--- trunk/iceiPaper/iceiPaper.tex	2004/09/16 21:38:10	1469
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/21 20:17:12	1488
@@ -33,17 +33,21 @@ The free energies of several ice polymorphs in the low
 %\doublespacing
 
 \begin{abstract}
-The free energies of several ice polymorphs in the low pressure regime
-were calculated using thermodynamic integration.  These integrations
-were done for most of the common water models. Ice-{\it i}, a
-structure we recently observed to be stable in one of the single-point
-water models, was determined to be the stable crystalline state (at 1
-atm) for {\it all} the water models investigated.  Phase diagrams were
+The absolute free energies of several ice polymorphs which are stable
+at low pressures were calculated using thermodynamic integration to a
+reference system (the Einstein crystal).  These integrations were
+performed for most of the common water models (SPC/E, TIP3P, TIP4P,
+TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed
+crystallizing at room temperature for 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.
+known low-pressure ice structures under all of these water models.
+Additionally, potential truncation was shown to have an effect on the
+calculated free energies, and can result in altered free energy
+landscapes.  Structure factor predictions for the new crystal were
+generated and we await experimental confirmation of the existence of
+this new polymorph.
 \end{abstract}
 
 %\narrowtext 
@@ -54,55 +58,46 @@ Molecular dynamics is a valuable tool for studying the
 
 \section{Introduction}
 
-Molecular dynamics is a valuable tool for studying the phase behavior
-of systems ranging from small or simple
-molecules\cite{Matsumoto02,andOthers} 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,
-leading to better understanding and depiction of physical processes
-and intricate molecular systems.
-
 Water has proven to be a challenging substance to depict in
 simulations, and a variety of models have been developed to describe
 its behavior under varying simulation
-conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
+conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,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.
+phenomena like phase transitions, transport properties, and the
+hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the
+choice of models available, it is only natural to compare the models
+under interesting thermodynamic conditions in an attempt to clarify
+the limitations of each of the
+models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two
+important properties 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 relevant transition temperatures and
+pressures for the model.
 
 In this paper, standard reference state methods were applied to known
-crystalline water polymorphs in the low pressure regime. This work is
-unique in the fact that one of the crystal lattices was arrived at
-through crystallization of a computationally efficient water model
-under constant pressure and temperature conditions. Crystallization
-events are interesting in and of
-themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure
-obtained in this case is different from any previously observed ice
-polymorphs in experiment or simulation.\cite{Fennell04} We have named
-this structure Ice-{\it i} to indicate its origin in computational
-simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
-water molecules that stack in rows of interlocking water
-tetramers. Proton ordering can be accomplished by orienting two of the
-molecules so that both of their donated hydrogen bonds are internal to
-their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
-constructed of water tetramers, the hydrogen bonds are not as linear
-as those observed in ice $I_h$, however the interlocking of these
-subunits appears to provide significant stabilization to the overall
+crystalline water polymorphs in the low pressure regime.  This work is
+unique in that one of the crystal lattices was arrived at through
+crystallization of a computationally efficient water model under
+constant pressure and temperature conditions. Crystallization events
+are interesting in and of themselves;\cite{Matsumoto02,Yamada02}
+however, the crystal structure obtained in this case is different from
+any previously observed ice polymorphs in experiment or
+simulation.\cite{Fennell04} We have named this structure Ice-{\it i}
+to indicate its origin in computational simulation. The unit cell
+(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in
+rows of interlocking water tetramers. Proton ordering can be
+accomplished by orienting two of the molecules so that both of their
+donated hydrogen bonds are internal to their tetramer
+(Fig. \ref{protOrder}). As expected in an ice crystal constructed of
+water tetramers, the hydrogen bonds are not as linear as those
+observed in ice $I_h$, however the interlocking of these subunits
+appears to provide significant stabilization to the overall
 crystal. The arrangement of these tetramers results in surrounding
 open octagonal cavities that are typically greater than 6.3 \AA\ in
 diameter. This relatively open overall structure leads to crystals
@@ -110,10 +105,11 @@ that are 0.07 g/cm$^3$ less dense on average than ice 
 
 \begin{figure}
 \includegraphics[width=\linewidth]{unitCell.eps}
-\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the
-elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$
-relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a =
-1.7850c$.} 
+\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, 
+the elongated variant of Ice-{\it i}.  The spheres represent the
+center-of-mass locations of the water molecules.  The $a$ to $c$
+ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by
+$a:2.1214c$ and $a:1.7850c$ respectively.}
 \label{iceiCell}
 \end{figure}
 
@@ -128,12 +124,12 @@ investigated (for discussions on these single point di
 
 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,Liu96,Bratko85}). Those results only
+had investigated (for discussions on these single point dipole models,
+see our previous work and related
+articles).\cite{Fennell04,Liu96,Bratko85} Those results only
 considered energetic stabilization and neglected entropic
-contributions to the overall free energy. To address this issue, the
-absolute free energy of this crystal was calculated using
+contributions to the overall free energy. To address this issue, we
+have calculated the absolute free energy of this crystal 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
@@ -142,11 +138,16 @@ be noted that a second version of Ice-{\it i} (Ice-$i^
 from which Ice-{\it i} was crystallized (SSD/E) in addition to several
 common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
 field parametrized single point dipole water model (SSD/RF). It should
-be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used
-in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of
-this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit
-it is extended in the direction of the (001) face and compressed along
-the other two faces.
+be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$)
+was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit
+cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it
+i} unit it is extended in the direction of the (001) face and
+compressed along the other two faces.  There is typically a small
+distortion of proton ordered Ice-{\it i}$^\prime$ that converts the
+normally square tetramer into a rhombus with alternating approximately
+85 and 95 degree angles.  The degree of this distortion is model
+dependent and significant enough to split the tetramer diagonal
+location peak in the radial distribution function.
 
 \section{Methods}
 
@@ -154,38 +155,55 @@ the implementation of these techniques can be found in
 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
+the implementation of this technique can be found in a recent
 publication.\cite{Dullweber1997}
 
-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.
+Thermodynamic integration is an established technique for
+determination of free energies of condensed phases of
+materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This
+method, implemented in the same manner illustrated by B\`{a}ez and
+Clancy, was utilized to calculate the free energy of several ice
+crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and
+SSD/E water models.\cite{Baez95a} Liquid state free energies at 300
+and 400 K for all of these water models were also determined using
+this same technique in order to determine melting points and to
+generate phase diagrams. All simulations were carried out at densities
+which correspond to a pressure of approximately 1 atm at their
+respective temperatures.
 
-A single thermodynamic integration involves a sequence of simulations
-over which the system of interest is converted into a reference system
-for which the free energy is known analytically. This transformation
-path is then integrated in order to determine the free energy
-difference between the two states:
+Thermodynamic integration involves a sequence of simulations during
+which the system of interest is converted into a reference system for
+which the free energy is known analytically. This transformation path
+is then integrated in order to determine the free energy difference
+between the two states:
 \begin{equation}
 \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda 
 )}{\partial\lambda}\right\rangle_\lambda d\lambda,
 \end{equation}
 where $V$ is the interaction potential and $\lambda$ is the
 transformation parameter that scales the overall
-potential. Simulations are distributed unevenly along this path in
-order to sufficiently sample the regions of greatest change in the
+potential. Simulations are distributed strategically along this path
+in order to sufficiently sample the regions of greatest change in the
 potential. Typical integrations in this study consisted of $\sim$25
 simulations ranging from 300 ps (for the unaltered system) to 75 ps
 (near the reference state) in length.
 
 For the thermodynamic integration of molecular crystals, the Einstein
-crystal was chosen as the reference state. In an Einstein crystal, the
-molecules are harmonically restrained at their ideal lattice locations
-and orientations. The partition function for a molecular crystal
+crystal was chosen as the reference system. In an Einstein crystal,
+the molecules are restrained at their ideal lattice locations and
+orientations. Using harmonic restraints, as applied by B\`{a}ez and
+Clancy, the total potential for this reference crystal
+($V_\mathrm{EC}$) is the sum of all the harmonic restraints,
+\begin{equation}
+V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} +
+\frac{K_\omega\omega^2}{2},
+\end{equation}
+where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are
+the spring constants restraining translational motion and deflection
+of and rotation around the principle axis of the molecule
+respectively.  It is clear from Fig. \ref{waterSpring} that the values
+of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from
+$-\pi$ to $\pi$.  The partition function for a molecular crystal
 restrained in this fashion can be evaluated analytically, and the
 Helmholtz Free Energy ({\it A}) is given by
 \begin{eqnarray}
@@ -199,14 +217,8 @@ where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a
 )^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
 \label{ecFreeEnergy}
 \end{eqnarray}
-where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation
-\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and
-$K_\mathrm{\omega}$ are the spring constants restraining translational
-motion and deflection of and rotation around the principle axis of the
-molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the
-minimum potential energy of the ideal crystal. In the case of
-molecular liquids, the ideal vapor is chosen as the target reference
-state.
+where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum
+potential energy of the ideal crystal.\cite{Baez95a}
 
 \begin{figure}
 \includegraphics[width=\linewidth]{rotSpring.eps}
@@ -219,6 +231,19 @@ Charge, dipole, and Lennard-Jones interactions were mo
 \label{waterSpring}
 \end{figure}
 
+In the case of molecular liquids, the ideal vapor is chosen as the
+target reference state.  There are several examples of liquid state
+free energy calculations of water models present in the
+literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods
+typically differ in regard to the path taken for switching off the
+interaction potential to convert the system to an ideal gas of water
+molecules.  In this study, we applied of one of the most convenient
+methods and integrated over the $\lambda^4$ path, where all
+interaction parameters are scaled equally by this transformation
+parameter.  This method has been shown to be reversible and provide
+results in excellent agreement with other established
+methods.\cite{Baez95b}
+
 Charge, dipole, and Lennard-Jones interactions were modified by a
 cubic switching between 100\% and 85\% of the cutoff value (9 \AA
 ). By applying this function, these interactions are smoothly
@@ -230,17 +255,17 @@ performed with longer cutoffs of 12 and 15 \AA\ to com
 simulations.\cite{Onsager36} For a series of the least computationally
 expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
 performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
-\AA\ cutoff results. Finally, results from the use of an Ewald
-summation were estimated for TIP3P and SPC/E by performing
-calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
-mechanics software package.\cite{Tinker} The calculated energy
-difference in the presence and absence of PME was applied to the
-previous results in order to predict changes to the free energy
-landscape.
+\AA\ cutoff results. Finally, the effects of utilizing an Ewald
+summation were estimated for TIP3P and SPC/E by performing single
+configuration calculations with Particle-Mesh Ewald (PME) in the
+TINKER molecular mechanics software package.\cite{Tinker} The
+calculated energy difference in the presence and absence of PME was
+applied to the previous results in order to predict changes to the
+free energy landscape.
 
 \section{Results and discussion}
 
-The free energy of proton ordered Ice-{\it i} was calculated and
+The free energy of proton-ordered Ice-{\it i} was calculated and
 compared with the free energies of proton ordered variants of the
 experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
 as well as the higher density ice B, observed by B\`{a}ez and Clancy
@@ -250,11 +275,13 @@ antiferroelectric version that has an 8 molecule unit
 $I_h$, 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.
+antiferroelectric version that we devised, and it has an 8 molecule
+unit cell similar to other predicted antiferroelectric $I_h$
+crystals.\cite{Davidson84} The crystals contained 648 or 1728
+molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000
+molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger
+crystal sizes were necessary for simulations involving larger cutoff
+values.
 
 \begin{table*}
 \begin{minipage}{\linewidth}
@@ -269,12 +296,12 @@ TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\
 \hline 
 Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
 \hline 
-TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\
-TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\
-TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\
-SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\
-SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\
-SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\
+TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\
+TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\
+TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\
+SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\
+SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\
+SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\
 \end{tabular}
 \label{freeEnergy}
 \end{center}
@@ -283,18 +310,18 @@ models studied. With the free energy at these state po
 
 The free energy values computed for the studied polymorphs indicate
 that Ice-{\it i} is the most stable state for all of the common water
-models studied. With the free energy at these state points, the
-Gibbs-Helmholtz equation was used to project to other state points and
-to build phase diagrams.  Figures
+models studied. With the calculated free energy at these state points,
+the Gibbs-Helmholtz equation was used to project to other state points
+and to build phase diagrams.  Figures
 \ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
 from the free energy results. All other models have similar structure,
-although the crossing points between the phases exist at slightly
+although the crossing points between the phases move to 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}
+TIP4P at higher pressures than those studied here.\cite{Sanz04}
 
 \begin{figure}
 \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
@@ -330,9 +357,9 @@ $T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - & 
 \hline 
 Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
 \hline 
-$T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\
-$T_b$ (K)  & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\
-$T_s$ (K)  & - & - & - & - & 355(3) & - & -\\
+$T_m$ (K)  & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\
+$T_b$ (K)  & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\
+$T_s$ (K)  & - & - & - & - & 355(2) & - & -\\
 \end{tabular}
 \label{meltandboil}
 \end{center}
@@ -357,11 +384,11 @@ conditions. While troubling, this behavior turned out 
 not exhibit a melting point at 1 atm, but it shows a sublimation point
 at 355 K. This is due to the significant stability of Ice-{\it i} over
 all other polymorphs for this particular model under these
-conditions. While troubling, this behavior turned out to be
-advantageous in that it facilitated the spontaneous crystallization of
-Ice-{\it i}. These observations provide a warning that simulations of
+conditions. While troubling, this behavior resulted in spontaneous
+crystallization of Ice-{\it i} and led us to investigate this
+structure. These observations provide a warning that simulations of
 SSD/E as a ``liquid'' near 300 K are actually metastable and run the
-risk of spontaneous crystallization. However, this risk changes when
+risk of spontaneous crystallization. However, this risk lessens when
 applying a longer cutoff.
 
 \begin{figure}
@@ -395,21 +422,21 @@ energy of identical crystals with and without PME usin
 was estimated by applying the potential energy difference do to its
 inclusion in systems in the presence and absence of the
 correction. This was accomplished by calculation of the potential
-energy of identical crystals with and without PME using TINKER. The
-free energies for the investigated polymorphs using the TIP3P and
-SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
-are not fully supported in TINKER, so the results for these models
-could not be estimated. The same trend pointed out through increase of
-cutoff radius is observed in these PME results. Ice-{\it i} is the
-preferred polymorph at ambient conditions for both the TIP3P and SPC/E
-water models; however, 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 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
-is a stable crystal structure that should be considered when studying
-the phase behavior of water models.
+energy of identical crystals both with and without PME. The free
+energies for the investigated polymorphs using the TIP3P and SPC/E
+water models are shown in Table \ref{pmeShift}. The same trend pointed
+out through increase of cutoff radius is observed in these PME
+results. Ice-{\it i} is the preferred polymorph at ambient conditions
+for both the TIP3P and SPC/E water models; however, the narrowing of
+the free energy differences between the various solid forms is
+significant enough that it becomes less clear that it is the most
+stable polymorph with the SPC/E model.  The free energies of Ice-{\it
+i} and ice B nearly overlap within error, with ice $I_c$ just outside
+as well, indicating that Ice-{\it i} might be metastable with respect
+to ice B and possibly ice $I_c$ with SPC/E. However, these results do
+not significantly alter the finding that the Ice-{\it i} polymorph is
+a stable crystal structure that should be considered when studying the
+phase behavior of water models.
 
 \begin{table*}
 \begin{minipage}{\linewidth}
@@ -422,8 +449,8 @@ TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)
 \hline 
 \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
 \hline 
-TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\
-SPC/E  & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\
+TIP3P  & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\
+SPC/E  & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\
 \end{tabular}
 \label{pmeShift}
 \end{center}
@@ -433,15 +460,15 @@ $I$, ice B, and recently discovered Ice-{\it i} were c
 \section{Conclusions}
 
 The free energy for proton ordered variants of hexagonal and cubic ice
-$I$, ice B, and recently discovered Ice-{\it i} were calculated under
-standard conditions for several common water models via thermodynamic
-integration. All the water models studied show Ice-{\it i} to be the
-minimum free energy crystal structure in the with a 9 \AA\ switching
-function cutoff. Calculated melting and boiling points show
-surprisingly good agreement with the experimental values; however, the
-solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
-interaction truncation was investigated through variation of the
-cutoff radius, use of a reaction field parameterized model, and
+$I$, ice B, and our recently discovered Ice-{\it i} structure 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 with a 9
+\AA\ switching function cutoff. Calculated melting and boiling points
+show surprisingly good agreement with the experimental values;
+however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The
+effect of interaction truncation was investigated through variation of
+the cutoff radius, use of a reaction field parameterized model, and
 estimation of the results in the presence of the Ewald
 summation. Interaction truncation has a significant effect on the
 computed free energy values, and may significantly alter the free
@@ -449,8 +476,8 @@ Due to this relative stability of Ice-{\it i} in all m
 these effects, these results show Ice-{\it i} to be an important ice
 polymorph that should be considered in simulation studies.
 
-Due to this relative stability of Ice-{\it i} in all manner of
-investigated simulation examples, the question arises as to possible
+Due to this relative stability of Ice-{\it i} in all of the
+investigated simulation conditions, the question arises as to possible
 experimental observation of this polymorph.  The rather extensive past
 and current experimental investigation of water in the low pressure
 regime makes us hesitant to ascribe any relevance of this work outside
@@ -462,23 +489,30 @@ and the structure factor ($S(\vec{q})$ for ice $I_c$ a
 deposition environments, and in clathrate structures involving small
 non-polar molecules.  Figs. \ref{fig:gofr} and \ref{fig:sofq} contain
 our predictions for both the pair distribution function ($g_{OO}(r)$)
-and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it
-i} at a temperature of 77K.  We will leave it to our experimental
-colleagues to determine whether this ice polymorph is named
-appropriately or if it should be promoted to Ice-0.
+and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for
+ice-{\it i} at a temperature of 77K.  In studies of the high and low
+density forms of amorphous ice, ``spurious'' diffraction peaks have
+been observed experimentally.\cite{Bizid87} It is possible that a
+variant of Ice-{\it i} could explain some of this behavior; however,
+we will leave it to our experimental colleagues to make the final
+determination on whether this ice polymorph is named appropriately
+(i.e. with an imaginary number) or if it can be promoted to Ice-0.
 
 \begin{figure}
 \includegraphics[width=\linewidth]{iceGofr.eps}
-\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.}
+\caption{Radial distribution functions of ice $I_h$, $I_c$, 
+Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations
+of the SSD/RF water model at 77 K.}
 \label{fig:gofr}
 \end{figure}
 
 \begin{figure}
 \includegraphics[width=\linewidth]{sofq.eps}
-\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at
-77 K.  The raw structure factors have been convoluted with a gaussian
-instrument function (0.075 \AA$^{-1}$ width) to compensate
-for the trunction effects in our finite size simulations.}
+\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i},
+ and Ice-{\it i}$^\prime$ at 77 K.  The raw structure factors have
+ been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$
+ width) to compensate for the trunction effects in our finite size
+ simulations.}
 \label{fig:sofq}
 \end{figure}
 
