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\begin{document}

\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more
stable than Ice $I_h$ for point-charge and point-dipole water models}

\author{Christopher J. Fennell and J. Daniel Gezelter \\
Department of Chemistry and Biochemistry\\ University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle
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\begin{abstract}
The absolute free energies of several ice polymorphs which are stable
at low pressures were calculated using thermodynamic integration to a
reference system (the Einstein crystal).  These integrations were
performed for most of the common water models (SPC/E, TIP3P, TIP4P,
TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed
crystallizing at room temperature for one of the single-point water
models, was determined to be the stable crystalline state (at 1 atm)
for {\it all} the water models investigated.  Phase diagrams were
generated, and phase coexistence lines were determined for all of the
known low-pressure ice structures under all of these water models.
Additionally, potential truncation was shown to have an effect on the
calculated free energies, and can result in altered free energy
landscapes.  Structure factor for the new crystal were generated and
we await experimental confirmation of the existence of this new
polymorph.  
\end{abstract}

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\section{Introduction}

Water has proven to be a challenging substance to depict in
simulations, and a variety of models have been developed to describe
its behavior under varying simulation
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04}
These models have been used to investigate important physical
phenomena like phase transitions, transport properties, and the
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the
choice of models available, it is only natural to compare the models
under interesting thermodynamic conditions in an attempt to clarify
the limitations of each of the
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two
important properties to quantify are the Gibbs and Helmholtz free
energies, particularly for the solid forms of water.  Difficulty in
these types of studies typically arises from the assortment of
possible crystalline polymorphs that water adopts over a wide range of
pressures and temperatures.  There are currently 13 recognized forms
of ice, and it is a challenging task to investigate the entire free
energy landscape.\cite{Sanz04} Ideally, research is focused on the
phases having the lowest free energy at a given state point, because
these phases will dictate the relevant transition temperatures and
pressures for the model.

In this paper, standard reference state methods were applied to known
crystalline water polymorphs in the low pressure regime.  This work is
unique in that one of the crystal lattices was arrived at through
crystallization of a computationally efficient water model under
constant pressure and temperature conditions. Crystallization events
are interesting in and of themselves;\cite{Matsumoto02,Yamada02}
however, the crystal structure obtained in this case is different from
any previously observed ice polymorphs in experiment or
simulation.\cite{Fennell04} We have named this structure Ice-{\it i}
to indicate its origin in computational simulation. The unit cell
(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in
rows of interlocking water tetramers. Proton ordering can be
accomplished by orienting two of the molecules so that both of their
donated hydrogen bonds are internal to their tetramer
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of
water tetramers, the hydrogen bonds are not as linear as those
observed in ice $I_h$, however the interlocking of these subunits
appears to provide significant stabilization to the overall
crystal. The arrangement of these tetramers results in surrounding
open octagonal cavities that are typically greater than 6.3 \AA\ in
diameter. This relatively open overall structure leads to crystals
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$.

\begin{figure}
\includegraphics[width=\linewidth]{unitCell.eps}
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, 
the elongated variant of Ice-{\it i}.  The spheres represent the
center-of-mass locations of the water molecules.  The $a$ to $c$
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by
$a:2.1214c$ and $a:1.7850c$ respectively.}
\label{iceiCell}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{orderedIcei.eps}
\caption{Image of a proton ordered crystal of Ice-{\it i} looking 
down the (001) crystal face. The rows of water tetramers surrounded by
octagonal pores leads to a crystal structure that is significantly
less dense than ice $I_h$.}
\label{protOrder}
\end{figure}

Results from our previous study indicated that Ice-{\it i} is the
minimum energy crystal structure for the single point water models we
had investigated (for discussions on these single point dipole models,
see our previous work and related
articles).\cite{Fennell04,Liu96,Bratko85} Those results only
considered energetic stabilization and neglected entropic
contributions to the overall free energy. To address this issue, we
have calculated the absolute free energy of this crystal using
thermodynamic integration and compared to the free energies of cubic
and hexagonal ice $I$ (the experimental low density ice polymorphs)
and ice B (a higher density, but very stable crystal structure
observed by B\`{a}ez and Clancy in free energy studies of
SPC/E).\cite{Baez95b} This work includes results for the water model
from which Ice-{\it i} was crystallized (SSD/E) in addition to several
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
field parametrized single point dipole water model (SSD/RF). It should
be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$)
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit
cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it
i} unit it is extended in the direction of the (001) face and
compressed along the other two faces.  There is typically a small
distortion of proton ordered Ice-{\it i}$^\prime$ that converts the
normally square tetramer into a rhombus with alternating approximately
85 and 95 degree angles.  The degree of this distortion is model
dependent and significant enough to split the tetramer diagonal
location peak in the radial distibution function.

\section{Methods}

Canonical ensemble (NVT) molecular dynamics calculations were
performed using the OOPSE molecular mechanics package.\cite{Meineke05}
All molecules were treated as rigid bodies, with orientational motion
propagated using the symplectic DLM integration method. Details about
the implementation of this technique can be found in a recent
publication.\cite{Dullweber1997}

Thermodynamic integration is an established technique for
determination of free energies of condensed phases of
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This
method, implemented in the same manner illustrated by B\`{a}ez and
Clancy, was utilized to calculate the free energy of several ice
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300
and 400 K for all of these water models were also determined using
this same technique in order to determine melting points and to
generate phase diagrams. All simulations were carried out at densities
which correspond to a pressure of approximately 1 atm at their
respective temperatures.

Thermodynamic integration involves a sequence of simulations during
which the system of interest is converted into a reference system for
which the free energy is known analytically. This transformation path
is then integrated in order to determine the free energy difference
between the two states:
\begin{equation}
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda 
)}{\partial\lambda}\right\rangle_\lambda d\lambda,
\end{equation}
where $V$ is the interaction potential and $\lambda$ is the
transformation parameter that scales the overall
potential. Simulations are distributed strategically along this path
in order to sufficiently sample the regions of greatest change in the
potential. Typical integrations in this study consisted of $\sim$25
simulations ranging from 300 ps (for the unaltered system) to 75 ps
(near the reference state) in length.

For the thermodynamic integration of molecular crystals, the Einstein
crystal was chosen as the reference system. In an Einstein crystal,
the molecules are restrained at their ideal lattice locations and
orientations. Using harmonic restraints, as applied by B\`{a}ez and
Clancy, the total potential for this reference crystal
($V_\mathrm{EC}$) is the sum of all the harmonic restraints,
\begin{equation}
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} +
\frac{K_\omega\omega^2}{2},
\end{equation}
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are
the spring constants restraining translational motion and deflection
of and rotation around the principle axis of the molecule
respectively.  It is clear from Fig. \ref{waterSpring} that the values
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from
$-\pi$ to $\pi$.  The partition function for a molecular crystal
restrained in this fashion can be evaluated analytically, and the
Helmholtz Free Energy ({\it A}) is given by
\begin{eqnarray}
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
\label{ecFreeEnergy}
\end{eqnarray}
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum
potential energy of the ideal crystal.\cite{Baez95a}

\begin{figure}
\includegraphics[width=\linewidth]{rotSpring.eps}
\caption{Possible orientational motions for a restrained molecule. 
$\theta$ angles correspond to displacement from the body-frame {\it
z}-axis, while $\omega$ angles correspond to rotation about the
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring
constants for the harmonic springs restraining motion in the $\theta$
and $\omega$ directions.}
\label{waterSpring}
\end{figure}

In the case of molecular liquids, the ideal vapor is chosen as the
target reference state.  There are several examples of liquid state
free energy calculations of water models present in the
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods
typically differ in regard to the path taken for switching off the
interaction potential to convert the system to an ideal gas of water
molecules.  In this study, we applied of one of the most convenient
methods and integrated over the $\lambda^4$ path, where all
interaction parameters are scaled equally by this transformation
parameter.  This method has been shown to be reversible and provide
results in excellent agreement with other established
methods.\cite{Baez95b}

Charge, dipole, and Lennard-Jones interactions were modified by a
cubic switching between 100\% and 85\% of the cutoff value (9 \AA
). By applying this function, these interactions are smoothly
truncated, thereby avoiding the poor energy conservation which results
from harsher truncation schemes. The effect of a long-range correction
was also investigated on select model systems in a variety of
manners. For the SSD/RF model, a reaction field with a fixed
dielectric constant of 80 was applied in all
simulations.\cite{Onsager36} For a series of the least computationally
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
\AA\ cutoff results. Finally, the effects of utilizing an Ewald
summation were estimated for TIP3P and SPC/E by performing single
configuration calculations with Particle-Mesh Ewald (PME) in the
TINKER molecular mechanics software package.\cite{Tinker} The
calculated energy difference in the presence and absence of PME was
applied to the previous results in order to predict changes to the
free energy landscape.

\section{Results and discussion}

The free energy of proton-ordered Ice-{\it i} was calculated and
compared with the free energies of proton ordered variants of the
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
as well as the higher density ice B, observed by B\`{a}ez and Clancy
and thought to be the minimum free energy structure for the SPC/E
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b}
Ice XI, the experimentally-observed proton-ordered variant of ice
$I_h$, was investigated initially, but was found to be not as stable
as proton disordered or antiferroelectric variants of ice $I_h$. The
proton ordered variant of ice $I_h$ used here is a simple
antiferroelectric version that we devised, and it has an 8 molecule
unit cell similar to other predicted antiferroelectric $I_h$
crystals.\cite{Davidson84} The crystals contained 648 or 1728
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger
crystal sizes were necessary for simulations involving larger cutoff
values.

\begin{table*}
\begin{minipage}{\linewidth}
\renewcommand{\thefootnote}{\thempfootnote}
\begin{center}
\caption{Calculated free energies for several ice polymorphs with a 
variety of common water models. All calculations used a cutoff radius
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
kcal/mol. Calculated error of the final digits is in parentheses. *Ice
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.}
\begin{tabular}{ l  c  c  c  c }
\hline 
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
\hline 
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\
\end{tabular}
\label{freeEnergy}
\end{center}
\end{minipage}
\end{table*}

The free energy values computed for the studied polymorphs indicate
that Ice-{\it i} is the most stable state for all of the common water
models studied. With the calculated free energy at these state points,
the Gibbs-Helmholtz equation was used to project to other state points
and to build phase diagrams.  Figures
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
from the free energy results. All other models have similar structure,
although the crossing points between the phases move to slightly
different temperatures and pressures. It is interesting to note that
ice $I$ does not exist in either cubic or hexagonal form in any of the
phase diagrams for any of the models. For purposes of this study, ice
B is representative of the dense ice polymorphs. A recent study by
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and
TIP4P at higher pressures than those studied here.\cite{Sanz04}

\begin{figure}
\includegraphics[width=\linewidth]{tp3PhaseDia.eps}
\caption{Phase diagram for the TIP3P water model in the low pressure 
regime. The displayed $T_m$ and $T_b$ values are good predictions of
the experimental values; however, the solid phases shown are not the
experimentally observed forms. Both cubic and hexagonal ice $I$ are
higher in energy and don't appear in the phase diagram.}
\label{tp3phasedia}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps}
\caption{Phase diagram for the SSD/RF water model in the low pressure 
regime. Calculations producing these results were done under an
applied reaction field. It is interesting to note that this
computationally efficient model (over 3 times more efficient than
TIP3P) exhibits phase behavior similar to the less computationally
conservative charge based models.}
\label{ssdrfphasedia}
\end{figure}

\begin{table*}
\begin{minipage}{\linewidth}
\renewcommand{\thefootnote}{\thempfootnote}
\begin{center}
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) 
temperatures at 1 atm for several common water models compared with
experiment. The $T_m$ and $T_s$ values from simulation correspond to a
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the
liquid or gas state.}
\begin{tabular}{ l  c  c  c  c  c  c  c }
\hline 
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
\hline 
$T_m$ (K)  & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\
$T_b$ (K)  & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\
$T_s$ (K)  & - & - & - & - & 355(2) & - & -\\
\end{tabular}
\label{meltandboil}
\end{center}
\end{minipage}
\end{table*}

Table \ref{meltandboil} lists the melting and boiling temperatures
calculated from this work. Surprisingly, most of these models have
melting points that compare quite favorably with experiment. The
unfortunate aspect of this result is that this phase change occurs
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the
liquid state. These results are actually not contrary to previous
studies in the literature. Earlier free energy studies of ice $I$
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences
being attributed to choice of interaction truncation and different
ordered and disordered molecular
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be
predicted from this work. However, the $T_m$ from Ice-{\it i} is
calculated at 265 K, significantly higher in temperature than the
previous studies. Also of interest in these results is that SSD/E does
not exhibit a melting point at 1 atm, but it shows a sublimation point
at 355 K. This is due to the significant stability of Ice-{\it i} over
all other polymorphs for this particular model under these
conditions. While troubling, this behavior resulted in spontaneous
crystallization of Ice-{\it i} and led us to investigate this
structure. These observations provide a warning that simulations of
SSD/E as a ``liquid'' near 300 K are actually metastable and run the
risk of spontaneous crystallization. However, this risk lessens when
applying a longer cutoff.

\begin{figure}
\includegraphics[width=\linewidth]{cutoffChange.eps}
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) 
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9
\AA . These crystals are unstable at 200 K and rapidly convert into
liquids. The connecting lines are qualitative visual aid.}
\label{incCutoff}
\end{figure}

Increasing the cutoff radius in simulations of the more
computationally efficient water models was done in order to evaluate
the trend in free energy values when moving to systems that do not
involve potential truncation. As seen in Fig. \ref{incCutoff}, the
free energy of all the ice polymorphs show a substantial dependence on
cutoff radius. In general, there is a narrowing of the free energy
differences while moving to greater cutoff radius. Interestingly, by
increasing the cutoff radius, the free energy gap was narrowed enough
in the SSD/E model that the liquid state is preferred under standard
simulation conditions (298 K and 1 atm). Thus, it is recommended that
simulations using this model choose interaction truncation radii
greater than 9 \AA\ . This narrowing trend is much more subtle in the
case of SSD/RF, indicating that the free energies calculated with a
reaction field present provide a more accurate picture of the free
energy landscape in the absence of potential truncation.

To further study the changes resulting to the inclusion of a
long-range interaction correction, the effect of an Ewald summation
was estimated by applying the potential energy difference do to its
inclusion in systems in the presence and absence of the
correction. This was accomplished by calculation of the potential
energy of identical crystals both with and without PME. The free
energies for the investigated polymorphs using the TIP3P and SPC/E
water models are shown in Table \ref{pmeShift}. The same trend pointed
out through increase of cutoff radius is observed in these PME
results. Ice-{\it i} is the preferred polymorph at ambient conditions
for both the TIP3P and SPC/E water models; however, the narrowing of
the free energy differences between the various solid forms is
significant enough that it becomes less clear that it is the most
stable polymorph with the SPC/E model.  The free energies of Ice-{\it
i} and ice B nearly overlap within error, with ice $I_c$ just outside
as well, indicating that Ice-{\it i} might be metastable with respect
to ice B and possibly ice $I_c$ with SPC/E. However, these results do
not significantly alter the finding that the Ice-{\it i} polymorph is
a stable crystal structure that should be considered when studying the
phase behavior of water models.

\begin{table*}
\begin{minipage}{\linewidth}
\renewcommand{\thefootnote}{\thempfootnote}
\begin{center}
\caption{The free energy of the studied ice polymorphs after applying 
the energy difference attributed to the inclusion of the PME
long-range interaction correction. Units are kcal/mol.}
\begin{tabular}{ l  c  c  c  c }
\hline 
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
\hline 
TIP3P  & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\
SPC/E  & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\
\end{tabular}
\label{pmeShift}
\end{center}
\end{minipage}
\end{table*}

\section{Conclusions}

The free energy for proton ordered variants of hexagonal and cubic ice
$I$, ice B, and our recently discovered Ice-{\it i} structure were
calculated under standard conditions for several common water models
via thermodynamic integration. All the water models studied show
Ice-{\it i} to be the minimum free energy crystal structure with a 9
\AA\ switching function cutoff. Calculated melting and boiling points
show surprisingly good agreement with the experimental values;
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The
effect of interaction truncation was investigated through variation of
the cutoff radius, use of a reaction field parameterized model, and
estimation of the results in the presence of the Ewald
summation. Interaction truncation has a significant effect on the
computed free energy values, and may significantly alter the free
energy landscape for the more complex multipoint water models. Despite
these effects, these results show Ice-{\it i} to be an important ice
polymorph that should be considered in simulation studies.

Due to this relative stability of Ice-{\it i} in all of the
investigated simulation conditions, the question arises as to possible
experimental observation of this polymorph.  The rather extensive past
and current experimental investigation of water in the low pressure
regime makes us hesitant to ascribe any relevance of this work outside
of the simulation community.  It is for this reason that we chose a
name for this polymorph which involves an imaginary quantity.  That
said, there are certain experimental conditions that would provide the
most ideal situation for possible observation. These include the
negative pressure or stretched solid regime, small clusters in vacuum
deposition environments, and in clathrate structures involving small
non-polar molecules.  Figs. \ref{fig:gofr} and \ref{fig:sofq} contain
our predictions for both the pair distribution function ($g_{OO}(r)$)
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for
ice-{\it i} at a temperature of 77K.  In studies of the high and low
density forms of amorphous ice, ``spurious'' diffraction peaks have
been observed experimentally.\cite{Bizid87} It is possible that a
variant of Ice-{\it i} could explain some of this behavior; however,
we will leave it to our experimental colleagues to make the final
determination on whether this ice polymorph is named appropriately
(i.e. with an imaginary number) or if it can be promoted to Ice-0.

\begin{figure}
\includegraphics[width=\linewidth]{iceGofr.eps}
\caption{Radial distribution functions of ice $I_h$, $I_c$, 
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations
of the SSD/RF water model at 77 K.}
\label{fig:gofr}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{sofq.eps}
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i},
 and Ice-{\it i}$^\prime$ at 77 K.  The raw structure factors have
 been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$
 width) to compensate for the trunction effects in our finite size
 simulations.}
\label{fig:sofq}
\end{figure}

\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 High Performance Computing Cluster and the Notre Dame
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).

\newpage

\bibliographystyle{jcp}
\bibliography{iceiPaper}


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