| 117 |
|
the three dimensional Ewald sum is appropriate for slab |
| 118 |
|
geometries.\cite{Parry:1975if} Modified Ewald methods that were |
| 119 |
|
developed to handle two-dimensional (2-D) electrostatic |
| 120 |
< |
interactions,\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
| 121 |
< |
but these methods were originally quite computationally |
| 120 |
> |
interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
| 121 |
> |
These methods were originally quite computationally |
| 122 |
|
expensive.\cite{Spohr97,Yeh99} There have been several successful |
| 123 |
< |
efforts that reduced the computational cost of 2-D lattice |
| 124 |
< |
summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} |
| 123 |
> |
efforts that reduced the computational cost of 2-D lattice summations, |
| 124 |
|
bringing them more in line with the scaling for the full 3-D |
| 125 |
< |
treatments. The inherent periodicity in the Ewald method can also |
| 126 |
< |
be problematic for interfacial molecular |
| 127 |
< |
systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
| 125 |
> |
treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The |
| 126 |
> |
inherent periodicity required by the Ewald method can also be |
| 127 |
> |
problematic in a number of protein/solvent and ionic solution |
| 128 |
> |
environments.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
| 129 |
|
|
| 130 |
|
\subsection{Real-space methods} |
| 131 |
|
Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
| 573 |
|
and have been compared with the values obtained from the multipolar |
| 574 |
|
Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
| 575 |
|
between two configurations is the primary quantity that governs how |
| 576 |
< |
the simulation proceeds. These differences are the most imporant |
| 576 |
> |
the simulation proceeds. These differences are the most important |
| 577 |
|
indicators of the reliability of a method even if the absolute |
| 578 |
|
energies are not exact. For each of the multipolar systems listed |
| 579 |
|
above, we have compared the change in electrostatic potential energy |
| 792 |
|
|
| 793 |
|
\begin{figure} |
| 794 |
|
\centering |
| 795 |
< |
\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
| 795 |
> |
\includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps} |
| 796 |
|
\caption{Statistical analysis of the quality of configurational |
| 797 |
|
energy differences for the real-space electrostatic methods |
| 798 |
|
compared with the reference Ewald sum. Results with a value equal |
| 865 |
|
|
| 866 |
|
\begin{figure} |
| 867 |
|
\centering |
| 868 |
< |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
| 868 |
> |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} |
| 869 |
|
\caption{Statistical analysis of the quality of the force vector |
| 870 |
|
magnitudes for the real-space electrostatic methods compared with |
| 871 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 879 |
|
|
| 880 |
|
\begin{figure} |
| 881 |
|
\centering |
| 882 |
< |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
| 882 |
> |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps} |
| 883 |
|
\caption{Statistical analysis of the quality of the torque vector |
| 884 |
|
magnitudes for the real-space electrostatic methods compared with |
| 885 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 937 |
|
|
| 938 |
|
\begin{figure} |
| 939 |
|
\centering |
| 940 |
< |
\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
| 940 |
> |
\includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps} |
| 941 |
|
\caption{The circular variance of the direction of the force and |
| 942 |
|
torque vectors obtained from the real-space methods around the |
| 943 |
|
reference Ewald vectors. A variance equal to 0 (dashed line) |
