| 12 |
|
\usepackage{braket} |
| 13 |
|
\usepackage{tabularx} |
| 14 |
|
\usepackage{rotating} |
| 15 |
+ |
\usepackage{multirow} |
| 16 |
+ |
\usepackage{booktabs} |
| 17 |
|
|
| 18 |
|
\newcolumntype{Y}{>{\centering\arraybackslash}X} |
| 19 |
|
|
| 167 |
|
\item potentials of mean force between solvated ions, |
| 168 |
|
\end{enumerate} |
| 169 |
|
|
| 170 |
+ |
\begin{figure} |
| 171 |
+ |
\includegraphics[width=\linewidth]{Schematic} |
| 172 |
+ |
\caption{Dielectric properties of a fluid measure the response to |
| 173 |
+ |
external electric fields and gradients (left panel), or internal |
| 174 |
+ |
fields and gradients generated by the molecules themselves (central |
| 175 |
+ |
panel), or fields produced by embedded ions (right panel). The |
| 176 |
+ |
dielectric constant measures all three responses in dipolar fluids. |
| 177 |
+ |
In quadrupolar liquids, the relevant bulk property is the |
| 178 |
+ |
quadrupolar susceptibility, and the geometry of the perturbation |
| 179 |
+ |
determines the effective dielectric screening.} |
| 180 |
+ |
\label{fig:schematic} |
| 181 |
+ |
\end{figure} |
| 182 |
+ |
|
| 183 |
|
Under the influence of weak external fields, the bulk polarization of |
| 184 |
|
the system is primarily a linear response to the perturbation, where |
| 185 |
|
proportionality constant depends on the electrostatic interactions |
| 400 |
|
$ \textbf{E} \rightarrow 0 $. |
| 401 |
|
|
| 402 |
|
\subsection{Correction Factors} |
| 403 |
< |
|
| 403 |
> |
\label{sec:corrFactor} |
| 404 |
|
In the presence of a uniform external field $ \mathbf{E}^\circ$, the |
| 405 |
|
total electric field at $\mathbf{r}$ depends on the polarization |
| 406 |
|
density at all other points in the system,\cite{NeumannI83} |
| 416 |
|
|
| 417 |
|
In simulations of dipolar fluids, the molecular dipoles may be |
| 418 |
|
represented either by closely-spaced point charges or by |
| 419 |
< |
point dipoles (see Fig. \ref{fig:stockmayer}). |
| 419 |
> |
point dipoles (see Fig. \ref{fig:tensor}). |
| 420 |
|
\begin{figure} |
| 421 |
< |
\includegraphics[width=3in]{DielectricFigure} |
| 422 |
< |
\caption{With the real-space electrostatic methods, the effective |
| 423 |
< |
dipole tensor, $\mathbf{T}$, governing interactions between |
| 424 |
< |
molecular dipoles is not the same for charge-charge interactions as |
| 425 |
< |
for point dipoles.} |
| 426 |
< |
\label{fig:stockmayer} |
| 421 |
> |
\includegraphics[width=\linewidth]{Tensors} |
| 422 |
> |
\caption{In the real-space electrostatic methods, the molecular dipole |
| 423 |
> |
tensor, $\mathbf{T}_{\alpha\beta}(r)$, is not the same for |
| 424 |
> |
charge-charge interactions as for point dipoles (left panel). The |
| 425 |
> |
same holds true for the molecular quadrupole tensor (right panel), |
| 426 |
> |
$\mathbf{T}_{\alpha\beta\gamma\delta}(r)$, which can have distinct |
| 427 |
> |
forms if the molecule is represented by charges, dipoles, or point |
| 428 |
> |
quadrupoles.} |
| 429 |
> |
\label{fig:tensor} |
| 430 |
|
\end{figure} |
| 431 |
|
In the case where point charges are interacting via an electrostatic |
| 432 |
|
kernel, $v(r)$, the effective {\it molecular} dipole tensor, |
| 491 |
|
\begin{equation} |
| 492 |
|
\epsilon_{CB} = 1 + \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 |
| 493 |
|
\epsilon_o V k_B T} |
| 494 |
< |
\label{correctionFormula} |
| 494 |
> |
\label{conductingBoundary} |
| 495 |
|
\end{equation} |
| 496 |
< |
Equation (\ref{correctionFormula}) allows estimation of the static |
| 497 |
< |
dielectric constant from fluctuations easily computed directly from |
| 498 |
< |
simulations. |
| 481 |
< |
|
| 496 |
> |
Eqs. (\ref{correctionFormula}) and (\ref{conductingBoundary}) allows |
| 497 |
> |
estimation of the static dielectric constant from fluctuations easily |
| 498 |
> |
computed directly from simulations. |
| 499 |
|
|
| 500 |
|
% Here $\chi^*_D$ is a dipolar susceptibility can be expressed in terms of dielectric constant as $ \chi^*_D = \epsilon - 1$ which different than macroscopic dipolar polarizability $\alpha_D$ in the sections \ref{subsec:perturbation} and \ref{subsec:fluctuation}. We can split integral into two parts: singular part i.e $|\textbf{r}-\textbf{r}'|\rightarrow 0 $ and non-singular part i.e $|\textbf{r}-\textbf{r}'| > 0 $ . The singular part of the integral can be written as,\cite{NeumannI83, Jackson98} |
| 501 |
|
% \begin{equation} |
| 539 |
|
for comparison using the Ewald convergence parameter ($\kappa$) |
| 540 |
|
and the real-space cutoff value ($r_c$). } |
| 541 |
|
\label{tab:A} |
| 542 |
< |
{% |
| 543 |
< |
\begin{tabular}{l|c|c|c} |
| 527 |
< |
|
| 542 |
> |
\begin{tabular}{l|c|c} |
| 543 |
> |
\toprule |
| 544 |
|
Method & point charges & point dipoles \\ |
| 545 |
< |
\hline |
| 545 |
> |
\colrule |
| 546 |
|
Spherical Cutoff (SC) & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
| 547 |
|
Shifted Potental (SP) & $ \mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(r_c \alpha) -\frac{2 \alpha r_c}{\sqrt{\pi}}\left(1+\frac{2\alpha^2 {r_c}^2}{3} \right)e^{-\alpha^2{r_c}^2} $\\ |
| 548 |
|
Gradient-shifted (GSF) & 1 & $\mathrm{erf}(\alpha r_c)-\frac{2 \alpha r_c}{\sqrt{\pi}} \left(1 + \frac{2 \alpha^2 r_c^2}{3} + \frac{\alpha^4 r_c^4}{3}\right)e^{-\alpha^2 r_c^2} $ \\ |
| 549 |
< |
Taylor-shifted (TSF) & 1 & 1 \\ |
| 550 |
< |
Ewald-Kornfeld (EK) & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa |
| 551 |
< |
r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ & \\ |
| 552 |
< |
\end{tabular}% |
| 553 |
< |
} |
| 549 |
> |
Taylor-shifted (TSF) & 1 & 1 \\ \colrule |
| 550 |
> |
Ewald-Kornfeld (EK) & \multicolumn{2}{c}{$\mathrm{erf}(r_c \kappa) - |
| 551 |
> |
\frac{2 \kappa r_c}{\sqrt{\pi}} e^{-\kappa^2 |
| 552 |
> |
r_c^2}$} \\ |
| 553 |
> |
\botrule |
| 554 |
> |
\end{tabular} |
| 555 |
|
\end{table} |
| 556 |
|
Note that for point charges, the GSF and TSF methods produce estimates |
| 557 |
|
of the dielectric that need no correction, and the TSF method likewise |
| 646 |
|
tensor connecting quadrupoles at $\mathbf{r}^\prime$ with the point of |
| 647 |
|
interest ($\mathbf{r}$). |
| 648 |
|
|
| 649 |
< |
\begin{figure} |
| 650 |
< |
\includegraphics[width=3in]{QuadrupoleFigure} |
| 651 |
< |
\caption{With the real-space electrostatic methods, the molecular |
| 652 |
< |
quadrupole tensor, $\mathbf{T}_{\alpha\beta\gamma\delta}(r)$, |
| 653 |
< |
governing interactions between molecules is not the same for |
| 654 |
< |
quadrupoles represented via sets of charges, point dipoles, or by |
| 655 |
< |
single point quadrupoles.} |
| 656 |
< |
\label{fig:quadrupolarFluid} |
| 657 |
< |
\end{figure} |
| 649 |
> |
% \begin{figure} |
| 650 |
> |
% \includegraphics[width=3in]{QuadrupoleFigure} |
| 651 |
> |
% \caption{With the real-space electrostatic methods, the molecular |
| 652 |
> |
% quadrupole tensor, $\mathbf{T}_{\alpha\beta\gamma\delta}(r)$, |
| 653 |
> |
% governing interactions between molecules is not the same for |
| 654 |
> |
% quadrupoles represented via sets of charges, point dipoles, or by |
| 655 |
> |
% single point quadrupoles.} |
| 656 |
> |
% \label{fig:quadrupolarFluid} |
| 657 |
> |
% \end{figure} |
| 658 |
|
|
| 659 |
|
In simulations of quadrupolar fluids, the molecular quadrupoles may be |
| 660 |
|
represented by closely-spaced point charges, by multiple point |
| 661 |
|
dipoles, or by a single point quadrupole (see |
| 662 |
< |
Fig. \ref{fig:quadrupolarFluid}). In the case where point charges are |
| 662 |
> |
Fig. \ref{fig:tensor}). In the case where point charges are |
| 663 |
|
interacting via an electrostatic kernel, $v(r)$, the effective |
| 664 |
|
molecular quadrupole tensor can obtained from four successive |
| 665 |
|
applications of the gradient operator to the electrostatic kernel, |
| 734 |
|
The integral in equation (\ref{gradMaxwell}) can be divided into two |
| 735 |
|
parts, $|\mathbf{r}-\mathbf{r}^\prime|\rightarrow 0 $ and |
| 736 |
|
$|\mathbf{r}-\mathbf{r}^\prime|> 0$. Since the self-contribution to |
| 737 |
< |
the field gradient vanishes at the singularity (see Appendix |
| 738 |
< |
\ref{singularQuad}), equation (\ref{gradMaxwell}) can be written as, |
| 737 |
> |
the field gradient vanishes at the singularity (see the supporting |
| 738 |
> |
information), equation (\ref{gradMaxwell}) can be written as, |
| 739 |
|
\begin{equation} |
| 740 |
|
\partial_\alpha E_\beta(\mathbf{r}) = \partial_\alpha {E}^\circ_\beta(\mathbf{r}) + |
| 741 |
|
\frac{1}{4\pi \epsilon_o}\int\limits_{|\mathbf{r}-\mathbf{r}^\prime|> 0 } |
| 852 |
|
susceptibility, |
| 853 |
|
\begin{equation} |
| 854 |
|
\chi_Q = \frac{\alpha_Q}{1 + B \alpha_Q}. |
| 855 |
+ |
\label{eq:quadrupolarSusceptiblity} |
| 856 |
|
\end{equation} |
| 857 |
|
If an electrostatic method produces $B \rightarrow 0$, the computed |
| 858 |
|
quadrupole polarizability and quadrupole susceptibility converge to |
| 865 |
|
of the correction factor are $ \mathrm{length}^{-2}$ for quadrupolar |
| 866 |
|
fluids.} |
| 867 |
|
\label{tab:B} |
| 868 |
< |
\begin{tabular}{l|c|c|c|c} |
| 869 |
< |
|
| 870 |
< |
Method & charges & dipoles & quadrupoles \\\hline |
| 868 |
> |
\begin{tabular}{l|c|c|c} |
| 869 |
> |
\toprule |
| 870 |
> |
Method & charges & dipoles & quadrupoles \\\colrule |
| 871 |
|
Spherical Cutoff (SC) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $ -\frac{8 {\alpha}^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
| 872 |
|
Shifted Potental (SP) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $- \frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$& $ -\frac{16 \alpha^7 {r_c}^5}{9\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
| 873 |
|
Gradient-shifted (GSF) & $- \frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & 0 & $-\frac{4{\alpha}^7{r_c}^5 }{9\sqrt{\pi}}e^{-\alpha^2 r_c^2}(-1+2\alpha ^2 r_c^2)$\\ |
| 874 |
|
Taylor-shifted (TSF) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} |
| 875 |
|
e^{-\alpha^2 r_c^2}$ & |
| 876 |
< |
$\frac{4\;\mathrm{erfc(\alpha r_c)}}{{r_c}^2} + \frac{8 \alpha}{3\sqrt{\pi}r_c}e^{-\alpha^2 {r_c}^2}\left(3+ 2 \alpha^2 {r_c}^2 + \alpha^4 {r_c}^4\right) $ & $\frac{10\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}}{9\sqrt{\pi}{r_c}}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)$ \\ |
| 877 |
< |
Ewald-Kornfeld (EK) & & & |
| 876 |
> |
$\frac{4\;\mathrm{erfc(\alpha |
| 877 |
> |
r_c)}}{{r_c}^2} + |
| 878 |
> |
\frac{8 |
| 879 |
> |
\alpha}{3\sqrt{\pi}r_c}e^{-\alpha^2 |
| 880 |
> |
{r_c}^2}\left(3+ 2 |
| 881 |
> |
\alpha^2 {r_c}^2 + |
| 882 |
> |
\alpha^4 {r_c}^4\right) |
| 883 |
> |
$ & |
| 884 |
> |
$\frac{10\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}}{9\sqrt{\pi}{r_c}}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)$ \\ |
| 885 |
> |
\colrule |
| 886 |
> |
Ewald-Kornfeld (EK) & \multicolumn{3}{c}{ Nothing Yet} \\ |
| 887 |
> |
\botrule |
| 888 |
|
\end{tabular} |
| 889 |
|
\end{sidewaystable} |
| 890 |
|
|
| 1045 |
|
0.5~ns in the canonical (NVT) ensemble. Data collection was carried |
| 1046 |
|
out over a 1~ns simulation in the microcanonical (NVE) ensemble. Box |
| 1047 |
|
dipole moments were sampled every fs. For simulations with external |
| 1048 |
< |
perturbations, field strengths ranging from |
| 1049 |
< |
$0 - 10 \times 10^{-4} \mathrm{~V/\r{A}}$ with increments of |
| 1050 |
< |
$ 10^{-4} \mathrm{~V/\r{A}}$ were carried out for each system. |
| 1048 |
> |
perturbations, field strengths ranging from $0 - 10 \times |
| 1049 |
> |
10^{-4}$~V/\AA with increments of $ 10^{-4}$~V/\AA were carried out |
| 1050 |
> |
for each system. |
| 1051 |
|
|
| 1052 |
|
Quadrupolar systems contained 4000 linear point quadrupoles with a |
| 1053 |
< |
density $2.338 \mathrm{~g/cm}^3$ at a temperature of 500~K. These |
| 1053 |
> |
density $2.338 \mathrm{~g/cm}^3$ at a temperature of 500~K. These |
| 1054 |
|
systems were equilibrated for 200~ps in a canonical (NVT) ensemble. |
| 1055 |
|
Data collection was carried out over a 500~ps simulation in the |
| 1056 |
< |
microcanonical (NVE) ensemble. Components of box quadrupole moments |
| 1057 |
< |
were sampled every 100 fs. For quadrupolar simulations with external |
| 1056 |
> |
microcanonical (NVE) ensemble. Components of box quadrupole moments |
| 1057 |
> |
were sampled every 100 fs. For quadrupolar simulations with external |
| 1058 |
|
field gradients, field strengths ranging from |
| 1059 |
< |
$0 - 9 \times 10^{-2} \mathrm{~V/\r{A}}^2$ with increments of |
| 1060 |
< |
$10^{-2} \mathrm{~V/\r{A}^2}$ were carried out for each system. |
| 1059 |
> |
$0 - 9 \times 10^{-2}$~V/\AA$^2$ with increments of |
| 1060 |
> |
$10^{-2}$~V/\AA$^2$ were carried out for each system. |
| 1061 |
|
|
| 1062 |
|
To carry out the PMF simulations, two of the multipolar molecules in |
| 1063 |
|
the test system were converted into \ce{q+} and \ce{q-} ions and |
| 1064 |
|
constrained to remain at a fixed distance for the duration of the |
| 1065 |
< |
simulation. The constrained distance was then varied from |
| 1066 |
< |
$5 - 12 \mathrm{~\r{A}}$. In the PMF calculations, all simulations were |
| 1067 |
< |
equilibrated for 500 ps in the NVT ensemble and run for 5 ns in the |
| 1068 |
< |
microcanonical (NVE) ensemble. Constraint forces were sampled every |
| 1041 |
< |
20~fs. |
| 1065 |
> |
simulation. The constrained distance was then varied from 5--12~\AA. |
| 1066 |
> |
In the PMF calculations, all simulations were equilibrated for 500 ps |
| 1067 |
> |
in the NVT ensemble and run for 5 ns in the microcanonical (NVE) |
| 1068 |
> |
ensemble. Constraint forces were sampled every 20~fs. |
| 1069 |
|
|
| 1070 |
|
\section{Results} |
| 1071 |
|
\subsection{Dipolar fluid} |
| 1083 |
|
\label{fig:dielectricDipole} |
| 1084 |
|
\end{figure} |
| 1085 |
|
The macroscopic polarizability ($\alpha_D$) for the dipolar fluid is |
| 1086 |
< |
shown in the upper panels in Fig. \ref{fig:dielectricDipole}. The |
| 1086 |
> |
shown in the upper panels in Fig. \ref{fig:dielectricDipole}. The |
| 1087 |
|
polarizability obtained from the both perturbation and fluctuation |
| 1088 |
|
approaches are in excellent agreement with each other. The data also |
| 1089 |
|
show a stong dependence on the damping parameter for both the Shifted |
| 1092 |
|
parameter. |
| 1093 |
|
|
| 1094 |
|
The calculated correction factors discussed in section |
| 1095 |
< |
\ref{sec:corrFactor} are shown in the middle panels. Because the TSF |
| 1095 |
> |
\ref{sec:corrFactor} are shown in the middle panels. Because the TSF |
| 1096 |
|
method has $A = 1$ for all values of the damping parameter, the |
| 1097 |
|
computed polarizabilities need no correction for the dielectric |
| 1098 |
|
calculation. The value of $A$ varies with the damping parameter in |
| 1099 |
|
both the SP and GSF methods, and inclusion of the correction yields |
| 1100 |
|
dielectric estimates (shown in the lower panel) that are generally too |
| 1101 |
< |
large until the damping reaches $\sim 0.25 \mathrm{~\r{A}}^{-1}$. |
| 1102 |
< |
Above this value, the dielectric constants are generally in good |
| 1103 |
< |
agreement with previous simulation results.\cite{NeumannI83} |
| 1101 |
> |
large until the damping reaches $\sim$~0.25~\AA$^{-1}$. Above this |
| 1102 |
> |
value, the dielectric constants are generally in good agreement with |
| 1103 |
> |
previous simulation results.\cite{NeumannI83} |
| 1104 |
|
|
| 1105 |
|
Figure \ref{fig:dielectricDipole} also contains back-calculations of |
| 1106 |
|
the polarizability using the reference (Ewald) simulation |
| 1112 |
|
the value of $\mathrm{A}$ deviates significantly from unity. |
| 1113 |
|
|
| 1114 |
|
These results also suggest an optimal value for the damping parameter |
| 1115 |
< |
of ($\alpha \sim 0.2-0.3 \mathrm{\r{A}}^{-1}$ when evaluating |
| 1116 |
< |
dielectric constants of point dipolar fluids using either the |
| 1117 |
< |
perturbation and fluctuation approaches for the new real-space |
| 1091 |
< |
methods. |
| 1115 |
> |
of ($\alpha \sim 0.2-0.3$~\AA$^{-1}$ when evaluating dielectric |
| 1116 |
> |
constants of point dipolar fluids using either the perturbation and |
| 1117 |
> |
fluctuation approaches for the new real-space methods. |
| 1118 |
|
|
| 1119 |
|
\begin{figure} |
| 1120 |
|
\includegraphics[width=4in]{ScreeningFactor_Dipole.pdf} |
| 1141 |
|
occurs when the ions are separated (or when the damping parameter is |
| 1142 |
|
large). In Fig. \ref{fig:ScreeningFactor_Dipole} we observe that for |
| 1143 |
|
the higher value of damping alpha \textit{i.e.} |
| 1144 |
< |
$\alpha = 0.2 \AA^{-1}$ and $0.3 \AA^{-1}$ and large separation |
| 1144 |
> |
$\alpha = 0.2$~\AA$^{-1}$ and $0.3$~\AA$^{-1}$ and large separation |
| 1145 |
|
between ions, the screening factor does indeed approach the correct |
| 1146 |
|
dielectric constant. |
| 1147 |
|
|
| 1270 |
|
The dielectric constant evaluated using the computed polarizability |
| 1271 |
|
and correction factors agrees well with the previous Ewald-based |
| 1272 |
|
simulation results \cite{Adams81,NeumannI83} for moderate damping |
| 1273 |
< |
parameters in the range 0.25 - 0.3$\AA^{-1}$. |
| 1273 |
> |
parameters in the range 0.25--0.3~\AA~$^{-1}$. |
| 1274 |
|
|
| 1275 |
|
Although the TSF method alters many dynamic and structural features in |
| 1276 |
|
multipolar liquids,\cite{PaperII} it is surprisingly good at computing |
