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\begin{document}

\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\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle

\begin{abstract}
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.  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}

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\section{Introduction}

One of the most important tasks in the simulation of biochemical
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.

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}
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
u_{ij}^{sp}
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
\end{equation}
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]
\ ,
\end{equation}
and
\begin{equation}
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}
({\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)]\ .
\end{equation}
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({\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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
\end{equation}
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}

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 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}

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.  Under the first 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}}} {\bf \mu}_{j} f(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 the case of liquid water), ${\bf \mu}_{j}$ is the dipole
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching
function.\cite{AllenTildesley} The reaction field contribution to the
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 to obtain the preferred density 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 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.

Integration of the equations of motion was carried out using the
symplectic splitting method proposed by Dullweber {\it et
al.}\cite{Dullweber1997} Our reason for selecting this integrator
centers on poor energy conservation of rigid body dynamics using
traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions
may work well for orientational motion under NVT or NPT integrators,
our limits on energy drift in the microcanonical ensemble were quite
strict, and the drift under quaternions was substantially greater than
in the symplectic splitting method.  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.  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 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 symplectic step 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 symplectic splitting 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 both quaternion based integration and
the symplectic step method proposed by Dullweber \emph{et al.} 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 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 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.

Energy drift in the symplectic step simulations 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
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.

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
isotropic volume fluctuations were performed under constant pressure,
so the ratio of edge lengths remained constant throughout the
simulations.

\section{Results and discussion}

Melting studies were performed on the randomized ice crystals using
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}

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 [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 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 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 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. 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 {\it et al.} to recently reparameterize
SSD.\cite{Ichiye03} Throughout the remainder of the paper our
reparamaterizations of SSD will be compared with the newer SSD1 model.

\subsection{Transport Behavior}

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,Mills73,Clancy94,Jorgensen01} 

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{betterDiffuse.epsi} 
\caption{Average self-diffusion constant as a function of temperature for 
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}],
and Experimental data [Refs. \citen{Gillen72} and \citen{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.}
\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 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 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
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs
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.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{corrDiag.eps} 
\caption{Two dimensional illustration of angles involved in the 
correlations observed in Fig. \ref{contour}.}
\label{corrAngle}
\end{center}
\end{figure}

\begin{figure}
\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 $g(r)$ values below 0.5 and black
areas have values above 1.5.}
\label{contour} 
\end{center}
\end{figure} 

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|{\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|{\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 $g(r)$ 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 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
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 $g_\mathrm{OO}(r)$.\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 ($s^\prime(r_{ij})$) 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 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 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
necessary to adjust the force field parameters for the primary
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 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}
({\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)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)],
\end{equation}
where $\nu_0$ scales the strength of the tetrahedral attraction and
$\nu_0^\prime$ scales the dipole repulsion term independently. The
separation was performed for purposes of the reparameterization, but
the final parameters were adjusted so that it is not necessary 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.  We are calling these
reparameterizations the 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 [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\\
\ \ \ $r_l$ (\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}
\label{params}
\end{center}
\end{table}

\begin{figure}
\begin{center} 
\epsfxsize=5in
\epsfbox{GofRCompare.epsi} 
\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} 

\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.}
\label{isosurface} 
\end{center}
\end{figure} 

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 our adjustments to SSD were
made while 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 $g(r)$ 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 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.

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 dielectric properties of
the fluid. Both theoretical and experimental measurements indicate a
liquid phase dipole moment ranging from 2.4 D 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.

In order to demonstrate the benefits of these reparameterizations, a
series of NPT and NVE simulations were performed to probe the density
and transport properties of the adapted models and compare the results
to the original SSD model. This comparison involved full NPT melting
sequences for both SSD/E and SSD/RF, as well as NVE transport
calculations at the calculated self-consistent densities. Again, the
results are obtained from five separate simulations of 1024 particle
systems, and the melting sequences were started from different ice
$I_h$ crystals constructed as described previously. Each NPT
simulation was equilibrated for 100 ps before a 200 ps data collection
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 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 [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} 

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 $C_p$
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 [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
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$, directly in line with experiment and
considerably better than the original SSD value of 0.941$\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
[Refs. \citen{Gillen72} and \citen{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.}
\label{ssdediffuse} 
\end{center}
\end{figure} 

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 the excellent 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, 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
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 increasing diffusion relative to
the experimental values is caused by the rapidly decreasing system
density with increasing temperature.  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.

\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
 [Refs. \citen{Gillen72} and \citen{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.}
\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 quantitatively, 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 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.

\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-{\it i} to
emphasize its simulation origins. This image was taken of the (001)
face of the crystal.}
\label{weirdice} 
\end{center}
\end{figure}

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.

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-{\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 the enthalpies of each were compared using all three water
models. With every model in the SSD family, ice-{\it i} had the lowest
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with
SSD/E, and 7.5\% lower with SSD/RF.  The enthalpy data is summarized
in Table \ref{iceenthalpy}.

\begin{table}
\begin{center}
\caption{Enthalpies (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 & -12.286 & -12.292 & -13.590 \\
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\
\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 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.

\section{Conclusions}

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
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 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 super-cooled or metastable liquids.  One
way to de-stabilize 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-0079647.

\newpage

\bibliographystyle{jcp}
\bibliography{nptSSD} 

%\pagebreak 

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