trunk/ssdePaper/nptSSD.tex

%\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4}
\documentclass[prb,aps,times,tabularx,preprint]{revtex4}
\usepackage{amsmath}
\usepackage{graphicx}

%\usepackage{endfloat}
%\usepackage{berkeley}
%\usepackage{epsf}
%\usepackage[ref]{overcite}
%\usepackage{setspace}
%\usepackage{tabularx}
%\usepackage{graphicx}
%\usepackage{curves}
%\usepackage{amsmath}
%\pagestyle{plain}
%\pagenumbering{arabic}
%\oddsidemargin 0.0cm \evensidemargin 0.0cm
%\topmargin -21pt \headsep 10pt
%\textheight 9.0in \textwidth 6.5in
%\brokenpenalty=10000
%\renewcommand{\baselinestretch}{1.2}
%\renewcommand\citemid{\ } % no comma in optional reference note

\begin{document}

\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models}

\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote}
\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}

\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\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
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.
\end{abstract}

\maketitle

%\narrowtext 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                              BODY OF TEXT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

One of the most important tasks in simulations 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
groups/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
bio-molecular 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
get specific properties correct or better than their predecessors, but
this is often at a cost of some other properties or of computer
time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P succeeds
in improving the structural and transport properties over its
predecessors, yet this comes at a greater than 50\% increase in
computational cost.\cite{Jorgensen01,Jorgensen00} One recently
developed model that succeeds in both retaining accuracy of system
properties and simplifying calculations to increase computational
efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}

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
\begin{equation}
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
u_{ij}^{sp}
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\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,
\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}
\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}\ ,
\end{equation}
\begin{equation}
\begin{split}
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)\\
& \quad \ +
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
\end{split}
\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, $s$ and
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
functions take the following forms,
\begin{equation}
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
\end{equation}
\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,
\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}

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 a
calculation speed up 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.

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. Toward the end, we suggest
alterations to the parameters that result in more water-like
behavior. It should be noted that in a recent publication, some 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 this new
model's behavior with the goal of improving upon the depiction of
water under conditions without the Ewald Sum.

\section{Methods}

As stated previously, in this study the long-range dipole-dipole
interactions were accounted for using the reaction field method. The
magnitude of the reaction field acting on dipole \emph{i} is given by
\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})\  ,
\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
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 system, 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 proposed on how to make
SSD more accurate both with and without a reaction field.

Simulations were performed in both the isobaric-isothermal and
microcanonical ensembles. The constant pressure simulations were
implemented using an integral thermostat and barostat as outlined by
Hoover.\cite{Hoover85,Hoover86} All particles 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 \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}

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 a 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.

The symplectic splitting method allows for Verlet style integration of
both linear and angular motion of rigid bodies. In the integration
method, the orientational propagation involves a sequence of matrix
evaluations to update the rotation matrix.\cite{Dullweber1997} These
matrix rotations end up being 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}.

\begin{figure}
\includegraphics[width=61mm, angle=-90]{timeStep.epsi}
\caption{Energy conservation using quaternion based integration versus 
the symplectic step method proposed by Dullweber \emph{et al.} with
increasing time step. For each time step, the dotted line is total
energy using the symplectic step integrator, and the solid line comes
from the quaternion integrator. The larger time step plots are shifted
up from the true energy baseline for clarity.}
\label{timestep}
\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 for
handling 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.

Energy drift in these SSD particle simulations was unnoticeable for
time steps up to three femtoseconds. 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
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 the simulations. The $I_h$ crystals were
formed by first arranging the center of masses 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 a 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 down to 25 K. The particles
were then allowed 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
constant pressure and temperature dynamics. By performing melting
simulations, the melting transition can be determined by monitoring
the heat capacity, in addition to determining the density maximum -
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).

\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; and 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 them to deform into a dense glassy state at lower
temperatures. This resulted in an overall low temperature density
maximum at 200 K, but they still retained a common liquid state
density maximum with the $I_h$ simulations.

\begin{figure}
\includegraphics[width=65mm,angle=-90]{dense2.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}}
\label{dense1}
\end{figure}

The density maximum for SSD actually 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.

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
the resulting densities overlapped within error and showed no
significant trend in lower or higher densities as a function of cutoff
radius, both for simulations 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 comforting in that the
use of a longer cutoff radius results in significant increases in the
time required to obtain a single trajectory.

The most important thing 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 with the use
of a reaction field for long-range electrostatic interactions, so the
most likely reason for these significantly lower densities 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, these 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
probably 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
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.

\subsection{Transport Behavior}
Of importance in these types of studies are the transport properties
of the particles and how they change when altering the 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 5 sets of these NVE simulations is displayed in
figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} 

\begin{figure}
\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi} 
\caption{Average diffusion coefficient over increasing temperature for 
SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental
data from Gillen \emph{et al.}\cite{Gillen72}, and from
Mills\cite{Mills73}.}
\label{diffuse}
\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 normal regimes. SPC/E
does a respectable job by getting similar values as 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 the colder temperatures and tend to diffuse too
rapidly at the higher temperatures. This type of trend at the higher
temperatures is not surprising in that the densities of both TIP5P and
SSD are lower than experimental water at temperatures higher than room
temperature. 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.

\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
phase behavior beyond the melting point. To locate the melting
transition, the constant pressure heat capacity (C$_\text{p}$) was
monitored 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, but this is not
surprising when considering the simplicity of the SSD model.

\begin{figure} 
\includegraphics[width=85mm]{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.}
\label{contour} 
\end{figure} 

\begin{figure}
\includegraphics[width=45mm]{corrDiag.eps} 
\caption{Two dimensional illustration of the angles involved in the 
correlations observed in figure \ref{contour}.}
\label{corrAngle}
\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:

\begin{multline}
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\ ,
\end{multline}
\begin{multline}
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\ ,
\end{multline}
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
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
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.

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 parts 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, and there is tightly
bunched group of axially arranged dipoles that most likely consist of
the smaller fraction aligned dipole pairs. The trailing part of the
split peak at 5 \AA\ is dominated by aligned dipoles that range
primarily within the axial to the chief 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, with the primary
energetically favorable dipole arrangements populating the region
immediately outside this repulsion region (around 4 \AA), and
arrangements that seek to ideally 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
correction is not recommended for improving the liquid structure,
because the second solvation shell will 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
quadrupole-quadrupole interactions for the water particles outside of
the first solvation shell, but this reduces 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 properties.

The possible parameters 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}
\begin{split}
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)]\\
& \quad \ + \frac{\nu_0^\prime}{2}
[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
\end{split}
\end{equation}

where $\nu_0$ scales the strength of the tetrahedral attraction and
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
term. For purposes of the reparameterization, the separation was
performed, 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 both the tetrahedral attractive and dipolar 
repulsive don't 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}
\caption{Parameters for the original and adjusted models}
\begin{tabular}{ l  c  c  c  c } 
\hline \\[-3mm]
\ \ \ Parameters\ \ \  & \ \ \ SSD$^\dagger$ \ \ \ & \ SSD1$^\ddagger$\ \  & \ 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\\
\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96}
\\$^\ddagger$ ref. \onlinecite{Ichiye03}
\end{tabular}
\label{params}
\end{table}

\begin{figure} 
\includegraphics[width=85mm]{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. Solid Line - experiment, dashed line - SSD/E
and SSD/RF, and dotted line - SSD1 (with and without reaction field).}
\label{grcompare} 
\end{figure} 

\begin{figure} 
\includegraphics[width=85mm]{dualsticky.ps} 
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& 
SSD/RF (right). Light areas correspond to the tetrahedral attractive
part, and the darker areas correspond to the dipolar repulsive part.}
\label{isosurface} 
\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 sharp first peak 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 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 experimental results. Both the 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 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 \cite{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 (the 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. Upon moving 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 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 finish bringing up the densities, the dipole
moments were increased in both the adjusted models. Being 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, so there is quite a range of available
values for a reasonable dipole
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of
the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF
respectively is moderate in this range; however, it 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 come from five separate simulations of 1024 particle systems,
and the melting sequences were started from different ice $I_h$
crystals constructed as stated earlier. Like before, each NPT
simulation was equilibrated for 100 ps before a 200 ps data collection
run at each temperature step, and they used the final configuration
from the previous temperature simulation as a starting point. All of
the NVE simulations had the same thermalization, equilibration, and
data collection times stated earlier in this paper.

\begin{figure} 
\includegraphics[width=62mm, angle=-90]{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
experiment when the long-range correction is neglected.}
\label{ssdedense} 
\end{figure} 

Figure \ref{ssdedense} shows the density profile for the SSD/E model
in comparison to SSD1 without a reaction field, experiment, and other
common water models. The calculated densities for both SSD/E and SSD1
have increased significantly over the original SSD model (see figure
\ref{dense1} and are in significantly better agreement with the
experimental values. At 298 K, the density 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$. The changes to the dipole moment
and sticky switching functions have improved the structuring of the
liquid (as seen in figure \ref{grcompare}, but they have shifted the
density maximum to much lower temperatures. This comes about via an
increase of the liquid disorder through the weakening of the sticky
potential and strengthening of the dipolar character. However, this
increasing disorder in the SSD/E model has little affect on the
melting transition. By monitoring C$\text{p}$ throughout these
simulations, the melting transition for SSD/E occurred at 235 K, the
same transition temperature observed with SSD and SSD1.

\begin{figure} 
\includegraphics[width=62mm, angle=-90]{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.}
\label{ssdrfdense} 
\end{figure} 

Including the reaction field long-range correction 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$, right in line with experiment and considerably better than
the SSD value of 0.941$\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
don't have that profound an effect on the melting point which is
observed at 245 K for SSD/RF, identical to SSD and only 5 K lower than
SSD1 with a reaction field. However, the difference in density maxima
is not quite as extreme with SSD/RF showing a density maximum at 255
K, fairly close to 260 and 265 K, the density maxima for SSD and SSD1
respectively. 

\begin{figure} 
\includegraphics[width=65mm, angle=-90]{ssdeDiffuse.epsi} 
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1,
 both without a reaction field, along with experimental results are
 from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{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{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 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. In the upper plot, the diffusion constant for SSD/E is
consistently a little faster than experiment, while SSD1 remains
slower than experiment until relatively high temperatures (greater
than 330 K). Both models follow the shape of the experimental trend
well below 300 K, but the trend leans toward diffusing 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 the water model.

\begin{figure} 
\includegraphics[width=65mm, angle=-90]{ssdrfDiffuse.epsi} 
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1,
 both with an active reaction field, along with experimental results
 from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{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{figure} 

In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
compared with 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 only showing 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 was stated in the SSD/E
comparisons, this deviation away from the ideal trend is due to a
rapid decrease in density at higher temperatures. SSD/RF doesn't
suffer from this problem as much as SSD1, because the calculated
densities are more true to experiment. These results again emphasize
the importance of careful reparameterization when using an altered
long-range correction.

\subsection{Additional Observations}

\begin{figure}
\includegraphics[width=85mm]{povIce.ps} 
\caption{A water lattice built from the crystal structure that SSD/E 
 assumed 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.}
\label{weirdice} 
\end{figure}

While performing restricted temperature melting sequences of SSD/E not
discussed earlier in this paper, 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 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. Looking back at figure
\ref{isosurface}, it is easy to see how these flexed hydrogen bonds
are allowed in that the attractive regions are conical in shape, with
the greatest attraction in the central region. 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$_h$.

These initial simulations indicated that ice 0 is the preferred ice
structure for at least SSD/E. 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. 

In addition to these low temperature comparisons, melting sequences
were performed with ice 0 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 had more reasonable melting transitions, down near 325 K. These
melting point observations emphasize how preferred this crystal
structure is over the most common types of ice when using these single
point water models.

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 all of these 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 $I_c$. From these initial
results, we would not be surprised if results from the other common
water models show ice 0 to be the lowest energy crystal structure. A
continuation on 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 very well in both the
normal 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 the appropriate
considerations, these simple water models are excellent choices for
representing explicit water in large scale simulations of biochemical
systems.

\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.

\bibliographystyle{jcp}

\bibliography{nptSSD} 

%\pagebreak 

\end{document}
Revision:	861
Committed:	Wed Nov 12 13:37:15 2003 UTC (21 years, 11 months ago) by chrisfen
Content type:	application/x-tex
File size:	50011 byte(s)
Log Message:	Massive changes to content. Declared as first draft.