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21   \begin{document}
22  
23 < \title{A Free Energy Study of Low Temperature and Anomolous Ice}
23 > \title{Ice-{\it i}: a simulation-predicted ice polymorph which is more
24 > stable than Ice $I_h$ for point-charge and point-dipole water models}
25  
26 < \author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote}
27 < \footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
27 <
28 < \address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
26 > \author{Christopher J. Fennell and J. Daniel Gezelter \\
27 > Department of Chemistry and Biochemistry\\ University of Notre Dame\\
28   Notre Dame, Indiana 46556}
29  
30   \date{\today}
31  
32 < %\maketitle
32 > \maketitle
33   %\doublespacing
34  
35   \begin{abstract}
36 + The free energies of several ice polymorphs in the low pressure regime
37 + were calculated using thermodynamic integration.  These integrations
38 + were done for most of the common water models. Ice-{\it i}, a
39 + structure we recently observed to be stable in one of the single-point
40 + water models, was determined to be the stable crystalline state (at 1
41 + atm) for {\it all} the water models investigated.  Phase diagrams were
42 + generated, and phase coexistence lines were determined for all of the
43 + known low-pressure ice structures under all of the common water
44 + models.  Additionally, potential truncation was shown to have an
45 + effect on the calculated free energies, and can result in altered free
46 + energy landscapes.
47   \end{abstract}
48  
39 – \maketitle
40 –
41 – \newpage
42 –
49   %\narrowtext
50  
51   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Line 48 | Line 54 | Notre Dame, Indiana 46556}
54  
55   \section{Introduction}
56  
57 + Computer simulations are a valuable tool for studying the phase
58 + behavior of systems ranging from small or simple molecules to complex
59 + biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques
60 + have been developed to investigate the thermodynamic properites of
61 + model substances, providing both qualitative and quantitative
62 + comparisons between simulations and
63 + experiment.\cite{Widom63,Frenkel84} Investigation of these properties
64 + leads to the development of new and more accurate models, leading to
65 + better understanding and depiction of physical processes and intricate
66 + molecular systems.
67 +
68 + Water has proven to be a challenging substance to depict in
69 + simulations, and a variety of models have been developed to describe
70 + its behavior under varying simulation
71 + conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04}
72 + These models have been used to investigate important physical
73 + phenomena like phase transitions, molecule transport, and the
74 + hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the
75 + choice of models available, it is only natural to compare the models
76 + under interesting thermodynamic conditions in an attempt to clarify
77 + the limitations of each of the
78 + models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two
79 + important property to quantify are the Gibbs and Helmholtz free
80 + energies, particularly for the solid forms of water.  Difficulty in
81 + these types of studies typically arises from the assortment of
82 + possible crystalline polymorphs that water adopts over a wide range of
83 + pressures and temperatures.  There are currently 13 recognized forms
84 + of ice, and it is a challenging task to investigate the entire free
85 + energy landscape.\cite{Sanz04} Ideally, research is focused on the
86 + phases having the lowest free energy at a given state point, because
87 + these phases will dictate the true transition temperatures and
88 + pressures for the model.
89 +
90 + In this paper, standard reference state methods were applied to known
91 + crystalline water polymorphs in the low pressure regime.  This work is
92 + unique in the fact that one of the crystal lattices was arrived at
93 + through crystallization of a computationally efficient water model
94 + under constant pressure and temperature conditions. Crystallization
95 + events are interesting in and of
96 + themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure
97 + obtained in this case is different from any previously observed ice
98 + polymorphs in experiment or simulation.\cite{Fennell04} We have named
99 + this structure Ice-{\it i} to indicate its origin in computational
100 + simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
101 + water molecules that stack in rows of interlocking water
102 + tetramers. Proton ordering can be accomplished by orienting two of the
103 + molecules so that both of their donated hydrogen bonds are internal to
104 + their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
105 + constructed of water tetramers, the hydrogen bonds are not as linear
106 + as those observed in ice $I_h$, however the interlocking of these
107 + subunits appears to provide significant stabilization to the overall
108 + crystal. The arrangement of these tetramers results in surrounding
109 + open octagonal cavities that are typically greater than 6.3 \AA\ in
110 + diameter. This relatively open overall structure leads to crystals
111 + that are 0.07 g/cm$^3$ less dense on average than ice $I_h$.
112 +
113 + \begin{figure}
114 + \includegraphics[width=\linewidth]{unitCell.eps}
115 + \caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the
116 + elongated variant of Ice-{\it i}.  The spheres represent the
117 + center-of-mass locations of the water molecules.  The $a$ to $c$
118 + ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by
119 + $a:2.1214c$ and $a:1.7850c$ respectively.}
120 + \label{iceiCell}
121 + \end{figure}
122 +
123 + \begin{figure}
124 + \includegraphics[width=\linewidth]{orderedIcei.eps}
125 + \caption{Image of a proton ordered crystal of Ice-{\it i} looking
126 + down the (001) crystal face. The rows of water tetramers surrounded by
127 + octagonal pores leads to a crystal structure that is significantly
128 + less dense than ice $I_h$.}
129 + \label{protOrder}
130 + \end{figure}
131 +
132 + Results from our previous study indicated that Ice-{\it i} is the
133 + minimum energy crystal structure for the single point water models we
134 + investigated (for discussions on these single point dipole models, see
135 + our previous work and related
136 + articles).\cite{Fennell04,Liu96,Bratko85} Those results only
137 + considered energetic stabilization and neglected entropic
138 + contributions to the overall free energy. To address this issue, the
139 + absolute free energy of this crystal was calculated using
140 + thermodynamic integration and compared to the free energies of cubic
141 + and hexagonal ice $I$ (the experimental low density ice polymorphs)
142 + and ice B (a higher density, but very stable crystal structure
143 + observed by B\`{a}ez and Clancy in free energy studies of
144 + SPC/E).\cite{Baez95b} This work includes results for the water model
145 + from which Ice-{\it i} was crystallized (SSD/E) in addition to several
146 + common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
147 + field parametrized single point dipole water model (SSD/RF). It should
148 + be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used
149 + in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of
150 + this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit
151 + it is extended in the direction of the (001) face and compressed along
152 + the other two faces.
153 +
154   \section{Methods}
155  
156 + Canonical ensemble (NVT) molecular dynamics calculations were
157 + performed using the OOPSE molecular mechanics package.\cite{Meineke05}
158 + All molecules were treated as rigid bodies, with orientational motion
159 + propagated using the symplectic DLM integration method. Details about
160 + the implementation of this technique can be found in a recent
161 + publication.\cite{Dullweber1997}
162 +
163 + Thermodynamic integration is an established technique for
164 + determination of free energies of condensed phases of
165 + materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This
166 + method, implemented in the same manner illustrated by B\`{a}ez and
167 + Clancy, was utilized to calculate the free energy of several ice
168 + crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and
169 + SSD/E water models.\cite{Baez95a} Liquid state free energies at 300
170 + and 400 K for all of these water models were also determined using
171 + this same technique in order to determine melting points and generate
172 + phase diagrams. All simulations were carried out at densities
173 + resulting in a pressure of approximately 1 atm at their respective
174 + temperatures.
175 +
176 + A single thermodynamic integration involves a sequence of simulations
177 + over which the system of interest is converted into a reference system
178 + for which the free energy is known analytically. This transformation
179 + path is then integrated in order to determine the free energy
180 + difference between the two states:
181 + \begin{equation}
182 + \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
183 + )}{\partial\lambda}\right\rangle_\lambda d\lambda,
184 + \end{equation}
185 + where $V$ is the interaction potential and $\lambda$ is the
186 + transformation parameter that scales the overall
187 + potential. Simulations are distributed strategically along this path
188 + in order to sufficiently sample the regions of greatest change in the
189 + potential. Typical integrations in this study consisted of $\sim$25
190 + simulations ranging from 300 ps (for the unaltered system) to 75 ps
191 + (near the reference state) in length.
192 +
193 + For the thermodynamic integration of molecular crystals, the Einstein
194 + crystal was chosen as the reference system. In an Einstein crystal,
195 + the molecules are restrained at their ideal lattice locations and
196 + orientations. Using harmonic restraints, as applied by B\`{a}ez and
197 + Clancy, the total potential for this reference crystal
198 + ($V_\mathrm{EC}$) is the sum of all the harmonic restraints,
199 + \begin{equation}
200 + V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} +
201 + \frac{K_\omega\omega^2}{2},
202 + \end{equation}
203 + where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are
204 + the spring constants restraining translational motion and deflection
205 + of and rotation around the principle axis of the molecule
206 + respectively.  It is clear from Fig. \ref{waterSpring} that the values
207 + of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from
208 + $-\pi$ to $\pi$.  The partition function for a molecular crystal
209 + restrained in this fashion can be evaluated analytically, and the
210 + Helmholtz Free Energy ({\it A}) is given by
211 + \begin{eqnarray}
212 + A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left
213 + [\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right
214 + )^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right
215 + )^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right
216 + )^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi
217 + K_\omega K_\theta)^{\frac{1}{2}}}\exp\left
218 + (-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right
219 + )^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
220 + \label{ecFreeEnergy}
221 + \end{eqnarray}
222 + where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum
223 + potential energy of the ideal crystal.\cite{Baez95a}
224 +
225 + \begin{figure}
226 + \includegraphics[width=\linewidth]{rotSpring.eps}
227 + \caption{Possible orientational motions for a restrained molecule.
228 + $\theta$ angles correspond to displacement from the body-frame {\it
229 + z}-axis, while $\omega$ angles correspond to rotation about the
230 + body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring
231 + constants for the harmonic springs restraining motion in the $\theta$
232 + and $\omega$ directions.}
233 + \label{waterSpring}
234 + \end{figure}
235 +
236 + In the case of molecular liquids, the ideal vapor is chosen as the
237 + target reference state.  There are several examples of liquid state
238 + free energy calculations of water models present in the
239 + literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods
240 + typically differ in regard to the path taken for switching off the
241 + interaction potential to convert the system to an ideal gas of water
242 + molecules.  In this study, we apply of one of the most convenient
243 + methods and integrate over the $\lambda^4$ path, where all interaction
244 + parameters are scaled equally by this transformation parameter.  This
245 + method has been shown to be reversible and provide results in
246 + excellent agreement with other established methods.\cite{Baez95b}
247 +
248 + Charge, dipole, and Lennard-Jones interactions were modified by a
249 + cubic switching between 100\% and 85\% of the cutoff value (9 \AA
250 + ). By applying this function, these interactions are smoothly
251 + truncated, thereby avoiding the poor energy conservation which results
252 + from harsher truncation schemes. The effect of a long-range correction
253 + was also investigated on select model systems in a variety of
254 + manners. For the SSD/RF model, a reaction field with a fixed
255 + dielectric constant of 80 was applied in all
256 + simulations.\cite{Onsager36} For a series of the least computationally
257 + expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
258 + performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
259 + \AA\ cutoff results. Finally, results from the use of an Ewald
260 + summation were estimated for TIP3P and SPC/E by performing
261 + calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
262 + mechanics software package.\cite{Tinker} The calculated energy
263 + difference in the presence and absence of PME was applied to the
264 + previous results in order to predict changes to the free energy
265 + landscape.
266 +
267   \section{Results and discussion}
268  
269 + The free energy of proton ordered Ice-{\it i} was calculated and
270 + compared with the free energies of proton ordered variants of the
271 + experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
272 + as well as the higher density ice B, observed by B\`{a}ez and Clancy
273 + and thought to be the minimum free energy structure for the SPC/E
274 + model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b}
275 + Ice XI, the experimentally-observed proton-ordered variant of ice
276 + $I_h$, was investigated initially, but was found to be not as stable
277 + as proton disordered or antiferroelectric variants of ice $I_h$. The
278 + proton ordered variant of ice $I_h$ used here is a simple
279 + antiferroelectric version that we divised, and it has an 8 molecule
280 + unit cell similar to other predicted antiferroelectric $I_h$
281 + crystals.\cite{Davidson84} The crystals contained 648 or 1728
282 + molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000
283 + molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger
284 + crystal sizes were necessary for simulations involving larger cutoff
285 + values.
286 +
287 + \begin{table*}
288 + \begin{minipage}{\linewidth}
289 + \renewcommand{\thefootnote}{\thempfootnote}
290 + \begin{center}
291 + \caption{Calculated free energies for several ice polymorphs with a
292 + variety of common water models. All calculations used a cutoff radius
293 + of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
294 + kcal/mol. Calculated error of the final digits is in parentheses. *Ice
295 + $I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.}
296 + \begin{tabular}{ l  c  c  c  c }
297 + \hline
298 + Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
299 + \hline
300 + TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\
301 + TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\
302 + TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\
303 + SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\
304 + SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\
305 + SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\
306 + \end{tabular}
307 + \label{freeEnergy}
308 + \end{center}
309 + \end{minipage}
310 + \end{table*}
311 +
312 + The free energy values computed for the studied polymorphs indicate
313 + that Ice-{\it i} is the most stable state for all of the common water
314 + models studied. With the free energy at these state points, the
315 + Gibbs-Helmholtz equation was used to project to other state points and
316 + to build phase diagrams.  Figures
317 + \ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
318 + from the free energy results. All other models have similar structure,
319 + although the crossing points between the phases exist at slightly
320 + different temperatures and pressures. It is interesting to note that
321 + ice $I$ does not exist in either cubic or hexagonal form in any of the
322 + phase diagrams for any of the models. For purposes of this study, ice
323 + B is representative of the dense ice polymorphs. A recent study by
324 + Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and
325 + TIP4P in the high pressure regime.\cite{Sanz04}
326 +
327 + \begin{figure}
328 + \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
329 + \caption{Phase diagram for the TIP3P water model in the low pressure
330 + regime. The displayed $T_m$ and $T_b$ values are good predictions of
331 + the experimental values; however, the solid phases shown are not the
332 + experimentally observed forms. Both cubic and hexagonal ice $I$ are
333 + higher in energy and don't appear in the phase diagram.}
334 + \label{tp3phasedia}
335 + \end{figure}
336 +
337 + \begin{figure}
338 + \includegraphics[width=\linewidth]{ssdrfPhaseDia.eps}
339 + \caption{Phase diagram for the SSD/RF water model in the low pressure
340 + regime. Calculations producing these results were done under an
341 + applied reaction field. It is interesting to note that this
342 + computationally efficient model (over 3 times more efficient than
343 + TIP3P) exhibits phase behavior similar to the less computationally
344 + conservative charge based models.}
345 + \label{ssdrfphasedia}
346 + \end{figure}
347 +
348 + \begin{table*}
349 + \begin{minipage}{\linewidth}
350 + \renewcommand{\thefootnote}{\thempfootnote}
351 + \begin{center}
352 + \caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$)
353 + temperatures at 1 atm for several common water models compared with
354 + experiment. The $T_m$ and $T_s$ values from simulation correspond to a
355 + transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the
356 + liquid or gas state.}
357 + \begin{tabular}{ l  c  c  c  c  c  c  c }
358 + \hline
359 + Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
360 + \hline
361 + $T_m$ (K)  & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\
362 + $T_b$ (K)  & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\
363 + $T_s$ (K)  & - & - & - & - & 355(2) & - & -\\
364 + \end{tabular}
365 + \label{meltandboil}
366 + \end{center}
367 + \end{minipage}
368 + \end{table*}
369 +
370 + Table \ref{meltandboil} lists the melting and boiling temperatures
371 + calculated from this work. Surprisingly, most of these models have
372 + melting points that compare quite favorably with experiment. The
373 + unfortunate aspect of this result is that this phase change occurs
374 + between Ice-{\it i} and the liquid state rather than ice $I_h$ and the
375 + liquid state. These results are actually not contrary to previous
376 + studies in the literature. Earlier free energy studies of ice $I$
377 + using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences
378 + being attributed to choice of interaction truncation and different
379 + ordered and disordered molecular
380 + arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and
381 + Ice-{\it i} were omitted, a $T_m$ value around 210 K would be
382 + predicted from this work. However, the $T_m$ from Ice-{\it i} is
383 + calculated at 265 K, significantly higher in temperature than the
384 + previous studies. Also of interest in these results is that SSD/E does
385 + not exhibit a melting point at 1 atm, but it shows a sublimation point
386 + at 355 K. This is due to the significant stability of Ice-{\it i} over
387 + all other polymorphs for this particular model under these
388 + conditions. While troubling, this behavior turned out to be
389 + advantageous in that it facilitated the spontaneous crystallization of
390 + Ice-{\it i}. These observations provide a warning that simulations of
391 + SSD/E as a ``liquid'' near 300 K are actually metastable and run the
392 + risk of spontaneous crystallization. However, this risk changes when
393 + applying a longer cutoff.
394 +
395 + \begin{figure}
396 + \includegraphics[width=\linewidth]{cutoffChange.eps}
397 + \caption{Free energy as a function of cutoff radius for (A) SSD/E, (B)
398 + TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12
399 + \AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9
400 + \AA . These crystals are unstable at 200 K and rapidly convert into
401 + liquids. The connecting lines are qualitative visual aid.}
402 + \label{incCutoff}
403 + \end{figure}
404 +
405 + Increasing the cutoff radius in simulations of the more
406 + computationally efficient water models was done in order to evaluate
407 + the trend in free energy values when moving to systems that do not
408 + involve potential truncation. As seen in Fig. \ref{incCutoff}, the
409 + free energy of all the ice polymorphs show a substantial dependence on
410 + cutoff radius. In general, there is a narrowing of the free energy
411 + differences while moving to greater cutoff radius. Interestingly, by
412 + increasing the cutoff radius, the free energy gap was narrowed enough
413 + in the SSD/E model that the liquid state is preferred under standard
414 + simulation conditions (298 K and 1 atm). Thus, it is recommended that
415 + simulations using this model choose interaction truncation radii
416 + greater than 9 \AA\ . This narrowing trend is much more subtle in the
417 + case of SSD/RF, indicating that the free energies calculated with a
418 + reaction field present provide a more accurate picture of the free
419 + energy landscape in the absence of potential truncation.
420 +
421 + To further study the changes resulting to the inclusion of a
422 + long-range interaction correction, the effect of an Ewald summation
423 + was estimated by applying the potential energy difference do to its
424 + inclusion in systems in the presence and absence of the
425 + correction. This was accomplished by calculation of the potential
426 + energy of identical crystals with and without PME using TINKER. The
427 + free energies for the investigated polymorphs using the TIP3P and
428 + SPC/E water models are shown in Table \ref{pmeShift}. The same trend
429 + pointed out through increase of cutoff radius is observed in these PME
430 + results. Ice-{\it i} is the preferred polymorph at ambient conditions
431 + for both the TIP3P and SPC/E water models; however, the narrowing of
432 + the free energy differences between the various solid forms is
433 + significant enough that it becomes less clear that it is the most
434 + stable polymorph with the SPC/E model.  The free energies of Ice-{\it
435 + i} and ice B nearly overlap within error, with ice $I_c$ just outside
436 + as well, indicating that Ice-{\it i} might be metastable with respect
437 + to ice B and possibly ice $I_c$ with SPC/E. However, these results do
438 + not significantly alter the finding that the Ice-{\it i} polymorph is
439 + a stable crystal structure that should be considered when studying the
440 + phase behavior of water models.
441 +
442 + \begin{table*}
443 + \begin{minipage}{\linewidth}
444 + \renewcommand{\thefootnote}{\thempfootnote}
445 + \begin{center}
446 + \caption{The free energy of the studied ice polymorphs after applying
447 + the energy difference attributed to the inclusion of the PME
448 + long-range interaction correction. Units are kcal/mol.}
449 + \begin{tabular}{ l  c  c  c  c }
450 + \hline
451 + \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
452 + \hline
453 + TIP3P  & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\
454 + SPC/E  & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\
455 + \end{tabular}
456 + \label{pmeShift}
457 + \end{center}
458 + \end{minipage}
459 + \end{table*}
460 +
461   \section{Conclusions}
462  
463 + The free energy for proton ordered variants of hexagonal and cubic ice
464 + $I$, ice B, and recently discovered Ice-{\it i} were calculated under
465 + standard conditions for several common water models via thermodynamic
466 + integration. All the water models studied show Ice-{\it i} to be the
467 + minimum free energy crystal structure in the with a 9 \AA\ switching
468 + function cutoff. Calculated melting and boiling points show
469 + surprisingly good agreement with the experimental values; however, the
470 + solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
471 + interaction truncation was investigated through variation of the
472 + cutoff radius, use of a reaction field parameterized model, and
473 + estimation of the results in the presence of the Ewald
474 + summation. Interaction truncation has a significant effect on the
475 + computed free energy values, and may significantly alter the free
476 + energy landscape for the more complex multipoint water models. Despite
477 + these effects, these results show Ice-{\it i} to be an important ice
478 + polymorph that should be considered in simulation studies.
479 +
480 + Due to this relative stability of Ice-{\it i} in all manner of
481 + investigated simulation examples, the question arises as to possible
482 + experimental observation of this polymorph.  The rather extensive past
483 + and current experimental investigation of water in the low pressure
484 + regime makes us hesitant to ascribe any relevance of this work outside
485 + of the simulation community.  It is for this reason that we chose a
486 + name for this polymorph which involves an imaginary quantity.  That
487 + said, there are certain experimental conditions that would provide the
488 + most ideal situation for possible observation. These include the
489 + negative pressure or stretched solid regime, small clusters in vacuum
490 + deposition environments, and in clathrate structures involving small
491 + non-polar molecules.  Figs. \ref{fig:gofr} and \ref{fig:sofq} contain
492 + our predictions for both the pair distribution function ($g_{OO}(r)$)
493 + and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it
494 + i} at a temperature of 77K.  In a quick comparison of the predicted
495 + S(q) for Ice-{\it i} and experimental studies of amorphous solid
496 + water, it is possible that some of the ``spurious'' peaks that could
497 + not be assigned in HDA could correspond to peaks labeled in this
498 + S(q).\cite{Bizid87} It should be noted that there is typically poor
499 + agreement on crystal densities between simulation and experiment, so
500 + such peak comparisons should be made with caution.  We will leave it
501 + to our experimental colleagues to determine whether this ice polymorph
502 + is named appropriately or if it should be promoted to Ice-0.
503 +
504 + \begin{figure}
505 + \includegraphics[width=\linewidth]{iceGofr.eps}
506 + \caption{Radial distribution functions of Ice-{\it i} and ice $I_c$
507 + calculated from from simulations of the SSD/RF water model at 77 K.}
508 + \label{fig:gofr}
509 + \end{figure}
510 +
511 + \begin{figure}
512 + \includegraphics[width=\linewidth]{sofq.eps}
513 + \caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at
514 + 77 K.  The raw structure factors have been convoluted with a gaussian
515 + instrument function (0.075 \AA$^{-1}$ width) to compensate for the
516 + trunction effects in our finite size simulations. The labeled peaks
517 + compared favorably with ``spurious'' peaks observed in experimental
518 + studies of amorphous solid water.\cite{Bizid87}}
519 + \label{fig:sofq}
520 + \end{figure}
521 +
522   \section{Acknowledgments}
523   Support for this project was provided by the National Science
524   Foundation under grant CHE-0134881. Computation time was provided by
525 < the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
526 < DMR-0079647.
525 > the Notre Dame High Performance Computing Cluster and the Notre Dame
526 > Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
527  
528   \newpage
529  

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