--- trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/19 02:48:19	2637
+++ trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/20 17:32:33	2643
@@ -65,39 +65,71 @@ In molecular simulations, proper accumulation of the e
 \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}
-The complete accumulation electrostatic interactions in a system with periodic boundary conditions (PBC) requires the consideration of the effect of all charges within a simulation box, as well as those in the periodic replicas,
+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 $\mathbf{n} = 0$.\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 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 discontiuous
-for non-neutral systems.
+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 $\mathbf{n} =
+0$.\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.
 
-This electrostatic summation problem was originally studied by Ewald
+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
@@ -109,23 +141,27 @@ where $\alpha$ is a damping parameter, or separation c
 \label{eq:EwaldSum}
 \end{equation}
 where $\alpha$ is a damping parameter, or separation constant, with
-units of \AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and equal
-$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric
-constant of the encompassing medium. The final two terms of
+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 and this dipole moment gets magnified through
-replication of the periodic images.\cite{deLeeuw80,Smith81} If this
-term is taken to be zero, the system is using conducting boundary
+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 size of systems, the entire
-simulation box was replicated to convergence.  Currently, we balance a
-spherical real-space cutoff with the reciprocal sum and consider the
-surrounding dielectric.
+time.  Initially, due to the small sizes of the systems that could be
+feasibly simulated, the entire simulation box was replicated to
+convergence.  In more modern simulations, the simulation boxes have
+grown large enough that a real-space cutoff could potentially give
+convergent behavior.  Indeed, it has often been observed that the
+reciprocal-space portion of the Ewald sum can be vanishingly
+small compared to the real-space portion.\cite{XXX}
+
 \begin{figure}
 \centering
 \includegraphics[width = \linewidth]{./ewaldProgression.pdf}
@@ -139,67 +175,65 @@ a surrounding dielectric term is included.}
 \label{fig:ewaldTime}
 \end{figure}
 
-The Ewald summation in the straight-forward form is an
-$\mathscr{O}(N^2)$ algorithm.  The separation constant $(\alpha)$
-plays an important role in the computational cost balance between the
-direct and reciprocal-space portions of the summation.  The choice of
-the magnitude of this value allows one to select whether the
-real-space or reciprocal space portion of the summation is an
-$\mathscr{O}(N^2)$ calcualtion (with the other being
-$\mathscr{O}(N)$).\cite{Sagui99} With appropriate choice of $\alpha$
-and thoughtful algorithm development, this cost can be brought down to
-$\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route taken to
-reduce the cost of the Ewald summation further is to set $\alpha$ such
-that the real-space interactions decay rapidly, allowing for a short
-spherical cutoff, and then optimize the reciprocal space summation.
-These optimizations usually involve the utilization of the fast
-Fourier transform (FFT),\cite{Hockney81} leading to the
+The original Ewald summation is an $\mathscr{O}(N^2)$ algorithm.  The
+separation constant $(\alpha)$ 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 from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N \log N)$.
+summation is reduced from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N
+\log N)$.
 
-These developments and optimizations have led the use of the Ewald
-summation to become routine in simulations with periodic boundary
-conditions. However, in certain systems the intrinsic three
-dimensional periodicity can prove to be problematic, such as two
-dimensional surfaces and membranes.  The Ewald sum has been
-reformulated to handle 2D
-systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89}, but the new
-methods have been found to be computationally
-expensive.\cite{Spohr97,Yeh99} Inclusion of a correction term in the
-full Ewald summation is a possible direction for enabling the handling
-of 2D systems and the inclusion of the optimizations described
-previously.\cite{Yeh99}
+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
+2D 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 systems that have the same
-dimensionality.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00}
-Good examples are solvated proteins kept at high relative
-concentration due to the periodicity of the electrostatics.  In these
+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 ought to be taken when
-considering the use of the Ewald summation where the intrinsic
-perodicity may negatively affect the system dynamics.
+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 the accurate accumulation of electrostatic interactions in an
-efficient pairwise fashion and 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 (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
+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
+(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
@@ -224,22 +258,20 @@ the potential are not commensurate.  Attempts to use b
 force expressions for use in simulations involving water.\cite{Zahn02}
 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
+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}).
+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.
 
@@ -270,7 +302,7 @@ shifted potential,
 \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},
@@ -278,7 +310,7 @@ and shifted force,
 \end{equation}
 and shifted force,
 \begin{equation}
-v_\textrm{SF}(r) = 	\begin{cases}
+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},
@@ -294,16 +326,16 @@ of the unshifted potential itself (when inside the cut
 The forces associated with the shifted potential are simply the forces
 of the unshifted potential itself (when inside the cutoff sphere),
 \begin{equation}
-f_{\textrm{SP}} = -\left( \frac{d v(r)}{dr} \right),
+F_{\textrm{SP}} = -\left( \frac{d v(r)}{dr} \right),
 \end{equation}
 and are zero outside.  Inside the cutoff sphere, the forces associated
 with the shifted force form can be written,
 \begin{equation}
-f_{\textrm{SF}} = -\left( \frac{d v(r)}{dr} \right) + \left(\frac{d
+F_{\textrm{SF}} = -\left( \frac{d v(r)}{dr} \right) + \left(\frac{d
 v(r)}{dr} \right)_{r=R_\textrm{c}}.
 \end{equation}
  
-If the potential ($v(r)$) is taken to be the normal Coulomb potential,
+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}
@@ -311,14 +343,14 @@ al.}'s undamped prescription:
 then the Shifted Potential ({\sc sp}) forms will give Wolf {\it et
 al.}'s undamped prescription:
 \begin{equation}
-v_\textrm{SP}(r) =
+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}.
+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
@@ -333,18 +365,18 @@ will give,
 The shifted force ({\sc sf}) form using the normal Coulomb potential
 will give,
 \begin{equation}
-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}.
+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}
 with associated forces,
 \begin{equation}
-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}.
+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}
 This formulation has the benefits that there are no discontinuities at
-the cutoff distance, 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
+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
@@ -352,13 +384,13 @@ Wolf \textit{et al.} originally discussed the energeti
 to gain functionality in dynamics simulations.
 
 Wolf \textit{et al.} originally discussed the energetics of the
-shifted Coulomb potential (Eq. \ref{eq:SPPot}), and they found that
-it was still insufficient for accurate determination of the energy
-with reasonable cutoff distances.  The calculated Madelung energies
-fluctuate around the expected value with increasing cutoff radius, but
-the oscillations converge toward the correct value.\cite{Wolf99} A
+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 functional forms could be
+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
@@ -367,37 +399,37 @@ v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r},
 v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r},
 \label{eq:dampCoulomb}
 \end{equation}
-the shifted potential (Eq. (\ref{eq:SPPot})) can be reacquired using
-eq. (\ref{eq:shiftingForm}),
+the shifted potential (eq. (\ref{eq:SPPot})) becomes
 \begin{equation}
-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},
+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}
 with associated forces,
 \begin{equation}
-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}.
+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 damped shifted potential suffers from a discontinuity and
-a lack of the image charges in the forces.  To remedy these concerns,
-one may derive a {\sc sf} variant by including  the derivative
-term in eq. (\ref{eq:shiftingForm}),
+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) = 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}.
+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 will lead to the following forces,
 \begin{equation}
 \begin{split}
-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}.
+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 chosen to be zero, the undamped
-case, eqs. (\ref{eq:SPPot}-\ref{eq:SFForces}) are correctly recovered
-from eqs. (\ref{eq:DSPPot}-\ref{eq:DSFForces}).
+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 {\sc sf} potential is similar to equation \ref{eq:ZahnPot}
 derived by Zahn \textit{et al.}; however, there are two important
@@ -409,55 +441,48 @@ would be expected to have sudden jumps as particle dis
 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 would be 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 $R_\textrm{c}$.
+$R_c$).  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 $R_\textrm{c}$.
 
 Eqs. (\ref{eq:DSFPot}) and (\ref{eq:DSFForces}) result in an
-electrostatic summation method that is continuous in both the
-potential and forces 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.
+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 will consider some
-other techniques that commonly get used in molecular simulations.  The
+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
-non-neutral molecules, collecting atoms into neutral groups takes
+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 groups, an orientational aspect is
-introduced to the interactions.  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
-switching function,
-\begin{equation}
-S(r) = \begin{cases} 1 &\quad r\leqslant r_\textrm{sw} \\
-\frac{(R_\textrm{c}+2r-3r_\textrm{sw})(R_\textrm{c}-r)^2}{(R_\textrm{c}-r_\textrm{sw})^3} &\quad r_\textrm{sw}<r\leqslant R_\textrm{c} \\
-0 &\quad r>R_\textrm{c}
-\end{cases},
-\end{equation}
-where the above form is for a cubic function.  If a smooth second
-derivative is desired, a fifth (or higher) order polynomial can be
-used.\cite{Andrea83} 
+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 their effect, 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 application, it 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.
+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}
 
@@ -542,7 +567,7 @@ between those computed from the particular method and 
 investigated through measurement of the angle ($\theta$) formed
 between those computed from the particular method and those from SPME,
 \begin{equation}
-\theta_f = \cos^{-1} \hat{f}_\textrm{SPME} \cdot \hat{f}_\textrm{Method},
+\theta_f = \cos^{-1} \left(\hat{f}_\textrm{SPME} \cdot \hat{f}_\textrm{Method}\right),
 \end{equation}
 where $\hat{f}_\textrm{M}$ is the unit vector pointing along the
 force vector computed using method $M$.  
@@ -572,31 +597,41 @@ when using the reference method (SPME).
 when using the reference method (SPME).
 
 \subsection{Short-time Dynamics}
-
-\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
+Evaluation of the short-time dynamics of charged systems was performed
+by considering the 1000 K 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,
+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
+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}
+Evaluation of the long-time dynamics of charged systems was performed
+by considering the NaCl crystal system, again while using a subset of
+the best performing pairwise methods.  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 information was saved every 5 fs over 100 ps
+trajectories.  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.
 
 \subsection{Representative Simulations}\label{sec:RepSims}
 A variety of common and representative simulations were analyzed to
@@ -620,17 +655,17 @@ snapshots were taken at regular intervals from higher 
 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
-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}).
+SPC/E water molecule trajectories and each equilibrated
+individually.\cite{Berendsen87} 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 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}).
 
 \begin{figure}
 \centering
@@ -685,19 +720,20 @@ realistic results using an unmodified cutoff.  This is
 
 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
+surprising since this results in 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} 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.
 
 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
+seen in the {\sc sp} 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
@@ -882,7 +918,7 @@ unnecessary when using the {\sc sf} method.
 up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably
 unnecessary when using the {\sc sf} method.
 
-\subsection{Collective Motion: Power Spectra of NaCl Crystals}
+\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals}
 
 In the previous studies using a {\sc sf} variant of the damped
 Wolf coulomb potential, the structure and dynamics of water were
@@ -897,30 +933,61 @@ summation methods from the above results.
 
 \begin{figure}
 \centering
+\includegraphics[width = \linewidth]{./vCorrPlot.pdf}
+\caption{Velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset is a magnification of the first trough. The times to first collision are nearly identical, but the differences can be seen in the peaks and troughs, where the undamped to weakly damped methods are stiffer than the moderately damped and SPME methods.}
+\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}
+
+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, {\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 {\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).  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})}{r}\right]S(r),
 \end{equation}
@@ -935,10 +1002,12 @@ blue-shifted such that the lowest frequency peak resid
 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 {\sc sf}
-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}
@@ -982,7 +1051,7 @@ today, the Ewald summation may no longer be required t
 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 
