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|
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\begin{document} |
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|
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\title{Is the Ewald Summation necessary? Pairwise alternatives to the accepted standard for long-range electrostatics} |
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> |
\title{Is the Ewald summation still necessary? \\ |
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Pairwise alternatives to the accepted standard for \\ |
| 30 |
> |
long-range electrostatics} |
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|
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\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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\maketitle |
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\doublespacing |
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|
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– |
\nobibliography{} |
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\begin{abstract} |
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< |
A new method for accumulating electrostatic interactions was derived |
| 45 |
< |
from the previous efforts described in \bibentry{Wolf99} and |
| 46 |
< |
\bibentry{Zahn02} as a possible replacement for lattice sum methods in |
| 47 |
< |
molecular simulations. Comparisons were performed with this and other |
| 48 |
< |
pairwise electrostatic summation techniques against the smooth |
| 49 |
< |
particle mesh Ewald (SPME) summation to see how well they reproduce |
| 50 |
< |
the energetics and dynamics of a variety of simulation types. The |
| 51 |
< |
newly derived Shifted-Force technique shows a remarkable ability to |
| 52 |
< |
reproduce the behavior exhibited in simulations using SPME with an |
| 53 |
< |
$\mathscr{O}(N)$ computational cost, equivalent to merely the |
| 54 |
< |
real-space portion of the lattice summation. |
| 55 |
< |
|
| 44 |
> |
We investigate pairwise electrostatic interaction methods and show |
| 45 |
> |
that there are viable and computationally efficient $(\mathscr{O}(N))$ |
| 46 |
> |
alternatives to the Ewald summation for typical modern molecular |
| 47 |
> |
simulations. These methods are extended from the damped and |
| 48 |
> |
cutoff-neutralized Coulombic sum originally proposed by |
| 49 |
> |
[D. Wolf, P. Keblinski, S.~R. Phillpot, and J. Eggebrecht, {\it J. Chem. Phys.} {\bf 110}, 8255 (1999)] One of these, the damped shifted force method, shows |
| 50 |
> |
a remarkable ability to reproduce the energetic and dynamic |
| 51 |
> |
characteristics exhibited by simulations employing lattice summation |
| 52 |
> |
techniques. Comparisons were performed with this and other pairwise |
| 53 |
> |
methods against the smooth particle mesh Ewald ({\sc spme}) summation |
| 54 |
> |
to see how well they reproduce the energetics and dynamics of a |
| 55 |
> |
variety of simulation types. |
| 56 |
|
\end{abstract} |
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|
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|
\newpage |
| 95 |
|
regarding possible artifacts caused by the inherent periodicity of the |
| 96 |
|
explicit Ewald summation.\cite{Tobias01} |
| 97 |
|
|
| 98 |
< |
In this paper, we focus on a new set of shifted methods devised by |
| 98 |
> |
In this paper, we focus on a new set of pairwise methods devised by |
| 99 |
|
Wolf {\it et al.},\cite{Wolf99} which we further extend. These |
| 100 |
|
methods along with a few other mixed methods (i.e. reaction field) are |
| 101 |
|
compared with the smooth particle mesh Ewald |
| 110 |
|
their usability in molecular simulations. |
| 111 |
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|
| 112 |
|
\subsection{The Ewald Sum} |
| 113 |
< |
The complete accumulation electrostatic interactions in a system with |
| 113 |
> |
The complete accumulation of the electrostatic interactions in a system with |
| 114 |
|
periodic boundary conditions (PBC) requires the consideration of the |
| 115 |
|
effect of all charges within a (cubic) simulation box as well as those |
| 116 |
|
in the periodic replicas, |
| 155 |
|
system is said to be using conducting (or ``tin-foil'') boundary |
| 156 |
|
conditions, $\epsilon_{\rm S} = \infty$. Figure |
| 157 |
|
\ref{fig:ewaldTime} shows how the Ewald sum has been applied over |
| 158 |
< |
time. Initially, due to the small sizes of the systems that could be |
| 159 |
< |
feasibly simulated, the entire simulation box was replicated to |
| 160 |
< |
convergence. In more modern simulations, the simulation boxes have |
| 161 |
< |
grown large enough that a real-space cutoff could potentially give |
| 162 |
< |
convergent behavior. Indeed, it has often been observed that the |
| 163 |
< |
reciprocal-space portion of the Ewald sum can be small and rapidly |
| 164 |
< |
convergent compared to the real-space portion with the choice of small |
| 165 |
< |
$\alpha$.\cite{Karasawa89,Kolafa92} |
| 158 |
> |
time. Initially, due to the small system sizes that could be |
| 159 |
> |
simulated feasibly, the entire simulation box was replicated to |
| 160 |
> |
convergence. In more modern simulations, the systems have grown large |
| 161 |
> |
enough that a real-space cutoff could potentially give convergent |
| 162 |
> |
behavior. Indeed, it has been observed that with the choice of a |
| 163 |
> |
small $\alpha$, the reciprocal-space portion of the Ewald sum can be |
| 164 |
> |
rapidly convergent and small relative to the real-space |
| 165 |
> |
portion.\cite{Karasawa89,Kolafa92} |
| 166 |
|
|
| 167 |
|
\begin{figure} |
| 168 |
|
\centering |
| 169 |
|
\includegraphics[width = \linewidth]{./ewaldProgression.pdf} |
| 170 |
< |
\caption{How the application of the Ewald summation has changed with |
| 171 |
< |
the increase in computer power. Initially, only small numbers of |
| 172 |
< |
particles could be studied, and the Ewald sum acted to replicate the |
| 173 |
< |
unit cell charge distribution out to convergence. Now, much larger |
| 174 |
< |
systems of charges are investigated with fixed distance cutoffs. The |
| 174 |
< |
calculated structure factor is used to sum out to great distance, and |
| 175 |
< |
a surrounding dielectric term is included.} |
| 170 |
> |
\caption{The change in the application of the Ewald sum with |
| 171 |
> |
increasing computational power. A:~Initially, only small systems could |
| 172 |
> |
be studied, and the Ewald sum replicated the simulation box to |
| 173 |
> |
convergence. B:~Now, much larger systems of charges can be |
| 174 |
> |
investigated with fixed-distance cutoffs.} |
| 175 |
|
\label{fig:ewaldTime} |
| 176 |
|
\end{figure} |
| 177 |
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|
| 227 |
|
charge contained within the cutoff radius is crucial for potential |
| 228 |
|
stability. They devised a pairwise summation method that ensures |
| 229 |
|
charge neutrality and gives results similar to those obtained with the |
| 230 |
< |
Ewald summation. The resulting shifted Coulomb potential |
| 231 |
< |
(Eq. \ref{eq:WolfPot}) includes image-charges subtracted out through |
| 232 |
< |
placement on the cutoff sphere and a distance-dependent damping |
| 233 |
< |
function (identical to that seen in the real-space portion of the |
| 235 |
< |
Ewald sum) to aid convergence |
| 230 |
> |
Ewald summation. The resulting shifted Coulomb potential includes |
| 231 |
> |
image-charges subtracted out through placement on the cutoff sphere |
| 232 |
> |
and a distance-dependent damping function (identical to that seen in |
| 233 |
> |
the real-space portion of the Ewald sum) to aid convergence |
| 234 |
|
\begin{equation} |
| 235 |
|
V_{\textrm{Wolf}}(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}. |
| 236 |
|
\label{eq:WolfPot} |
| 516 |
|
The pairwise summation techniques (outlined in section |
| 517 |
|
\ref{sec:ESMethods}) were evaluated for use in MC simulations by |
| 518 |
|
studying the energy differences between conformations. We took the |
| 519 |
< |
SPME-computed energy difference between two conformations to be the |
| 519 |
> |
{\sc spme}-computed energy difference between two conformations to be the |
| 520 |
|
correct behavior. An ideal performance by an alternative method would |
| 521 |
|
reproduce these energy differences exactly (even if the absolute |
| 522 |
|
energies calculated by the methods are different). Since none of the |
| 524 |
|
regressions of energy gap data to evaluate how closely the methods |
| 525 |
|
mimicked the Ewald energy gaps. Unitary results for both the |
| 526 |
|
correlation (slope) and correlation coefficient for these regressions |
| 527 |
< |
indicate perfect agreement between the alternative method and SPME. |
| 527 |
> |
indicate perfect agreement between the alternative method and {\sc spme}. |
| 528 |
|
Sample correlation plots for two alternate methods are shown in |
| 529 |
|
Fig. \ref{fig:linearFit}. |
| 530 |
|
|
| 553 |
|
We evaluated the pairwise methods (outlined in section |
| 554 |
|
\ref{sec:ESMethods}) for use in MD simulations by |
| 555 |
|
comparing the force and torque vectors with those obtained using the |
| 556 |
< |
reference Ewald summation (SPME). Both the magnitude and the |
| 556 |
> |
reference Ewald summation ({\sc spme}). Both the magnitude and the |
| 557 |
|
direction of these vectors on each of the bodies in the system were |
| 558 |
|
analyzed. For the magnitude of these vectors, linear least squares |
| 559 |
|
regression analyses were performed as described previously for |
| 568 |
|
|
| 569 |
|
The {\it directionality} of the force and torque vectors was |
| 570 |
|
investigated through measurement of the angle ($\theta$) formed |
| 571 |
< |
between those computed from the particular method and those from SPME, |
| 571 |
> |
between those computed from the particular method and those from {\sc spme}, |
| 572 |
|
\begin{equation} |
| 573 |
|
\theta_f = \cos^{-1} \left(\hat{F}_\textrm{SPME} \cdot \hat{F}_\textrm{M}\right), |
| 574 |
|
\end{equation} |
| 575 |
< |
where $\hat{f}_\textrm{M}$ is the unit vector pointing along the force |
| 576 |
< |
vector computed using method M. |
| 577 |
< |
|
| 578 |
< |
Each of these $\theta$ values was accumulated in a distribution |
| 581 |
< |
function and weighted by the area on the unit sphere. Non-linear |
| 582 |
< |
Gaussian fits were used to measure the width of the resulting |
| 583 |
< |
distributions. |
| 584 |
< |
|
| 585 |
< |
\begin{figure} |
| 586 |
< |
\centering |
| 587 |
< |
\includegraphics[width = \linewidth]{./gaussFit.pdf} |
| 588 |
< |
\caption{Sample fit of the angular distribution of the force vectors |
| 589 |
< |
accumulated using all of the studied systems. Gaussian fits were used |
| 590 |
< |
to obtain values for the variance in force and torque vectors.} |
| 591 |
< |
\label{fig:gaussian} |
| 592 |
< |
\end{figure} |
| 593 |
< |
|
| 594 |
< |
Figure \ref{fig:gaussian} shows an example distribution with applied |
| 595 |
< |
non-linear fits. The solid line is a Gaussian profile, while the |
| 596 |
< |
dotted line is a Voigt profile, a convolution of a Gaussian and a |
| 597 |
< |
Lorentzian. Since this distribution is a measure of angular error |
| 575 |
> |
where $\hat{F}_\textrm{M}$ is the unit vector pointing along the force |
| 576 |
> |
vector computed using method M. Each of these $\theta$ values was |
| 577 |
> |
accumulated in a distribution function and weighted by the area on the |
| 578 |
> |
unit sphere. Since this distribution is a measure of angular error |
| 579 |
|
between two different electrostatic summation methods, there is no |
| 580 |
< |
{\it a priori} reason for the profile to adhere to any specific shape. |
| 581 |
< |
Gaussian fits was used to compare all the tested methods. The |
| 582 |
< |
variance ($\sigma^2$) was extracted from each of these fits and was |
| 583 |
< |
used to compare distribution widths. Values of $\sigma^2$ near zero |
| 584 |
< |
indicate vector directions indistinguishable from those calculated |
| 585 |
< |
when using the reference method (SPME). |
| 580 |
> |
{\it a priori} reason for the profile to adhere to any specific |
| 581 |
> |
shape. Thus, gaussian fits were used to measure the width of the |
| 582 |
> |
resulting distributions. The variance ($\sigma^2$) was extracted from |
| 583 |
> |
each of these fits and was used to compare distribution widths. |
| 584 |
> |
Values of $\sigma^2$ near zero indicate vector directions |
| 585 |
> |
indistinguishable from those calculated when using the reference |
| 586 |
> |
method ({\sc spme}). |
| 587 |
|
|
| 588 |
|
\subsection{Short-time Dynamics} |
| 589 |
|
|
| 598 |
|
autocorrelation functions (Eq. \ref{eq:vCorr}) were computed for each |
| 599 |
|
of the trajectories, |
| 600 |
|
\begin{equation} |
| 601 |
< |
C_v(t) = \frac{\langle v_i(0)\cdot v_i(t)\rangle}{\langle v_i(0)\cdot v_i(0)\rangle}. |
| 601 |
> |
C_v(t) = \frac{\langle v(0)\cdot v(t)\rangle}{\langle v^2\rangle}. |
| 602 |
|
\label{eq:vCorr} |
| 603 |
|
\end{equation} |
| 604 |
|
Velocity autocorrelation functions require detailed short time data, |
| 611 |
|
|
| 612 |
|
The effects of the same subset of alternative electrostatic methods on |
| 613 |
|
the {\it long-time} dynamics of charged systems were evaluated using |
| 614 |
< |
the same model system (NaCl crystals at 1000K). The power spectrum |
| 614 |
> |
the same model system (NaCl crystals at 1000~K). The power spectrum |
| 615 |
|
($I(\omega)$) was obtained via Fourier transform of the velocity |
| 616 |
|
autocorrelation function, \begin{equation} I(\omega) = |
| 617 |
|
\frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt, |
| 626 |
|
\subsection{Representative Simulations}\label{sec:RepSims} |
| 627 |
|
A variety of representative simulations were analyzed to determine the |
| 628 |
|
relative effectiveness of the pairwise summation techniques in |
| 629 |
< |
reproducing the energetics and dynamics exhibited by SPME. We wanted |
| 629 |
> |
reproducing the energetics and dynamics exhibited by {\sc spme}. We wanted |
| 630 |
|
to span the space of modern simulations (i.e. from liquids of neutral |
| 631 |
|
molecules to ionic crystals), so the systems studied were: |
| 632 |
|
\begin{enumerate} |
| 651 |
|
the crystal). The solid and liquid NaCl systems consisted of 500 |
| 652 |
|
$\textrm{Na}^{+}$ and 500 $\textrm{Cl}^{-}$ ions. Configurations for |
| 653 |
|
these systems were selected and equilibrated in the same manner as the |
| 654 |
< |
water systems. The equilibrated temperatures were 1000~K for the NaCl |
| 655 |
< |
crystal and 7000~K for the liquid. The ionic solutions were made by |
| 656 |
< |
solvating 4 (or 40) ions in a periodic box containing 1000 SPC/E water |
| 657 |
< |
molecules. Ion and water positions were then randomly swapped, and |
| 658 |
< |
the resulting configurations were again equilibrated individually. |
| 659 |
< |
Finally, for the Argon / Water ``charge void'' systems, the identities |
| 660 |
< |
of all the SPC/E waters within 6 \AA\ of the center of the |
| 661 |
< |
equilibrated water configurations were converted to argon |
| 662 |
< |
(Fig. \ref{fig:argonSlice}). |
| 654 |
> |
water systems. In order to introduce measurable fluctuations in the |
| 655 |
> |
configuration energy differences, the crystalline simulations were |
| 656 |
> |
equilibrated at 1000~K, near the $T_\textrm{m}$ for NaCl. The liquid |
| 657 |
> |
NaCl configurations needed to represent a fully disordered array of |
| 658 |
> |
point charges, so the high temperature of 7000~K was selected for |
| 659 |
> |
equilibration. The ionic solutions were made by solvating 4 (or 40) |
| 660 |
> |
ions in a periodic box containing 1000 SPC/E water molecules. Ion and |
| 661 |
> |
water positions were then randomly swapped, and the resulting |
| 662 |
> |
configurations were again equilibrated individually. Finally, for the |
| 663 |
> |
Argon / Water ``charge void'' systems, the identities of all the SPC/E |
| 664 |
> |
waters within 6 \AA\ of the center of the equilibrated water |
| 665 |
> |
configurations were converted to argon. |
| 666 |
|
|
| 667 |
|
These procedures guaranteed us a set of representative configurations |
| 668 |
< |
from chemically-relevant systems sampled from an appropriate |
| 669 |
< |
ensemble. Force field parameters for the ions and Argon were taken |
| 668 |
> |
from chemically-relevant systems sampled from appropriate |
| 669 |
> |
ensembles. Force field parameters for the ions and Argon were taken |
| 670 |
|
from the force field utilized by {\sc oopse}.\cite{Meineke05} |
| 671 |
|
|
| 687 |
– |
\begin{figure} |
| 688 |
– |
\centering |
| 689 |
– |
\includegraphics[width = \linewidth]{./slice.pdf} |
| 690 |
– |
\caption{A slice from the center of a water box used in a charge void |
| 691 |
– |
simulation. The darkened region represents the boundary sphere within |
| 692 |
– |
which the water molecules were converted to argon atoms.} |
| 693 |
– |
\label{fig:argonSlice} |
| 694 |
– |
\end{figure} |
| 695 |
– |
|
| 672 |
|
\subsection{Comparison of Summation Methods}\label{sec:ESMethods} |
| 673 |
|
We compared the following alternative summation methods with results |
| 674 |
< |
from the reference method (SPME): |
| 674 |
> |
from the reference method ({\sc spme}): |
| 675 |
|
\begin{itemize} |
| 676 |
|
\item {\sc sp} with damping parameters ($\alpha$) of 0.0, 0.1, 0.2, |
| 677 |
|
and 0.3 \AA$^{-1}$, |
| 682 |
|
\end{itemize} |
| 683 |
|
Group-based cutoffs with a fifth-order polynomial switching function |
| 684 |
|
were utilized for the reaction field simulations. Additionally, we |
| 685 |
< |
investigated the use of these cutoffs with the SP, SF, and pure |
| 686 |
< |
cutoff. The SPME electrostatics were performed using the TINKER |
| 687 |
< |
implementation of SPME,\cite{Ponder87} while all other method |
| 688 |
< |
calculations were performed using the OOPSE molecular mechanics |
| 685 |
> |
investigated the use of these cutoffs with the {\sc sp}, {\sc sf}, and pure |
| 686 |
> |
cutoff. The {\sc spme} electrostatics were performed using the {\sc tinker} |
| 687 |
> |
implementation of {\sc spme},\cite{Ponder87} while all other calculations |
| 688 |
> |
were performed using the {\sc oopse} molecular mechanics |
| 689 |
|
package.\cite{Meineke05} All other portions of the energy calculation |
| 690 |
|
(i.e. Lennard-Jones interactions) were handled in exactly the same |
| 691 |
|
manner across all systems and configurations. |
| 692 |
|
|
| 693 |
< |
The althernative methods were also evaluated with three different |
| 693 |
> |
The alternative methods were also evaluated with three different |
| 694 |
|
cutoff radii (9, 12, and 15 \AA). As noted previously, the |
| 695 |
|
convergence parameter ($\alpha$) plays a role in the balance of the |
| 696 |
|
real-space and reciprocal-space portions of the Ewald calculation. |
| 697 |
|
Typical molecular mechanics packages set this to a value dependent on |
| 698 |
|
the cutoff radius and a tolerance (typically less than $1 \times |
| 699 |
|
10^{-4}$ kcal/mol). Smaller tolerances are typically associated with |
| 700 |
< |
increased accuracy at the expense of increased time spent calculating |
| 701 |
< |
the reciprocal-space portion of the |
| 702 |
< |
summation.\cite{Perram88,Essmann95} The default TINKER tolerance of $1 |
| 703 |
< |
\times 10^{-8}$ kcal/mol was used in all SPME calculations, resulting |
| 704 |
< |
in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for |
| 705 |
< |
cutoff radii of 9, 12, and 15 \AA\ respectively. |
| 700 |
> |
increasing accuracy at the expense of computational time spent on the |
| 701 |
> |
reciprocal-space portion of the summation.\cite{Perram88,Essmann95} |
| 702 |
> |
The default {\sc tinker} tolerance of $1 \times 10^{-8}$ kcal/mol was used |
| 703 |
> |
in all {\sc spme} calculations, resulting in Ewald coefficients of 0.4200, |
| 704 |
> |
0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ |
| 705 |
> |
respectively. |
| 706 |
|
|
| 707 |
|
\section{Results and Discussion} |
| 708 |
|
|
| 710 |
|
In order to evaluate the performance of the pairwise electrostatic |
| 711 |
|
summation methods for Monte Carlo simulations, the energy differences |
| 712 |
|
between configurations were compared to the values obtained when using |
| 713 |
< |
SPME. The results for the subsequent regression analysis are shown in |
| 713 |
> |
{\sc spme}. The results for the subsequent regression analysis are shown in |
| 714 |
|
figure \ref{fig:delE}. |
| 715 |
|
|
| 716 |
|
\begin{figure} |
| 720 |
|
differences for a given electrostatic method compared with the |
| 721 |
|
reference Ewald sum. Results with a value equal to 1 (dashed line) |
| 722 |
|
indicate $\Delta E$ values indistinguishable from those obtained using |
| 723 |
< |
SPME. Different values of the cutoff radius are indicated with |
| 723 |
> |
{\sc spme}. Different values of the cutoff radius are indicated with |
| 724 |
|
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = |
| 725 |
|
inverted triangles).} |
| 726 |
|
\label{fig:delE} |
| 744 |
|
readers can consult the accompanying supporting information for a |
| 745 |
|
comparison where all groups are neutral. |
| 746 |
|
|
| 747 |
< |
For the {\sc sp} method, inclusion of potential damping improves the |
| 748 |
< |
agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows |
| 749 |
< |
an excellent correlation and quality of fit with the SPME results, |
| 750 |
< |
particularly with a cutoff radius greater than 12 |
| 747 |
> |
For the {\sc sp} method, inclusion of electrostatic damping improves |
| 748 |
> |
the agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$ |
| 749 |
> |
shows an excellent correlation and quality of fit with the {\sc spme} |
| 750 |
> |
results, particularly with a cutoff radius greater than 12 |
| 751 |
|
\AA . Use of a larger damping parameter is more helpful for the |
| 752 |
|
shortest cutoff shown, but it has a detrimental effect on simulations |
| 753 |
|
with larger cutoffs. |
| 754 |
|
|
| 755 |
< |
In the {\sc sf} sets, increasing damping results in progressively |
| 756 |
< |
worse correlation with Ewald. Overall, the undamped case is the best |
| 755 |
> |
In the {\sc sf} sets, increasing damping results in progressively {\it |
| 756 |
> |
worse} correlation with Ewald. Overall, the undamped case is the best |
| 757 |
|
performing set, as the correlation and quality of fits are |
| 758 |
|
consistently superior regardless of the cutoff distance. The undamped |
| 759 |
|
case is also less computationally demanding (because no evaluation of |
| 768 |
|
|
| 769 |
|
Evaluation of pairwise methods for use in Molecular Dynamics |
| 770 |
|
simulations requires consideration of effects on the forces and |
| 771 |
< |
torques. Investigation of the force and torque vector magnitudes |
| 772 |
< |
provides a measure of the strength of these values relative to SPME. |
| 773 |
< |
Figures \ref{fig:frcMag} and \ref{fig:trqMag} respectively show the |
| 774 |
< |
force and torque vector magnitude regression results for the |
| 799 |
< |
accumulated analysis over all the system types. |
| 771 |
> |
torques. Figures \ref{fig:frcMag} and \ref{fig:trqMag} show the |
| 772 |
> |
regression results for the force and torque vector magnitudes, |
| 773 |
> |
respectively. The data in these figures was generated from an |
| 774 |
> |
accumulation of the statistics from all of the system types. |
| 775 |
|
|
| 776 |
|
\begin{figure} |
| 777 |
|
\centering |
| 778 |
|
\includegraphics[width=5.5in]{./frcMagplot.pdf} |
| 779 |
< |
\caption{Statistical analysis of the quality of the force vector magnitudes for a given electrostatic method compared with the reference Ewald sum. Results with a value equal to 1 (dashed line) indicate force magnitude values indistinguishable from those obtained using SPME. Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).} |
| 779 |
> |
\caption{Statistical analysis of the quality of the force vector |
| 780 |
> |
magnitudes for a given electrostatic method compared with the |
| 781 |
> |
reference Ewald sum. Results with a value equal to 1 (dashed line) |
| 782 |
> |
indicate force magnitude values indistinguishable from those obtained |
| 783 |
> |
using {\sc spme}. Different values of the cutoff radius are indicated with |
| 784 |
> |
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = |
| 785 |
> |
inverted triangles).} |
| 786 |
|
\label{fig:frcMag} |
| 787 |
|
\end{figure} |
| 788 |
|
|
| 789 |
+ |
Again, it is striking how well the Shifted Potential and Shifted Force |
| 790 |
+ |
methods are doing at reproducing the {\sc spme} forces. The undamped and |
| 791 |
+ |
weakly-damped {\sc sf} method gives the best agreement with Ewald. |
| 792 |
+ |
This is perhaps expected because this method explicitly incorporates a |
| 793 |
+ |
smooth transition in the forces at the cutoff radius as well as the |
| 794 |
+ |
neutralizing image charges. |
| 795 |
+ |
|
| 796 |
|
Figure \ref{fig:frcMag}, for the most part, parallels the results seen |
| 797 |
|
in the previous $\Delta E$ section. The unmodified cutoff results are |
| 798 |
|
poor, but using group based cutoffs and a switching function provides |
| 799 |
< |
a improvement much more significant than what was seen with $\Delta |
| 800 |
< |
E$. Looking at the {\sc sp} sets, the slope and $R^2$ |
| 801 |
< |
improve with the use of damping to an optimal result of 0.2 \AA |
| 802 |
< |
$^{-1}$ for the 12 and 15 \AA\ cutoffs. Further increases in damping, |
| 799 |
> |
an improvement much more significant than what was seen with $\Delta |
| 800 |
> |
E$. |
| 801 |
> |
|
| 802 |
> |
With moderate damping and a large enough cutoff radius, the {\sc sp} |
| 803 |
> |
method is generating usable forces. Further increases in damping, |
| 804 |
|
while beneficial for simulations with a cutoff radius of 9 \AA\ , is |
| 805 |
< |
detrimental to simulations with larger cutoff radii. The undamped |
| 806 |
< |
{\sc sf} method gives forces in line with those obtained using |
| 807 |
< |
SPME, and use of a damping function results in minor improvement. The |
| 819 |
< |
reaction field results are surprisingly good, considering the poor |
| 805 |
> |
detrimental to simulations with larger cutoff radii. |
| 806 |
> |
|
| 807 |
> |
The reaction field results are surprisingly good, considering the poor |
| 808 |
|
quality of the fits for the $\Delta E$ results. There is still a |
| 809 |
< |
considerable degree of scatter in the data, but it correlates well in |
| 810 |
< |
general. To be fair, we again note that the reaction field |
| 811 |
< |
calculations do not encompass NaCl crystal and melt systems, so these |
| 809 |
> |
considerable degree of scatter in the data, but the forces correlate |
| 810 |
> |
well with the Ewald forces in general. We note that the reaction |
| 811 |
> |
field calculations do not include the pure NaCl systems, so these |
| 812 |
|
results are partly biased towards conditions in which the method |
| 813 |
|
performs more favorably. |
| 814 |
|
|
| 815 |
|
\begin{figure} |
| 816 |
|
\centering |
| 817 |
|
\includegraphics[width=5.5in]{./trqMagplot.pdf} |
| 818 |
< |
\caption{Statistical analysis of the quality of the torque vector magnitudes for a given electrostatic method compared with the reference Ewald sum. Results with a value equal to 1 (dashed line) indicate torque magnitude values indistinguishable from those obtained using SPME. Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).} |
| 818 |
> |
\caption{Statistical analysis of the quality of the torque vector |
| 819 |
> |
magnitudes for a given electrostatic method compared with the |
| 820 |
> |
reference Ewald sum. Results with a value equal to 1 (dashed line) |
| 821 |
> |
indicate torque magnitude values indistinguishable from those obtained |
| 822 |
> |
using {\sc spme}. Different values of the cutoff radius are indicated with |
| 823 |
> |
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = |
| 824 |
> |
inverted triangles).} |
| 825 |
|
\label{fig:trqMag} |
| 826 |
|
\end{figure} |
| 827 |
|
|
| 828 |
< |
To evaluate the torque vector magnitudes, the data set from which |
| 829 |
< |
values are drawn is limited to rigid molecules in the systems |
| 830 |
< |
(i.e. water molecules). In spite of this smaller sampling pool, the |
| 837 |
< |
torque vector magnitude results in figure \ref{fig:trqMag} are still |
| 838 |
< |
similar to those seen for the forces; however, they more clearly show |
| 839 |
< |
the improved behavior that comes with increasing the cutoff radius. |
| 840 |
< |
Moderate damping is beneficial to the {\sc sp} and helpful |
| 841 |
< |
yet possibly unnecessary with the {\sc sf} method, and they also |
| 842 |
< |
show that over-damping adversely effects all cutoff radii rather than |
| 843 |
< |
showing an improvement for systems with short cutoffs. The reaction |
| 844 |
< |
field method performs well when calculating the torques, better than |
| 845 |
< |
the Shifted Force method over this limited data set. |
| 828 |
> |
Molecular torques were only available from the systems which contained |
| 829 |
> |
rigid molecules (i.e. the systems containing water). The data in |
| 830 |
> |
fig. \ref{fig:trqMag} is taken from this smaller sampling pool. |
| 831 |
|
|
| 832 |
+ |
Torques appear to be much more sensitive to charges at a longer |
| 833 |
+ |
distance. The striking feature in comparing the new electrostatic |
| 834 |
+ |
methods with {\sc spme} is how much the agreement improves with increasing |
| 835 |
+ |
cutoff radius. Again, the weakly damped and undamped {\sc sf} method |
| 836 |
+ |
appears to be reproducing the {\sc spme} torques most accurately. |
| 837 |
+ |
|
| 838 |
+ |
Water molecules are dipolar, and the reaction field method reproduces |
| 839 |
+ |
the effect of the surrounding polarized medium on each of the |
| 840 |
+ |
molecular bodies. Therefore it is not surprising that reaction field |
| 841 |
+ |
performs best of all of the methods on molecular torques. |
| 842 |
+ |
|
| 843 |
|
\subsection{Directionality of the Force and Torque Vectors} |
| 844 |
|
|
| 845 |
< |
Having force and torque vectors with magnitudes that are well |
| 846 |
< |
correlated to SPME is good, but if they are not pointing in the proper |
| 847 |
< |
direction the results will be incorrect. These vector directions were |
| 848 |
< |
investigated through measurement of the angle formed between them and |
| 849 |
< |
those from SPME. The results (Fig. \ref{fig:frcTrqAng}) are compared |
| 850 |
< |
through the variance ($\sigma^2$) of the Gaussian fits of the angle |
| 851 |
< |
error distributions of the combined set over all system types. |
| 845 |
> |
It is clearly important that a new electrostatic method can reproduce |
| 846 |
> |
the magnitudes of the force and torque vectors obtained via the Ewald |
| 847 |
> |
sum. However, the {\it directionality} of these vectors will also be |
| 848 |
> |
vital in calculating dynamical quantities accurately. Force and |
| 849 |
> |
torque directionalities were investigated by measuring the angles |
| 850 |
> |
formed between these vectors and the same vectors calculated using |
| 851 |
> |
{\sc spme}. The results (Fig. \ref{fig:frcTrqAng}) are compared through the |
| 852 |
> |
variance ($\sigma^2$) of the Gaussian fits of the angle error |
| 853 |
> |
distributions of the combined set over all system types. |
| 854 |
|
|
| 855 |
|
\begin{figure} |
| 856 |
|
\centering |
| 857 |
|
\includegraphics[width=5.5in]{./frcTrqAngplot.pdf} |
| 858 |
< |
\caption{Statistical analysis of the quality of the Gaussian fit of the force and torque vector angular distributions for a given electrostatic method compared with the reference Ewald sum. Results with a variance ($\sigma^2$) equal to zero (dashed line) indicate force and torque directions indistinguishable from those obtained using SPME. Different values of the cutoff radius are indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).} |
| 858 |
> |
\caption{Statistical analysis of the width of the angular distribution |
| 859 |
> |
that the force and torque vectors from a given electrostatic method |
| 860 |
> |
make with their counterparts obtained using the reference Ewald sum. |
| 861 |
> |
Results with a variance ($\sigma^2$) equal to zero (dashed line) |
| 862 |
> |
indicate force and torque directions indistinguishable from those |
| 863 |
> |
obtained using {\sc spme}. Different values of the cutoff radius are |
| 864 |
> |
indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, |
| 865 |
> |
and 15\AA\ = inverted triangles).} |
| 866 |
|
\label{fig:frcTrqAng} |
| 867 |
|
\end{figure} |
| 868 |
|
|
| 869 |
|
Both the force and torque $\sigma^2$ results from the analysis of the |
| 870 |
|
total accumulated system data are tabulated in figure |
| 871 |
< |
\ref{fig:frcTrqAng}. All of the sets, aside from the over-damped case |
| 872 |
< |
show the improvement afforded by choosing a longer simulation cutoff. |
| 873 |
< |
Increasing the cutoff from 9 to 12 \AA\ typically results in a halving |
| 874 |
< |
of the distribution widths, with a similar improvement going from 12 |
| 875 |
< |
to 15 \AA . The undamped {\sc sf}, Group Based Cutoff, and |
| 876 |
< |
Reaction Field methods all do equivalently well at capturing the |
| 877 |
< |
direction of both the force and torque vectors. Using damping |
| 878 |
< |
improves the angular behavior significantly for the {\sc sp} |
| 879 |
< |
and moderately for the {\sc sf} methods. Increasing the damping |
| 880 |
< |
too far is destructive for both methods, particularly to the torque |
| 881 |
< |
vectors. Again it is important to recognize that the force vectors |
| 882 |
< |
cover all particles in the systems, while torque vectors are only |
| 883 |
< |
available for neutral molecular groups. Damping appears to have a |
| 884 |
< |
more beneficial effect on non-neutral bodies, and this observation is |
| 885 |
< |
investigated further in the accompanying supporting information. |
| 871 |
> |
\ref{fig:frcTrqAng}. Here it is clear that the Shifted Potential ({\sc |
| 872 |
> |
sp}) method would be essentially unusable for molecular dynamics |
| 873 |
> |
unless the damping function is added. The Shifted Force ({\sc sf}) |
| 874 |
> |
method, however, is generating force and torque vectors which are |
| 875 |
> |
within a few degrees of the Ewald results even with weak (or no) |
| 876 |
> |
damping. |
| 877 |
> |
|
| 878 |
> |
All of the sets (aside from the over-damped case) show the improvement |
| 879 |
> |
afforded by choosing a larger cutoff radius. Increasing the cutoff |
| 880 |
> |
from 9 to 12 \AA\ typically results in a halving of the width of the |
| 881 |
> |
distribution, with a similar improvement when going from 12 to 15 |
| 882 |
> |
\AA . |
| 883 |
> |
|
| 884 |
> |
The undamped {\sc sf}, group-based cutoff, and reaction field methods |
| 885 |
> |
all do equivalently well at capturing the direction of both the force |
| 886 |
> |
and torque vectors. Using the electrostatic damping improves the |
| 887 |
> |
angular behavior significantly for the {\sc sp} and moderately for the |
| 888 |
> |
{\sc sf} methods. Overdamping is detrimental to both methods. Again |
| 889 |
> |
it is important to recognize that the force vectors cover all |
| 890 |
> |
particles in all seven systems, while torque vectors are only |
| 891 |
> |
available for neutral molecular groups. Damping is more beneficial to |
| 892 |
> |
charged bodies, and this observation is investigated further in the |
| 893 |
> |
accompanying supporting information. |
| 894 |
|
|
| 895 |
+ |
Although not discussed previously, group based cutoffs can be applied |
| 896 |
+ |
to both the {\sc sp} and {\sc sf} methods. The group-based cutoffs |
| 897 |
+ |
will reintroduce small discontinuities at the cutoff radius, but the |
| 898 |
+ |
effects of these can be minimized by utilizing a switching function. |
| 899 |
+ |
Though there are no significant benefits or drawbacks observed in |
| 900 |
+ |
$\Delta E$ and the force and torque magnitudes when doing this, there |
| 901 |
+ |
is a measurable improvement in the directionality of the forces and |
| 902 |
+ |
torques. Table \ref{tab:groupAngle} shows the angular variances |
| 903 |
+ |
obtained using group based cutoffs along with the results seen in |
| 904 |
+ |
figure \ref{fig:frcTrqAng}. The {\sc sp} (with an $\alpha$ of 0.2 |
| 905 |
+ |
\AA$^{-1}$ or smaller) shows much narrower angular distributions when |
| 906 |
+ |
using group-based cutoffs. The {\sc sf} method likewise shows |
| 907 |
+ |
improvement in the undamped and lightly damped cases. |
| 908 |
+ |
|
| 909 |
|
\begin{table}[htbp] |
| 910 |
< |
\centering |
| 911 |
< |
\caption{Variance ($\sigma^2$) of the force (top set) and torque (bottom set) vector angle difference distributions for the Shifted Potential and Shifted Force methods. Calculations were performed both with (Y) and without (N) group based cutoffs and a switching function. The $\alpha$ values have units of \AA$^{-1}$ and the variance values have units of degrees$^2$.} |
| 910 |
> |
\centering |
| 911 |
> |
\caption{Statistical analysis of the angular |
| 912 |
> |
distributions that the force (upper) and torque (lower) vectors |
| 913 |
> |
from a given electrostatic method make with their counterparts |
| 914 |
> |
obtained using the reference Ewald sum. Calculations were |
| 915 |
> |
performed both with (Y) and without (N) group based cutoffs and a |
| 916 |
> |
switching function. The $\alpha$ values have units of \AA$^{-1}$ |
| 917 |
> |
and the variance values have units of degrees$^2$.} |
| 918 |
> |
|
| 919 |
|
\begin{tabular}{@{} ccrrrrrrrr @{}} |
| 920 |
|
\\ |
| 921 |
|
\toprule |
| 946 |
|
\label{tab:groupAngle} |
| 947 |
|
\end{table} |
| 948 |
|
|
| 949 |
< |
Although not discussed previously, group based cutoffs can be applied |
| 950 |
< |
to both the {\sc sp} and {\sc sf} methods. Use off a |
| 951 |
< |
switching function corrects for the discontinuities that arise when |
| 952 |
< |
atoms of a group exit the cutoff before the group's center of mass. |
| 953 |
< |
Though there are no significant benefit or drawbacks observed in |
| 954 |
< |
$\Delta E$ and vector magnitude results when doing this, there is a |
| 955 |
< |
measurable improvement in the vector angle results. Table |
| 956 |
< |
\ref{tab:groupAngle} shows the angular variance values obtained using |
| 957 |
< |
group based cutoffs and a switching function alongside the standard |
| 958 |
< |
results seen in figure \ref{fig:frcTrqAng} for comparison purposes. |
| 959 |
< |
The {\sc sp} shows much narrower angular distributions for |
| 960 |
< |
both the force and torque vectors when using an $\alpha$ of 0.2 |
| 961 |
< |
\AA$^{-1}$ or less, while {\sc sf} shows improvements in the |
| 962 |
< |
undamped and lightly damped cases. Thus, by calculating the |
| 963 |
< |
electrostatic interactions in terms of molecular pairs rather than |
| 964 |
< |
atomic pairs, the direction of the force and torque vectors are |
| 965 |
< |
determined more accurately. |
| 949 |
> |
One additional trend in table \ref{tab:groupAngle} is that the |
| 950 |
> |
$\sigma^2$ values for both {\sc sp} and {\sc sf} converge as $\alpha$ |
| 951 |
> |
increases, something that is more obvious with group-based cutoffs. |
| 952 |
> |
The complimentary error function inserted into the potential weakens |
| 953 |
> |
the electrostatic interaction as the value of $\alpha$ is increased. |
| 954 |
> |
However, at larger values of $\alpha$, it is possible to overdamp the |
| 955 |
> |
electrostatic interaction and to remove it completely. Kast |
| 956 |
> |
\textit{et al.} developed a method for choosing appropriate $\alpha$ |
| 957 |
> |
values for these types of electrostatic summation methods by fitting |
| 958 |
> |
to $g(r)$ data, and their methods indicate optimal values of 0.34, |
| 959 |
> |
0.25, and 0.16 \AA$^{-1}$ for cutoff values of 9, 12, and 15 \AA\ |
| 960 |
> |
respectively.\cite{Kast03} These appear to be reasonable choices to |
| 961 |
> |
obtain proper MC behavior (Fig. \ref{fig:delE}); however, based on |
| 962 |
> |
these findings, choices this high would introduce error in the |
| 963 |
> |
molecular torques, particularly for the shorter cutoffs. Based on our |
| 964 |
> |
observations, empirical damping up to 0.2 \AA$^{-1}$ is beneficial, |
| 965 |
> |
but damping may be unnecessary when using the {\sc sf} method. |
| 966 |
|
|
| 933 |
– |
One additional trend to recognize in table \ref{tab:groupAngle} is |
| 934 |
– |
that the $\sigma^2$ values for both {\sc sp} and |
| 935 |
– |
{\sc sf} converge as $\alpha$ increases, something that is easier |
| 936 |
– |
to see when using group based cutoffs. Looking back on figures |
| 937 |
– |
\ref{fig:delE}, \ref{fig:frcMag}, and \ref{fig:trqMag}, show this |
| 938 |
– |
behavior clearly at large $\alpha$ and cutoff values. The reason for |
| 939 |
– |
this is that the complimentary error function inserted into the |
| 940 |
– |
potential weakens the electrostatic interaction as $\alpha$ increases. |
| 941 |
– |
Thus, at larger values of $\alpha$, both the summation method types |
| 942 |
– |
progress toward non-interacting functions, so care is required in |
| 943 |
– |
choosing large damping functions lest one generate an undesirable loss |
| 944 |
– |
in the pair interaction. Kast \textit{et al.} developed a method for |
| 945 |
– |
choosing appropriate $\alpha$ values for these types of electrostatic |
| 946 |
– |
summation methods by fitting to $g(r)$ data, and their methods |
| 947 |
– |
indicate optimal values of 0.34, 0.25, and 0.16 \AA$^{-1}$ for cutoff |
| 948 |
– |
values of 9, 12, and 15 \AA\ respectively.\cite{Kast03} These appear |
| 949 |
– |
to be reasonable choices to obtain proper MC behavior |
| 950 |
– |
(Fig. \ref{fig:delE}); however, based on these findings, choices this |
| 951 |
– |
high would introduce error in the molecular torques, particularly for |
| 952 |
– |
the shorter cutoffs. Based on the above findings, empirical damping |
| 953 |
– |
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 954 |
– |
unnecessary when using the {\sc sf} method. |
| 955 |
– |
|
| 967 |
|
\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals} |
| 968 |
|
|
| 969 |
< |
In the previous studies using a {\sc sf} variant of the damped |
| 970 |
< |
Wolf coulomb potential, the structure and dynamics of water were |
| 971 |
< |
investigated rather extensively.\cite{Zahn02,Kast03} Their results |
| 972 |
< |
indicated that the damped {\sc sf} method results in properties |
| 973 |
< |
very similar to those obtained when using the Ewald summation. |
| 974 |
< |
Considering the statistical results shown above, the good performance |
| 975 |
< |
of this method is not that surprising. Rather than consider the same |
| 976 |
< |
systems and simply recapitulate their results, we decided to look at |
| 977 |
< |
the solid state dynamical behavior obtained using the best performing |
| 978 |
< |
summation methods from the above results. |
| 969 |
> |
Zahn {\it et al.} investigated the structure and dynamics of water |
| 970 |
> |
using eqs. (\ref{eq:ZahnPot}) and |
| 971 |
> |
(\ref{eq:WolfForces}).\cite{Zahn02,Kast03} Their results indicated |
| 972 |
> |
that a method similar (but not identical with) the damped {\sc sf} |
| 973 |
> |
method resulted in properties very similar to those obtained when |
| 974 |
> |
using the Ewald summation. The properties they studied (pair |
| 975 |
> |
distribution functions, diffusion constants, and velocity and |
| 976 |
> |
orientational correlation functions) may not be particularly sensitive |
| 977 |
> |
to the long-range and collective behavior that governs the |
| 978 |
> |
low-frequency behavior in crystalline systems. Additionally, the |
| 979 |
> |
ionic crystals are the worst case scenario for the pairwise methods |
| 980 |
> |
because they lack the reciprocal space contribution contained in the |
| 981 |
> |
Ewald summation. |
| 982 |
|
|
| 983 |
+ |
We are using two separate measures to probe the effects of these |
| 984 |
+ |
alternative electrostatic methods on the dynamics in crystalline |
| 985 |
+ |
materials. For short- and intermediate-time dynamics, we are |
| 986 |
+ |
computing the velocity autocorrelation function, and for long-time |
| 987 |
+ |
and large length-scale collective motions, we are looking at the |
| 988 |
+ |
low-frequency portion of the power spectrum. |
| 989 |
+ |
|
| 990 |
|
\begin{figure} |
| 991 |
|
\centering |
| 992 |
|
\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
| 993 |
< |
\caption{Velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset is a magnification of the first trough. The times to first collision are nearly identical, but the differences can be seen in the peaks and troughs, where the undamped to weakly damped methods are stiffer than the moderately damped and SPME methods.} |
| 993 |
> |
\caption{Velocity autocorrelation functions of NaCl crystals at |
| 994 |
> |
1000 K using {\sc spme}, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc |
| 995 |
> |
sp} ($\alpha$ = 0.2). The inset is a magnification of the area around |
| 996 |
> |
the first minimum. The times to first collision are nearly identical, |
| 997 |
> |
but differences can be seen in the peaks and troughs, where the |
| 998 |
> |
undamped and weakly damped methods are stiffer than the moderately |
| 999 |
> |
damped and {\sc spme} methods.} |
| 1000 |
|
\label{fig:vCorrPlot} |
| 1001 |
|
\end{figure} |
| 1002 |
|
|
| 1003 |
< |
The short-time decays through the first collision are nearly identical |
| 1004 |
< |
in figure \ref{fig:vCorrPlot}, but the peaks and troughs of the |
| 1005 |
< |
functions show how the methods differ. The undamped {\sc sf} method |
| 1006 |
< |
has deeper troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher |
| 1007 |
< |
peaks than any of the other methods. As the damping function is |
| 1008 |
< |
increased, these peaks are smoothed out, and approach the SPME |
| 1009 |
< |
curve. The damping acts as a distance dependent Gaussian screening of |
| 1010 |
< |
the point charges for the pairwise summation methods; thus, the |
| 1011 |
< |
collisions are more elastic in the undamped {\sc sf} potential, and the |
| 1012 |
< |
stiffness of the potential is diminished as the electrostatic |
| 1013 |
< |
interactions are softened by the damping function. With $\alpha$ |
| 1014 |
< |
values of 0.2 \AA$^{-1}$, the {\sc sf} and {\sc sp} functions are |
| 1015 |
< |
nearly identical and track the SPME features quite well. This is not |
| 1016 |
< |
too surprising in that the differences between the {\sc sf} and {\sc |
| 1017 |
< |
sp} potentials are mitigated with increased damping. However, this |
| 991 |
< |
appears to indicate that once damping is utilized, the form of the |
| 992 |
< |
potential seems to play a lesser role in the crystal dynamics. |
| 1003 |
> |
The short-time decay of the velocity autocorrelation function through |
| 1004 |
> |
the first collision are nearly identical in figure |
| 1005 |
> |
\ref{fig:vCorrPlot}, but the peaks and troughs of the functions show |
| 1006 |
> |
how the methods differ. The undamped {\sc sf} method has deeper |
| 1007 |
> |
troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher peaks than |
| 1008 |
> |
any of the other methods. As the damping parameter ($\alpha$) is |
| 1009 |
> |
increased, these peaks are smoothed out, and the {\sc sf} method |
| 1010 |
> |
approaches the {\sc spme} results. With $\alpha$ values of 0.2 \AA$^{-1}$, |
| 1011 |
> |
the {\sc sf} and {\sc sp} functions are nearly identical and track the |
| 1012 |
> |
{\sc spme} features quite well. This is not surprising because the {\sc sf} |
| 1013 |
> |
and {\sc sp} potentials become nearly identical with increased |
| 1014 |
> |
damping. However, this appears to indicate that once damping is |
| 1015 |
> |
utilized, the details of the form of the potential (and forces) |
| 1016 |
> |
constructed out of the damped electrostatic interaction are less |
| 1017 |
> |
important. |
| 1018 |
|
|
| 1019 |
|
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 1020 |
|
|
| 1021 |
< |
The short time dynamics were extended to evaluate how the differences |
| 1022 |
< |
between the methods affect the collective long-time motion. The same |
| 1023 |
< |
electrostatic summation methods were used as in the short time |
| 1024 |
< |
velocity autocorrelation function evaluation, but the trajectories |
| 1025 |
< |
were sampled over a much longer time. The power spectra of the |
| 1026 |
< |
resulting velocity autocorrelation functions were calculated and are |
| 1027 |
< |
displayed in figure \ref{fig:methodPS}. |
| 1021 |
> |
To evaluate how the differences between the methods affect the |
| 1022 |
> |
collective long-time motion, we computed power spectra from long-time |
| 1023 |
> |
traces of the velocity autocorrelation function. The power spectra for |
| 1024 |
> |
the best-performing alternative methods are shown in |
| 1025 |
> |
fig. \ref{fig:methodPS}. Apodization of the correlation functions via |
| 1026 |
> |
a cubic switching function between 40 and 50 ps was used to reduce the |
| 1027 |
> |
ringing resulting from data truncation. This procedure had no |
| 1028 |
> |
noticeable effect on peak location or magnitude. |
| 1029 |
|
|
| 1030 |
|
\begin{figure} |
| 1031 |
|
\centering |
| 1032 |
|
\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
| 1033 |
< |
\caption{Power spectra obtained from the velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, {\sc sf} ($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). Apodization of the correlation functions via a cubic switching function between 40 and 50 ps was used to clear up the spectral noise resulting from data truncation, and had no noticeable effect on peak location or magnitude. The inset shows the frequency region below 100 cm$^{-1}$ to highlight where the spectra begin to differ.} |
| 1033 |
> |
\caption{Power spectra obtained from the velocity auto-correlation |
| 1034 |
> |
functions of NaCl crystals at 1000 K while using {\sc spme}, {\sc sf} |
| 1035 |
> |
($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset |
| 1036 |
> |
shows the frequency region below 100 cm$^{-1}$ to highlight where the |
| 1037 |
> |
spectra differ.} |
| 1038 |
|
\label{fig:methodPS} |
| 1039 |
|
\end{figure} |
| 1040 |
|
|
| 1041 |
< |
While high frequency peaks of the spectra in this figure overlap, |
| 1042 |
< |
showing the same general features, the low frequency region shows how |
| 1043 |
< |
the summation methods differ. Considering the low-frequency inset |
| 1044 |
< |
(expanded in the upper frame of figure \ref{fig:dampInc}), at |
| 1045 |
< |
frequencies below 100 cm$^{-1}$, the correlated motions are |
| 1046 |
< |
blue-shifted when using undamped or weakly damped {\sc sf}. When |
| 1047 |
< |
using moderate damping ($\alpha = 0.2$ \AA$^{-1}$) both the {\sc sf} |
| 1048 |
< |
and {\sc sp} methods give near identical correlated motion behavior as |
| 1049 |
< |
the Ewald method (which has a damping value of 0.3119). This |
| 1050 |
< |
weakening of the electrostatic interaction with increased damping |
| 1051 |
< |
explains why the long-ranged correlated motions are at lower |
| 1052 |
< |
frequencies for the moderately damped methods than for undamped or |
| 1053 |
< |
weakly damped methods. To see this effect more clearly, we show how |
| 1054 |
< |
damping strength alone affects a simple real-space electrostatic |
| 1055 |
< |
potential, |
| 1056 |
< |
\begin{equation} |
| 1057 |
< |
V_\textrm{damped}=\sum^N_i\sum^N_{j\ne i}q_iq_j\left[\frac{\textrm{erfc}({\alpha r})}{r}\right]S(r), |
| 1058 |
< |
\end{equation} |
| 1059 |
< |
where $S(r)$ is a switching function that smoothly zeroes the |
| 1060 |
< |
potential at the cutoff radius. Figure \ref{fig:dampInc} shows how |
| 1061 |
< |
the low frequency motions are dependent on the damping used in the |
| 1062 |
< |
direct electrostatic sum. As the damping increases, the peaks drop to |
| 1063 |
< |
lower frequencies. Incidentally, use of an $\alpha$ of 0.25 |
| 1064 |
< |
\AA$^{-1}$ on a simple electrostatic summation results in low |
| 1065 |
< |
frequency correlated dynamics equivalent to a simulation using SPME. |
| 1066 |
< |
When the coefficient lowers to 0.15 \AA$^{-1}$ and below, these peaks |
| 1067 |
< |
shift to higher frequency in exponential fashion. Though not shown, |
| 1068 |
< |
the spectrum for the simple undamped electrostatic potential is |
| 1039 |
< |
blue-shifted such that the lowest frequency peak resides near 325 |
| 1040 |
< |
cm$^{-1}$. In light of these results, the undamped {\sc sf} method |
| 1041 |
< |
producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is quite |
| 1042 |
< |
respectable and shows that the shifted force procedure accounts for |
| 1043 |
< |
most of the effect afforded through use of the Ewald summation. |
| 1044 |
< |
However, it appears as though moderate damping is required for |
| 1045 |
< |
accurate reproduction of crystal dynamics. |
| 1041 |
> |
While the high frequency regions of the power spectra for the |
| 1042 |
> |
alternative methods are quantitatively identical with Ewald spectrum, |
| 1043 |
> |
the low frequency region shows how the summation methods differ. |
| 1044 |
> |
Considering the low-frequency inset (expanded in the upper frame of |
| 1045 |
> |
figure \ref{fig:dampInc}), at frequencies below 100 cm$^{-1}$, the |
| 1046 |
> |
correlated motions are blue-shifted when using undamped or weakly |
| 1047 |
> |
damped {\sc sf}. When using moderate damping ($\alpha = 0.2$ |
| 1048 |
> |
\AA$^{-1}$) both the {\sc sf} and {\sc sp} methods give nearly identical |
| 1049 |
> |
correlated motion to the Ewald method (which has a convergence |
| 1050 |
> |
parameter of 0.3119 \AA$^{-1}$). This weakening of the electrostatic |
| 1051 |
> |
interaction with increased damping explains why the long-ranged |
| 1052 |
> |
correlated motions are at lower frequencies for the moderately damped |
| 1053 |
> |
methods than for undamped or weakly damped methods. |
| 1054 |
> |
|
| 1055 |
> |
To isolate the role of the damping constant, we have computed the |
| 1056 |
> |
spectra for a single method ({\sc sf}) with a range of damping |
| 1057 |
> |
constants and compared this with the {\sc spme} spectrum. |
| 1058 |
> |
Fig. \ref{fig:dampInc} shows more clearly that increasing the |
| 1059 |
> |
electrostatic damping red-shifts the lowest frequency phonon modes. |
| 1060 |
> |
However, even without any electrostatic damping, the {\sc sf} method |
| 1061 |
> |
has at most a 10 cm$^{-1}$ error in the lowest frequency phonon mode. |
| 1062 |
> |
Without the {\sc sf} modifications, an undamped (pure cutoff) method |
| 1063 |
> |
would predict the lowest frequency peak near 325 cm$^{-1}$. {\it |
| 1064 |
> |
Most} of the collective behavior in the crystal is accurately captured |
| 1065 |
> |
using the {\sc sf} method. Quantitative agreement with Ewald can be |
| 1066 |
> |
obtained using moderate damping in addition to the shifting at the |
| 1067 |
> |
cutoff distance. |
| 1068 |
> |
|
| 1069 |
|
\begin{figure} |
| 1070 |
|
\centering |
| 1071 |
< |
\includegraphics[width = \linewidth]{./comboSquare.pdf} |
| 1072 |
< |
\caption{Regions of spectra showing the low-frequency correlated motions for NaCl crystals at 1000 K using various electrostatic summation methods. The upper plot is a zoomed inset from figure \ref{fig:methodPS}. As the damping value for the {\sc sf} potential increases, the low-frequency peaks red-shift. The lower plot is of spectra when using SPME and a simple damped Coulombic sum with damping coefficients ($\alpha$) ranging from 0.15 to 0.3 \AA$^{-1}$. As $\alpha$ increases, the peaks are red-shifted toward and eventually beyond the values given by SPME. The larger $\alpha$ values weaken the real-space electrostatics, explaining this shift towards less strongly correlated motions in the crystal.} |
| 1071 |
> |
\includegraphics[width = \linewidth]{./increasedDamping.pdf} |
| 1072 |
> |
\caption{Effect of damping on the two lowest-frequency phonon modes in |
| 1073 |
> |
the NaCl crystal at 1000~K. The undamped shifted force ({\sc sf}) |
| 1074 |
> |
method is off by less than 10 cm$^{-1}$, and increasing the |
| 1075 |
> |
electrostatic damping to 0.25 \AA$^{-1}$ gives quantitative agreement |
| 1076 |
> |
with the power spectrum obtained using the Ewald sum. Overdamping can |
| 1077 |
> |
result in underestimates of frequencies of the long-wavelength |
| 1078 |
> |
motions.} |
| 1079 |
|
\label{fig:dampInc} |
| 1080 |
|
\end{figure} |
| 1081 |
|
|
| 1082 |
|
\section{Conclusions} |
| 1083 |
|
|
| 1084 |
|
This investigation of pairwise electrostatic summation techniques |
| 1085 |
< |
shows that there are viable and more computationally efficient |
| 1086 |
< |
electrostatic summation techniques than the Ewald summation, chiefly |
| 1087 |
< |
methods derived from the damped Coulombic sum originally proposed by |
| 1088 |
< |
Wolf \textit{et al.}\cite{Wolf99,Zahn02} In particular, the |
| 1089 |
< |
{\sc sf} method, reformulated above as eq. (\ref{eq:DSFPot}), |
| 1090 |
< |
shows a remarkable ability to reproduce the energetic and dynamic |
| 1091 |
< |
characteristics exhibited by simulations employing lattice summation |
| 1092 |
< |
techniques. The cumulative energy difference results showed the |
| 1093 |
< |
undamped {\sc sf} and moderately damped {\sc sp} methods |
| 1094 |
< |
produced results nearly identical to SPME. Similarly for the dynamic |
| 1095 |
< |
features, the undamped or moderately damped {\sc sf} and |
| 1096 |
< |
moderately damped {\sc sp} methods produce force and torque |
| 1097 |
< |
vector magnitude and directions very similar to the expected values. |
| 1098 |
< |
These results translate into long-time dynamic behavior equivalent to |
| 1099 |
< |
that produced in simulations using SPME. |
| 1085 |
> |
shows that there are viable and computationally efficient alternatives |
| 1086 |
> |
to the Ewald summation. These methods are derived from the damped and |
| 1087 |
> |
cutoff-neutralized Coulombic sum originally proposed by Wolf |
| 1088 |
> |
\textit{et al.}\cite{Wolf99} In particular, the {\sc sf} |
| 1089 |
> |
method, reformulated above as eqs. (\ref{eq:DSFPot}) and |
| 1090 |
> |
(\ref{eq:DSFForces}), shows a remarkable ability to reproduce the |
| 1091 |
> |
energetic and dynamic characteristics exhibited by simulations |
| 1092 |
> |
employing lattice summation techniques. The cumulative energy |
| 1093 |
> |
difference results showed the undamped {\sc sf} and moderately damped |
| 1094 |
> |
{\sc sp} methods produced results nearly identical to {\sc spme}. Similarly |
| 1095 |
> |
for the dynamic features, the undamped or moderately damped {\sc sf} |
| 1096 |
> |
and moderately damped {\sc sp} methods produce force and torque vector |
| 1097 |
> |
magnitude and directions very similar to the expected values. These |
| 1098 |
> |
results translate into long-time dynamic behavior equivalent to that |
| 1099 |
> |
produced in simulations using {\sc spme}. |
| 1100 |
|
|
| 1101 |
+ |
As in all purely-pairwise cutoff methods, these methods are expected |
| 1102 |
+ |
to scale approximately {\it linearly} with system size, and they are |
| 1103 |
+ |
easily parallelizable. This should result in substantial reductions |
| 1104 |
+ |
in the computational cost of performing large simulations. |
| 1105 |
+ |
|
| 1106 |
|
Aside from the computational cost benefit, these techniques have |
| 1107 |
|
applicability in situations where the use of the Ewald sum can prove |
| 1108 |
< |
problematic. Primary among them is their use in interfacial systems, |
| 1109 |
< |
where the unmodified lattice sum techniques artificially accentuate |
| 1110 |
< |
the periodicity of the system in an undesirable manner. There have |
| 1111 |
< |
been alterations to the standard Ewald techniques, via corrections and |
| 1112 |
< |
reformulations, to compensate for these systems; but the pairwise |
| 1113 |
< |
techniques discussed here require no modifications, making them |
| 1114 |
< |
natural tools to tackle these problems. Additionally, this |
| 1115 |
< |
transferability gives them benefits over other pairwise methods, like |
| 1116 |
< |
reaction field, because estimations of physical properties (e.g. the |
| 1117 |
< |
dielectric constant) are unnecessary. |
| 1108 |
> |
problematic. Of greatest interest is their potential use in |
| 1109 |
> |
interfacial systems, where the unmodified lattice sum techniques |
| 1110 |
> |
artificially accentuate the periodicity of the system in an |
| 1111 |
> |
undesirable manner. There have been alterations to the standard Ewald |
| 1112 |
> |
techniques, via corrections and reformulations, to compensate for |
| 1113 |
> |
these systems; but the pairwise techniques discussed here require no |
| 1114 |
> |
modifications, making them natural tools to tackle these problems. |
| 1115 |
> |
Additionally, this transferability gives them benefits over other |
| 1116 |
> |
pairwise methods, like reaction field, because estimations of physical |
| 1117 |
> |
properties (e.g. the dielectric constant) are unnecessary. |
| 1118 |
|
|
| 1119 |
< |
We are not suggesting any flaw with the Ewald sum; in fact, it is the |
| 1120 |
< |
standard by which these simple pairwise sums are judged. However, |
| 1121 |
< |
these results do suggest that in the typical simulations performed |
| 1122 |
< |
today, the Ewald summation may no longer be required to obtain the |
| 1123 |
< |
level of accuracy most researchers have come to expect |
| 1119 |
> |
If a researcher is using Monte Carlo simulations of large chemical |
| 1120 |
> |
systems containing point charges, most structural features will be |
| 1121 |
> |
accurately captured using the undamped {\sc sf} method or the {\sc sp} |
| 1122 |
> |
method with an electrostatic damping of 0.2 \AA$^{-1}$. These methods |
| 1123 |
> |
would also be appropriate for molecular dynamics simulations where the |
| 1124 |
> |
data of interest is either structural or short-time dynamical |
| 1125 |
> |
quantities. For long-time dynamics and collective motions, the safest |
| 1126 |
> |
pairwise method we have evaluated is the {\sc sf} method with an |
| 1127 |
> |
electrostatic damping between 0.2 and 0.25 |
| 1128 |
> |
\AA$^{-1}$. |
| 1129 |
|
|
| 1130 |
+ |
We are not suggesting that there is any flaw with the Ewald sum; in |
| 1131 |
+ |
fact, it is the standard by which these simple pairwise sums have been |
| 1132 |
+ |
judged. However, these results do suggest that in the typical |
| 1133 |
+ |
simulations performed today, the Ewald summation may no longer be |
| 1134 |
+ |
required to obtain the level of accuracy most researchers have come to |
| 1135 |
+ |
expect. |
| 1136 |
+ |
|
| 1137 |
|
\section{Acknowledgments} |
| 1138 |
+ |
Support for this project was provided by the National Science |
| 1139 |
+ |
Foundation under grant CHE-0134881. The authors would like to thank |
| 1140 |
+ |
Steve Corcelli and Ed Maginn for helpful discussions and comments. |
| 1141 |
+ |
|
| 1142 |
|
\newpage |
| 1143 |
|
|
| 1144 |
|
\bibliographystyle{jcp2} |
