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# Line 33 | Line 33 | The free energies of several ice polymorphs in the low
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
36 > The absolute free energies of several ice polymorphs which are stable
37 > at low pressures were calculated using thermodynamic integration to a
38 > reference system (the Einstein crystal).  These integrations were
39 > performed for most of the common water models (SPC/E, TIP3P, TIP4P,
40 > TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed
41 > crystallizing at room temperature for one of the single-point water
42 > models, was determined to be the stable crystalline state (at 1 atm)
43 > for {\it all} the water models investigated.  Phase diagrams were
44   generated, and phase coexistence lines were determined for all of the
45 < known low-pressure ice structures under all of the common water
46 < models.  Additionally, potential truncation was shown to have an
47 < effect on the calculated free energies, and can result in altered free
48 < energy landscapes.
45 > known low-pressure ice structures under all of these water models.
46 > Additionally, potential truncation was shown to have an effect on the
47 > calculated free energies, and can result in altered free energy
48 > landscapes.  Structure factor for the new crystal were generated and
49 > we await experimental confirmation of the existence of this new
50 > polymorph.  
51   \end{abstract}
52  
53   %\narrowtext
# Line 54 | Line 58 | Molecular dynamics is a valuable tool for studying the
58  
59   \section{Introduction}
60  
57 – Molecular dynamics is a valuable tool for studying the phase behavior
58 – of systems ranging from small or simple
59 – molecules\cite{Matsumoto02andOthers} to complex biological
60 – species.\cite{bigStuff} Many techniques have been developed to
61 – investigate the thermodynamic properites of model substances,
62 – providing both qualitative and quantitative comparisons between
63 – simulations and experiment.\cite{thermMethods} Investigation of these
64 – properties leads to the development of new and more accurate models,
65 – leading to better understanding and depiction of physical processes
66 – and intricate molecular systems.
67 –
61   Water has proven to be a challenging substance to depict in
62   simulations, and a variety of models have been developed to describe
63   its behavior under varying simulation
64 < conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
64 > conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04,Amoeba,POL3}
65   These models have been used to investigate important physical
66 < phenomena like phase transitions and the hydrophobic
67 < effect.\cite{Yamada02} With the choice of models available, it
68 < is only natural to compare the models under interesting thermodynamic
69 < conditions in an attempt to clarify the limitations of each of the
70 < models.\cite{modelProps} Two important property to quantify are the
71 < Gibbs and Helmholtz free energies, particularly for the solid forms of
72 < water.  Difficulty in these types of studies typically arises from the
73 < assortment of possible crystalline polymorphs that water adopts over a
74 < wide range of pressures and temperatures. There are currently 13
75 < recognized forms of ice, and it is a challenging task to investigate
76 < the entire free energy landscape.\cite{Sanz04} Ideally, research is
77 < focused on the phases having the lowest free energy at a given state
78 < point, because these phases will dictate the true transition
79 < temperatures and pressures for their respective model.
66 > phenomena like phase transitions, transport properties, and the
67 > hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the
68 > choice of models available, it is only natural to compare the models
69 > under interesting thermodynamic conditions in an attempt to clarify
70 > the limitations of each of the
71 > models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two
72 > important properties to quantify are the Gibbs and Helmholtz free
73 > energies, particularly for the solid forms of water.  Difficulty in
74 > these types of studies typically arises from the assortment of
75 > possible crystalline polymorphs that water adopts over a wide range of
76 > pressures and temperatures.  There are currently 13 recognized forms
77 > of ice, and it is a challenging task to investigate the entire free
78 > energy landscape.\cite{Sanz04} Ideally, research is focused on the
79 > phases having the lowest free energy at a given state point, because
80 > these phases will dictate the relevant transition temperatures and
81 > pressures for the model.
82  
83   In this paper, standard reference state methods were applied to known
84 < crystalline water polymorphs in the low pressure regime. This work is
85 < unique in the fact that one of the crystal lattices was arrived at
86 < through crystallization of a computationally efficient water model
87 < under constant pressure and temperature conditions. Crystallization
88 < events are interesting in and of
89 < themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure
90 < obtained in this case is different from any previously observed ice
91 < polymorphs in experiment or simulation.\cite{Fennell04} We have named
92 < this structure Ice-{\it i} to indicate its origin in computational
93 < simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
94 < water molecules that stack in rows of interlocking water
95 < tetramers. Proton ordering can be accomplished by orienting two of the
96 < molecules so that both of their donated hydrogen bonds are internal to
97 < their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
98 < constructed of water tetramers, the hydrogen bonds are not as linear
99 < as those observed in ice $I_h$, however the interlocking of these
100 < subunits appears to provide significant stabilization to the overall
84 > crystalline water polymorphs in the low pressure regime.  This work is
85 > unique in that one of the crystal lattices was arrived at through
86 > crystallization of a computationally efficient water model under
87 > constant pressure and temperature conditions. Crystallization events
88 > are interesting in and of themselves;\cite{Matsumoto02,Yamada02}
89 > however, the crystal structure obtained in this case is different from
90 > any previously observed ice polymorphs in experiment or
91 > simulation.\cite{Fennell04} We have named this structure Ice-{\it i}
92 > to indicate its origin in computational simulation. The unit cell
93 > (Fig. \ref{iceiCell}A) consists of eight water molecules that stack in
94 > rows of interlocking water tetramers. Proton ordering can be
95 > accomplished by orienting two of the molecules so that both of their
96 > donated hydrogen bonds are internal to their tetramer
97 > (Fig. \ref{protOrder}). As expected in an ice crystal constructed of
98 > water tetramers, the hydrogen bonds are not as linear as those
99 > observed in ice $I_h$, however the interlocking of these subunits
100 > appears to provide significant stabilization to the overall
101   crystal. The arrangement of these tetramers results in surrounding
102   open octagonal cavities that are typically greater than 6.3 \AA\ in
103   diameter. This relatively open overall structure leads to crystals
# Line 111 | Line 106 | elongated variant of Ice-{\it i}.  For Ice-{\it i}, th
106   \begin{figure}
107   \includegraphics[width=\linewidth]{unitCell.eps}
108   \caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the
109 < elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$
110 < relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a =
111 < 1.7850c$.}
109 > elongated variant of Ice-{\it i}.  The spheres represent the
110 > center-of-mass locations of the water molecules.  The $a$ to $c$
111 > ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by
112 > $a:2.1214c$ and $a:1.7850c$ respectively.}
113   \label{iceiCell}
114   \end{figure}
115  
# Line 128 | Line 124 | investigated (for discussions on these single point di
124  
125   Results from our previous study indicated that Ice-{\it i} is the
126   minimum energy crystal structure for the single point water models we
127 < investigated (for discussions on these single point dipole models, see
128 < the previous work and related
129 < articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
127 > had investigated (for discussions on these single point dipole models,
128 > see our previous work and related
129 > articles).\cite{Fennell04,Liu96,Bratko85} Those results only
130   considered energetic stabilization and neglected entropic
131 < contributions to the overall free energy. To address this issue, the
132 < absolute free energy of this crystal was calculated using
131 > contributions to the overall free energy. To address this issue, we
132 > have calculated the absolute free energy of this crystal using
133   thermodynamic integration and compared to the free energies of cubic
134   and hexagonal ice $I$ (the experimental low density ice polymorphs)
135   and ice B (a higher density, but very stable crystal structure
# Line 142 | Line 138 | be noted that a second version of Ice-{\it i} (Ice-$i^
138   from which Ice-{\it i} was crystallized (SSD/E) in addition to several
139   common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
140   field parametrized single point dipole water model (SSD/RF). It should
141 < be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used
142 < in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of
143 < this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit
144 < it is extended in the direction of the (001) face and compressed along
145 < the other two faces.
141 > be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was
142 > used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell
143 > of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i}
144 > unit it is extended in the direction of the (001) face and compressed
145 > along the other two faces.
146  
147   \section{Methods}
148  
# Line 154 | Line 150 | the implementation of these techniques can be found in
150   performed using the OOPSE molecular mechanics package.\cite{Meineke05}
151   All molecules were treated as rigid bodies, with orientational motion
152   propagated using the symplectic DLM integration method. Details about
153 < the implementation of these techniques can be found in a recent
154 < publication.\cite{DLM}
153 > the implementation of this technique can be found in a recent
154 > publication.\cite{Dullweber1997}
155  
156 < Thermodynamic integration was utilized to calculate the free energy of
157 < several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E,
158 < SSD/RF, and SSD/E water models. Liquid state free energies at 300 and
159 < 400 K for all of these water models were also determined using this
160 < same technique in order to determine melting points and generate phase
161 < diagrams. All simulations were carried out at densities resulting in a
162 < pressure of approximately 1 atm at their respective temperatures.
156 > Thermodynamic integration is an established technique for
157 > determination of free energies of condensed phases of
158 > materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This
159 > method, implemented in the same manner illustrated by B\`{a}ez and
160 > Clancy, was utilized to calculate the free energy of several ice
161 > crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and
162 > SSD/E water models.\cite{Baez95a} Liquid state free energies at 300
163 > and 400 K for all of these water models were also determined using
164 > this same technique in order to determine melting points and to
165 > generate phase diagrams. All simulations were carried out at densities
166 > which correspond to a pressure of approximately 1 atm at their
167 > respective temperatures.
168  
169 < A single thermodynamic integration involves a sequence of simulations
170 < over which the system of interest is converted into a reference system
171 < for which the free energy is known analytically. This transformation
172 < path is then integrated in order to determine the free energy
173 < difference between the two states:
169 > Thermodynamic integration involves a sequence of simulations during
170 > which the system of interest is converted into a reference system for
171 > which the free energy is known analytically. This transformation path
172 > is then integrated in order to determine the free energy difference
173 > between the two states:
174   \begin{equation}
175   \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
176   )}{\partial\lambda}\right\rangle_\lambda d\lambda,
177   \end{equation}
178   where $V$ is the interaction potential and $\lambda$ is the
179   transformation parameter that scales the overall
180 < potential. Simulations are distributed unevenly along this path in
181 < order to sufficiently sample the regions of greatest change in the
180 > potential. Simulations are distributed strategically along this path
181 > in order to sufficiently sample the regions of greatest change in the
182   potential. Typical integrations in this study consisted of $\sim$25
183   simulations ranging from 300 ps (for the unaltered system) to 75 ps
184   (near the reference state) in length.
185  
186   For the thermodynamic integration of molecular crystals, the Einstein
187 < crystal was chosen as the reference state. In an Einstein crystal, the
188 < molecules are harmonically restrained at their ideal lattice locations
189 < and orientations. The partition function for a molecular crystal
187 > crystal was chosen as the reference system. In an Einstein crystal,
188 > the molecules are restrained at their ideal lattice locations and
189 > orientations. Using harmonic restraints, as applied by B\`{a}ez and
190 > Clancy, the total potential for this reference crystal
191 > ($V_\mathrm{EC}$) is the sum of all the harmonic restraints,
192 > \begin{equation}
193 > V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} +
194 > \frac{K_\omega\omega^2}{2},
195 > \end{equation}
196 > where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are
197 > the spring constants restraining translational motion and deflection
198 > of and rotation around the principle axis of the molecule
199 > respectively.  It is clear from Fig. \ref{waterSpring} that the values
200 > of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from
201 > $-\pi$ to $\pi$.  The partition function for a molecular crystal
202   restrained in this fashion can be evaluated analytically, and the
203   Helmholtz Free Energy ({\it A}) is given by
204   \begin{eqnarray}
# Line 199 | Line 212 | where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a
212   )^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
213   \label{ecFreeEnergy}
214   \end{eqnarray}
215 < where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation
216 < \ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and
204 < $K_\mathrm{\omega}$ are the spring constants restraining translational
205 < motion and deflection of and rotation around the principle axis of the
206 < molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the
207 < minimum potential energy of the ideal crystal. In the case of
208 < molecular liquids, the ideal vapor is chosen as the target reference
209 < state.
215 > where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum
216 > potential energy of the ideal crystal.\cite{Baez95a}
217  
218   \begin{figure}
219   \includegraphics[width=\linewidth]{rotSpring.eps}
# Line 219 | Line 226 | Charge, dipole, and Lennard-Jones interactions were mo
226   \label{waterSpring}
227   \end{figure}
228  
229 + In the case of molecular liquids, the ideal vapor is chosen as the
230 + target reference state.  There are several examples of liquid state
231 + free energy calculations of water models present in the
232 + literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods
233 + typically differ in regard to the path taken for switching off the
234 + interaction potential to convert the system to an ideal gas of water
235 + molecules.  In this study, we applied of one of the most convenient
236 + methods and integrated over the $\lambda^4$ path, where all
237 + interaction parameters are scaled equally by this transformation
238 + parameter.  This method has been shown to be reversible and provide
239 + results in excellent agreement with other established
240 + methods.\cite{Baez95b}
241 +
242   Charge, dipole, and Lennard-Jones interactions were modified by a
243   cubic switching between 100\% and 85\% of the cutoff value (9 \AA
244   ). By applying this function, these interactions are smoothly
# Line 230 | Line 250 | performed with longer cutoffs of 12 and 15 \AA\ to com
250   simulations.\cite{Onsager36} For a series of the least computationally
251   expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
252   performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
253 < \AA\ cutoff results. Finally, results from the use of an Ewald
254 < summation were estimated for TIP3P and SPC/E by performing
255 < calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
256 < mechanics software package.\cite{Tinker} The calculated energy
257 < difference in the presence and absence of PME was applied to the
258 < previous results in order to predict changes to the free energy
259 < landscape.
253 > \AA\ cutoff results. Finally, the effects of utilizing an Ewald
254 > summation were estimated for TIP3P and SPC/E by performing single
255 > configuration calculations with Particle-Mesh Ewald (PME) in the
256 > TINKER molecular mechanics software package.\cite{Tinker} The
257 > calculated energy difference in the presence and absence of PME was
258 > applied to the previous results in order to predict changes to the
259 > free energy landscape.
260  
261   \section{Results and discussion}
262  
263 < The free energy of proton ordered Ice-{\it i} was calculated and
263 > The free energy of proton-ordered Ice-{\it i} was calculated and
264   compared with the free energies of proton ordered variants of the
265   experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
266   as well as the higher density ice B, observed by B\`{a}ez and Clancy
# Line 250 | Line 270 | antiferroelectric version that has an 8 molecule unit
270   $I_h$, was investigated initially, but was found to be not as stable
271   as proton disordered or antiferroelectric variants of ice $I_h$. The
272   proton ordered variant of ice $I_h$ used here is a simple
273 < antiferroelectric version that has an 8 molecule unit
274 < cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules
275 < for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for
276 < ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes
277 < were necessary for simulations involving larger cutoff values.
273 > antiferroelectric version that we devised, and it has an 8 molecule
274 > unit cell similar to other predicted antiferroelectric $I_h$
275 > crystals.\cite{Davidson84} The crystals contained 648 or 1728
276 > molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000
277 > molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger
278 > crystal sizes were necessary for simulations involving larger cutoff
279 > values.
280  
281   \begin{table*}
282   \begin{minipage}{\linewidth}
# Line 269 | Line 291 | TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\
291   \hline
292   Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
293   \hline
294 < TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\
295 < TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\
296 < TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\
297 < SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\
298 < SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\
299 < SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\
294 > TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\
295 > TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\
296 > TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\
297 > SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\
298 > SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\
299 > SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\
300   \end{tabular}
301   \label{freeEnergy}
302   \end{center}
# Line 283 | Line 305 | models studied. With the free energy at these state po
305  
306   The free energy values computed for the studied polymorphs indicate
307   that Ice-{\it i} is the most stable state for all of the common water
308 < models studied. With the free energy at these state points, the
309 < Gibbs-Helmholtz equation was used to project to other state points and
310 < to build phase diagrams.  Figures
308 > models studied. With the calculated free energy at these state points,
309 > the Gibbs-Helmholtz equation was used to project to other state points
310 > and to build phase diagrams.  Figures
311   \ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
312   from the free energy results. All other models have similar structure,
313 < although the crossing points between the phases exist at slightly
313 > although the crossing points between the phases move to slightly
314   different temperatures and pressures. It is interesting to note that
315   ice $I$ does not exist in either cubic or hexagonal form in any of the
316   phase diagrams for any of the models. For purposes of this study, ice
317   B is representative of the dense ice polymorphs. A recent study by
318   Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and
319 < TIP4P in the high pressure regime.\cite{Sanz04}
319 > TIP4P at higher pressures than those studied here.\cite{Sanz04}
320  
321   \begin{figure}
322   \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
# Line 330 | Line 352 | $T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - &
352   \hline
353   Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
354   \hline
355 < $T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\
356 < $T_b$ (K)  & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\
357 < $T_s$ (K)  & - & - & - & - & 355(3) & - & -\\
355 > $T_m$ (K)  & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\
356 > $T_b$ (K)  & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\
357 > $T_s$ (K)  & - & - & - & - & 355(2) & - & -\\
358   \end{tabular}
359   \label{meltandboil}
360   \end{center}
# Line 357 | Line 379 | conditions. While troubling, this behavior turned out
379   not exhibit a melting point at 1 atm, but it shows a sublimation point
380   at 355 K. This is due to the significant stability of Ice-{\it i} over
381   all other polymorphs for this particular model under these
382 < conditions. While troubling, this behavior turned out to be
383 < advantageous in that it facilitated the spontaneous crystallization of
384 < Ice-{\it i}. These observations provide a warning that simulations of
382 > conditions. While troubling, this behavior resulted in spontaneous
383 > crystallization of Ice-{\it i} and led us to investigate this
384 > structure. These observations provide a warning that simulations of
385   SSD/E as a ``liquid'' near 300 K are actually metastable and run the
386 < risk of spontaneous crystallization. However, this risk changes when
386 > risk of spontaneous crystallization. However, this risk lessens when
387   applying a longer cutoff.
388  
389   \begin{figure}
# Line 385 | Line 407 | greater than 9 \AA\. This narrowing trend is much more
407   in the SSD/E model that the liquid state is preferred under standard
408   simulation conditions (298 K and 1 atm). Thus, it is recommended that
409   simulations using this model choose interaction truncation radii
410 < greater than 9 \AA\. This narrowing trend is much more subtle in the
410 > greater than 9 \AA\ . This narrowing trend is much more subtle in the
411   case of SSD/RF, indicating that the free energies calculated with a
412   reaction field present provide a more accurate picture of the free
413   energy landscape in the absence of potential truncation.
# Line 395 | Line 417 | energy of identical crystals with and without PME usin
417   was estimated by applying the potential energy difference do to its
418   inclusion in systems in the presence and absence of the
419   correction. This was accomplished by calculation of the potential
420 < energy of identical crystals with and without PME using TINKER. The
421 < free energies for the investigated polymorphs using the TIP3P and
422 < SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
423 < are not fully supported in TINKER, so the results for these models
424 < could not be estimated. The same trend pointed out through increase of
425 < cutoff radius is observed in these PME results. Ice-{\it i} is the
426 < preferred polymorph at ambient conditions for both the TIP3P and SPC/E
427 < water models; however, there is a narrowing of the free energy
428 < differences between the various solid forms. In the case of SPC/E this
429 < narrowing is significant enough that it becomes less clear that
430 < Ice-{\it i} is the most stable polymorph, and is possibly metastable
431 < with respect to ice B and possibly ice $I_c$. However, these results
432 < do not significantly alter the finding that the Ice-{\it i} polymorph
433 < is a stable crystal structure that should be considered when studying
434 < the phase behavior of water models.
420 > energy of identical crystals both with and without PME. The free
421 > energies for the investigated polymorphs using the TIP3P and SPC/E
422 > water models are shown in Table \ref{pmeShift}. The same trend pointed
423 > out through increase of cutoff radius is observed in these PME
424 > results. Ice-{\it i} is the preferred polymorph at ambient conditions
425 > for both the TIP3P and SPC/E water models; however, the narrowing of
426 > the free energy differences between the various solid forms is
427 > significant enough that it becomes less clear that it is the most
428 > stable polymorph with the SPC/E model.  The free energies of Ice-{\it
429 > i} and ice B nearly overlap within error, with ice $I_c$ just outside
430 > as well, indicating that Ice-{\it i} might be metastable with respect
431 > to ice B and possibly ice $I_c$ with SPC/E. However, these results do
432 > not significantly alter the finding that the Ice-{\it i} polymorph is
433 > a stable crystal structure that should be considered when studying the
434 > phase behavior of water models.
435  
436   \begin{table*}
437   \begin{minipage}{\linewidth}
# Line 422 | Line 444 | TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)
444   \hline
445   \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
446   \hline
447 < TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\
448 < SPC/E  & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\
447 > TIP3P  & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\
448 > SPC/E  & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\
449   \end{tabular}
450   \label{pmeShift}
451   \end{center}
# Line 433 | Line 455 | $I$, ice B, and recently discovered Ice-{\it i} were c
455   \section{Conclusions}
456  
457   The free energy for proton ordered variants of hexagonal and cubic ice
458 < $I$, ice B, and recently discovered Ice-{\it i} were calculated under
459 < standard conditions for several common water models via thermodynamic
460 < integration. All the water models studied show Ice-{\it i} to be the
461 < minimum free energy crystal structure in the with a 9 \AA\ switching
462 < function cutoff. Calculated melting and boiling points show
463 < surprisingly good agreement with the experimental values; however, the
464 < solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
465 < interaction truncation was investigated through variation of the
466 < cutoff radius, use of a reaction field parameterized model, and
458 > $I$, ice B, and our recently discovered Ice-{\it i} structure were
459 > calculated under standard conditions for several common water models
460 > via thermodynamic integration. All the water models studied show
461 > Ice-{\it i} to be the minimum free energy crystal structure with a 9
462 > \AA\ switching function cutoff. Calculated melting and boiling points
463 > show surprisingly good agreement with the experimental values;
464 > however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The
465 > effect of interaction truncation was investigated through variation of
466 > the cutoff radius, use of a reaction field parameterized model, and
467   estimation of the results in the presence of the Ewald
468   summation. Interaction truncation has a significant effect on the
469   computed free energy values, and may significantly alter the free
# Line 449 | Line 471 | Due to this relative stability of Ice-{\it i} in all m
471   these effects, these results show Ice-{\it i} to be an important ice
472   polymorph that should be considered in simulation studies.
473  
474 < Due to this relative stability of Ice-{\it i} in all manner of
475 < investigated simulation examples, the question arises as to possible
474 > Due to this relative stability of Ice-{\it i} in all of the
475 > investigated simulation conditions, the question arises as to possible
476   experimental observation of this polymorph.  The rather extensive past
477   and current experimental investigation of water in the low pressure
478   regime makes us hesitant to ascribe any relevance of this work outside
# Line 460 | Line 482 | non-polar molecules.  Fig. \ref{fig:gofr} contains our
482   most ideal situation for possible observation. These include the
483   negative pressure or stretched solid regime, small clusters in vacuum
484   deposition environments, and in clathrate structures involving small
485 < non-polar molecules.  Fig. \ref{fig:gofr} contains our predictions
486 < of both the pair distribution function ($g_{OO}(r)$) and the structure
487 < factor ($S(\vec{q})$ for this polymorph at a temperature of 77K.  We
488 < will leave it to our experimental colleagues to determine whether this
489 < ice polymorph should really be called Ice-{\it i} or if it should be
490 < promoted to Ice-0.
485 > non-polar molecules.  Figs. \ref{fig:gofr} and \ref{fig:sofq} contain
486 > our predictions for both the pair distribution function ($g_{OO}(r)$)
487 > and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it
488 > i} at a temperature of 77K.  In our initial comparison of the
489 > predicted S(q) for Ice-{\it i} and experimental studies of amorphous
490 > solid water, it is possible that some of the ``spurious'' peaks that
491 > could not be assigned in an early report on high-density amorphous
492 > (HDA) ice could correspond to peaks labeled in this
493 > S(q).\cite{Bizid87} It should be noted that there is typically poor
494 > agreement on crystal densities between simulation and experiment, so
495 > such peak comparisons should be made with caution.  We will leave it
496 > to our experimental colleagues to make the final determination on
497 > whether this ice polymorph is named appropriately (i.e. with an
498 > imaginary number) or if it can be promoted to Ice-0.
499  
500   \begin{figure}
501   \includegraphics[width=\linewidth]{iceGofr.eps}
502 < \caption{Radial distribution functions of (A) Ice-{\it i} and (B) ice $I_c$ at 77 K from simulations of the SSD/RF water model.}
502 > \caption{Radial distribution functions of Ice-{\it i} and ice $I_c$
503 > calculated from from simulations of the SSD/RF water model at 77 K.}
504   \label{fig:gofr}
505   \end{figure}
506  
507 + \begin{figure}
508 + \includegraphics[width=\linewidth]{sofq.eps}
509 + \caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at
510 + 77 K.  The raw structure factors have been convoluted with a gaussian
511 + instrument function (0.075 \AA$^{-1}$ width) to compensate for the
512 + trunction effects in our finite size simulations. The labeled peaks
513 + compared favorably with ``spurious'' peaks observed in experimental
514 + studies of amorphous solid water.\cite{Bizid87}}
515 + \label{fig:sofq}
516 + \end{figure}
517 +
518   \section{Acknowledgments}
519   Support for this project was provided by the National Science
520   Foundation under grant CHE-0134881. Computation time was provided by

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