--- trunk/iceiPaper/iceiPaper.tex	2004/09/16 23:40:02	1470
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/17 06:47:18	1471
@@ -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{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.
+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,Sanz04,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,Liu96,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
+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
@@ -397,19 +423,19 @@ 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.  The free energies of Ice-{\it i} and ice B overlap
+within error, with ice $I_c$ just outside, indicating that Ice-{\it i}
+might be metastable with respect to ice B and possibly ice $I_c$ in
+the SPC/E water model. 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}
