--- trunk/iceiPaper/iceiPaper.tex	2004/09/21 20:17:12	1488
+++ trunk/iceiPaper/iceiPaper.tex	2005/03/24 19:35:08	2133
@@ -1,10 +1,10 @@
 %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
-\documentclass[11pt]{article}
-%\documentclass[11pt]{article}
+\documentclass[12pt]{article}
 \usepackage{endfloat}
 \usepackage{amsmath}
 \usepackage{epsf}
-\usepackage{berkeley}
+\usepackage{times}
+\usepackage{mathptm}
 \usepackage{setspace}
 \usepackage{tabularx}
 \usepackage{graphicx}
@@ -20,11 +20,12 @@
 
 \begin{document}
 
-\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}
+\title{Computational free energy studies of a new ice polymorph which
+exhibits greater stability than Ice I$_h$}
 
 \author{Christopher J. Fennell and J. Daniel Gezelter \\
-Department of Chemistry and Biochemistry\\ University of Notre Dame\\
+Department of Chemistry and Biochemistry\\ 
+University of Notre Dame\\
 Notre Dame, Indiana 46556}
 
 \date{\today}
@@ -33,21 +34,16 @@ The absolute free energies of several ice polymorphs w
 %\doublespacing
 
 \begin{abstract}
-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 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.
+The absolute free energies of several ice polymorphs were calculated
+using thermodynamic integration.  These polymorphs are predicted by
+computer simulations using a variety of common water models to be
+stable at low pressures.  A recently discovered ice polymorph that has
+as yet {\it only} been observed in computer simulations (Ice-{\it i}),
+was determined to be the stable crystalline state 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.  Additionally, potential truncation was shown to play
+a role in the resulting shape of the free energy landscape.
 \end{abstract}
 
 %\narrowtext 
@@ -67,133 +63,126 @@ the limitations of each of 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
+the limitations of
+each.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two important
+properties to quantify are the Gibbs and Helmholtz free energies,
+particularly for the solid forms of water as these predict the
+thermodynamic stability of the various phases.  Water has a
+particularly rich phase diagram and takes on a number of different and
+stable crystalline structures as the temperature and pressure are
+varied.  It is a challenging task to investigate the entire free
+energy landscape\cite{Sanz04}; and 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.
+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 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
-that are 0.07 g/cm$^3$ less dense on average than ice $I_h$.
+The high-pressure phases of water (ice II - ice X as well as ice XII)
+have been studied extensively both experimentally and
+computationally. In this paper, standard reference state methods were
+applied in the {\it low} pressure regime to evaluate the free energies
+for a few known crystalline water polymorphs that might be stable at
+these pressures.  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 of Ice-{\it i} and an axially-elongated
+variant named Ice-{\it i}$^\prime$ both consist of eight water
+molecules that stack in rows of interlocking water tetramers as
+illustrated in figures \ref{unitcell}A and \ref{unitcell}B.  These
+tetramers form a crystal structure similar in appearance to a recent
+two-dimensional surface tessellation simulated on silica.\cite{Yang04}
+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 octagonal cavities that are typically
+greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}).  This open
+structure leads to crystals that are typically 0.07 g/cm$^3$ less
+dense than ice I$_h$.
 
 \begin{figure}
+\centering
 \includegraphics[width=\linewidth]{unitCell.eps}
-\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}
+\caption{(A) Unit cells for Ice-{\it i} and (B) Ice-{\it i}$^\prime$.  
+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 2.1214 and 1.785 respectively.}
+\label{unitcell}
 \end{figure}
 
 \begin{figure}
+\centering
 \includegraphics[width=\linewidth]{orderedIcei.eps}
-\caption{Image of a proton ordered crystal of Ice-{\it i} looking 
-down the (001) crystal face. The rows of water tetramers surrounded by
-octagonal pores leads to a crystal structure that is significantly
-less dense than ice $I_h$.}
-\label{protOrder}
+\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking
+down the (001) crystal face.  The presence of large octagonal pores
+leads to a polymorph that is less dense than ice I$_h$.}
+\label{iCrystal}
 \end{figure}
 
 Results from our previous study indicated that Ice-{\it i} is the
-minimum energy crystal structure for the single point water models we
-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, we
+minimum energy crystal structure for the single point water models
+investigated (for discussions on these single point dipole models, see
+our previous work and related
+articles).\cite{Fennell04,Liu96,Bratko85} Our earlier results
+considered only energetic stabilization and neglected entropic
+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
-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-{\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.
+thermodynamic integration and compared it to the free energies of ice
+I$_c$ and ice I$_h$ (the common 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).  The axially-elongated
+variant, Ice-{\it i}$^\prime$, was used in calculations involving
+SPC/E, TIP4P, and TIP5P.  The square tetramers in Ice-{\it i} distort
+in Ice-{\it i}$^\prime$ to form a rhombus with alternating 85 and 95
+degree angles.  Under SPC/E, TIP4P, and TIP5P, this geometry is better
+at forming favorable hydrogen bonds.  The degree of rhomboid
+distortion depends on the water model used, but is significant enough
+to split a peak in the radial distribution function which corresponds
+to diagonal sites in the tetramers.
 
 \section{Methods}
 
 Canonical ensemble (NVT) molecular dynamics calculations were
-performed using the OOPSE molecular mechanics package.\cite{Meineke05}
+performed using the OOPSE molecular mechanics program.\cite{Meineke05}
 All molecules were treated as rigid bodies, with orientational motion
-propagated using the symplectic DLM integration method. Details about
+propagated using the symplectic DLM integration method.  Details about
 the implementation of this technique can be found in a recent
 publication.\cite{Dullweber1997}
 
-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.
-
-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:
+Thermodynamic integration was utilized to calculate the Helmholtz free
+energies ($A$) of the listed water models at various state points
+using the OOPSE molecular dynamics program.\cite{Meineke05}
+Thermodynamic integration is an established technique that has been
+used extensively in the calculation of free energies for condensed
+phases of
+materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}.  This
+method uses a sequence of simulations during which the system of
+interest is converted into a reference system for which the free
+energy is known analytically ($A_0$).  The difference in potential
+energy between the reference system and the system of interest
+($\Delta V$) 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,
+ A = A_0 + \int_0^1 \left\langle \Delta V \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 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.
+Here, $\lambda$ is the parameter that governs the transformation
+between the reference system and the system of interest.  For
+crystalline phases, an harmonically-restrained (Einsten) crystal is
+chosen as the reference state, while for liquid phases, the ideal gas
+is taken as the reference state.  
 
-For the thermodynamic integration of molecular crystals, the Einstein
-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,
+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},
@@ -201,11 +190,16 @@ respectively.  It is clear from Fig. \ref{waterSpring}
 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
+respectively.  These spring constants are typically calculated from
+the mean-square displacements of water molecules in an unrestrained
+ice crystal at 200 K.  For these studies, $K_\mathrm{v} = 4.29$ kcal
+mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$,
+and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$.  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}
 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
@@ -221,11 +215,12 @@ potential energy of the ideal crystal.\cite{Baez95a}
 potential energy of the ideal crystal.\cite{Baez95a}
 
 \begin{figure}
-\includegraphics[width=\linewidth]{rotSpring.eps}
+\centering
+\includegraphics[width=4in]{rotSpring.eps}
 \caption{Possible orientational motions for a restrained molecule. 
 $\theta$ angles correspond to displacement from the body-frame {\it
 z}-axis, while $\omega$ angles correspond to rotation about the
-body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring
+body-frame {\it z}-axis.  $K_\theta$ and $K_\omega$ are spring
 constants for the harmonic springs restraining motion in the $\theta$
 and $\omega$ directions.}
 \label{waterSpring}
@@ -237,278 +232,229 @@ molecules.  In this study, we applied of one of the mo
 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
+molecules.  In this study, we applied 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
+Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and
+Lennard-Jones interactions were gradually reduced by a cubic switching
+function.  By applying this function, these interactions are smoothly
 truncated, thereby avoiding the poor energy conservation which results
-from harsher truncation schemes. The effect of a long-range correction
-was also investigated on select model systems in a variety of
-manners. For the SSD/RF model, a reaction field with a fixed
+from harsher truncation schemes.  The effect of a long-range
+correction was also investigated on select model systems in a variety
+of manners.  For the SSD/RF model, a reaction field with a fixed
 dielectric constant of 80 was applied in all
 simulations.\cite{Onsager36} For a series of the least computationally
-expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
-performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
-\AA\ cutoff results. Finally, 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
+expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were
+performed with longer cutoffs of 10.5, 12, 13.5, and 15 \AA\ to
+compare with the 9 \AA\ cutoff results.  Finally, the effects of using
+the Ewald summation were estimated for TIP3P and SPC/E by performing
+single configuration Particle-Mesh Ewald (PME)
+calculations~\cite{Tinker} for each of the ice polymorphs.  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}
+\section{Results and Discussion}
 
-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
-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 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 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.
+The calculated free energies of proton-ordered variants of three low
+density polymorphs (I$_h$, I$_c$, and Ice-{\it i} or Ice-{\it
+i}$^\prime$) and the stable higher density ice B are listed in Table
+\ref{freeEnergy}.  Ice B was included because it has been 
+shown to be a minimum free energy structure for SPC/E at ambient
+conditions.\cite{Baez95b} In addition to the free energies, the
+relevant transition temperatures at standard pressure are also
+displayed in Table \ref{freeEnergy}.  These free energy values
+indicate that Ice-{\it i} is the most stable state for all of the
+investigated water models.  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.  Figure \ref{tp3PhaseDia} is
+an example diagram built from the results for the TIP3P water model.
+All other models have similar structure, although the crossing points
+between the phases move to different temperatures and pressures as
+indicated from the transition temperatures in Table \ref{freeEnergy}.
+It is interesting to note that ice I$_h$ (and ice I$_c$ for that
+matter) do not appear 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.} provides
+details on the phase diagrams for SPC/E and TIP4P at higher pressures
+than those studied here.\cite{Sanz04}
 
 \begin{table*}
 \begin{minipage}{\linewidth}
-\renewcommand{\thefootnote}{\thempfootnote}
 \begin{center}
-\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$1 atm. Units are
-kcal/mol. Calculated error of the final digits is in parentheses. *Ice
-$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.}
-\begin{tabular}{ l  c  c  c  c }
+\caption{Calculated free energies for several ice polymorphs along 
+with the calculated melting (or sublimation) and boiling points for
+the investigated water models.  All free energy calculations used a
+cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm.
+Units of free energy are kcal/mol, while transition temperature are in
+Kelvin.  Calculated error of the final digits is in parentheses.}
+\begin{tabular}{lccccccc}
 \hline 
-Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
+Water Model & I$_h$ & I$_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\
 \hline 
-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)\\
+TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\
+TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\
+TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\
+SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\
+SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\
+SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 287(4) & 382(2)\\
 \end{tabular}
 \label{freeEnergy}
 \end{center}
 \end{minipage}
 \end{table*}
 
-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 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 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 at higher pressures than those studied here.\cite{Sanz04}
-
 \begin{figure}
 \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
 \caption{Phase diagram for the TIP3P water model in the low pressure 
-regime. The displayed $T_m$ and $T_b$ values are good predictions of
+regime.  The displayed $T_m$ and $T_b$ values are good predictions of
 the experimental values; however, the solid phases shown are not the
-experimentally observed forms. Both cubic and hexagonal ice $I$ are
+experimentally observed forms.  Both cubic and hexagonal ice $I$ are
 higher in energy and don't appear in the phase diagram.}
-\label{tp3phasedia}
+\label{tp3PhaseDia}
 \end{figure}
 
-\begin{figure}
-\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps}
-\caption{Phase diagram for the SSD/RF water model in the low pressure 
-regime. Calculations producing these results were done under an
-applied reaction field. It is interesting to note that this
-computationally efficient model (over 3 times more efficient than
-TIP3P) exhibits phase behavior similar to the less computationally
-conservative charge based models.}
-\label{ssdrfphasedia}
-\end{figure}
-
-\begin{table*}
-\begin{minipage}{\linewidth}
-\renewcommand{\thefootnote}{\thempfootnote}
-\begin{center}
-\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) 
-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 & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
-\hline 
-$T_m$ (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}
-\end{minipage}
-\end{table*}
-
-Table \ref{meltandboil} lists the melting and boiling temperatures
-calculated from this work. Surprisingly, most of these models have
-melting points that compare quite favorably with experiment. The
-unfortunate aspect of this result is that this phase change occurs
-between Ice-{\it i} and the liquid state rather than ice $I_h$ and the
-liquid state. These results are actually not contrary to previous
-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
+Most of the water models have melting points that compare quite
+favorably with the experimental value of 273 K.  The unfortunate
+aspect of this result is that this phase change occurs between
+Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid
+state.  These results do not contradict other studies.  Studies of ice
+I$_h$ 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).\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
-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 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 lessens when
-applying a longer cutoff.
+predicted from this work.  However, the $T_m$ from Ice-{\it i} is
+calculated to be 265 K, indicating that these simulation based
+structures ought to be included in studies probing phase transitions
+with this model.  Also of interest in these results is that SSD/E does
+not exhibit a melting point at 1 atm but does sublime 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 resulted in the spontaneous crystallization
+of Ice-{\it i} which 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, when a longer cutoff radius is
+used, SSD/E prefers the liquid state under standard temperature and
+pressure.
 
 \begin{figure}
 \includegraphics[width=\linewidth]{cutoffChange.eps}
-\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
-liquids. The connecting lines are qualitative visual aid.}
+\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, 
+SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models
+with an added Ewald correction term.  Error for the larger cutoff
+points is equivalent to that observed at 9.0\AA\ (see Table
+\ref{freeEnergy}).  Data for ice I$_c$ with TIP3P using both 12 and
+13.5 \AA\ cutoffs were omitted because the crystal was prone to
+distortion and melting at 200 K.  Ice-{\it i}$^\prime$ is the form of
+Ice-{\it i} used in the SPC/E simulations.}
 \label{incCutoff}
 \end{figure}
 
-Increasing the cutoff radius in simulations of the more
-computationally efficient water models was done in order to evaluate
-the trend in free energy values when moving to systems that do not
-involve potential truncation. As seen in Fig. \ref{incCutoff}, the
-free energy of all the ice polymorphs show a substantial dependence on
-cutoff radius. In general, there is a narrowing of the free energy
-differences while moving to greater cutoff radius. Interestingly, by
-increasing the cutoff radius, the free energy gap was narrowed enough
-in the SSD/E model that the liquid state is preferred under standard
-simulation conditions (298 K and 1 atm). Thus, it is recommended that
-simulations using this model choose interaction truncation radii
-greater than 9 \AA\ . This narrowing trend is much more subtle in the
-case of SSD/RF, indicating that the free energies calculated with a
-reaction field present provide a more accurate picture of the free
-energy landscape in the absence of potential truncation.
+For the more computationally efficient water models, we have also
+investigated the effect of potential trunctaion on the computed free
+energies as a function of the cutoff radius.  As seen in
+Fig. \ref{incCutoff}, the free energies of the ice polymorphs with
+water models lacking a long-range correction show significant cutoff
+dependence.  In general, there is a narrowing of the free energy
+differences while moving to greater cutoff radii.  As the free
+energies for the polymorphs converge, the stability advantage that
+Ice-{\it i} exhibits is reduced.  Adjacent to each of these plots are
+results for systems with applied or estimated long-range corrections.
+SSD/RF was parametrized for use with a reaction field, and the benefit
+provided by this computationally inexpensive correction is apparent.
+The free energies are largely independent of the size of the reaction
+field cavity in this model, so small cutoff radii mimic bulk
+calculations quite well under SSD/RF.
+ 
+Although TIP3P was paramaterized for use without the Ewald summation,
+we have estimated the effect of this method for computing long-range
+electrostatics for both TIP3P and SPC/E.  This was accomplished by
+calculating the potential energy of identical crystals both with and
+without particle mesh Ewald (PME).  Similar behavior to that observed
+with reaction field is seen for both of these models.  The free
+energies show reduced dependence on cutoff radius and span a narrower
+range for the various polymorphs.  Like the dipolar water models,
+TIP3P displays a relatively constant preference for the Ice-{\it i}
+polymorph.  Crystal preference is much more difficult to determine for
+SPC/E.  Without a long-range correction, each of the polymorphs
+studied assumes the role of the preferred polymorph under different
+cutoff radii.  The inclusion of the Ewald correction flattens and
+narrows the gap in free energies such that the polymorphs are
+isoenergetic within statistical uncertainty.  This suggests that other
+conditions, such as the density in fixed-volume simulations, can
+influence the polymorph expressed upon crystallization.
 
-To further study the changes resulting to the inclusion of a
-long-range interaction correction, the effect of an Ewald summation
-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 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.
+\section{Conclusions}
 
-\begin{table*}
-\begin{minipage}{\linewidth}
-\renewcommand{\thefootnote}{\thempfootnote}
-\begin{center}
-\caption{The free energy of the studied ice polymorphs after applying 
-the energy difference attributed to the inclusion of the PME
-long-range interaction correction. Units are kcal/mol.}
-\begin{tabular}{ l  c  c  c  c }
-\hline 
-\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
-\hline 
-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}
-\end{minipage}
-\end{table*}
+In this work, thermodynamic integration was used to determine the
+absolute free energies of several ice polymorphs.  The new polymorph,
+Ice-{\it i} was observed to be the stable crystalline state for {\it
+all} the water models when using a 9.0 \AA\ cutoff.  However, the free
+energy partially depends on simulation conditions (particularly on the
+choice of long range correction method). Regardless, Ice-{\it i} was
+still observered to be a stable polymorph for all of the studied water
+models.
 
-\section{Conclusions}
+So what is the preferred solid polymorph for simulated water?  As
+indicated above, the answer appears to be dependent both on the
+conditions and the model used.  In the case of short cutoffs without a
+long-range interaction correction, Ice-{\it i} and Ice-{\it
+i}$^\prime$ have the lowest free energy of the studied polymorphs with
+all the models.  Ideally, crystallization of each model under constant
+pressure conditions, as was done with SSD/E, would aid in the
+identification of their respective preferred structures.  This work,
+however, helps illustrate how studies involving one specific model can
+lead to insight about important behavior of others. 
 
-The free energy for proton ordered variants of hexagonal and cubic ice
-$I$, 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
-energy landscape for the more complex multipoint water models. Despite
-these effects, these results show Ice-{\it i} to be an important ice
-polymorph that should be considered in simulation studies.
+We also note that none of the water models used in this study are
+polarizable or flexible models.  It is entirely possible that the
+polarizability of real water makes Ice-{\it i} substantially less
+stable than ice I$_h$.  However, the calculations presented above seem
+interesting enough to communicate before the role of polarizability
+(or flexibility) has been thoroughly investigated. 
 
-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
-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
+Finally, due to the stability of Ice-{\it i} in 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 to 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.  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 $I_h$, $I_c$, 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.
+non-polar molecules.  For the purpose of comparison with experimental
+results, we have calculated the oxygen-oxygen pair correlation
+function, $g_{OO}(r)$, and the structure factor, $S(\vec{q})$ for the
+two Ice-{\it i} variants (along with example ice I$_h$ and I$_c$
+plots) at 77K, and they are shown in figures \ref{fig:gofr} and
+\ref{fig:sofq} respectively.  It is interesting to note that the
+structure factors for Ice-{\it i}$^\prime$ and Ice-I$_c$ are quite similar.
+The primary differences are small peaks at 1.125, 2.29, and 2.53
+\AA${-1}$, so particular attention to these regions would be needed
+to identify the new {\it i}$^\prime$ variant from the I$_{c}$ variant.
 
 \begin{figure}
+\centering
 \includegraphics[width=\linewidth]{iceGofr.eps}
-\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.}
+\caption{Radial distribution functions of ice I$_h$, I$_c$, and
+Ice-{\it i} calculated from from simulations of the SSD/RF water model
+at 77 K.  The Ice-{\it i} distribution function was obtained from
+simulations composed of TIP4P water.}
 \label{fig:gofr}
 \end{figure}
 
 \begin{figure}
+\centering
 \includegraphics[width=\linewidth]{sofq.eps}
-\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i},
+\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
@@ -524,7 +470,7 @@ Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR
 
 \newpage
 
-\bibliographystyle{jcp}
+\bibliographystyle{achemso}
 \bibliography{iceiPaper}
 
 
