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%\preprint{AIP/123-QED} |
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\title{Real space alternatives to the Ewald |
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Sum. II. Comparison of Methods} % Force line breaks with \\ |
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\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods} |
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556}%Lines break automatically or can be forced with \\ |
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\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556} |
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|
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\author{Kathie E. Newman} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556} |
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|
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\author{J. Daniel Gezelter}% |
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\email{gezelter@nd.edu.} |
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\affiliation{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556%\\This line break forced with \textbackslash\textbackslash |
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}% |
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\affiliation{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556 |
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} |
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|
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\date{\today} |
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\begin{abstract} |
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We have tested the real-space shifted potential (SP), |
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gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods |
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for multipole interactions that were developed in the first paper in |
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this series, using the multipolar Ewald sum as a reference |
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method. The tests were carried out in a variety of condensed-phase |
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environments which were designed to test all levels of the |
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multipole-multipole interactions. Comparisons of the energy |
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differences between configurations, molecular forces, and torques |
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were used to analyze how well the real-space models perform relative |
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to the more computationally expensive Ewald treatment. We have also |
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investigated the energy conservation properties of the new methods |
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in molecular dynamics simulations. The SP method shows excellent |
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agreement with configurational energy differences, forces, and |
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torques, and would be suitable for use in Monte Carlo calculations. |
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Of the two new shifted-force methods, the GSF approach shows the |
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best agreement with Ewald-derived energies, forces, and torques and |
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exhibits energy conservation properties that make it an excellent |
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choice for efficient computation of electrostatic interactions in |
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molecular dynamics simulations. |
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We report on tests of the shifted potential (SP), gradient shifted |
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force (GSF), and Taylor shifted force (TSF) real-space methods for |
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multipole interactions developed in the first paper in this series, |
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using the multipolar Ewald sum as a reference method. The tests were |
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carried out in a variety of condensed-phase environments designed to |
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test up to quadrupole-quadrupole interactions. Comparisons of the |
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energy differences between configurations, molecular forces, and |
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torques were used to analyze how well the real-space models perform |
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relative to the more computationally expensive Ewald treatment. We |
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have also investigated the energy conservation properties of the new |
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methods in molecular dynamics simulations. The SP method shows |
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excellent agreement with configurational energy differences, forces, |
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and torques, and would be suitable for use in Monte Carlo |
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calculations. Of the two new shifted-force methods, the GSF |
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approach shows the best agreement with Ewald-derived energies, |
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forces, and torques and also exhibits energy conservation properties |
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that make it an excellent choice for efficient computation of |
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electrostatic interactions in molecular dynamics simulations. |
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\end{abstract} |
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|
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%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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\maketitle |
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|
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– |
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\section{\label{sec:intro}Introduction} |
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Computing the interactions between electrostatic sites is one of the |
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most expensive aspects of molecular simulations. There have been |
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the conditionally convergent electrostatic energy is converted into |
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two absolutely convergent contributions, one which is carried out in |
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real space with a cutoff radius, and one in reciprocal |
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space. BETTER CITATIONS\cite{Clarke:1986eu,Woodcock75} |
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space.\cite{Ewald21,deLeeuw80,Smith81,Allen87} |
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|
| 104 |
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When carried out as originally formulated, the reciprocal-space |
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portion of the Ewald sum exhibits relatively poor computational |
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< |
scaling, making it prohibitive for large systems. By utilizing |
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particle meshes and three dimensional fast Fourier transforms (FFT), |
| 108 |
< |
the particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
| 109 |
< |
(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) methods can decrease |
| 110 |
< |
the computational cost from $O(N^2)$ down to $O(N \log |
| 111 |
< |
N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}. |
| 106 |
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scaling, making it prohibitive for large systems. By utilizing a |
| 107 |
> |
particle mesh and three dimensional fast Fourier transforms (FFT), the |
| 108 |
> |
particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
| 109 |
> |
(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) |
| 110 |
> |
methods can decrease the computational cost from $O(N^2)$ down to $O(N |
| 111 |
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\log |
| 112 |
> |
N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb} |
| 113 |
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|
| 114 |
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Because of the artificial periodicity required for the Ewald sum, |
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interfacial molecular systems such as membranes and liquid-vapor |
| 116 |
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interfaces require modifications to the |
| 117 |
< |
method.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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Parry's extension of the three dimensional Ewald sum is appropriate |
| 119 |
< |
for slab geometries.\cite{Parry:1975if} Modified Ewald methods that |
| 120 |
< |
were developed to handle two-dimensional (2D) electrostatic |
| 121 |
< |
interactions in interfacial systems have not seen similar |
| 122 |
< |
particle-mesh treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77, |
| 123 |
< |
Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly |
| 124 |
< |
with system size. The inherent periodicity in the Ewald’s method can |
| 125 |
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also be problematic for interfacial molecular |
| 126 |
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systems.\cite{Fennell:2006lq} |
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> |
interfaces require modifications to the method. Parry's extension of |
| 117 |
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the three dimensional Ewald sum is appropriate for slab |
| 118 |
> |
geometries.\cite{Parry:1975if} Modified Ewald methods that were |
| 119 |
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developed to handle two-dimensional (2-D) electrostatic |
| 120 |
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interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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These methods were originally quite computationally |
| 122 |
> |
expensive.\cite{Spohr97,Yeh99} There have been several successful |
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efforts that reduced the computational cost of 2-D lattice summations, |
| 124 |
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bringing them more in line with the scaling for the full 3-D |
| 125 |
> |
treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The |
| 126 |
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inherent periodicity required by the Ewald method can also be |
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problematic in a number of protein/solvent and ionic solution |
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environments.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
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|
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\subsection{Real-space methods} |
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Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
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what is effectively a set of octupoles at large distances. These facts |
| 148 |
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suggest that the Madelung constants are relatively short ranged for |
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perfect ionic crystals.\cite{Wolf:1999dn} For this reason, careful |
| 150 |
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application of Wolf's method are able to obtain accurate estimates of |
| 150 |
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application of Wolf's method can provide accurate estimates of |
| 151 |
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Madelung constants using relatively short cutoff radii. |
| 152 |
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|
| 153 |
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Direct truncation of interactions at a cutoff radius creates numerical |
| 154 |
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errors. Wolf \textit{et al.} argued that truncation errors are due |
| 154 |
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errors. Wolf \textit{et al.} suggest that truncation errors are due |
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to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To |
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neutralize this charge they proposed placing an image charge on the |
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surface of the cutoff sphere for every real charge inside the cutoff. |
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These charges are present for the evaluation of both the pair |
| 159 |
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interaction energy and the force, although the force expression |
| 160 |
< |
maintained a discontinuity at the cutoff sphere. In the original Wolf |
| 160 |
> |
maintains a discontinuity at the cutoff sphere. In the original Wolf |
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formulation, the total energy for the charge and image were not equal |
| 162 |
< |
to the integral of their force expression, and as a result, the total |
| 162 |
> |
to the integral of the force expression, and as a result, the total |
| 163 |
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energy would not be conserved in molecular dynamics (MD) |
| 164 |
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simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and |
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Gezelter later proposed shifted force variants of the Wolf method with |
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commensurate force and energy expressions that do not exhibit this |
| 167 |
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problem.\cite{Fennell:2006lq} Related real-space methods were also |
| 168 |
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proposed by Chen \textit{et |
| 167 |
> |
problem.\cite{Zahn:2002hc,Fennell:2006lq} Related real-space methods |
| 168 |
> |
were also proposed by Chen \textit{et |
| 169 |
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al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
| 170 |
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and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has used |
| 171 |
< |
neutralization of the higher order moments for the calculation of the |
| 172 |
< |
electrostatic interaction of the point charge |
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< |
systems.\cite{Fukuda:2013sf} |
| 170 |
> |
and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has successfuly |
| 171 |
> |
used additional neutralization of higher order moments for systems of |
| 172 |
> |
point charges.\cite{Fukuda:2013sf} |
| 173 |
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|
| 174 |
|
\begin{figure} |
| 175 |
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\centering |
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\includegraphics[width=\linewidth]{schematic.pdf} |
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> |
\includegraphics[width=\linewidth]{schematic.eps} |
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\caption{Top: Ionic systems exhibit local clustering of dissimilar |
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charges (in the smaller grey circle), so interactions are |
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effectively charge-multipole at longer distances. With hard |
| 191 |
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One can make a similar effective range argument for crystals of point |
| 192 |
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\textit{multipoles}. The Luttinger and Tisza treatment of energy |
| 193 |
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constants for dipolar lattices utilizes 24 basis vectors that contain |
| 194 |
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dipoles at the eight corners of a unit cube. Only three of these |
| 195 |
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basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole |
| 194 |
> |
dipoles at the eight corners of a unit cube.\cite{LT} Only three of |
| 195 |
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these basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole |
| 196 |
|
moments, while the rest have zero net dipole and retain contributions |
| 197 |
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only from higher order multipoles. The lowest energy crystalline |
| 197 |
> |
only from higher order multipoles. The lowest-energy crystalline |
| 198 |
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structures are built out of basis vectors that have only residual |
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quadrupolar moments (e.g. the $Z_5$ array). In these low energy |
| 200 |
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structures, the effective interaction between a dipole at the center |
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|
| 218 |
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The shorter effective range of electrostatic interactions is not |
| 219 |
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limited to perfect crystals, but can also apply in disordered fluids. |
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Even at elevated temperatures, there is, on average, local charge |
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< |
balance in an ionic liquid, where each positive ion has surroundings |
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< |
dominated by negaitve ions and vice versa. The reversed-charge images |
| 223 |
< |
on the cutoff sphere that are integral to the Wolf and DSF approaches |
| 224 |
< |
retain the effective multipolar interactions as the charges traverse |
| 225 |
< |
the cutoff boundary. |
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> |
Even at elevated temperatures, there is local charge balance in an |
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> |
ionic liquid, where each positive ion has surroundings dominated by |
| 222 |
> |
negaitve ions and vice versa. The reversed-charge images on the |
| 223 |
> |
cutoff sphere that are integral to the Wolf and DSF approaches retain |
| 224 |
> |
the effective multipolar interactions as the charges traverse the |
| 225 |
> |
cutoff boundary. |
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|
| 227 |
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In multipolar fluids (see Fig. \ref{fig:schematic}) there is |
| 228 |
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significant orientational averaging that additionally reduces the |
| 241 |
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% to the non-neutralized value of the higher order moments within the |
| 242 |
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% cutoff sphere. |
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|
| 244 |
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The forces and torques acting on atomic sites are the fundamental |
| 245 |
< |
factors driving dynamics in molecular simulations. Fennell and |
| 246 |
< |
Gezelter proposed the damped shifted force (DSF) energy kernel to |
| 247 |
< |
obtain consistent energies and forces on the atoms within the cutoff |
| 248 |
< |
sphere. Both the energy and the force go smoothly to zero as an atom |
| 249 |
< |
aproaches the cutoff radius. The comparisons of the accuracy these |
| 250 |
< |
quantities between the DSF kernel and SPME was surprisingly |
| 251 |
< |
good.\cite{Fennell:2006lq} The DSF method has seen increasing use for |
| 252 |
< |
calculating electrostatic interactions in molecular systems with |
| 253 |
< |
relatively uniform charge |
| 258 |
< |
densities.\cite{Shi:2013ij,Kannam:2012rr,Acevedo13,Space12,English08,Lawrence13,Vergne13} |
| 244 |
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Forces and torques acting on atomic sites are fundamental in driving |
| 245 |
> |
dynamics in molecular simulations, and the damped shifted force (DSF) |
| 246 |
> |
energy kernel provides consistent energies and forces on charged atoms |
| 247 |
> |
within the cutoff sphere. Both the energy and the force go smoothly to |
| 248 |
> |
zero as an atom aproaches the cutoff radius. The comparisons of the |
| 249 |
> |
accuracy these quantities between the DSF kernel and SPME was |
| 250 |
> |
surprisingly good.\cite{Fennell:2006lq} As a result, the DSF method |
| 251 |
> |
has seen increasing use in molecular systems with relatively uniform |
| 252 |
> |
charge |
| 253 |
> |
densities.\cite{English08,Kannam:2012rr,Space12,Lawrence13,Acevedo13,Shi:2013ij,Vergne13} |
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|
| 255 |
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\subsection{The damping function} |
| 256 |
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The damping function has been discussed in detail in the first paper |
| 277 |
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required to compute configurational |
| 278 |
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energies.\cite{Ren06,Essex10,Essex11} |
| 279 |
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|
| 280 |
< |
Because electrons in a molecule are not localized at specific points, |
| 281 |
< |
the assignment of partial charges to atomic centers is always an |
| 282 |
< |
approximation. Atomic sites can also be assigned point multipoles and |
| 283 |
< |
polarizabilities to increase the accuracy of the molecular model. |
| 284 |
< |
Recently, water has been modeled with point multipoles up to octupolar |
| 285 |
< |
order using the soft sticky dipole-quadrupole-octupole (SSDQO) |
| 280 |
> |
Additionally, because electrons in a molecule are not localized at |
| 281 |
> |
specific points, the assignment of partial charges to atomic centers |
| 282 |
> |
is always an approximation. For increased accuracy, atomic sites can |
| 283 |
> |
also be assigned point multipoles and polarizabilities. Recently, |
| 284 |
> |
water has been modeled with point multipoles up to octupolar order |
| 285 |
> |
using the soft sticky dipole-quadrupole-octupole (SSDQO) |
| 286 |
|
model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point |
| 287 |
|
multipoles up to quadrupolar order have also been coupled with point |
| 288 |
|
polarizabilities in the high-quality AMOEBA and iAMOEBA water |
| 289 |
|
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However, |
| 290 |
|
truncating point multipoles without smoothing the forces and torques |
| 291 |
< |
will create energy conservation issues in molecular dynamics simulations. |
| 291 |
> |
can create energy conservation issues in molecular dynamics |
| 292 |
> |
simulations. |
| 293 |
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|
| 294 |
|
In this paper we test a set of real-space methods that were developed |
| 295 |
|
for point multipolar interactions. These methods extend the damped |
| 296 |
|
shifted force (DSF) and Wolf methods originally developed for |
| 297 |
|
charge-charge interactions and generalize them for higher order |
| 298 |
< |
multipoles. The detailed mathematical development of these methods has |
| 299 |
< |
been presented in the first paper in this series, while this work |
| 300 |
< |
covers the testing the energies, forces, torques, and energy |
| 298 |
> |
multipoles. The detailed mathematical development of these methods |
| 299 |
> |
has been presented in the first paper in this series, while this work |
| 300 |
> |
covers the testing of energies, forces, torques, and energy |
| 301 |
|
conservation properties of the methods in realistic simulation |
| 302 |
|
environments. In all cases, the methods are compared with the |
| 303 |
< |
reference method, a full multipolar Ewald treatment. |
| 303 |
> |
reference method, a full multipolar Ewald treatment.\cite{Smith82,Smith98} |
| 304 |
|
|
| 305 |
|
|
| 306 |
|
%\subsection{Conservation of total energy } |
| 573 |
|
and have been compared with the values obtained from the multipolar |
| 574 |
|
Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
| 575 |
|
between two configurations is the primary quantity that governs how |
| 576 |
< |
the simulation proceeds. These differences are the most imporant |
| 576 |
> |
the simulation proceeds. These differences are the most important |
| 577 |
|
indicators of the reliability of a method even if the absolute |
| 578 |
|
energies are not exact. For each of the multipolar systems listed |
| 579 |
|
above, we have compared the change in electrostatic potential energy |
| 585 |
|
\subsection{Implementation} |
| 586 |
|
The real-space methods developed in the first paper in this series |
| 587 |
|
have been implemented in our group's open source molecular simulation |
| 588 |
< |
program, OpenMD,\cite{openmd} which was used for all calculations in |
| 588 |
> |
program, OpenMD,\cite{Meineke05,openmd} which was used for all calculations in |
| 589 |
|
this work. The complementary error function can be a relatively slow |
| 590 |
|
function on some processors, so all of the radial functions are |
| 591 |
|
precomputed on a fine grid and are spline-interpolated to provide |
| 792 |
|
|
| 793 |
|
\begin{figure} |
| 794 |
|
\centering |
| 795 |
< |
\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
| 795 |
> |
\includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps} |
| 796 |
|
\caption{Statistical analysis of the quality of configurational |
| 797 |
|
energy differences for the real-space electrostatic methods |
| 798 |
|
compared with the reference Ewald sum. Results with a value equal |
| 865 |
|
|
| 866 |
|
\begin{figure} |
| 867 |
|
\centering |
| 868 |
< |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
| 868 |
> |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} |
| 869 |
|
\caption{Statistical analysis of the quality of the force vector |
| 870 |
|
magnitudes for the real-space electrostatic methods compared with |
| 871 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 879 |
|
|
| 880 |
|
\begin{figure} |
| 881 |
|
\centering |
| 882 |
< |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
| 882 |
> |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps} |
| 883 |
|
\caption{Statistical analysis of the quality of the torque vector |
| 884 |
|
magnitudes for the real-space electrostatic methods compared with |
| 885 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
| 937 |
|
|
| 938 |
|
\begin{figure} |
| 939 |
|
\centering |
| 940 |
< |
\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
| 940 |
> |
\includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps} |
| 941 |
|
\caption{The circular variance of the direction of the force and |
| 942 |
|
torque vectors obtained from the real-space methods around the |
| 943 |
|
reference Ewald vectors. A variance equal to 0 (dashed line) |
| 984 |
|
|
| 985 |
|
\begin{figure} |
| 986 |
|
\centering |
| 987 |
< |
\includegraphics[width=\textwidth]{newDrift_12.pdf} |
| 987 |
> |
\includegraphics[width=\textwidth]{newDrift_12.eps} |
| 988 |
|
\label{fig:energyDrift} |
| 989 |
|
\caption{Analysis of the energy conservation of the real-space |
| 990 |
|
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
