--- trunk/multipole/multipole_2/multipole2.tex	2014/06/11 21:03:11	4180
+++ trunk/multipole/multipole_2/multipole2.tex	2014/06/15 02:31:11	4186
@@ -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}
@@ -47,158 +47,156 @@ preprint,
 
 %\preprint{AIP/123-QED}
 
-\title{Real space alternatives to the Ewald
-Sum. II. Comparison of Methods} % Force line breaks with \\
+\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods}
 
 \author{Madan Lamichhane}
- \affiliation{Department of Physics, University
-of Notre Dame, Notre Dame, IN 46556}%Lines break automatically or can be forced with \\
+ \affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}
 
 \author{Kathie E. Newman}
-\affiliation{Department of Physics, University
-of Notre Dame, Notre Dame, IN 46556}
+\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}
 
 \author{J. Daniel Gezelter}%
  \email{gezelter@nd.edu.}
-\affiliation{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556%\\This line break forced with \textbackslash\textbackslash
-}%
+\affiliation{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556
+}
 
-\date{\today}% It is always \today, today,
-             %  but any date may be explicitly specified
+\date{\today}
 
 \begin{abstract}
-  We have tested the real-space shifted potential (SP),
+  We report on tests of the real-space shifted potential (SP),
   gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods
-  for multipoles that were developed in the first paper in this series
-  against a reference method. The tests were carried out in a variety
-  of condensed-phase environments which were designed to test all
-  levels of the multipole-multipole interactions.  Comparisons of the
-  energy differences between configurations, molecular forces, and
-  torques were used to analyze how well the real-space models perform
-  relative to the more computationally expensive Ewald sum.  We have
-  also investigated the energy conservation properties of the new
-  methods in molecular dynamics simulations using all of these
-  methods. The SP method shows excellent agreement with
+  for multipole interactions developed in the first paper in this
+  series, using the multipolar Ewald sum as a reference method. The
+  tests were carried out in a variety of condensed-phase environments
+  designed to test up to quadrupole-quadrupole interactions.
+  Comparisons of the energy differences between configurations,
+  molecular forces, and torques were used to analyze how well the
+  real-space models perform relative to the more computationally
+  expensive Ewald treatment.  We have also investigated the energy
+  conservation properties of the new methods in molecular dynamics
+  simulations. The SP method shows excellent agreement with
   configurational energy differences, forces, and torques, and would
   be suitable for use in Monte Carlo calculations.  Of the two new
   shifted-force methods, the GSF approach shows the best agreement
-  with Ewald-derived energies, forces, and torques and exhibits energy
-  conservation properties that make it an excellent choice for
-  efficiently computing electrostatic interactions in molecular
+  with Ewald-derived energies, forces, and torques and also exhibits
+  energy conservation properties that make it an excellent choice for
+  efficient computation of electrostatic interactions in molecular
   dynamics simulations.
 \end{abstract}
 
 %\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
                              % Classification Scheme.
-\keywords{Electrostatics, Multipoles, Real-space}
+%\keywords{Electrostatics, Multipoles, Real-space}
 
 \maketitle
 
-
 \section{\label{sec:intro}Introduction}
 Computing the interactions between electrostatic sites is one of the
-most expensive aspects of molecular simulations, which is why there
-have been significant efforts to develop practical, efficient and
-convergent methods for handling these interactions. Ewald's method is
-perhaps the best known and most accurate method for evaluating
-energies, forces, and torques in explicitly-periodic simulation
-cells. In this approach, the conditionally convergent electrostatic
-energy is converted into two absolutely convergent contributions, one
-which is carried out in real space with a cutoff radius, and one in
-reciprocal space.\cite{Clarke:1986eu,Woodcock75}
+most expensive aspects of molecular simulations. There have been
+significant efforts to develop practical, efficient and convergent
+methods for handling these interactions. Ewald's method is perhaps the
+best known and most accurate method for evaluating energies, forces,
+and torques in explicitly-periodic simulation cells. In this approach,
+the conditionally convergent electrostatic energy is converted into
+two absolutely convergent contributions, one which is carried out in
+real space with a cutoff radius, and one in reciprocal
+space. BETTER CITATIONS\cite{Clarke:1986eu,Woodcock75}
 
 When carried out as originally formulated, the reciprocal-space
 portion of the Ewald sum exhibits relatively poor computational
 scaling, making it prohibitive for large systems. By utilizing
 particle meshes and three dimensional fast Fourier transforms (FFT),
 the particle-mesh Ewald (PME), particle-particle particle-mesh Ewald
-(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) methods can decrease
-the computational cost from $O(N^2)$ down to $O(N \log
+(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME)
+methods can decrease the computational cost from $O(N^2)$ down to $O(N
+\log
 N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}.
 
-Because of the artificial periodicity required for the Ewald sum, the
-method may require modification to compute interactions for
+Because of the artificial periodicity required for the Ewald sum,
 interfacial molecular systems such as membranes and liquid-vapor
-interfaces.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl}
-To simulate interfacial systems, Parry's extension of the 3D Ewald sum
-is appropriate for slab geometries.\cite{Parry:1975if} The inherent
-periodicity in the Ewald’s method can also be problematic for
-interfacial molecular systems.\cite{Fennell:2006lq} Modified Ewald
-methods that were developed to handle two-dimensional (2D)
-electrostatic interactions in interfacial systems have not had similar
-particle-mesh treatments.\cite{Parry:1975if, Parry:1976fq, Clarke77,
-  Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq}
+interfaces require modifications to the
+method.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl}
+Parry's extension of the three dimensional Ewald sum is appropriate
+for slab geometries.\cite{Parry:1975if} Modified Ewald methods that
+were developed to handle two-dimensional (2D) electrostatic
+interactions in interfacial systems have not seen similar
+particle-mesh treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77,
+  Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly
+with system size. The inherent periodicity in the Ewald’s method can
+also be problematic for interfacial molecular
+systems.\cite{Fennell:2006lq}
 
 \subsection{Real-space methods}
 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
-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
-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}
+charges. They argued that the effective Coulomb interaction in most
+condensed phase systems is effectively 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:schematic}). 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} 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} For this reason, careful
+application of Wolf's method can provide accurate estimates of
+Madelung constants using relatively short cutoff radii.
 
-\begin{figure}[h!]
-  \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}
-\end{figure}
-
-The direct truncation of interactions at a cutoff radius creates
-truncation defects. Wolf \textit{et al.} further argued that
-truncation errors are due to net charge remaining inside the cutoff
-sphere.\cite{Wolf:1999dn} To neutralize this charge they proposed
-placing an image charge on the surface of the cutoff sphere for every
-real charge inside the cutoff.  These charges are present for the
-evaluation of both the pair interaction energy and the force, although
-the force expression maintained a discontinuity at the cutoff sphere.
-In the original Wolf formulation, the total energy for the charge and
-image were not equal to the integral of their force expression, and as
-a result, the total energy would not be conserved in molecular
-dynamics (MD) simulations.\cite{Zahn:2002hc} Zahn \textit{et al.} and
-Fennel and Gezelter later proposed shifted force variants of the Wolf
-method with commensurate force and energy expressions that do not
-exhibit this problem.\cite{Fennell:2006lq}   Related real-space
-methods were also proposed by Chen \textit{et
+Direct truncation of interactions at a cutoff radius creates numerical
+errors.  Wolf \textit{et al.} suggest that truncation errors are due
+to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To
+neutralize this charge they proposed placing an image charge on the
+surface of the cutoff sphere for every real charge inside the cutoff.
+These charges are present for the evaluation of both the pair
+interaction energy and the force, although the force expression
+maintains a discontinuity at the cutoff sphere.  In the original Wolf
+formulation, the total energy for the charge and image were not equal
+to the integral of the force expression, and as a result, the total
+energy would not be conserved in molecular dynamics (MD)
+simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and
+Gezelter later proposed shifted force variants of the Wolf method with
+commensurate force and energy expressions that do not exhibit this
+problem.\cite{Zahn:2002hc,Fennell:2006lq} Related real-space methods
+were also proposed by Chen \textit{et
   al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw}
-and by Wu and Brooks.\cite{Wu:044107}
+and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has successfuly
+used additional neutralization of higher order moments for systems of
+point charges.\cite{Fukuda:2013sf}
 
-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
-\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}
+\begin{figure}
+  \centering
+  \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 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}
 
-One can make a similar argument for crystals of point multipoles. The
-Luttinger and Tisza treatment of energy constants for dipolar lattices
-utilizes 24 basis vectors that contain dipoles at the eight corners of
-a unit cube.  Only three of these basis vectors, $X_1, Y_1,
-\mathrm{~and~} Z_1,$ retain net dipole moments, while the rest have
-zero net dipole and retain contributions only from higher order
-multipoles.  The effective interaction between a dipole at the center
+One can make a similar effective range argument for crystals of point
+\textit{multipoles}. The Luttinger and Tisza treatment of energy
+constants for dipolar lattices utilizes 24 basis vectors that contain
+dipoles at the eight corners of a unit cube.\cite{LT} Only three of
+these basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole
+moments, while the rest have zero net dipole and retain contributions
+only from higher order multipoles.  The lowest-energy crystalline
+structures are built out of basis vectors that have only residual
+quadrupolar moments (e.g. the $Z_5$ array). In these low energy
+structures, the effective interaction between a dipole at the center
 of a crystal and a group of eight dipoles farther away is
 significantly shorter ranged than the $r^{-3}$ that one would expect
 for raw dipole-dipole interactions.  Only in crystals which retain a
@@ -208,14 +206,30 @@ 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 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
 % fluctuation in the electrostatic energy for the perfect crystal. In
@@ -226,72 +240,66 @@ The forces and torques acting on atomic sites are the 
 % to the non-neutralized value of the higher order moments within the
 % cutoff sphere.
 
-The forces and torques acting on atomic sites are the fundamental
-factors driving dynamics in molecular simulations. Fennell and
-Gezelter proposed the damped shifted force (DSF) energy kernel to
-obtain consistent energies and forces on the atoms within the cutoff
-sphere. Both the energy and the force go smoothly to zero as an atom
-aproaches the cutoff radius. The comparisons of the accuracy these
-quantities between the DSF kernel and SPME was surprisingly
-good.\cite{Fennell:2006lq} The DSF method has seen increasing use for
-calculating electrostatic interactions in molecular systems with
-relatively uniform charge
-densities.\cite{Shi:2013ij,Kannam:2012rr,Acevedo13,Space12,English08,Lawrence13,Vergne13}
+Forces and torques acting on atomic sites are fundamental in driving
+dynamics in molecular simulations, and the damped shifted force (DSF)
+energy kernel provides consistent energies and forces on charged atoms
+within the cutoff sphere. Both the energy and the force go smoothly to
+zero as an atom aproaches the cutoff radius. The comparisons of the
+accuracy these quantities between the DSF kernel and SPME was
+surprisingly good.\cite{Fennell:2006lq} As a result, the DSF method
+has seen increasing use in molecular systems with relatively uniform
+charge
+densities.\cite{English08,Kannam:2012rr,Space12,Lawrence13,Acevedo13,Shi:2013ij,Vergne13}
 
 \subsection{The damping function}
-The damping function used in our research has been discussed in detail
-in the first paper of this series.\cite{PaperI} The radial kernel
-$1/r$ for the interactions between point charges can be replaced by
-the complementary error function $\mathrm{erfc}(\alpha r)/r$ to
-accelerate the rate of convergence, where $\alpha$ is a damping
-parameter with units of inverse distance.  Altering the value of
-$\alpha$ is equivalent to changing the width of Gaussian charge
-distributions that replace each point charge -- Gaussian overlap
-integrals yield complementary error functions when truncated at a
-finite distance.
+The damping function has been discussed in detail in the first paper
+of this series.\cite{PaperI} The $1/r$ Coulombic kernel for the
+interactions between point charges can be replaced by the
+complementary error function $\mathrm{erfc}(\alpha r)/r$ to accelerate
+convergence, where $\alpha$ is a damping parameter with units of
+inverse distance.  Altering the value of $\alpha$ is equivalent to
+changing the width of Gaussian charge distributions that replace each
+point charge, as Coulomb integrals with Gaussian charge distributions
+produce complementary error functions when truncated at a finite
+distance.
 
-By using suitable value of damping alpha ($\alpha \sim 0.2$) for a
-cutoff radius ($r_{­c}=9 A$), Fennel and Gezelter produced very good
-agreement with SPME for the interaction energies, forces and torques
-for charge-charge interactions.\cite{Fennell:2006lq}
+With moderate damping coefficients, $\alpha \sim 0.2$, the DSF method
+produced very good agreement with SPME for interaction energies,
+forces and torques for charge-charge
+interactions.\cite{Fennell:2006lq}
 
 \subsection{Point multipoles in molecular modeling}
 Coarse-graining approaches which treat entire molecular subsystems as
 a single rigid body are now widely used. A common feature of many
 coarse-graining approaches is simplification of the electrostatic
 interactions between bodies so that fewer site-site interactions are
-required to compute configurational energies.  Many coarse-grained
-molecular structures would normally consist of equal positive and
-negative charges, and rather than use multiple site-site interactions,
-the interaction between higher order multipoles can also be used to
-evaluate a single molecule-molecule
-interaction.\cite{Ren06,Essex10,Essex11}
+required to compute configurational
+energies.\cite{Ren06,Essex10,Essex11}
 
-Because electrons in a molecule are not localized at specific points,
-the assignment of partial charges to atomic centers is a relatively
-rough approximation.  Atomic sites can also be assigned point
-multipoles and polarizabilities to increase the accuracy of the
-molecular model.  Recently, water has been modeled with point
-multipoles up to octupolar order using the soft sticky
-dipole-quadrupole-octupole (SSDQO)
+Additionally, because electrons in a molecule are not localized at
+specific points, the assignment of partial charges to atomic centers
+is always an approximation.  For increased accuracy, atomic sites can
+also be assigned point multipoles and polarizabilities.  Recently,
+water has been modeled with point multipoles up to octupolar order
+using the soft sticky dipole-quadrupole-octupole (SSDQO)
 model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point
 multipoles up to quadrupolar order have also been coupled with point
 polarizabilities in the high-quality AMOEBA and iAMOEBA water
-models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} But
-using point multipole with the real space truncation without
-accounting for multipolar neutrality will create energy conservation
-issues in molecular dynamics (MD) simulations.
+models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However,
+truncating point multipoles without smoothing the forces and torques
+can create energy conservation issues in molecular dynamics
+simulations.
 
 In this paper we test a set of real-space methods that were developed
 for point multipolar interactions.  These methods extend the damped
 shifted force (DSF) and Wolf methods originally developed for
 charge-charge interactions and generalize them for higher order
-multipoles. The detailed mathematical development of these methods has
-been presented in the first paper in this series, while this work
-covers the testing the energies, forces, torques, and energy
+multipoles.  The detailed mathematical development of these methods
+has been presented in the first paper in this series, while this work
+covers the testing of energies, forces, torques, and energy
 conservation properties of the methods in realistic simulation
 environments.  In all cases, the methods are compared with the
-reference method, a full multipolar Ewald treatment.
+reference method, a full multipolar Ewald treatment.\cite{Smith82,Smith98}
 
 
 %\subsection{Conservation of total energy } 
@@ -320,7 +328,7 @@ $\bf a$.
 where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is
 expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf
     a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object
-$\bf a$.
+$\bf a$, etc.
 
 % Interactions between multipoles can be expressed as higher derivatives
 % of the bare Coulomb potential, so one way of ensuring that the forces
@@ -348,12 +356,12 @@ 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 $\text{erfc}(\alpha r)/r$.
@@ -393,7 +401,7 @@ to another site within cutoff sphere are derived from
 connection to unmodified electrostatics as well as the smooth
 transition to zero in both these functions as $r\rightarrow r_c$.  The
 electrostatic forces and torques acting on the central multipole due
-to another site within cutoff sphere are derived from
+to another site within the cutoff sphere are derived from
 Eq.~\ref{generic}, accounting for the appropriate number of
 derivatives. Complete energy, force, and torque expressions are
 presented in the first paper in this series (Reference
@@ -407,12 +415,12 @@ 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}
+  \nabla U_{D_{\bf a}D_{\bf b}}(r_c).
+\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
@@ -435,18 +443,21 @@ U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathb
 In general, the gradient shifted potential between a central multipole
 and any multipolar site inside the cutoff radius is given by,
 \begin{equation}
-U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
-U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r}
-\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert  _{r_c} \right]
+  U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
+    U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}}
+    \cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
 \label{generic2}
 \end{equation}
 where the sum describes a separate force-shifting that is applied to
-each orientational contribution to the energy. 
+each orientational contribution to the energy.  In this expression,
+$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles
+($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$
+represent the orientations the multipoles.
 
 The third term converges more rapidly than the first two terms as a
 function of radius, hence the contribution of the third term is very
 small for large cutoff radii.  The force and torque derived from
-equation \ref{generic2} are consistent with the energy expression and
+Eq. \ref{generic2} are consistent with the energy expression and
 approach zero as $r \rightarrow r_c$.  Both the GSF and TSF methods
 can be considered generalizations of the original DSF method for
 higher order multipole interactions. GSF and TSF are also identical up
@@ -454,7 +465,7 @@ GSF potential are presented in the first paper in this
 the energy, force and torque for higher order multipole-multipole
 interactions. Complete energy, force, and torque expressions for the
 GSF potential are presented in the first paper in this series
-(Reference~\onlinecite{PaperI})
+(Reference~\onlinecite{PaperI}).
 
 
 \subsection{Shifted potential (SP) }
@@ -468,7 +479,7 @@ U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \ri
 effectively shifts the total potential to zero at the cutoff radius,
 \begin{equation}
 U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
-U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
+U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
 \label{eq:SP}
 \end{equation}      	
 where the sum describes separate potential shifting that is done for
@@ -957,7 +968,7 @@ 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
