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%\preprint{AIP/123-QED} |
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\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods} |
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\title{Real space electrostatics for multipoles. II. Comparisons with |
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the Ewald Sum} |
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|
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556} |
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the three dimensional Ewald sum is appropriate for slab |
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geometries.\cite{Parry:1975if} Modified Ewald methods that were |
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developed to handle two-dimensional (2-D) electrostatic |
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interactions,\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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but these methods were originally quite computationally |
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interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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These methods were originally quite computationally |
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expensive.\cite{Spohr97,Yeh99} There have been several successful |
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efforts that reduced the computational cost of 2-D lattice |
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summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} |
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efforts that reduced the computational cost of 2-D lattice summations, |
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bringing them more in line with the scaling for the full 3-D |
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treatments. The inherent periodicity in the Ewald’s method can also |
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be problematic for interfacial molecular |
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systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
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treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The |
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inherent periodicity required by the Ewald method can also be |
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problematic in a number of protein/solvent and ionic solution |
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environments.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
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|
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\subsection{Real-space methods} |
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Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{schematic.pdf} |
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\includegraphics[width=\linewidth]{schematic.eps} |
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\caption{Top: Ionic systems exhibit local clustering of dissimilar |
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charges (in the smaller grey circle), so interactions are |
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effectively charge-multipole at longer distances. With hard |
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and have been compared with the values obtained from the multipolar |
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Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
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between two configurations is the primary quantity that governs how |
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the simulation proceeds. These differences are the most imporant |
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the simulation proceeds. These differences are the most important |
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indicators of the reliability of a method even if the absolute |
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energies are not exact. For each of the multipolar systems listed |
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above, we have compared the change in electrostatic potential energy |
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of configurational |
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energy differences for the real-space electrostatic methods |
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compared with the reference Ewald sum. Results with a value equal |
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of the force vector |
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magnitudes for the real-space electrostatic methods compared with |
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the reference Ewald sum. Results with a value equal to 1 (dashed |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of the torque vector |
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magnitudes for the real-space electrostatic methods compared with |
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the reference Ewald sum. Results with a value equal to 1 (dashed |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
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\includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps} |
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\caption{The circular variance of the direction of the force and |
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torque vectors obtained from the real-space methods around the |
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reference Ewald vectors. A variance equal to 0 (dashed line) |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\textwidth]{newDrift_12.pdf} |
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\includegraphics[width=\textwidth]{newDrift_12.eps} |
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\label{fig:energyDrift} |
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\caption{Analysis of the energy conservation of the real-space |
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electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
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of a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
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300 K starting from the same initial configuration. All runs |
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utilized the same real-space cutoff, $r_c = 12$\AA.} |
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\end{figure} |
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|
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\subsection{Reproduction of Structural Features\label{sec:structure}} |
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One of the best tests of modified interaction potentials is the |
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fidelity with which they can reproduce structural features in a |
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liquid. One commonly-utilized measure of structural ordering is the |
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pair distribution function, $g(r)$, which measures local density |
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deviations in relation to the bulk density. In the electrostatic |
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approaches studied here, the short-range repulsion from the |
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Lennard-Jones potential is identical for the various electrostatic |
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methods, and since short range repulsion determines much of the local |
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liquid ordering, one would not expect to see any differences in |
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$g(r)$. Indeed, the pair distributions are essentially identical for |
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all of the electrostatic methods studied (for each of the different |
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systems under investigation). Interested readers may consult the |
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supplementary information for plots of these pair distribution |
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functions. |
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|
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A direct measure of the structural features that is a more |
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enlightening test of the modified electrostatic methods is the average |
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value of the electrostatic energy $\langle U_\mathrm{elect} \rangle$ |
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which is obtained by sampling the liquid-state configurations |
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experienced by a liquid evolving entirely under the influence of the |
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methods being investigated. In figure \ref{fig:Uelect} we show how |
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$\langle U_\mathrm{elect} \rangle$ for varies with the damping parameter, |
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$\alpha$, for each of the methods. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\textwidth]{averagePotentialEnergy_r9_12.eps} |
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\label{fig:Uelect} |
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\caption{The average electrostatic potential energy, |
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$\langle U_\mathrm{elect} \rangle$ for the SSDQ water with ions as a function |
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of the damping parameter, $\alpha$, for each of the real-space |
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electrostatic methods. Top panel: simulations run with a real-space |
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cutoff, $r_c = 9$\AA. Bottom panel: the same quantity, but with a |
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larger cutoff, $r_c = 12$\AA.} |
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\end{figure} |
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|
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It is clear that moderate damping is important for converging the mean |
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potential energy values, particularly for the two shifted force |
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methods (GSF and TSF). A value of $\alpha \approx 0.18$ \AA$^{-1}$ is |
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sufficient to converge the SP and GSF energies with a cutoff of 12 |
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\AA, while shorter cutoffs require more dramatic damping ($\alpha |
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\approx 0.36$ \AA$^{-1}$ for $r_c = 9$ \AA). It is also clear from |
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fig. \ref{fig:Uelect} that it is possible to overdamp the real-space |
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electrostatic methods, causing the estimate of the energy to drop |
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below the Ewald results. |
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|
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These ``optimal'' values of the damping coefficient are slightly |
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larger than what were observed for DSF electrostatics for purely |
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point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for |
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$r_c = 12$\AA appears to be an excellent compromise for mixed charge |
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multipole systems. |
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|
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\subsection{Reproduction of Dynamic Properties\label{sec:structure}} |
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To test the fidelity of the electrostatic methods at reproducing |
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dynamics in a multipolar liquid, it is also useful to look at |
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transport properties, particularly the diffusion constant, |
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\begin{equation} |
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D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left| |
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\mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle |
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\label{eq:diff} |
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\end{equation} |
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which measures long-time behavior and is sensitive to the forces on |
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the multipoles. For the soft dipolar fluid, and the SSDQ liquid |
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systems, the self-diffusion constants (D) were calculated from linear |
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fits to the long-time portion of the mean square displacement |
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($\langle r^{2}(t) \rangle$).\cite{Allen87} |
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|
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In addition to translational diffusion, orientational relaxation times |
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were calculated for comparisons with the Ewald simulations and with |
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experiments. These values were determined from the same 1~ns $NVE$ |
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trajectories used for translational diffusion by calculating the |
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orientational time correlation function, |
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\begin{equation} |
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C_l^\alpha(t) = \left\langle P_l\left[\hat{\mathbf{u}}_i^\gamma(t) |
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\cdot\hat{\mathbf{u}}_i^\gamma(0)\right]\right\rangle, |
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\label{eq:OrientCorr} |
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\end{equation} |
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where $P_l$ is the Legendre polynomial of order $l$ and |
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$\hat{\mathbf{u}}_i^\alpha$ is the unit vector of molecule $i$ along |
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axis $\gamma$. The body-fixed reference frame used for our |
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orientational correlation functions has the $z$-axis running along the |
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dipoles, and for the SSDQ water model, the $y$-axis connects the two |
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implied hydrogen atoms. |
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|
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From the orientation autocorrelation functions, we can obtain time |
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constants for rotational relaxation either by fitting an exponential |
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function or by integrating the entire correlation function. These |
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decay times are directly comparable to water orientational relaxation |
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times from nuclear magnetic resonance (NMR). The relaxation constant |
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obtained from $C_2^y(t)$ is normally of experimental interest because |
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it describes the relaxation of the principle axis connecting the |
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hydrogen atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular |
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portion of the dipole-dipole relaxation from a proton NMR signal and |
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should provide an estimate of the NMR relaxation time |
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constant.\cite{Impey82} |
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|
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Results for the diffusion constants and orientational relaxation times |
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are shown in figure \ref{fig:dynamics}. From this data, it is apparent |
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that the values for both $D$ and $\tau_2$ using the Ewald sum are |
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reproduced with high fidelity by the GSF method. |
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|
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The $\tau_2$ results in \ref{fig:dynamics} show a much greater |
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difference between the real-space and the Ewald results. |
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|
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|
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\section{CONCLUSION} |
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In the first paper in this series, we generalized the |
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charge-neutralized electrostatic energy originally developed by Wolf |
