--- trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/19 19:34:53	2638
+++ trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/20 13:31:52	2640
@@ -77,25 +77,56 @@ accurately incorporate their effect, and since the com
 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.
+increases proportionally with the cutoff sphere, it quickly becomes
+very time-consuming to perform these calculations.
 
+There have been many efforts to address this issue of both proper and
+practical handling of electrostatic interactions, and these have
+resulted in the availability of 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 incorporate dynamic 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 this cost, there has been some question of the inherent
+periodicity of the explicit Ewald summation artificially influencing
+systems dynamics.\cite{Tobias01}
+
+In this paper, we focus on the common mixed and explicit methods of
+reaction filed and smooth particle mesh
+Ewald\cite{Onsager36,Essmann99} and a new set of shifted methods
+devised by Wolf {\it et al.} which we further extend.\cite{Wolf99}
+These new methods for handling electrostatics are quite
+computationally efficient, since they involve only a simple
+modification to the direct pairwise sum, and they lack the added
+periodicity of the Ewald sum. Below, these methods are evaluated using
+a variety of model systems and comparison methodologies 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 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 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 discontinuous for non-neutral systems.
 
 This electrostatic summation problem was originally studied by Ewald
 for the case of an infinite crystal.\cite{Ewald21}. The approach he
@@ -145,7 +176,7 @@ real-space or reciprocal space portion of the summatio
 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
+$\mathscr{O}(N^2)$ calculation (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
@@ -181,7 +212,7 @@ considering the use of the Ewald summation where the i
 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.
+periodicity may negatively affect the system dynamics.
 
 
 \subsection{The Wolf and Zahn Methods}
@@ -199,7 +230,7 @@ the real-space portion of the Ewald sum) to aid conver
 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
@@ -542,7 +573,7 @@ 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 $M$.  
@@ -707,7 +738,7 @@ al.},\cite{Wolf99} and this correction indeed improves
 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
@@ -920,7 +951,7 @@ the point charges for the pairwise summation methods; 
 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} potental, and 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