--- trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/18 05:31:17	2636
+++ trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/03/19 02:48:19	2637
@@ -104,21 +104,23 @@ and a long-ranged reciprocal-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}{3L^3}\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2,
+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 a damping parameter, or separation constant, with
-units of \AA$^{-1}$, and $\mathbf{k}$ are the reciprocal vectors and
-equal $2\pi\mathbf{n}/L^2$. The final two terms of
+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
 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} This term is zero
-for systems where $\epsilon_{\rm S} = \infty$. Figure
+replication of the periodic images.\cite{deLeeuw80,Smith81} If this
+term is taken to be zero, the system is using conducting 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
@@ -141,28 +143,61 @@ direct and reciprocal-space portions of the summation.
 $\mathscr{O}(N^2)$ 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 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 
+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
-accelerate the Ewald summation is to se
+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
+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 $\mathscr{O}(N \log N)$.
 
+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}
+
+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
+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.
+
+
 \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.\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 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
 \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\}.
 \label{eq:WolfPot}
