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reproduce the behavior exhibited in simulations using SPME with an |
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$\mathscr{O}(N)$ computational cost, equivalent to merely the |
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real-space portion of the lattice summation. |
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+ |
|
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\end{abstract} |
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\newpage |
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\begin{figure} |
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\centering |
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< |
\includegraphics[width = \linewidth]{./linearFit.pdf} |
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\caption{Example least squares regression of the $\Delta E$ between configurations for the SF method against SPME in the pure water system. } |
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> |
\includegraphics[width = \linewidth]{./dualLinear.pdf} |
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> |
\caption{Example least squares regressions of the configuration energy differences for SPC/E water systems. The upper plot shows a data set with a poor correlation coefficient ($R^2$), while the lower plot shows a data set with a good correlation coefficient.} |
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\label{fig:linearFit} |
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\end{figure} |
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|
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< |
Each system type (detailed in section \ref{sec:RepSims}) studied consisted of 500 independent configurations, each equilibrated from higher temperature trajectories. Thus, 124,750 $\Delta E$ data points are used in a regression of a single system type. Results and discussion for the individual analysis of each of the system types appear in the supporting information, while the cumulative results over all the investigated systems appears below in section \ref{sec:EnergyResults}. |
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> |
Each system type (detailed in section \ref{sec:RepSims}) studied |
| 276 |
> |
consisted of 500 independent configurations, each equilibrated from |
| 277 |
> |
higher temperature trajectories. Thus, 124,750 $\Delta E$ data points |
| 278 |
> |
are used in a regression of a single system type. Results and |
| 279 |
> |
discussion for the individual analysis of each of the system types |
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> |
appear in the supporting information, while the cumulative results |
| 281 |
> |
over all the investigated systems appears below in section |
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> |
\ref{sec:EnergyResults}. |
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|
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\subsection{Molecular Dynamics and the Force and Torque Vectors}\label{sec:MDMethods} |
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Evaluation of the pairwise methods (outlined in section \ref{sec:ESMethods}) for use in MD simulations was performed through comparison of the force and torque vectors obtained with those from SPME. Both the magnitude and the direction of these vectors on each of the bodies in the system were analyzed. For the magnitude of these vectors, linear least squares regression analysis can be performed as described previously for comparing $\Delta E$ values. Instead of a single value between two system configurations, there is a value for each particle in each configuration. For a system of 1000 water molecules and 40 ions, there are 1040 force vectors and 1000 torque vectors. With 500 configurations, this results in 520,000 force and 500,000 torque vector comparisons samples for each system type. |
