--- trunk/electrostaticMethodsPaper/SupportingInfo.tex 2006/03/22 21:00:07 2658 +++ trunk/electrostaticMethodsPaper/SupportingInfo.tex 2006/03/23 15:46:45 2666 @@ -1,5 +1,5 @@ %\documentclass[prb,aps,twocolumn,tabularx]{revtex4} -\documentclass[12pt]{article} +\documentclass[11pt]{article} %\usepackage{endfloat} \usepackage{amsmath} \usepackage{amssymb} @@ -24,21 +24,22 @@ This document includes individual system-based compari \begin{document} This document includes individual system-based comparisons of the -studied methods with smooth particle-mesh Ewald. Each of the seven -systems comprises its own section and has its own discussion and -tabular listing of the results for the $\Delta E$, force and torque -vector magnitude, and force and torque vector direction comparisons. +studied methods with smooth particle mesh Ewald {\sc spme}. Each of +the seven systems comprises its own section and has its own discussion +and tabular listing of the results for the $\Delta E$, force and +torque vector magnitude, and force and torque vector direction +comparisons. -\section{\label{app-water}Liquid Water} +\section{\label{app:water}Liquid Water} -500 liquid state configurations were generated as described in the -Methods section using the SPC/E model of water.\cite{Berendsen87} The -results for the energy gap comparisons and the force and torque vector -magnitude comparisons are shown in table \ref{tab:spce}. The force -and torque vector directionality results are displayed separately in -table \ref{tab:spceAng}, where the effect of group-based cutoffs and +The first system considered was liquid water at 300K using the SPC/E +model of water.\cite{Berendsen87} The results for the energy gap +comparisons and the force and torque vector magnitude comparisons are +shown in table \ref{tab:spce}. The force and torque vector +directionality results are displayed separately in table +\ref{tab:spceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are -investigated. +investigated. \begin{table}[htbp] \centering \caption{Regression results for the liquid water system. Tabulated @@ -66,10 +67,8 @@ GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0. & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ -RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ - +RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ \midrule - PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ @@ -81,9 +80,7 @@ RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1. & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ - \midrule - PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ @@ -141,49 +138,49 @@ GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & \label{tab:spceAng} \end{table} -For the most parts, the water results appear to parallel the combined -results seen in the discussion in the main paper. There is good -agreement with SPME in both energetic and dynamic behavior when using -the {\sc sf} method with and without damping. The {\sc sp} method does -well with an $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff -radii greater than 12 \AA. The results for both of these methods also -begin to decay as damping gets too large. - -The pure cutoff (PC) method performs poorly, as seen in the main -discussion section. In contrast to the combined values, however, the -use of a switching function and group based cutoffs really improves -the results for these neutral water molecules. The group switched -cutoff (GSC) shows mimics the energetics of SPME more poorly than the -{\sc sp} (with moderate damping) and {\sc sf} methods, but the -dynamics are quite good. The switching functions corrects -discontinuities in the potential and forces, leading to the improved -results. Such improvements with the use of a switching function has -been recognized in previous studies,\cite{Andrea83,Steinbach94} and it -is a useful tactic for stably incorporating local area electrostatic -effects. - -The reaction field (RF) method simply extends the results observed in -the GSC case. Both methods are similar in form (i.e. neutral groups, -switching function), but RF incorporates an added effect from the -external dielectric. This similarity translates into the same good -dynamic results and improved energetic results. These still fall -short of the moderately damped {\sc sp} and {\sc sf} methods, but they -display how incorporating some implicit properties of the surroundings -(i.e. $\epsilon_\textrm{S}$) can improve results. +The water results appear to parallel the combined results seen in the +discussion section of the main paper. There is good agreement with +{\sc spme} in both energetic and dynamic behavior when using the {\sc sf} +method with and without damping. The {\sc sp} method does well with an +$\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater +than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}. +The pure cutoff ({\sc pc}) method performs poorly, again mirroring the +observations in the main portion of this paper. In contrast to the +combined values, however, the use of a switching function and group +based cutoffs really improves the results for these neutral water +molecules. The group switched cutoff ({\sc gsc}) does not mimic the +energetics of {\sc spme} as well as the {\sc sp} (with moderate +damping) and {\sc sf} methods, but the dynamics are quite good. The +switching functions corrects discontinuities in the potential and +forces, leading to these improved results. Such improvements with the +use of a switching function has been recognized in previous +studies,\cite{Andrea83,Steinbach94} and this proves to be a useful +tactic for stably incorporating local area electrostatic effects. + +The reaction field ({\sc rf}) method simply extends upon the results +observed in the {\sc gsc} case. Both methods are similar in form +(i.e. neutral groups, switching function), but {\sc rf} incorporates +an added effect from the external dielectric. This similarity +translates into the same good dynamic results and improved energetic +agreement with {\sc spme}. Though this agreement is not to the level +of the moderately damped {\sc sp} and {\sc sf} methods, these results +show how incorporating some implicit properties of the surroundings +(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction. + A final note for the liquid water system, use of group cutoffs and a -switching function also leads to noticeable improvements in the {\sc -sp} and {\sc sf} methods, primarily in directionality of the force and -torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant -narrowing of the angle distribution in the cases with little to no -damping and only modest improvement for the ideal conditions ($\alpha$ -= 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The {\sc sf} -method simply shows modest narrowing across all damping and cutoff -ranges of interest. Group cutoffs and the switching function do -nothing for cases were error is introduced by overdamping the -potentials. +switching function leads to noticeable improvements in the {\sc sp} +and {\sc sf} methods, primarily in directionality of the force and +torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows +significant narrowing of the angle distribution when using little to +no damping and only modest improvement for the recommended conditions +($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The +{\sc sf} method shows modest narrowing across all damping and cutoff +ranges of interest. When overdamping these methods, group cutoffs and +the switching function do not improve the force and torque +directionalities. -\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} +\section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$} In addition to the disordered molecular system above, the ordered molecular system of ice I$_\textrm{c}$ was also considered. The @@ -287,35 +284,35 @@ GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & \label{tab:iceAng} \end{table} -Highly ordered systems are a difficult test for the pairwise systems -in that they lack the periodicity inherent to the Ewald summation. As -expected, the energy gap agreement with SPME reduces for the {\sc sp} -and {\sc sf} with parameters that were perfectly acceptable for the -disordered liquid system. Moving to higher $R_\textrm{c}$ remedies -this degraded performance, though at increase in computational cost. -However, the dynamics of this crystalline system (both in magnitude -and direction) are little affected. Both methods still reproduce the -Ewald behavior with the same parameter recommendations from the -previous section. +Highly ordered systems are a difficult test for the pairwise methods +in that they lack the periodicity term of the Ewald summation. As +expected, the energy gap agreement with {\sc spme} reduces for the +{\sc sp} and {\sc sf} methods with parameters that were acceptable for +the disordered liquid system. Moving to higher $R_\textrm{c}$ helps +improve the agreement, though at an increase in computational cost. +The dynamics of this crystalline system (both in magnitude and +direction) are little affected. Both methods still reproduce the Ewald +behavior with the same parameter recommendations from the previous +section. -It is also worth noting that RF exhibits a slightly improved energy -gap results over the liquid water system. One possible explanation is +It is also worth noting that {\sc rf} exhibits improved energy gap +results over the liquid water system. One possible explanation is that the ice I$_\textrm{c}$ crystal is ordered such that the net dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = \infty$, the reaction field incorporates this structural organization by actively enforcing a zeroed dipole moment within each cutoff sphere. -\section{\label{app-melt}NaCl Melt} +\section{\label{app:melt}NaCl Melt} A high temperature NaCl melt was tested to gauge the accuracy of the -pairwise summation methods in a highly charge disordered system. The -results for the energy gap comparisons and the force and torque vector +pairwise summation methods in a charged disordered system. The results +for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:melt}. The force and torque vector directionality results are displayed separately in table \ref{tab:meltAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are -investigated. +investigated. \begin{table}[htbp] \centering @@ -378,9 +375,18 @@ SF & 0.0 & 1.693 & 0.603 & 0.256 \\ \label{tab:meltAng} \end{table} -The molten NaCl system shows the a +The molten NaCl system shows more sensitivity to the electrostatic +damping than the water systems. The most noticeable point is that the +undamped {\sc sf} method does very well at replicating the {\sc spme} +configurational energy differences and forces. Light damping appears +to minimally improve the dynamics, but this comes with a deterioration +of the energy gap results. In contrast, this light damping improves +the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic +damping reduce the agreement with {\sc spme} for both methods. From +these observations, the undamped {\sc sf} method is the best choice +for disordered systems of charges. -\section{\label{app-salt}NaCl Crystal} +\section{\label{app:salt}NaCl Crystal} A 1000K NaCl crystal was used to investigate the accuracy of the pairwise summation methods in an ordered system of charged @@ -459,8 +465,28 @@ SF & 0.0 & 10.025 & 3.555 & 1.648 \\ \label{tab:saltAng} \end{table} -\section{\label{app-sol1}Weak NaCl Solution} +The crystalline NaCl system is the most challenging test case for the +pairwise summation methods, as evidenced by the results in tables +\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped +{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best +choices. These methods match well with {\sc spme} across the energy +gap, force magnitude, and force directionality tests. The {\sc sp} +method struggles in all cases, with the exception of good dynamics +reproduction when using weak electrostatic damping with a large cutoff +radius. +The moderate electrostatic damping case is not as good as we would +expect given the good long-time dynamics results observed for this +system. Since the data tabulated in table \ref{tab:salt} and +\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates +that good long-time dynamics comes in part at the expense of +short-time dynamics. Further indication of this comes from the full +power spectra shown in the main text. It appears as though a +distortion is introduced between 200 to 350 cm$^{-1}$ with increased +$\alpha$. + +\section{\label{app:solnWeak}Weak NaCl Solution} + In an effort to bridge the charged atomic and neutral molecular systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into the liquid water system. This low ionic strength system consists of 4 @@ -571,17 +597,28 @@ GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & \label{tab:solnWeakAng} \end{table} -\section{\label{app-sol10}Strong NaCl Solution} +Because this system is a perturbation of the pure liquid water system, +comparisons are best drawn between these two sets. The {\sc sp} and +{\sc sf} methods are not significantly affected by the inclusion of a +few ions. The aspect of cutoff sphere neutralization aids in the +smooth incorporation of these ions; thus, all of the observations +regarding these methods carry over from section \ref{app:water}. The +differences between these systems are more visible for the {\sc rf} +method. Though good force agreement is still maintained, the energy +gaps show a significant increase in the data scatter. This foreshadows +the breakdown of the method as we introduce charged inhomogeneities. +\section{\label{app:solnStr}Strong NaCl Solution} + The bridging of the charged atomic and neutral molecular systems was -furthered by considering a high ionic strength system consisting of 40 -ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for -the energy gap comparisons and the force and torque vector magnitude -comparisons are shown in table \ref{tab:solnWeak}. The force and -torque vector directionality results are displayed separately in table -\ref{tab:solnWeakAng}, where the effect of group-based cutoffs and -switching functions on the {\sc sp} and {\sc sf} potentials are -investigated. +further developed by considering a high ionic strength system +consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 +M). The results for the energy gap comparisons and the force and +torque vector magnitude comparisons are shown in table +\ref{tab:solnWeak}. The force and torque vector directionality +results are displayed separately in table \ref{tab:solnWeakAng}, where +the effect of group-based cutoffs and switching functions on the {\sc +sp} and {\sc sf} potentials are investigated. \begin{table}[htbp] \centering @@ -676,8 +713,20 @@ GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & \label{tab:solnStrAng} \end{table} -\section{\label{app-argon}Argon Sphere in Water} +The {\sc rf} method struggles with the jump in ionic strength. The +configuration energy difference degrade to unusable levels while the +forces and torques show a more modest reduction in the agreement with +{\sc spme}. The {\sc rf} method was designed for homogeneous systems, +and this attribute is apparent in these results. +The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain +their agreement with {\sc spme}. With these results, we still +recommend no to moderate damping for the {\sc sf} method and moderate +damping for the {\sc sp} method, both with cutoffs greater than 12 +\AA. + +\section{\label{app:argon}Argon Sphere in Water} + The final model system studied was 6 \AA\ sphere of Argon solvated by SPC/E water. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table @@ -784,6 +833,18 @@ GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & \label{tab:argonAng} \end{table} +This system appears not to show in any significant deviation in the +previously observed results. The {\sc sp} and {\sc sf} methods give +result qualities similar to those observed in section +\ref{app:water}. The only significant difference is the improvement +for the configuration energy differences for the {\sc rf} method. This +is surprising in that we are introducing an inhomogeneity to the +system; however, this inhomogeneity is charge-neutral and does not +result in charged cutoff spheres. The charge-neutrality of the cutoff +spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce, +seems to play a greater role in the stability of the {\sc rf} method +than the required homogeneity of the environment. + \newpage \bibliographystyle{jcp2}