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1 < \chapter{\label{chap:electrostatics}ELECTROSTATIC INTERACTION CORRECTION
2 < TECHNIQUES}
1 > \chapter{\label{chap:electrostatics}ELECTROSTATIC INTERACTION CORRECTION \\ TECHNIQUES}
2  
3   In molecular simulations, proper accumulation of the electrostatic
4   interactions is essential and is one of the most
# Line 123 | Line 122 | convergence for the larger systems of charges that are
122   \label{fig:ewaldTime}
123   \end{figure}
124  
125 < The original Ewald summation is an $\mathscr{O}(N^2)$ algorithm.  The
125 > The original Ewald summation is an $\mathcal{O}(N^2)$ algorithm.  The
126   convergence parameter $(\alpha)$ plays an important role in balancing
127   the computational cost between the direct and reciprocal-space
128   portions of the summation.  The choice of this value allows one to
129   select whether the real-space or reciprocal space portion of the
130 < summation is an $\mathscr{O}(N^2)$ calculation (with the other being
131 < $\mathscr{O}(N)$).\cite{Sagui99} With the appropriate choice of
130 > summation is an $\mathcal{O}(N^2)$ calculation (with the other being
131 > $\mathcal{O}(N)$).\cite{Sagui99} With the appropriate choice of
132   $\alpha$ and thoughtful algorithm development, this cost can be
133 < reduced to $\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route
133 > reduced to $\mathcal{O}(N^{3/2})$.\cite{Perram88} The typical route
134   taken to reduce the cost of the Ewald summation even further is to set
135   $\alpha$ such that the real-space interactions decay rapidly, allowing
136   for a short spherical cutoff. Then the reciprocal space summation is
# Line 140 | Line 139 | methods, the cost of the reciprocal-space portion of t
139   particle-particle particle-mesh (P3M) and particle mesh Ewald (PME)
140   methods.\cite{Shimada93,Luty94,Luty95,Darden93,Essmann95} In these
141   methods, the cost of the reciprocal-space portion of the Ewald
142 < summation is reduced from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N
142 > summation is reduced from $\mathcal{O}(N^2)$ down to $\mathcal{O}(N
143   \log N)$.
144  
145   These developments and optimizations have made the use of the Ewald
# Line 453 | Line 452 | summation performed by the Ewald sum.
452  
453   \section{Evaluating Pairwise Summation Techniques}
454  
455 < In classical molecular mechanics simulations, there are two primary
456 < techniques utilized to obtain information about the system of
457 < interest: Monte Carlo (MC) and Molecular Dynamics (MD).  Both of these
458 < techniques utilize pairwise summations of interactions between
459 < particle sites, but they use these summations in different ways.
455 > As mentioned in the introduction, there are two primary techniques
456 > utilized to obtain information about the system of interest in
457 > classical molecular mechanics simulations: Monte Carlo (MC) and
458 > Molecular Dynamics (MD).  Both of these techniques utilize pairwise
459 > summations of interactions between particle sites, but they use these
460 > summations in different ways.
461  
462   In MC, the potential energy difference between configurations dictates
463   the progression of MC sampling.  Going back to the origins of this
# Line 513 | Line 513 | Results and discussion for the individual analysis of
513   differences.
514  
515   Results and discussion for the individual analysis of each of the
516 < system types appear in sections \ref{sec:IndividualResults}, while the
516 > system types appear in appendix \ref{app:IndividualResults}, while the
517   cumulative results over all the investigated systems appear below in
518   sections \ref{sec:EnergyResults}.
519  
# Line 714 | Line 714 | significant improvement using the group-switched cutof
714   some degree by using group based cutoffs with a switching
715   function.\cite{Adams79,Steinbach94,Leach01} However, we do not see
716   significant improvement using the group-switched cutoff because the
717 < salt and salt solution systems contain non-neutral groups.  Section
718 < \ref{sec:IndividualResults} includes results for systems comprised entirely
719 < of neutral groups.
717 > salt and salt solution systems contain non-neutral groups.  Appendix
718 > \ref{app:IndividualResults} includes results for systems comprised
719 > entirely of neutral groups.
720  
721   For the {\sc sp} method, inclusion of electrostatic damping improves
722   the agreement with Ewald, and using an $\alpha$ of 0.2~\AA $^{-1}$
# Line 864 | Line 864 | charged bodies, and this observation is investigated f
864   particles in all seven systems, while torque vectors are only
865   available for neutral molecular groups.  Damping is more beneficial to
866   charged bodies, and this observation is investigated further in
867 < section \ref{sec:IndividualResults}.
867 > appendix \ref{app:IndividualResults}.
868  
869   Although not discussed previously, group based cutoffs can be applied
870   to both the {\sc sp} and {\sc sf} methods.  The group-based cutoffs
# Line 940 | Line 940 | but damping may be unnecessary when using the {\sc sf}
940   molecular torques, particularly for the shorter cutoffs.  Based on our
941   observations, empirical damping up to 0.2~\AA$^{-1}$ is beneficial,
942   but damping may be unnecessary when using the {\sc sf} method.
943 –
944 – \section{Individual System Analysis Results}\label{sec:IndividualResults}
945 –
946 – The combined results of the previous sections show how the pairwise
947 – methods compare to the Ewald summation in the general sense over all
948 – of the system types.  It is also useful to consider each of the
949 – studied systems in an individual fashion, so that we can identify
950 – conditions that are particularly difficult for a selected pairwise
951 – method to address. This allows us to further establish the limitations
952 – of these pairwise techniques.  Below, the energy difference, force
953 – vector, and torque vector analyses are presented on an individual
954 – system basis.
955 –
956 – \subsection{SPC/E Water Results}\label{sec:WaterResults}
957 –
958 – The first system considered was liquid water at 300~K using the SPC/E
959 – model of water.\cite{Berendsen87} The results for the energy gap
960 – comparisons and the force and torque vector magnitude comparisons are
961 – shown in table \ref{tab:spce}.  The force and torque vector
962 – directionality results are displayed separately in table
963 – \ref{tab:spceAng}, where the effect of group-based cutoffs and
964 – switching functions on the {\sc sp} and {\sc sf} potentials are also
965 – investigated.  In all of the individual results table, the method
966 – abbreviations are as follows:
967 –
968 – \begin{itemize}[itemsep=0pt]
969 – \item PC = Pure Cutoff,
970 – \item SP = Shifted Potential,
971 – \item SF = Shifted Force,
972 – \item GSC = Group Switched Cutoff,
973 – \item RF = Reaction Field (where $\varepsilon \approx\infty$),
974 – \item GSSP = Group Switched Shifted Potential, and
975 – \item GSSF = Group Switched Shifted Force.
976 – \end{itemize}
977 –
978 – \begin{table}[htbp]
979 – \centering
980 – \caption{REGRESSION RESULTS OF THE LIQUID WATER SYSTEM FOR THE
981 – $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it middle})
982 – AND TORQUE VECTOR MAGNITUDES ({\it lower})}
983 –
984 – \footnotesize
985 – \begin{tabular}{@{} ccrrrrrr @{}}
986 – \toprule
987 – \toprule
988 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
989 – \cmidrule(lr){3-4}
990 – \cmidrule(lr){5-6}
991 – \cmidrule(l){7-8}
992 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
993 – \midrule
994 – PC  &     & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
995 – SP  & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
996 –    & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
997 –    & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
998 –    & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
999 – SF  & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
1000 –    & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
1001 –    & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
1002 –    & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
1003 – GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
1004 – RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                
1005 – \midrule
1006 – PC  &     & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
1007 – SP  & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
1008 –    & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
1009 –    & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
1010 –    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
1011 – SF  & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
1012 –    & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
1013 –    & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
1014 –    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
1015 – GSC &     & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
1016 – RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\          
1017 – \midrule
1018 – PC  &     & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
1019 – SP  & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
1020 –    & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
1021 –    & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
1022 –    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
1023 – SF  & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
1024 –    & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
1025 –    & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
1026 –    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
1027 – GSC &     & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
1028 – RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
1029 – \bottomrule
1030 – \end{tabular}
1031 – \label{tab:spce}
1032 – \end{table}
1033 –
1034 – \begin{table}[htbp]
1035 – \centering
1036 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
1037 – DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE LIQUID WATER
1038 – SYSTEM}
1039 –
1040 – \footnotesize
1041 – \begin{tabular}{@{} ccrrrrrr @{}}
1042 – \toprule
1043 – \toprule
1044 – & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
1045 – \cmidrule(lr){3-5}
1046 – \cmidrule(l){6-8}
1047 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
1048 – \midrule
1049 – PC  &     & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
1050 – SP  & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
1051 –    & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
1052 –    & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
1053 –    & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
1054 – SF  & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
1055 –    & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
1056 –    & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
1057 –    & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
1058 – GSC &     & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
1059 – RF  &     & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\      
1060 – \midrule
1061 – GSSP  & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
1062 –      & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
1063 –      & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
1064 –      & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
1065 – GSSF  & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
1066 –      & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
1067 –      & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
1068 –      & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
1069 – \bottomrule
1070 – \end{tabular}
1071 – \label{tab:spceAng}
1072 – \end{table}
1073 –
1074 – The water results parallel the combined results seen in sections
1075 – \ref{sec:EnergyResults} through \ref{sec:FTDirResults}.  There is good
1076 – agreement with {\sc spme} in both energetic and dynamic behavior when
1077 – using the {\sc sf} method with and without damping. The {\sc sp}
1078 – method does well with an $\alpha$ around 0.2~\AA$^{-1}$, particularly
1079 – with cutoff radii greater than 12~\AA. Over-damping the electrostatics
1080 – reduces the agreement between both these methods and {\sc spme}.
1081 –
1082 – The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
1083 – observations from the combined results.  In contrast to these results, however, the use of a switching function and group
1084 – based cutoffs greatly improves the results for these neutral water
1085 – molecules.  The group switched cutoff ({\sc gsc}) does not mimic the
1086 – energetics of {\sc spme} as well as the {\sc sp} (with moderate
1087 – damping) and {\sc sf} methods, but the dynamics are quite good.  The
1088 – switching functions correct discontinuities in the potential and
1089 – forces, leading to these improved results.  Such improvements with the
1090 – use of a switching function have been recognized in previous
1091 – studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
1092 – tactic for stably incorporating local area electrostatic effects.
1093 –
1094 – The reaction field ({\sc rf}) method simply extends upon the results
1095 – observed in the {\sc gsc} case.  Both methods are similar in form
1096 – (i.e. neutral groups, switching function), but {\sc rf} incorporates
1097 – an added effect from the external dielectric. This similarity
1098 – translates into the same good dynamic results and improved energetic
1099 – agreement with {\sc spme}.  Though this agreement is not to the level
1100 – of the moderately damped {\sc sp} and {\sc sf} methods, these results
1101 – show how incorporating some implicit properties of the surroundings
1102 – (i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.
1103 –
1104 – As a final note for the liquid water system, use of group cutoffs and a
1105 – switching function leads to noticeable improvements in the {\sc sp}
1106 – and {\sc sf} methods, primarily in directionality of the force and
1107 – torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
1108 – significant narrowing of the angle distribution when using little to
1109 – no damping and only modest improvement for the recommended conditions
1110 – ($\alpha = 0.2$~\AA$^{-1}$ and $R_\textrm{c}~\geqslant~12$~\AA).  The
1111 – {\sc sf} method shows modest narrowing across all damping and cutoff
1112 – ranges of interest.  When over-damping these methods, group cutoffs and
1113 – the switching function do not improve the force and torque
1114 – directionalities.
1115 –
1116 – \subsection{SPC/E Ice I$_\textrm{c}$ Results}\label{sec:IceResults}
1117 –
1118 – In addition to the disordered molecular system above, the ordered
1119 – molecular system of ice I$_\textrm{c}$ was also considered.  Ice
1120 – polymorph could have been used to fit this role; however, ice
1121 – I$_\textrm{c}$ was chosen because it can form an ideal periodic
1122 – lattice with the same number of water molecules used in the disordered
1123 – liquid state case.  The results for the energy gap comparisons and the
1124 – force and torque vector magnitude comparisons are shown in table
1125 – \ref{tab:ice}.  The force and torque vector directionality results are
1126 – displayed separately in table \ref{tab:iceAng}, where the effect of
1127 – group-based cutoffs and switching functions on the {\sc sp} and {\sc
1128 – sf} potentials are also displayed.
1129 –
1130 – \begin{table}[htbp]
1131 – \centering
1132 – \caption{REGRESSION RESULTS OF THE ICE I$_\textrm{c}$ SYSTEM FOR
1133 – $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it
1134 – middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
1135 –
1136 – \footnotesize
1137 – \begin{tabular}{@{} ccrrrrrr @{}}
1138 – \toprule
1139 – \toprule
1140 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1141 – \cmidrule(lr){3-4}
1142 – \cmidrule(lr){5-6}
1143 – \cmidrule(l){7-8}
1144 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1145 – \midrule
1146 – PC  &     & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
1147 – SP  & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
1148 –    & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
1149 –    & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
1150 –    & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
1151 – SF  & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
1152 –    & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
1153 –    & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
1154 –    & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
1155 – GSC &     & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
1156 – RF  &     & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
1157 – \midrule
1158 – PC  &     & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
1159 – SP  & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
1160 –    & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
1161 –    & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
1162 –    & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
1163 – SF  & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
1164 –    & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
1165 –    & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
1166 –    & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
1167 – GSC &     & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
1168 – RF  &     & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
1169 – \midrule
1170 – PC  &     & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
1171 – SP  & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
1172 –    & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
1173 –    & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
1174 –    & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
1175 – SF  & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
1176 –    & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
1177 –    & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
1178 –    & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
1179 – GSC &     & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
1180 – RF  &     & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
1181 – \bottomrule
1182 – \end{tabular}
1183 – \label{tab:ice}
1184 – \end{table}
1185 –
1186 – \begin{table}[htbp]
1187 – \centering
1188 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
1189 – OF THE FORCE AND TORQUE VECTORS IN THE ICE I$_\textrm{c}$ SYSTEM}      
1190 –
1191 – \footnotesize
1192 – \begin{tabular}{@{} ccrrrrrr @{}}
1193 – \toprule
1194 – \toprule
1195 – & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque
1196 – $\sigma^2$} \\
1197 – \cmidrule(lr){3-5}
1198 – \cmidrule(l){6-8}
1199 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
1200 – \midrule
1201 – PC  &     & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
1202 – SP  & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
1203 –    & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
1204 –    & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
1205 –    & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
1206 – SF  & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
1207 –    & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
1208 –    & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
1209 –    & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
1210 – GSC &     & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
1211 – RF  &     & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
1212 – \midrule
1213 – GSSP  & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
1214 –      & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
1215 –      & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
1216 –      & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
1217 – GSSF  & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
1218 –      & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
1219 –      & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
1220 –      & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
1221 – \bottomrule
1222 – \end{tabular}
1223 – \label{tab:iceAng}
1224 – \end{table}
1225 –
1226 – Highly ordered systems are a difficult test for the pairwise methods
1227 – in that they lack the implicit periodicity of the Ewald summation.  As
1228 – expected, the energy gap agreement with {\sc spme} is reduced for the
1229 – {\sc sp} and {\sc sf} methods with parameters that were ideal for the
1230 – disordered liquid system.  Moving to higher $R_\textrm{c}$ helps
1231 – improve the agreement, though at an increase in computational cost.
1232 – The dynamics of this crystalline system (both in magnitude and
1233 – direction) are little affected. Both methods still reproduce the Ewald
1234 – behavior with the same parameter recommendations from the previous
1235 – section.
1236 –
1237 – It is also worth noting that {\sc rf} exhibits improved energy gap
1238 – results over the liquid water system.  One possible explanation is
1239 – that the ice I$_\textrm{c}$ crystal is ordered such that the net
1240 – dipole moment of the crystal is zero.  With $\epsilon_\textrm{S} =
1241 – \infty$, the reaction field incorporates this structural organization
1242 – by actively enforcing a zeroed dipole moment within each cutoff
1243 – sphere.
1244 –
1245 – \subsection{NaCl Melt Results}\label{sec:SaltMeltResults}
1246 –
1247 – A high temperature NaCl melt was tested to gauge the accuracy of the
1248 – pairwise summation methods in a disordered system of charges. The
1249 – results for the energy gap comparisons and the force vector magnitude
1250 – comparisons are shown in table \ref{tab:melt}.  The force vector
1251 – directionality results are displayed separately in table
1252 – \ref{tab:meltAng}.
1253 –
1254 – \begin{table}[htbp]
1255 – \centering
1256 – \caption{REGRESSION RESULTS OF THE MOLTEN SODIUM CHLORIDE SYSTEM FOR
1257 – $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES ({\it
1258 – lower})}
1259 –
1260 – \footnotesize
1261 – \begin{tabular}{@{} ccrrrrrr @{}}
1262 – \toprule
1263 – \toprule
1264 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1265 – \cmidrule(lr){3-4}
1266 – \cmidrule(lr){5-6}
1267 – \cmidrule(l){7-8}
1268 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1269 – \midrule
1270 – PC  &     & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
1271 – SP  & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
1272 –    & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
1273 –    & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
1274 –    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
1275 – SF  & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
1276 –    & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
1277 –    & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
1278 –    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
1279 –            \midrule
1280 – PC  &     & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
1281 – SP  & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
1282 –    & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
1283 –    & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
1284 –    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
1285 – SF  & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
1286 –    & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
1287 –    & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
1288 –    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
1289 – \bottomrule
1290 – \end{tabular}
1291 – \label{tab:melt}
1292 – \end{table}
1293 –
1294 – \begin{table}[htbp]
1295 – \centering
1296 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
1297 – OF THE FORCE VECTORS IN THE MOLTEN SODIUM CHLORIDE SYSTEM}      
1298 –
1299 – \footnotesize
1300 – \begin{tabular}{@{} ccrrrrrr @{}}
1301 – \toprule
1302 – \toprule
1303 – & & \multicolumn{3}{c}{Force $\sigma^2$} \\
1304 – \cmidrule(lr){3-5}
1305 – \cmidrule(l){6-8}
1306 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA \\
1307 – \midrule
1308 – PC  &     & 13.294 & 8.035 & 5.366 \\
1309 – SP  & 0.0 & 13.316 & 8.037 & 5.385 \\
1310 –    & 0.1 & 5.705 & 1.391 & 0.360 \\
1311 –    & 0.2 & 2.415 & 7.534 & 13.927 \\
1312 –    & 0.3 & 23.769 & 67.306 & 57.252 \\
1313 – SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
1314 –    & 0.1 & 1.687 & 0.653 & 0.272 \\
1315 –    & 0.2 & 2.598 & 7.523 & 13.930 \\
1316 –    & 0.3 & 23.734 & 67.305 & 57.252 \\
1317 – \bottomrule
1318 – \end{tabular}
1319 – \label{tab:meltAng}
1320 – \end{table}
1321 –
1322 – The molten NaCl system shows more sensitivity to the electrostatic
1323 – damping than the water systems. The most noticeable point is that the
1324 – undamped {\sc sf} method does very well at replicating the {\sc spme}
1325 – configurational energy differences and forces. Light damping appears
1326 – to minimally improve the dynamics, but this comes with a deterioration
1327 – of the energy gap results. In contrast, this light damping improves
1328 – the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
1329 – damping reduce the agreement with {\sc spme} for both methods. From
1330 – these observations, the undamped {\sc sf} method is the best choice
1331 – for disordered systems of charges.
1332 –
1333 – \subsection{NaCl Crystal Results}\label{sec:SaltCrystalResults}
1334 –
1335 – Similar to the use of ice I$_\textrm{c}$ to investigate the role of
1336 – order in molecular systems on the effectiveness of the pairwise
1337 – methods, the 1000~K NaCl crystal system was used to investigate the
1338 – accuracy of the pairwise summation methods in an ordered system of
1339 – charged particles. The results for the energy gap comparisons and the
1340 – force vector magnitude comparisons are shown in table \ref{tab:salt}.
1341 – The force vector directionality results are displayed separately in
1342 – table \ref{tab:saltAng}.
1343 –
1344 – \begin{table}[htbp]
1345 – \centering
1346 – \caption{REGRESSION RESULTS OF THE CRYSTALLINE SODIUM CHLORIDE
1347 – SYSTEM FOR $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES
1348 – ({\it lower})}
1349 –
1350 – \footnotesize
1351 – \begin{tabular}{@{} ccrrrrrr @{}}
1352 – \toprule
1353 – \toprule
1354 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1355 – \cmidrule(lr){3-4}
1356 – \cmidrule(lr){5-6}
1357 – \cmidrule(l){7-8}
1358 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1359 – \midrule
1360 – PC  &     & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
1361 – SP  & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
1362 –    & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
1363 –    & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
1364 –    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
1365 – SF  & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
1366 –    & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
1367 –    & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
1368 –    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
1369 – \midrule
1370 – PC  &     & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
1371 – SP  & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
1372 –    & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
1373 –    & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
1374 –    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
1375 – SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
1376 –    & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
1377 –    & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
1378 –    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
1379 – \bottomrule
1380 – \end{tabular}
1381 – \label{tab:salt}
1382 – \end{table}
1383 –
1384 – \begin{table}[htbp]
1385 – \centering
1386 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
1387 – DISTRIBUTIONS OF THE FORCE VECTORS IN THE CRYSTALLINE SODIUM CHLORIDE
1388 – SYSTEM}
1389 –
1390 – \footnotesize
1391 – \begin{tabular}{@{} ccrrrrrr @{}}
1392 – \toprule
1393 – \toprule
1394 – & & \multicolumn{3}{c}{Force $\sigma^2$} \\
1395 – \cmidrule(lr){3-5}
1396 – \cmidrule(l){6-8}
1397 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA \\
1398 – \midrule
1399 – PC  &     & 111.945 & 111.824 & 111.866 \\
1400 – SP  & 0.0 & 112.414 & 152.215 & 38.087 \\
1401 –    & 0.1 & 52.361 & 42.574 & 2.819 \\
1402 –    & 0.2 & 10.847 & 9.709 & 9.686 \\
1403 –    & 0.3 & 31.128 & 31.104 & 31.029 \\
1404 – SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
1405 –    & 0.1 & 9.462 & 3.303 & 1.721 \\
1406 –    & 0.2 & 11.454 & 9.813 & 9.701 \\
1407 –    & 0.3 & 31.120 & 31.105 & 31.029 \\
1408 – \bottomrule
1409 – \end{tabular}
1410 – \label{tab:saltAng}
1411 – \end{table}
1412 –
1413 – The crystalline NaCl system is the most challenging test case for the
1414 – pairwise summation methods, as evidenced by the results in tables
1415 – \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
1416 – {\sc sf} methods seem to be the best choices. These methods match well
1417 – with {\sc spme} across the energy gap, force magnitude, and force
1418 – directionality tests.  The {\sc sp} method struggles in all cases,
1419 – with the exception of good dynamics reproduction when using weak
1420 – electrostatic damping with a large cutoff radius.
1421 –
1422 – The moderate electrostatic damping case is not as good as we would
1423 – expect given the long-time dynamics results observed for this system
1424 – (see section \ref{sec:LongTimeDynamics}). Since the data tabulated in
1425 – tables \ref{tab:salt} and \ref{tab:saltAng} are a test of
1426 – instantaneous dynamics, this indicates that good long-time dynamics
1427 – comes in part at the expense of short-time dynamics.
1428 –
1429 – \subsection{0.11M NaCl Solution Results}
1430 –
1431 – In an effort to bridge the charged atomic and neutral molecular
1432 – systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
1433 – the liquid water system. This low ionic strength system consists of 4
1434 – ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
1435 – for the energy gap comparisons and the force and torque vector
1436 – magnitude comparisons are shown in table \ref{tab:solnWeak}.  The
1437 – force and torque vector directionality results are displayed
1438 – separately in table \ref{tab:solnWeakAng}, where the effect of
1439 – group-based cutoffs and switching functions on the {\sc sp} and {\sc
1440 – sf} potentials are investigated.
1441 –
1442 – \begin{table}[htbp]
1443 – \centering
1444 – \caption{REGRESSION RESULTS OF THE WEAK SODIUM CHLORIDE SOLUTION
1445 – SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
1446 – ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
1447 –
1448 – \footnotesize
1449 – \begin{tabular}{@{} ccrrrrrr @{}}
1450 – \toprule
1451 – \toprule
1452 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1453 – \cmidrule(lr){3-4}
1454 – \cmidrule(lr){5-6}
1455 – \cmidrule(l){7-8}
1456 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1457 – \midrule
1458 – PC  &     & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
1459 – SP  & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
1460 –    & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
1461 –    & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
1462 –    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
1463 – SF  & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
1464 –    & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
1465 –    & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
1466 –    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
1467 – GSC &     & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
1468 – RF  &     & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
1469 – \midrule
1470 – PC  &     & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
1471 – SP  & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
1472 –    & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
1473 –    & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
1474 –    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
1475 – SF  & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
1476 –    & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
1477 –    & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
1478 –    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
1479 – GSC &     & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
1480 – RF  &     & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
1481 – \midrule
1482 – PC  &     & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
1483 – SP  & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
1484 –    & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
1485 –    & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
1486 –    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
1487 – SF  & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
1488 –    & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
1489 –    & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
1490 –    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
1491 – GSC &     & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
1492 – RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
1493 – \bottomrule
1494 – \end{tabular}
1495 – \label{tab:solnWeak}
1496 – \end{table}
1497 –
1498 – \begin{table}[htbp]
1499 – \centering
1500 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
1501 – DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE WEAK SODIUM
1502 – CHLORIDE SOLUTION SYSTEM}
1503 –
1504 – \footnotesize
1505 – \begin{tabular}{@{} ccrrrrrr @{}}
1506 – \toprule
1507 – \toprule
1508 – & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
1509 – \cmidrule(lr){3-5}
1510 – \cmidrule(l){6-8}
1511 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
1512 – \midrule
1513 – PC  &     & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
1514 – SP  & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
1515 –    & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
1516 –    & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
1517 –    & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
1518 – SF  & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
1519 –    & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
1520 –    & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
1521 –    & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
1522 – GSC &     & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
1523 – RF  &     & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
1524 – \midrule
1525 – GSSP  & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
1526 –      & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
1527 –      & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
1528 –      & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
1529 – GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
1530 –      & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
1531 –      & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
1532 –      & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
1533 – \bottomrule
1534 – \end{tabular}
1535 – \label{tab:solnWeakAng}
1536 – \end{table}
1537 –
1538 – Because this system is a perturbation of the pure liquid water system,
1539 – comparisons are best drawn between these two sets. The {\sc sp} and
1540 – {\sc sf} methods are not significantly affected by the inclusion of a
1541 – few ions. The aspect of cutoff sphere neutralization aids in the
1542 – smooth incorporation of these ions; thus, all of the observations
1543 – regarding these methods carry over from section
1544 – \ref{sec:WaterResults}. The differences between these systems are more
1545 – visible for the {\sc rf} method. Though good force agreement is still
1546 – maintained, the energy gaps show a significant increase in the scatter
1547 – of the data.
1548 –
1549 – \subsection{1.1M NaCl Solution Results}
1550 –
1551 – The bridging of the charged atomic and neutral molecular systems was
1552 – further developed by considering a high ionic strength system
1553 – consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
1554 – M). The results for the energy gap comparisons and the force and
1555 – torque vector magnitude comparisons are shown in table
1556 – \ref{tab:solnStr}.  The force and torque vector directionality
1557 – results are displayed separately in table \ref{tab:solnStrAng}, where
1558 – the effect of group-based cutoffs and switching functions on the {\sc
1559 – sp} and {\sc sf} potentials are investigated.
1560 –
1561 – \begin{table}[htbp]
1562 – \centering
1563 – \caption{REGRESSION RESULTS OF THE STRONG SODIUM CHLORIDE SOLUTION
1564 – SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
1565 – ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
1566 –
1567 – \footnotesize
1568 – \begin{tabular}{@{} ccrrrrrr @{}}
1569 – \toprule
1570 – \toprule
1571 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1572 – \cmidrule(lr){3-4}
1573 – \cmidrule(lr){5-6}
1574 – \cmidrule(l){7-8}
1575 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1576 – \midrule
1577 – PC  &     & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
1578 – SP  & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
1579 –    & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
1580 –    & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
1581 –    & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
1582 – SF  & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
1583 –    & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
1584 –    & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
1585 –    & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
1586 – GSC &     & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
1587 – RF  &     & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
1588 – \midrule
1589 – PC  &     & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
1590 – SP  & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
1591 –    & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
1592 –    & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
1593 –    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
1594 – SF  & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
1595 –    & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
1596 –    & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
1597 –    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
1598 – GSC &     & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
1599 – RF  &     & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
1600 – \midrule
1601 – PC  &     & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
1602 – SP  & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
1603 –    & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
1604 –    & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
1605 –    & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
1606 – SF  & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
1607 –    & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
1608 –    & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
1609 –    & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
1610 – GSC &     & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
1611 – RF  &     & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
1612 – \bottomrule
1613 – \end{tabular}
1614 – \label{tab:solnStr}
1615 – \end{table}
943  
1617 – \begin{table}[htbp]
1618 – \centering
1619 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
1620 – OF THE FORCE AND TORQUE VECTORS IN THE STRONG SODIUM CHLORIDE SOLUTION
1621 – SYSTEM}
944  
1623 – \footnotesize
1624 – \begin{tabular}{@{} ccrrrrrr @{}}
1625 – \toprule
1626 – \toprule
1627 – & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
1628 – \cmidrule(lr){3-5}
1629 – \cmidrule(l){6-8}
1630 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
1631 – \midrule
1632 – PC  &     & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
1633 – SP  & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
1634 –    & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
1635 –    & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
1636 –    & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
1637 – SF  & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
1638 –    & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
1639 –    & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
1640 –    & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
1641 – GSC &     & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
1642 – RF  &     & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
1643 – \midrule
1644 – GSSP  & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
1645 –      & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
1646 –      & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
1647 –      & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
1648 – GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
1649 –      & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
1650 –      & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
1651 –      & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
1652 – \bottomrule
1653 – \end{tabular}
1654 – \label{tab:solnStrAng}
1655 – \end{table}
1656 –
1657 – The {\sc rf} method struggles with the jump in ionic strength. The
1658 – configuration energy differences degrade to unusable levels while the
1659 – forces and torques show a more modest reduction in the agreement with
1660 – {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
1661 – and this attribute is apparent in these results.
1662 –
1663 – The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
1664 – their agreement with {\sc spme}. With these results, we still
1665 – recommend undamped to moderate damping for the {\sc sf} method and
1666 – moderate damping for the {\sc sp} method, both with cutoffs greater
1667 – than 12~\AA.
1668 –
1669 – \subsection{6~\AA\ Argon Sphere in SPC/E Water Results}
1670 –
1671 – The final model system studied was a 6~\AA\ sphere of Argon solvated
1672 – by SPC/E water. This serves as a test case of a specifically sized
1673 – electrostatic defect in a disordered molecular system. The results for
1674 – the energy gap comparisons and the force and torque vector magnitude
1675 – comparisons are shown in table \ref{tab:argon}.  The force and torque
1676 – vector directionality results are displayed separately in table
1677 – \ref{tab:argonAng}, where the effect of group-based cutoffs and
1678 – switching functions on the {\sc sp} and {\sc sf} potentials are
1679 – investigated.
1680 –
1681 – \begin{table}[htbp]
1682 – \centering
1683 – \caption{REGRESSION RESULTS OF THE 6~\AA\ ARGON SPHERE IN LIQUID
1684 – WATER SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR
1685 – MAGNITUDES ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
1686 –
1687 – \footnotesize
1688 – \begin{tabular}{@{} ccrrrrrr @{}}
1689 – \toprule
1690 – \toprule
1691 – & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
1692 – \cmidrule(lr){3-4}
1693 – \cmidrule(lr){5-6}
1694 – \cmidrule(l){7-8}
1695 – Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
1696 – \midrule
1697 – PC  &     & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
1698 – SP  & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
1699 –    & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
1700 –    & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
1701 –    & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
1702 – SF  & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
1703 –    & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
1704 –    & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
1705 –    & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
1706 – GSC &     & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
1707 – RF  &     & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
1708 – \midrule
1709 – PC  &     & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
1710 – SP  & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
1711 –    & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
1712 –    & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
1713 –    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
1714 – SF  & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
1715 –    & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
1716 –    & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
1717 –    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
1718 – GSC &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
1719 – RF  &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
1720 – \midrule
1721 – PC  &     & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
1722 – SP  & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
1723 –    & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
1724 –    & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
1725 –    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
1726 – SF  & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
1727 –    & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
1728 –    & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
1729 –    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
1730 – GSC &     & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
1731 – RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
1732 – \bottomrule
1733 – \end{tabular}
1734 – \label{tab:argon}
1735 – \end{table}
1736 –
1737 – \begin{table}[htbp]
1738 – \centering
1739 – \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
1740 – DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE 6~\AA\ SPHERE OF
1741 – ARGON IN LIQUID WATER SYSTEM}  
1742 –
1743 – \footnotesize
1744 – \begin{tabular}{@{} ccrrrrrr @{}}
1745 – \toprule
1746 – \toprule
1747 – & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
1748 – \cmidrule(lr){3-5}
1749 – \cmidrule(l){6-8}
1750 – Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
1751 – \midrule
1752 – PC  &     & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
1753 – SP  & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
1754 –    & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
1755 –    & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
1756 –    & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
1757 – SF  & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
1758 –    & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
1759 –    & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
1760 –    & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
1761 – GSC &     & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
1762 – RF  &     & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
1763 – \midrule
1764 – GSSP  & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
1765 –      & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
1766 –      & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
1767 –      & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
1768 – GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
1769 –      & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
1770 –      & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
1771 –      & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
1772 – \bottomrule
1773 – \end{tabular}
1774 – \label{tab:argonAng}
1775 – \end{table}
1776 –
1777 – This system does not appear to show any significant deviations from
1778 – the previously observed results. The {\sc sp} and {\sc sf} methods
1779 – have agreements similar to those observed in section
1780 – \ref{sec:WaterResults}. The only significant difference is the
1781 – improvement in the configuration energy differences for the {\sc rf}
1782 – method. This is surprising in that we are introducing an inhomogeneity
1783 – to the system; however, this inhomogeneity is charge-neutral and does
1784 – not result in charged cutoff spheres. The charge-neutrality of the
1785 – cutoff spheres, which the {\sc sp} and {\sc sf} methods explicitly
1786 – enforce, seems to play a greater role in the stability of the {\sc rf}
1787 – method than the required homogeneity of the environment.
1788 –
1789 –
945   \section{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals}\label{sec:ShortTimeDynamics}
946  
947   Zahn {\it et al.} investigated the structure and dynamics of water
# Line 2635 | Line 1790 | these Ewald examples, the results discussed in section
1790  
1791   Although it is tempting to choose damping parameters equivalent to
1792   these Ewald examples, the results discussed in sections
1793 < \ref{sec:EnergyResults} through \ref{sec:IndividualResults} indicate
1794 < that values this high are destructive to both the energetics and
1795 < dynamics. Ideally, $\alpha$ should not exceed 0.3~\AA$^{-1}$ for any of
1796 < the cutoff values in this range. If the optimal damping parameter is
1797 < chosen to be midway between 0.275 and 0.3~\AA$^{-1}$ (0.2875~\AA$^{-1}$)
1798 < at the 9~\AA\ cutoff, then the linear trend with $R_\textrm{c}$ will
1799 < always keep $\alpha$ below 0.3~\AA$^{-1}$. This linear progression
1800 < would give values of 0.2875, 0.2625, 0.2375, and 0.2125~\AA$^{-1}$ for
1801 < cutoff radii of 9, 10, 11, and 12~\AA. Setting this to be the default
1802 < behavior for the damped {\sc sf} technique will result in consistent
1803 < dielectric behavior for these and other condensed molecular systems,
1804 < regardless of the chosen cutoff radius. The static dielectric
1805 < constants for TIP5P-E, TIP4P-Ew, SPC/E, and SSD/RF will be
1806 < approximately fixed at 74, 52, 58, and 89 respectively. These values
1807 < are generally lower than the values reported in the literature;
1808 < however, the relative dielectric behavior scales as expected when
1809 < comparing the models to one another.
1793 > \ref{sec:EnergyResults} through \ref{sec:FTDirResults} and appendix
1794 > \ref{app:IndividualResults} indicate that values this high are
1795 > destructive to both the energetics and dynamics. Ideally, $\alpha$
1796 > should not exceed 0.3~\AA$^{-1}$ for any of the cutoff values in this
1797 > range. If the optimal damping parameter is chosen to be midway between
1798 > 0.275 and 0.3~\AA$^{-1}$ (0.2875~\AA$^{-1}$) at the 9~\AA\ cutoff,
1799 > then the linear trend with $R_\textrm{c}$ will always keep $\alpha$
1800 > below 0.3~\AA$^{-1}$. This linear progression would give values of
1801 > 0.2875, 0.2625, 0.2375, and 0.2125~\AA$^{-1}$ for cutoff radii of 9,
1802 > 10, 11, and 12~\AA. Setting this to be the default behavior for the
1803 > damped {\sc sf} technique will result in consistent dielectric
1804 > behavior for these and other condensed molecular systems, regardless
1805 > of the chosen cutoff radius. The static dielectric constants for
1806 > TIP5P-E, TIP4P-Ew, SPC/E, and SSD/RF will be fixed at approximately
1807 > 74, 52, 58, and 89 respectively. These values are generally lower than
1808 > the values reported in the literature; however, the relative
1809 > dielectric behavior scales as expected when comparing the models to
1810 > one another.
1811  
1812   \section{Conclusions}\label{sec:PairwiseConclusions}
1813  

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