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%\linenumbers\relax % Commence numbering lines |
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\usepackage{amsmath} |
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\usepackage{times} |
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\usepackage{mathptm} |
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\usepackage{mathptmx} |
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\usepackage{tabularx} |
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\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
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\usepackage{url} |
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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 multipoles that were developed in the first paper in this series |
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against a reference method. The tests were carried out in a variety |
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of condensed-phase environments which were designed to test all |
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levels of the multipole-multipole 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 sum. We have |
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also investigated the energy conservation properties of the new |
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methods in molecular dynamics simulations using all of these |
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methods. The SP method shows excellent agreement with |
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configurational energy differences, forces, and torques, and would |
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be suitable for use in Monte Carlo calculations. Of the two new |
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shifted-force methods, the GSF approach shows the best agreement |
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with Ewald-derived energies, forces, and torques and exhibits energy |
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conservation properties that make it an excellent choice for |
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efficiently computing electrostatic interactions in molecular |
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dynamics simulations. |
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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 |
75 |
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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 |
77 |
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differences between configurations, molecular forces, and torques |
78 |
> |
were used to analyze how well the real-space models perform relative |
79 |
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to the more computationally expensive Ewald treatment. We have also |
80 |
> |
investigated the energy conservation properties of the new methods |
81 |
> |
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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\end{abstract} |
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%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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% Classification Scheme. |
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\keywords{Electrostatics, Multipoles, Real-space} |
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%\keywords{Electrostatics, Multipoles, Real-space} |
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|
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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, which is why there |
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have been significant efforts to develop practical, efficient and |
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convergent methods for handling these interactions. Ewald's method is |
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perhaps the best known and most accurate method for evaluating |
104 |
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energies, forces, and torques in explicitly-periodic simulation |
105 |
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cells. In this approach, the conditionally convergent electrostatic |
106 |
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energy is converted into two absolutely convergent contributions, one |
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which is carried out in real space with a cutoff radius, and one in |
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reciprocal space.\cite{Clarke:1986eu,Woodcock75} |
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most expensive aspects of molecular simulations. There have been |
101 |
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significant efforts to develop practical, efficient and convergent |
102 |
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methods for handling these interactions. Ewald's method is perhaps the |
103 |
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best known and most accurate method for evaluating energies, forces, |
104 |
> |
and torques in explicitly-periodic simulation cells. In this approach, |
105 |
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the conditionally convergent electrostatic energy is converted into |
106 |
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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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|
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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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the computational cost from $O(N^2)$ down to $O(N \log |
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N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}. |
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|
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Because of the artificial periodicity required for the Ewald sum, the |
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method may require modification to compute interactions for |
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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 |
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interfaces.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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To simulate interfacial systems, Parry’s extension of the 3D Ewald sum |
123 |
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is appropriate for slab geometries.\cite{Parry:1975if} The inherent |
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periodicity in the Ewald’s method can also be problematic for |
125 |
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interfacial molecular systems.\cite{Fennell:2006lq} Modified Ewald |
126 |
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methods that were developed to handle two-dimensional (2D) |
127 |
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electrostatic interactions in interfacial systems have not had similar |
128 |
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particle-mesh treatments.\cite{Parry:1975if, Parry:1976fq, Clarke77, |
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Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} |
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interfaces require modifications to the |
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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 |
124 |
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for slab geometries.\cite{Parry:1975if} Modified Ewald methods that |
125 |
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were developed to handle two-dimensional (2D) electrostatic |
126 |
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interactions in interfacial systems have not seen similar |
127 |
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particle-mesh treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77, |
128 |
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Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly |
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with system size. The inherent periodicity in the Ewald’s method can |
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also be problematic for interfacial molecular |
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systems.\cite{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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method for calculating electrostatic interactions between point |
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charges. They argued that the effective Coulomb interaction in |
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condensed systems is actually short ranged.\cite{Wolf92,Wolf95}. For |
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an ordered lattice (e.g. when computing the Madelung constant of an |
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ionic solid), the material can be considered as a set of ions |
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interacting with neutral dipolar or quadrupolar ``molecules'' giving |
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an effective distance dependence for the electrostatic interactions of |
142 |
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$r^{-5}$ (see figure \ref{fig:NaCl}. For this reason, careful |
143 |
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applications of Wolf's method are able to obtain accurate estimates of |
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Madelung constants using relatively short cutoff radii. Recently, |
145 |
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Fukuda used neutralization of the higher order moments for the |
146 |
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calculation of the electrostatic interaction of the point charges |
147 |
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system.\cite{Fukuda:2013sf} |
136 |
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charges. They argued that the effective Coulomb interaction in most |
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condensed phase systems is effectively short |
138 |
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ranged.\cite{Wolf92,Wolf95} For an ordered lattice (e.g., when |
139 |
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computing the Madelung constant of an ionic solid), the material can |
140 |
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be considered as a set of ions interacting with neutral dipolar or |
141 |
> |
quadrupolar ``molecules'' giving an effective distance dependence for |
142 |
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the electrostatic interactions of $r^{-5}$ (see figure |
143 |
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\ref{fig:schematic}). If one views the \ce{NaCl} crystal as a simple |
144 |
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cubic (SC) structure with an octupolar \ce{(NaCl)4} basis, the |
145 |
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electrostatic energy per ion converges more rapidly to the Madelung |
146 |
> |
energy than the dipolar approximation.\cite{Wolf92} To find the |
147 |
> |
correct Madelung constant, Lacman suggested that the NaCl structure |
148 |
> |
could be constructed in a way that the finite crystal terminates with |
149 |
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complete \ce{(NaCl)4} molecules.\cite{Lacman65} The central ion sees |
150 |
> |
what is effectively a set of octupoles at large distances. These facts |
151 |
> |
suggest that the Madelung constants are relatively short ranged for |
152 |
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perfect ionic crystals.\cite{Wolf:1999dn} For this reason, careful |
153 |
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application of Wolf's method are able to obtain accurate estimates of |
154 |
> |
Madelung constants using relatively short cutoff radii. |
155 |
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|
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\begin{figure}[h!] |
156 |
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Direct truncation of interactions at a cutoff radius creates numerical |
157 |
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errors. Wolf \textit{et al.} argued that truncation errors are due |
158 |
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to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To |
159 |
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neutralize this charge they proposed placing an image charge on the |
160 |
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surface of the cutoff sphere for every real charge inside the cutoff. |
161 |
> |
These charges are present for the evaluation of both the pair |
162 |
> |
interaction energy and the force, although the force expression |
163 |
> |
maintained a discontinuity at the cutoff sphere. In the original Wolf |
164 |
> |
formulation, the total energy for the charge and image were not equal |
165 |
> |
to the integral of their force expression, and as a result, the total |
166 |
> |
energy would not be conserved in molecular dynamics (MD) |
167 |
> |
simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and |
168 |
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Gezelter later proposed shifted force variants of the Wolf method with |
169 |
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commensurate force and energy expressions that do not exhibit this |
170 |
> |
problem.\cite{Fennell:2006lq} Related real-space methods were also |
171 |
> |
proposed by Chen \textit{et |
172 |
> |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
173 |
> |
and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has used |
174 |
> |
neutralization of the higher order moments for the calculation of the |
175 |
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electrostatic interaction of the point charge |
176 |
> |
systems.\cite{Fukuda:2013sf} |
177 |
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|
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\begin{figure} |
179 |
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\centering |
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\includegraphics[width=0.50 \textwidth]{chargesystem.pdf} |
181 |
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\caption{Top: NaCl crystal showing how spherical truncation can |
182 |
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breaking effective charge ordering, and how complete \ce{(NaCl)4} |
183 |
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molecules interact with the central ion. Bottom: A dipolar |
184 |
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crystal exhibiting similar behavior and illustrating how the |
185 |
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effective dipole-octupole interactions can be disrupted by |
186 |
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spherical truncation.} |
187 |
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\label{fig:NaCl} |
180 |
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\includegraphics[width=\linewidth]{schematic.pdf} |
181 |
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\caption{Top: Ionic systems exhibit local clustering of dissimilar |
182 |
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charges (in the smaller grey circle), so interactions are |
183 |
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effectively charge-multipole at longer distances. With hard |
184 |
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cutoffs, motion of individual charges in and out of the cutoff |
185 |
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sphere can break the effective multipolar ordering. Bottom: |
186 |
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dipolar crystals and fluids have a similar effective |
187 |
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\textit{quadrupolar} ordering (in the smaller grey circles), and |
188 |
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orientational averaging helps to reduce the effective range of the |
189 |
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interactions in the fluid. Placement of reversed image multipoles |
190 |
> |
on the surface of the cutoff sphere recovers the effective |
191 |
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higher-order multipole behavior.} |
192 |
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\label{fig:schematic} |
193 |
|
\end{figure} |
194 |
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|
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The direct truncation of interactions at a cutoff radius creates |
196 |
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truncation defects. Wolf \textit{et al.} further argued that |
197 |
< |
truncation errors are due to net charge remaining inside the cutoff |
198 |
< |
sphere.\cite{Wolf:1999dn} To neutralize this charge they proposed |
199 |
< |
placing an image charge on the surface of the cutoff sphere for every |
200 |
< |
real charge inside the cutoff. These charges are present for the |
201 |
< |
evaluation of both the pair interaction energy and the force, although |
202 |
< |
the force expression maintained a discontinuity at the cutoff sphere. |
203 |
< |
In the original Wolf formulation, the total energy for the charge and |
204 |
< |
image were not equal to the integral of their force expression, and as |
170 |
< |
a result, the total energy would not be conserved in molecular |
171 |
< |
dynamics (MD) simulations.\cite{Zahn:2002hc} Zahn \textit{et al.} and |
172 |
< |
Fennel and Gezelter later proposed shifted force variants of the Wolf |
173 |
< |
method with commensurate force and energy expressions that do not |
174 |
< |
exhibit this problem.\cite{Fennell:2006lq} Related real-space |
175 |
< |
methods were also proposed by Chen \textit{et |
176 |
< |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
177 |
< |
and by Wu and Brooks.\cite{Wu:044107} |
178 |
< |
|
179 |
< |
Considering the interaction of one central ion in an ionic crystal |
180 |
< |
with a portion of the crystal at some distance, the effective Columbic |
181 |
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potential is found to be decreasing as $r^{-5}$. If one views the |
182 |
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\ce{NaCl} crystal as simple cubic (SC) structure with an octupolar |
183 |
< |
\ce{(NaCl)4} basis, the electrostatic energy per ion converges more |
184 |
< |
rapidly to the Madelung energy than the dipolar |
185 |
< |
approximation.\cite{Wolf92} To find the correct Madelung constant, |
186 |
< |
Lacman suggested that the NaCl structure could be constructed in a way |
187 |
< |
that the finite crystal terminates with complete \ce{(NaCl)4} |
188 |
< |
molecules.\cite{Lacman65} Any charge in a NaCl crystal is surrounded |
189 |
< |
by opposite charges. Similarly for each pair of charges, there is an |
190 |
< |
opposite pair of charge adjacent to it. The central ion sees what is |
191 |
< |
effectively a set of octupoles at large distances. These facts suggest |
192 |
< |
that the Madelung constants are relatively short ranged for perfect |
193 |
< |
ionic crystals.\cite{Wolf:1999dn} |
194 |
< |
|
195 |
< |
One can make a similar argument for crystals of point multipoles. The |
196 |
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Luttinger and Tisza treatment of energy constants for dipolar lattices |
197 |
< |
utilizes 24 basis vectors that contain dipoles at the eight corners of |
198 |
< |
a unit cube. Only three of these basis vectors, $X_1, Y_1, |
199 |
< |
\mathrm{~and~} Z_1,$ retain net dipole moments, while the rest have |
200 |
< |
zero net dipole and retain contributions only from higher order |
201 |
< |
multipoles. The effective interaction between a dipole at the center |
195 |
> |
One can make a similar effective range argument for crystals of point |
196 |
> |
\textit{multipoles}. The Luttinger and Tisza treatment of energy |
197 |
> |
constants for dipolar lattices utilizes 24 basis vectors that contain |
198 |
> |
dipoles at the eight corners of a unit cube. Only three of these |
199 |
> |
basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole |
200 |
> |
moments, while the rest have zero net dipole and retain contributions |
201 |
> |
only from higher order multipoles. The lowest energy crystalline |
202 |
> |
structures are built out of basis vectors that have only residual |
203 |
> |
quadrupolar moments (e.g. the $Z_5$ array). In these low energy |
204 |
> |
structures, the effective interaction between a dipole at the center |
205 |
|
of a crystal and a group of eight dipoles farther away is |
206 |
|
significantly shorter ranged than the $r^{-3}$ that one would expect |
207 |
|
for raw dipole-dipole interactions. Only in crystals which retain a |
211 |
|
unstable. |
212 |
|
|
213 |
|
In ionic crystals, real-space truncation can break the effective |
214 |
< |
multipolar arrangements (see Fig. \ref{fig:NaCl}), causing significant |
215 |
< |
swings in the electrostatic energy as the cutoff radius is increased |
216 |
< |
(or as individual ions move back and forth across the boundary). This |
217 |
< |
is why the image charges were necessary for the Wolf sum to exhibit |
218 |
< |
rapid convergence. Similarly, the real-space truncation of point |
219 |
< |
multipole interactions breaks higher order multipole arrangements, and |
220 |
< |
image multipoles are required for real-space treatments of |
218 |
< |
electrostatic energies. |
214 |
> |
multipolar arrangements (see Fig. \ref{fig:schematic}), causing |
215 |
> |
significant swings in the electrostatic energy as individual ions move |
216 |
> |
back and forth across the boundary. This is why the image charges are |
217 |
> |
necessary for the Wolf sum to exhibit rapid convergence. Similarly, |
218 |
> |
the real-space truncation of point multipole interactions breaks |
219 |
> |
higher order multipole arrangements, and image multipoles are required |
220 |
> |
for real-space treatments of electrostatic energies. |
221 |
|
|
222 |
+ |
The shorter effective range of electrostatic interactions is not |
223 |
+ |
limited to perfect crystals, but can also apply in disordered fluids. |
224 |
+ |
Even at elevated temperatures, there is, on average, local charge |
225 |
+ |
balance in an ionic liquid, where each positive ion has surroundings |
226 |
+ |
dominated by negaitve ions and vice versa. The reversed-charge images |
227 |
+ |
on the cutoff sphere that are integral to the Wolf and DSF approaches |
228 |
+ |
retain the effective multipolar interactions as the charges traverse |
229 |
+ |
the cutoff boundary. |
230 |
+ |
|
231 |
+ |
In multipolar fluids (see Fig. \ref{fig:schematic}) there is |
232 |
+ |
significant orientational averaging that additionally reduces the |
233 |
+ |
effect of long-range multipolar interactions. The image multipoles |
234 |
+ |
that are introduced in the TSF, GSF, and SP methods mimic this effect |
235 |
+ |
and reduce the effective range of the multipolar interactions as |
236 |
+ |
interacting molecules traverse each other's cutoff boundaries. |
237 |
+ |
|
238 |
|
% Because of this reason, although the nature of electrostatic |
239 |
|
% interaction short ranged, the hard cutoff sphere creates very large |
240 |
|
% fluctuation in the electrostatic energy for the perfect crystal. In |
258 |
|
densities.\cite{Shi:2013ij,Kannam:2012rr,Acevedo13,Space12,English08,Lawrence13,Vergne13} |
259 |
|
|
260 |
|
\subsection{The damping function} |
261 |
< |
The damping function used in our research has been discussed in detail |
262 |
< |
in the first paper of this series.\cite{PaperI} The radial kernel |
263 |
< |
$1/r$ for the interactions between point charges can be replaced by |
264 |
< |
the complementary error function $\mathrm{erfc}(\alpha r)/r$ to |
265 |
< |
accelerate the rate of convergence, where $\alpha$ is a damping |
266 |
< |
parameter with units of inverse distance. Altering the value of |
267 |
< |
$\alpha$ is equivalent to changing the width of Gaussian charge |
268 |
< |
distributions that replace each point charge -- Gaussian overlap |
269 |
< |
integrals yield complementary error functions when truncated at a |
270 |
< |
finite distance. |
261 |
> |
The damping function has been discussed in detail in the first paper |
262 |
> |
of this series.\cite{PaperI} The $1/r$ Coulombic kernel for the |
263 |
> |
interactions between point charges can be replaced by the |
264 |
> |
complementary error function $\mathrm{erfc}(\alpha r)/r$ to accelerate |
265 |
> |
convergence, where $\alpha$ is a damping parameter with units of |
266 |
> |
inverse distance. Altering the value of $\alpha$ is equivalent to |
267 |
> |
changing the width of Gaussian charge distributions that replace each |
268 |
> |
point charge, as Coulomb integrals with Gaussian charge distributions |
269 |
> |
produce complementary error functions when truncated at a finite |
270 |
> |
distance. |
271 |
|
|
272 |
< |
By using suitable value of damping alpha ($\alpha \sim 0.2$) for a |
273 |
< |
cutoff radius ($r_{c}=9 A$), Fennel and Gezelter produced very good |
274 |
< |
agreement with SPME for the interaction energies, forces and torques |
275 |
< |
for charge-charge interactions.\cite{Fennell:2006lq} |
272 |
> |
With moderate damping coefficients, $\alpha \sim 0.2$, the DSF method |
273 |
> |
produced very good agreement with SPME for interaction energies, |
274 |
> |
forces and torques for charge-charge |
275 |
> |
interactions.\cite{Fennell:2006lq} |
276 |
|
|
277 |
|
\subsection{Point multipoles in molecular modeling} |
278 |
|
Coarse-graining approaches which treat entire molecular subsystems as |
279 |
|
a single rigid body are now widely used. A common feature of many |
280 |
|
coarse-graining approaches is simplification of the electrostatic |
281 |
|
interactions between bodies so that fewer site-site interactions are |
282 |
< |
required to compute configurational energies. Many coarse-grained |
283 |
< |
molecular structures would normally consist of equal positive and |
266 |
< |
negative charges, and rather than use multiple site-site interactions, |
267 |
< |
the interaction between higher order multipoles can also be used to |
268 |
< |
evaluate a single molecule-molecule |
269 |
< |
interaction.\cite{Ren06,Essex10,Essex11} |
282 |
> |
required to compute configurational |
283 |
> |
energies.\cite{Ren06,Essex10,Essex11} |
284 |
|
|
285 |
|
Because electrons in a molecule are not localized at specific points, |
286 |
< |
the assignment of partial charges to atomic centers is a relatively |
287 |
< |
rough approximation. Atomic sites can also be assigned point |
288 |
< |
multipoles and polarizabilities to increase the accuracy of the |
289 |
< |
molecular model. Recently, water has been modeled with point |
290 |
< |
multipoles up to octupolar |
291 |
< |
order.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point |
286 |
> |
the assignment of partial charges to atomic centers is always an |
287 |
> |
approximation. Atomic sites can also be assigned point multipoles and |
288 |
> |
polarizabilities to increase the accuracy of the molecular model. |
289 |
> |
Recently, water has been modeled with point multipoles up to octupolar |
290 |
> |
order using the soft sticky dipole-quadrupole-octupole (SSDQO) |
291 |
> |
model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point |
292 |
|
multipoles up to quadrupolar order have also been coupled with point |
293 |
|
polarizabilities in the high-quality AMOEBA and iAMOEBA water |
294 |
< |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk}. But |
295 |
< |
using point multipole with the real space truncation without |
296 |
< |
accounting for multipolar neutrality will create energy conservation |
283 |
< |
issues in molecular dynamics (MD) simulations. |
294 |
> |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However, |
295 |
> |
truncating point multipoles without smoothing the forces and torques |
296 |
> |
will create energy conservation issues in molecular dynamics simulations. |
297 |
|
|
298 |
|
In this paper we test a set of real-space methods that were developed |
299 |
|
for point multipolar interactions. These methods extend the damped |
330 |
|
\begin{equation} |
331 |
|
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
332 |
|
\end{equation} |
333 |
< |
where the multipole operator for site $\bf a$, |
334 |
< |
\begin{equation} |
335 |
< |
\hat{M}_{\bf a} = C_{\bf a} - D_{{\bf a}\alpha} \frac{\partial}{\partial r_{\alpha}} |
336 |
< |
+ Q_{{\bf a}\alpha\beta} |
324 |
< |
\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
325 |
< |
\end{equation} |
326 |
< |
is expressed in terms of the point charge, $C_{\bf a}$, dipole, |
327 |
< |
$D_{{\bf a}\alpha}$, and quadrupole, $Q_{{\bf a}\alpha\beta}$, for |
328 |
< |
object $\bf a$. Note that in this work, we use the primitive |
329 |
< |
quadrupole tensor, $Q_{a\alpha,\beta}=\frac{1}{2}\sum_{k\;in\;a}q_k |
330 |
< |
r_{k\alpha}r_{k\beta}$ to represent point quadrupoles on a site. |
333 |
> |
where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is |
334 |
> |
expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf |
335 |
> |
a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object |
336 |
> |
$\bf a$, etc. |
337 |
|
|
338 |
< |
Interactions between multipoles can be expressed as higher derivatives |
339 |
< |
of the bare Coulomb potential, so one way of ensuring that the forces |
340 |
< |
and torques vanish at the cutoff distance is to include a larger |
341 |
< |
number of terms in the truncated Taylor expansion, e.g., |
342 |
< |
% |
343 |
< |
\begin{equation} |
344 |
< |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
345 |
< |
\end{equation} |
346 |
< |
% |
347 |
< |
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
348 |
< |
Thus, for $f(r)=1/r$, we find |
349 |
< |
% |
350 |
< |
\begin{equation} |
351 |
< |
f_1(r)=\frac{1}{r}- \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} - \frac{(r-r_c)^2}{r_c^3} . |
352 |
< |
\end{equation} |
353 |
< |
This function is an approximate electrostatic potential that has |
354 |
< |
vanishing second derivatives at the cutoff radius, making it suitable |
355 |
< |
for shifting the forces and torques of charge-dipole interactions. |
338 |
> |
% Interactions between multipoles can be expressed as higher derivatives |
339 |
> |
% of the bare Coulomb potential, so one way of ensuring that the forces |
340 |
> |
% and torques vanish at the cutoff distance is to include a larger |
341 |
> |
% number of terms in the truncated Taylor expansion, e.g., |
342 |
> |
% % |
343 |
> |
% \begin{equation} |
344 |
> |
% f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
345 |
> |
% \end{equation} |
346 |
> |
% % |
347 |
> |
% The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
348 |
> |
% Thus, for $f(r)=1/r$, we find |
349 |
> |
% % |
350 |
> |
% \begin{equation} |
351 |
> |
% f_1(r)=\frac{1}{r}- \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} - \frac{(r-r_c)^2}{r_c^3} . |
352 |
> |
% \end{equation} |
353 |
> |
% This function is an approximate electrostatic potential that has |
354 |
> |
% vanishing second derivatives at the cutoff radius, making it suitable |
355 |
> |
% for shifting the forces and torques of charge-dipole interactions. |
356 |
|
|
357 |
< |
In general, the TSF potential for any multipole-multipole interaction |
358 |
< |
can be written |
357 |
> |
The TSF potential for any multipole-multipole interaction can be |
358 |
> |
written |
359 |
|
\begin{equation} |
360 |
|
U^{\text{TSF}}= (\text{prefactor}) (\text{derivatives}) f_n(r) |
361 |
|
\label{generic} |
362 |
|
\end{equation} |
363 |
< |
with $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for |
364 |
< |
charge-quadrupole and dipole-dipole, $n=3$ for dipole-quadrupole, and |
365 |
< |
$n=4$ for quadrupole-quadrupole. To ensure smooth convergence of the |
366 |
< |
energy, force, and torques, the required number of terms from Taylor |
367 |
< |
series expansion in $f_n(r)$ must be performed for different |
368 |
< |
multipole-multipole interactions. |
363 |
> |
where $f_n(r)$ is a shifted kernel that is appropriate for the order |
364 |
> |
of the interaction (see Ref. \onlinecite{PaperI}), with $n=0$ for |
365 |
> |
charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole |
366 |
> |
and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for |
367 |
> |
quadrupole-quadrupole. To ensure smooth convergence of the energy, |
368 |
> |
force, and torques, a Taylor expansion with $n$ terms must be |
369 |
> |
performed at cutoff radius ($r_c$) to obtain $f_n(r)$. |
370 |
|
|
371 |
< |
To carry out the same procedure for a damped electrostatic kernel, we |
372 |
< |
replace $1/r$ in the Coulomb potential with $\text{erfc}(\alpha r)/r$. |
373 |
< |
Many of the derivatives of the damped kernel are well known from |
374 |
< |
Smith's early work on multipoles for the Ewald |
375 |
< |
summation.\cite{Smith82,Smith98} |
371 |
> |
% To carry out the same procedure for a damped electrostatic kernel, we |
372 |
> |
% replace $1/r$ in the Coulomb potential with $\text{erfc}(\alpha r)/r$. |
373 |
> |
% Many of the derivatives of the damped kernel are well known from |
374 |
> |
% Smith's early work on multipoles for the Ewald |
375 |
> |
% summation.\cite{Smith82,Smith98} |
376 |
|
|
377 |
< |
Note that increasing the value of $n$ will add additional terms to the |
378 |
< |
electrostatic potential, e.g., $f_2(r)$ includes orders up to |
379 |
< |
$(r-r_c)^3/r_c^4$, and so on. Successive derivatives of the $f_n(r)$ |
380 |
< |
functions are denoted $g_2(r) = f^\prime_2(r)$, $h_2(r) = |
381 |
< |
f^{\prime\prime}_2(r)$, etc. These higher derivatives are required |
382 |
< |
for computing multipole energies, forces, and torques, and smooth |
383 |
< |
cutoffs of these quantities can be guaranteed as long as the number of |
384 |
< |
terms in the Taylor series exceeds the derivative order required. |
377 |
> |
% Note that increasing the value of $n$ will add additional terms to the |
378 |
> |
% electrostatic potential, e.g., $f_2(r)$ includes orders up to |
379 |
> |
% $(r-r_c)^3/r_c^4$, and so on. Successive derivatives of the $f_n(r)$ |
380 |
> |
% functions are denoted $g_2(r) = f^\prime_2(r)$, $h_2(r) = |
381 |
> |
% f^{\prime\prime}_2(r)$, etc. These higher derivatives are required |
382 |
> |
% for computing multipole energies, forces, and torques, and smooth |
383 |
> |
% cutoffs of these quantities can be guaranteed as long as the number of |
384 |
> |
% terms in the Taylor series exceeds the derivative order required. |
385 |
|
|
386 |
|
For multipole-multipole interactions, following this procedure results |
387 |
< |
in separate radial functions for each distinct orientational |
388 |
< |
contribution to the potential, and ensures that the forces and torques |
389 |
< |
from {\it each} of these contributions will vanish at the cutoff |
390 |
< |
radius. For example, the direct dipole dot product ($\mathbf{D}_{i} |
391 |
< |
\cdot \mathbf{D}_{j}$) is treated differently than the dipole-distance |
387 |
> |
in separate radial functions for each of the distinct orientational |
388 |
> |
contributions to the potential, and ensures that the forces and |
389 |
> |
torques from each of these contributions will vanish at the cutoff |
390 |
> |
radius. For example, the direct dipole dot product |
391 |
> |
($\mathbf{D}_{\bf a} |
392 |
> |
\cdot \mathbf{D}_{\bf b}$) is treated differently than the dipole-distance |
393 |
|
dot products: |
394 |
|
\begin{equation} |
395 |
< |
U_{D_{i}D_{j}}(r)= -\frac{1}{4\pi \epsilon_0} \left( \mathbf{D}_{i} \cdot |
396 |
< |
\mathbf{D}_{j} \right) \frac{g_2(r)}{r} |
397 |
< |
-\frac{1}{4\pi \epsilon_0} |
398 |
< |
\left( \mathbf{D}_{i} \cdot \hat{r} \right) |
399 |
< |
\left( \mathbf{D}_{j} \cdot \hat{r} \right) \left(h_2(r) - |
392 |
< |
\frac{g_2(r)}{r} \right) |
395 |
> |
U_{D_{\bf a}D_{\bf b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left( |
396 |
> |
\mathbf{D}_{\bf a} \cdot |
397 |
> |
\mathbf{D}_{\bf b} \right) v_{21}(r) + |
398 |
> |
\left( \mathbf{D}_{\bf a} \cdot \hat{r} \right) |
399 |
> |
\left( \mathbf{D}_{\bf b} \cdot \hat{r} \right) v_{22}(r) \right] |
400 |
|
\end{equation} |
401 |
|
|
402 |
< |
The electrostatic forces and torques acting on the central multipole |
403 |
< |
site due to another site within cutoff sphere are derived from |
402 |
> |
For the Taylor shifted (TSF) method with the undamped kernel, |
403 |
> |
$v_{21}(r) = -\frac{1}{r^3} + \frac{3 r}{r_c^4} - \frac{8}{r_c^3} + |
404 |
> |
\frac{6}{r r_c^2}$ and $v_{22}(r) = \frac{3}{r^3} + \frac{3 r}{r_c^4} |
405 |
> |
- \frac{6}{r r_c^2}$. In these functions, one can easily see the |
406 |
> |
connection to unmodified electrostatics as well as the smooth |
407 |
> |
transition to zero in both these functions as $r\rightarrow r_c$. The |
408 |
> |
electrostatic forces and torques acting on the central multipole due |
409 |
> |
to another site within the cutoff sphere are derived from |
410 |
|
Eq.~\ref{generic}, accounting for the appropriate number of |
411 |
|
derivatives. Complete energy, force, and torque expressions are |
412 |
|
presented in the first paper in this series (Reference |
414 |
|
|
415 |
|
\subsection{Gradient-shifted force (GSF)} |
416 |
|
|
417 |
< |
A second (and significantly simpler) method involves shifting the |
418 |
< |
gradient of the raw coulomb potential for each particular multipole |
417 |
> |
A second (and conceptually simpler) method involves shifting the |
418 |
> |
gradient of the raw Coulomb potential for each particular multipole |
419 |
|
order. For example, the raw dipole-dipole potential energy may be |
420 |
|
shifted smoothly by finding the gradient for two interacting dipoles |
421 |
|
which have been projected onto the surface of the cutoff sphere |
422 |
|
without changing their relative orientation, |
423 |
< |
\begin{displaymath} |
424 |
< |
U_{D_{i}D_{j}}(r_{ij}) = U_{D_{i}D_{j}}(r_{ij}) - U_{D_{i}D_{j}}(r_c) |
425 |
< |
- (r_{ij}-r_c) \hat{r}_{ij} \cdot |
426 |
< |
\vec{\nabla} V_{D_{i}D_{j}}(r) \Big \lvert _{r_c} |
427 |
< |
\end{displaymath} |
428 |
< |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{i}$ |
429 |
< |
and $\mathbf{D}_{j}$, are retained at the cutoff distance (although |
430 |
< |
the signs are reversed for the dipole that has been projected onto the |
431 |
< |
cutoff sphere). In many ways, this simpler approach is closer in |
432 |
< |
spirit to the original shifted force method, in that it projects a |
433 |
< |
neutralizing multipole (and the resulting forces from this multipole) |
434 |
< |
onto a cutoff sphere. The resulting functional forms for the |
435 |
< |
potentials, forces, and torques turn out to be quite similar in form |
436 |
< |
to the Taylor-shifted approach, although the radial contributions are |
437 |
< |
significantly less perturbed by the Gradient-shifted approach than |
438 |
< |
they are in the Taylor-shifted method. |
423 |
> |
\begin{equation} |
424 |
> |
U_{D_{\bf a}D_{\bf b}}(r) = U_{D_{\bf a}D_{\bf b}}(r) - |
425 |
> |
U_{D_{\bf a} D_{\bf b}}(r_c) |
426 |
> |
- (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot |
427 |
> |
\nabla U_{D_{\bf a}D_{\bf b}}(r_c). |
428 |
> |
\end{equation} |
429 |
> |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf |
430 |
> |
a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance |
431 |
> |
(although the signs are reversed for the dipole that has been |
432 |
> |
projected onto the cutoff sphere). In many ways, this simpler |
433 |
> |
approach is closer in spirit to the original shifted force method, in |
434 |
> |
that it projects a neutralizing multipole (and the resulting forces |
435 |
> |
from this multipole) onto a cutoff sphere. The resulting functional |
436 |
> |
forms for the potentials, forces, and torques turn out to be quite |
437 |
> |
similar in form to the Taylor-shifted approach, although the radial |
438 |
> |
contributions are significantly less perturbed by the gradient-shifted |
439 |
> |
approach than they are in the Taylor-shifted method. |
440 |
|
|
441 |
+ |
For the gradient shifted (GSF) method with the undamped kernel, |
442 |
+ |
$v_{21}(r) = -\frac{1}{r^3} + \frac{3 r}{r_c^4} + \frac{4}{r_c^3}$ and |
443 |
+ |
$v_{22}(r) = \frac{3}{r^3} + \frac{9 r}{r_c^4} - \frac{12}{r_c^3}$. |
444 |
+ |
Again, these functions go smoothly to zero as $r\rightarrow r_c$, and |
445 |
+ |
because the Taylor expansion retains only one term, they are |
446 |
+ |
significantly less perturbed than the TSF functions. |
447 |
+ |
|
448 |
|
In general, the gradient shifted potential between a central multipole |
449 |
|
and any multipolar site inside the cutoff radius is given by, |
450 |
|
\begin{equation} |
451 |
< |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
452 |
< |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
453 |
< |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} \right] |
451 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
452 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}} |
453 |
> |
\cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
454 |
|
\label{generic2} |
455 |
|
\end{equation} |
456 |
|
where the sum describes a separate force-shifting that is applied to |
457 |
< |
each orientational contribution to the energy. |
457 |
> |
each orientational contribution to the energy. In this expression, |
458 |
> |
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles |
459 |
> |
($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ |
460 |
> |
represent the orientations the multipoles. |
461 |
|
|
462 |
|
The third term converges more rapidly than the first two terms as a |
463 |
|
function of radius, hence the contribution of the third term is very |
464 |
|
small for large cutoff radii. The force and torque derived from |
465 |
< |
equation \ref{generic2} are consistent with the energy expression and |
465 |
> |
Eq. \ref{generic2} are consistent with the energy expression and |
466 |
|
approach zero as $r \rightarrow r_c$. Both the GSF and TSF methods |
467 |
|
can be considered generalizations of the original DSF method for |
468 |
|
higher order multipole interactions. GSF and TSF are also identical up |
470 |
|
the energy, force and torque for higher order multipole-multipole |
471 |
|
interactions. Complete energy, force, and torque expressions for the |
472 |
|
GSF potential are presented in the first paper in this series |
473 |
< |
(Reference~\onlinecite{PaperI}) |
473 |
> |
(Reference~\onlinecite{PaperI}). |
474 |
|
|
475 |
|
|
476 |
|
\subsection{Shifted potential (SP) } |
483 |
|
interactions with the central multipole and the image. This |
484 |
|
effectively shifts the total potential to zero at the cutoff radius, |
485 |
|
\begin{equation} |
486 |
< |
U_{SP}(\vec r)=\sum U(\vec r) - U(\vec r_c) |
486 |
> |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
487 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
488 |
|
\label{eq:SP} |
489 |
|
\end{equation} |
490 |
|
where the sum describes separate potential shifting that is done for |
496 |
|
multipoles that reorient after leaving the cutoff sphere can re-enter |
497 |
|
the cutoff sphere without perturbing the total energy. |
498 |
|
|
499 |
< |
The potential energy between a central multipole and other multipolar |
500 |
< |
sites then goes smoothly to zero as $r \rightarrow r_c$. However, the |
501 |
< |
force and torque obtained from the shifted potential (SP) are |
502 |
< |
discontinuous at $r_c$. Therefore, MD simulations will still |
503 |
< |
experience energy drift while operating under the SP potential, but it |
504 |
< |
may be suitable for Monte Carlo approaches where the configurational |
505 |
< |
energy differences are the primary quantity of interest. |
499 |
> |
For the shifted potential (SP) method with the undamped kernel, |
500 |
> |
$v_{21}(r) = -\frac{1}{r^3} + \frac{1}{r_c^3}$ and $v_{22}(r) = |
501 |
> |
\frac{3}{r^3} - \frac{3}{r_c^3}$. The potential energy between a |
502 |
> |
central multipole and other multipolar sites goes smoothly to zero as |
503 |
> |
$r \rightarrow r_c$. However, the force and torque obtained from the |
504 |
> |
shifted potential (SP) are discontinuous at $r_c$. MD simulations |
505 |
> |
will still experience energy drift while operating under the SP |
506 |
> |
potential, but it may be suitable for Monte Carlo approaches where the |
507 |
> |
configurational energy differences are the primary quantity of |
508 |
> |
interest. |
509 |
|
|
510 |
< |
\subsection{The Self term} |
510 |
> |
\subsection{The Self Term} |
511 |
|
In the TSF, GSF, and SP methods, a self-interaction is retained for |
512 |
|
the central multipole interacting with its own image on the surface of |
513 |
|
the cutoff sphere. This self interaction is nearly identical with the |
529 |
|
used the multipolar Ewald sum as a reference method for comparing |
530 |
|
energies, forces, and torques for molecular models that mimic |
531 |
|
disordered and ordered condensed-phase systems. The parameters used |
532 |
< |
in the test-cases are given in table~\ref{tab:pars}. |
532 |
> |
in the test cases are given in table~\ref{tab:pars}. |
533 |
|
|
534 |
|
\begin{table} |
535 |
|
\label{tab:pars} |
547 |
|
& (\AA) & (kcal/mol) & (e) & (debye) & \multicolumn{3}{c|}{(debye \AA)} & (amu) & \multicolumn{3}{c}{(amu |
548 |
|
\AA\textsuperscript{2})} \\ \hline |
549 |
|
Soft Dipolar fluid & 3.051 & 0.152 & & 2.35 & & & & 18.0153 & 1.77&0.6145& 1.155 \\ |
550 |
< |
Soft Dipolar solid & 2.837 & 1.0 & & 2.35 & & & & 10,000 & 17.6 &17.6 & 0 \\ |
550 |
> |
Soft Dipolar solid & 2.837 & 1.0 & & 2.35 & & & & $10^4$ & 17.6 &17.6 & 0 \\ |
551 |
|
Soft Quadrupolar fluid & 3.051 & 0.152 & & & -1&-1&-2.5 & 18.0153 & 1.77&0.6145&1.155 \\ |
552 |
< |
Soft Quadrupolar solid & 2.837 & 1.0 & & & -1&-1&-2.5 & 10,000 & 17.6&17.6&0 \\ |
552 |
> |
Soft Quadrupolar solid & 2.837 & 1.0 & & & -1&-1&-2.5 & $10^4$ & 17.6&17.6&0 \\ |
553 |
|
SSDQ water & 3.051 & 0.152 & & 2.35 & -1.35&0&-0.68 & 18.0153 & 1.77&0.6145&1.155 \\ |
554 |
|
\ce{Na+} & 2.579 & 0.118 & +1& & & & & 22.99 & & &\\ |
555 |
|
\ce{Cl-} & 4.445 & 0.1 & -1& & & & & 35.4527& & & \\ \hline |
574 |
|
electrostatic energy, as well as the electrostatic contributions to |
575 |
|
the force and torque on each molecule. These quantities have been |
576 |
|
computed using the SP, TSF, and GSF methods, as well as a hard cutoff, |
577 |
< |
and have been compared with the values obtaine from the multipolar |
578 |
< |
Ewald sum. In Mote Carlo (MC) simulations, the energy differences |
577 |
> |
and have been compared with the values obtained from the multipolar |
578 |
> |
Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
579 |
|
between two configurations is the primary quantity that governs how |
580 |
|
the simulation proceeds. These differences are the most imporant |
581 |
|
indicators of the reliability of a method even if the absolute |
622 |
|
recomputed at each time step. |
623 |
|
|
624 |
|
\subsection{Model systems} |
625 |
< |
To sample independent configurations of multipolar crystals, a body |
626 |
< |
centered cubic (bcc) crystal which is a minimum energy structure for |
627 |
< |
point dipoles was generated using 3,456 molecules. The multipoles |
628 |
< |
were translationally locked in their respective crystal sites for |
629 |
< |
equilibration at a relatively low temperature (50K), so that dipoles |
630 |
< |
or quadrupoles could freely explore all accessible orientations. The |
631 |
< |
translational constraints were removed, and the crystals were |
632 |
< |
simulated for 10 ps in the microcanonical (NVE) ensemble with an |
633 |
< |
average temperature of 50 K. Configurations were sampled at equal |
634 |
< |
time intervals for the comparison of the configurational energy |
635 |
< |
differences. The crystals were not simulated close to the melting |
636 |
< |
points in order to avoid translational deformation away of the ideal |
637 |
< |
lattice geometry. |
625 |
> |
To sample independent configurations of the multipolar crystals, body |
626 |
> |
centered cubic (bcc) crystals, which exhibit the minimum energy |
627 |
> |
structures for point dipoles, were generated using 3,456 molecules. |
628 |
> |
The multipoles were translationally locked in their respective crystal |
629 |
> |
sites for equilibration at a relatively low temperature (50K) so that |
630 |
> |
dipoles or quadrupoles could freely explore all accessible |
631 |
> |
orientations. The translational constraints were then removed, the |
632 |
> |
systems were re-equilibrated, and the crystals were simulated for an |
633 |
> |
additional 10 ps in the microcanonical (NVE) ensemble with an average |
634 |
> |
temperature of 50 K. The balance between moments of inertia and |
635 |
> |
particle mass were chosen to allow orientational sampling without |
636 |
> |
significant translational motion. Configurations were sampled at |
637 |
> |
equal time intervals in order to compare configurational energy |
638 |
> |
differences. The crystals were simulated far from the melting point |
639 |
> |
in order to avoid translational deformation away of the ideal lattice |
640 |
> |
geometry. |
641 |
|
|
642 |
< |
For dipolar, quadrupolar, and mixed-multipole liquid simulations, each |
643 |
< |
system was created with 2048 molecules oriented randomly. These were |
644 |
< |
|
645 |
< |
system with 2,048 molecules simulated for 1ns in NVE ensemble at 300 K |
646 |
< |
temperature after equilibration. We collected 250 different |
647 |
< |
configurations in equal interval of time. For the ions mixed liquid |
648 |
< |
system, we converted 48 different molecules into 24 \ce{Na+} and 24 |
649 |
< |
\ce{Cl-} ions and equilibrated. After equilibration, the system was run |
650 |
< |
at the same environment for 1ns and 250 configurations were |
651 |
< |
collected. While comparing energies, forces, and torques with Ewald |
652 |
< |
method, Lennard-Jones potentials were turned off and purely |
653 |
< |
electrostatic interaction had been compared. |
642 |
> |
For dipolar, quadrupolar, and mixed-multipole \textit{liquid} |
643 |
> |
simulations, each system was created with 2,048 randomly-oriented |
644 |
> |
molecules. These were equilibrated at a temperature of 300K for 1 ns. |
645 |
> |
Each system was then simulated for 1 ns in the microcanonical (NVE) |
646 |
> |
ensemble. We collected 250 different configurations at equal time |
647 |
> |
intervals. For the liquid system that included ionic species, we |
648 |
> |
converted 48 randomly-distributed molecules into 24 \ce{Na+} and 24 |
649 |
> |
\ce{Cl-} ions and re-equilibrated. After equilibration, the system was |
650 |
> |
run under the same conditions for 1 ns. A total of 250 configurations |
651 |
> |
were collected. In the following comparisons of energies, forces, and |
652 |
> |
torques, the Lennard-Jones potentials were turned off and only the |
653 |
> |
purely electrostatic quantities were compared with the same values |
654 |
> |
obtained via the Ewald sum. |
655 |
|
|
656 |
|
\subsection{Accuracy of Energy Differences, Forces and Torques} |
657 |
|
The pairwise summation techniques (outlined above) were evaluated for |
665 |
|
should be identical for all methods. |
666 |
|
|
667 |
|
Since none of the real-space methods provide exact energy differences, |
668 |
< |
we used least square regressions analysiss for the six different |
668 |
> |
we used least square regressions analysis for the six different |
669 |
|
molecular systems to compare $\Delta E$ from Hard, SP, GSF, and TSF |
670 |
|
with the multipolar Ewald reference method. Unitary results for both |
671 |
|
the correlation (slope) and correlation coefficient for these |
676 |
|
configurations and 250 configurations were recorded for comparison. |
677 |
|
Each system provided 31,125 energy differences for a total of 186,750 |
678 |
|
data points. Similarly, the magnitudes of the forces and torques have |
679 |
< |
also been compared by using least squares regression analyses. In the |
679 |
> |
also been compared using least squares regression analysis. In the |
680 |
|
forces and torques comparison, the magnitudes of the forces acting in |
681 |
|
each molecule for each configuration were evaluated. For example, our |
682 |
|
dipolar liquid simulation contains 2048 molecules and there are 250 |
861 |
|
molecules inside each other's cutoff spheres in order to correct the |
862 |
|
energy conservation issues, and this perturbation is evident in the |
863 |
|
statistics accumulated for the molecular forces. The GSF |
864 |
< |
perturbations are minimal, particularly for moderate damping and and |
864 |
> |
perturbations are minimal, particularly for moderate damping and |
865 |
|
commonly-used cutoff values ($r_c = 12$\AA). The TSF method shows |
866 |
|
reasonable agreement in the correlation coefficient but again the |
867 |
|
systematic error in the forces is concerning if replication of Ewald |
918 |
|
these quantities. Force and torque vectors for all six systems were |
919 |
|
analyzed using Fisher statistics, and the quality of the vector |
920 |
|
directionality is shown in terms of circular variance |
921 |
< |
($\mathrm{Var}(\theta$) in figure |
921 |
> |
($\mathrm{Var}(\theta)$) in figure |
922 |
|
\ref{fig:slopeCorr_circularVariance}. The force and torque vectors |
923 |
|
from the new real-space methods exhibit nearly-ideal Fisher probability |
924 |
|
distributions (Eq.~\ref{eq:pdf}). Both the hard and SP cutoff methods |
973 |
|
energy over time, $\delta E_1$, and the standard deviation of energy |
974 |
|
fluctuations around this drift $\delta E_0$. Both of the |
975 |
|
shifted-force methods (GSF and TSF) provide excellent energy |
976 |
< |
conservation (drift less than $10^{-6}$ kcal / mol / ns / particle), |
976 |
> |
conservation (drift less than $10^{-5}$ kcal / mol / ns / particle), |
977 |
|
while the hard cutoff is essentially unusable for molecular dynamics. |
978 |
|
SP provides some benefit over the hard cutoff because the energetic |
979 |
|
jumps that happen as particles leave and enter the cutoff sphere are |
988 |
|
|
989 |
|
\begin{figure} |
990 |
|
\centering |
991 |
< |
\includegraphics[width=\textwidth]{newDrift.pdf} |
991 |
> |
\includegraphics[width=\textwidth]{newDrift_12.pdf} |
992 |
|
\label{fig:energyDrift} |
993 |
|
\caption{Analysis of the energy conservation of the real-space |
994 |
|
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
995 |
< |
energy over time and $\delta \mathrm{E}_0$ is the standard deviation |
996 |
< |
of energy fluctuations around this drift. All simulations were of a |
997 |
< |
2000-molecule simulation of SSDQ water with 48 ionic charges at 300 |
998 |
< |
K starting from the same initial configuration. All runs utilized |
999 |
< |
the same real-space cutoff, $r_c = 12$\AA.} |
995 |
> |
energy over time (in kcal / mol / particle / ns) and $\delta |
996 |
> |
\mathrm{E}_0$ is the standard deviation of energy fluctuations |
997 |
> |
around this drift (in kcal / mol / particle). All simulations were |
998 |
> |
of a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
999 |
> |
300 K starting from the same initial configuration. All runs |
1000 |
> |
utilized the same real-space cutoff, $r_c = 12$\AA.} |
1001 |
|
\end{figure} |
1002 |
|
|
1003 |
|
|
1077 |
|
handling of energies, forces, and torques as multipoles cross the |
1078 |
|
real-space cutoff boundary. |
1079 |
|
|
1080 |
+ |
\begin{acknowledgments} |
1081 |
+ |
JDG acknowledges helpful discussions with Christopher |
1082 |
+ |
Fennell. Support for this project was provided by the National |
1083 |
+ |
Science Foundation under grant CHE-1362211. Computational time was |
1084 |
+ |
provided by the Center for Research Computing (CRC) at the |
1085 |
+ |
University of Notre Dame. |
1086 |
+ |
\end{acknowledgments} |
1087 |
+ |
|
1088 |
|
%\bibliographystyle{aip} |
1089 |
|
\newpage |
1090 |
|
\bibliography{references} |