trunk/electrostaticMethodsPaper/electrostaticMethods.tex

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\begin{document}

\title{On the necessity of the Ewald Summation in molecular simulations.}

\author{Christopher J. Fennell and J. Daniel Gezelter \\
Department of Chemistry and Biochemistry\\ 
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle
%\doublespacing

\begin{abstract}
\end{abstract}

%\narrowtext 

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%                              BODY OF TEXT
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\section{Introduction}

In this paper, a variety of simulation situations were analyzed to determine the relative effectiveness of the adapted Wolf spherical truncation schemes at reproducing the results obtained using a smooth particle mesh Ewald (SPME) summation technique.  In addition to the Shifted-Potential (SP) and Shifted-Force (SF) adapted Wolf methods, both reaction field and uncorrected cutoff methods were included for comparison purposes.  The general usability of these methods in both Monte Carlo (MC) and Molecular Dynamics (MD) calculations was assessed through statistical analysis over the combined results from all of the following studied systems:
\begin{list}{-}{}
\item Liquid Water
\item Crystalline Water (Ice I$_\textrm{c}$)
\item 1 M Solution of NaCl in Water 
\item 10 M Solution of NaCl in Water 
\item 6 \AA\  Radius Sphere of Argon in Water 
\item NaCl Crystal
\item NaCl Melt
\end{list}
Additional discussion on the results from the individual systems was also performed to identify limitations of the considered methods in specific systems.

\section{Methods}

In each of the simulated systems, 500 distinct configurations were generated, and the electrostatic summation methods were compared via sequential application on each of these fixed configurations.  The methods compared include SPME, the aforementioned SP and SF methods - both with damping parameters ($\alpha$) of 0, 0.1, 0.2, and 0.3 \AA$^{-1}$, reaction field with an infinite dielectric constant, and an unmodified cutoff.  Group-based cutoffs with a fifth-order polynomial switching function were necessary for the reaction field simulations and were utilized in the SP, SF, and pure cutoff methods for comparison to the standard lack of group-based cutoffs with a hard truncation.  

Generation of the system configurations was dependent on the system type.  For the solid and liquid water configurations, configuration snapshots were taken at regular intervals from higher temperature 1000 SPC/E water molecule trajectories and individually equilibrated.  The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl- ions and were selected and equilibrated in the same fashion as the water systems.  For the 1 and 10 M NaCl solutions, 4 and 40 ions, respectively, were first solvated in a 1000 water molecule boxes.  Ion and water positions were then randomly swapped, and the resulting configurations were again individually equilibrated.  Finally, for the Argon/Water "charge void" systems, the identities of all the SPC/E waters within 6 \AA\ of the center of the equilibrated water configurations were converted to argon (Fig. \ref{argonSlice}).

\begin{figure}
\centering
\includegraphics[width=3.25in]{./slice.pdf}
\caption{A slice from the center of a water box used in a charge void simulation.  The darkened region represents the boundary sphere within which the water molecules were converted to argon atoms.}
\label{argonSlice}
\end{figure}

All of these comparisons were performed with three different cutoff radii (9, 12, and 15 \AA) to investigate the cutoff radius dependence of the various techniques.  It should be noted that the damping parameter chosen in SPME, or so called ``Ewald Coefficient", has a significant effect on the energies and forces calculated.  Typical molecular mechanics packages default this to a value dependent on the cutoff radius and a tolerance (typically less than $1 \times 10^{-5}$ kcal/mol).  We chose a tolerance of $1 \times 10^{-8}$, resulting in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ respectively.

\section{Results and Discussion}

In order to evaluate the performance of the adapted Wolf SP and SF electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME.  Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated.  Linear least squares regression of the $\Delta$E values between configurations using SPME against $\Delta$E values using tested methods provides a quantitative comparison of this agreement.  Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods.  The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}).

\begin{figure}
\centering
\includegraphics[width=3.25in]{./linearFit.pdf}
\caption{Example least squares regression of the $\Delta$E between configurations for the SF method against SPME in the pure water system.  }
\label{linearFit}
\end{figure}

With 500 independent configurations, 124,750 $\Delta$E data points are used in a regression of a single system.  A table with the results for analysis of To gauge the applicability of each method in the general case, all the different system types were included in a separate Figure \ref{delEplot} shows the results for analysis of all the simulation types 

\section{Conclusions}

\section{Acknowledgments}

\newpage

\bibliographystyle{achemso}
\bibliography{electrostaticMethods}


\end{document}
Revision:	2590
Committed:	Fri Feb 10 22:35:53 2006 UTC (18 years, 4 months ago) by chrisfen
Content type:	application/x-tex
File size:	6469 byte(s)
Log Message:	added some plots and text
#	Content
1	%\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
2	\documentclass[12pt]{article}
3	\usepackage{endfloat}
4	\usepackage{amsmath}
5	\usepackage{epsf}
6	\usepackage{times}
7	\usepackage{mathptm}
8	\usepackage{setspace}
9	\usepackage{tabularx}
10	\usepackage{graphicx}
11	%\usepackage{berkeley}
12	\usepackage[ref]{overcite}
13	\pagestyle{plain}
14	\pagenumbering{arabic}
15	\oddsidemargin 0.0cm \evensidemargin 0.0cm
16	\topmargin -21pt \headsep 10pt
17	\textheight 9.0in \textwidth 6.5in
18	\brokenpenalty=10000
19	\renewcommand{\baselinestretch}{1.2}
20	\renewcommand\citemid{\ } % no comma in optional reference note
21
22	\begin{document}
23
24	\title{On the necessity of the Ewald Summation in molecular simulations.}
25
26	\author{Christopher J. Fennell and J. Daniel Gezelter \\
27	Department of Chemistry and Biochemistry\\
28	University of Notre Dame\\
29	Notre Dame, Indiana 46556}
30
31	\date{\today}
32
33	\maketitle
34	%\doublespacing
35
36	\begin{abstract}
37	\end{abstract}
38
39	%\narrowtext
40
41	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
42	% BODY OF TEXT
43	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
44
45	\section{Introduction}
46
47	In this paper, a variety of simulation situations were analyzed to determine the relative effectiveness of the adapted Wolf spherical truncation schemes at reproducing the results obtained using a smooth particle mesh Ewald (SPME) summation technique. In addition to the Shifted-Potential (SP) and Shifted-Force (SF) adapted Wolf methods, both reaction field and uncorrected cutoff methods were included for comparison purposes. The general usability of these methods in both Monte Carlo (MC) and Molecular Dynamics (MD) calculations was assessed through statistical analysis over the combined results from all of the following studied systems:
48	\begin{list}{-}{}
49	\item Liquid Water
50	\item Crystalline Water (Ice I$_\textrm{c}$)
51	\item 1 M Solution of NaCl in Water
52	\item 10 M Solution of NaCl in Water
53	\item 6 \AA\ Radius Sphere of Argon in Water
54	\item NaCl Crystal
55	\item NaCl Melt
56	\end{list}
57	Additional discussion on the results from the individual systems was also performed to identify limitations of the considered methods in specific systems.
58
59	\section{Methods}
60
61	In each of the simulated systems, 500 distinct configurations were generated, and the electrostatic summation methods were compared via sequential application on each of these fixed configurations. The methods compared include SPME, the aforementioned SP and SF methods - both with damping parameters ($\alpha$) of 0, 0.1, 0.2, and 0.3 \AA$^{-1}$, reaction field with an infinite dielectric constant, and an unmodified cutoff. Group-based cutoffs with a fifth-order polynomial switching function were necessary for the reaction field simulations and were utilized in the SP, SF, and pure cutoff methods for comparison to the standard lack of group-based cutoffs with a hard truncation.
62
63	Generation of the system configurations was dependent on the system type. For the solid and liquid water configurations, configuration snapshots were taken at regular intervals from higher temperature 1000 SPC/E water molecule trajectories and individually equilibrated. The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl- ions and were selected and equilibrated in the same fashion as the water systems. For the 1 and 10 M NaCl solutions, 4 and 40 ions, respectively, were first solvated in a 1000 water molecule boxes. Ion and water positions were then randomly swapped, and the resulting configurations were again individually equilibrated. Finally, for the Argon/Water "charge void" systems, the identities of all the SPC/E waters within 6 \AA\ of the center of the equilibrated water configurations were converted to argon (Fig. \ref{argonSlice}).
64
65	\begin{figure}
66	\centering
67	\includegraphics[width=3.25in]{./slice.pdf}
68	\caption{A slice from the center of a water box used in a charge void simulation. The darkened region represents the boundary sphere within which the water molecules were converted to argon atoms.}
69	\label{argonSlice}
70	\end{figure}
71
72	All of these comparisons were performed with three different cutoff radii (9, 12, and 15 \AA) to investigate the cutoff radius dependence of the various techniques. It should be noted that the damping parameter chosen in SPME, or so called ``Ewald Coefficient", has a significant effect on the energies and forces calculated. Typical molecular mechanics packages default this to a value dependent on the cutoff radius and a tolerance (typically less than $1 \times 10^{-5}$ kcal/mol). We chose a tolerance of $1 \times 10^{-8}$, resulting in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ respectively.
73
74	\section{Results and Discussion}
75
76	In order to evaluate the performance of the adapted Wolf SP and SF electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME. Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated. Linear least squares regression of the $\Delta$E values between configurations using SPME against $\Delta$E values using tested methods provides a quantitative comparison of this agreement. Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods. The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}).
77
78	\begin{figure}
79	\centering
80	\includegraphics[width=3.25in]{./linearFit.pdf}
81	\caption{Example least squares regression of the $\Delta$E between configurations for the SF method against SPME in the pure water system. }
82	\label{linearFit}
83	\end{figure}
84
85	With 500 independent configurations, 124,750 $\Delta$E data points are used in a regression of a single system. A table with the results for analysis of To gauge the applicability of each method in the general case, all the different system types were included in a separate Figure \ref{delEplot} shows the results for analysis of all the simulation types
86
87	\section{Conclusions}
88
89	\section{Acknowledgments}
90
91	\newpage
92
93	\bibliographystyle{achemso}
94	\bibliography{electrostaticMethods}
95
96
97	\end{document}