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V^\mathrm{SF}\left(r_{ij}\right)=q_iq_j\left\{\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}}-\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}}+\left[\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2}+\frac{2\alpha}{\pi^{1/2}}\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}}\right]\left(r_{ij}-R_\mathrm{c}\right)\right\}. |
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\label{eq:SFPot} |
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\end{equation} |
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< |
Equation \ref{eq:SFPot} is similar to equation \ref{eq:ZahnPot} derived by Zahn \textit{et al.}; however, there are two important differences.\cite{Zahn02} First, the $v_\textrm{c}$ term is simply equation \ref{eq:dampCoulomb} with $R_\textrm{c}$ supplied for $r_{ij}$. This term is not present in equation \ref{eq:ZahnPot}, resulting in a discontinuity in the potential as particles cross $R_\textrm{c}$. Second, the sign of the derivative portion is different. The constant $v_\textrm{c}$ term is not going to have a presence in the forces after performing the derivative, but the negative sign results in a discontinuity in the forces at the cutoff, since the function is pushed in the opposite direction and doesn't cross to zero at $R_\textrm{c}$. |
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Equation \ref{eq:SFPot} is similar to equation \ref{eq:ZahnPot} derived by Zahn \textit{et al.}; however, there are two important differences.\cite{Zahn02} First, the $v_\textrm{c}$ term is simply equation \ref{eq:dampCoulomb} with $R_\textrm{c}$ supplied for $r_{ij}$. This term is not present in equation \ref{eq:ZahnPot}, resulting in a discontinuity in the potential as particles cross $R_\textrm{c}$. Second, the sign of the derivative portion is different. The constant $v_\textrm{c}$ term is not going to have a presence in the forces after performing the derivative, but the negative sign does effect the derivative. In fact, it introduces a discontinuity in the forces at the cutoff, because the force function is shifted in the wrong direction and doesn't cross zero at $R_\textrm{c}$. Thus, these alterations make for an electrostatic summation method that is continuous in both the potential and forces and incorporates the pairwise sum considerations stressed by Wolf \textit{et al.}\cite{Wolf99} |
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< |
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.\cite{Essmann95} In addition to the Shifted-Potential and Shifted-Force 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 and Molecular Dynamics calculations was assessed through statistical analysis over the combined results from all of the following studied systems: |
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It is important to note that shifted force techniques have a drawback in that they alter the shape of the original potential. We thereby lose a degree of clarity about the original formulation of the potential in order to gain functionality in dynamics simulations. An alternative direction would be use the derivatives of the original potential for the forces. This was addressed by Wolf \textit{et al.} as undesirable, because the effect of the image charges is neglected in the forces when they would expect there to be some pressure exerted due to their presence.\cite{Wolf99} As a consequence the forces, though mathematically valid, may not be physically correct due to this missing component. In Monte Carlo simulations, this argument is mute, because forces are not evaluated. With this in mind, we decided to consider both the Shifted-Force technique described above and this Shifted-Potential technique to determine their usability in the evaluation of both energetic and dynamic results in simulations with electrostatics. |
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| 86 |
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To this end, 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.\cite{Essmann95} In addition to the Shifted-Potential and Shifted-Force 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 and Molecular Dynamics calculations was assessed through statistical analysis over the combined results from all of the following studied systems: |
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\begin{enumerate} |
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\item Liquid Water |
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\item Crystalline Water (Ice I$_\textrm{c}$) |
