--- trunk/fennellDissertation/waterChapter.tex	2006/08/30 21:13:57	2985
+++ trunk/fennellDissertation/waterChapter.tex	2006/08/30 22:14:37	2986
@@ -856,6 +856,7 @@ C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{
 by:
 \begin{equation}
 C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle,
+\label{eq:reorientCorr}
 \end{equation}
 where $P_l$ are Legendre polynomials of order $l$ and
 $\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular
@@ -950,7 +951,7 @@ nature of contour plotting.
 the visual contours in the figure. This highlights the qualitative
 nature of contour plotting.
 
-\section{Tetrahedrally Restructured Elongated Dipole (TRED) Water Model}
+\section{Tetrahedrally Restructured Elongated Dipole (TRED) Water Model}\label{sec:tredWater}
 
 The SSD/RF model works well with damped electrostatics, but because of
 its point multipole character, there is no charge neutralization
@@ -1016,7 +1017,10 @@ and $\omega^\circ$ unaltered. Finally, the strength of
 to the tenths place. We also unified the sticky parameters for the
 switching functions on the repulsive and attractive interactions in
 the interest of simplicity, and we left the quadrupole moment elements
-and $\omega^\circ$ unaltered. Finally, the strength of the sticky
+and $\omega^\circ$ unaltered. It should be noted that additional logic
+needs to be included into the electrostatic code when using TRED to
+insure that the charges of each water do not interact with the other
+water's quadrupole moment. Finally, the strength of the sticky
 interaction ($v_0$) was modified to improve the shape of the first
 peaks in $g_\textrm{OO}(r)$ and $g_\textrm{OH}(r)$, while the $\sigma$
 and $\epsilon$ values were varied to adjust the location of the first
@@ -1079,9 +1083,9 @@ the first minimum value of the experimental $g_\textrm
 \ref{fig:tredGofR}), $n_C$ and $n_H$ counts increase because of the
 further first minimum distance locations. This results in the
 integration extending over a larger water volume. If we integrate to
-the first minimum value of the experimental $g_\textrm{OO}(r)$ (3.42
-\AA ) in both the SSD/RF and TRED plots, the $n_C$ values for both are
-much closer to experiment (4.7 for SSD/RF and 4.9 for TRED).
+the first minimum value of the experimental $g_\textrm{OO}(r)$
+(3.42~\AA ) in both the SSD/RF and TRED plots, the $n_C$ values for
+both are much closer to experiment (4.7 for SSD/RF and 4.9 for TRED).
 
 \begin{figure}
 \centering
@@ -1097,8 +1101,30 @@ interaction strength, and identical applied damping co
 bottom of table \ref{tab:tredProperties}. The static dielectric
 constant results for SSD/RF and TRED are identical within error. This
 is not surprising given the similar dipole moment, similar sticky
-interaction strength, and identical applied damping constant. Comparing the static dielectric constant contour map (figure \ref{fig:tredDielectric}) with the dielectric map for SSD/RF 
+interaction strength, and identical applied damping
+constant. Comparing the static dielectric constant contour map (figure
+\ref{fig:tredDielectric}) with the dielectric map for SSD/RF (figure
+\ref{fig:dielectricMap}D) highlights the similarities in how these
+models respond to dielectric damping and how the dipolar and monopolar
+electrostatic damping act in an equivalent fashion. Both these
+dielectric maps span a larger range than the 3, 4, and 5 point-charge
+water models; however, the SSD/RF range is greater than TRED,
+indicating that multipoles are a little more sensitive to damping than
+monopoles. 
 
+The final dielectric comparison comes through the Debye relaxation
+time ($\tau_D$) or the collective dipolar relaxation time when
+assuming a Debye treatment for the dielectric
+relaxation.\cite{Chandra99,Kindt96} This value is calculated through
+equation (\ref{eq:reorientCorr}) applied to the total system dipole
+moment. The values for both of the models are a slower than the
+experimental relaxation; however, they compare compare very well to
+experiment considering the Debye relaxation times calculated for the
+original SSD (11.95~ps) and the SPC/E (6.95~ps) and TIP3P (6.1~ps)
+values. The $\tau_D$ for TRED is about 1.5~ps slower than the $\tau_D$
+for SSD/RF, most likely due to the slower decay of the charge-charge
+interaction, even when screened by the same damping constant.
+
 \section{Conclusions}
 
 In the above sections, the density maximum and temperature dependence
@@ -1120,10 +1146,17 @@ The simple water models investigated here are excellen
 when changing the method of calculating long-range electrostatic
 interactions.  
 
+We also showed that SSD/RF performs well under the alternative damped
+electrostatic conditions, validating the multipolar damping work in
+the previous chapter. To improve the modeling of water when {\sc sf},
+the TRED water model was developed. This model maintains improves upon
+the thermodynamic properties of SSD/RF using damped electrostatics
+while maintaining the exceptional depiction of water dynamics.
+
 The simple water models investigated here are excellent choices for
 representing explicit water in large scale simulations of biochemical
 systems. They are more computationally efficient than the common
 charge based water models, and, in many cases, exhibit more realistic
 bulk phase fluid properties. These models are one of the few cases in
 which maximizing efficiency does not result in a loss in realistic
-representation.
+liquid water representation.
