| 676 |
|
\begin{figure} |
| 677 |
|
\centering |
| 678 |
|
\includegraphics[width=0.5 \textwidth]{Variance_forceNtorque_modified-crop.pdf} |
| 679 |
< |
\caption{The circular variance of the data sets of the direction of the force and torque vectors obtained from a given method about reference Ewald method. The result equal to 0 (dashed line) indicates direction of the vectors are indistinguishable from the Ewald method. Here different symbols represent different value of the cutoff radius (9 $A^o$ = circle, 12 $A^o$ = square 15 $A^o$ = inverted triangle)} |
| 679 |
> |
\caption{The circular variance of the data sets of the |
| 680 |
> |
direction of the force and torque vectors obtained from a |
| 681 |
> |
given method about reference Ewald method. The result equal |
| 682 |
> |
to 0 (dashed line) indicates direction of the vectors are |
| 683 |
> |
indistinguishable from the Ewald method. Here different |
| 684 |
> |
symbols represent different value of the cutoff radius (9 |
| 685 |
> |
\AA\ = circle, 12 \AA\ = square, 15 \AA\ = inverted triangle)} |
| 686 |
|
\label{fig:slopeCorr_circularVariance} |
| 687 |
|
\end{figure} |
| 688 |
|
\subsection{Total energy conservation} |
| 689 |
< |
We have tested the conservation of energy in the SSDQC liquid system by running system for 1ns in the Hard, SP, GSF and TSF method. The Hard cutoff method shows very high energy drifts 433.53 KCal/Mol/ns/particle and energy fluctuation 32574.53 Kcal/Mol (measured by the SD from the slope) for the undamped case, which makes it completely unusable in MD simulations. The SP method also shows large value of energy drift 1.289 Kcal/Mol/ns/particle and energy fluctuation 0.01953 Kcal/Mol. The energy fluctuation in the SP method is due to the non-vanishing nature of the torque and force at the cutoff radius. We can improve the energy conservation in some extent by the proper selection of the damping alpha but the improvement is not good enough, which can be observed in Figure 9a and 9b .The GSF and TSF shows very low value of energy drift 0.09016, 0.07371 KCal/Mol/ns/particle and fluctuation 3.016E-6, 1.217E-6 Kcal/Mol respectively for the undamped case. Since the absolute value of the evaluated electrostatic energy, force and torque from TSF method are deviated from the Ewald, it does not mimic MD simulations appropriately. The electrostatic energy, force and torque from the GSF method have very good agreement with the Ewald. In addition, the energy drift and energy fluctuation from the GSF method is much better than Ewald’s method for reciprocal space vector value ($k_f$) equal to 7 as shown in Figure~\ref{fig:energyDrift} and ~\ref{fig:fluctuation}. We can improve the total energy fluctuation and drift for the Ewald’s method by increasing size of the reciprocal space, which extremely increseses the simulation time. In our current simulation, the simulation time for the Hard, SP, and GSF methods are about 5.5 times faster than the Ewald method. |
| 690 |
< |
\begin{figure} |
| 691 |
< |
\centering |
| 692 |
< |
\includegraphics[width=0.5 \textwidth]{log(energyDrift)-crop.pdf} |
| 693 |
< |
\label{fig:energyDrift} |
| 694 |
< |
\end{figure} |
| 689 |
> |
We have tested the conservation of energy in the SSDQC liquid system |
| 690 |
> |
by running system for 1ns in the Hard, SP, GSF and TSF method. The |
| 691 |
> |
Hard cutoff method shows very high energy drifts 433.53 |
| 692 |
> |
KCal/Mol/ns/particle and energy fluctuation 32574.53 Kcal/Mol |
| 693 |
> |
(measured by the SD from the slope) for the undamped case, which makes |
| 694 |
> |
it completely unusable in MD simulations. The SP method also shows |
| 695 |
> |
large value of energy drift 1.289 Kcal/Mol/ns/particle and energy |
| 696 |
> |
fluctuation 0.01953 Kcal/Mol. The energy fluctuation in the SP method |
| 697 |
> |
is due to the non-vanishing nature of the torque and force at the |
| 698 |
> |
cutoff radius. We can improve the energy conservation in some extent |
| 699 |
> |
by the proper selection of the damping alpha but the improvement is |
| 700 |
> |
not good enough, which can be observed in Figure 9a and 9b .The GSF |
| 701 |
> |
and TSF shows very low value of energy drift 0.09016, 0.07371 |
| 702 |
> |
KCal/Mol/ns/particle and fluctuation 3.016E-6, 1.217E-6 Kcal/Mol |
| 703 |
> |
respectively for the undamped case. Since the absolute value of the |
| 704 |
> |
evaluated electrostatic energy, force and torque from TSF method are |
| 705 |
> |
deviated from the Ewald, it does not mimic MD simulations |
| 706 |
> |
appropriately. The electrostatic energy, force and torque from the GSF |
| 707 |
> |
method have very good agreement with the Ewald. In addition, the |
| 708 |
> |
energy drift and energy fluctuation from the GSF method is much better |
| 709 |
> |
than Ewald’s method for reciprocal space vector value ($k_f$) equal to |
| 710 |
> |
7 as shown in Figure~\ref{fig:energyDrift} and |
| 711 |
> |
~\ref{fig:fluctuation}. We can improve the total energy fluctuation |
| 712 |
> |
and drift for the Ewald’s method by increasing size of the reciprocal |
| 713 |
> |
space, which extremely increseses the simulation time. In our current |
| 714 |
> |
simulation, the simulation time for the Hard, SP, and GSF methods are |
| 715 |
> |
about 5.5 times faster than the Ewald method. |
| 716 |
> |
|
| 717 |
> |
In Fig.~\ref{fig:energyDrift}, $\delta \mbox{E}_1$ is a measure of the |
| 718 |
> |
linear energy drift in units of $\mbox{kcal mol}^{-1}$ per particle |
| 719 |
> |
over a nanosecond of simulation time, and $\delta \mbox{E}_0$ is the |
| 720 |
> |
standard deviation of the energy fluctuations in units of $\mbox{kcal |
| 721 |
> |
mol}^{-1}$ per particle. In the bottom plot, it is apparent that the |
| 722 |
> |
energy drift is reduced by a significant amount (2 to 3 orders of |
| 723 |
> |
magnitude improvement at all values of the damping coefficient) by |
| 724 |
> |
chosing either of the shifted-force methods over the hard or SP |
| 725 |
> |
methods. We note that the two shifted-force method can give |
| 726 |
> |
significantly better energy conservation than the multipolar Ewald sum |
| 727 |
> |
with the same choice of real-space cutoffs. |
| 728 |
> |
|
| 729 |
|
\begin{figure} |
| 730 |
< |
\centering |
| 731 |
< |
\includegraphics[width=0.5 \textwidth]{logSD-crop.pdf} |
| 732 |
< |
\caption{The plot showing (a) standard deviation, and (b) total energy drift in the total energy conservation plot for different values of the damping alpha for different cut off methods. } |
| 733 |
< |
\label{fig:fluctuation} |
| 734 |
< |
\end{figure} |
| 730 |
> |
\centering |
| 731 |
> |
\includegraphics[width=\textwidth]{newDrift.pdf} |
| 732 |
> |
\label{fig:energyDrift} |
| 733 |
> |
\caption{Analysis of the energy conservation of the real space |
| 734 |
> |
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
| 735 |
> |
energy over time and $\delta \mathrm{E}_0$ is the standard deviation |
| 736 |
> |
of energy fluctuations around this drift. All simulations were of a |
| 737 |
> |
2000-molecule simulation of SSDQ water with 48 ionic charges at 300 |
| 738 |
> |
K starting from the same initial configuration.} |
| 739 |
> |
\end{figure} |
| 740 |
> |
|
| 741 |
|
\section{CONCLUSION} |
| 742 |
|
We have generalized the charged neutralized potential energy originally developed by the Wolf et al.\cite{Wolf:1999dn} for the charge-charge interaction to the charge-multipole and multipole-multipole interaction in the SP method for higher order multipoles. Also, we have developed GSF and TSF methods by implementing the modification purposed by Fennel and Gezelter\cite{Fennell:2006lq} for the charge-charge interaction to the higher order multipoles to ensure consistency and smooth truncation of the electrostatic energy, force, and torque for the spherical truncation. The SP methods for multipoles proved its suitability in MC simulations. On the other hand, the results from the GSF method produced good agreement with the Ewald's energy, force, and torque. Also, it shows very good energy conservation in MD simulations. |
| 743 |
|
The direct truncation of any molecular system without multipole neutralization creates the fluctuation in the electrostatic energy. This fluctuation in the energy is very large for the case of crystal because of long range of multipole ordering (Refer paper I).\cite{PaperI} This is also significant in the case of the liquid because of the local multipole ordering in the molecules. If the net multipole within cutoff radius neutralized within cutoff sphere by placing image multiples on the surface of the sphere, this fluctuation in the energy reduced significantly. Also, the multipole neutralization in the generalized SP method showed very good agreement with the Ewald as compared to direct truncation for the evaluation of the $\triangle E$ between the configurations. |
