--- trunk/iceiPaper/iceiPaper.tex	2004/09/16 20:56:49	1467
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/17 14:56:05	1474
@@ -54,39 +54,41 @@ 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{Matsumoto02andOthers} 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.
+Computer simulations are a valuable tool for studying the phase
+behavior of systems ranging from small or simple molecules to complex
+biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques
+have been developed to investigate the thermodynamic properites of
+model substances, providing both qualitative and quantitative
+comparisons between simulations and
+experiment.\cite{Widom63,Frenkel84} 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{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,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, molecule transport, 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 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 the model.
 
 In this paper, standard reference state methods were applied to known
-crystalline water polymorphs in the low pressure regime. This work is
+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
@@ -111,9 +113,10 @@ elongated variant of Ice-{\it i}.  For Ice-{\it i}, th
 \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-$i^\prime$, $a =
-1.7850c$.} 
+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}
 
@@ -129,8 +132,8 @@ 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
+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
@@ -154,16 +157,21 @@ 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
-publication.\cite{DLM}
+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 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
@@ -176,16 +184,28 @@ potential. Simulations are distributed unevenly along 
 \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 +219,8 @@ 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 +233,18 @@ 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 apply of one of the most convenient
+methods and integrate 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
@@ -250,11 +276,13 @@ 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 divised, 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 +297,12 @@ 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}
@@ -330,9 +358,9 @@ $T_m$ (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}
@@ -385,7 +413,7 @@ greater than 9 \AA\. This narrowing trend is much more
 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
+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.
@@ -397,19 +425,19 @@ SPC/E water models are shown in Table \ref{pmeShift}. 
 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.
+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 +450,8 @@ 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}
@@ -460,19 +488,37 @@ non-polar molecules.  Fig. \ref{fig:gofr} contains our
 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.
+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.  In a quick comparison of the predicted
+S(q) for Ice-{\it i} and experimental studies of amorphous solid
+water, it is possible that some of the ``spurious'' peaks that could
+not be assigned in HDA could correspond to peaks labeled in this
+S(q).\cite{Bizid87} It should be noted that there is typically poor
+agreement on crystal densities between simulation and experiment, so
+such peak comparisons should be made with caution.  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.
 
 \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-{\it i} and ice $I_c$ 
+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. The labeled peaks
+compared favorably with ``spurious'' peaks observed in experimental
+studies of amorphous solid water.\cite{Bizid87}}
+\label{fig:sofq}
+\end{figure}
+
 \section{Acknowledgments}
 Support for this project was provided by the National Science
 Foundation under grant CHE-0134881. Computation time was provided by
