--- 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 @@ - %\documentclass[prb,aps,twocolumn,tabularx]{revtex4} -\documentclass[preprint,aps,endfloats]{revtex4} +\documentclass[11pt]{article} %\documentclass[11pt]{article} -%\usepackage{endfloat} +\usepackage{endfloat} \usepackage{amsmath} \usepackage{epsf} \usepackage{berkeley} -%\usepackage{setspace} -%\usepackage{tabularx} +\usepackage{setspace} +\usepackage{tabularx} \usepackage{graphicx} -%\usepackage[ref]{overcite} -%\pagestyle{plain} -%\pagenumbering{arabic} -%\oddsidemargin 0.0cm \evensidemargin 0.0cm -%\topmargin -21pt \headsep 10pt -%\textheight 9.0in \textwidth 6.5in -%\brokenpenalty=10000 +\usepackage[ref]{overcite} +\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 -%\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