--- trunk/iceiPaper/iceiPaper.tex	2004/09/15 21:11:19	1463
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/16 21:15:38	1468
@@ -20,7 +20,8 @@
 
 \begin{document}
 
-\title{Ice-{\it i}: a novel ice polymorph predicted via computer simulation}
+\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more
+stable than Ice $I_h$ for point-charge and point-dipole water models}
 
 \author{Christopher J. Fennell and J. Daniel Gezelter \\
 Department of Chemistry and Biochemistry\\ University of Notre Dame\\
@@ -70,7 +71,7 @@ effect.\cite{evenMorePapers} With the choice of models
 conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
 These models have been used to investigate important physical
 phenomena like phase transitions and the hydrophobic
-effect.\cite{evenMorePapers} With the choice of models available, it
+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
@@ -84,20 +85,20 @@ In this paper, standard reference state methods were a
 point, because these phases will dictate the true transition
 temperatures and pressures for their respective model.
 
-In this paper, standard reference state methods were applied to the
-study of crystalline water polymorphs in the low pressure regime. This
-work is unique in the fact that one of the crystal lattices was
-arrived at through crystallization of a computationally efficient
-water model under constant pressure and temperature
-conditions. Crystallization events are interesting in and of
-themselves\cite{Matsumoto02,Yamada02}; however, the crystal structure
-obtained in this case was different from any previously observed ice
-polymorphs, in experiment or simulation.\cite{Fennell04} This crystal
-was termed Ice-{\it i} in homage to its origin in computational
+In this paper, standard reference state methods were applied to known
+crystalline water polymorphs in the low pressure regime. This work is
+unique in the fact that one of the crystal lattices was arrived at
+through crystallization of a computationally efficient water model
+under constant pressure and temperature conditions. Crystallization
+events are interesting in and of
+themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure
+obtained in this case is different from any previously observed ice
+polymorphs in experiment or simulation.\cite{Fennell04} We have named
+this structure Ice-{\it i} to indicate its origin in computational
 simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
 water molecules that stack in rows of interlocking water
 tetramers. Proton ordering can be accomplished by orienting two of the
-waters so that both of their donated hydrogen bonds are internal to
+molecules so that both of their donated hydrogen bonds are internal to
 their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
 constructed of water tetramers, the hydrogen bonds are not as linear
 as those observed in ice $I_h$, however the interlocking of these
@@ -109,7 +110,10 @@ 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-2{\it i}, 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-2{\it i}, $a = 1.7850c$.}
+\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the
+elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$
+relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a =
+1.7850c$.} 
 \label{iceiCell}
 \end{figure}
 
@@ -122,51 +126,50 @@ Results in the previous study indicated that Ice-{\it 
 \label{protOrder}
 \end{figure}
 
-Results in the previous study indicated that Ice-{\it i} is the
-minimum energy crystal structure for the single point water models
-being studied (for discussions on these single point dipole models,
-see the previous work and related
+Results from our previous study indicated that Ice-{\it i} is the
+minimum energy crystal structure for the single point water models we
+investigated (for discussions on these single point dipole models, see
+the previous work and related
 articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
-consider energetic stabilization and neglect entropic contributions to
-the overall free energy. To address this issue, the absolute free
-energy of this crystal was calculated using thermodynamic integration
-and compared to the free energies of cubic and hexagonal ice $I$ (the
-experimental low density ice polymorphs) and ice B (a higher density,
-but very stable crystal structure observed by B\`{a}ez and Clancy in
-free energy studies of SPC/E).\cite{Baez95b} This work includes
-results for the water model from which Ice-{\it i} was crystallized
-(soft sticky dipole extended, SSD/E) in addition to several common
-water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field
-parametrized single point dipole water model (soft sticky dipole
-reaction field, SSD/RF). In should be noted that a second version of
-Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E,
-TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B)
-is similar to the Ice-{\it i} unit it is extended in the direction of
-the (001) face and compressed along the other two faces.
+considered energetic stabilization and neglected entropic
+contributions to the overall free energy. To address this issue, the
+absolute free energy of this crystal was calculated using
+thermodynamic integration and compared to the free energies of cubic
+and hexagonal ice $I$ (the experimental low density ice polymorphs)
+and ice B (a higher density, but very stable crystal structure
+observed by B\`{a}ez and Clancy in free energy studies of
+SPC/E).\cite{Baez95b} This work includes results for the water model
+from which Ice-{\it i} was crystallized (SSD/E) in addition to several
+common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
+field parametrized single point dipole water model (SSD/RF). It should
+be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used
+in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of
+this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit
+it is extended in the direction of the (001) face and compressed along
+the other two faces.
 
 \section{Methods}
 
 Canonical ensemble (NVT) molecular dynamics calculations were
-performed using the OOPSE (Object-Oriented Parallel Simulation Engine)
-molecular mechanics package. All molecules were treated as rigid
-bodies, with orientational motion propagated using the symplectic DLM
-integration method. Details about the implementation of these
-techniques can be found in a recent publication.\cite{Meineke05} 
+performed using the OOPSE molecular mechanics package.\cite{Meineke05}
+All molecules were treated as rigid bodies, with orientational motion
+propagated using the symplectic DLM integration method. Details about
+the implementation of these techniques can be found in a recent
+publication.\cite{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.
+same technique in order to determine melting points and generate phase
+diagrams. All simulations were carried out at densities resulting in a
+pressure of approximately 1 atm at their respective temperatures.
 
 A single thermodynamic integration involves a sequence of simulations
 over which the system of interest is converted into a reference system
-for which the free energy is known. This transformation path is then
-integrated in order to determine the free energy difference between
-the two states:
+for which the free energy is known analytically. This transformation
+path is then integrated in order to determine the free energy
+difference between the two states:
 \begin{equation}
 \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda 
 )}{\partial\lambda}\right\rangle_\lambda d\lambda,
@@ -180,12 +183,11 @@ Crystal is chosen as the reference state that the syst
 (near the reference state) in length.
 
 For the thermodynamic integration of molecular crystals, the Einstein
-Crystal is chosen as the reference state that the system is converted
-to over the course of the simulation. In an Einstein Crystal, the
+crystal was chosen as the reference state. In an Einstein crystal, the
 molecules are harmonically restrained at their ideal lattice locations
 and orientations. The partition function for a molecular crystal
-restrained in this fashion has been evaluated, and the Helmholtz Free
-Energy ({\it A}) is given by
+restrained in this fashion can be evaluated analytically, and the
+Helmholtz Free Energy ({\it A}) is given by
 \begin{eqnarray}
 A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left
 [\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right
@@ -220,7 +222,7 @@ truncated, thereby avoiding poor energy conserving dyn
 Charge, dipole, and Lennard-Jones interactions were modified by a
 cubic switching between 100\% and 85\% of the cutoff value (9 \AA
 ). By applying this function, these interactions are smoothly
-truncated, thereby avoiding poor energy conserving dynamics resulting
+truncated, thereby avoiding the poor energy conservation which results
 from harsher truncation schemes. The effect of a long-range correction
 was also investigated on select model systems in a variety of
 manners. For the SSD/RF model, a reaction field with a fixed
@@ -231,11 +233,9 @@ mechanics software package.\cite{Tinker} TINKER was ch
 \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} TINKER was chosen because it
-can also propagate the motion of rigid-bodies, and provides the most
-direct comparison to the results from OOPSE. The calculated energy
+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 in the free energy
+previous results in order to predict changes to the free energy
 landscape.
 
 \section{Results and discussion}
@@ -246,14 +246,14 @@ Ice XI, the experimentally observed proton ordered var
 as well as the higher density ice B, observed by B\`{a}ez and Clancy
 and thought to be the minimum free energy structure for the SPC/E
 model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b}
-Ice XI, the experimentally observed proton ordered variant of ice
-$I_h$, was investigated initially, but it was found not to be as
-stable as antiferroelectric variants of proton ordered or even proton
-disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of
-ice $I_h$ used here is a simple antiferroelectric version that has an
-8 molecule unit cell. The crystals contained 648 or 1728 molecules for
-ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice
-$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes
+Ice XI, the experimentally-observed proton-ordered variant of ice
+$I_h$, was investigated initially, but was found to be not as stable
+as proton disordered or antiferroelectric variants of ice $I_h$. The
+proton ordered variant of ice $I_h$ used here is a simple
+antiferroelectric version that has an 8 molecule unit
+cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules
+for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for
+ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes
 were necessary for simulations involving larger cutoff values.
 
 \begin{table*}
@@ -263,17 +263,18 @@ kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF
 \caption{Calculated free energies for several ice polymorphs with a 
 variety of common water models. All calculations used a cutoff radius
 of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
-kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.}
+kcal/mol. Calculated error of the final digits is in parentheses. *Ice
+$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.}
 \begin{tabular}{ l  c  c  c  c }
 \hline 
-\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \  & \ \quad \ \ \ \ B \ \  & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\
+Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
 \hline 
-\ \quad \ TIP3P  & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\
-\ \quad \ TIP4P  & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\
-\ \quad \ TIP5P  & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\
-\ \quad \ SPC/E  & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\
-\ \quad \ SSD/E  & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\
-\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\
+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)\\
 \end{tabular}
 \label{freeEnergy}
 \end{center}
@@ -283,17 +284,17 @@ temperature and pressure dependence of the free energy
 The free energy values computed for the studied polymorphs indicate
 that Ice-{\it i} is the most stable state for all of the common water
 models studied. With the free energy at these state points, the
-temperature and pressure dependence of the free energy was used to
-project to other state points and build phase diagrams. Figures
+Gibbs-Helmholtz equation was used to project to other state points and
+to build phase diagrams.  Figures
 \ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
 from the free energy results. All other models have similar structure,
-only the crossing points between these phases exist at different
-temperatures and pressures. It is interesting to note that ice $I$
-does not exist in either cubic or hexagonal form in any of the phase
-diagrams for any of the models. For purposes of this study, ice B is
-representative of the dense ice polymorphs. A recent study by Sanz
-{\it et al.} goes into detail on the phase diagrams for SPC/E and
-TIP4P in the high pressure regime.\cite{Sanz04} 
+although the crossing points between the phases exist at slightly
+different temperatures and pressures. It is interesting to note that
+ice $I$ does not exist in either cubic or hexagonal form in any of the
+phase diagrams for any of the models. For purposes of this study, ice
+B is representative of the dense ice polymorphs. A recent study by
+Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and
+TIP4P in the high pressure regime.\cite{Sanz04}
 
 \begin{figure}
 \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
@@ -321,14 +322,17 @@ temperatures of several common water models compared w
 \renewcommand{\thefootnote}{\thempfootnote}
 \begin{center}
 \caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) 
-temperatures of several common water models compared with experiment.}
+temperatures at 1 atm for several common water models compared with
+experiment. The $T_m$ and $T_s$ values from simulation correspond to a
+transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the
+liquid or gas state.}
 \begin{tabular}{ l  c  c  c  c  c  c  c }
 \hline 
-\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\
+Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
 \hline 
-\ \ $T_m$ (K)  & \ \ 269 & \ \ 265 & \ \ 271 &  297 & \ \ - & \ \ 278 & \ \ 273\\
-\ \ $T_b$ (K)  & \ \ 357 & \ \ 354 & \ \ 337 &  396 & \ \ - & \ \ 349 & \ \ 373\\
-\ \ $T_s$ (K)  & \ \ - & \ \ - & \ \ - &  - & \ \ 355 & \ \ - & \ \ -\\
+$T_m$ (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) & - & -\\
 \end{tabular}
 \label{meltandboil}
 \end{center}
@@ -344,8 +348,9 @@ ordered and disordered molecular arrangements). If the
 studies in the literature. Earlier free energy studies of ice $I$
 using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences
 being attributed to choice of interaction truncation and different
-ordered and disordered molecular arrangements). If the presence of ice
-B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be
+ordered and disordered molecular
+arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and
+Ice-{\it i} were omitted, a $T_m$ value around 210 K would be
 predicted from this work. However, the $T_m$ from Ice-{\it i} is
 calculated at 265 K, significantly higher in temperature than the
 previous studies. Also of interest in these results is that SSD/E does
@@ -364,8 +369,8 @@ TIP3P, and (C) SSD/RF. Data points omitted include SSD
 \caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) 
 TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12
 \AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9
-\AA\. These crystals are unstable at 200 K and rapidly convert into a
-liquid. The connecting lines are qualitative visual aid.}
+\AA . These crystals are unstable at 200 K and rapidly convert into
+liquids. The connecting lines are qualitative visual aid.}
 \label{incCutoff}
 \end{figure}
 
@@ -399,7 +404,7 @@ narrowing is significant enough that it becomes less c
 preferred polymorph at ambient conditions for both the TIP3P and SPC/E
 water models; however, there is a narrowing of the free energy
 differences between the various solid forms. In the case of SPC/E this
-narrowing is significant enough that it becomes less clear cut that
+narrowing is significant enough that it becomes less clear that
 Ice-{\it i} is the most stable polymorph, and is possibly metastable
 with respect to ice B and possibly ice $I_c$. However, these results
 do not significantly alter the finding that the Ice-{\it i} polymorph
@@ -417,8 +422,8 @@ long-range interaction correction. Units are kcal/mol.
 \hline 
 \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
 \hline 
-\ \ TIP3P  & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\
-\ \ SPC/E  & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\
+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)\\
 \end{tabular}
 \label{pmeShift}
 \end{center}
@@ -428,7 +433,7 @@ $I$, ice B, and recently discovered Ice-{\it i} where 
 \section{Conclusions}
 
 The free energy for proton ordered variants of hexagonal and cubic ice
-$I$, ice B, and recently discovered Ice-{\it i} where calculated under
+$I$, ice B, and recently discovered Ice-{\it i} were calculated under
 standard conditions for several common water models via thermodynamic
 integration. All the water models studied show Ice-{\it i} to be the
 minimum free energy crystal structure in the with a 9 \AA\ switching
@@ -437,8 +442,8 @@ estimation of the results in the presence of the Ewald
 solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
 interaction truncation was investigated through variation of the
 cutoff radius, use of a reaction field parameterized model, and
-estimation of the results in the presence of the Ewald summation
-correction. Interaction truncation has a significant effect on the
+estimation of the results in the presence of the Ewald
+summation. Interaction truncation has a significant effect on the
 computed free energy values, and may significantly alter the free
 energy landscape for the more complex multipoint water models. Despite
 these effects, these results show Ice-{\it i} to be an important ice
@@ -446,16 +451,28 @@ experimental observation of this polymorph. The rather
 
 Due to this relative stability of Ice-{\it i} in all manner of
 investigated simulation examples, the question arises as to possible
-experimental observation of this polymorph. The rather extensive past
+experimental observation of this polymorph.  The rather extensive past
 and current experimental investigation of water in the low pressure
-regime leads the authors to be hesitant in ascribing relevance outside
-of computational models, hence the descriptive name presented. That
-being said, there are certain experimental conditions that would
-provide the most ideal situation for possible observation. These
-include the negative pressure or stretched solid regime, small
-clusters in vacuum deposition environments, and in clathrate
-structures involving small non-polar molecules.
+regime makes us hesitant to ascribe any relevance of this work outside
+of the simulation community.  It is for this reason that we chose a
+name for this polymorph which involves an imaginary quantity.  That
+said, there are certain experimental conditions that would provide the
+most ideal situation for possible observation. These include the
+negative pressure or stretched solid regime, small clusters in vacuum
+deposition environments, and in clathrate structures involving small
+non-polar molecules.  Fig. \ref{fig:gofr} contains our predictions
+of both the pair distribution function ($g_{OO}(r)$) and the structure
+factor ($S(\vec{q})$ for this polymorph at a temperature of 77K.  We
+will leave it to our experimental colleagues to determine whether this
+ice polymorph should really be called Ice-{\it i} or if it should be
+promoted to Ice-0.
 
+\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.}
+\label{fig:gofr}
+\end{figure}
+
 \section{Acknowledgments}
 Support for this project was provided by the National Science
 Foundation under grant CHE-0134881. Computation time was provided by