| 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} This work uses the |
| 481 |
> |
multipolar Ewald sum as a reference method for comparing energies, |
| 482 |
> |
forces, and torques for molecular models that mimic disordered and |
| 483 |
> |
ordered condensed-phase systems. These test-cases include: |
| 484 |
|
|
| 485 |
< |
\subsection{Statistical analysis} |
| 486 |
< |
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. |
| 485 |
> |
\begin{itemize} |
| 486 |
> |
\item Soft Dipolar fluids ($\sigma = , \epsilon = , |D| = $) |
| 487 |
> |
\item Soft Dipolar solids ($\sigma = , \epsilon = , |D| = $) |
| 488 |
> |
\item Soft Quadrupolar fluids ($\sigma = , \epsilon = , Q_{xx} = ...$) |
| 489 |
> |
\item Soft Quadrupolar solids ($\sigma = , \epsilon = , Q_{xx} = ...$) |
| 490 |
> |
\item A mixed multipole model for water |
| 491 |
> |
\item A mixed multipole models for water with dissolved ions |
| 492 |
> |
\end{itemize} |
| 493 |
> |
This last test case exercises all levels of the multipole-multipole |
| 494 |
> |
interactions we have derived so far and represents the most complete |
| 495 |
> |
test of the new methods. |
| 496 |
|
|
| 497 |
< |
\subsection{Analysis of vector quantities} |
| 498 |
< |
R.A. Fisher has developed a probablity density function to analyse directional data sets is expressed as below,\cite{fisher53} |
| 499 |
< |
\begin{equation} |
| 500 |
< |
p_f(\theta) = \frac{\kappa}{2 \sinh\kappa}\sin\theta \exp(\kappa \cos\theta) |
| 501 |
< |
\label{eq:pdf} |
| 502 |
< |
\end{equation} |
| 503 |
< |
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, |
| 497 |
> |
In the following section, we present results for the total |
| 498 |
> |
electrostatic energy, as well as the electrostatic contributions to |
| 499 |
> |
the force and torque on each molecule. These quantities have been |
| 500 |
> |
computed using the SP, TSF, and GSF methods, as well as a hard cutoff, |
| 501 |
> |
and have been compared with the values obtaine from the multipolar |
| 502 |
> |
Ewald sum. In Mote Carlo (MC) simulations, the energy differences |
| 503 |
> |
between two configurations is the primary quantity that governs how |
| 504 |
> |
the simulation proceeds. These differences are the most imporant |
| 505 |
> |
indicators of the reliability of a method even if the absolute |
| 506 |
> |
energies are not exact. For each of the multipolar systems listed |
| 507 |
> |
above, we have compared the change in electrostatic potential energy |
| 508 |
> |
($\Delta E$) between 250 statistically-independent configurations. In |
| 509 |
> |
molecular dynamics (MD) simulations, the forces and torques govern the |
| 510 |
> |
behavior of the simulation, so we also compute the electrostatic |
| 511 |
> |
contributions to the forces and torques. |
| 512 |
> |
|
| 513 |
> |
\subsection{Model systems} |
| 514 |
> |
To sample independent configurations of multipolar crystals, a body |
| 515 |
> |
centered cubic (BCC) crystal which is a minimum energy structure for |
| 516 |
> |
point dipoles was generated using 3,456 molecules. The multipoles |
| 517 |
> |
were translationally locked in their respective crystal sites for |
| 518 |
> |
equilibration at a relatively low temperature (50K), so that dipoles |
| 519 |
> |
or quadrupoles could freely explore all accessible orientations. The |
| 520 |
> |
translational constraints were removed, and the crystals were |
| 521 |
> |
simulated for 10 ps in the microcanonical (NVE) ensemble with an |
| 522 |
> |
average temperature of 50 K. Configurations were sampled at equal |
| 523 |
> |
time intervals for the comparison of the configurational energy |
| 524 |
> |
differences. The crystals were not simulated close to the melting |
| 525 |
> |
points in order to avoid translational deformation away of the ideal |
| 526 |
> |
lattice geometry. |
| 527 |
> |
|
| 528 |
> |
For dipolar, quadrupolar, and mixed-multipole liquid simulations, each |
| 529 |
> |
system was created with 2048 molecules oriented randomly. These were |
| 530 |
> |
|
| 531 |
> |
system with 2,048 molecules simulated for 1ns in NVE ensemble at 300 K |
| 532 |
> |
temperature after equilibration. We collected 250 different |
| 533 |
> |
configurations in equal interval of time. For the ions mixed liquid |
| 534 |
> |
system, we converted 48 different molecules into 24 \ce{Na+} and 24 |
| 535 |
> |
\ce{Cl-} ions and equilibrated. After equilibration, the system was run |
| 536 |
> |
at the same environment for 1ns and 250 configurations were |
| 537 |
> |
collected. While comparing energies, forces, and torques with Ewald |
| 538 |
> |
method, Lennard-Jones potentials were turned off and purely |
| 539 |
> |
electrostatic interaction had been compared. |
| 540 |
> |
|
| 541 |
> |
\subsection{Accuracy of Energy Differences, Forces and Torques} |
| 542 |
> |
The pairwise summation techniques (outlined above) were evaluated for |
| 543 |
> |
use in MC simulations by studying the energy differences between |
| 544 |
> |
different configurations. We took the Ewald-computed energy |
| 545 |
> |
difference between two conformations to be the correct behavior. An |
| 546 |
> |
ideal performance by one of the new methods would reproduce these |
| 547 |
> |
energy differences exactly. The configurational energies being used |
| 548 |
> |
here contain only contributions from electrostatic interactions. |
| 549 |
> |
Lennard-Jones interactions were omitted from the comparison as they |
| 550 |
> |
should be identical for all methods. |
| 551 |
> |
|
| 552 |
> |
Since none of the real-space methods provide exact energy differences, |
| 553 |
> |
we used least square regressions analysiss for the six different |
| 554 |
> |
molecular systems to compare $\Delta E$ from Hard, SP, GSF, and TSF |
| 555 |
> |
with the multipolar Ewald reference method. Unitary results for both |
| 556 |
> |
the correlation (slope) and correlation coefficient for these |
| 557 |
> |
regressions indicate perfect agreement between the real-space method |
| 558 |
> |
and the multipolar Ewald sum. |
| 559 |
> |
|
| 560 |
> |
Molecular systems were run long enough to explore independent |
| 561 |
> |
configurations and 250 configurations were recorded for comparison. |
| 562 |
> |
Each system provided 31,125 energy differences for a total of 186,750 |
| 563 |
> |
data points. Similarly, the magnitudes of the forces and torques have |
| 564 |
> |
also been compared by using least squares regression analyses. In the |
| 565 |
> |
forces and torques comparison, the magnitudes of the forces acting in |
| 566 |
> |
each molecule for each configuration were evaluated. For example, our |
| 567 |
> |
dipolar liquid simulation contains 2048 molecules and there are 250 |
| 568 |
> |
different configurations for each system resulting in 3,072,000 data |
| 569 |
> |
points for comparison of forces and torques. |
| 570 |
> |
|
| 571 |
> |
\subsection{Analysis of vector quantities} |
| 572 |
> |
Getting the magnitudes of the force and torque vectors correct is only |
| 573 |
> |
part of the issue for carrying out accurate molecular dynamics |
| 574 |
> |
simulations. Because the real space methods reweight the different |
| 575 |
> |
orientational contributions to the energies, it is also important to |
| 576 |
> |
understand how the methods impact the \textit{directionality} of the |
| 577 |
> |
force and torque vectors. Fisher developed a probablity density |
| 578 |
> |
function to analyse directional data sets, |
| 579 |
|
\begin{equation} |
| 580 |
< |
R = \sqrt{(\sum\limits_{i=1}^N \sin\theta_i)^2 + (\sum\limits_{i=1}^N \sin\theta_i)^2} |
| 580 |
> |
p_f(\theta) = \frac{\kappa}{2 \sinh\kappa}\sin\theta e^{\kappa \cos\theta} |
| 581 |
> |
\label{eq:pdf} |
| 582 |
> |
\end{equation} |
| 583 |
> |
where $\kappa$ measures directional dispersion of the data around the |
| 584 |
> |
mean direction.\cite{fisher53} This quantity $(\kappa)$ can be |
| 585 |
> |
estimated as a reciprocal of the circular variance.\cite{Allen91} To |
| 586 |
> |
quantify the directional error, forces obtained from the Ewald sum |
| 587 |
> |
were taken as the mean (or correct) direction and the angle between |
| 588 |
> |
the forces obtained via the Ewald sum and the real-space methods were |
| 589 |
> |
evaluated, |
| 590 |
> |
\begin{equation} |
| 591 |
> |
\cos\theta_i = \frac{\vec{f}_i^\mathrm{~Ewald} \cdot |
| 592 |
> |
\vec{f}_i^\mathrm{~GSF}}{\left|\vec{f}_i^\mathrm{~Ewald}\right| \left|\vec{f}_i^\mathrm{~GSF}\right|} |
| 593 |
> |
\end{equation} |
| 594 |
> |
The total angular displacement of the vectors was calculated as, |
| 595 |
> |
\begin{equation} |
| 596 |
> |
R = \sqrt{\left(\sum\limits_{i=1}^N \cos\theta_i\right)^2 + \left(\sum\limits_{i=1}^N \sin\theta_i\right)^2} |
| 597 |
|
\label{eq:displacement} |
| 598 |
|
\end{equation} |
| 599 |
< |
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. |
| 599 |
> |
where $N$ is number of force vectors. The circular variance is |
| 600 |
> |
defined as |
| 601 |
> |
\begin{equation} |
| 602 |
> |
\mathrm{Var}(\theta) \approx 1/\kappa = 1 - R/N |
| 603 |
> |
\end{equation} |
| 604 |
> |
The circular variance takes on values between from 0 to 1, with 0 |
| 605 |
> |
indicating a perfect directional match between the Ewald force vectors |
| 606 |
> |
and the real-space forces. Lower values of $\mathrm{Var}(\theta)$ |
| 607 |
> |
correspond to higher values of $\kappa$, which indicates tighter |
| 608 |
> |
clustering of the real-space force vectors around the Ewald forces. |
| 609 |
|
|
| 610 |
+ |
A similar analysis was carried out for the electrostatic contribution |
| 611 |
+ |
to the molecular torques as well as forces. |
| 612 |
+ |
|
| 613 |
|
\subsection{Energy conservation} |
| 614 |
< |
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. |
| 614 |
> |
To test conservation the energy for the methods, the mixed molecular |
| 615 |
> |
system of 2000 SSDQ water molecules with 24 \ce{Na+} and 24 \ce{Cl-} |
| 616 |
> |
ions was run for 1 ns in the microcanonical ensemble at an average |
| 617 |
> |
temperature of 300K. Each of the different electrostatic methods |
| 618 |
> |
(Ewald, Hard, SP, GSF, and TSF) was tested for a range of different |
| 619 |
> |
damping values. The molecular system was started with same initial |
| 620 |
> |
positions and velocities for all cutoff methods. The energy drift |
| 621 |
> |
($\delta E_1$) and standard deviation of the energy about the slope |
| 622 |
> |
($\delta E_0$) were evaluated from the total energy of the system as a |
| 623 |
> |
function of time. Although both measures are valuable at |
| 624 |
> |
investigating new methods for molecular dynamics, a useful interaction |
| 625 |
> |
model must allow for long simulation times with minimal energy drift. |
| 626 |
|
|
| 627 |
|
\section{\label{sec:result}RESULTS} |
| 628 |
|
\subsection{Configurational energy differences} |
| 695 |
|
\section{CONCLUSION} |
| 696 |
|
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. |
| 697 |
|
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. |
| 698 |
< |
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. |
| 698 |
> |
In MD simulations, the energy conservation is very important. The |
| 699 |
> |
conservation of the total energy can be ensured by i) enforcing the |
| 700 |
> |
smooth truncation of the energy, force and torque in the cutoff radius |
| 701 |
> |
and ii) making the energy, force and torque consistent with each |
| 702 |
> |
other. The GSF and TSF methods ensure the consistency and smooth |
| 703 |
> |
truncation of the energy, force and torque at the cutoff radius, as a |
| 704 |
> |
result show very good total energy conservation. But the TSF method |
| 705 |
> |
does not show good agreement in the absolute value of the |
| 706 |
> |
electrostatic energy, force and torque with the Ewald. The GSF method |
| 707 |
> |
has mimicked Ewald’s force, energy and torque accurately and also |
| 708 |
> |
conserved energy. Therefore, the GSF method is the suitable method for |
| 709 |
> |
evaluating required force field in MD simulations. In addition, the |
| 710 |
> |
energy drift and fluctuation from the GSF method is much better than |
| 711 |
> |
Ewald’s method for finite-sized reciprocal space. |
| 712 |
> |
|
| 713 |
> |
Note that the TSF, GSF, and SP models are $\mathcal{O}(N)$ methods |
| 714 |
> |
that can be made extremely efficient using spline interpolations of |
| 715 |
> |
the radial functions. They require no Fourier transforms or $k$-space |
| 716 |
> |
sums, and guarantee the smooth handling of energies, forces, and |
| 717 |
> |
torques as multipoles cross the real-space cutoff boundary. |
| 718 |
> |
|
| 719 |
|
%\bibliographystyle{aip} |
| 720 |
|
\newpage |
| 721 |
|
\bibliography{references} |
