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+++ trunk/iceiPaper/iceiPaper.tex	2004/09/17 14:16:25	1472
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 \begin{document}
 
-\title{A Free Energy Study of Low Temperature and Anomolous Ice}
+\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{\thefootnote}
-\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
-
-\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
+\author{Christopher J. Fennell and J. Daniel Gezelter \\
+Department of Chemistry and Biochemistry\\ University of Notre Dame\\
 Notre Dame, Indiana 46556}
 
 \date{\today}
 
-%\maketitle
+\maketitle
 %\doublespacing
 
 \begin{abstract}
+The free energies of several ice polymorphs in the low pressure regime
+were calculated using thermodynamic integration.  These integrations
+were done for most of the common water models. Ice-{\it i}, a
+structure we recently observed to be stable in 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 the common water
+models.  Additionally, potential truncation was shown to have an
+effect on the calculated free energies, and can result in altered free
+energy landscapes.
 \end{abstract}
 
-\maketitle
-
-\newpage
-
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@@ -48,17 +54,474 @@ Notre Dame, Indiana 46556}
 
 \section{Introduction}
 
+Computer simulations are a valuable tool for studying the phase
+behavior of systems ranging from small or simple molecules to complex
+biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques
+have been developed to investigate the thermodynamic properites of
+model substances, providing both qualitative and quantitative
+comparisons between simulations and
+experiment.\cite{Widom63,Frenkel84} Investigation of these properties
+leads to the development of new and more accurate models, leading to
+better understanding and depiction of physical processes and intricate
+molecular systems.
+
+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{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04}
+These models have been used to investigate important physical
+phenomena like phase transitions, molecule transport, 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 property 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 true 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 the fact 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-$i^\prime$, 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
+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, the
+absolute free energy of this crystal was calculated 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-$i^\prime$) 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.
+
 \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 generate
+phase diagrams. All simulations were carried out at densities
+resulting in a pressure of approximately 1 atm at their respective
+temperatures.
+
+A single thermodynamic integration involves a sequence of simulations
+over 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 apply of one of the most convenient
+methods and integrate 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, results from the use of an Ewald
+summation were estimated for TIP3P and SPC/E by performing
+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 has an 8 molecule unit
+cell.\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(4) & -11.23(6) & -11.82(5) & -12.30(5)\\
+TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\
+TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\
+SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\
+SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\
+SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\
+\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 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 exist at 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 in the high pressure regime.\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(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\
+$T_b$ (K)  & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\
+$T_s$ (K)  & - & - & - & - & 355(3) & - & -\\
+\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 turned out to be
+advantageous in that it facilitated the spontaneous crystallization of
+Ice-{\it i}. 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 changes 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 with and without PME using TINKER. 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.  The free energies of Ice-{\it i} and ice B overlap
+within error, with ice $I_c$ just outside, indicating that Ice-{\it i}
+might be metastable with respect to ice B and possibly ice $I_c$ in
+the SPC/E water model. 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(4) & -11.24(6) & -11.51(5) & -11.67(5)\\
+SPC/E  & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\
+\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 recently discovered Ice-{\it i} 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 in the 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 manner of
+investigated simulation examples, 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 $I_c$ and for ice-{\it
+i} at a temperature of 77K.  In a quick comparison of the predicted
+S(q) for Ice-{\it i} and experimental studies of amorphous solid
+water, it is possible that some of the ``spurious'' peaks that could
+not be assigned in HDA could correspond to peaks labeled in this
+S(q).\cite{Bizid87} It should be noted that there is typically poor
+agreement on crystal densities between simulation and experiment, so
+such peak comparisons should be made with caution.  We will leave it
+to our experimental colleagues to determine whether this ice polymorph
+is named appropriately or if it should be promoted to Ice-0.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{iceGofr.eps}
+\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ 
+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-{\it i} and ice $I_c$ 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. The labeled peaks
+compared favorably with ``spurious'' peaks observed in experimental
+studies of amorphous solid water.\cite{Bizid87}}
+\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 Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
-DMR-0079647.
+the Notre Dame High Performance Computing Cluster and the Notre Dame
+Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
 
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