--- trunk/iceiPaper/iceiPaper.tex	2004/09/14 23:03:53	1457
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/15 06:34:49	1458
@@ -66,6 +66,24 @@ For the thermodynamic integration of molecular crystal
 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:
+\begin{equation}
+\begin{center}
+\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda 
+)}{\partial\lambda}\right\rangle_\lambda d\lambda,
+\end{center}
+\end{equation}
+where $V$ is the interaction potential and $\lambda$ is the
+transformation parameter. Simulations are distributed unevenly 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 is chosen as the reference state that the system is converted
 to over the course of the simulation. In an Einstein Crystal, the
@@ -241,6 +259,16 @@ Increasing the cutoff radius in simulations of the mor
 risk of spontaneous crystallization. However, this risk changes when
 applying a longer cutoff.
 
+\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 a
+liquid. The connecting lines are qualitative visual aid.}
+\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
@@ -278,7 +306,9 @@ the phase behavior of water models.
 \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.}
+\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 \\[-7mm]
 \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
@@ -293,11 +323,26 @@ the phase behavior of water models.
 
 \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
+standard conditions for several common water models via thermodynamic
+integration. All the water models studied show Ice-{\it i} to be the
+minimum free energy crystal structure in the with a 9 \AA\ switching
+function cutoff. Calculated melting and boiling points show
+surprisingly good agreement with the experimental values; however, the
+solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
+interaction truncation was investigated through variation of the
+cutoff radius, use of a reaction field parameterized model, and
+estimation of the results in the presence of the Ewald summation
+correction. Interaction truncation has a significant effect on the
+computed free energy values, but Ice-{\it i} is still observed to be a
+relavent ice polymorph in simulation studies.
+
 \section{Acknowledgments}
 Support for this project was provided by the National Science
 Foundation under grant CHE-0134881. Computation time was provided by
-the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
-DMR-0079647.
+the Notre Dame High Performance Computing Cluster and the Notre Dame
+Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
 
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