--- trunk/ssdePaper/nptSSD.tex	2003/11/12 22:23:14	862
+++ trunk/ssdePaper/nptSSD.tex	2004/02/05 22:54:57	1030
@@ -21,7 +21,8 @@
 
 \begin{document}
 
-\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models}
+\title{On the structural and transport properties of the soft sticky
+dipole (SSD) and related single point water models}
 
 \author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
 Department of Chemistry and Biochemistry\\ University of Notre Dame\\
@@ -32,20 +33,29 @@ NVE and NPT molecular dynamics simulations were perfor
 \maketitle
 
 \begin{abstract}
-NVE and NPT molecular dynamics simulations were performed in order to
-investigate the density maximum and temperature dependent transport
-for SSD and related water models, both with and without the use of
-reaction field. The constant pressure simulations of the melting of
-both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most
-cases, the calculated densities were significantly lower than the
-densities calculated in simulations of other water models. Analysis of
-particle diffusion showed SSD to capture the transport properties of
+The density maximum and temperature dependence of the self-diffusion
+constant were investigated for the soft sticky dipole (SSD) water
+model and two related re-parameterizations of this single-point model.
+A combination of microcanonical and isobaric-isothermal molecular
+dynamics simulations were used to calculate these properties, both
+with and without the use of reaction field to handle long-range
+electrostatics.  The isobaric-isothermal (NPT) simulations of the
+melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near
+260 K.  In most cases, the use of the reaction field resulted in
+calculated densities which were were significantly lower than
+experimental densities.  Analysis of self-diffusion constants shows
+that the original SSD model captures the transport properties of
 experimental water very well in both the normal and super-cooled
-liquid regimes. In order to correct the density behavior, SSD was
-reparameterized for use both with and without a long-range interaction
-correction, SSD/RF and SSD/E respectively. Compared to the density
-corrected version of SSD (SSD1), these modified models were shown to
-maintain or improve upon the structural and transport properties.
+liquid regimes.  We also present our re-parameterized versions of SSD
+for use both with the reaction field or without any long-range
+electrostatic corrections.  These are called the SSD/RF and SSD/E
+models respectively.  These modified models were shown to maintain or
+improve upon the experimental agreement with the structural and
+transport properties that can be obtained with either the original SSD
+or the density corrected version of the original model (SSD1).
+Additionally, a novel low-density ice structure is presented
+which appears to be the most stable ice structure for the entire SSD
+family.
 \end{abstract}
 
 \newpage
@@ -60,233 +70,267 @@ systems is the proper depiction of water and water sol
 \section{Introduction}
 
 One of the most important tasks in the simulation of biochemical
-systems is the proper depiction of water and water solvation. In fact,
-the bulk of the calculations performed in solvated simulations are of
-interactions with or between solvent molecules. Thus, the outcomes of
-these types of simulations are highly dependent on the physical
-properties of water, both as individual molecules and in clusters or
-bulk. Due to the fact that explicit solvent accounts for a massive
-portion of the calculations, it necessary to simplify the solvent to
-some extent in order to complete simulations in a reasonable amount of
-time. In the case of simulating water in biomolecular studies, the
-balance between accurate properties and computational efficiency is
-especially delicate, and it has resulted in a variety of different
-water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these
-models predict specific properties more accurately than their
-predecessors, but often at the cost of other properties or of computer
-time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves
-upon the structural and transport properties of water relative to the
-previous TIP models, yet this comes at a greater than 50\% increase in
-computational cost.\cite{Jorgensen01,Jorgensen00} One recently
-developed model that succeeds in both retaining the accuracy of system
-properties and simplifying calculations to increase computational
-efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}
+systems is the proper depiction of the aqueous environment of the
+molecules of interest.  In some cases (such as in the simulation of 
+phospholipid bilayers), the majority of the calculations that are
+performed involve interactions with or between solvent molecules.
+Thus, the properties one may observe in biochemical simulations are
+going to be highly dependent on the physical properties of the water
+model that is chosen.
 
-The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye
-\emph{et al.} as a modified form of the hard-sphere water model
-proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD
-consists of a single point dipole with a Lennard-Jones core and a
-sticky potential that directs the particles to assume the proper
-hydrogen bond orientation in the first solvation shell. Thus, the
-interaction between two SSD water molecules \emph{i} and \emph{j} is
-given by the potential
+There is an especially delicate balance between computational
+efficiency and the ability of the water model to accurately predict
+the properties of bulk
+water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the
+TIP5P model improves on the structural and transport properties of
+water relative to the previous TIP models, yet this comes at a greater
+than 50\% increase in computational
+cost.\cite{Jorgensen01,Jorgensen00}
+
+One recently developed model that largely succeeds in retaining the
+accuracy of bulk properties while greatly reducing the computational
+cost is the Soft Sticky Dipole (SSD) water
+model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model
+was developed by Ichiye \emph{et al.} as a modified form of the
+hard-sphere water model proposed by Bratko, Blum, and
+Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model
+which has an interaction site that is both a point dipole along with a
+Lennard-Jones core.  However, since the normal aligned and
+anti-aligned geometries favored by point dipoles are poor mimics of
+local structure in liquid water, a short ranged ``sticky'' potential
+is also added.  The sticky potential directs the molecules to assume
+the proper hydrogen bond orientation in the first solvation shell.
+
+The interaction between two SSD water molecules \emph{i} and \emph{j}
+is given by the potential
 \begin{equation}
 u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
-(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
+({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
 u_{ij}^{sp}
-(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
+({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
 \end{equation}
-where the $\mathbf{r}_{ij}$ is the position vector between molecules
-\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and
-$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
-orientations of the respective molecules. The Lennard-Jones, dipole,
-and sticky parts of the potential are giving by the following
-equations:
+where the ${\bf r}_{ij}$ is the position vector between molecules
+\emph{i} and \emph{j} with magnitude $r_{ij}$, and
+${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of
+the two molecules. The Lennard-Jones and dipole interactions are given
+by the following familiar forms:
 \begin{equation}
-u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
+u_{ij}^{LJ}(r_{ij}) = 4\epsilon
+\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]
+\ ,
 \end{equation}
+and
 \begin{equation}
-u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ ,
+u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left(
+\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf
+r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ ,
 \end{equation}
+where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along
+the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and
+$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf
+r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule
+$i$.
+
+The sticky potential is somewhat less familiar:
 \begin{equation}
 u_{ij}^{sp}
-(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
-\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
+({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
+\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)
++ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf
+\Omega}_j)]\ .
+\label{stickyfunction}
 \end{equation}
-where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
-unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
-$\nu_0$ scales the strength of the overall sticky potential, and $s$
-and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
-functions take the following forms:
+Here, $\nu_0$ is a strength parameter for the sticky potential, and
+$s$ and $s^\prime$ are cubic switching functions which turn off the
+sticky interaction beyond the first solvation shell. The $w$ function
+can be thought of as an attractive potential with tetrahedral
+geometry:
 \begin{equation}
-w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
+w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
 \end{equation}
+while the $w^\prime$ function counters the normal aligned and
+anti-aligned structures favored by point dipoles:
 \begin{equation}
-w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
+w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ,
 \end{equation}
-where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
-term that promotes hydrogen bonding orientations within the first
-solvation shell, and $w^\prime$ is a dipolar repulsion term that
-repels unrealistic dipolar arrangements within the first solvation
-shell. A more detailed description of the functional parts and
-variables in this potential can be found in other
-articles.\cite{Ichiye96,Ichiye99}
+It should be noted that $w$ is proportional to the sum of the $Y_3^2$
+and $Y_3^{-2}$ spherical harmonics (a linear combination which
+enhances the tetrahedral geometry for hydrogen bonded structures),
+while $w^\prime$ is a purely empirical function.  A more detailed
+description of the functional parts and variables in this potential
+can be found in the original SSD
+articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
 
-Being that this is a one-site point dipole model, the actual force
-calculations are simplified significantly. In the original Monte Carlo
-simulations using this model, Ichiye \emph{et al.} reported an
-increase in calculation efficiency of up to an order of magnitude over
-other comparable models, while maintaining the structural behavior of
-water.\cite{Ichiye96} In the original molecular dynamics studies, it
-was shown that SSD improves on the prediction of many of water's
-dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This
-attractive combination of speed and accurate depiction of solvent
-properties makes SSD a model of interest for the simulation of large
-scale biological systems, such as membrane phase behavior.
+Since SSD is a single-point {\it dipolar} model, the force
+calculations are simplified significantly relative to the standard
+{\it charged} multi-point models. In the original Monte Carlo
+simulations using this model, Ichiye {\it et al.} reported that using
+SSD decreased computer time by a factor of 6-7 compared to other
+models.\cite{Ichiye96} What is most impressive is that this savings
+did not come at the expense of accurate depiction of the liquid state
+properties.  Indeed, SSD maintains reasonable agreement with the
+Soper data for the structural features of liquid
+water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
+exhibited by SSD agree with experiment better than those of more
+computationally expensive models (like TIP3P and
+SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
+of solvent properties makes SSD a very attractive model for the
+simulation of large scale biochemical simulations.
 
-One of the key limitations of this water model, however, is that it
-has been parameterized for use with the Ewald Sum technique for the
-handling of long-ranged interactions.  When studying very large
-systems, the Ewald summation and even particle-mesh Ewald become
-computational burdens, with their respective ideal $N^\frac{3}{2}$ and
-$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99}
-In applying this water model in these types of systems, it would be
-useful to know its properties and behavior with the more
-computationally efficient reaction field (RF) technique, and even with
-a cutoff that lacks any form of long-range correction. This study
-addresses these issues by looking at the structural and transport
-behavior of SSD over a variety of temperatures with the purpose of
-utilizing the RF correction technique. We then suggest alterations to
-the parameters that result in more water-like behavior. It should be
-noted that in a recent publication, some of the original investigators of
-the SSD water model have put forth adjustments to the SSD water model
-to address abnormal density behavior (also observed here), calling the
-corrected model SSD1.\cite{Ichiye03} This study will make comparisons
-with SSD1's behavior with the goal of improving upon the
-depiction of water under conditions without the Ewald Sum.
+One feature of the SSD model is that it was parameterized for
+use with the Ewald sum to handle long-range interactions.  This would
+normally be the best way of handling long-range interactions in
+systems that contain other point charges.  However, our group has
+recently become interested in systems with point dipoles as mimics for
+neutral, but polarized regions on molecules (e.g. the zwitterionic
+head group regions of phospholipids).  If the system of interest does
+not contain point charges, the Ewald sum and even particle-mesh Ewald
+become computational bottlenecks.  Their respective ideal
+$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$
+particles can become prohibitive when $N$ becomes
+large.\cite{Darden99} In applying this water model in these types of
+systems, it would be useful to know its properties and behavior under
+the more computationally efficient reaction field (RF) technique, or
+even with a simple cutoff. This study addresses these issues by
+looking at the structural and transport behavior of SSD over a
+variety of temperatures with the purpose of utilizing the RF
+correction technique.  We then suggest modifications to the parameters
+that result in more realistic bulk phase behavior.  It should be noted
+that in a recent publication, some of the original investigators of
+the SSD water model have suggested adjustments to the SSD
+water model to address abnormal density behavior (also observed here),
+calling the corrected model SSD1.\cite{Ichiye03} In what
+follows, we compare our reparamaterization of SSD with both the
+original SSD and SSD1 models with the goal of improving
+the bulk phase behavior of an SSD-derived model in simulations
+utilizing the Reaction Field.
 
 \section{Methods}
 
-As stated previously, the long-range dipole-dipole interactions were
-accounted for in this study by using the reaction field method. The
-magnitude of the reaction field acting on dipole \emph{i} is given by
+Long-range dipole-dipole interactions were accounted for in this study
+by using either the reaction field method or by resorting to a simple
+cubic switching function at a cutoff radius.  The reaction field
+method was actually first used in Monte Carlo simulations of liquid
+water.\cite{Barker73} Under this method, the magnitude of the reaction
+field acting on dipole $i$ is
 \begin{equation}
 \mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
-\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\  ,
+\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}),
 \label{rfequation}
 \end{equation}
 where $\mathcal{R}$ is the cavity defined by the cutoff radius
 ($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
-system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment
-vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching
+system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole
+moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching
 function.\cite{AllenTildesley} The reaction field contribution to the
-total energy by particle \emph{i} is given by
-$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
-on dipole \emph{i} by
-$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use
-of reaction field is known to alter the orientational dynamic
-properties, such as the dielectric relaxation time, based on changes
-in the length of the cutoff radius.\cite{Berendsen98} This variable
-behavior makes reaction field a less attractive method than other
-methods, like the Ewald summation; however, for the simulation of
-large-scale systems, the computational cost benefit of reaction field
-is dramatic. To address some of the dynamical property alterations due
-to the use of reaction field, simulations were also performed without
-a surrounding dielectric and suggestions are presented on how to make
-SSD more accurate both with and without a reaction field.
+total energy by particle $i$ is given by $-\frac{1}{2}{\bf
+\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf
+\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley}  Use of the reaction
+field is known to alter the bulk orientational properties, such as the
+dielectric relaxation time.  There is particular sensitivity of this
+property on changes in the length of the cutoff
+radius.\cite{Berendsen98} This variable behavior makes reaction field
+a less attractive method than the Ewald sum.  However, for very large
+systems, the computational benefit of reaction field is dramatic.
+ 
+We have also performed a companion set of simulations {\it without} a
+surrounding dielectric (i.e. using a simple cubic switching function
+at the cutoff radius), and as a result we have two reparamaterizations
+of SSD which could be used either with or without the reaction
+field turned on.
 
-Simulations were performed in both the isobaric-isothermal and
-microcanonical ensembles. The constant pressure simulations were
+Simulations to obtain the preferred densities of the models were
+performed in the isobaric-isothermal (NPT) ensemble, while all
+dynamical properties were obtained from microcanonical (NVE)
+simulations done at densities matching the NPT density for a
+particular target temperature.  The constant pressure simulations were
 implemented using an integral thermostat and barostat as outlined by
-Hoover.\cite{Hoover85,Hoover86} All particles were treated as
+Hoover.\cite{Hoover85,Hoover86} All molecules were treated as
 non-linear rigid bodies. Vibrational constraints are not necessary in
-simulations of SSD, because there are no explicit hydrogen atoms, and
-thus no molecular vibrational modes need to be considered.
+simulations of SSD, because there are no explicit hydrogen
+atoms, and thus no molecular vibrational modes need to be considered.
 
 Integration of the equations of motion was carried out using the
-symplectic splitting method proposed by Dullweber \emph{et
-al.}\cite{Dullweber1997} The reason for this integrator selection
-deals with poor energy conservation of rigid body systems using
-quaternions. While quaternions work well for orientational motion in
-alternate ensembles, the microcanonical ensemble has a constant energy
-requirement that is quite sensitive to errors in the equations of
-motion. The original implementation of this code utilized quaternions
-for rotational motion propagation; however, a detailed investigation
-showed that they resulted in a steady drift in the total energy,
-something that has been observed by others.\cite{Laird97}
+symplectic splitting method proposed by Dullweber, Leimkuhler, and
+McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting
+this integrator centers on poor energy conservation of rigid body
+dynamics using traditional quaternion
+integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble
+simulations, the energy drift when using quaternions was substantially
+greater than when using the {\sc dlm} method (fig. \ref{timestep}).
+This steady drift in the total energy has also been observed by Kol
+{\it et al.}\cite{Laird97}
 
 The key difference in the integration method proposed by Dullweber
 \emph{et al.} is that the entire rotation matrix is propagated from
-one time step to the next. In the past, this would not have been as
-feasible an option, being that the rotation matrix for a single body is
-nine elements long as opposed to 3 or 4 elements for Euler angles and
-quaternions respectively. System memory has become much less of an
-issue in recent times, and this has resulted in substantial benefits
-in energy conservation. There is still the issue of 5 or 6 additional
-elements for describing the orientation of each particle, which will
-increase dump files substantially. Simply translating the rotation
-matrix into its component Euler angles or quaternions for storage
-purposes relieves this burden.
+one time step to the next.  The additional memory required by the
+algorithm is inconsequential on modern computers, and translating the
+rotation matrix into quaternions for storage purposes makes trajectory
+data quite compact.
 
-The symplectic splitting method allows for Verlet style integration of
-both linear and angular motion of rigid bodies. In this integration
-method, the orientational propagation involves a sequence of matrix
-evaluations to update the rotation matrix.\cite{Dullweber1997} These
-matrix rotations are more costly computationally than the simpler
-arithmetic quaternion propagation. With the same time step, a 1000 SSD
-particle simulation shows an average 7\% increase in computation time
-using the symplectic step method in place of quaternions. This cost is
-more than justified when comparing the energy conservation of the two
-methods as illustrated in figure \ref{timestep}.
+The {\sc dlm} method allows for Verlet style integration of both
+translational and orientational motion of rigid bodies. In this
+integration method, the orientational propagation involves a sequence
+of matrix evaluations to update the rotation
+matrix.\cite{Dullweber1997} These matrix rotations are more costly
+than the simpler arithmetic quaternion propagation. With the same time
+step, a 1000 SSD particle simulation shows an average 7\%
+increase in computation time using the {\sc dlm} method in place of
+quaternions. The additional expense per step is justified when one
+considers the ability to use time steps that are nearly twice as large
+under {\sc dlm} than would be usable under quaternion dynamics.  The
+energy conservation of the two methods using a number of different
+time steps is illustrated in figure
+\ref{timestep}.
 
 \begin{figure}
 \begin{center}
 \epsfxsize=6in
 \epsfbox{timeStep.epsi}
-\caption{Energy conservation using quaternion based integration versus 
-the symplectic step method proposed by Dullweber \emph{et al.} with
-increasing time step. The larger time step plots are shifted up from
-the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
+\caption{Energy conservation using both quaternion-based integration and
+the {\sc dlm} method with increasing time step. The larger time step plots
+are shifted from the true energy baseline (that of $\Delta t$ = 0.1
+fs) for clarity.}
 \label{timestep}
 \end{center}
 \end{figure}
 
 In figure \ref{timestep}, the resulting energy drift at various time
-steps for both the symplectic step and quaternion integration schemes
-is compared. All of the 1000 SSD particle simulations started with the
-same configuration, and the only difference was the method used to
-handle rotational motion. At time steps of 0.1 and 0.5 fs, both
-methods for propagating particle rotation conserve energy fairly well,
-with the quaternion method showing a slight energy drift over time in
-the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
-energy conservation benefits of the symplectic step method are clearly
-demonstrated. Thus, while maintaining the same degree of energy
-conservation, one can take considerably longer time steps, leading to
-an overall reduction in computation time.
+steps for both the {\sc dlm} and quaternion integration schemes is
+compared.  All of the 1000 SSD particle simulations started with
+the same configuration, and the only difference was the method used to
+handle orientational motion. At time steps of 0.1 and 0.5 fs, both
+methods for propagating the orientational degrees of freedom conserve
+energy fairly well, with the quaternion method showing a slight energy
+drift over time in the 0.5 fs time step simulation. At time steps of 1
+and 2 fs, the energy conservation benefits of the {\sc dlm} method are
+clearly demonstrated. Thus, while maintaining the same degree of
+energy conservation, one can take considerably longer time steps,
+leading to an overall reduction in computation time.
 
-Energy drift in the symplectic step simulations was unnoticeable for
-time steps up to three femtoseconds. A slight energy drift on the
+Energy drift in the simulations using {\sc dlm} integration was
+unnoticeable for time steps up to 3 fs. A slight energy drift on the
 order of 0.012 kcal/mol per nanosecond was observed at a time step of
-four femtoseconds, and as expected, this drift increases dramatically
-with increasing time step. To insure accuracy in the constant energy
+4 fs, and as expected, this drift increases dramatically with
+increasing time step. To insure accuracy in our microcanonical
 simulations, time steps were set at 2 fs and kept at this value for
 constant pressure simulations as well.
 
-Ice crystals in both the $I_h$ and $I_c$ lattices were generated as
-starting points for all simulations. The $I_h$ crystals were formed by
-first arranging the centers of mass of the SSD particles into a
-``hexagonal'' ice lattice of 1024 particles. Because of the crystal
-structure of $I_h$ ice, the simulation box assumed a rectangular shape
-with an edge length ratio of approximately
-1.00$\times$1.06$\times$1.23. The particles were then allowed to
-orient freely about fixed positions with angular momenta randomized at
-400 K for varying times. The rotational temperature was then scaled
-down in stages to slowly cool the crystals to 25 K. The particles were
-then allowed to translate with fixed orientations at a constant
-pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were
-removed and the ice crystals were allowed to equilibrate for 50 ps at
-25 K and a constant pressure of 1 atm.  This procedure resulted in
-structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
+Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices
+were generated as starting points for all simulations. The $I_h$
+crystals were formed by first arranging the centers of mass of the
+SSD particles into a ``hexagonal'' ice lattice of 1024
+particles. Because of the crystal structure of $I_h$ ice, the
+simulation box assumed an orthorhombic shape with an edge length ratio
+of approximately 1.00$\times$1.06$\times$1.23. The particles were then
+allowed to orient freely about fixed positions with angular momenta
+randomized at 400 K for varying times. The rotational temperature was
+then scaled down in stages to slowly cool the crystals to 25 K. The
+particles were then allowed to translate with fixed orientations at a
+constant pressure of 1 atm for 50 ps at 25 K. Finally, all constraints
+were removed and the ice crystals were allowed to equilibrate for 50
+ps at 25 K and a constant pressure of 1 atm.  This procedure resulted
+in structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
 rules.\cite{Bernal33,Rahman72} This method was also utilized in the
 making of diamond lattice $I_c$ ice crystals, with each cubic
 simulation box consisting of either 512 or 1000 particles. Only
@@ -297,152 +341,151 @@ constant pressure and temperature dynamics. During mel
 \section{Results and discussion}
 
 Melting studies were performed on the randomized ice crystals using
-constant pressure and temperature dynamics. During melting
-simulations, the melting transition and the density maximum can both
-be observed, provided that the density maximum occurs in the liquid
-and not the supercooled regime. An ensemble average from five separate
-melting simulations was acquired, each starting from different ice
-crystals generated as described previously. All simulations were
-equilibrated for 100 ps prior to a 200 ps data collection run at each
-temperature setting. The temperature range of study spanned from 25 to
-400 K, with a maximum degree increment of 25 K. For regions of
-interest along this stepwise progression, the temperature increment
-was decreased from 25 K to 10 and 5 K. The above equilibration and
-production times were sufficient in that the system volume
-fluctuations dampened out in all but the very cold simulations (below
-225 K).
+isobaric-isothermal (NPT) dynamics. During melting simulations, the
+melting transition and the density maximum can both be observed,
+provided that the density maximum occurs in the liquid and not the
+supercooled regime. An ensemble average from five separate melting
+simulations was acquired, each starting from different ice crystals
+generated as described previously. All simulations were equilibrated
+for 100 ps prior to a 200 ps data collection run at each temperature
+setting. The temperature range of study spanned from 25 to 400 K, with
+a maximum degree increment of 25 K. For regions of interest along this
+stepwise progression, the temperature increment was decreased from 25
+K to 10 and 5 K.  The above equilibration and production times were
+sufficient in that fluctuations in the volume autocorrelation function
+were damped out in all simulations in under 20 ps.
 
 \subsection{Density Behavior}
-Initial simulations focused on the original SSD water model, and an
-average density versus temperature plot is shown in figure
-\ref{dense1}. Note that the density maximum when using a reaction 
-field appears between 255 and 265 K, where the calculated densities
-within this range were nearly indistinguishable. The greater certainty
-of the average value at 260 K makes a good argument for the actual
-density maximum residing at this midpoint value. Figure \ref{dense1}
-was constructed using ice $I_h$ crystals for the initial
-configuration; though not pictured, the simulations starting from ice
-$I_c$ crystal configurations showed similar results, with a
-liquid-phase density maximum in this same region (between 255 and 260
-K). In addition, the $I_c$ crystals are more fragile than the $I_h$
-crystals, leading to deformation into a dense glassy state at lower
-temperatures. This resulted in an overall low temperature density
-maximum at 200 K, while still retaining a liquid state density maximum
-in common with the $I_h$ simulations.
 
+Our initial simulations focused on the original SSD water model,
+and an average density versus temperature plot is shown in figure
+\ref{dense1}. Note that the density maximum when using a reaction
+field appears between 255 and 265 K.  There were smaller fluctuations
+in the density at 260 K than at either 255 or 265, so we report this
+value as the location of the density maximum. Figure \ref{dense1} was
+constructed using ice $I_h$ crystals for the initial configuration;
+though not pictured, the simulations starting from ice $I_c$ crystal
+configurations showed similar results, with a liquid-phase density
+maximum in this same region (between 255 and 260 K). 
+
 \begin{figure}
 \begin{center}
 \epsfxsize=6in
-\epsfbox{denseSSD.eps}
-\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b} 
- TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction
- Field, SSD, and experiment.\cite{CRC80} The arrows indicate the
- change in densities observed when turning off the reaction field. The
- the lower than expected densities for the SSD model were what
- prompted the original reparameterization.\cite{Ichiye03}}
+\epsfbox{denseSSDnew.eps}
+\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}],
+ TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD
+ without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The
+ arrows indicate the change in densities observed when turning off the
+ reaction field. The the lower than expected densities for the SSD
+ model were what prompted the original reparameterization of SSD1
+ [Ref. \citen{Ichiye03}].}
 \label{dense1}
 \end{center}
 \end{figure}
 
-The density maximum for SSD actually compares quite favorably to other
+The density maximum for SSD compares quite favorably to other
 simple water models. Figure \ref{dense1} also shows calculated
 densities of several other models and experiment obtained from other
 sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
-models, SSD has results closest to the experimentally observed water
-density maximum. Of the listed water models, TIP4P has a density
-maximum behavior most like that seen in SSD. Though not included in
-this particular plot, it is useful to note that TIP5P has a water
-density maximum nearly identical to experiment.
+models, SSD has a temperature closest to the experimentally
+observed density maximum. Of the {\it charge-based} models in
+Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that
+seen in SSD. Though not included in this plot, it is useful to
+note that TIP5P has a density maximum nearly identical to the
+experimentally measured temperature.
 
-It has been observed that densities are dependent on the cutoff radius
-used for a variety of water models in simulations both with and
-without the use of reaction field.\cite{Berendsen98} In order to
-address the possible affect of cutoff radius, simulations were
-performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the
-previous SSD simulations, all performed with a cutoff of 9.0 \AA. All
-of the resulting densities overlapped within error and showed no
-significant trend toward lower or higher densities as a function of
-cutoff radius, for simulations both with and without reaction
-field. These results indicate that there is no major benefit in
-choosing a longer cutoff radius in simulations using SSD. This is
-advantageous in that the use of a longer cutoff radius results in
-significant increases in the time required to obtain a single
-trajectory.
+It has been observed that liquid state densities in water are
+dependent on the cutoff radius used both with and without the use of
+reaction field.\cite{Berendsen98} In order to address the possible
+effect of cutoff radius, simulations were performed with a dipolar
+cutoff radius of 12.0 \AA\ to complement the previous SSD
+simulations, all performed with a cutoff of 9.0 \AA. All of the
+resulting densities overlapped within error and showed no significant
+trend toward lower or higher densities as a function of cutoff radius,
+for simulations both with and without reaction field. These results
+indicate that there is no major benefit in choosing a longer cutoff
+radius in simulations using SSD. This is advantageous in that
+the use of a longer cutoff radius results in a significant increase in
+the time required to obtain a single trajectory.
 
 The key feature to recognize in figure \ref{dense1} is the density
 scaling of SSD relative to other common models at any given
-temperature. Note that the SSD model assumes a lower density than any
-of the other listed models at the same pressure, behavior which is
-especially apparent at temperatures greater than 300 K. Lower than
-expected densities have been observed for other systems using a
-reaction field for long-range electrostatic interactions, so the most
-likely reason for the significantly lower densities seen in these
-simulations is the presence of the reaction
-field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the
-reaction field on the density of the systems, the simulations were
-repeated without a reaction field present. The results of these
-simulations are also displayed in figure \ref{dense1}. Without
-reaction field, the densities increase considerably to more
-experimentally reasonable values, especially around the freezing point
-of liquid water. The shape of the curve is similar to the curve
-produced from SSD simulations using reaction field, specifically the
-rapidly decreasing densities at higher temperatures; however, a shift
-in the density maximum location, down to 245 K, is observed. This is a
-more accurate comparison to the other listed water models, in that no
-long range corrections were applied in those
-simulations.\cite{Clancy94,Jorgensen98b} However, even without a
+temperature. SSD assumes a lower density than any of the other
+listed models at the same pressure, behavior which is especially
+apparent at temperatures greater than 300 K. Lower than expected
+densities have been observed for other systems using a reaction field
+for long-range electrostatic interactions, so the most likely reason
+for the significantly lower densities seen in these simulations is the
+presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order
+to test the effect of the reaction field on the density of the
+systems, the simulations were repeated without a reaction field
+present. The results of these simulations are also displayed in figure
+\ref{dense1}. Without the reaction field, the densities increase
+to more experimentally reasonable values, especially around the
+freezing point of liquid water. The shape of the curve is similar to
+the curve produced from SSD simulations using reaction field,
+specifically the rapidly decreasing densities at higher temperatures;
+however, a shift in the density maximum location, down to 245 K, is
+observed. This is a more accurate comparison to the other listed water
+models, in that no long range corrections were applied in those
+simulations.\cite{Clancy94,Jorgensen98b} However, even without the
 reaction field, the density around 300 K is still significantly lower
 than experiment and comparable water models. This anomalous behavior
-was what lead Ichiye \emph{et al.} to recently reparameterize SSD and
-make SSD1.\cite{Ichiye03} In discussing potential adjustments later in
-this paper, all comparisons were performed with this new model.
+was what lead Tan {\it et al.} to recently reparameterize
+SSD.\cite{Ichiye03} Throughout the remainder of the paper our
+reparamaterizations of SSD will be compared with their newer SSD1
+model.
 
 \subsection{Transport Behavior}
-Of importance in these types of studies are the transport properties
-of the particles and their change in responce to altering
-environmental conditions. In order to probe transport, constant energy
-simulations were performed about the average density uncovered by the
-constant pressure simulations. Simulations started with randomized
-velocities and underwent 50 ps of temperature scaling and 50 ps of
-constant energy equilibration before obtaining a 200 ps
-trajectory. Diffusion constants were calculated via root-mean square
-deviation analysis. The averaged results from five sets of NVE
-simulations are displayed in figure \ref{diffuse}, alongside
-experimental, SPC/E, and TIP5P
-results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
 
+Accurate dynamical properties of a water model are particularly
+important when using the model to study permeation or transport across
+biological membranes.  In order to probe transport in bulk water,
+constant energy (NVE) simulations were performed at the average
+density obtained by the NPT simulations at an identical target
+temperature. Simulations started with randomized velocities and
+underwent 50 ps of temperature scaling and 50 ps of constant energy
+equilibration before a 200 ps data collection run. Diffusion constants
+were calculated via linear fits to the long-time behavior of the
+mean-square displacement as a function of time. The averaged results
+from five sets of NVE simulations are displayed in figure
+\ref{diffuse}, alongside experimental, SPC/E, and TIP5P
+results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01} 
+
 \begin{figure}
 \begin{center}
 \epsfxsize=6in
 \epsfbox{betterDiffuse.epsi} 
-\caption{Average diffusion coefficient over increasing temperature for 
-SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental
-data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has
-the least deviation from the experimental values. The rapidly
-increasing diffusion constants for TIP5P and SSD correspond to
-significant decrease in density at the higher temperatures.}
+\caption{Average self-diffusion constant as a function of temperature for 
+SSD, SPC/E [Ref. \citen{Clancy94}], and TIP5P
+[Ref. \citen{Jorgensen01}] compared with experimental data
+[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models
+shown, SSD has the least deviation from the experimental values. The
+rapidly increasing diffusion constants for TIP5P and SSD correspond to
+significant decreases in density at the higher temperatures.}
 \label{diffuse}
 \end{center}
 \end{figure} 
 
 The observed values for the diffusion constant point out one of the
-strengths of the SSD model. Of the three experimental models shown,
-the SSD model has the most accurate depiction of the diffusion trend
-seen in experiment in both the supercooled and liquid temperature
-regimes. SPC/E does a respectable job by producing values similar to
-SSD and experiment around 290 K; however, it deviates at both higher
-and lower temperatures, failing to predict the experimental
-trend. TIP5P and SSD both start off low at colder temperatures and
-tend to diffuse too rapidly at higher temperatures. This trend at
-higher temperatures is not surprising in that the densities of both
-TIP5P and SSD are lower than experimental water at these higher
-temperatures. When calculating the diffusion coefficients for SSD at
-experimental densities, the resulting values fall more in line with
-experiment at these temperatures, albeit not at standard pressure.
+strengths of the SSD model. Of the three models shown, the SSD model
+has the most accurate depiction of self-diffusion in both the
+supercooled and liquid regimes.  SPC/E does a respectable job by
+reproducing values similar to experiment around 290 K; however, it
+deviates at both higher and lower temperatures, failing to predict the
+correct thermal trend. TIP5P and SSD both start off low at colder
+temperatures and tend to diffuse too rapidly at higher temperatures.
+This behavior at higher temperatures is not particularly surprising
+since the densities of both TIP5P and SSD are lower than experimental
+water densities at higher temperatures.  When calculating the
+diffusion coefficients for SSD at experimental densities
+(instead of the densities from the NPT simulations), the resulting
+values fall more in line with experiment at these temperatures.
 
 \subsection{Structural Changes and Characterization}
+
 By starting the simulations from the crystalline state, the melting
-transition and the ice structure can be studied along with the liquid
+transition and the ice structure can be obtained along with the liquid
 phase behavior beyond the melting point. The constant pressure heat
 capacity (C$_\text{p}$) was monitored to locate the melting transition
 in each of the simulations. In the melting simulations of the 1024
@@ -450,15 +493,13 @@ considerably lower than the experimental value, but th
 at 245 K, indicating a first order phase transition for the melting of
 these ice crystals. When the reaction field is turned off, the melting
 transition occurs at 235 K.  These melting transitions are
-considerably lower than the experimental value, but this is not a
-surprise considering the simplicity of the SSD model.
+considerably lower than the experimental value.
 
 \begin{figure}
 \begin{center}
 \epsfxsize=6in
 \epsfbox{corrDiag.eps} 
-\caption{Two dimensional illustration of angles involved in the 
-correlations observed in figure \ref{contour}.}
+\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.}
 \label{corrAngle}
 \end{center}
 \end{figure}
@@ -467,83 +508,84 @@ correlations observed in figure \ref{contour}.}
 \begin{center} 
 \epsfxsize=6in
 \epsfbox{fullContours.eps} 
-\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at 
-100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for
-clarity: dark areas signify peaks while light areas signify
-depressions. White areas have g(\emph{r}) values below 0.5 and black
-areas have values above 1.5.}
+\caption{Contour plots of 2D angular pair correlation functions for
+512 SSD molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas
+signify regions of enhanced density while light areas signify
+depletion relative to the bulk density. White areas have pair
+correlation values below 0.5 and black areas have values above 1.5.}
 \label{contour} 
 \end{center}
 \end{figure} 
 
-Additional analysis of the melting phase-transition process was
-performed by using two-dimensional structure and dipole angle
-correlations. Expressions for these correlations are as follows:
+Additional analysis of the melting process was performed using
+two-dimensional structure and dipole angle correlations. Expressions
+for these correlations are as follows:
 
 \begin{equation}
-g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ ,
+g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
 \end{equation}
 \begin{equation}
 g_{\text{AB}}(r,\cos\omega) = 
-\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ ,
+\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
 \end{equation}
 where $\theta$ and $\omega$ refer to the angles shown in figure
 \ref{corrAngle}. By binning over both distance and the cosine of the
-desired angle between the two dipoles, the g(\emph{r}) can be
-dissected to determine the common dipole arrangements that constitute
-the peaks and troughs. Frames A and B of figure \ref{contour} show a
-relatively crystalline state of an ice $I_c$ simulation. The first
-peak of the g(\emph{r}) consists primarily of the preferred hydrogen
+desired angle between the two dipoles, the $g(r)$ can be analyzed to
+determine the common dipole arrangements that constitute the peaks and
+troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
+of figure \ref{contour} show results from an ice $I_c$ simulation. The
+first peak in the $g(r)$ consists primarily of the preferred hydrogen
 bonding arrangements as dictated by the tetrahedral sticky potential -
-one peak for the donating and the other for the accepting hydrogen
-bonds. Due to the high degree of crystallinity of the sample, the
-second and third solvation shells show a repeated peak arrangement
+one peak for the hydrogen bond donor and the other for the hydrogen
+bond acceptor.  Due to the high degree of crystallinity of the sample,
+the second and third solvation shells show a repeated peak arrangement
 which decays at distances around the fourth solvation shell, near the
 imposed cutoff for the Lennard-Jones and dipole-dipole interactions.
 In the higher temperature simulation shown in frames C and D, these
-longer-ranged repeated peak features deteriorate rapidly. The first
-solvation shell still shows the strong effect of the sticky-potential,
-although it covers a larger area, extending to include a fraction of
-aligned dipole peaks within the first solvation shell. The latter
-peaks lose definition as thermal motion and the competing dipole force
-overcomes the sticky potential's tight tetrahedral structuring of the
-fluid.
+long-range features deteriorate rapidly. The first solvation shell
+still shows the strong effect of the sticky-potential, although it
+covers a larger area, extending to include a fraction of aligned
+dipole peaks within the first solvation shell. The latter peaks lose
+due to thermal motion and as the competing dipole force overcomes the
+sticky potential's tight tetrahedral structuring of the crystal.
 
 This complex interplay between dipole and sticky interactions was
 remarked upon as a possible reason for the split second peak in the
-oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the
-second solvation shell peak appears to have two distinct components
-that blend together to form one observable peak. At higher
-temperatures, this split character alters to show the leading 4 \AA\
-peak dominated by equatorial anti-parallel dipole orientations. There
-is also a tightly bunched group of axially arranged dipoles that most
-likely consist of the smaller fraction of aligned dipole pairs. The
-trailing component of the split peak at 5 \AA\ is dominated by aligned
-dipoles that assume hydrogen bond arrangements similar to those seen
-in the first solvation shell. This evidence indicates that the dipole
-pair interaction begins to dominate outside of the range of the
-dipolar repulsion term. Primary energetically favorable dipole
-arrangements populate the region immediately outside this repulsion
-region (around 4 \AA), while arrangements that seek to ideally satisfy
-both the sticky and dipole forces locate themselves just beyond this
-initial buildup (around 5 \AA).
+oxygen-oxygen pair correlation function,
+$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second
+solvation shell peak appears to have two distinct components that
+blend together to form one observable peak. At higher temperatures,
+this split character alters to show the leading 4 \AA\ peak dominated
+by equatorial anti-parallel dipole orientations. There is also a
+tightly bunched group of axially arranged dipoles that most likely
+consist of the smaller fraction of aligned dipole pairs. The trailing
+component of the split peak at 5 \AA\ is dominated by aligned dipoles
+that assume hydrogen bond arrangements similar to those seen in the
+first solvation shell. This evidence indicates that the dipole pair
+interaction begins to dominate outside of the range of the dipolar
+repulsion term.  The energetically favorable dipole arrangements
+populate the region immediately outside this repulsion region (around
+4 \AA), while arrangements that seek to satisfy both the sticky and
+dipole forces locate themselves just beyond this initial buildup
+(around 5 \AA).
 
 From these findings, the split second peak is primarily the product of
 the dipolar repulsion term of the sticky potential. In fact, the inner
 peak can be pushed out and merged with the outer split peak just by
-extending the switching function cutoff ($s^\prime(r_{ij})$) from its
-normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of
+extending the switching function ($s^\prime(r_{ij})$) from its normal
+4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of
 correction is not recommended for improving the liquid structure,
 since the second solvation shell would still be shifted too far
 out. In addition, this would have an even more detrimental effect on
 the system densities, leading to a liquid with a more open structure
-and a density considerably lower than the normal SSD behavior shown
-previously. A better correction would be to include the
+and a density considerably lower than the already low SSD
+density.  A better correction would be to include the
 quadrupole-quadrupole interactions for the water particles outside of
-the first solvation shell, but this reduces the simplicity and speed
-advantage of SSD.
+the first solvation shell, but this would remove the simplicity and
+speed advantage of SSD.
 
 \subsection{Adjusted Potentials: SSD/RF and SSD/E}
+
 The propensity of SSD to adopt lower than expected densities under
 varying conditions is troubling, especially at higher temperatures. In
 order to correct this model for use with a reaction field, it is
@@ -551,47 +593,34 @@ densities while maintaining the accurate transport pro
 intermolecular interactions. In undergoing a reparameterization, it is
 important not to focus on just one property and neglect the other
 important properties. In this case, it would be ideal to correct the
-densities while maintaining the accurate transport properties.
+densities while maintaining the accurate transport behavior.
 
-The parameters available for tuning include the $\sigma$ and $\epsilon$
-Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
-attractive and dipole repulsive terms with their respective
-cutoffs. To alter the attractive and repulsive terms of the sticky
-potential independently, it is necessary to separate the terms as
-follows:
-\begin{equation}
-u_{ij}^{sp}
-(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
-\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
-\end{equation}
+The parameters available for tuning include the $\sigma$ and
+$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the
+strength of the sticky potential ($\nu_0$), and the cutoff distances
+for the sticky attractive and dipole repulsive cubic switching
+function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$
+respectively). The results of the reparameterizations are shown in
+table \ref{params}. We are calling these reparameterizations the Soft
+Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction
+field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve
+the liquid structure in simulations without a long-range correction).
 
-where $\nu_0$ scales the strength of the tetrahedral attraction and
-$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
-term. The separation was performed for purposes of the
-reparameterization, but the final parameters were adjusted so that it
-is unnecessary to separate the terms when implementing the adjusted
-water potentials. The results of the reparameterizations are shown in
-table \ref{params}. Note that the tetrahedral attractive and dipolar
-repulsive terms do not share the same lower cutoff ($r_l$) in the
-newly parameterized potentials - soft sticky dipole reaction field
-(SSD/RF - for use with a reaction field) and soft sticky dipole
-enhanced (SSD/E - an attempt to improve the liquid structure in
-simulations without a long-range correction).
-
 \begin{table}
 \begin{center}
 \caption{Parameters for the original and adjusted models}
 \begin{tabular}{ l  c  c  c  c } 
 \hline \\[-3mm]
-\ \ \ Parameters\ \ \  & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \  & \ SSD/E\ \  & \ SSD/RF \\
+\ \ \ Parameters\ \ \  & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \
+& \ SSD1 [Ref. \citen{Ichiye03}]\ \  & \ SSD/E\ \  & \ SSD/RF \\
 \hline \\[-3mm]
 \ \ \ $\sigma$ (\AA)  & 3.051 & 3.016 & 3.035 & 3.019\\
 \ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
 \ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\
 \ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
+\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\
 \ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\
 \ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\
-\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
 \ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
 \ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
 \end{tabular}
@@ -603,11 +632,12 @@ simulations without a long-range correction).
 \begin{center} 
 \epsfxsize=5in
 \epsfbox{GofRCompare.epsi} 
-\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E 
-and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with
-reaction field turned on (bottom). The insets show the respective
-first peaks in detail. Note how the changes in parameters have lowered
-and broadened the first peak of SSD/E and SSD/RF.}
+\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with 
+SSD/E and SSD1 without reaction field (top), as well as
+SSD/RF and SSD1 with reaction field turned on
+(bottom). The insets show the respective first peaks in detail. Note
+how the changes in parameters have lowered and broadened the first
+peak of SSD/E and SSD/RF.}
 \label{grcompare} 
 \end{center}
 \end{figure} 
@@ -615,59 +645,59 @@ and broadened the first peak of SSD/E and SSD/RF.}
 \begin{figure}
 \begin{center} 
 \epsfxsize=6in
-\epsfbox{dualsticky.ps} 
-\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& 
-SSD/RF (right). Light areas correspond to the tetrahedral attractive
-component, and darker areas correspond to the dipolar repulsive
-component.}
+\epsfbox{dualsticky_bw.eps} 
+\caption{Positive and negative isosurfaces of the sticky potential for
+SSD1 (left) and SSD/E \& SSD/RF (right). Light areas
+correspond to the tetrahedral attractive component, and darker areas
+correspond to the dipolar repulsive component.}
 \label{isosurface} 
 \end{center}
 \end{figure} 
 
-In the paper detailing the development of SSD, Liu and Ichiye placed
-particular emphasis on an accurate description of the first solvation
-shell. This resulted in a somewhat tall and narrow first peak in the
-g(\emph{r}) that integrated to give similar coordination numbers to
+In the original paper detailing the development of SSD, Liu and Ichiye
+placed particular emphasis on an accurate description of the first
+solvation shell. This resulted in a somewhat tall and narrow first
+peak in $g(r)$ that integrated to give similar coordination numbers to
 the experimental data obtained by Soper and
 Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering
 data from the Head-Gordon lab indicates a slightly lower and shifted
-first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made
-while taking into consideration the new experimental
+first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were
+made after taking into consideration the new experimental
 findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the
-relocation of the first peak of the oxygen-oxygen g(\emph{r}) by
-comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new
+relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
+the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
 experimental results. Both modified water models have shorter peaks
-that are brought in more closely to the experimental peak (as seen in
-the insets of figure \ref{grcompare}).  This structural alteration was
+that match more closely to the experimental peak (as seen in the
+insets of figure \ref{grcompare}).  This structural alteration was
 accomplished by the combined reduction in the Lennard-Jones $\sigma$
-variable and adjustment of the sticky potential strength and
-cutoffs. As can be seen in table \ref{params}, the cutoffs for the
-tetrahedral attractive and dipolar repulsive terms were nearly swapped
-with each other. Isosurfaces of the original and modified sticky
-potentials are shown in figure \ref{isosurface}. In these isosurfaces,
-it is easy to see how altering the cutoffs changes the repulsive and
-attractive character of the particles. With a reduced repulsive
-surface (darker region), the particles can move closer to one another,
-increasing the density for the overall system. This change in
-interaction cutoff also results in a more gradual orientational motion
-by allowing the particles to maintain preferred dipolar arrangements
-before they begin to feel the pull of the tetrahedral
-restructuring. As the particles move closer together, the dipolar
-repulsion term becomes active and excludes unphysical nearest-neighbor
-arrangements. This compares with how SSD and SSD1 exclude preferred
-dipole alignments before the particles feel the pull of the ``hydrogen
-bonds''. Aside from improving the shape of the first peak in the
-g(\emph{r}), this modification improves the densities considerably by
-allowing the persistence of full dipolar character below the previous
-4.0 \AA\ cutoff.
+variable and adjustment of the sticky potential strength and cutoffs.
+As can be seen in table \ref{params}, the cutoffs for the tetrahedral
+attractive and dipolar repulsive terms were nearly swapped with each
+other.  Isosurfaces of the original and modified sticky potentials are
+shown in figure \ref{isosurface}. In these isosurfaces, it is easy to
+see how altering the cutoffs changes the repulsive and attractive
+character of the particles. With a reduced repulsive surface (darker
+region), the particles can move closer to one another, increasing the
+density for the overall system.  This change in interaction cutoff
+also results in a more gradual orientational motion by allowing the
+particles to maintain preferred dipolar arrangements before they begin
+to feel the pull of the tetrahedral restructuring. As the particles
+move closer together, the dipolar repulsion term becomes active and
+excludes unphysical nearest-neighbor arrangements. This compares with
+how SSD and SSD1 exclude preferred dipole alignments before the
+particles feel the pull of the ``hydrogen bonds''. Aside from
+improving the shape of the first peak in the g(\emph{r}), this
+modification improves the densities considerably by allowing the
+persistence of full dipolar character below the previous 4.0 \AA\
+cutoff.
 
-While adjusting the location and shape of the first peak of
-g(\emph{r}) improves the densities, these changes alone are
-insufficient to bring the system densities up to the values observed
-experimentally. To further increase the densities, the dipole moments
-were increased in both of the adjusted models. Since SSD is a dipole
-based model, the structure and transport are very sensitive to changes
-in the dipole moment. The original SSD simply used the dipole moment
+While adjusting the location and shape of the first peak of $g(r)$
+improves the densities, these changes alone are insufficient to bring
+the system densities up to the values observed experimentally.  To
+further increase the densities, the dipole moments were increased in
+both of our adjusted models. Since SSD is a dipole based model,
+the structure and transport are very sensitive to changes in the
+dipole moment. The original SSD simply used the dipole moment
 calculated from the TIP3P water model, which at 2.35 D is
 significantly greater than the experimental gas phase value of 1.84
 D. The larger dipole moment is a more realistic value and improves the
@@ -676,9 +706,9 @@ increasing the dipole moments to 2.42 and 2.48 D for S
 to values as high as 3.11 D, providing a substantial range of
 reasonable values for a dipole
 moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
-increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF,
-respectively, leads to significant changes in the density and
-transport of the water models.
+increasing the dipole moments to 2.42 and 2.48 D for SSD/E and
+SSD/RF, respectively, leads to significant changes in the
+density and transport of the water models.
 
 In order to demonstrate the benefits of these reparameterizations, a
 series of NPT and NVE simulations were performed to probe the density
@@ -693,86 +723,91 @@ collection times as stated earlier in this paper.
 run at each temperature step, and the final configuration from the
 previous temperature simulation was used as a starting point. All NVE
 simulations had the same thermalization, equilibration, and data
-collection times as stated earlier in this paper.
+collection times as stated previously.
 
 \begin{figure}
 \begin{center} 
 \epsfxsize=6in
 \epsfbox{ssdeDense.epsi} 
-\caption{Comparison of densities calculated with SSD/E to SSD1 without a 
-reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
-SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a
-expansion around 300 K with error bars included to clarify this region
-of interest. Note that both SSD1 and SSD/E show good agreement with
+\caption{Comparison of densities calculated with SSD/E to 
+SSD1 without a reaction field, TIP3P [Ref. \citen{Jorgensen98b}],
+TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and
+experiment [Ref. \citen{CRC80}]. The window shows a expansion around
+300 K with error bars included to clarify this region of
+interest. Note that both SSD1 and SSD/E show good agreement with
 experiment when the long-range correction is neglected.}
 \label{ssdedense} 
 \end{center}
 \end{figure} 
 
-Figure \ref{ssdedense} shows the density profile for the SSD/E model
-in comparison to SSD1 without a reaction field, other common water
-models, and experimental results. The calculated densities for both
-SSD/E and SSD1 have increased significantly over the original SSD
-model (see figure \ref{dense1}) and are in better agreement with the
-experimental values. At 298 K, the densities of SSD/E and SSD1 without
+Fig. \ref{ssdedense} shows the density profile for the SSD/E
+model in comparison to SSD1 without a reaction field, other
+common water models, and experimental results. The calculated
+densities for both SSD/E and SSD1 have increased
+significantly over the original SSD model (see
+fig. \ref{dense1}) and are in better agreement with the experimental
+values. At 298 K, the densities of SSD/E and SSD1 without
 a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
 0.999$\pm$0.001 g/cm$^3$ respectively.  These both compare well with
 the experimental value of 0.997 g/cm$^3$, and they are considerably
-better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to
-the dipole moment and sticky switching functions have improved the
-structuring of the liquid (as seen in figure \ref{grcompare}, but they
-have shifted the density maximum to much lower temperatures. This
-comes about via an increase in the liquid disorder through the
-weakening of the sticky potential and strengthening of the dipolar
-character. However, this increasing disorder in the SSD/E model has
-little effect on the melting transition. By monitoring C$\text{p}$
-throughout these simulations, the melting transition for SSD/E was
-shown to occur at 235 K, the same transition temperature observed with
-SSD and SSD1.
+better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The
+changes to the dipole moment and sticky switching functions have
+improved the structuring of the liquid (as seen in figure
+\ref{grcompare}, but they have shifted the density maximum to much
+lower temperatures. This comes about via an increase in the liquid
+disorder through the weakening of the sticky potential and
+strengthening of the dipolar character. However, this increasing
+disorder in the SSD/E model has little effect on the melting
+transition. By monitoring $C_p$ throughout these simulations, the
+melting transition for SSD/E was shown to occur at 235 K.  The
+same transition temperature observed with SSD and SSD1.
 
 \begin{figure}
 \begin{center} 
 \epsfxsize=6in
 \epsfbox{ssdrfDense.epsi} 
-\caption{Comparison of densities calculated with SSD/RF to SSD1 with a 
-reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
-SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the
-necessity of reparameterization when utilizing a reaction field
-long-ranged correction - SSD/RF provides significantly more accurate
-densities than SSD1 when performing room temperature simulations.}
+\caption{Comparison of densities calculated with SSD/RF to 
+SSD1 with a reaction field, TIP3P [Ref. \citen{Jorgensen98b}],
+TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and
+experiment [Ref. \citen{CRC80}]. The inset shows the necessity of
+reparameterization when utilizing a reaction field long-ranged
+correction - SSD/RF provides significantly more accurate
+densities than SSD1 when performing room temperature
+simulations.}
 \label{ssdrfdense} 
 \end{center}
 \end{figure} 
 
 Including the reaction field long-range correction in the simulations
-results in a more interesting comparison. A density profile including
+results in a more interesting comparison.  A density profile including
 SSD/RF and SSD1 with an active reaction field is shown in figure
 \ref{ssdrfdense}.  As observed in the simulations without a reaction
 field, the densities of SSD/RF and SSD1 show a dramatic increase over
 normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
 of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
-considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and
-the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further
-emphasize the importance of reparameterization in order to model the
-density properly under different simulation conditions. Again, these
-changes have only a minor effect on the melting point, which observed
-at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1
-with a reaction field. Additionally, the difference in density maxima
-is not as extreme, with SSD/RF showing a density maximum at 255 K,
-fairly close to the density maxima of 260 K and 265 K, shown by SSD
-and SSD1 respectively.
+considerably better than the original SSD value of 0.941$\pm$0.001
+g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results
+further emphasize the importance of reparameterization in order to
+model the density properly under different simulation conditions.
+Again, these changes have only a minor effect on the melting point,
+which observed at 245 K for SSD/RF, is identical to SSD and only 5 K
+lower than SSD1 with a reaction field. Additionally, the difference in
+density maxima is not as extreme, with SSD/RF showing a density
+maximum at 255 K, fairly close to the density maxima of 260 K and 265
+K, shown by SSD and SSD1 respectively.
 
 \begin{figure}
 \begin{center} 
 \epsfxsize=6in
 \epsfbox{ssdeDiffuse.epsi} 
-\caption{Plots of the diffusion constants calculated from SSD/E and SSD1,
- both without a reaction field, along with experimental
- results.\cite{Gillen72,Mills73} The NVE calculations were performed
- at the average densities observed in the 1 atm NPT simulations for
- the respective models. SSD/E is slightly more fluid than experiment
- at all of the temperatures, but it is closer than SSD1 without a
- long-range correction.}
+\caption{The diffusion constants calculated from SSD/E and 
+SSD1 (both without a reaction field) along with experimental results
+[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were
+performed at the average densities observed in the 1 atm NPT
+simulations for the respective models. SSD/E is slightly more mobile
+than experiment at all of the temperatures, but it is closer to
+experiment at biologically relevant temperatures than SSD1 without a
+long-range correction.}
 \label{ssdediffuse} 
 \end{center}
 \end{figure} 
@@ -780,159 +815,272 @@ the densities, it is important that particle transport
 The reparameterization of the SSD water model, both for use with and
 without an applied long-range correction, brought the densities up to
 what is expected for simulating liquid water. In addition to improving
-the densities, it is important that particle transport be maintained
-or improved. Figure \ref{ssdediffuse} compares the temperature
-dependence of the diffusion constant of SSD/E to SSD1 without an
-active reaction field, both at the densities calculated at 1 atm and
-at the experimentally calculated densities for super-cooled and liquid
-water. The diffusion constant for SSD/E is consistently a little
-higher than experiment, while SSD1 remains lower than experiment until
-relatively high temperatures (greater than 330 K). Both models follow
-the shape of the experimental curve well below 300 K but tend to
-diffuse too rapidly at higher temperatures, something that is
-especially apparent with SSD1. This accelerated increasing of
-diffusion is caused by the rapidly decreasing system density with
-increasing temperature. Though it is difficult to see in figure
-\ref{ssdedense}, the densities of SSD1 decay more rapidly with
-temperature than do those of SSD/E, leading to more visible deviation
-from the experimental diffusion trend. Thus, the changes made to
-improve the liquid structure may have had an adverse affect on the
-density maximum, but they improve the transport behavior of SSD/E
-relative to SSD1.
+the densities, it is important that the diffusive behavior of SSD be
+maintained or improved. Figure \ref{ssdediffuse} compares the
+temperature dependence of the diffusion constant of SSD/E to SSD1
+without an active reaction field at the densities calculated from
+their respective NPT simulations at 1 atm. The diffusion constant for
+SSD/E is consistently higher than experiment, while SSD1 remains lower
+than experiment until relatively high temperatures (around 360
+K). Both models follow the shape of the experimental curve well below
+300 K but tend to diffuse too rapidly at higher temperatures, as seen
+in SSD1's crossing above 360 K.  This increasing diffusion relative to
+the experimental values is caused by the rapidly decreasing system
+density with increasing temperature.  Both SSD1 and SSD/E show this
+deviation in particle mobility, but this trend has different
+implications on the diffusive behavior of the models.  While SSD1
+shows more experimentally accurate diffusive behavior in the high
+temperature regimes, SSD/E shows more accurate behavior in the
+supercooled and biologically relevant temperature ranges.  Thus, the
+changes made to improve the liquid structure may have had an adverse
+affect on the density maximum, but they improve the transport behavior
+of SSD/E relative to SSD1 under the most commonly simulated
+conditions.
 
 \begin{figure}
 \begin{center} 
 \epsfxsize=6in
 \epsfbox{ssdrfDiffuse.epsi} 
-\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1,
- both with an active reaction field, along with experimental
- results.\cite{Gillen72,Mills73} The NVE calculations were performed
- at the average densities observed in the 1 atm NPT simulations for
- both of the models. Note how accurately SSD/RF simulates the
- diffusion of water throughout this temperature range. The more
- rapidly increasing diffusion constants at high temperatures for both
- models is attributed to the significantly lower densities than
- observed in experiment.}
+\caption{The diffusion constants calculated from SSD/RF and 
+SSD1 (both with an active reaction field) along with
+experimental results [Refs. \citen{Gillen72} and \citen{Holz00}]. The
+NVE calculations were performed at the average densities observed in
+the 1 atm NPT simulations for both of the models. SSD/RF
+simulates the diffusion of water throughout this temperature range
+very well. The rapidly increasing diffusion constants at high
+temperatures for both models can be attributed to lower calculated
+densities than those observed in experiment.}
 \label{ssdrfdiffuse} 
 \end{center}
 \end{figure} 
 
 In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
 compared to SSD1 with an active reaction field. Note that SSD/RF
-tracks the experimental results incredibly well, identical within
-error throughout the temperature range shown and with only a slight
-increasing trend at higher temperatures. SSD1 tends to diffuse more
-slowly at low temperatures and deviates to diffuse too rapidly at
-temperatures greater than 330 K. As stated in relation to SSD/E, this
-deviation away from the ideal trend is due to a rapid decrease in
-density at higher temperatures. SSD/RF does not suffer from this
-problem as much as SSD1, because the calculated densities are closer
-to the experimental value. These results again emphasize the
-importance of careful reparameterization when using an altered
-long-range correction.
+tracks the experimental results quantitatively, identical within error
+throughout most of the temperature range shown and exhibiting only a
+slight increasing trend at higher temperatures. SSD1 tends to diffuse
+more slowly at low temperatures and deviates to diffuse too rapidly at
+temperatures greater than 330 K.  As stated above, this deviation away
+from the ideal trend is due to a rapid decrease in density at higher
+temperatures. SSD/RF does not suffer from this problem as much as SSD1
+because the calculated densities are closer to the experimental
+values. These results again emphasize the importance of careful
+reparameterization when using an altered long-range correction.
 
+\begin{table}
+\begin{minipage}{\linewidth}
+\renewcommand{\thefootnote}{\thempfootnote}
+\begin{center}
+\caption{Properties of the single-point water models compared with
+experimental data at ambient conditions} 
+\begin{tabular}{ l  c  c  c  c  c } 
+\hline \\[-3mm]
+\ \ \ \ \ \  & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ 
+\ & \ SSD/RF \ \ \ & \ Expt. \\
+ \hline \\[-3mm]
+\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\
+\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\
+\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 &
+2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\
+\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 &
+4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in
+Ref. \citen{Head-Gordon00_1}} \\ 
+\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 &
+3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in
+Ref. \citen{Soper86}}  \\
+\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 &
+7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\
+\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2
+$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in
+Ref. \citen{Krynicki66}}
+\end{tabular}
+\label{liquidproperties}
+\end{center}
+\end{minipage}
+\end{table}
+
+Table \ref{liquidproperties} gives a synopsis of the liquid state
+properties of the water models compared in this study along with the
+experimental values for liquid water at ambient conditions. The
+coordination number ($n_C$) and number of hydrogen bonds per particle
+($n_H$) were calculated by integrating the following relations:
+\begin{equation}
+n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr,
+\end{equation}
+\begin{equation}
+n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr,
+\end{equation}
+where $\rho$ is the number density of the specified pair interactions,
+$a$ and $b$ are the radial locations of the minima following the first
+peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number
+of hydrogen bonds stays relatively constant across all of the models,
+but the coordination numbers of SSD/E and SSD/RF show an
+improvement over SSD1.  This improvement is primarily due to
+extension of the first solvation shell in the new parameter sets.
+Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears
+that all molecules in the first solvation shell are involved in
+hydrogen bonds.  Since $n_H$ and $n_C$ differ in the newly
+parameterized models, the orientations in the first solvation shell
+are a bit more ``fluid''.  Therefore SSD1 overstructures the
+first solvation shell and our reparameterizations have returned this
+shell to more realistic liquid-like behavior.
+
+The time constants for the orientational autocorrelation functions
+are also displayed in Table \ref{liquidproperties}. The dipolar
+orientational time correlation functions ($C_{l}$) are described
+by:
+\begin{equation}
+C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle,
+\end{equation}
+where $P_l$ are Legendre polynomials of order $l$ and
+$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular
+dipole.\cite{Rahman71} From these correlation functions, the
+orientational relaxation time of the dipole vector can be calculated
+from an exponential fit in the long-time regime ($t >
+\tau_l$).\cite{Rothschild84} Calculation of these time constants were
+averaged over five detailed NVE simulations performed at the ambient
+conditions for each of the respective models. It should be noted that
+the commonly cited value of 1.9 ps for $\tau_2$ was determined from
+the NMR data in Ref. \citen{Krynicki66} at a temperature near
+34$^\circ$C.\cite{Rahman71} Because of the strong temperature
+dependence of $\tau_2$, it is necessary to recalculate it at 298 K to
+make proper comparisons. The value shown in Table
+\ref{liquidproperties} was calculated from the same NMR data in the 
+fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was
+recomputed for 298 K from the data in Ref. \citen{Eisenberg69}.
+Again, SSD/E and SSD/RF show improved behavior over SSD1, both with
+and without an active reaction field. Turning on the reaction field
+leads to much improved time constants for SSD1; however, these results
+also include a corresponding decrease in system density.
+Orientational relaxation times published in the original SSD dynamics
+papers are smaller than the values observed here, and this difference
+can be attributed to the use of the Ewald sum.\cite{Ichiye99}
+
 \subsection{Additional Observations}
 
 \begin{figure}
 \begin{center}
 \epsfxsize=6in
-\epsfbox{povIce.ps} 
-\caption{A water lattice built from the crystal structure assumed by 
- SSD/E when undergoing an extremely restricted temperature NPT
- simulation. This form of ice is referred to as ice 0 to emphasize its
- simulation origins. This image was taken of the (001) face of the
- crystal.}
+\epsfbox{icei_bw.eps} 
+\caption{The most stable crystal structure assumed by the SSD family
+of water models.  We refer to this structure as Ice-{\it i} to
+indicate its origins in computer simulation.  This image was taken of
+the (001) face of the crystal.}
 \label{weirdice} 
 \end{center}
 \end{figure}
 
-While performing restricted temperature melting sequences of SSD/E not
-previously discussed, some interesting observations were made. After
-melting at 235 K, two of five systems underwent crystallization events
-near 245 K. As the heating process continued, the two systems remained
-crystalline until finally melting between 320 and 330 K. The final
-configurations of these two melting sequences show an expanded
-zeolite-like crystal structure that does not correspond to any known
-form of ice. For convenience, and to help distinguish it from the
-experimentally observed forms of ice, this crystal structure will
-henceforth be referred to as ice-zero (ice 0). The crystallinity was
-extensive enough that a near ideal crystal structure of ice 0 could be
-obtained. Figure \ref{weirdice} shows the repeating crystal structure
-of a typical crystal at 5 K. Each water molecule is hydrogen bonded to
-four others; however, the hydrogen bonds are flexed rather than
-perfectly straight. This results in a skewed tetrahedral geometry
-about the central molecule. Referring to figure \ref{isosurface},
-these flexed hydrogen bonds are allowed due to the conical shape of
-the attractive regions, with the greatest attraction along the direct
-hydrogen bond configuration. Though not ideal, these flexed hydrogen
-bonds are favorable enough to stabilize an entire crystal generated
-around them. In fact, the imperfect ice 0 crystals were so stable that
-they melted at temperatures nearly 100 K greater than both ice I$_c$
-and I$_h$.
+While performing a series of melting simulations on an early iteration
+of SSD/E not discussed in this paper, we observed
+recrystallization into a novel structure not previously known for
+water.  After melting at 235 K, two of five systems underwent
+crystallization events near 245 K.  The two systems remained
+crystalline up to 320 and 330 K, respectively.  The crystal exhibits
+an expanded zeolite-like structure that does not correspond to any
+known form of ice.  This appears to be an artifact of the point
+dipolar models, so to distinguish it from the experimentally observed
+forms of ice, we have denoted the structure
+Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}).  A large enough
+portion of the sample crystallized that we have been able to obtain a
+near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice}
+shows the repeating crystal structure of a typical crystal at 5
+K. Each water molecule is hydrogen bonded to four others; however, the
+hydrogen bonds are bent rather than perfectly straight. This results
+in a skewed tetrahedral geometry about the central molecule.  In
+figure \ref{isosurface}, it is apparent that these flexed hydrogen
+bonds are allowed due to the conical shape of the attractive regions,
+with the greatest attraction along the direct hydrogen bond
+configuration. Though not ideal, these flexed hydrogen bonds are
+favorable enough to stabilize an entire crystal generated around them.
 
-These initial simulations indicated that ice 0 is the preferred ice
+Initial simulations indicated that Ice-{\it i} is the preferred ice
 structure for at least the SSD/E model. To verify this, a comparison
-was made between near ideal crystals of ice $I_h$, ice $I_c$, and ice
-0 at constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal
-versions of the three types of crystals were cooled to 1 K, and the
-potential energies of each were compared using all three water
-models. With every water model, ice 0 turned out to have the lowest
-potential energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with
-SSD/E, and 7.5\% lower with SSD/RF.
+was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and
+Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and
+SSD1. Near-ideal versions of the three types of crystals were cooled
+to 1 K, and enthalpies of formation of each were compared using all
+three water models.  Enthalpies were estimated from the
+isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where
+$P_{\text ext}$ is the applied pressure.  A constant value of -60.158
+kcal / mol has been added to place our zero for the enthalpies of
+formation for these systems at the traditional state (elemental forms
+at standard temperature and pressure).  With every model in the SSD
+family, Ice-{\it i} had the lowest calculated enthalpy of formation.
 
-In addition to these low temperature comparisons, melting sequences
-were performed with ice 0 as the initial configuration using SSD/E,
+\begin{table}
+\begin{center}
+\caption{Enthalpies of Formation (in kcal / mol) of the three crystal
+structures (at 1 K) exhibited by the SSD family of water models} 
+\begin{tabular}{ l  c  c  c  } 
+\hline \\[-3mm]
+\ \ \ Water Model \ \ \  & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \  & \
+Ice-{\it i} \\
+\hline \\[-3mm]
+\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\
+\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\
+\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\
+\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\
+\end{tabular}
+\label{iceenthalpy}
+\end{center}
+\end{table}
+
+In addition to these energetic comparisons, melting simulations were
+performed with ice-{\it i} as the initial configuration using SSD/E,
 SSD/RF, and SSD1 both with and without a reaction field. The melting
-transitions for both SSD/E and SSD1 without a reaction field occurred
-at temperature in excess of 375 K. SSD/RF and SSD1 with a reaction
-field showed more reasonable melting transitions near 325 K. These
-melting point observations emphasize the preference for this crystal
-structure over the most common types of ice when using these single
-point water models.
+transitions for both SSD/E and SSD1 without reaction field occurred at
+temperature in excess of 375~K.  SSD/RF and SSD1 with a reaction field
+showed more reasonable melting transitions near 325~K.  These melting
+point observations clearly show that all of the SSD-derived models
+prefer the ice-{\it i} structure.
 
-Recognizing that the above tests show ice 0 to be both the most stable
-and lowest density crystal structure for these single point water
-models, it is interesting to speculate on the relative stability of
-this crystal structure with charge based water models. As a quick
-test, these 3 crystal types were converted from SSD type particles to
-TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
-minimizations were performed on the crystals to compare the system
-energies. Again, ice 0 was observed to have the lowest total system
-energy. The total energy of ice 0 was ~2\% lower than ice $I_h$, which
-was in turn ~3\% lower than ice $I_c$. Based on these initial studies,
-it would not be surprising if results from the other common water
-models show ice 0 to be the lowest energy crystal structure. A
-continuation of this work studying ice 0 with multi-point water models
-will be published in a coming article.
-
 \section{Conclusions}
-The density maximum and temperature dependent transport for the SSD
-water model, both with and without the use of reaction field, were
-studied via a series of NPT and NVE simulations. The constant pressure
-simulations of the melting of both $I_h$ and $I_c$ ice showed a
-density maximum near 260 K. In most cases, the calculated densities
-were significantly lower than the densities calculated in simulations
-of other water models. Analysis of particle diffusion showed SSD to
-capture the transport properties of experimental water well in both
-the liquid and super-cooled liquid regimes. In order to correct the
-density behavior, the original SSD model was reparameterized for use
-both with and without a reaction field (SSD/RF and SSD/E), and
-comparison simulations were performed with SSD1, the density corrected
-version of SSD. Both models improve the liquid structure, density
-values, and diffusive properties under their respective conditions,
-indicating the necessity of reparameterization when altering the
-long-range correction specifics. When taking into account the
-appropriate considerations, these simple water models are excellent
-choices for representing explicit water in large scale simulations of
-biochemical systems.
 
+The density maximum and temperature dependence of the self-diffusion
+constant were studied for the SSD water model, both with and
+without the use of reaction field, via a series of NPT and NVE
+simulations. The constant pressure simulations showed a density
+maximum near 260 K. In most cases, the calculated densities were
+significantly lower than the densities obtained from other water
+models (and experiment). Analysis of self-diffusion showed SSD
+to capture the transport properties of water well in both the liquid
+and supercooled liquid regimes.
+
+In order to correct the density behavior, the original SSD model was
+reparameterized for use both with and without a reaction field (SSD/RF
+and SSD/E), and comparisons were made with SSD1, Ichiye's density
+corrected version of SSD. Both models improve the liquid structure,
+densities, and diffusive properties under their respective simulation
+conditions, indicating the necessity of reparameterization when
+changing the method of calculating long-range electrostatic
+interactions.  In general, however, these simple water models are
+excellent choices for representing explicit water in large scale
+simulations of biochemical systems.
+
+The existence of a novel low-density ice structure that is preferred
+by the SSD family of water models is somewhat troubling, since
+liquid simulations on this family of water models at room temperature
+are effectively simulations of supercooled or metastable liquids.  One
+way to destabilize this unphysical ice structure would be to make the
+range of angles preferred by the attractive part of the sticky
+potential much narrower.  This would require extensive
+reparameterization to maintain the same level of agreement with the
+experiments.
+
+Additionally, our initial calculations show that the Ice-{\it i}
+structure may also be a preferred crystal structure for at least one
+other popular multi-point water model (TIP3P), and that much of the
+simulation work being done using this popular model could also be at
+risk for crystallization into this unphysical structure.  A future
+publication will detail the relative stability of the known ice
+structures for a wide range of popular water models.
+
 \section{Acknowledgments}
 Support for this project was provided by the National Science
 Foundation under grant CHE-0134881. Computation time was provided by
 the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
-DMR 00 79647.
+DMR-0079647.
 
-
 \newpage
 
 \bibliographystyle{jcp}
