--- trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/22 21:20:40	2659
+++ trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/24 16:54:13	2669
@@ -25,7 +25,7 @@
 
 \begin{document}
 
-\title{Is the Ewald Summation necessary? \\
+\title{Is the Ewald summation still necessary? \\
 Pairwise alternatives to the accepted standard for \\
 long-range electrostatics}
 
@@ -40,20 +40,19 @@ Notre Dame, Indiana 46556}
 \maketitle
 \doublespacing
 
-\nobibliography{}
 \begin{abstract}
 We investigate pairwise electrostatic interaction methods and show
 that there are viable and computationally efficient $(\mathscr{O}(N))$
 alternatives to the Ewald summation for typical modern molecular
 simulations.  These methods are extended from the damped and
-cutoff-neutralized Coulombic sum originally proposed by Wolf
-\textit{et al.}  One of these, the damped shifted force method, shows
+cutoff-neutralized Coulombic sum originally proposed by
+[D. Wolf, P. Keblinski, S.~R. Phillpot, and J. Eggebrecht, {\it J. Chem. Phys.} {\bf 110}, 8255 (1999)] One of these, the damped shifted force method, shows
 a remarkable ability to reproduce the energetic and dynamic
 characteristics exhibited by simulations employing lattice summation
 techniques.  Comparisons were performed with this and other pairwise
-methods against the smooth particle mesh Ewald ({\sc spme}) summation to see
-how well they reproduce the energetics and dynamics of a variety of
-simulation types.
+methods against the smooth particle mesh Ewald ({\sc spme}) summation
+to see how well they reproduce the energetics and dynamics of a
+variety of simulation types.
 \end{abstract}
 
 \newpage
@@ -96,7 +95,7 @@ explicit Ewald summation.\cite{Tobias01}
 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
+In this paper, we focus on a new set of pairwise 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
@@ -111,7 +110,7 @@ their usability in molecular simulations.
 their usability in molecular simulations.
 
 \subsection{The Ewald Sum}
-The complete accumulation electrostatic interactions in a system with
+The complete accumulation of the 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,
@@ -168,11 +167,11 @@ portion.\cite{Karasawa89,Kolafa92}
 \begin{figure}
 \centering
 \includegraphics[width = \linewidth]{./ewaldProgression.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.}
+\caption{The change in the need for the Ewald sum with
+increasing computational power.  A:~Initially, only small systems
+could be studied, and the Ewald sum replicated the simulation box to
+convergence.  B:~Now, radial cutoff methods should be able to reach
+convergence for the larger systems of charges that are common today.}
 \label{fig:ewaldTime}
 \end{figure}
 
@@ -228,11 +227,10 @@ charge neutrality and gives results similar to those o
 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
+Ewald summation.  The resulting shifted Coulomb potential 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_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}
@@ -581,29 +579,11 @@ shape. Thus, gaussian fits were used to measure the wi
 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 ({\sc spme}).
+resulting distributions. 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 ({\sc spme}).
 
 \subsection{Short-time Dynamics}
 
@@ -631,7 +611,7 @@ the {\it long-time} dynamics of charged systems were e
 
 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
+the same model system (NaCl crystals at 1000~K).  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,
@@ -671,29 +651,23 @@ these systems were selected and equilibrated in the sa
 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}).
+water systems. In order to introduce measurable fluctuations in the
+configuration energy differences, the crystalline simulations were
+equilibrated at 1000~K, near the $T_\textrm{m}$ for NaCl. The liquid
+NaCl configurations needed to represent a fully disordered array of
+point charges, so the high temperature of 7000~K was selected for
+equilibration. 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.
 
 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}
-
-%\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}
 
 \subsection{Comparison of Summation Methods}\label{sec:ESMethods}
 We compared the following alternative summation methods with results
@@ -716,7 +690,7 @@ manner across all systems and configurations.
 (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
+The alternative 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.
@@ -1096,7 +1070,7 @@ cutoff distance.
 \centering
 \includegraphics[width = \linewidth]{./increasedDamping.pdf}
 \caption{Effect of damping on the two lowest-frequency phonon modes in
-the NaCl crystal at 1000K.  The undamped shifted force ({\sc sf})
+the NaCl crystal at 1000~K.  The undamped shifted force ({\sc sf})
 method is off by less than 10 cm$^{-1}$, and increasing the
 electrostatic damping to 0.25 \AA$^{-1}$ gives quantitative agreement
 with the power spectrum obtained using the Ewald sum.  Overdamping can
