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% some packages for things like equations and graphics |
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\usepackage{amsmath,bm} |
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\usepackage{tabularx} |
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\usepackage{graphicx} |
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\usepackage{booktabs} |
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\usepackage{cite} |
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\begin{document} |
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structural and dynamic properties of water compared the same |
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properties obtained using the Ewald sum. |
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|
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\section{Simple Forms for Pairwise Electrostatics} |
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\section{Simple Forms for Pairwise Electrostatics}\label{sec:PairwiseDerivation} |
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|
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The potentials proposed by Wolf \textit{et al.} and Zahn \textit{et |
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al.} are constructed using two different (and separable) computational |
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the complementary error function is required). |
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|
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The reaction field results illustrates some of that method's |
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limitations, primarily that it was developed for use in homogenous |
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limitations, primarily that it was developed for use in homogeneous |
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systems; although it does provide results that are an improvement over |
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those from an unmodified cutoff. |
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|
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all do equivalently well at capturing the direction of both the force |
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and torque vectors. Using the electrostatic damping improves the |
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angular behavior significantly for the {\sc sp} and moderately for the |
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{\sc sf} methods. Overdamping is detrimental to both methods. Again |
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{\sc sf} methods. Over-damping is detrimental to both methods. Again |
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it is important to recognize that the force vectors cover all |
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particles in all seven systems, while torque vectors are only |
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available for neutral molecular groups. Damping is more beneficial to |
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charged bodies, and this observation is investigated further in |
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section \ref{IndividualResults}. |
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section \ref{sec:IndividualResults}. |
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Although not discussed previously, group based cutoffs can be applied |
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to both the {\sc sp} and {\sc sf} methods. The group-based cutoffs |
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increases, something that is more obvious with group-based cutoffs. |
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The complimentary error function inserted into the potential weakens |
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the electrostatic interaction as the value of $\alpha$ is increased. |
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However, at larger values of $\alpha$, it is possible to overdamp the |
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However, at larger values of $\alpha$, it is possible to over-damp the |
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electrostatic interaction and to remove it completely. Kast |
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\textit{et al.} developed a method for choosing appropriate $\alpha$ |
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values for these types of electrostatic summation methods by fitting |
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agreement with {\sc spme} in both energetic and dynamic behavior when |
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using the {\sc sf} method with and without damping. The {\sc sp} |
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method does well with an $\alpha$ around 0.2\AA$^{-1}$, particularly |
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with cutoff radii greater than 12\AA. Overdamping the electrostatics |
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with cutoff radii greater than 12\AA. Over-damping the electrostatics |
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reduces the agreement between both these methods and {\sc spme}. |
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The pure cutoff ({\sc pc}) method performs poorly, again mirroring the |
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no damping and only modest improvement for the recommended conditions |
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($\alpha = 0.2$\AA$^{-1}$ and $R_\textrm{c}~\geqslant~12$\AA). The |
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{\sc sf} method shows modest narrowing across all damping and cutoff |
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ranges of interest. When overdamping these methods, group cutoffs and |
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ranges of interest. When over-damping these methods, group cutoffs and |
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the switching function do not improve the force and torque |
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directionalities. |
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|
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This system does not appear to show any significant deviations from |
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the previously observed results. The {\sc sp} and {\sc sf} methods |
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have aggrements similar to those observed in section |
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have agreements similar to those observed in section |
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\ref{sec:WaterResults}. The only significant difference is the |
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improvement in the configuration energy differences for the {\sc rf} |
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method. This is surprising in that we are introducing an inhomogeneity |
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the NaCl crystal at 1000K. The undamped shifted force ({\sc sf}) |
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method is off by less than 10 cm$^{-1}$, and increasing the |
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electrostatic damping to 0.25\AA$^{-1}$ gives quantitative agreement |
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with the power spectrum obtained using the Ewald sum. Overdamping can |
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with the power spectrum obtained using the Ewald sum. Over-damping can |
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result in underestimates of frequencies of the long-wavelength |
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motions.} |
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\label{fig:dampInc} |
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\end{figure} |
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|
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\section{Synopsis of the Pairwise Method Evaluation}\label{sec:PairwiseSynopsis} |
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\section{An Application: TIP5P-E Water}\label{sec:t5peApplied} |
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|
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The above sections focused on the energetics and dynamics of a variety |
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of systems when utilizing the {\sc sp} and {\sc sf} pairwise |
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techniques. A unitary correlation with results obtained using the |
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Ewald summation should result in a successful reproduction of both the |
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static and dynamic properties of any selected system. To test this, |
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we decided to calculate a series of properties for the TIP5P-E water |
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model when using the {\sc sf} technique. |
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|
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The TIP5P-E water model is a variant of Mahoney and Jorgensen's |
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five-point transferable intermolecular potential (TIP5P) model for |
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water.\cite{Mahoney00} TIP5P was developed to reproduce the density |
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maximum anomaly present in liquid water near 4$^\circ$C. As with many |
| 1977 |
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previous point charge water models (such as ST2, TIP3P, TIP4P, SPC, |
| 1978 |
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and SPC/E), TIP5P was parametrized using a simple cutoff with no |
| 1979 |
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long-range electrostatic |
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correction.\cite{Stillinger74,Jorgensen83,Berendsen81,Berendsen87} |
| 1981 |
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Without this correction, the pressure term on the central particle |
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from the surroundings is missing. Because they expand to compensate |
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for this added pressure term when this correction is included, systems |
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composed of these particles tend to underpredict the density of water |
| 1985 |
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under standard conditions. When using any form of long-range |
| 1986 |
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electrostatic correction, it has become common practice to develop or |
| 1987 |
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utilize a reparametrized water model that corrects for this |
| 1988 |
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effect.\cite{vanderSpoel98,Fennell04,Horn04} The TIP5P-E model follows |
| 1989 |
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this practice and was optimized specifically for use with the Ewald |
| 1990 |
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summation.\cite{Rick04} In his publication, Rick preserved the |
| 1991 |
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geometry and point charge magnitudes in TIP5P and focused on altering |
| 1992 |
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the Lennard-Jones parameters to correct the density at |
| 1993 |
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298K.\cite{Rick04} With the density corrected, he compared common |
| 1994 |
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water properties for TIP5P-E using the Ewald sum with TIP5P using a |
| 1995 |
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9\AA\ cutoff. |
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|
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In the following sections, we compared these same water properties |
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calculated from TIP5P-E using the Ewald sum with TIP5P-E using the |
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{\sc sf} technique. In the above evaluation of the pairwise |
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techniques, we observed some flexibility in the choice of parameters. |
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Because of this, the following comparisons include the {\sc sf} |
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technique with a 12\AA\ cutoff and an $\alpha$ = 0.0, 0.1, and |
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0.2\AA$^{-1}$, as well as a 9\AA\ cutoff with an $\alpha$ = |
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0.2\AA$^{-1}$. |
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|
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\subsection{Density}\label{sec:t5peDensity} |
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|
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As stated previously, the property that prompted the development of |
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TIP5P-E was the density at 1 atm. The density depends upon the |
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internal pressure of the system in the $NPT$ ensemble, and the |
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calculation of the pressure includes a components from both the |
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kinetic energy and the virial. More specifically, the instantaneous |
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molecular pressure ($P(t)$) is given by |
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\begin{equation} |
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P(t) = \frac{1}{\textrm{d}V}\sum_\mu |
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\left[\frac{\mathbf{P}_{\mu}^2}{M_{\mu}} |
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+ \mathbf{R}_{\mu}\cdot\sum_i\mathbf{F}_{\mu i}\right], |
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\label{eq:MolecularPressure} |
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\end{equation} |
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where $V$ is the volume, $\mathbf{P}_{\mu}$ is the momentum of |
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molecule $\mu$, $\mathbf{R}_\mu$ is the position of the center of mass |
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($M_\mu$) of molecule $\mu$, and $\mathbf{F}_{\mu i}$ is the force on |
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atom $i$ of molecule $\mu$.\cite{Melchionna93} The virial term (the |
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right term in the brackets of eq. \ref{eq:MolecularPressure}) is |
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directly dependent on the interatomic forces. Since the {\sc sp} |
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method does not modify the forces (see |
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sec. \ref{sec:PairwiseDerivation}), the pressure using {\sc sp} will |
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be identical to that obtained without an electrostatic correction. |
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The {\sc sf} method does alter the virial component and, by way of the |
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modified pressures, should provide densities more in line with those |
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obtained using the Ewald summation. |
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|
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To compare densities, $NPT$ simulations were performed with the same |
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temperatures as those selected by Rick in his Ewald summation |
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simulations.\cite{Rick04} In order to improve statistics around the |
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density maximum, 3ns trajectories were accumulated at 0, 12.5, and |
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25$^\circ$C, while 2ns trajectories were obtained at all other |
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temperatures. The average densities were calculated from the later |
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three-fourths of each trajectory. Similar to Mahoney and Jorgensen's |
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method for accumulating statistics, these sequences were spliced into |
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200 segments to calculate the average density and standard deviation |
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at each temperature.\cite{Mahoney00} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/tip5peDensities.pdf} |
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\caption{Density versus temperature for the TIP5P-E water model when |
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using the Ewald summation (Ref. \cite{Rick04}) and the {\sc sf} method |
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with various parameters. The pressure term from the image-charge shell |
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is larger than that provided by the reciprocal-space portion of the |
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Ewald summation, leading to slightly lower densities. This effect is |
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more visible with the 9\AA\ cutoff, where the image charges exert a |
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greater force on the central particle. The error bars for the {\sc sf} |
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methods show plus or minus the standard deviation of the density |
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measurement at each temperature.} |
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\label{fig:t5peDensities} |
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\end{figure} |
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|
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Figure \ref{fig:t5peDensities} shows the densities calculated for |
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TIP5P-E using differing electrostatic corrections overlaid on the |
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experimental values.\cite{CRC80} The densities when using the {\sc sf} |
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technique are close to, though typically lower than, those calculated |
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while using the Ewald summation. These slightly reduced densities |
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indicate that the pressure component from the image charges at |
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R$_\textrm{c}$ is larger than that exerted by the reciprocal-space |
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portion of the Ewald summation. Bringing the image charges closer to |
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the central particle by choosing a 9\AA\ R$_\textrm{c}$ (rather than |
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the preferred 12\AA\ R$_\textrm{c}$) increases the strength of their |
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interactions, resulting in a further reduction of the densities. |
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|
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Because the strength of the image charge interactions has a noticable |
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effect on the density, we would expect the use of electrostatic |
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damping to also play a role in these calculations. Larger values of |
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$\alpha$ weaken the pair-interactions; and since electrostatic damping |
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is distance-dependent, force components from the image charges will be |
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reduced more than those from particles close the the central |
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charge. This effect is visible in figure \ref{fig:t5peDensities} with |
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the damped {\sc sf} sums showing slightly higher densities; however, |
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it is apparent that the choice of cutoff radius plays a much more |
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important role in the resulting densities. |
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|
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As a final note, all of the above density calculations were performed |
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with systems of 512 water molecules. Rick observed a system sized |
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dependence of the computed densities when using the Ewald summation, |
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most likely due to his tying of the convergence parameter to the box |
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dimensions.\cite{Rick04} For systems of 256 water molecules, the |
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calculated densities were on average 0.002 to 0.003 g/cm$^3$ lower. A |
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system size of 256 molecules would force the use of a shorter |
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R$_\textrm{c}$ when using the {\sc sf} technique, and this would also |
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lower the densities. Moving to larger systems, as long as the |
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R$_\textrm{c}$ remains at a fixed value, we would expect the densities |
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to remain constant. |
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|
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\subsection{Liquid Structure}\label{sec:t5peLiqStructure} |
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|
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A common function considered when developing and comparing water |
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models is the oxygen-oxygen radial distribution function |
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($g_\textrm{OO}(r)$). The $g_\textrm{OO}(r)$ is the probability of |
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finding a pair of oxygen atoms some distance ($r$) apart relative to a |
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random distribution at the same density.\cite{Allen87} It is |
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calculated via |
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\begin{equation} |
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g_\textrm{OO}(r) = \frac{V}{N^2}\left\langle\sum_i\sum_{j\ne i} |
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\delta(\mathbf{r}-\mathbf{r}_{ij})\right\rangle, |
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\label{eq:GOOofR} |
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\end{equation} |
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where the double sum is over all $i$ and $j$ pairs of $N$ oxygen |
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atoms. The $g_\textrm{OO}(r)$ can be directly compared to X-ray or |
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neutron scattering experiments through the oxygen-oxygen structure |
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factor ($S_\textrm{OO}(k)$) by the following relationship: |
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\begin{equation} |
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S_\textrm{OO}(k) = 1 + 4\pi\rho |
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\int_0^\infty r^2\frac{\sin kr}{kr}g_\textrm{OO}(r)\textrm{d}r. |
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\label{eq:SOOofK} |
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\end{equation} |
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Thus, $S_\textrm{OO}(k)$ is related to the Fourier transform of |
| 2116 |
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$g_\textrm{OO}(r)$. |
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|
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The expermentally determined $g_\textrm{OO}(r)$ for liquid water has |
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been compared in great detail with the various common water models, |
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and TIP5P was found to be in better agreement than other rigid, |
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non-polarizable models.\cite{Sorenson00} This excellent agreement with |
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experiment was maintained when Rick developed TIP5P-E.\cite{Rick04} To |
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check whether the choice of using the Ewald summation or the {\sc sf} |
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technique alters the liquid structure, the $g_\textrm{OO}(r)$s at 298K |
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and 1atm were determined for the systems compared in the previous |
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section. |
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|
| 2128 |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/tip5peGofR.pdf} |
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\caption{The $g_\textrm{OO}(r)$s calculated for TIP5P-E at 298K and |
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1atm while using the Ewald summation (Ref. \cite{Rick04}) and the {\sc |
| 2132 |
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sf} technique with varying parameters. Even with the reduced densities |
| 2133 |
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using the {\sc sf} technique, the $g_\textrm{OO}(r)$s are essentially |
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identical.} |
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\label{fig:t5peGofRs} |
| 2136 |
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\end{figure} |
| 2137 |
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|
| 2138 |
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The $g_\textrm{OO}(r)$s calculated for TIP5P-E while using the {\sc |
| 2139 |
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sf} technique with a various parameters are overlaid on the |
| 2140 |
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$g_\textrm{OO}(r)$ while using the Ewald summation. The differences in |
| 2141 |
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density do not appear to have any effect on the liquid structure as |
| 2142 |
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the $g_\textrm{OO}(r)$s are indistinquishable. These results indicate |
| 2143 |
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that the $g_\textrm{OO}(r)$ is insensitive to the choice of |
| 2144 |
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electrostatic correction. |
| 2145 |
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|
| 2146 |
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\subsection{Thermodynamic Properties} |
| 2147 |
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|
| 2148 |
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\subsection{Dynamic Properties} |
| 2149 |
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|
| 2150 |
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\section{Damping of Point Multipoles} |
| 2151 |
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|
| 2152 |
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\section{Damping and Dielectric Constants} |
| 2153 |
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|
| 2154 |
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\section{Conclusions}\label{sec:PairwiseConclusions} |
| 2155 |
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|
| 2156 |
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The above investigation of pairwise electrostatic summation techniques |
| 2157 |
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shows that there are viable and computationally efficient alternatives |
| 2158 |
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to the Ewald summation. These methods are derived from the damped and |
| 2205 |
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required to obtain the level of accuracy most researchers have come to |
| 2206 |
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expect. |
| 2207 |
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|
| 2016 |
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\section{An Application: TIP5P-E Water} |
| 2017 |
– |
|
| 2018 |
– |
|
| 2208 |
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\chapter{\label{chap:water}SIMPLE MODELS FOR WATER} |
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|
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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS} |
