--- trunk/multipole/multipole_2/multipole2.tex	2014/06/15 00:18:18	4185
+++ trunk/multipole/multipole_2/multipole2.tex	2014/06/15 16:25:42	4187
@@ -47,45 +47,40 @@ 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),
-  gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods
-  for multipole interactions that were 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 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 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 efficient computation of electrostatic interactions in
-  molecular dynamics simulations.
+  We report on tests of the shifted potential (SP), gradient shifted
+  force (GSF), and Taylor shifted force (TSF) real-space methods 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 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
@@ -94,7 +89,6 @@ of Notre Dame, Notre Dame, IN 46556}
 
 \maketitle
 
-
 \section{\label{sec:intro}Introduction}
 Computing the interactions between electrostatic sites is one of the
 most expensive aspects of molecular simulations. There have been
@@ -105,30 +99,33 @@ space. BETTER CITATIONS\cite{Clarke:1986eu,Woodcock75}
 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}
+space.\cite{Ewald21,deLeeuw80,Smith81,Allen87} 
 
 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 $O(N \log
-N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}.
+scaling, making it prohibitive for large systems. By utilizing a
+particle mesh 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 $O(N
+\log
+N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}
 
 Because of the artificial periodicity required for the Ewald sum,
 interfacial molecular systems such as membranes and liquid-vapor
-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}
+interfaces require modifications to the method.  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 (2-D) electrostatic
+interactions,\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl}
+but these methods were originally quite computationally
+expensive.\cite{Spohr97,Yeh99} There have been several successful
+efforts that reduced the computational cost of 2-D lattice
+summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04}
+bringing them more in line with the scaling for the full 3-D
+treatments.  The inherent periodicity in the Ewald’s method can also
+be problematic for interfacial molecular
+systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq}
 
 \subsection{Real-space methods}
 Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$
@@ -150,30 +147,29 @@ application of Wolf's method are able to obtain accura
 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 are able to obtain accurate estimates of
+application of Wolf's method can provide accurate estimates of
 Madelung constants using relatively short cutoff radii.
 
 Direct truncation of interactions at a cutoff radius creates numerical
-errors.  Wolf \textit{et al.}  argued that truncation errors are due
+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
-maintained a discontinuity at the cutoff sphere.  In the original Wolf
+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 their force expression, and as a result, the total
+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{Fennell:2006lq} Related real-space methods were also
-proposed by Chen \textit{et
+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} Recently, Fukuda has used
-neutralization of the higher order moments for the calculation of the
-electrostatic interaction of the point charge
-systems.\cite{Fukuda:2013sf}
+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}
 
 \begin{figure}
   \centering
@@ -195,10 +191,10 @@ dipoles at the eight corners of a unit cube.  Only thr
 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.  Only three of these
-basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole
+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
+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
@@ -221,12 +217,12 @@ Even at elevated temperatures, there is, on average, l
 
 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.
+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
@@ -245,17 +241,16 @@ 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 has been discussed in detail in the first paper
@@ -282,29 +277,30 @@ Because electrons in a molecule are not localized at s
 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 always an
-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} However,
 truncating point multipoles without smoothing the forces and torques
-will create energy conservation issues in molecular dynamics simulations.
+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 } 
@@ -589,7 +585,7 @@ program, OpenMD,\cite{openmd} which was used for all c
 \subsection{Implementation}
 The real-space methods developed in the first paper in this series
 have been implemented in our group's open source molecular simulation
-program, OpenMD,\cite{openmd} which was used for all calculations in
+program, OpenMD,\cite{Meineke05,openmd} which was used for all calculations in
 this work.  The complementary error function can be a relatively slow
 function on some processors, so all of the radial functions are
 precomputed on a fine grid and are spline-interpolated to provide
