--- trunk/multipole/multipole_2/multipole2.tex	2014/06/17 16:08:03	4191
+++ trunk/multipole/multipole_2/multipole2.tex	2014/08/06 19:10:04	4203
@@ -47,7 +47,8 @@ preprint,
 
 %\preprint{AIP/123-QED}
 
-\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods}
+\title{Real space electrostatics for multipoles. II. Comparisons with
+  the Ewald Sum}
 
 \author{Madan Lamichhane}
  \affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}
@@ -994,9 +995,114 @@ $k$-space cutoff values.
   of a 2000-molecule simulation of SSDQ water with 48 ionic charges at
   300 K starting from the same initial configuration. All runs
   utilized the same real-space cutoff, $r_c = 12$\AA.}
+\end{figure} 
+
+\subsection{Reproduction of Structural Features\label{sec:structure}}
+One of the best tests of modified interaction potentials is the
+fidelity with which they can reproduce structural features in a
+liquid.  One commonly-utilized measure of structural ordering is the
+pair distribution function, $g(r)$, which measures local density
+deviations in relation to the bulk density.  In the electrostatic
+approaches studied here, the short-range repulsion from the
+Lennard-Jones potential is identical for the various electrostatic
+methods, and since short range repulsion determines much of the local
+liquid ordering, one would not expect to see any differences in
+$g(r)$.  Indeed, the pair distributions are essentially identical for
+all of the electrostatic methods studied (for each of the different
+systems under investigation).  Interested readers may consult the
+supplementary information for plots of these pair distribution
+functions.
+
+A direct measure of the structural features that is a more
+enlightening test of the modified electrostatic methods is the average
+value of the electrostatic energy $\langle U_\mathrm{elect} \rangle$
+which is obtained by sampling the liquid-state configurations
+experienced by a liquid evolving entirely under the influence of the
+methods being investigated.  In figure \ref{fig:Uelect} we show how
+$\langle U_\mathrm{elect} \rangle$ for varies with the damping parameter,
+$\alpha$, for each of the methods.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\textwidth]{averagePotentialEnergy_r9_12.eps}
+\label{fig:Uelect}        
+\caption{The average electrostatic potential energy, 
+  $\langle U_\mathrm{elect} \rangle$ for the SSDQ water with ions as a function
+  of the damping parameter, $\alpha$, for each of the real-space
+  electrostatic methods. Top panel: simulations run with a real-space
+  cutoff, $r_c = 9$\AA.  Bottom panel: the same quantity, but with a
+  larger cutoff, $r_c = 12$\AA.}
 \end{figure} 
+
+It is clear that moderate damping is important for converging the mean
+potential energy values, particularly for the two shifted force
+methods (GSF and TSF).  A value of $\alpha \approx 0.18$ \AA$^{-1}$ is
+sufficient to converge the SP and GSF energies with a cutoff of 12
+\AA, while shorter cutoffs require more dramatic damping ($\alpha
+\approx 0.36$ \AA$^{-1}$ for $r_c = 9$ \AA).  It is also clear from
+fig. \ref{fig:Uelect} that it is possible to overdamp the real-space
+electrostatic methods, causing the estimate of the energy to drop
+below the Ewald results.
+
+These ``optimal'' values of the damping coefficient are slightly
+larger than what were observed for DSF electrostatics for purely
+point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for
+$r_c = 12$\AA appears to be an excellent compromise for mixed charge
+multipole systems.
 
+\subsection{Reproduction of Dynamic Properties\label{sec:structure}}
+To test the fidelity of the electrostatic methods at reproducing
+dynamics in a multipolar liquid, it is also useful to look at
+transport properties, particularly the diffusion constant,
+\begin{equation}
+D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left|
+  \mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle
+\label{eq:diff}
+\end{equation}
+which measures long-time behavior and is sensitive to the forces on
+the multipoles.  For the soft dipolar fluid, and the SSDQ liquid
+systems, the self-diffusion constants (D) were calculated from linear
+fits to the long-time portion of the mean square displacement
+($\langle r^{2}(t) \rangle$).\cite{Allen87}
 
+In addition to translational diffusion, orientational relaxation times
+were calculated for comparisons with the Ewald simulations and with
+experiments. These values were determined from the same 1~ns $NVE$
+trajectories used for translational diffusion by calculating the
+orientational time correlation function,
+\begin{equation}
+C_l^\alpha(t) = \left\langle P_l\left[\hat{\mathbf{u}}_i^\gamma(t)
+                \cdot\hat{\mathbf{u}}_i^\gamma(0)\right]\right\rangle,
+\label{eq:OrientCorr}
+\end{equation}
+where $P_l$ is the Legendre polynomial of order $l$ and
+$\hat{\mathbf{u}}_i^\alpha$ is the unit vector of molecule $i$ along
+axis $\gamma$.  The body-fixed reference frame used for our
+orientational correlation functions has the $z$-axis running along the
+dipoles, and for the SSDQ water model, the $y$-axis connects the two
+implied hydrogen atoms.
+
+From the orientation autocorrelation functions, we can obtain time
+constants for rotational relaxation either by fitting an exponential
+function or by integrating the entire correlation function.  These
+decay times are directly comparable to water orientational relaxation
+times from nuclear magnetic resonance (NMR). The relaxation constant
+obtained from $C_2^y(t)$ is normally of experimental interest because
+it describes the relaxation of the principle axis connecting the
+hydrogen atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular
+portion of the dipole-dipole relaxation from a proton NMR signal and
+should provide an estimate of the NMR relaxation time
+constant.\cite{Impey82}
+
+Results for the diffusion constants and orientational relaxation times
+are shown in figure \ref{fig:dynamics}. From this data, it is apparent
+that the values for both $D$ and $\tau_2$ using the Ewald sum are
+reproduced with high fidelity by the GSF method. 
+
+The $\tau_2$ results in \ref{fig:dynamics} show a much greater
+difference between the real-space and the Ewald results.
+
+
 \section{CONCLUSION}
 In the first paper in this series, we generalized the
 charge-neutralized electrostatic energy originally developed by Wolf