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%\linenumbers\relax % Commence numbering lines |
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\usepackage{amsmath} |
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\usepackage{times} |
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\usepackage{mathptm} |
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\usepackage{mathptmx} |
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\usepackage{tabularx} |
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\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
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\usepackage{url} |
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Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
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method for calculating electrostatic interactions between point |
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charges. They argued that the effective Coulomb interaction in |
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condensed systems is actually short ranged.\cite{Wolf92,Wolf95}. For |
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an ordered lattice (e.g., when computing the Madelung constant of an |
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condensed systems is actually short ranged.\cite{Wolf92,Wolf95} For an |
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ordered lattice (e.g., when computing the Madelung constant of an |
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ionic solid), the material can be considered as a set of ions |
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interacting with neutral dipolar or quadrupolar ``molecules'' giving |
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an effective distance dependence for the electrostatic interactions of |
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$r^{-5}$ (see figure \ref{fig:NaCl}). For this reason, careful |
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$r^{-5}$ (see figure \ref{fig:schematic}). For this reason, careful |
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applications of Wolf's method are able to obtain accurate estimates of |
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Madelung constants using relatively short cutoff radii. Recently, |
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Fukuda used neutralization of the higher order moments for the |
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calculation of the electrostatic interaction of the point charges |
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system.\cite{Fukuda:2013sf} |
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|
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\begin{figure}[h!] |
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\begin{figure} |
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\centering |
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\includegraphics[width=0.50 \textwidth]{chargesystem.pdf} |
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\caption{Top: NaCl crystal showing how spherical truncation can |
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breaking effective charge ordering, and how complete \ce{(NaCl)4} |
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molecules interact with the central ion. Bottom: A dipolar |
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crystal exhibiting similar behavior and illustrating how the |
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effective dipole-octupole interactions can be disrupted by |
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spherical truncation.} |
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\label{fig:NaCl} |
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\includegraphics[width=\linewidth]{schematic.pdf} |
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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 in order at longer distances. With |
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hard cutoffs, motion of individual charges in and out of the |
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cutoff sphere can break the effective multipolar ordering. |
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Bottom: dipolar crystals and fluids have a similar effective |
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\textit{quadrupolar} ordering (in the smaller grey circles), and |
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orientational averaging helps to reduce the effective range of the |
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interactions in the fluid. Placement of reversed image multipoles |
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on the surface of the cutoff sphere recovers the effective |
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higher-order multipole behavior.} |
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\label{fig:schematic} |
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\end{figure} |
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|
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The direct truncation of interactions at a cutoff radius creates |
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|
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Considering the interaction of one central ion in an ionic crystal |
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with a portion of the crystal at some distance, the effective Columbic |
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potential is found to be decreasing as $r^{-5}$. If one views the |
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\ce{NaCl} crystal as simple cubic (SC) structure with an octupolar |
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potential is found to decrease as $r^{-5}$. If one views the \ce{NaCl} |
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crystal as a simple cubic (SC) structure with an octupolar |
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\ce{(NaCl)4} basis, the electrostatic energy per ion converges more |
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rapidly to the Madelung energy than the dipolar |
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approximation.\cite{Wolf92} To find the correct Madelung constant, |
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Lacman suggested that the NaCl structure could be constructed in a way |
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that the finite crystal terminates with complete \ce{(NaCl)4} |
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molecules.\cite{Lacman65} Any charge in a NaCl crystal is surrounded |
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by opposite charges. Similarly for each pair of charges, there is an |
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opposite pair of charge adjacent to it. The central ion sees what is |
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effectively a set of octupoles at large distances. These facts suggest |
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that the Madelung constants are relatively short ranged for perfect |
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ionic crystals.\cite{Wolf:1999dn} |
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molecules.\cite{Lacman65} The central ion sees what is effectively a |
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set of octupoles at large distances. These facts suggest that the |
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Madelung constants are relatively short ranged for perfect ionic |
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crystals.\cite{Wolf:1999dn} |
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|
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One can make a similar argument for crystals of point multipoles. The |
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Luttinger and Tisza treatment of energy constants for dipolar lattices |
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unstable. |
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|
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In ionic crystals, real-space truncation can break the effective |
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multipolar arrangements (see Fig. \ref{fig:NaCl}), causing significant |
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swings in the electrostatic energy as individual ions move back and |
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forth across the boundary. This is why the image charges are |
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multipolar arrangements (see Fig. \ref{fig:schematic}), causing |
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significant swings in the electrostatic energy as individual ions move |
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back and forth across the boundary. This is why the image charges are |
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necessary for the Wolf sum to exhibit rapid convergence. Similarly, |
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the real-space truncation of point multipole interactions breaks |
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higher order multipole arrangements, and image multipoles are required |
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for real-space treatments of electrostatic energies. |
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|
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The shorter effective range of electrostatic interactions is not |
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limited to perfect crystals, but can also apply in disordered fluids. |
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Even at elevated temperatures, there is, on average, local charge |
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balance in an ionic liquid, where each positive ion has surroundings |
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dominated by negaitve ions and vice versa. The reversed-charge images |
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on the cutoff sphere that are integral to the Wolf and DSF approaches |
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retain the effective multipolar interactions as the charges traverse |
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the cutoff boundary. |
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|
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In multipolar fluids (see Fig. \ref{fig:schematic}) there is |
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significant orientational averaging that additionally reduces the |
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effect of long-range multipolar interactions. The image multipoles |
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that are introduced in the TSF, GSF, and SP methods mimic this effect |
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and reduce the effective range of the multipolar interactions as |
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interacting molecules traverse each other's cutoff boundaries. |
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|
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% Because of this reason, although the nature of electrostatic |
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% interaction short ranged, the hard cutoff sphere creates very large |
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\label{generic} |
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\end{equation} |
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where $f_n(r)$ is a shifted kernel that is appropriate for the order |
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of the interaction, with $n=0$ for charge-charge, $n=1$ for |
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charge-dipole, $n=2$ for charge-quadrupole and dipole-dipole, $n=3$ |
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for dipole-quadrupole, and $n=4$ for quadrupole-quadrupole. To ensure |
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smooth convergence of the energy, force, and torques, a Taylor |
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expansion with $n$ terms must be performed at cutoff radius ($r_c$) to |
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obtain $f_n(r)$. |
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of the interaction (see Ref. \onlinecite{PaperI}), with $n=0$ for |
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charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole |
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and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for |
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quadrupole-quadrupole. To ensure smooth convergence of the energy, |
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force, and torques, a Taylor expansion with $n$ terms must be |
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performed at cutoff radius ($r_c$) to obtain $f_n(r)$. |
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|
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% To carry out the same procedure for a damped electrostatic kernel, we |
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% replace $1/r$ in the Coulomb potential with $\text{erfc}(\alpha r)/r$. |
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shifted smoothly by finding the gradient for two interacting dipoles |
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which have been projected onto the surface of the cutoff sphere |
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without changing their relative orientation, |
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\begin{displaymath} |
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\begin{equation} |
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U_{D_{\bf a}D_{\bf b}}(r) = U_{D_{\bf a}D_{\bf b}}(r) - |
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U_{D_{\bf a} D_{\bf b}}(r_c) |
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- (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot |
| 433 |
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\vec{\nabla} U_{D_{\bf a}D_{\bf b}}(r) \Big \lvert _{r_c} |
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\end{displaymath} |
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\end{equation} |
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Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf |
| 436 |
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a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance |
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(although the signs are reversed for the dipole that has been |
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energy over time, $\delta E_1$, and the standard deviation of energy |
| 977 |
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fluctuations around this drift $\delta E_0$. Both of the |
| 978 |
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shifted-force methods (GSF and TSF) provide excellent energy |
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conservation (drift less than $10^{-6}$ kcal / mol / ns / particle), |
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conservation (drift less than $10^{-5}$ kcal / mol / ns / particle), |
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while the hard cutoff is essentially unusable for molecular dynamics. |
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SP provides some benefit over the hard cutoff because the energetic |
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jumps that happen as particles leave and enter the cutoff sphere are |
