| 802 |
|
\label{fourierQuadZeroK} |
| 803 |
|
\end{equation} |
| 804 |
|
The quadrupolar tensor |
| 805 |
< |
$\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ is a rank-4 tensor |
| 805 |
> |
$\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ is a rank 4 tensor |
| 806 |
|
with 81 elements. The only non-zero elements, however, are those with |
| 807 |
|
two doubly-repeated indices, \textit{i.e.} |
| 808 |
|
$\tilde{T}_{aabb}(\mathrm{0})$ and all permutations of these indices. |
| 820 |
|
\end{equation} |
| 821 |
|
Here $\mathrm{B} = \tilde{T}_{abab}(\mathrm{0}) / 4 \pi$ for |
| 822 |
|
$a \neq b$. Using this quadrupolar contraction we can solve equation |
| 823 |
< |
\ref{quadContraction} as follows |
| 824 |
< |
|
| 823 |
> |
\ref{fourierQuadZeroK} as follows |
| 824 |
|
\begin{eqnarray} |
| 825 |
|
\frac{1}{3}\tilde{\Theta}_{\alpha\beta}(\mathrm{0}) &=& \epsilon_o |
| 826 |
|
{\chi}_Q |
| 832 |
|
\nonumber \\ |
| 833 |
|
&=& \left[\frac{\epsilon_o {\chi}_Q} {1-{\chi}_Q \mathrm{B}}\right] |
| 834 |
|
{\partial_\alpha \tilde{E}^\circ_\beta}(\mathrm{0}). |
| 835 |
< |
\label{fourierQuad} |
| 835 |
> |
\label{fourierQuad2} |
| 836 |
|
\end{eqnarray} |
| 837 |
|
In real space, the correction factor, |
| 838 |
|
\begin{equation} |
| 862 |
|
which has been integrated over the interaction volume $V$ and has |
| 863 |
|
units of $\mathrm{length}^{-2}$. |
| 864 |
|
|
| 865 |
< |
In terms of the traced quadrupole moment, equation (\ref{fourierQuad}) |
| 865 |
> |
In terms of the traced quadrupole moment, equation (\ref{fourierQuad2}) |
| 866 |
|
can be written, |
| 867 |
|
\begin{equation} |
| 868 |
|
\mathsf{Q} - \frac{\mathbf{I}}{3} \mathrm{Tr}(\mathsf{Q}) |
| 900 |
|
the same value. |
| 901 |
|
|
| 902 |
|
\begin{sidewaystable} |
| 903 |
< |
\caption{Expressions for the quadrupolar correction factor ($\mathrm{B}$) for |
| 904 |
< |
the real-space electrostatic methods in terms of the damping |
| 905 |
< |
parameter ($\alpha$) and the cutoff radius ($r_c$). The units |
| 906 |
< |
of the correction factor are $ \mathrm{length}^{-2}$ for quadrupolar |
| 907 |
< |
fluids.} |
| 908 |
< |
\label{tab:B} |
| 909 |
< |
\begin{tabular}{l|c|c|c} |
| 911 |
< |
\toprule |
| 903 |
> |
\caption{Expressions for the quadrupolar correction factor |
| 904 |
> |
($\mathrm{B}$) for the real-space electrostatic methods in terms |
| 905 |
> |
of the damping parameter ($\alpha$) and the cutoff radius |
| 906 |
> |
($r_c$). The units of the correction factor are |
| 907 |
> |
$ \mathrm{length}^{-2}$ for quadrupolar fluids.} |
| 908 |
> |
\label{tab:B} |
| 909 |
> |
\begin{tabular}{|l|c|c|c|} |
| 910 |
|
Method & charges & dipoles & quadrupoles \\\colrule |
| 911 |
< |
Spherical Cutoff (SC) & \multicolumn{3}{c}{$ -\frac{8 \alpha^5 |
| 911 |
> |
Spherical Cutoff (SC) & \multicolumn{3}{c|}{$ -\frac{8 \alpha^5 |
| 912 |
|
{r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $}\\ \colrule |
| 913 |
|
Shifted Potental (SP) & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $- \frac{3 \mathrm{erfc(r_c\alpha)}}{5{r_c}^2}- \frac{2 \alpha e^{-\alpha^2 r_c^2}(9+6\alpha^2 r_c^2+4\alpha^4 r_c^4)}{15{\sqrt{\pi}r_c}}$& $ -\frac{16 \alpha^7 {r_c}^5 e^{-\alpha^2 r_c^2 }}{45\sqrt{\pi}}$ \\ |
| 914 |
|
Gradient-shifted (GSF) & $- \frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & 0 & $-\frac{4{\alpha}^7{r_c}^5 e^{-\alpha^2 r_c^2}(-1+2\alpha ^2 r_c^2)}{45\sqrt{\pi}}$\\ |
| 915 |
|
Taylor-shifted (TSF) & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $\frac{4\;\mathrm{erfc(\alpha r_c)}}{{5r_c}^2} + \frac{8\alpha e^{-\alpha^2{r_c}^2}\left(3+ 2\alpha^2 {r_c}^2 +\alpha^4{r_c}^4 \right)}{15\sqrt{\pi}r_c}$ & $\frac{2\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)}{45\sqrt{\pi}{r_c}}$ \\ |
| 916 |
|
\colrule |
| 917 |
< |
Ewald-Kornfeld (EK) & \multicolumn{3}{c}{$ -\frac{8 \kappa^5 {r_c}^3e^{-\kappa^2 r_c^2}}{15\sqrt{\pi}}$} \\ |
| 917 |
> |
Ewald-Kornfeld (EK) & \multicolumn{3}{c|}{$ -\frac{8 \kappa^5 {r_c}^3e^{-\kappa^2 r_c^2}}{15\sqrt{\pi}}$} \\ |
| 918 |
|
\botrule |
| 919 |
|
\end{tabular} |
| 920 |
|
\end{sidewaystable} |
| 1077 |
|
out over a 1~ns simulation in the microcanonical (NVE) ensemble. Box |
| 1078 |
|
dipole moments were sampled every fs. For simulations with external |
| 1079 |
|
perturbations, field strengths ranging from $0 - 10 \times |
| 1080 |
< |
10^{-4}$~V/\AA with increments of $ 10^{-4}$~V/\AA were carried out |
| 1080 |
> |
10^{-4}$~V/\AA\ with increments of $ 10^{-4}$~V/\AA\ were carried out |
| 1081 |
|
for each system. |
| 1082 |
|
|
| 1083 |
|
Quadrupolar systems contained 4000 linear point quadrupoles with a |
| 1154 |
|
shown in separate panels, and different values of the damping |
| 1155 |
|
parameter ($\alpha$) are indicated with different symbols. All of |
| 1156 |
|
the methods appear to be converging to the bulk dielectric constant |
| 1157 |
< |
($\sim 65$) at large ion separations. CHECK PLOT} |
| 1157 |
> |
($\sim 65$) at large ion separations.} |
| 1158 |
|
\label{fig:ScreeningFactor_Dipole} |
| 1159 |
|
\end{figure} |
| 1160 |
|
We have also evaluated the distance-dependent screening factor, |
| 1453 |
|
\end{eqnarray} |
| 1454 |
|
where the first term is the reciprocal space portion. Since the |
| 1455 |
|
quadrupolar correction factor $B = \tilde{T}_{abab}(0) / 4\pi$ and |
| 1456 |
< |
$\mathbf{k} \neq 0 $ is excluded from the reciprocal space sum, |
| 1456 |
> |
$\mathbf{k} = 0 $ is excluded from the reciprocal space sum, |
| 1457 |
|
$\mathbf{T}^\mathrm{K}$ will not contribute.\cite{NeumannII83} The |
| 1458 |
|
remaining terms, |
| 1459 |
|
\begin{equation} |
