system types appear in the supporting information, while the
-+
+cumulative results over all the investigated systems appears below in
-+
+section \ref{sec:EnergyResults}.
-+
+\subsection{Molecular Dynamics and the Force and Torque Vectors}\label{sec:MDMethods}
-<
+Evaluation of the pairwise methods (outlined in section
-<
+\ref{sec:ESMethods}) for use in MD simulations was performed through
-<
+comparison of the force and torque vectors obtained with those from
-<
+SPME.  Both the magnitude and the direction of these vectors on each
-<
+of the bodies in the system were analyzed.  For the magnitude of these
-<
+vectors, linear least squares regression analysis can be performed as
-<
+described previously for comparing $\Delta E$ values. Instead of a
-<
+single value between two system configurations, there is a value for
-<
+each particle in each configuration.  For a system of 1000 water
-<
+molecules and 40 ions, there are 1040 force vectors and 1000 torque
-<
+vectors.  With 500 configurations, this results in 520,000 force and
-<
+,000 torque vector comparisons samples for each system type.
->
+We evaluated the pairwise methods (outlined in section
->
+\ref{sec:ESMethods}) for use in MD simulations by
->
+comparing the force and torque vectors with those obtained using the
->
+reference Ewald summation (SPME).  Both the magnitude and the
->
+direction of these vectors on each of the bodies in the system were
->
+analyzed.  For the magnitude of these vectors, linear least squares
->
+regression analyses were performed as described previously for
->
+comparing $\Delta E$ values.  Instead of a single energy difference
->
+between two system configurations, we compared the magnitudes of the
->
+forces (and torques) on each molecule in each configuration.  For a
->
+system of 1000 water molecules and 40 ions, there are 1040 force
->
+vectors and 1000 torque vectors.  With 500 configurations, this
->
+results in 520,000 force and 500,000 torque vector comparisons.
->
+Additionally, data from seven different system types was aggregated
->
+before the comparison was made.
-<
+The force and torque vector directions were investigated through
-<
+measurement of the angle ($\theta$) formed between those from the
-<
+particular method and those from SPME
->
+The {\it directionality} of the force and torque vectors was
->
+investigated through measurement of the angle ($\theta$) formed
->
+between those computed from the particular method and those from SPME,
+\begin{equation}
-<
+\theta_F = \frac{\vec{F}_\textrm{SPME}}{|\vec{F}_\textrm{SPME}|}\cdot\frac{\vec{F}_\textrm{Method}}{|\vec{F}_\textrm{Method}|}.
->
+\theta_f = \cos^{-1} \left(\hat{F}_\textrm{SPME} \cdot \hat{F}_\textrm{M}\right),
+\end{equation}
-<
+Each of these $\theta$ values was accumulated in a distribution
-<
+function, weighted by the area on the unit sphere.  Non-linear fits
-<
+were used to measure the shape of the resulting distributions.
->
+where $\hat{f}_\textrm{M}$ is the unit vector pointing along the force
->
+vector computed using method M.  Each of these $\theta$ values was
->
+accumulated in a distribution function and weighted by the area on the
->
+unit sphere.  Since this distribution is a measure of angular error
->
+between two different electrostatic summation methods, there is no
->
+{\it a priori} reason for the profile to adhere to any specific
->
+shape. Thus, gaussian fits were used to measure the width of the
->
+resulting distributions.
->
+%
->
+%\begin{figure}
->
+%\centering
->
+%\includegraphics[width = \linewidth]{./gaussFit.pdf}
->
+%\caption{Sample fit of the angular distribution of the force vectors
->
+%accumulated using all of the studied systems.  Gaussian fits were used
->
+%to obtain values for the variance in force and torque vectors.}
->
+%\label{fig:gaussian}
->
+%\end{figure}
->
+%
->
+%Figure \ref{fig:gaussian} shows an example distribution with applied
->
+%non-linear fits.  The solid line is a Gaussian profile, while the
->
+%dotted line is a Voigt profile, a convolution of a Gaussian and a
->
+%Lorentzian.
->
+%Since this distribution is a measure of angular error between two
->
+%different electrostatic summation methods, there is no {\it a priori}
->
+%reason for the profile to adhere to any specific shape.
->
+%Gaussian fits was used to compare all the tested methods.
->
+The variance ($\sigma^2$) was extracted from each of these fits and
->
+was used to compare distribution widths.  Values of $\sigma^2$ near
->
+zero indicate vector directions indistinguishable from those
->
+calculated when using the reference method (SPME).
-<
+\begin{figure}
-<
+\centering
-<
+\includegraphics[width = \linewidth]{./gaussFit.pdf}
-<
+\caption{Sample fit of the angular distribution of the force vectors over all of the studied systems.  Gaussian fits were used to obtain values for the variance in force and torque vectors used in the following figure.}
-<
+\label{fig:gaussian}
-<
+\end{figure}
->
+\subsection{Short-time Dynamics}
-<
+Figure \ref{fig:gaussian} shows an example distribution with applied
-<
+non-linear fits.  The solid line is a Gaussian profile, while the
-<
+dotted line is a Voigt profile, a convolution of a Gaussian and a
-<
+Lorentzian.  Since this distribution is a measure of angular error
-<
+between two different electrostatic summation methods, there is
-<
+particular reason for the profile to adhere to a specific shape.
-<
+Because of this and the Gaussian profile's more statistically
-<
+meaningful properties, Gaussian fits was used to compare all the
-<
+tested methods.  The variance ($\sigma^2$) was extracted from each of
-<
+these fits and was used to compare distribution widths.  Values of
-<
+$\sigma^2$ near zero indicate vector directions indistinguishable from
-<
+those calculated when using SPME.
-<
-<
+\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods}
-<
+Evaluation of the long-time dynamics of charged systems was performed
-<
+by considering the NaCl crystal system while using a subset of the
-<
+best performing pairwise methods.  The NaCl crystal was chosen to
-<
+avoid possible complications involving the propagation techniques of
-<
+orientational motion in molecular systems.  To enhance the atomic
-<
+motion, these crystals were equilibrated at 1000 K, near the
-<
+experimental $T_m$ for NaCl.  Simulations were performed under the
-<
+microcanonical ensemble, and velocity autocorrelation functions
-<
+(Eq. \ref{eq:vCorr}) were computed for each of the trajectories,
->
+The effects of the alternative electrostatic summation methods on the
->
+short-time dynamics of charged systems were evaluated by considering a
->
+NaCl crystal at a temperature of 1000 K.  A subset of the best
->
+performing pairwise methods was used in this comparison.  The NaCl
->
+crystal was chosen to avoid possible complications from the treatment
->
+of orientational motion in molecular systems.  All systems were
->
+started with the same initial positions and velocities.  Simulations
->
+were performed under the microcanonical ensemble, and velocity
->
+autocorrelation functions (Eq. \ref{eq:vCorr}) were computed for each
->
+of the trajectories,
+\begin{equation}
-<
+C_v(t) = \langle v_i(0)\cdot v_i(t)\rangle.
->
+C_v(t) = \frac{\langle v_i(0)\cdot v_i(t)\rangle}{\langle v_i(0)\cdot v_i(0)\rangle}.
+\label{eq:vCorr}
+\end{equation}
-<
+Velocity autocorrelation functions require detailed short time data
-<
+and long trajectories for good statistics, thus velocity information
-<
+was saved every 5 fs over 100 ps trajectories.  The power spectrum
-<
+($I(\omega)$) is obtained via Fourier transform of the autocorrelation
-<
+function
-<
+\begin{equation}
-<
+I(\omega) = \frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt,
->
+Velocity autocorrelation functions require detailed short time data,
->
+thus velocity information was saved every 2 fs over 10 ps
->
+trajectories. Because the NaCl crystal is composed of two different
->
+atom types, the average of the two resulting velocity autocorrelation
->
+functions was used for comparisons.
->
->
+\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods}
->
->
+The effects of the same subset of alternative electrostatic methods on
->
+the {\it long-time} dynamics of charged systems were evaluated using
->
+the same model system (NaCl crystals at 1000K).  The power spectrum
->
+($I(\omega)$) was obtained via Fourier transform of the velocity
->
+autocorrelation function, \begin{equation} I(\omega) =
->
+\frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt,
+\label{eq:powerSpec}
+\end{equation}
-<
+where the frequency, $\omega=0,\ 1,\ ...,\ N-1$.
->
+where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. Again, because the
->
+NaCl crystal is composed of two different atom types, the average of
->
+the two resulting power spectra was used for comparisons. Simulations
->
+were performed under the microcanonical ensemble, and velocity
->
+information was saved every 5 fs over 100 ps trajectories.
+\subsection{Representative Simulations}\label{sec:RepSims}
-<
+A variety of common and representative simulations were analyzed to
-<
+determine the relative effectiveness of the pairwise summation
-<
+techniques in reproducing the energetics and dynamics exhibited by
-<
+SPME.  The studied systems were as follows:
->
+A variety of representative simulations were analyzed to determine the
->
+relative effectiveness of the pairwise summation techniques in
->
+reproducing the energetics and dynamics exhibited by SPME.  We wanted
->
+to span the space of modern simulations (i.e. from liquids of neutral
->
+molecules to ionic crystals), so the systems studied were:
+\begin{enumerate}
-<
+\item Liquid Water
-<
+\item Crystalline Water (Ice I$_\textrm{c}$)
-<
+\item NaCl Crystal
-<
+\item NaCl Melt
-<
+\item Low Ionic Strength Solution of NaCl in Water
-<
+\item High Ionic Strength Solution of NaCl in Water
-<
+\item 6 \AA\  Radius Sphere of Argon in Water
->
+\item liquid water (SPC/E),\cite{Berendsen87}
->
+\item crystalline water (Ice I$_\textrm{c}$ crystals of SPC/E),
->
+\item NaCl crystals,
->
+\item NaCl melts,
->
+\item a low ionic strength solution of NaCl in water (0.11 M),
->
+\item a high ionic strength solution of NaCl in water (1.1 M), and
->
+\item a 6 \AA\  radius sphere of Argon in water.
+\end{enumerate}
+By utilizing the pairwise techniques (outlined in section
+\ref{sec:ESMethods}) in systems composed entirely of neutral groups,
-<
+charged particles, and mixtures of the two, we can comment on possible
-<
+system dependence and/or universal applicability of the techniques.
->
+charged particles, and mixtures of the two, we hope to discern under
->
+which conditions it will be possible to use one of the alternative
->
+summation methodologies instead of the Ewald sum.
-<
+Generation of the system configurations was dependent on the system
-<
+type.  For the solid and liquid water configurations, configuration
-<
+snapshots were taken at regular intervals from higher temperature 1000
-<
+SPC/E water molecule trajectories and each equilibrated individually.
-<
+The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl-
-<
+ions and were selected and equilibrated in the same fashion as the
-<
+water systems.  For the low and high ionic strength NaCl solutions, 4
-<
+and 40 ions were first solvated in a 1000 water molecule boxes
-<
+respectively.  Ion and water positions were then randomly swapped, and
->
+For the solid and liquid water configurations, configurations were
->
+taken at regular intervals from high temperature trajectories of 1000
->
+SPC/E water molecules.  Each configuration was equilibrated
->
+independently at a lower temperature (300~K for the liquid, 200~K for
->
+the crystal).  The solid and liquid NaCl systems consisted of 500
->
+$\textrm{Na}^{+}$ and 500 $\textrm{Cl}^{-}$ ions.  Configurations for
->
+these systems were selected and equilibrated in the same manner as the
->
+water systems.  The equilibrated temperatures were 1000~K for the NaCl
->
+crystal and 7000~K for the liquid. The ionic solutions were made by
->
+solvating 4 (or 40) ions in a periodic box containing 1000 SPC/E water
->
+molecules.  Ion and water positions were then randomly swapped, and
+the resulting configurations were again equilibrated individually.
-<
+Finally, for the Argon/Water "charge void" systems, the identities of
-<
+all the SPC/E waters within 6 \AA\ of the center of the equilibrated
-<
+water configurations were converted to argon
-<
+(Fig. \ref{fig:argonSlice}).
->
+Finally, for the Argon / Water ``charge void'' systems, the identities
->
+of all the SPC/E waters within 6 \AA\ of the center of the
->
+equilibrated water configurations were converted to argon.
->
+%(Fig. \ref{fig:argonSlice}).
-<
+\begin{figure}
-<
+\centering
-<
+\includegraphics[width = \linewidth]{./slice.pdf}
-<
+\caption{A slice from the center of a water box used in a charge void simulation.  The darkened region represents the boundary sphere within which the water molecules were converted to argon atoms.}
-<
+\label{fig:argonSlice}
-<
+\end{figure}
->
+These procedures guaranteed us a set of representative configurations
->
+from chemically-relevant systems sampled from appropriate
->
+ensembles. Force field parameters for the ions and Argon were taken
->
+from the force field utilized by {\sc oopse}.\cite{Meineke05}
-<
+\subsection{Electrostatic Summation Methods}\label{sec:ESMethods}
-<
+Electrostatic summation method comparisons were performed using SPME,
-<
+the Shifted-Potential and Shifted-Force methods - both with damping
-<
+parameters ($\alpha$) of 0.0, 0.1, 0.2, and 0.3 \AA$^{-1}$ (no, weak,
-<
+moderate, and strong damping respectively), reaction field with an
-<
+infinite dielectric constant, and an unmodified cutoff.  Group-based
-<
+cutoffs with a fifth-order polynomial switching function were
-<
+necessary for the reaction field simulations and were utilized in the
-<
+SP, SF, and pure cutoff methods for comparison to the standard lack of
-<
+group-based cutoffs with a hard truncation.  The SPME calculations
-<
+were performed using the TINKER implementation of SPME,\cite{Ponder87}
-<
+while all other method calculations were performed using the OOPSE
-<
+molecular mechanics package.\cite{Meineke05}
->
+%\begin{figure}
->
+%\centering
->
+%\includegraphics[width = \linewidth]{./slice.pdf}
->
+%\caption{A slice from the center of a water box used in a charge void
->
+%simulation.  The darkened region represents the boundary sphere within
->
+%which the water molecules were converted to argon atoms.}
->
+%\label{fig:argonSlice}
->
+%\end{figure}
-<
+These methods were additionally evaluated with three different cutoff
-<
+radii (9, 12, and 15 \AA) to investigate possible cutoff radius
-<
+dependence.  It should be noted that the damping parameter chosen in
-<
+SPME, or so called ``Ewald Coefficient", has a significant effect on
-<
+the energies and forces calculated.  Typical molecular mechanics
-<
+packages default this to a value dependent on the cutoff radius and a
-<
+tolerance (typically less than $1 \times 10^{-4}$ kcal/mol).  Smaller
-<
+tolerances are typically associated with increased accuracy in the
-<
+real-space portion of the summation.\cite{Essmann95} The default
-<
+TINKER tolerance of $1 \times 10^{-8}$ kcal/mol was used in all SPME
-<
+calculations, resulting in Ewald Coefficients of 0.4200, 0.3119, and
-<
+.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ respectively.
->
+\subsection{Comparison of Summation Methods}\label{sec:ESMethods}
->
+We compared the following alternative summation methods with results
->
+from the reference method (SPME):
->
+\begin{itemize}
->
+\item {\sc sp} with damping parameters ($\alpha$) of 0.0, 0.1, 0.2,
->
+and 0.3 \AA$^{-1}$,
->
+\item {\sc sf} with damping parameters ($\alpha$) of 0.0, 0.1, 0.2,
->
+and 0.3 \AA$^{-1}$,
->
+\item reaction field with an infinite dielectric constant, and
->
+\item an unmodified cutoff.
->
+\end{itemize}
->
+Group-based cutoffs with a fifth-order polynomial switching function
->
+were utilized for the reaction field simulations.  Additionally, we
->
+investigated the use of these cutoffs with the SP, SF, and pure
->
+cutoff.  The SPME electrostatics were performed using the TINKER
->
+implementation of SPME,\cite{Ponder87} while all other calculations
->
+were performed using the {\sc oopse} molecular mechanics
->
+package.\cite{Meineke05} All other portions of the energy calculation
->
+(i.e. Lennard-Jones interactions) were handled in exactly the same
->
+manner across all systems and configurations.
-+
+The althernative methods were also evaluated with three different
-+
+cutoff radii (9, 12, and 15 \AA).  As noted previously, the
-+
+convergence parameter ($\alpha$) plays a role in the balance of the
-+
+real-space and reciprocal-space portions of the Ewald calculation.
-+
+Typical molecular mechanics packages set this to a value dependent on
-+
+the cutoff radius and a tolerance (typically less than $1 \times
-+
+^{-4}$ kcal/mol).  Smaller tolerances are typically associated with
-+
+increasing accuracy at the expense of computational time spent on the
-+
+reciprocal-space portion of the summation.\cite{Perram88,Essmann95}
-+
+The default TINKER tolerance of $1 \times 10^{-8}$ kcal/mol was used
-+
+in all SPME calculations, resulting in Ewald coefficients of 0.4200,
-+
+.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\
-+
+respectively.
-+
+\section{Results and Discussion}
+\subsection{Configuration Energy Differences}\label{sec:EnergyResults}
+\begin{figure}
+\centering
+\includegraphics[width=5.5in]{./delEplot.pdf}
-<
+\caption{Statistical analysis of the quality of configurational energy differences for a given electrostatic method compared with the reference Ewald sum.  Results with a value equal to 1 (dashed line) indicate $\Delta E$ values indistinguishable from those obtained using SPME.  Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).}
->
+\caption{Statistical analysis of the quality of configurational energy
->
+differences for a given electrostatic method compared with the
->
+reference Ewald sum.  Results with a value equal to 1 (dashed line)
->
+indicate $\Delta E$ values indistinguishable from those obtained using
->
+SPME.  Different values of the cutoff radius are indicated with
->
+different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
->
+inverted triangles).}
+\label{fig:delE}
+\end{figure}
-<
+In this figure, it is apparent that it is unreasonable to expect
-<
+realistic results using an unmodified cutoff.  This is not all that
-<
+surprising since this results in large energy fluctuations as atoms
-<
+move in and out of the cutoff radius.  These fluctuations can be
-<
+alleviated to some degree by using group based cutoffs with a
-<
+switching function.\cite{Steinbach94} The Group Switch Cutoff row
-<
+doesn't show a significant improvement in this plot because the salt
-<
+and salt solution systems contain non-neutral groups, see the
-<
+accompanying supporting information for a comparison where all groups
-<
+are neutral.
->
+The most striking feature of this plot is how well the Shifted Force
->
+({\sc sf}) and Shifted Potential ({\sc sp}) methods capture the energy
->
+differences.  For the undamped {\sc sf} method, and the
->
+moderately-damped {\sc sp} methods, the results are nearly
->
+indistinguishable from the Ewald results.  The other common methods do
->
+significantly less well.
-<
+Correcting the resulting charged cutoff sphere is one of the purposes
-<
+of the damped Coulomb summation proposed by Wolf \textit{et
-<
+al.},\cite{Wolf99} and this correction indeed improves the results as
-<
+seen in the Shifted-Potental rows.  While the undamped case of this
-<
+method is a significant improvement over the pure cutoff, it still
-<
+doesn't correlate that well with SPME.  Inclusion of potential damping
-<
+improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows
-<
+an excellent correlation and quality of fit with the SPME results,
-<
+particularly with a cutoff radius greater than 12 \AA .  Use of a
-<
+larger damping parameter is more helpful for the shortest cutoff
-<
+shown, but it has a detrimental effect on simulations with larger
-<
+cutoffs.  In the Shifted-Force sets, increasing damping results in
-<
+progressively poorer correlation.  Overall, the undamped case is the
-<
+best performing set, as the correlation and quality of fits are
-<
+consistently superior regardless of the cutoff distance.  This result
-<
+is beneficial in that the undamped case is less computationally
-<
+prohibitive do to the lack of complimentary error function calculation
-<
+when performing the electrostatic pair interaction.  The reaction
-<
+field results illustrates some of that method's limitations, primarily
-<
+that it was developed for use in homogenous systems; although it does
-<
+provide results that are an improvement over those from an unmodified
-<
+cutoff.
->
+The unmodified cutoff method is essentially unusable.  This is not
->
+surprising since hard cutoffs give large energy fluctuations as atoms
->
+or molecules move in and out of the cutoff
->
+radius.\cite{Rahman71,Adams79} These fluctuations can be alleviated to
->
+some degree by using group based cutoffs with a switching
->
+function.\cite{Adams79,Steinbach94,Leach01} However, we do not see
->
+significant improvement using the group-switched cutoff because the
->
+salt and salt solution systems contain non-neutral groups.  Interested
->
+readers can consult the accompanying supporting information for a
->
+comparison where all groups are neutral.
-+
+For the {\sc sp} method, inclusion of electrostatic damping improves
-+
+the agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$
-+
+shows an excellent correlation and quality of fit with the SPME
-+
+results, particularly with a cutoff radius greater than 12
-+
+\AA .  Use of a larger damping parameter is more helpful for the
-+
+shortest cutoff shown, but it has a detrimental effect on simulations
-+
+with larger cutoffs.
-+
-+
+In the {\sc sf} sets, increasing damping results in progressively {\it
-+
+worse} correlation with Ewald.  Overall, the undamped case is the best
-+
+performing set, as the correlation and quality of fits are
-+
+consistently superior regardless of the cutoff distance.  The undamped
-+
+case is also less computationally demanding (because no evaluation of
-+
+the complementary error function is required).
-+
-+
+The reaction field results illustrates some of that method's
-+
+limitations, primarily that it was developed for use in homogenous
-+
+systems; although it does provide results that are an improvement over
-+
+those from an unmodified cutoff.
-+
+\subsection{Magnitudes of the Force and Torque Vectors}
+Evaluation of pairwise methods for use in Molecular Dynamics
+simulations requires consideration of effects on the forces and
-<
+torques.  Investigation of the force and torque vector magnitudes
-<
+provides a measure of the strength of these values relative to SPME.
-<
+Figures \ref{fig:frcMag} and \ref{fig:trqMag} respectively show the
-<
+force and torque vector magnitude regression results for the
-<
+accumulated analysis over all the system types.
->
+torques.  Figures \ref{fig:frcMag} and \ref{fig:trqMag} show the
->
+regression results for the force and torque vector magnitudes,
->
+respectively.  The data in these figures was generated from an
->
+accumulation of the statistics from all of the system types.
+\begin{figure}
+\centering
+\includegraphics[width=5.5in]{./frcMagplot.pdf}
-<
+\caption{Statistical analysis of the quality of the force vector magnitudes for a given electrostatic method compared with the reference Ewald sum.  Results with a value equal to 1 (dashed line) indicate force magnitude values indistinguishable from those obtained using SPME.  Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).}
->
+\caption{Statistical analysis of the quality of the force vector
->
+magnitudes for a given electrostatic method compared with the
->
+reference Ewald sum.  Results with a value equal to 1 (dashed line)
->
+indicate force magnitude values indistinguishable from those obtained
->
+using SPME.  Different values of the cutoff radius are indicated with
->
+different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
->
+inverted triangles).}
+\label{fig:frcMag}
+\end{figure}
-+
+Again, it is striking how well the Shifted Potential and Shifted Force
-+
+methods are doing at reproducing the SPME forces.  The undamped and
-+
+weakly-damped {\sc sf} method gives the best agreement with Ewald.
-+
+This is perhaps expected because this method explicitly incorporates a
-+
+smooth transition in the forces at the cutoff radius as well as the
-+
+neutralizing image charges.
-+
+Figure \ref{fig:frcMag}, for the most part, parallels the results seen
+in the previous $\Delta E$ section.  The unmodified cutoff results are
+poor, but using group based cutoffs and a switching function provides
-<
+a improvement much more significant than what was seen with $\Delta
-<
+E$.  Looking at the Shifted-Potential sets, the slope and $R^2$
-<
+improve with the use of damping to an optimal result of 0.2 \AA
-<
+$^{-1}$ for the 12 and 15 \AA\ cutoffs.  Further increases in damping,
->
+an improvement much more significant than what was seen with $\Delta
->
+E$.
->
->
+With moderate damping and a large enough cutoff radius, the {\sc sp}
->
+method is generating usable forces.  Further increases in damping,
+while beneficial for simulations with a cutoff radius of 9 \AA\ , is
-<
+detrimental to simulations with larger cutoff radii.  The undamped
-<
+Shifted-Force method gives forces in line with those obtained using
-<
+SPME, and use of a damping function results in minor improvement.  The
-<
+reaction field results are surprisingly good, considering the poor
->
+detrimental to simulations with larger cutoff radii.
->
->
+The reaction field results are surprisingly good, considering the poor
+quality of the fits for the $\Delta E$ results.  There is still a
-<
+considerable degree of scatter in the data, but it correlates well in
-<
+general.  To be fair, we again note that the reaction field
-<
+calculations do not encompass NaCl crystal and melt systems, so these
->
+considerable degree of scatter in the data, but the forces correlate
->
+well with the Ewald forces in general.  We note that the reaction
->
+field calculations do not include the pure NaCl systems, so these
+results are partly biased towards conditions in which the method
+performs more favorably.
+\begin{figure}
+\centering
+\includegraphics[width=5.5in]{./trqMagplot.pdf}
-<
+\caption{Statistical analysis of the quality of the torque vector magnitudes for a given electrostatic method compared with the reference Ewald sum.  Results with a value equal to 1 (dashed line) indicate torque magnitude values indistinguishable from those obtained using SPME.  Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).}
->
+\caption{Statistical analysis of the quality of the torque vector
->
+magnitudes for a given electrostatic method compared with the
->
+reference Ewald sum.  Results with a value equal to 1 (dashed line)
->
+indicate torque magnitude values indistinguishable from those obtained
->
+using SPME.  Different values of the cutoff radius are indicated with
->
+different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
->
+inverted triangles).}
+\label{fig:trqMag}
+\end{figure}
-<
+To evaluate the torque vector magnitudes, the data set from which
-<
+values are drawn is limited to rigid molecules in the systems
-<
+(i.e. water molecules).  In spite of this smaller sampling pool, the
-<
+torque vector magnitude results in figure \ref{fig:trqMag} are still
-<
+similar to those seen for the forces; however, they more clearly show
-<
+the improved behavior that comes with increasing the cutoff radius.
-<
+Moderate damping is beneficial to the Shifted-Potential and helpful
-<
+yet possibly unnecessary with the Shifted-Force method, and they also
-<
+show that over-damping adversely effects all cutoff radii rather than
-<
+showing an improvement for systems with short cutoffs.  The reaction
-<
+field method performs well when calculating the torques, better than
-<
+the Shifted Force method over this limited data set.
->
+Molecular torques were only available from the systems which contained
->
+rigid molecules (i.e. the systems containing water).  The data in
->
+fig. \ref{fig:trqMag} is taken from this smaller sampling pool.
-+
+Torques appear to be much more sensitive to charges at a longer
-+
+distance.   The striking feature in comparing the new electrostatic
-+
+methods with SPME is how much the agreement improves with increasing
-+
+cutoff radius.  Again, the weakly damped and undamped {\sc sf} method
-+
+appears to be reproducing the SPME torques most accurately.
-+
-+
+Water molecules are dipolar, and the reaction field method reproduces
-+
+the effect of the surrounding polarized medium on each of the
-+
+molecular bodies. Therefore it is not surprising that reaction field
-+
+performs best of all of the methods on molecular torques.
-+
+\subsection{Directionality of the Force and Torque Vectors}
-<
+Having force and torque vectors with magnitudes that are well
-<
+correlated to SPME is good, but if they are not pointing in the proper
-<
+direction the results will be incorrect.  These vector directions were
-<
+investigated through measurement of the angle formed between them and
-<
+those from SPME.  The results (Fig. \ref{fig:frcTrqAng}) are compared
-<
+through the variance ($\sigma^2$) of the Gaussian fits of the angle
-<
+error distributions of the combined set over all system types.
->
+It is clearly important that a new electrostatic method can reproduce
->
+the magnitudes of the force and torque vectors obtained via the Ewald
->
+sum. However, the {\it directionality} of these vectors will also be
->
+vital in calculating dynamical quantities accurately.  Force and
->
+torque directionalities were investigated by measuring the angles
->
+formed between these vectors and the same vectors calculated using
->
+SPME.  The results (Fig. \ref{fig:frcTrqAng}) are compared through the
->
+variance ($\sigma^2$) of the Gaussian fits of the angle error
->
+distributions of the combined set over all system types.
+\begin{figure}
+\centering
+\includegraphics[width=5.5in]{./frcTrqAngplot.pdf}
-<
+\caption{Statistical analysis of the quality of the Gaussian fit of the force and torque vector angular distributions for a given electrostatic method compared with the reference Ewald sum.  Results with a variance ($\sigma^2$) equal to zero (dashed line) indicate force and torque directions indistinguishable from those obtained using SPME.  Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).}
->
+\caption{Statistical analysis of the width of the angular distribution
->
+that the force and torque vectors from a given electrostatic method
->
+make with their counterparts obtained using the reference Ewald sum.
->
+Results with a variance ($\sigma^2$) equal to zero (dashed line)
->
+indicate force and torque directions indistinguishable from those
->
+obtained using SPME.  Different values of the cutoff radius are
->
+indicated with different symbols (9\AA\ = circles, 12\AA\ = squares,
->
+and 15\AA\ = inverted triangles).}
+\label{fig:frcTrqAng}
+\end{figure}
+Both the force and torque $\sigma^2$ results from the analysis of the
+total accumulated system data are tabulated in figure
-<
+\ref{fig:frcTrqAng}.  All of the sets, aside from the over-damped case
-<
+show the improvement afforded by choosing a longer simulation cutoff.
-<
+Increasing the cutoff from 9 to 12 \AA\ typically results in a halving
-<
+of the distribution widths, with a similar improvement going from 12
-<
+to 15 \AA .  The undamped Shifted-Force, Group Based Cutoff, and
-<
+Reaction Field methods all do equivalently well at capturing the
-<
+direction of both the force and torque vectors.  Using damping
-<
+improves the angular behavior significantly for the Shifted-Potential
-<
+and moderately for the Shifted-Force methods.  Increasing the damping
-<
+too far is destructive for both methods, particularly to the torque
-<
+vectors.  Again it is important to recognize that the force vectors
-<
+cover all particles in the systems, while torque vectors are only
-<
+available for neutral molecular groups.  Damping appears to have a
-<
+more beneficial effect on non-neutral bodies, and this observation is
-<
+investigated further in the accompanying supporting information.
->
+\ref{fig:frcTrqAng}. Here it is clear that the Shifted Potential ({\sc
->
+sp}) method would be essentially unusable for molecular dynamics until
->
+the damping function is added.  The Shifted Force ({\sc sf}) method,
->
+however, is generating force and torque vectors which are within a few
->
+degrees of the Ewald results even with weak (or no) damping.
-+
+All of the sets (aside from the over-damped case) show the improvement
-+
+afforded by choosing a larger cutoff radius.  Increasing the cutoff
-+
+from 9 to 12 \AA\ typically results in a halving of the width of the
-+
+distribution, with a similar improvement going from 12 to 15
-+
+\AA .
-+
-+
+The undamped {\sc sf}, group-based cutoff, and reaction field methods
-+
+all do equivalently well at capturing the direction of both the force
-+
+and torque vectors.  Using damping improves the angular behavior
-+
+significantly for the {\sc sp} and moderately for the {\sc sf}
-+
+methods.  Overdamping is detrimental to both methods.  Again it is
-+
+important to recognize that the force vectors cover all particles in
-+
+the systems, while torque vectors are only available for neutral
-+
+molecular groups.  Damping appears to have a more beneficial effect on
-+
+charged bodies, and this observation is investigated further in the
-+
+accompanying supporting information.
-+
-+
+Although not discussed previously, group based cutoffs can be applied
-+
+to both the {\sc sp} and {\sc sf} methods.  Use of a switching
-+
+function corrects for the discontinuities that arise when atoms of the
-+
+two groups exit the cutoff radius before the group centers leave each
-+
+other's cutoff. Though there are no significant benefits or drawbacks
-+
+observed in $\Delta E$ and vector magnitude results when doing this,
-+
+there is a measurable improvement in the vector angle results.  Table
-+
+\ref{tab:groupAngle} shows the angular variance values obtained using
-+
+group based cutoffs and a switching function alongside the results
-+
+seen in figure \ref{fig:frcTrqAng}.  The {\sc sp} shows much narrower
-+
+angular distributions for both the force and torque vectors when using
-+
+an $\alpha$ of 0.2 \AA$^{-1}$ or less, while {\sc sf} shows
-+
+improvements in the undamped and lightly damped cases.  Thus, by
-+
+calculating the electrostatic interactions in terms of molecular pairs
-+
+rather than atomic pairs, the direction of the force and torque
-+
+vectors can be determined more accurately.
-+
+\begin{table}[htbp]
+   \centering
-<
+   \caption{Variance ($\sigma^2$) of the force (top set) and torque (bottom set) vector angle difference distributions for the Shifted Potential and Shifted Force methods.  Calculations were performed both with (Y) and without (N) group based cutoffs and a switching function.  The $\alpha$ values have units of \AA$^{-1}$ and the variance values have units of degrees$^2$.}
->
+   \caption{Variance ($\sigma^2$) of the force (top set) and torque
->
+(bottom set) vector angle difference distributions for the Shifted Potential and Shifted Force methods.  Calculations were performed both with (Y) and without (N) group based cutoffs and a switching function.  The $\alpha$ values have units of \AA$^{-1}$ and the variance values have units of degrees$^2$.}
+   \begin{tabular}{@{} ccrrrrrrrr @{}}
+      \\
+      \toprule
+   \label{tab:groupAngle}
+\end{table}
-–
+Although not discussed previously, group based cutoffs can be applied
-–
+to both the Shifted-Potential and Shifted-Force methods.  Use off a
-–
+switching function corrects for the discontinuities that arise when
-–
+atoms of a group exit the cutoff before the group's center of mass.
-–
+Though there are no significant benefit or drawbacks observed in
-–
+$\Delta E$ and vector magnitude results when doing this, there is a
-–
+measurable improvement in the vector angle results.  Table
-–
+\ref{tab:groupAngle} shows the angular variance values obtained using
-–
+group based cutoffs and a switching function alongside the standard
-–
+results seen in figure \ref{fig:frcTrqAng} for comparison purposes.
-–
+The Shifted-Potential shows much narrower angular distributions for
-–
+both the force and torque vectors when using an $\alpha$ of 0.2
-–
+\AA$^{-1}$ or less, while Shifted-Force shows improvements in the
-–
+undamped and lightly damped cases.  Thus, by calculating the
-–
+electrostatic interactions in terms of molecular pairs rather than
-–
+atomic pairs, the direction of the force and torque vectors are
-–
+determined more accurately.
-–
+One additional trend to recognize in table \ref{tab:groupAngle} is
-<
+that the $\sigma^2$ values for both Shifted-Potential and
-<
+Shifted-Force converge as $\alpha$ increases, something that is easier
-<
+to see when using group based cutoffs.  Looking back on figures
-<
+\ref{fig:delE}, \ref{fig:frcMag}, and \ref{fig:trqMag}, show this
-<
+behavior clearly at large $\alpha$ and cutoff values.  The reason for
-<
+this is that the complimentary error function inserted into the
-<
+potential weakens the electrostatic interaction as $\alpha$ increases.
-<
+Thus, at larger values of $\alpha$, both the summation method types
-<
+progress toward non-interacting functions, so care is required in
-<
+choosing large damping functions lest one generate an undesirable loss
-<
+in the pair interaction.  Kast \textit{et al.}  developed a method for
-<
+choosing appropriate $\alpha$ values for these types of electrostatic
-<
+summation methods by fitting to $g(r)$ data, and their methods
-<
+indicate optimal values of 0.34, 0.25, and 0.16 \AA$^{-1}$ for cutoff
-<
+values of 9, 12, and 15 \AA\ respectively.\cite{Kast03} These appear
-<
+to be reasonable choices to obtain proper MC behavior
-<
+(Fig. \ref{fig:delE}); however, based on these findings, choices this
-<
+high would introduce error in the molecular torques, particularly for
-<
+the shorter cutoffs.  Based on the above findings, empirical damping
-<
+up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably
-<
+unnecessary when using the Shifted-Force method.
->
+that the $\sigma^2$ values for both {\sc sp} and {\sc sf} converge as
->
+$\alpha$ increases, something that is easier to see when using group
->
+based cutoffs.  The reason for this is that the complimentary error
->
+function inserted into the potential weakens the electrostatic
->
+interaction as $\alpha$ increases.  Thus, at larger values of
->
+$\alpha$, both summation methods progress toward non-interacting
->
+functions, so care is required in choosing large damping functions
->
+lest one generate an undesirable loss in the pair interaction.  Kast
->
+\textit{et al.}  developed a method for choosing appropriate $\alpha$
->
+values for these types of electrostatic summation methods by fitting
->
+to $g(r)$ data, and their methods indicate optimal values of 0.34,
->
+.25, and 0.16 \AA$^{-1}$ for cutoff values of 9, 12, and 15 \AA\
->
+respectively.\cite{Kast03} These appear to be reasonable choices to
->
+obtain proper MC behavior (Fig. \ref{fig:delE}); however, based on
->
+these findings, choices this high would introduce error in the
->
+molecular torques, particularly for the shorter cutoffs.  Based on the
->
+above findings, empirical damping up to 0.2 \AA$^{-1}$ proves to be
->
+beneficial, but damping may be unnecessary when using the {\sc sf}
->
+method.
-+
+\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals}
-+
-+
+Zahn {\it et al.} investigated the structure and dynamics of water
-+
+using eqs. (\ref{eq:ZahnPot}) and
-+
+(\ref{eq:WolfForces}).\cite{Zahn02,Kast03} Their results indicated
-+
+that a method similar (but not identical with) the damped {\sc sf}
-+
+method resulted in properties very similar to those obtained when
-+
+using the Ewald summation.  The properties they studied (pair
-+
+distribution functions, diffusion constants, and velocity and
-+
+orientational correlation functions) may not be particularly sensitive
-+
+to the long-range and collective behavior that governs the
-+
+low-frequency behavior in crystalline systems.
-+
-+
+We are using two separate measures to probe the effects of these
-+
+alternative electrostatic methods on the dynamics in crystalline
-+
+materials.  For short- and intermediate-time dynamics, we are
-+
+computing the velocity autocorrelation function, and for long-time
-+
+and large length-scale collective motions, we are looking at the
-+
+low-frequency portion of the power spectrum.
-+
-+
+\begin{figure}
-+
+\centering
-+
+\includegraphics[width = \linewidth]{./vCorrPlot.pdf}
-+
+\caption{Velocity auto-correlation functions of NaCl crystals at
-+
+K using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc
-+
+sp} ($\alpha$ = 0.2). The inset is a magnification of the area around
-+
+the first minimum.  The times to first collision are nearly identical,
-+
+but differences can be seen in the peaks and troughs, where the
-+
+undamped and weakly damped methods are stiffer than the moderately
-+
+damped and SPME methods.}
-+
+\label{fig:vCorrPlot}
-+
+\end{figure}
-+
-+
+The short-time decays through the first collision are nearly identical
-+
+in figure \ref{fig:vCorrPlot}, but the peaks and troughs of the
-+
+functions show how the methods differ.  The undamped {\sc sf} method
-+
+has deeper troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher
-+
+peaks than any of the other methods.  As the damping function is
-+
+increased, these peaks are smoothed out, and approach the SPME
-+
+curve. The damping acts as a distance dependent Gaussian screening of
-+
+the point charges for the pairwise summation methods; thus, the
-+
+collisions are more elastic in the undamped {\sc sf} potential, and the
-+
+stiffness of the potential is diminished as the electrostatic
-+
+interactions are softened by the damping function.  With $\alpha$
-+
+values of 0.2 \AA$^{-1}$, the {\sc sf} and {\sc sp} functions are
-+
+nearly identical and track the SPME features quite well.  This is not
-+
+too surprising in that the differences between the {\sc sf} and {\sc
-+
+sp} potentials are mitigated with increased damping.  However, this
-+
+appears to indicate that once damping is utilized, the form of the
-+
+potential seems to play a lesser role in the crystal dynamics.
-+
+\subsection{Collective Motion: Power Spectra of NaCl Crystals}
-<
+In the previous studies using a Shifted-Force variant of the damped
-<
+Wolf coulomb potential, the structure and dynamics of water were
-<
+investigated rather extensively.\cite{Zahn02,Kast03} Their results
-<
+indicated that the damped Shifted-Force method results in properties
-<
+very similar to those obtained when using the Ewald summation.
-<
+Considering the statistical results shown above, the good performance
-<
+of this method is not that surprising.  Rather than consider the same
-<
+systems and simply recapitulate their results, we decided to look at
-<
+the solid state dynamical behavior obtained using the best performing
-<
+summation methods from the above results.
->
+The short time dynamics were extended to evaluate how the differences
->
+between the methods affect the collective long-time motion.  The same
->
+electrostatic summation methods were used as in the short time
->
+velocity autocorrelation function evaluation, but the trajectories
->
+were sampled over a much longer time. The power spectra of the
->
+resulting velocity autocorrelation functions were calculated and are
->
+displayed in figure \ref{fig:methodPS}.
+\begin{figure}
+\centering
+\includegraphics[width = \linewidth]{./spectraSquare.pdf}
-<
+\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.}
->
+\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.}
+\label{fig:methodPS}
+\end{figure}
-<
+Figure \ref{fig:methodPS} shows the power spectra for the NaCl
-<
+crystals (from averaged Na and Cl ion velocity autocorrelation
-<
+functions) using the stated electrostatic summation methods.  While
-<
+high frequency peaks of all the spectra overlap, showing the same
-<
+general features, the low frequency region shows how the summation
-<
+methods differ.  Considering the low-frequency inset (expanded in the
-<
+upper frame of figure \ref{fig:dampInc}), at frequencies below 100
-<
+cm$^{-1}$, the correlated motions are blue-shifted when using undamped
-<
+or weakly damped Shifted-Force.  When using moderate damping ($\alpha
-<
+= 0.2$ \AA$^{-1}$) both the Shifted-Force and Shifted-Potential
-<
+methods give near identical correlated motion behavior as the Ewald
-<
+method (which has a damping value of 0.3119).  The damping acts as a
-<
+distance dependent Gaussian screening of the point charges for the
-<
+pairwise summation methods.  This weakening of the electrostatic
-<
+interaction with distance explains why the long-ranged correlated
-<
+motions are at lower frequencies for the moderately damped methods
-<
+than for undamped or weakly damped methods.  To see this effect more
-<
+clearly, we show how damping strength affects a simple real-space
-<
+electrostatic potential,
->
+While high frequency peaks of the spectra in this figure overlap,
->
+showing the same general features, the low frequency region shows how
->
+the summation methods differ.  Considering the low-frequency inset
->
+(expanded in the upper frame of figure \ref{fig:dampInc}), at
->
+frequencies below 100 cm$^{-1}$, the correlated motions are
->
+blue-shifted when using undamped or weakly damped {\sc sf}.  When
->
+using moderate damping ($\alpha = 0.2$ \AA$^{-1}$) both the {\sc sf}
->
+and {\sc sp} methods give near identical correlated motion behavior as
->
+the Ewald method (which has a damping value of 0.3119).  This
->
+weakening of the electrostatic interaction with increased damping
->
+explains why the long-ranged correlated motions are at lower
->
+frequencies for the moderately damped methods than for undamped or
->
+weakly damped methods.  To see this effect more clearly, we show how
->
+damping strength alone affects a simple real-space electrostatic
->
+potential,
+\begin{equation}
-<
+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),
->
+V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r})}{r}\right]S(r),
+\end{equation}
+where $S(r)$ is a switching function that smoothly zeroes the
+potential at the cutoff radius.  Figure \ref{fig:dampInc} shows how
+shift to higher frequency in exponential fashion.  Though not shown,
+the spectrum for the simple undamped electrostatic potential is
+blue-shifted such that the lowest frequency peak resides near 325
-<
+cm$^{-1}$.  In light of these results, the undamped Shifted-Force
-<
+method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is
-<
+quite respectable; however, it appears as though moderate damping is
-<
+required for accurate reproduction of crystal dynamics.
->
+cm$^{-1}$.  In light of these results, the undamped {\sc sf} method
->
+producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is quite
->
+respectable and shows that the shifted force procedure accounts for
->
+most of the effect afforded through use of the Ewald summation.
->
+However, it appears as though moderate damping is required for
->
+accurate reproduction of crystal dynamics.
+\begin{figure}
+\centering
+\includegraphics[width = \linewidth]{./comboSquare.pdf}
-<
+\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.}
->
+\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.}
+\label{fig:dampInc}
+\end{figure}
+electrostatic summation techniques than the Ewald summation, chiefly
+methods derived from the damped Coulombic sum originally proposed by
+Wolf \textit{et al.}\cite{Wolf99,Zahn02} In particular, the
-<
+Shifted-Force method, reformulated above as equation \ref{eq:SFPot},
->
+{\sc sf} method, reformulated above as eq. (\ref{eq:DSFPot}),
+shows a remarkable ability to reproduce the energetic and dynamic
+characteristics exhibited by simulations employing lattice summation
+techniques.  The cumulative energy difference results showed the
-<
+undamped Shifted-Force and moderately damped Shifted-Potential methods
->
+undamped {\sc sf} and moderately damped {\sc sp} methods
+produced results nearly identical to SPME.  Similarly for the dynamic
-<
+features, the undamped or moderately damped Shifted-Force and
-<
+moderately damped Shifted-Potential methods produce force and torque
->
+features, the undamped or moderately damped {\sc sf} and
->
+moderately damped {\sc sp} methods produce force and torque
+vector magnitude and directions very similar to the expected values.
+These results translate into long-time dynamic behavior equivalent to
+that produced in simulations using SPME.
+standard by which these simple pairwise sums are judged.  However,
+these results do suggest that in the typical simulations performed
+today, the Ewald summation may no longer be required to obtain the
-<
+level of accuracy most researcher have come to expect
->
+level of accuracy most researchers have come to expect
+\section{Acknowledgments}
+\newpage

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-–
+Removed lines
-+
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-<
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Comparing trunk/electrostaticMethodsPaper/electrostaticMethods.tex (file contents): Revision 2620 by chrisfen, Wed Mar 15 04:34:51 2006 UTC vs. Revision 2653 by gezelter, Tue Mar 21 20:46:55 2006 UTC

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Comparing trunk/electrostaticMethodsPaper/electrostaticMethods.tex (file contents):
Revision 2620 by chrisfen, Wed Mar 15 04:34:51 2006 UTC vs.
Revision 2653 by gezelter, Tue Mar 21 20:46:55 2006 UTC