--- trunk/multipole/multipole_2/multipole2.tex	2014/06/11 21:03:11	4180
+++ trunk/multipole/multipole_2/multipole2.tex	2014/06/12 14:58:06	4181
@@ -35,7 +35,7 @@ preprint,
 %\linenumbers\relax % Commence numbering lines
 \usepackage{amsmath}
 \usepackage{times}
-\usepackage{mathptm}
+\usepackage{mathptmx}
 \usepackage{tabularx}
 \usepackage[version=3]{mhchem}  % this is a great package for formatting chemical reactions
 \usepackage{url}
@@ -133,28 +133,33 @@ condensed systems is actually short ranged.\cite{Wolf9
 Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$
 method for calculating electrostatic interactions between point
 charges. They argued that the effective Coulomb interaction in
-condensed systems is actually short ranged.\cite{Wolf92,Wolf95}.  For
-an ordered lattice (e.g., when computing the Madelung constant of an
+condensed systems is actually short ranged.\cite{Wolf92,Wolf95} For an
+ordered lattice (e.g., when computing the Madelung constant of an
 ionic solid), the material can be considered as a set of ions
 interacting with neutral dipolar or quadrupolar ``molecules'' giving
 an effective distance dependence for the electrostatic interactions of
-$r^{-5}$ (see figure \ref{fig:NaCl}).  For this reason, careful
+$r^{-5}$ (see figure \ref{fig:schematic}).  For this reason, careful
 applications of Wolf's method are able to obtain accurate estimates of
 Madelung constants using relatively short cutoff radii.  Recently,
 Fukuda used neutralization of the higher order moments for the
 calculation of the electrostatic interaction of the point charges
 system.\cite{Fukuda:2013sf}
 
-\begin{figure}[h!]
+\begin{figure}
   \centering
-  \includegraphics[width=0.50 \textwidth]{chargesystem.pdf}
-  \caption{Top: NaCl crystal showing how spherical truncation can
-    breaking effective charge ordering, and how complete \ce{(NaCl)4}
-    molecules interact with the central ion.  Bottom: A dipolar
-    crystal exhibiting similar behavior and illustrating how the
-    effective dipole-octupole interactions can be disrupted by
-    spherical truncation.}
-  \label{fig:NaCl}
+  \includegraphics[width=\linewidth]{schematic.pdf}
+  \caption{Top: Ionic systems exhibit local clustering of dissimilar
+    charges (in the smaller grey circle), so interactions are
+    effectively charge-multipole in order at longer distances.  With
+    hard cutoffs, motion of individual charges in and out of the
+    cutoff sphere can break the effective multipolar ordering.
+    Bottom: dipolar crystals and fluids have a similar effective
+    \textit{quadrupolar} ordering (in the smaller grey circles), and
+    orientational averaging helps to reduce the effective range of the
+    interactions in the fluid.  Placement of reversed image multipoles
+    on the surface of the cutoff sphere recovers the effective
+    higher-order multipole behavior.}
+  \label{fig:schematic}
 \end{figure}
 
 The direct truncation of interactions at a cutoff radius creates
@@ -178,19 +183,17 @@ potential is found to be decreasing as $r^{-5}$. If on
 
 Considering the interaction of one central ion in an ionic crystal
 with a portion of the crystal at some distance, the effective Columbic
-potential is found to be decreasing as $r^{-5}$. If one views the
-\ce{NaCl} crystal as simple cubic (SC) structure with an octupolar
+potential is found to decrease as $r^{-5}$. If one views the \ce{NaCl}
+crystal as a simple cubic (SC) structure with an octupolar
 \ce{(NaCl)4} basis, the electrostatic energy per ion converges more
 rapidly to the Madelung energy than the dipolar
 approximation.\cite{Wolf92} To find the correct Madelung constant,
 Lacman suggested that the NaCl structure could be constructed in a way
 that the finite crystal terminates with complete \ce{(NaCl)4}
-molecules.\cite{Lacman65} Any charge in a NaCl crystal is surrounded
-by opposite charges. Similarly for each pair of charges, there is an
-opposite pair of charge adjacent to it.  The central ion sees what is
-effectively a set of octupoles at large distances. These facts suggest
-that the Madelung constants are relatively short ranged for perfect
-ionic crystals.\cite{Wolf:1999dn}
+molecules.\cite{Lacman65} The central ion sees what is effectively a
+set of octupoles at large distances. These facts suggest that the
+Madelung constants are relatively short ranged for perfect ionic
+crystals.\cite{Wolf:1999dn}
 
 One can make a similar argument for crystals of point multipoles. The
 Luttinger and Tisza treatment of energy constants for dipolar lattices
@@ -208,13 +211,29 @@ multipolar arrangements (see Fig. \ref{fig:NaCl}), cau
 unstable.
 
 In ionic crystals, real-space truncation can break the effective
-multipolar arrangements (see Fig. \ref{fig:NaCl}), causing significant
-swings in the electrostatic energy as individual ions move back and
-forth across the boundary.  This is why the image charges are
+multipolar arrangements (see Fig. \ref{fig:schematic}), causing
+significant swings in the electrostatic energy as individual ions move
+back and forth across the boundary.  This is why the image charges are
 necessary for the Wolf sum to exhibit rapid convergence.  Similarly,
 the real-space truncation of point multipole interactions breaks
 higher order multipole arrangements, and image multipoles are required
 for real-space treatments of electrostatic energies.
+
+The shorter effective range of electrostatic interactions is not
+limited to perfect crystals, but can also apply in disordered fluids.
+Even at elevated temperatures, there is, on average, local charge
+balance in an ionic liquid, where each positive ion has surroundings
+dominated by negaitve ions and vice versa.  The reversed-charge images
+on the cutoff sphere that are integral to the Wolf and DSF approaches
+retain the effective multipolar interactions as the charges traverse
+the cutoff boundary.
+
+In multipolar fluids (see Fig. \ref{fig:schematic}) there is
+significant orientational averaging that additionally reduces the
+effect of long-range multipolar interactions.  The image multipoles
+that are introduced in the TSF, GSF, and SP methods mimic this effect
+and reduce the effective range of the multipolar interactions as
+interacting molecules traverse each other's cutoff boundaries.
 
 % Because of this reason, although the nature of electrostatic
 % interaction short ranged, the hard cutoff sphere creates very large
@@ -348,12 +367,12 @@ of the interaction, with $n=0$ for charge-charge, $n=1
 \label{generic}
 \end{equation}
 where $f_n(r)$ is a shifted kernel that is appropriate for the order
-of the interaction, with $n=0$ for charge-charge, $n=1$ for
-charge-dipole, $n=2$ for charge-quadrupole and dipole-dipole, $n=3$
-for dipole-quadrupole, and $n=4$ for quadrupole-quadrupole.  To ensure
-smooth convergence of the energy, force, and torques, a Taylor
-expansion with $n$ terms must be performed at cutoff radius ($r_c$) to
-obtain $f_n(r)$.
+of the interaction (see Ref. \onlinecite{PaperI}), with $n=0$ for
+charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole
+and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for
+quadrupole-quadrupole.  To ensure smooth convergence of the energy,
+force, and torques, a Taylor expansion with $n$ terms must be
+performed at cutoff radius ($r_c$) to obtain $f_n(r)$.
 
 % To carry out the same procedure for a damped electrostatic kernel, we
 % replace $1/r$ in the Coulomb potential with $\text{erfc}(\alpha r)/r$.
@@ -407,12 +426,12 @@ without changing their relative orientation,
 shifted smoothly by finding the gradient for two interacting dipoles
 which have been projected onto the surface of the cutoff sphere
 without changing their relative orientation,
-\begin{displaymath}
+\begin{equation}
 U_{D_{\bf a}D_{\bf b}}(r)  = U_{D_{\bf a}D_{\bf b}}(r) -
 U_{D_{\bf a} D_{\bf b}}(r_c)
    - (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot
   \vec{\nabla} U_{D_{\bf a}D_{\bf b}}(r) \Big \lvert _{r_c}
-\end{displaymath}
+\end{equation}
 Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf
   a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance
 (although the signs are reversed for the dipole that has been
@@ -957,7 +976,7 @@ conservation (drift less than $10^{-6}$ kcal / mol / n
 energy over time, $\delta E_1$, and the standard deviation of energy
 fluctuations around this drift $\delta E_0$.  Both of the
 shifted-force methods (GSF and TSF) provide excellent energy
-conservation (drift less than $10^{-6}$ kcal / mol / ns / particle),
+conservation (drift less than $10^{-5}$ kcal / mol / ns / particle),
 while the hard cutoff is essentially unusable for molecular dynamics.
 SP provides some benefit over the hard cutoff because the energetic
 jumps that happen as particles leave and enter the cutoff sphere are
