| 377 |
|
Eq.~\ref{generic}, accounting for the appropriate number of |
| 378 |
|
derivatives. Complete energy, force, and torque expressions are |
| 379 |
|
presented in the first paper in this series (Reference |
| 380 |
< |
\bibpunct{}{}{,}{n}{}{,} \protect\citep{PaperI}). |
| 380 |
> |
\citep{PaperI}). |
| 381 |
|
|
| 382 |
|
\subsection{Gradient-shifted force (GSF)} |
| 383 |
|
|
| 426 |
|
the energy, force and torque for higher order multipole-multipole |
| 427 |
|
interactions. Complete energy, force, and torque expressions for the |
| 428 |
|
GSF potential are presented in the first paper in this series |
| 429 |
< |
(Reference \bibpunct{}{}{,}{n}{}{,} \protect\citep{PaperI}) |
| 429 |
> |
(Reference \citep{PaperI}) |
| 430 |
|
|
| 431 |
|
|
| 432 |
|
\subsection{Shifted potential (SP) } |
| 459 |
|
may be suitable for Monte Carlo approaches where the configurational |
| 460 |
|
energy differences are the primary quantity of interest. |
| 461 |
|
|
| 462 |
< |
\subsection{Self term} |
| 462 |
> |
\subsection{The Self term} |
| 463 |
|
In the TSF, GSF, and SP methods, a self-interaction is retained for |
| 464 |
|
the central multipole interacting with its own image on the surface of |
| 465 |
|
the cutoff sphere. This self interaction is nearly identical with the |
| 466 |
|
self-terms that arise in the Ewald sum for multipoles. Complete |
| 467 |
|
expressions for the self terms are presented in the first paper in |
| 468 |
< |
this series (Reference \bibpunct{}{}{,}{n}{}{,} |
| 469 |
< |
\protect\citep{PaperI}) |
| 468 |
> |
this series (Reference \citep{PaperI}) |
| 469 |
|
|
| 471 |
– |
Note that the TSF, GSF, and SP models are $\mathcal{O}(N)$ methods |
| 472 |
– |
that can be made extremely efficient using spline interpolations of |
| 473 |
– |
the radial functions. They require no Fourier transforms or $k$-space |
| 474 |
– |
sums, and guarantee the smooth handling of energies, forces, and |
| 475 |
– |
torques as multipoles cross the real-space cutoff boundary. |
| 470 |
|
|
| 471 |
< |
\section{\label{sec:test}Test systems} |
| 478 |
< |
We have compared the electrostatic force and torque of each molecule from SP, TSF and GSF method with the multipolar-Ewald method. Furthermore, total electrostatic energies of a molecular system from the different methods have also been compared with total energy from the Ewald. In Mote Carlo (MC) simulation, the energy difference between different configurations of the molecular system is important, even though absolute energies are not accurate. We have compared the change in electrostatic potential energy ($\triangle E$) of 250 different configurations of the various multipolar molecular systems (Section IV B) calculated from the Hard, SP, GSF, and TSF methods with the well-known Ewald method. In MD simulations, the force and torque acting on the molecules drives the whole dynamics of the molecules in a system. The magnitudes of the electrostatic force, torque and their direction for each molecule of the all 250 configurations have also been compared against the Ewald’s method. |
| 471 |
> |
\section{\label{sec:methodology}Methodology} |
| 472 |
|
|
| 473 |
< |
\subsection{Modeled systems} |
| 474 |
< |
We studied the comparison of the energy differences, forces and torques for six different systems; i) dipolar liquid, ii) quadrupolar liquid, iii) dipolar crystal, iv) quadrupolar crystal v) dipolar-quadrupolar liquid(SSDQ), and vi) ions in dipolar-qudrupolar liquid(SSDQC). To simulate different configurations of the crystals, the body centered cubic (BCC) minimum energy crystal with 3,456 molecules was taken and translationally locked in their respective crystal sites. The thermal energy was supplied to the rotational motion so that dipoles or quadrupoles can freely explore all possible orientation. The crystals were simulated for 10,000 fs in NVE ensemble at 50 K and 250 different configurations was taken in equal time interval for the comparative study. The crystals were not simulated at high temperature and for a long run time to avoid possible translational deformation of the crystal sites. |
| 475 |
< |
For dipolar, quadrupolar, and dipolar-quadrupolar liquids simulation, each molecular system with 2,048 molecules simulated for 1ns in NVE ensemble at 300 K temperature after equilibration. We collected 250 different configurations in equal interval of time. For the ions mixed liquid system, we converted 48 different molecules into 24 $Na^+$ and $24 Cl^-$ ions and equilibrated. After equilibration, the system was run at the same environment for 1ns and 250 configurations were collected. While comparing energies, forces, and torques with Ewald method, Lennad Jone’s potentials were turned off and purely electrostatic interaction had been compared. |
| 473 |
> |
To understand how the real-space multipole methods behave in computer |
| 474 |
> |
simulations, it is vital to test against established methods for |
| 475 |
> |
computing electrostatic interactions in periodic systems, and to |
| 476 |
> |
evaluate the size and sources of any errors that arise from the |
| 477 |
> |
real-space cutoffs. In the first paper of this series, we compared |
| 478 |
> |
the dipolar and quadrupolar energy expressions against analytic |
| 479 |
> |
expressions for ordered dipolar and quadrupolar |
| 480 |
> |
arrays.\cite{Sauer,LT,Nagai01081960,Nagai01091963} In this work, we |
| 481 |
> |
used the multipolar Ewald sum as a reference method for comparing |
| 482 |
> |
energies, forces, and torques for molecular models that mimic |
| 483 |
> |
disordered and ordered condensed-phase systems. These test-cases |
| 484 |
> |
include: |
| 485 |
> |
\begin{itemize} |
| 486 |
> |
\item Soft Dipolar fluids ($\sigma = 3.051$, $\epsilon =0.152$, $|D| = 2.35$) |
| 487 |
> |
\item Soft Dipolar solids ($\sigma = 2.837$, $\epsilon =1.0$, $|D| = 2.35$) |
| 488 |
> |
\item Soft Quadrupolar fluids ($\sigma = 3.051$, $\epsilon =0.152$, $Q_{\alpha\alpha} =\left\{-1,-1,-2.5\right\}$) |
| 489 |
> |
\item Soft Quadrupolar solids ($\sigma = 2.837$, $\epsilon = 1.0$, $Q_{\alpha\alpha} =\left\{-1,-1,-2.5\right\}$) |
| 490 |
> |
\item A mixed multipole model (SSDQ) for water ($\sigma = 3.051$, $\epsilon = 0.152$, $D_z = 2.35$, $Q_{\alpha\alpha} =\left\{-1.35,0,-0.68\right\}$) |
| 491 |
> |
\item A mixed multipole models for water with 48 dissolved ions, 24 |
| 492 |
> |
\ce{Na+}: ($\sigma = 2.579$, $\epsilon =0.118$, $q = 1e$) and 24 |
| 493 |
> |
\ce{Cl-}: ($\sigma = 4.445$, $\epsilon =0.1$l, $q = -1e$) |
| 494 |
> |
\end{itemize} |
| 495 |
> |
All Lennard-Jones parameters are in units of \AA\ $(\sigma)$ and kcal |
| 496 |
> |
/ mole $(\epsilon)$. Partial charges are reported in electrons, while |
| 497 |
> |
dipoles are in Debye units, and quadrupoles are in units of Debye-\AA. |
| 498 |
|
|
| 499 |
< |
\subsection{Statistical analysis} |
| 500 |
< |
We have used least square regression analyses for six different molecular systems to compare $\triangle E$ from Hard, SP, GSF, and TSF with the reference method. Molecular systems were run longer enough to explore various configurations and 250 independent configurations were recorded for comparison. The total numbers of 31,125 energy differences from the proposed methods have been compared with the Ewald. Similarly, the magnitudes of the forces and torques have also been compared by using least square regression analyses. In the forces and torques comparison, the magnitudes of the forces acting in each molecule for each configuration were evaluated. For example, our dipolar liquid simulation contains 2048 molecules and there are 250 different configurations for each system thus there are 512,000 force and torque comparisons. The correlation coefficient and correlation slope varies from 0 to 1, where 1 is the best agreement between the two methods. |
| 499 |
> |
The last test case exercises all levels of the multipole-multipole |
| 500 |
> |
interactions we have derived so far and represents the most complete |
| 501 |
> |
test of the new methods. In the following section, we present results |
| 502 |
> |
for the total electrostatic energy, as well as the electrostatic |
| 503 |
> |
contributions to the force and torque on each molecule. These |
| 504 |
> |
quantities have been computed using the SP, TSF, and GSF methods, as |
| 505 |
> |
well as a hard cutoff, and have been compared with the values obtaine |
| 506 |
> |
from the multipolar Ewald sum. In Mote Carlo (MC) simulations, the |
| 507 |
> |
energy differences between two configurations is the primary quantity |
| 508 |
> |
that governs how the simulation proceeds. These differences are the |
| 509 |
> |
most imporant indicators of the reliability of a method even if the |
| 510 |
> |
absolute energies are not exact. For each of the multipolar systems |
| 511 |
> |
listed above, we have compared the change in electrostatic potential |
| 512 |
> |
energy ($\Delta E$) between 250 statistically-independent |
| 513 |
> |
configurations. In molecular dynamics (MD) simulations, the forces |
| 514 |
> |
and torques govern the behavior of the simulation, so we also compute |
| 515 |
> |
the electrostatic contributions to the forces and torques. |
| 516 |
|
|
| 517 |
+ |
\subsection{Model systems} |
| 518 |
+ |
To sample independent configurations of multipolar crystals, a body |
| 519 |
+ |
centered cubic (bcc) crystal which is a minimum energy structure for |
| 520 |
+ |
point dipoles was generated using 3,456 molecules. The multipoles |
| 521 |
+ |
were translationally locked in their respective crystal sites for |
| 522 |
+ |
equilibration at a relatively low temperature (50K), so that dipoles |
| 523 |
+ |
or quadrupoles could freely explore all accessible orientations. The |
| 524 |
+ |
translational constraints were removed, and the crystals were |
| 525 |
+ |
simulated for 10 ps in the microcanonical (NVE) ensemble with an |
| 526 |
+ |
average temperature of 50 K. Configurations were sampled at equal |
| 527 |
+ |
time intervals for the comparison of the configurational energy |
| 528 |
+ |
differences. The crystals were not simulated close to the melting |
| 529 |
+ |
points in order to avoid translational deformation away of the ideal |
| 530 |
+ |
lattice geometry. |
| 531 |
+ |
|
| 532 |
+ |
For dipolar, quadrupolar, and mixed-multipole liquid simulations, each |
| 533 |
+ |
system was created with 2048 molecules oriented randomly. These were |
| 534 |
+ |
|
| 535 |
+ |
system with 2,048 molecules simulated for 1ns in NVE ensemble at 300 K |
| 536 |
+ |
temperature after equilibration. We collected 250 different |
| 537 |
+ |
configurations in equal interval of time. For the ions mixed liquid |
| 538 |
+ |
system, we converted 48 different molecules into 24 \ce{Na+} and 24 |
| 539 |
+ |
\ce{Cl-} ions and equilibrated. After equilibration, the system was run |
| 540 |
+ |
at the same environment for 1ns and 250 configurations were |
| 541 |
+ |
collected. While comparing energies, forces, and torques with Ewald |
| 542 |
+ |
method, Lennard-Jones potentials were turned off and purely |
| 543 |
+ |
electrostatic interaction had been compared. |
| 544 |
+ |
|
| 545 |
+ |
\subsection{Accuracy of Energy Differences, Forces and Torques} |
| 546 |
+ |
The pairwise summation techniques (outlined above) were evaluated for |
| 547 |
+ |
use in MC simulations by studying the energy differences between |
| 548 |
+ |
different configurations. We took the Ewald-computed energy |
| 549 |
+ |
difference between two conformations to be the correct behavior. An |
| 550 |
+ |
ideal performance by one of the new methods would reproduce these |
| 551 |
+ |
energy differences exactly. The configurational energies being used |
| 552 |
+ |
here contain only contributions from electrostatic interactions. |
| 553 |
+ |
Lennard-Jones interactions were omitted from the comparison as they |
| 554 |
+ |
should be identical for all methods. |
| 555 |
+ |
|
| 556 |
+ |
Since none of the real-space methods provide exact energy differences, |
| 557 |
+ |
we used least square regressions analysiss for the six different |
| 558 |
+ |
molecular systems to compare $\Delta E$ from Hard, SP, GSF, and TSF |
| 559 |
+ |
with the multipolar Ewald reference method. Unitary results for both |
| 560 |
+ |
the correlation (slope) and correlation coefficient for these |
| 561 |
+ |
regressions indicate perfect agreement between the real-space method |
| 562 |
+ |
and the multipolar Ewald sum. |
| 563 |
+ |
|
| 564 |
+ |
Molecular systems were run long enough to explore independent |
| 565 |
+ |
configurations and 250 configurations were recorded for comparison. |
| 566 |
+ |
Each system provided 31,125 energy differences for a total of 186,750 |
| 567 |
+ |
data points. Similarly, the magnitudes of the forces and torques have |
| 568 |
+ |
also been compared by using least squares regression analyses. In the |
| 569 |
+ |
forces and torques comparison, the magnitudes of the forces acting in |
| 570 |
+ |
each molecule for each configuration were evaluated. For example, our |
| 571 |
+ |
dipolar liquid simulation contains 2048 molecules and there are 250 |
| 572 |
+ |
different configurations for each system resulting in 3,072,000 data |
| 573 |
+ |
points for comparison of forces and torques. |
| 574 |
+ |
|
| 575 |
|
\subsection{Analysis of vector quantities} |
| 576 |
< |
R.A. Fisher has developed a probablity density function to analyse directional data sets is expressed as below,\cite{fisher53} |
| 577 |
< |
\begin{equation} |
| 578 |
< |
p_f(\theta) = \frac{\kappa}{2 \sinh\kappa}\sin\theta \exp(\kappa \cos\theta) |
| 579 |
< |
\label{eq:pdf} |
| 580 |
< |
\end{equation} |
| 581 |
< |
where $\kappa$ measures directional dispersion of the data about mean direction can be estimated as a reciprocal of the circular variance for large number of directional data sets.\cite{Allen91} In our calculation, the unit vector from the Ewald method was considered as mean direction and the angle between the vectors from Ewald and the purposed method were evaluated.The total displacement of the unit vectors from the purposed method was calculated as, |
| 576 |
> |
Getting the magnitudes of the force and torque vectors correct is only |
| 577 |
> |
part of the issue for carrying out accurate molecular dynamics |
| 578 |
> |
simulations. Because the real space methods reweight the different |
| 579 |
> |
orientational contributions to the energies, it is also important to |
| 580 |
> |
understand how the methods impact the \textit{directionality} of the |
| 581 |
> |
force and torque vectors. Fisher developed a probablity density |
| 582 |
> |
function to analyse directional data sets, |
| 583 |
|
\begin{equation} |
| 584 |
< |
R = \sqrt{(\sum\limits_{i=1}^N \sin\theta_i)^2 + (\sum\limits_{i=1}^N \sin\theta_i)^2} |
| 584 |
> |
p_f(\theta) = \frac{\kappa}{2 \sinh\kappa}\sin\theta e^{\kappa \cos\theta} |
| 585 |
> |
\label{eq:pdf} |
| 586 |
> |
\end{equation} |
| 587 |
> |
where $\kappa$ measures directional dispersion of the data around the |
| 588 |
> |
mean direction.\cite{fisher53} This quantity $(\kappa)$ can be |
| 589 |
> |
estimated as a reciprocal of the circular variance.\cite{Allen91} To |
| 590 |
> |
quantify the directional error, forces obtained from the Ewald sum |
| 591 |
> |
were taken as the mean (or correct) direction and the angle between |
| 592 |
> |
the forces obtained via the Ewald sum and the real-space methods were |
| 593 |
> |
evaluated, |
| 594 |
> |
\begin{equation} |
| 595 |
> |
\cos\theta_i = \frac{\vec{f}_i^\mathrm{~Ewald} \cdot |
| 596 |
> |
\vec{f}_i^\mathrm{~GSF}}{\left|\vec{f}_i^\mathrm{~Ewald}\right| \left|\vec{f}_i^\mathrm{~GSF}\right|} |
| 597 |
> |
\end{equation} |
| 598 |
> |
The total angular displacement of the vectors was calculated as, |
| 599 |
> |
\begin{equation} |
| 600 |
> |
R = \sqrt{\left(\sum\limits_{i=1}^N \cos\theta_i\right)^2 + \left(\sum\limits_{i=1}^N \sin\theta_i\right)^2} |
| 601 |
|
\label{eq:displacement} |
| 602 |
|
\end{equation} |
| 603 |
< |
where N is number of directional data sets and $theta_i$ are the angles between unit vectors evaluated from the Ewald and the purposed methods. The circular variance is defined as $ Var(\theta) = 1 -R/N$. The value of circular variance varies from 0 to 1. The lower the value of $Var{\theta}$ is higher the value of $\kappa$, which expresses tighter clustering of the direction sets around Ewald direction. |
| 603 |
> |
where $N$ is number of force vectors. The circular variance is |
| 604 |
> |
defined as |
| 605 |
> |
\begin{equation} |
| 606 |
> |
\mathrm{Var}(\theta) \approx 1/\kappa = 1 - R/N |
| 607 |
> |
\end{equation} |
| 608 |
> |
The circular variance takes on values between from 0 to 1, with 0 |
| 609 |
> |
indicating a perfect directional match between the Ewald force vectors |
| 610 |
> |
and the real-space forces. Lower values of $\mathrm{Var}(\theta)$ |
| 611 |
> |
correspond to higher values of $\kappa$, which indicates tighter |
| 612 |
> |
clustering of the real-space force vectors around the Ewald forces. |
| 613 |
|
|
| 614 |
+ |
A similar analysis was carried out for the electrostatic contribution |
| 615 |
+ |
to the molecular torques as well as forces. |
| 616 |
+ |
|
| 617 |
|
\subsection{Energy conservation} |
| 618 |
< |
To test conservation of the energy, the mixed molecular system of 2000 dipolar-quadrupolar molecules with 24 $Na^+$, and 24 $Cl^-$ was run for 1ns in the microcanonical ensemble at 300 K temperature for different cutoff methods (Ewald, Hard, SP, GSF, and TSF). The molecular system was run in 12 parallel computers and started with same initial positions and velocities for all cutoff methods. The slope and Standard Deviation of the energy about the slope (SD) were evaluated in the total energy versus time plot, where the slope evaluates the total energy drift and SD calculates the energy fluctuation in MD simulations. Also, the time duration for the simulation was recorded to compare efficiency of the purposed methods with the Ewald. |
| 618 |
> |
To test conservation the energy for the methods, the mixed molecular |
| 619 |
> |
system of 2000 SSDQ water molecules with 24 \ce{Na+} and 24 \ce{Cl-} |
| 620 |
> |
ions was run for 1 ns in the microcanonical ensemble at an average |
| 621 |
> |
temperature of 300K. Each of the different electrostatic methods |
| 622 |
> |
(Ewald, Hard, SP, GSF, and TSF) was tested for a range of different |
| 623 |
> |
damping values. The molecular system was started with same initial |
| 624 |
> |
positions and velocities for all cutoff methods. The energy drift |
| 625 |
> |
($\delta E_1$) and standard deviation of the energy about the slope |
| 626 |
> |
($\delta E_0$) were evaluated from the total energy of the system as a |
| 627 |
> |
function of time. Although both measures are valuable at |
| 628 |
> |
investigating new methods for molecular dynamics, a useful interaction |
| 629 |
> |
model must allow for long simulation times with minimal energy drift. |
| 630 |
|
|
| 631 |
|
\section{\label{sec:result}RESULTS} |
| 632 |
|
\subsection{Configurational energy differences} |
| 651 |
|
|
| 652 |
|
% \label{fig:barGraph2} |
| 653 |
|
% \end{figure} |
| 654 |
< |
%The correlation coefficient ($R^2$) and slope of the linear regression plots for the energy differences for all six different molecular systems is shown in figure 4a and 4b.The plot shows that the correlation coefficient improves for the SP cutoff method as compared to the undamped hard cutoff method in the case of SSDQC, SSDQ, dipolar crystal, and dipolar liquid. For the quadrupolar crystal and liquid, the correlation coefficient is almost unchanged and close to 1. The correlation coefficient is smallest (0.696276 for $r_c$ = 9 $A^o$) for the SSDQC liquid because of the presence of charge-charge and charge-multipole interactions. Since the charge-charge and charge-multipole interaction is long ranged, there is huge deviation of correlation coefficient from 1. Similarly, the quarupole–quadrupole interaction is short ranged ($\sim 1/r^6$) with compared to interactions in the other multipolar systems, thus the correlation coefficient very close to 1 even for hard cutoff method. The idea of placing image multipole on the surface of the cutoff sphere improves the correlation coefficient and makes it close to 1 for all types of multipolar systems. Similarly the slope is hugely deviated from the correct value for the lower order multipole-multipole interaction and slightly deviated for higher order multipole – multipole interaction. The SP method improves both correlation coefficient ($R^2$) and slope significantly in SSDQC and dipolar systems. The Slope is found to be deviated more in dipolar crystal as compared to liquid which is associated with the large fluctuation in the electrostatic energy in crystal. The GSF also produced better values of correlation coefficient and slope with the proper selection of the damping alpha (Interested reader can consult accompanying supporting material). The TSF method gives good value of correlation coefficient for the dipolar crystal, dipolar liquid, SSDQ, and SSDQC (not for the quadrupolar crystal and liquid) but the regression slopes are significantly deviated. |
| 654 |
> |
%The correlation coefficient ($R^2$) and slope of the linear |
| 655 |
> |
%regression plots for the energy differences for all six different |
| 656 |
> |
%molecular systems is shown in figure 4a and 4b.The plot shows that |
| 657 |
> |
%the correlation coefficient improves for the SP cutoff method as |
| 658 |
> |
%compared to the undamped hard cutoff method in the case of SSDQC, |
| 659 |
> |
%SSDQ, dipolar crystal, and dipolar liquid. For the quadrupolar |
| 660 |
> |
%crystal and liquid, the correlation coefficient is almost unchanged |
| 661 |
> |
%and close to 1. The correlation coefficient is smallest (0.696276 |
| 662 |
> |
%for $r_c$ = 9 $A^\circ$) for the SSDQC liquid because of the presence of |
| 663 |
> |
%charge-charge and charge-multipole interactions. Since the |
| 664 |
> |
%charge-charge and charge-multipole interaction is long ranged, there |
| 665 |
> |
%is huge deviation of correlation coefficient from 1. Similarly, the |
| 666 |
> |
%quarupole–quadrupole interaction is short ranged ($\sim 1/r^6$) with |
| 667 |
> |
%compared to interactions in the other multipolar systems, thus the |
| 668 |
> |
%correlation coefficient very close to 1 even for hard cutoff |
| 669 |
> |
%method. The idea of placing image multipole on the surface of the |
| 670 |
> |
%cutoff sphere improves the correlation coefficient and makes it close |
| 671 |
> |
%to 1 for all types of multipolar systems. Similarly the slope is |
| 672 |
> |
%hugely deviated from the correct value for the lower order |
| 673 |
> |
%multipole-multipole interaction and slightly deviated for higher |
| 674 |
> |
%order multipole – multipole interaction. The SP method improves both |
| 675 |
> |
%correlation coefficient ($R^2$) and slope significantly in SSDQC and |
| 676 |
> |
%dipolar systems. The Slope is found to be deviated more in dipolar |
| 677 |
> |
%crystal as compared to liquid which is associated with the large |
| 678 |
> |
%fluctuation in the electrostatic energy in crystal. The GSF also |
| 679 |
> |
%produced better values of correlation coefficient and slope with the |
| 680 |
> |
%proper selection of the damping alpha (Interested reader can consult |
| 681 |
> |
%accompanying supporting material). The TSF method gives good value of |
| 682 |
> |
%correlation coefficient for the dipolar crystal, dipolar liquid, |
| 683 |
> |
%SSDQ, and SSDQC (not for the quadrupolar crystal and liquid) but the |
| 684 |
> |
%regression slopes are significantly deviated. |
| 685 |
> |
|
| 686 |
|
\begin{figure} |
| 687 |
< |
\centering |
| 688 |
< |
\includegraphics[width=0.50 \textwidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
| 689 |
< |
\caption{The correlation coefficient and regression slope of configurational energy differences for a given method with compared with the reference Ewald method. The value of result equal to 1(dashed line) indicates energy difference is indistinguishable from the Ewald method. Here different symbols represent different value of the cutoff radius (9 \AA\ = circle, 12 \AA\ = square 15 \AA\ = inverted triangle)} |
| 690 |
< |
\label{fig:slopeCorr_energy} |
| 691 |
< |
\end{figure} |
| 692 |
< |
The combined correlation coefficient and slope for all six systems is shown in Figure ~\ref{fig:slopeCorr_energy}. The correlation coefficient for the undamped hard cutoff method is does not have good agreement with the Ewald because of the fluctuation of the electrostatic energy in the direct truncation method. This deviation in correlation coefficient is improved by using SP, GSF, and TSF method. But the TSF method worsens the regression slope stating that this method produces statistically more biased result as compared to Ewald. Also the GSF method slightly deviate slope but it can be alleviated by using proper value of damping alpha and cutoff radius. The SP method shows good agreement with Ewald method for all values of damping alpha and radii. |
| 693 |
< |
\subsection{Magnitude of the force and torque vectors} |
| 694 |
< |
The comparison of the magnitude of the combined forces and torques for the data accumulated from all system types are shown in Figure ~\ref{fig:slopeCorr_force}. The correlation and slope for the forces agree with the Ewald even for the hard cutoff method. For the system of molecules with higher order multipoles, the interaction is short ranged. Moreover, the force decays more rapidly than the electrostatic energy hence the hard cutoff method also produces good results. Although the pure cutoff gives the good match of the electrostatic force, the discontinuity in the force at the cutoff radius causes problem in the total energy conservation in MD simulations, which will be discussed in detail in subsection D. The correlation coefficient for GSF method also perfectly matches with Ewald but the slope is slightly deviated (due to extra term obtained from the angular differentiation). This deviation in the slope can be alleviated with proper selection of the damping alpha and radii ($\alpha = 0.2$ and $r_c = 12 A^o$ are good choice). The TSF method shows good agreement in the correlation coefficient but the slope is not good as compared to the Ewald. |
| 695 |
< |
\begin{figure} |
| 696 |
< |
\centering |
| 697 |
< |
\includegraphics[width=0.50 \textwidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
| 698 |
< |
\caption{The correlation coefficient and regression slope of the magnitude of the force for a given method with compared to the reference Ewald method. The value of result equal to 1(dashed line) indicates, the magnitude of the force from a method is indistinguishable from the Ewald method. Here different symbols represent different value of the cutoff radius (9 \AA\ = circle, 12 \AA\ = square 15 \AA\ = inverted triangle). } |
| 540 |
< |
\label{fig:slopeCorr_force} |
| 541 |
< |
\end{figure} |
| 542 |
< |
The torques appears to be very influenced because of extra term generated when the potential energy is modified to get consistent force and torque. The result shows that the torque from the hard cutoff method has good agreement with Ewald. As the potential is modified to make it consistent with the force and torque, the correlation and slope is deviated as shown in Figure~\ref{fig:slopeCorr_torque} for SP, GSF and TSF cutoff methods. But the proper value of the damping alpha and radius can improve the agreement of the GSF with the Ewald method. The TSF method shows worst agreement in the slope as compared to Ewald even for larger cutoff radii. |
| 543 |
< |
\begin{figure} |
| 544 |
< |
\centering |
| 545 |
< |
\includegraphics[width=0.5 \textwidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
| 546 |
< |
\caption{The correlation coefficient and regression slope of the magnitude of the torque for a given method with compared to the reference Ewald method. The value of result equal to 1(dashed line) indicates, the magnitude of the force from a method is 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).} |
| 547 |
< |
\label{fig:slopeCorr_torque} |
| 548 |
< |
\end{figure} |
| 549 |
< |
\subsection{Directionality of the force and torque vectors} |
| 550 |
< |
The accurate evaluation of the direction of the force and torques are also important for the dynamic simulation.In our research, the direction data sets were computed from the purposed method and compared with Ewald using Fisher statistics and results are expressed in terms of circular variance ($Var(\theta$).The force and torque vectors from the purposed method followed Fisher probability distribution function expressed in equation~\ref{eq:pdf}. The circular variance for the force and torque vectors of each molecule in the 250 configurations for all system types is shown in Figure~\ref{fig:slopeCorr_circularVariance}. The direction of the force and torque vectors from hard and SP cutoff methods showed best directional agreement with the Ewald. The force and torque vectors from GSF method also showed good agreement with the Ewald method, which can also be improved by varying damping alpha and cutoff radius.For $\alpha = 0.2$ and $r_c = 12 A^o$, $ Var(\theta) $ for direction of the force was found to be 0.002061 and corresponding value of $\kappa $ was 485.20. Integration of equation ~\ref{eq:pdf} for that corresponding value of $\kappa$ showed that 95\% of force vectors are with in $6.37^o$. The TSF method is the poorest in evaluating accurate direction with compared to Hard, SP, and GSF methods. The circular variance for the direction of the torques is larger as compared to force. For same $\alpha = 0.2, r_c = 12 A^o$ and GSF method, the circular variance was 0.01415, which showed 95\% of torque vectors are within $16.75^o$.The direction of the force and torque vectors can be improved by varying $\alpha$ and $r_c$. |
| 687 |
> |
\centering |
| 688 |
> |
\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
| 689 |
> |
\caption{Statistical analysis of the quality of configurational |
| 690 |
> |
energy differences for the real-space electrostatic methods |
| 691 |
> |
compared with the reference Ewald sum. Results with a value equal |
| 692 |
> |
to 1 (dashed line) indicate $\Delta E$ values indistinguishable |
| 693 |
> |
from those obtained using the multipolar Ewald sum. Different |
| 694 |
> |
values of the cutoff radius are indicated with different symbols |
| 695 |
> |
(9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted |
| 696 |
> |
triangles).} |
| 697 |
> |
\label{fig:slopeCorr_energy} |
| 698 |
> |
\end{figure} |
| 699 |
|
|
| 700 |
+ |
The combined correlation coefficient and slope for all six systems is |
| 701 |
+ |
shown in Figure ~\ref{fig:slopeCorr_energy}. Most of the methods |
| 702 |
+ |
reproduce the Ewald-derived configurational energy differences with |
| 703 |
+ |
remarkable fidelity. Undamped hard cutoffs introduce a significant |
| 704 |
+ |
amount of random scatter in the energy differences which is apparent |
| 705 |
+ |
in the reduced value of the correlation coefficient for this method. |
| 706 |
+ |
This can be understood easily as configurations which exhibit only |
| 707 |
+ |
small traversals of a few dipoles or quadrupoles out of the cutoff |
| 708 |
+ |
sphere will see large energy jumps when hard cutoffs are used. The |
| 709 |
+ |
orientations of the multipoles (particularly in the ordered crystals) |
| 710 |
+ |
mean that these jumps can go either up or down in energy, producing a |
| 711 |
+ |
significant amount of random scatter. |
| 712 |
+ |
|
| 713 |
+ |
The TSF method produces energy differences that are highly correlated |
| 714 |
+ |
with the Ewald results, but it also introduces a significant |
| 715 |
+ |
systematic bias in the values of the energies, particularly for |
| 716 |
+ |
smaller cutoff values. The TSF method alters the distance dependence |
| 717 |
+ |
of different orientational contributions to the energy in a |
| 718 |
+ |
non-uniform way, so the size of the cutoff sphere can have a large |
| 719 |
+ |
effect on crystalline systems. |
| 720 |
+ |
|
| 721 |
+ |
Both the SP and GSF methods appear to reproduce the Ewald results with |
| 722 |
+ |
excellent fidelity, particularly for moderate damping ($\alpha = |
| 723 |
+ |
0.1-0.2$\AA$^{-1}$) and commonly-used cutoff values ($r_c = 12$\AA). |
| 724 |
+ |
With the exception of the undamped hard cutoff, and the TSF method |
| 725 |
+ |
with short cutoffs, all of the methods would be appropriate for use in |
| 726 |
+ |
Monte Carlo simulations. |
| 727 |
+ |
|
| 728 |
+ |
\subsection{Magnitude of the force and torque vectors} |
| 729 |
+ |
|
| 730 |
+ |
The comparison of the magnitude of the combined forces and torques for |
| 731 |
+ |
the data accumulated from all system types are shown in Figures |
| 732 |
+ |
~\ref{fig:slopeCorr_force} and \ref{fig:slopeCorr_torque}. The |
| 733 |
+ |
correlation and slope for the forces agree well with the Ewald sum |
| 734 |
+ |
even for the hard cutoff method. |
| 735 |
+ |
|
| 736 |
+ |
For the system of molecules with higher order multipoles, the |
| 737 |
+ |
interaction is quite short ranged. Moreover, the force decays more |
| 738 |
+ |
rapidly than the electrostatic energy hence the hard cutoff method can |
| 739 |
+ |
also produces reasonable agreement. Although the pure cutoff gives |
| 740 |
+ |
the good match of the electrostatic force for pairs of molecules |
| 741 |
+ |
included within the cutoff sphere, the discontinuity in the force at |
| 742 |
+ |
the cutoff radius can potentially cause problems the total energy |
| 743 |
+ |
conservation as molecules enter and leave the cutoff sphere. This is |
| 744 |
+ |
discussed in detail in section \ref{sec:}. |
| 745 |
+ |
|
| 746 |
+ |
The two shifted-force methods (GSF and TSF) exhibit a small amount of |
| 747 |
+ |
systematic variation and scatter compared with the Ewald forces. The |
| 748 |
+ |
shifted-force models intentionally perturb the forces between pairs of |
| 749 |
+ |
molecules inside the cutoff sphere in order to correct the energy |
| 750 |
+ |
conservation issues, so it is not particularly surprising that this |
| 751 |
+ |
perturbation is evident in these same molecular forces. The GSF |
| 752 |
+ |
perturbations are minimal, particularly for moderate damping and and |
| 753 |
+ |
commonly-used cutoff values ($r_c = 12$\AA). The TSF method shows |
| 754 |
+ |
reasonable agreement in the correlation coefficient but again the |
| 755 |
+ |
systematic error in the forces is concerning if replication of Ewald |
| 756 |
+ |
forces is desired. |
| 757 |
+ |
|
| 758 |
|
\begin{figure} |
| 759 |
< |
\centering |
| 760 |
< |
\includegraphics[width=0.5 \textwidth]{Variance_forceNtorque_modified-crop.pdf} |
| 761 |
< |
\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)} |
| 762 |
< |
\label{fig:slopeCorr_circularVariance} |
| 763 |
< |
\end{figure} |
| 764 |
< |
\subsection{Total energy conservation} |
| 765 |
< |
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. |
| 759 |
> |
\centering |
| 760 |
> |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
| 761 |
> |
\caption{Statistical analysis of the quality of the force vector |
| 762 |
> |
magnitudes for the real-space electrostatic methods compared with |
| 763 |
> |
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 764 |
> |
line) indicate force magnitude values indistinguishable from those |
| 765 |
> |
obtained using the multipolar Ewald sum. Different values of the |
| 766 |
> |
cutoff radius are indicated with different symbols (9\AA\ = |
| 767 |
> |
circles, 12\AA\ = squares, and 15\AA\ = inverted triangles). } |
| 768 |
> |
\label{fig:slopeCorr_force} |
| 769 |
> |
\end{figure} |
| 770 |
> |
|
| 771 |
> |
|
| 772 |
|
\begin{figure} |
| 773 |
< |
\centering |
| 774 |
< |
\includegraphics[width=0.5 \textwidth]{log(energyDrift)-crop.pdf} |
| 775 |
< |
\label{fig:energyDrift} |
| 776 |
< |
\end{figure} |
| 773 |
> |
\centering |
| 774 |
> |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
| 775 |
> |
\caption{Statistical analysis of the quality of the torque vector |
| 776 |
> |
magnitudes for the real-space electrostatic methods compared with |
| 777 |
> |
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 778 |
> |
line) indicate force magnitude values indistinguishable from those |
| 779 |
> |
obtained using the multipolar Ewald sum. Different values of the |
| 780 |
> |
cutoff radius are indicated with different symbols (9\AA\ = |
| 781 |
> |
circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).} |
| 782 |
> |
\label{fig:slopeCorr_torque} |
| 783 |
> |
\end{figure} |
| 784 |
> |
|
| 785 |
> |
The torques (Fig. \ref{fig:slopeCorr_torque}) appear to be |
| 786 |
> |
significantly influenced by the choice of real-space method. The |
| 787 |
> |
torque expressions have the same distance dependence as the energies, |
| 788 |
> |
which are naturally longer-ranged expressions than the inter-site |
| 789 |
> |
forces. Torques are also quite sensitive to orientations of |
| 790 |
> |
neighboring molecules, even those that are near the cutoff distance. |
| 791 |
> |
|
| 792 |
> |
The results shows that the torque from the hard cutoff method |
| 793 |
> |
reproduces the torques in quite good agreement with the Ewald sum. |
| 794 |
> |
The other real-space methods can cause some significant deviations, |
| 795 |
> |
but excellent agreement with the Ewald sum torques is recovered at |
| 796 |
> |
moderate values of the damping coefficient ($\alpha = |
| 797 |
> |
0.1-0.2$\AA$^{-1}$) and cutoff radius ($r_c \ge 12$\AA). The TSF |
| 798 |
> |
method exhibits the only fair agreement in the slope as compared to |
| 799 |
> |
Ewald even for larger cutoff radii. It appears that the severity of |
| 800 |
> |
the perturbations in the TSF method are most apparent in the torques. |
| 801 |
> |
|
| 802 |
> |
\subsection{Directionality of the force and torque vectors} |
| 803 |
> |
|
| 804 |
> |
The accurate evaluation of force and torque directions is just as |
| 805 |
> |
important for molecular dynamics simulations as the magnitudes of |
| 806 |
> |
these quantities. Force and torque vectors for all six systems were |
| 807 |
> |
analyzed using Fisher statistics, and the quality of the vector |
| 808 |
> |
directionality is shown in terms of circular variance |
| 809 |
> |
($\mathrm{Var}(\theta$) in figure |
| 810 |
> |
\ref{fig:slopeCorr_circularVariance}. The force and torque vectors |
| 811 |
> |
from the new real-space method exhibit nearly-ideal Fisher probability |
| 812 |
> |
distributions (Eq.~\ref{eq:pdf}). Both the hard and SP cutoff methods |
| 813 |
> |
exhibit the best vectorial agreement with the Ewald sum. The force and |
| 814 |
> |
torque vectors from GSF method also show good agreement with the Ewald |
| 815 |
> |
method, which can also be systematically improved by using moderate |
| 816 |
> |
damping and a reasonable cutoff radius. For $\alpha = 0.2$ and $r_c = |
| 817 |
> |
12$\AA, we observe $\mathrm{Var}(\theta) = 0.00206$, which corresponds |
| 818 |
> |
to a distribution with 95\% of force vectors within $6.37^\circ$ of the |
| 819 |
> |
corresponding Ewald forces. The TSF method produces the poorest |
| 820 |
> |
agreement with the Ewald force directions. |
| 821 |
> |
|
| 822 |
> |
Torques are again more perturbed by the new real-space methods, than |
| 823 |
> |
forces, but even here the variance is reasonably small. For the same |
| 824 |
> |
method (GSF) with the same parameters ($\alpha = 0.2, r_c = 12$\AA), |
| 825 |
> |
the circular variance was 0.01415, corresponds to a distribution which |
| 826 |
> |
has 95\% of torque vectors are within $16.75^\circ$ of the Ewald |
| 827 |
> |
results. Again, the direction of the force and torque vectors can be |
| 828 |
> |
systematically improved by varying $\alpha$ and $r_c$. |
| 829 |
> |
|
| 830 |
|
\begin{figure} |
| 831 |
< |
\centering |
| 832 |
< |
\includegraphics[width=0.5 \textwidth]{logSD-crop.pdf} |
| 833 |
< |
\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. } |
| 834 |
< |
\label{fig:fluctuation} |
| 835 |
< |
\end{figure} |
| 831 |
> |
\centering |
| 832 |
> |
\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
| 833 |
> |
\caption{The circular variance of the direction of the force and |
| 834 |
> |
torque vectors obtained from the real-space methods around the |
| 835 |
> |
reference Ewald vectors. A variance equal to 0 (dashed line) |
| 836 |
> |
indicates direction of the force or torque vectors are |
| 837 |
> |
indistinguishable from those obtained from the Ewald sum. Here |
| 838 |
> |
different symbols represent different values of the cutoff radius |
| 839 |
> |
(9 \AA\ = circle, 12 \AA\ = square, 15 \AA\ = inverted triangle)} |
| 840 |
> |
\label{fig:slopeCorr_circularVariance} |
| 841 |
> |
\end{figure} |
| 842 |
> |
|
| 843 |
> |
\subsection{Energy conservation} |
| 844 |
> |
|
| 845 |
> |
We have tested the conservation of energy one can expect to see with |
| 846 |
> |
the new real-space methods using the SSDQ water model with a small |
| 847 |
> |
fraction of solvated ions. This is a test system which exercises all |
| 848 |
> |
orders of multipole-multipole interactions derived in the first paper |
| 849 |
> |
in this series and provides the most comprehensive test of the new |
| 850 |
> |
methods. A liquid-phase system was created with 2000 water molecules |
| 851 |
> |
and 48 dissolved ions at a density of 0.98 g cm${-3}$ and a |
| 852 |
> |
temperature of 300K. After equilibration, this liquid-phase system |
| 853 |
> |
was run for 1 ns under the Ewald, Hard, SP, GSF, and TSF methods with |
| 854 |
> |
a cutoff radius of 9\AA. The value of the damping coefficient was |
| 855 |
> |
also varied from the undamped case ($\alpha = 0$) to a heavily damped |
| 856 |
> |
case ($\alpha = 0.3$ \AA$^{-1}$) for the real space methods. A sample |
| 857 |
> |
was also run using the multipolar Ewald sum. |
| 858 |
> |
|
| 859 |
> |
In figure~\ref{fig:energyDrift} we show the both the linear drift in |
| 860 |
> |
energy over time, $\delta E_1$, and the standard deviation of energy |
| 861 |
> |
fluctuations around this drift $\delta E_0$. Both of the |
| 862 |
> |
shifted-force methods (GSF and TSF) provide excellent energy |
| 863 |
> |
conservation (drift less than $10^{-6}$ kcal / mol / ns / particle), |
| 864 |
> |
while the hard cutoff is essentially unusable for molecular dynamics. |
| 865 |
> |
SP provides some benefit over the hard cutoff because the energetic |
| 866 |
> |
jumps that happen as particles leave and enter the cutoff sphere are |
| 867 |
> |
somewhat reduced. |
| 868 |
> |
|
| 869 |
> |
We note that for all tested values of the cutoff radius, the new |
| 870 |
> |
real-space methods can provide better energy conservation behavior |
| 871 |
> |
than the multipolar Ewald sum, even when utilizing a relatively large |
| 872 |
> |
$k$-space cutoff values. |
| 873 |
> |
|
| 874 |
> |
\begin{figure} |
| 875 |
> |
\centering |
| 876 |
> |
\includegraphics[width=\textwidth]{newDrift.pdf} |
| 877 |
> |
\label{fig:energyDrift} |
| 878 |
> |
\caption{Analysis of the energy conservation of the real-space |
| 879 |
> |
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
| 880 |
> |
energy over time and $\delta \mathrm{E}_0$ is the standard deviation |
| 881 |
> |
of energy fluctuations around this drift. All simulations were of a |
| 882 |
> |
2000-molecule simulation of SSDQ water with 48 ionic charges at 300 |
| 883 |
> |
K starting from the same initial configuration.} |
| 884 |
> |
\end{figure} |
| 885 |
> |
|
| 886 |
> |
|
| 887 |
|
\section{CONCLUSION} |
| 888 |
< |
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. |
| 889 |
< |
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. |
| 890 |
< |
In MD simulations, the energy conservation is very important. The conservation of the total energy can be ensured by i) enforcing the smooth truncation of the energy, force and torque in the cutoff radius and ii) making the energy, force and torque consistent with each other. The GSF and TSF methods ensure the consistency and smooth truncation of the energy, force and torque at the cutoff radius, as a result show very good total energy conservation. But the TSF method does not show good agreement in the absolute value of the electrostatic energy, force and torque with the Ewald. The GSF method has mimicked Ewald’s force, energy and torque accurately and also conserved energy. Therefore, the GSF method is the suitable method for evaluating required force field in MD simulations. In addition, the energy drift and fluctuation from the GSF method is much better than Ewald’s method for finite-sized reciprocal space. |
| 888 |
> |
We have generalized the charged neutralized potential energy |
| 889 |
> |
originally developed by the Wolf et al.\cite{Wolf:1999dn} for the |
| 890 |
> |
charge-charge interaction to the charge-multipole and |
| 891 |
> |
multipole-multipole interaction in the SP method for higher order |
| 892 |
> |
multipoles. Also, we have developed GSF and TSF methods by |
| 893 |
> |
implementing the modification purposed by Fennel and |
| 894 |
> |
Gezelter\cite{Fennell:2006lq} for the charge-charge interaction to the |
| 895 |
> |
higher order multipoles to ensure consistency and smooth truncation of |
| 896 |
> |
the electrostatic energy, force, and torque for the spherical |
| 897 |
> |
truncation. The SP methods for multipoles proved its suitability in MC |
| 898 |
> |
simulations. On the other hand, the results from the GSF method |
| 899 |
> |
produced good agreement with the Ewald's energy, force, and |
| 900 |
> |
torque. Also, it shows very good energy conservation in MD |
| 901 |
> |
simulations. The direct truncation of any molecular system without |
| 902 |
> |
multipole neutralization creates the fluctuation in the electrostatic |
| 903 |
> |
energy. This fluctuation in the energy is very large for the case of |
| 904 |
> |
crystal because of long range of multipole ordering (Refer paper |
| 905 |
> |
I).\cite{PaperI} This is also significant in the case of the liquid |
| 906 |
> |
because of the local multipole ordering in the molecules. If the net |
| 907 |
> |
multipole within cutoff radius neutralized within cutoff sphere by |
| 908 |
> |
placing image multiples on the surface of the sphere, this fluctuation |
| 909 |
> |
in the energy reduced significantly. Also, the multipole |
| 910 |
> |
neutralization in the generalized SP method showed very good agreement |
| 911 |
> |
with the Ewald as compared to direct truncation for the evaluation of |
| 912 |
> |
the $\triangle E$ between the configurations. In MD simulations, the |
| 913 |
> |
energy conservation is very important. The conservation of the total |
| 914 |
> |
energy can be ensured by i) enforcing the smooth truncation of the |
| 915 |
> |
energy, force and torque in the cutoff radius and ii) making the |
| 916 |
> |
energy, force and torque consistent with each other. The GSF and TSF |
| 917 |
> |
methods ensure the consistency and smooth truncation of the energy, |
| 918 |
> |
force and torque at the cutoff radius, as a result show very good |
| 919 |
> |
total energy conservation. But the TSF method does not show good |
| 920 |
> |
agreement in the absolute value of the electrostatic energy, force and |
| 921 |
> |
torque with the Ewald. The GSF method has mimicked Ewald’s force, |
| 922 |
> |
energy and torque accurately and also conserved energy. Therefore, the |
| 923 |
> |
GSF method is the suitable method for evaluating required force field |
| 924 |
> |
in MD simulations. In addition, the energy drift and fluctuation from |
| 925 |
> |
the GSF method is much better than Ewald’s method for finite-sized |
| 926 |
> |
reciprocal space. |
| 927 |
> |
|
| 928 |
> |
Note that the TSF, GSF, and SP models are $\mathcal{O}(N)$ methods |
| 929 |
> |
that can be made extremely efficient using spline interpolations of |
| 930 |
> |
the radial functions. They require no Fourier transforms or $k$-space |
| 931 |
> |
sums, and guarantee the smooth handling of energies, forces, and |
| 932 |
> |
torques as multipoles cross the real-space cutoff boundary. |
| 933 |
> |
|
| 934 |
|
%\bibliographystyle{aip} |
| 935 |
|
\newpage |
| 936 |
|
\bibliography{references} |
