--- trunk/iceiPaper/iceiPaper.tex	2004/09/15 20:12:03	1460
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/15 22:33:18	1465
@@ -1,57 +1,51 @@
-
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+\renewcommand\citemid{\ } % no comma in optional reference note
 
-%\renewcommand\citemid{\ } % no comma in optional reference note
-
 \begin{document}
 
-\title{A Free Energy Study of Low Temperature and Anomalous 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 of systems consisting
-of a variety of common water models. Ice-{\it i}, a recent
-computationally observed solid structure, was determined to be the
-stable state with the lowest free energy for all the water models
-investigated. Phase diagrams were generated, and melting and boiling
-points for all the models were determined and show relatively good
-agreement with experiment, although the solid phase is different
-between simulation and experiment. In addition, potential truncation
-was shown to have an effect on the calculated free energies, and may
-result in altered free energy landscapes.
+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
-
 %\narrowtext 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -60,11 +54,11 @@ Molecular dynamics has developed into a valuable tool 
 
 \section{Introduction}
 
-Molecular dynamics has developed into a valuable tool for studying the
-phase behavior of systems ranging from small or simple
-molecules\cite{smallStuff} to complex biological
-species.\cite{bigStuff} Many techniques have been developed in order
-to investigate the thermodynamic properites of model substances,
+Molecular dynamics is a valuable tool for studying the phase behavior
+of systems ranging from small or simple
+molecules\cite{Matsumoto02andOthers} to complex biological
+species.\cite{bigStuff} Many techniques have been developed to
+investigate the thermodynamic properites of model substances,
 providing both qualitative and quantitative comparisons between
 simulations and experiment.\cite{thermMethods} Investigation of these
 properties leads to the development of new and more accurate models,
@@ -72,39 +66,39 @@ simulations, and has resulted in a variety of models t
 and intricate molecular systems.
 
 Water has proven to be a challenging substance to depict in
-simulations, and has resulted in a variety of models that attempt to
-describe its behavior under a varying simulation
-conditions.\cite{lotsOfWaterPapers} Many of these models have been
-used to investigate important physical phenomena like phase
-transitions and the hydrophobic effect.\cite{evenMorePapers} With the
-advent of numerous differing models, it is only natural that attention
-is placed on the properties of the models themselves in an attempt to
-clarify their benefits and limitations when applied to a system of
-interest.\cite{modelProps} One important but challenging property to
-quantify is the free energy, particularly of the solid forms of
-water. Difficulty in these types of studies typically arises from the
-assortment of possible crystalline polymorphs that water 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,
-because these phases will dictate the true transition temperatures and
-pressures for their respective model. 
+simulations, and a variety of models have been developed to describe
+its behavior under varying simulation
+conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
+These models have been used to investigate important physical
+phenomena like phase transitions and the hydrophobic
+effect.\cite{Yamada02} 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{modelProps} 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 their respective model.
 
-In this paper, standard reference state methods were applied to the
-study of 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{nucleationStudies}; however, the crystal structure
-obtained in this case was different from any previously observed ice
-polymorphs, in experiment or simulation.\cite{Fennell04} This crystal
-was termed Ice-{\it i} in homage to its origin in computational
+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
-waters so that both of their donated hydrogen bonds are internal to
+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
@@ -113,13 +107,18 @@ that are 0.07 g/cm$^3$ less dense on average than ice 
 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[scale=1.0]{unitCell.eps}
-\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-2{\it i}, the elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$ relation is given by $a = 1.0607c$, while for Ice-2{\it i}, $a = 0.8925c$.}
+\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}.  For Ice-{\it i}, the $a$ to $c$
+relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a =
+1.7850c$.} 
 \label{iceiCell}
 \end{figure}
+
 \begin{figure}
-\includegraphics[scale=1.0]{orderedIcei.eps}
+\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
@@ -127,51 +126,50 @@ Results in the previous study indicated that Ice-{\it 
 \label{protOrder}
 \end{figure}
 
-Results in the previous study indicated that Ice-{\it i} is the
-minimum energy crystal structure for the single point water models
-being studied (for discussions on these single point dipole models,
-see the previous work and related
+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
+the previous work and related
 articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
-consider energetic stabilization and neglect 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
-(soft sticky dipole extended, 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 (soft sticky dipole
-reaction field, SSD/RF). In should be noted that a second version of
-Ice-{\it i} (Ice-2{\it i}) 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.
+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 (Object-Oriented Parallel Simulation Engine)
-molecular mechanics package. All molecules were treated as rigid
-bodies, with orientational motion propagated using the symplectic DLM
-integration method. Details about the implementation of these
-techniques can be found in a recent publication.\cite{Meineke05} 
+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 these techniques can be found in a recent
+publication.\cite{DLM}
 
 Thermodynamic integration 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. 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.
+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. This transformation path is then
-integrated in order to determine the free energy difference between
-the two states:
+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,
@@ -185,12 +183,11 @@ Crystal is chosen as the reference state that the syst
 (near the reference state) in length.
 
 For the thermodynamic integration of molecular crystals, the Einstein
-Crystal is chosen as the reference state that the system is converted
-to over the course of the simulation. In an Einstein Crystal, the
+crystal was chosen as the reference state. In an Einstein crystal, the
 molecules are harmonically restrained at their ideal lattice locations
 and orientations. The partition function for a molecular crystal
-restrained in this fashion has been evaluated, and the Helmholtz Free
-Energy ({\it A}) is given by
+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
@@ -210,8 +207,9 @@ state.
 minimum potential energy of the ideal crystal. In the case of
 molecular liquids, the ideal vapor is chosen as the target reference
 state.
+
 \begin{figure}
-\includegraphics[scale=1.0]{rotSpring.eps}
+\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
@@ -222,23 +220,23 @@ cubic switching between 100\% and 85\% of the cutoff v
 \end{figure}
 
 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 poor energy conserving dynamics resulting 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
+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. TINKER was chosen because it can also
-propagate the motion of rigid-bodies, and provides the most direct
-comparison to the results from OOPSE. The calculated energy difference
-in the presence and absence of PME was applied to the previous results
-in order to predict changes in the free energy landscape.
+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}
 
@@ -248,14 +246,14 @@ Ice XI, the experimentally observed proton ordered var
 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 it was found not to be as
-stable as antiferroelectric variants of proton ordered or even proton
-disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of
-ice $I_h$ used here is a simple antiferroelectric version that has an
-8 molecule unit cell. 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
+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*}
@@ -267,9 +265,9 @@ kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF
 of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
 kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.}
 \begin{tabular}{ l  c  c  c  c }
-\hline \\[-7mm]
+\hline 
 \ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \  & \ \quad \ \ \ \ B \ \  & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\
-\hline \\[-3mm]
+\hline 
 \ \quad \ TIP3P  & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\
 \ \quad \ TIP4P  & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\
 \ \quad \ TIP5P  & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\
@@ -285,17 +283,18 @@ temperature and pressure dependence of the free energy
 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
-temperature and pressure dependence of the free energy was used to
-project to other state points and build phase diagrams. Figures
+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,
-only the crossing points between these phases exist at 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} 
+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 
@@ -305,6 +304,7 @@ higher in energy and don't appear in the phase diagram
 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 
@@ -323,9 +323,9 @@ temperatures of several common water models compared w
 \caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) 
 temperatures of several common water models compared with experiment.}
 \begin{tabular}{ l  c  c  c  c  c  c  c }
-\hline \\[-7mm]
+\hline 
 \ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\
-\hline \\[-3mm]
+\hline 
 \ \ $T_m$ (K)  & \ \ 269 & \ \ 265 & \ \ 271 &  297 & \ \ - & \ \ 278 & \ \ 273\\
 \ \ $T_b$ (K)  & \ \ 357 & \ \ 354 & \ \ 337 &  396 & \ \ - & \ \ 349 & \ \ 373\\
 \ \ $T_s$ (K)  & \ \ - & \ \ - & \ \ - &  - & \ \ 355 & \ \ - & \ \ -\\
@@ -399,7 +399,7 @@ narrowing is significant enough that it becomes less c
 preferred polymorph at ambient conditions for both the TIP3P and SPC/E
 water models; however, there is a narrowing of the free energy
 differences between the various solid forms. In the case of SPC/E this
-narrowing is significant enough that it becomes less clear cut that
+narrowing is significant enough that it becomes less clear that
 Ice-{\it i} is the most stable polymorph, and is possibly metastable
 with respect to ice B and possibly ice $I_c$. However, these results
 do not significantly alter the finding that the Ice-{\it i} polymorph
@@ -414,9 +414,9 @@ long-range interaction correction. Units are kcal/mol.
 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 \\[-7mm]
+\hline 
 \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
-\hline \\[-3mm]
+\hline 
 \ \ TIP3P  & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\
 \ \ SPC/E  & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\
 \end{tabular}
@@ -428,7 +428,7 @@ $I$, ice B, and recently discovered Ice-{\it i} where 
 \section{Conclusions}
 
 The free energy for proton ordered variants of hexagonal and cubic ice
-$I$, ice B, and recently discovered Ice-{\it i} where calculated under
+$I$, 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
@@ -437,8 +437,8 @@ estimation of the results in the presence of the Ewald
 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
-correction. Interaction truncation has a significant effect on the
+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
@@ -446,15 +446,21 @@ experimental observation of this polymorph. The rather
 
 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
+experimental observation of this polymorph.  The rather extensive past
 and current experimental investigation of water in the low pressure
-regime leads the authors to be hesitant in ascribing relevance outside
-of computational models, hence the descriptive name presented. That
-being 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.
+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.  Fig. \ref{fig:sofkgofr} contains our predictions
+of both the pair distribution function ($g_{OO}(r)$) and the structure
+factor ($S(\vec{q})$ for this polymorph at a temperature of 77K.  We
+will leave it to our experimental colleagues to determine whether this
+ice polymorph should really be called Ice-{\it i} or if it should be
+promoted to Ice-0.
 
 \section{Acknowledgments}
 Support for this project was provided by the National Science
