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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 electrostatics for multipoles. II. Comparisons with |
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the Ewald Sum} |
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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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\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 multipoles that were developed in the first paper in this series |
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against the multipolar Ewald sum as a reference method. The tests |
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were carried out in a variety of condensed-phase environments which |
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were designed to test all levels of the multipole-multipole |
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interactions. Comparisons of the energy differences between |
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configurations, molecular forces, and torques were used to analyze |
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how well the real-space models perform relative to the more |
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computationally expensive Ewald treatment. We have also investigated the |
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energy conservation properties of the new methods in molecular |
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dynamics simulations using all of these methods. The SP method shows |
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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 exhibits energy conservation properties that |
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make it an excellent choice for efficiently computing electrostatic |
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interactions in molecular dynamics simulations. |
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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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%\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, which is why there |
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have been significant efforts to develop practical, efficient and |
97 |
< |
convergent methods for handling these interactions. Ewald's method is |
98 |
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perhaps the best known and most accurate method for evaluating |
99 |
< |
energies, forces, and torques in explicitly-periodic simulation |
100 |
< |
cells. In this approach, the conditionally convergent electrostatic |
101 |
< |
energy is converted into two absolutely convergent contributions, one |
102 |
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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} |
95 |
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most expensive aspects of molecular simulations. There have been |
96 |
> |
significant efforts to develop practical, efficient and convergent |
97 |
> |
methods for handling these interactions. Ewald's method is perhaps the |
98 |
> |
best known and most accurate method for evaluating energies, forces, |
99 |
> |
and torques in explicitly-periodic simulation cells. In this approach, |
100 |
> |
the conditionally convergent electrostatic energy is converted into |
101 |
> |
two absolutely convergent contributions, one which is carried out in |
102 |
> |
real space with a cutoff radius, and one in reciprocal |
103 |
> |
space.\cite{Ewald21,deLeeuw80,Smith81,Allen87} |
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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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scaling, making it prohibitive for large systems. By utilizing |
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< |
particle meshes and three dimensional fast Fourier transforms (FFT), |
109 |
< |
the particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
110 |
< |
(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) methods can decrease |
111 |
< |
the computational cost from $O(N^2)$ down to $O(N \log |
112 |
< |
N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}. |
107 |
> |
scaling, making it prohibitive for large systems. By utilizing a |
108 |
> |
particle mesh and three dimensional fast Fourier transforms (FFT), the |
109 |
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particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
110 |
> |
(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) |
111 |
> |
methods can decrease the computational cost from $O(N^2)$ down to $O(N |
112 |
> |
\log |
113 |
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N)$.\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 |
115 |
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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} |
118 |
< |
To simulate interfacial systems, Parry's extension of the 3D Ewald sum |
119 |
< |
is appropriate for slab geometries.\cite{Parry:1975if} The inherent |
120 |
< |
periodicity in the Ewald method can also be problematic for molecular |
121 |
< |
interfaces.\cite{Fennell:2006lq} Modified Ewald methods that were |
122 |
< |
developed to handle two-dimensional (2D) electrostatic interactions in |
123 |
< |
interfacial systems have not seen similar particle-mesh |
124 |
< |
treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77, |
125 |
< |
Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly |
126 |
< |
with system size. |
117 |
> |
interfaces require modifications to the method. Parry's extension of |
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the three dimensional Ewald sum is appropriate for slab |
119 |
> |
geometries.\cite{Parry:1975if} Modified Ewald methods that were |
120 |
> |
developed to handle two-dimensional (2-D) electrostatic |
121 |
> |
interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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These methods were originally quite computationally |
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expensive.\cite{Spohr97,Yeh99} There have been several successful |
124 |
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efforts that reduced the computational cost of 2-D lattice summations, |
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bringing them more in line with the scaling for the full 3-D |
126 |
> |
treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The |
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inherent periodicity required by the Ewald method can also be |
128 |
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problematic in a number of protein/solvent and ionic solution |
129 |
> |
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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method for calculating electrostatic interactions between point |
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charges. They argued that the effective Coulomb interaction in |
135 |
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condensed phase systems is actually short ranged.\cite{Wolf92,Wolf95} |
136 |
< |
For an ordered lattice (e.g., when computing the Madelung constant of |
137 |
< |
an ionic solid), the material can be considered as a set of ions |
138 |
< |
interacting with neutral dipolar or quadrupolar ``molecules'' giving |
139 |
< |
an effective distance dependence for the electrostatic interactions of |
140 |
< |
$r^{-5}$ (see figure \ref{fig:schematic}). For this reason, careful |
141 |
< |
application of Wolf's method can obtain accurate estimates of Madelung |
142 |
< |
constants using relatively short cutoff radii. Recently, Fukuda used |
143 |
< |
neutralization of the higher order moments for calculation of the |
144 |
< |
electrostatic interactions in point charge |
145 |
< |
systems.\cite{Fukuda:2013sf} |
134 |
> |
charges. They argued that the effective Coulomb interaction in most |
135 |
> |
condensed phase systems is effectively short |
136 |
> |
ranged.\cite{Wolf92,Wolf95} For an ordered lattice (e.g., when |
137 |
> |
computing the Madelung constant of an ionic solid), the material can |
138 |
> |
be considered as a set of ions interacting with neutral dipolar or |
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> |
quadrupolar ``molecules'' giving an effective distance dependence for |
140 |
> |
the electrostatic interactions of $r^{-5}$ (see figure |
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> |
\ref{fig:schematic}). If one views the \ce{NaCl} crystal as a simple |
142 |
> |
cubic (SC) structure with an octupolar \ce{(NaCl)4} basis, the |
143 |
> |
electrostatic energy per ion converges more rapidly to the Madelung |
144 |
> |
energy than the dipolar approximation.\cite{Wolf92} To find the |
145 |
> |
correct Madelung constant, Lacman suggested that the NaCl structure |
146 |
> |
could be constructed in a way that the finite crystal terminates with |
147 |
> |
complete \ce{(NaCl)4} molecules.\cite{Lacman65} The central ion sees |
148 |
> |
what is effectively a set of octupoles at large distances. These facts |
149 |
> |
suggest that the Madelung constants are relatively short ranged for |
150 |
> |
perfect ionic crystals.\cite{Wolf:1999dn} For this reason, careful |
151 |
> |
application of Wolf's method can provide accurate estimates of |
152 |
> |
Madelung constants using relatively short cutoff radii. |
153 |
|
|
154 |
+ |
Direct truncation of interactions at a cutoff radius creates numerical |
155 |
+ |
errors. Wolf \textit{et al.} suggest that truncation errors are due |
156 |
+ |
to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To |
157 |
+ |
neutralize this charge they proposed placing an image charge on the |
158 |
+ |
surface of the cutoff sphere for every real charge inside the cutoff. |
159 |
+ |
These charges are present for the evaluation of both the pair |
160 |
+ |
interaction energy and the force, although the force expression |
161 |
+ |
maintains a discontinuity at the cutoff sphere. In the original Wolf |
162 |
+ |
formulation, the total energy for the charge and image were not equal |
163 |
+ |
to the integral of the force expression, and as a result, the total |
164 |
+ |
energy would not be conserved in molecular dynamics (MD) |
165 |
+ |
simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and |
166 |
+ |
Gezelter later proposed shifted force variants of the Wolf method with |
167 |
+ |
commensurate force and energy expressions that do not exhibit this |
168 |
+ |
problem.\cite{Zahn:2002hc,Fennell:2006lq} Related real-space methods |
169 |
+ |
were also proposed by Chen \textit{et |
170 |
+ |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
171 |
+ |
and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has successfuly |
172 |
+ |
used additional neutralization of higher order moments for systems of |
173 |
+ |
point charges.\cite{Fukuda:2013sf} |
174 |
+ |
|
175 |
|
\begin{figure} |
176 |
|
\centering |
177 |
< |
\includegraphics[width=\linewidth]{schematic.pdf} |
177 |
> |
\includegraphics[width=\linewidth]{schematic.eps} |
178 |
|
\caption{Top: Ionic systems exhibit local clustering of dissimilar |
179 |
|
charges (in the smaller grey circle), so interactions are |
180 |
|
effectively charge-multipole at longer distances. With hard |
189 |
|
\label{fig:schematic} |
190 |
|
\end{figure} |
191 |
|
|
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< |
The direct truncation of interactions at a cutoff radius creates |
193 |
< |
truncation defects. Wolf \textit{et al.} argued that |
194 |
< |
truncation errors are due to net charge remaining inside the cutoff |
195 |
< |
sphere.\cite{Wolf:1999dn} To neutralize this charge they proposed |
196 |
< |
placing an image charge on the surface of the cutoff sphere for every |
197 |
< |
real charge inside the cutoff. These charges are present for the |
198 |
< |
evaluation of both the pair interaction energy and the force, although |
199 |
< |
the force expression maintained a discontinuity at the cutoff sphere. |
200 |
< |
In the original Wolf formulation, the total energy for the charge and |
201 |
< |
image were not equal to the integral of their force expression, and as |
176 |
< |
a result, the total energy would not be conserved in molecular |
177 |
< |
dynamics (MD) simulations.\cite{Zahn:2002hc} Zahn \textit{et al.} and |
178 |
< |
Fennel and Gezelter later proposed shifted force variants of the Wolf |
179 |
< |
method with commensurate force and energy expressions that do not |
180 |
< |
exhibit this problem.\cite{Fennell:2006lq} Related real-space |
181 |
< |
methods were also proposed by Chen \textit{et |
182 |
< |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
183 |
< |
and by Wu and Brooks.\cite{Wu:044107} |
184 |
< |
|
185 |
< |
Considering the interaction of one central ion in an ionic crystal |
186 |
< |
with a portion of the crystal at some distance, the effective Columbic |
187 |
< |
potential is found to decrease as $r^{-5}$. If one views the \ce{NaCl} |
188 |
< |
crystal as a simple cubic (SC) structure with an octupolar |
189 |
< |
\ce{(NaCl)4} basis, the electrostatic energy per ion converges more |
190 |
< |
rapidly to the Madelung energy than the dipolar |
191 |
< |
approximation.\cite{Wolf92} To find the correct Madelung constant, |
192 |
< |
Lacman suggested that the NaCl structure could be constructed in a way |
193 |
< |
that the finite crystal terminates with complete \ce{(NaCl)4} |
194 |
< |
molecules.\cite{Lacman65} The central ion sees what is effectively a |
195 |
< |
set of octupoles at large distances. These facts suggest that the |
196 |
< |
Madelung constants are relatively short ranged for perfect ionic |
197 |
< |
crystals.\cite{Wolf:1999dn} |
198 |
< |
|
199 |
< |
One can make a similar argument for crystals of point multipoles. The |
200 |
< |
Luttinger and Tisza treatment of energy constants for dipolar lattices |
201 |
< |
utilizes 24 basis vectors that contain dipoles at the eight corners of |
202 |
< |
a unit cube. Only three of these basis vectors, $X_1, Y_1, |
203 |
< |
\mathrm{~and~} Z_1,$ retain net dipole moments, while the rest have |
204 |
< |
zero net dipole and retain contributions only from higher order |
205 |
< |
multipoles. The effective interaction between a dipole at the center |
192 |
> |
One can make a similar effective range argument for crystals of point |
193 |
> |
\textit{multipoles}. The Luttinger and Tisza treatment of energy |
194 |
> |
constants for dipolar lattices utilizes 24 basis vectors that contain |
195 |
> |
dipoles at the eight corners of a unit cube.\cite{LT} Only three of |
196 |
> |
these basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole |
197 |
> |
moments, while the rest have zero net dipole and retain contributions |
198 |
> |
only from higher order multipoles. The lowest-energy crystalline |
199 |
> |
structures are built out of basis vectors that have only residual |
200 |
> |
quadrupolar moments (e.g. the $Z_5$ array). In these low energy |
201 |
> |
structures, the effective interaction between a dipole at the center |
202 |
|
of a crystal and a group of eight dipoles farther away is |
203 |
|
significantly shorter ranged than the $r^{-3}$ that one would expect |
204 |
|
for raw dipole-dipole interactions. Only in crystals which retain a |
218 |
|
|
219 |
|
The shorter effective range of electrostatic interactions is not |
220 |
|
limited to perfect crystals, but can also apply in disordered fluids. |
221 |
< |
Even at elevated temperatures, there is, on average, local charge |
222 |
< |
balance in an ionic liquid, where each positive ion has surroundings |
223 |
< |
dominated by negaitve ions and vice versa. The reversed-charge images |
224 |
< |
on the cutoff sphere that are integral to the Wolf and DSF approaches |
225 |
< |
retain the effective multipolar interactions as the charges traverse |
226 |
< |
the cutoff boundary. |
221 |
> |
Even at elevated temperatures, there is local charge balance in an |
222 |
> |
ionic liquid, where each positive ion has surroundings dominated by |
223 |
> |
negative ions and vice versa. The reversed-charge images on the |
224 |
> |
cutoff sphere that are integral to the Wolf and DSF approaches retain |
225 |
> |
the effective multipolar interactions as the charges traverse the |
226 |
> |
cutoff boundary. |
227 |
|
|
228 |
|
In multipolar fluids (see Fig. \ref{fig:schematic}) there is |
229 |
|
significant orientational averaging that additionally reduces the |
242 |
|
% to the non-neutralized value of the higher order moments within the |
243 |
|
% cutoff sphere. |
244 |
|
|
245 |
< |
The forces and torques acting on atomic sites are the fundamental |
246 |
< |
factors driving dynamics in molecular simulations. Fennell and |
247 |
< |
Gezelter proposed the damped shifted force (DSF) energy kernel to |
248 |
< |
obtain consistent energies and forces on the atoms within the cutoff |
249 |
< |
sphere. Both the energy and the force go smoothly to zero as an atom |
250 |
< |
aproaches the cutoff radius. The comparisons of the accuracy these |
251 |
< |
quantities between the DSF kernel and SPME was surprisingly |
252 |
< |
good.\cite{Fennell:2006lq} The DSF method has seen increasing use for |
253 |
< |
calculating electrostatic interactions in molecular systems with |
254 |
< |
relatively uniform charge |
259 |
< |
densities.\cite{Shi:2013ij,Kannam:2012rr,Acevedo13,Space12,English08,Lawrence13,Vergne13} |
245 |
> |
Forces and torques acting on atomic sites are fundamental in driving |
246 |
> |
dynamics in molecular simulations, and the damped shifted force (DSF) |
247 |
> |
energy kernel provides consistent energies and forces on charged atoms |
248 |
> |
within the cutoff sphere. Both the energy and the force go smoothly to |
249 |
> |
zero as an atom aproaches the cutoff radius. The comparisons of the |
250 |
> |
accuracy these quantities between the DSF kernel and SPME was |
251 |
> |
surprisingly good.\cite{Fennell:2006lq} As a result, the DSF method |
252 |
> |
has seen increasing use in molecular systems with relatively uniform |
253 |
> |
charge |
254 |
> |
densities.\cite{English08,Kannam:2012rr,Space12,Lawrence13,Acevedo13,Shi:2013ij,Vergne13} |
255 |
|
|
256 |
|
\subsection{The damping function} |
257 |
< |
The damping function used in our research has been discussed in detail |
258 |
< |
in the first paper of this series.\cite{PaperI} The radial kernel |
259 |
< |
$1/r$ for the interactions between point charges can be replaced by |
260 |
< |
the complementary error function $\mathrm{erfc}(\alpha r)/r$ to |
261 |
< |
accelerate the rate of convergence, where $\alpha$ is a damping |
262 |
< |
parameter with units of inverse distance. Altering the value of |
263 |
< |
$\alpha$ is equivalent to changing the width of Gaussian charge |
264 |
< |
distributions that replace each point charge -- Gaussian overlap |
265 |
< |
integrals yield complementary error functions when truncated at a |
266 |
< |
finite distance. |
257 |
> |
The damping function has been discussed in detail in the first paper |
258 |
> |
of this series.\cite{PaperI} The $1/r$ Coulombic kernel for the |
259 |
> |
interactions between point charges can be replaced by the |
260 |
> |
complementary error function $\mathrm{erfc}(\alpha r)/r$ to accelerate |
261 |
> |
convergence, where $\alpha$ is a damping parameter with units of |
262 |
> |
inverse distance. Altering the value of $\alpha$ is equivalent to |
263 |
> |
changing the width of Gaussian charge distributions that replace each |
264 |
> |
point charge, as Coulomb integrals with Gaussian charge distributions |
265 |
> |
produce complementary error functions when truncated at a finite |
266 |
> |
distance. |
267 |
|
|
268 |
< |
By using suitable value of damping alpha ($\alpha \sim 0.2$) for a |
269 |
< |
cutoff radius ($r_{Âc}=9 A$), Fennel and Gezelter produced very good |
270 |
< |
agreement with SPME for the interaction energies, forces and torques |
271 |
< |
for charge-charge interactions.\cite{Fennell:2006lq} |
268 |
> |
With moderate damping coefficients, $\alpha \sim 0.2$, the DSF method |
269 |
> |
produced very good agreement with SPME for interaction energies, |
270 |
> |
forces and torques for charge-charge |
271 |
> |
interactions.\cite{Fennell:2006lq} |
272 |
|
|
273 |
|
\subsection{Point multipoles in molecular modeling} |
274 |
|
Coarse-graining approaches which treat entire molecular subsystems as |
275 |
|
a single rigid body are now widely used. A common feature of many |
276 |
|
coarse-graining approaches is simplification of the electrostatic |
277 |
|
interactions between bodies so that fewer site-site interactions are |
278 |
< |
required to compute configurational energies. Many coarse-grained |
279 |
< |
molecular structures would normally consist of equal positive and |
285 |
< |
negative charges, and rather than use multiple site-site interactions, |
286 |
< |
the interaction between higher order multipoles can also be used to |
287 |
< |
evaluate a single molecule-molecule |
288 |
< |
interaction.\cite{Ren06,Essex10,Essex11} |
278 |
> |
required to compute configurational |
279 |
> |
energies.\cite{Ren06,Essex10,Essex11} |
280 |
|
|
281 |
< |
Because electrons in a molecule are not localized at specific points, |
282 |
< |
the assignment of partial charges to atomic centers is a relatively |
283 |
< |
rough approximation. Atomic sites can also be assigned point |
284 |
< |
multipoles and polarizabilities to increase the accuracy of the |
285 |
< |
molecular model. Recently, water has been modeled with point |
286 |
< |
multipoles up to octupolar order using the soft sticky |
296 |
< |
dipole-quadrupole-octupole (SSDQO) |
281 |
> |
Additionally, because electrons in a molecule are not localized at |
282 |
> |
specific points, the assignment of partial charges to atomic centers |
283 |
> |
is always an approximation. For increased accuracy, atomic sites can |
284 |
> |
also be assigned point multipoles and polarizabilities. Recently, |
285 |
> |
water has been modeled with point multipoles up to octupolar order |
286 |
> |
using the soft sticky dipole-quadrupole-octupole (SSDQO) |
287 |
|
model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point |
288 |
|
multipoles up to quadrupolar order have also been coupled with point |
289 |
|
polarizabilities in the high-quality AMOEBA and iAMOEBA water |
290 |
< |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} But |
291 |
< |
using point multipole with the real space truncation without |
292 |
< |
accounting for multipolar neutrality will create energy conservation |
293 |
< |
issues in molecular dynamics (MD) simulations. |
290 |
> |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However, |
291 |
> |
truncating point multipoles without smoothing the forces and torques |
292 |
> |
can create energy conservation issues in molecular dynamics |
293 |
> |
simulations. |
294 |
|
|
295 |
|
In this paper we test a set of real-space methods that were developed |
296 |
|
for point multipolar interactions. These methods extend the damped |
297 |
|
shifted force (DSF) and Wolf methods originally developed for |
298 |
|
charge-charge interactions and generalize them for higher order |
299 |
< |
multipoles. The detailed mathematical development of these methods has |
300 |
< |
been presented in the first paper in this series, while this work |
301 |
< |
covers the testing the energies, forces, torques, and energy |
299 |
> |
multipoles. The detailed mathematical development of these methods |
300 |
> |
has been presented in the first paper in this series, while this work |
301 |
> |
covers the testing of energies, forces, torques, and energy |
302 |
|
conservation properties of the methods in realistic simulation |
303 |
|
environments. In all cases, the methods are compared with the |
304 |
< |
reference method, a full multipolar Ewald treatment. |
304 |
> |
reference method, a full multipolar Ewald treatment.\cite{Smith82,Smith98} |
305 |
|
|
306 |
|
|
307 |
|
%\subsection{Conservation of total energy } |
325 |
|
expressed as the product of two multipole operators and a Coulombic |
326 |
|
kernel, |
327 |
|
\begin{equation} |
328 |
< |
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
328 |
> |
U_{ab}(r)= M_{a} M_{b} \frac{1}{r} \label{kernel}. |
329 |
|
\end{equation} |
330 |
< |
where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is |
331 |
< |
expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf |
332 |
< |
a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object |
343 |
< |
$\bf a$, etc. |
330 |
> |
where the multipole operator for site $a$, $M_{a}$, is |
331 |
> |
expressed in terms of the point charge, $C_{a}$, dipole, ${\bf D}_{a}$, and quadrupole, $\mathsf{Q}_{a}$, for object |
332 |
> |
$a$, etc. |
333 |
|
|
334 |
|
% Interactions between multipoles can be expressed as higher derivatives |
335 |
|
% of the bare Coulomb potential, so one way of ensuring that the forces |
384 |
|
contributions to the potential, and ensures that the forces and |
385 |
|
torques from each of these contributions will vanish at the cutoff |
386 |
|
radius. For example, the direct dipole dot product |
387 |
< |
($\mathbf{D}_{\bf a} |
388 |
< |
\cdot \mathbf{D}_{\bf b}$) is treated differently than the dipole-distance |
387 |
> |
($\mathbf{D}_{a} |
388 |
> |
\cdot \mathbf{D}_{b}$) is treated differently than the dipole-distance |
389 |
|
dot products: |
390 |
|
\begin{equation} |
391 |
< |
U_{D_{\bf a}D_{\bf b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left( |
392 |
< |
\mathbf{D}_{\bf a} \cdot |
393 |
< |
\mathbf{D}_{\bf b} \right) v_{21}(r) + |
394 |
< |
\left( \mathbf{D}_{\bf a} \cdot \hat{r} \right) |
395 |
< |
\left( \mathbf{D}_{\bf b} \cdot \hat{r} \right) v_{22}(r) \right] |
391 |
> |
U_{D_{a}D_{b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left( |
392 |
> |
\mathbf{D}_{a} \cdot |
393 |
> |
\mathbf{D}_{b} \right) v_{21}(r) + |
394 |
> |
\left( \mathbf{D}_{a} \cdot \hat{\mathbf{r}} \right) |
395 |
> |
\left( \mathbf{D}_{b} \cdot \hat{\mathbf{r}} \right) v_{22}(r) \right] |
396 |
|
\end{equation} |
397 |
|
|
398 |
|
For the Taylor shifted (TSF) method with the undamped kernel, |
417 |
|
which have been projected onto the surface of the cutoff sphere |
418 |
|
without changing their relative orientation, |
419 |
|
\begin{equation} |
420 |
< |
U_{D_{\bf a}D_{\bf b}}(r) = U_{D_{\bf a}D_{\bf b}}(r) - |
421 |
< |
U_{D_{\bf a} D_{\bf b}}(r_c) |
422 |
< |
- (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot |
423 |
< |
\nabla U_{D_{\bf a}D_{\bf b}}(r_c). |
420 |
> |
U_{D_{a}D_{b}}(r) = U_{D_{a}D_{b}}(r) - |
421 |
> |
U_{D_{a}D_{b}}(r_c) |
422 |
> |
- (r_{ab}-r_c) ~~~\hat{\mathbf{r}}_{ab} \cdot |
423 |
> |
\nabla U_{D_{a}D_{b}}(r_c). |
424 |
|
\end{equation} |
425 |
< |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf |
437 |
< |
a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance |
425 |
> |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{a}$ and $\mathbf{D}_{b}$, are retained at the cutoff distance |
426 |
|
(although the signs are reversed for the dipole that has been |
427 |
|
projected onto the cutoff sphere). In many ways, this simpler |
428 |
|
approach is closer in spirit to the original shifted force method, in |
443 |
|
In general, the gradient shifted potential between a central multipole |
444 |
|
and any multipolar site inside the cutoff radius is given by, |
445 |
|
\begin{equation} |
446 |
< |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
447 |
< |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}} |
448 |
< |
\cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
446 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \mathsf{A}, \mathsf{B}) - |
447 |
> |
U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) - (r-r_c) |
448 |
> |
\hat{\mathbf{r}} \cdot \nabla U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) \right] |
449 |
|
\label{generic2} |
450 |
|
\end{equation} |
451 |
|
where the sum describes a separate force-shifting that is applied to |
452 |
|
each orientational contribution to the energy. In this expression, |
453 |
|
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles |
454 |
< |
($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ |
454 |
> |
($a$ and $b$) in space, and $\mathsf{A}$ and $\mathsf{B}$ |
455 |
|
represent the orientations the multipoles. |
456 |
|
|
457 |
|
The third term converges more rapidly than the first two terms as a |
478 |
|
interactions with the central multipole and the image. This |
479 |
|
effectively shifts the total potential to zero at the cutoff radius, |
480 |
|
\begin{equation} |
481 |
< |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
482 |
< |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
481 |
> |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \mathsf{A}, \mathsf{B}) - |
482 |
> |
U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) \right] |
483 |
|
\label{eq:SP} |
484 |
|
\end{equation} |
485 |
|
where the sum describes separate potential shifting that is done for |
524 |
|
used the multipolar Ewald sum as a reference method for comparing |
525 |
|
energies, forces, and torques for molecular models that mimic |
526 |
|
disordered and ordered condensed-phase systems. The parameters used |
527 |
< |
in the test cases are given in table~\ref{tab:pars}. |
527 |
> |
in the test cases are given in table~\ref{tab:pars}. |
528 |
|
|
529 |
|
\begin{table} |
530 |
|
\label{tab:pars} |
572 |
|
and have been compared with the values obtained from the multipolar |
573 |
|
Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
574 |
|
between two configurations is the primary quantity that governs how |
575 |
< |
the simulation proceeds. These differences are the most imporant |
575 |
> |
the simulation proceeds. These differences are the most important |
576 |
|
indicators of the reliability of a method even if the absolute |
577 |
|
energies are not exact. For each of the multipolar systems listed |
578 |
|
above, we have compared the change in electrostatic potential energy |
584 |
|
\subsection{Implementation} |
585 |
|
The real-space methods developed in the first paper in this series |
586 |
|
have been implemented in our group's open source molecular simulation |
587 |
< |
program, OpenMD,\cite{openmd} which was used for all calculations in |
587 |
> |
program, OpenMD,\cite{Meineke05,openmd} which was used for all calculations in |
588 |
|
this work. The complementary error function can be a relatively slow |
589 |
|
function on some processors, so all of the radial functions are |
590 |
|
precomputed on a fine grid and are spline-interpolated to provide |
594 |
|
with a reciprocal space cutoff, $k_\mathrm{max} = 7$. Our version of |
595 |
|
the Ewald sum is a re-implementation of the algorithm originally |
596 |
|
proposed by Smith that does not use the particle mesh or smoothing |
597 |
< |
approximations.\cite{Smith82,Smith98} In all cases, the quantities |
598 |
< |
being compared are the electrostatic contributions to energies, force, |
599 |
< |
and torques. All other contributions to these quantities (i.e. from |
600 |
< |
Lennard-Jones interactions) are removed prior to the comparisons. |
597 |
> |
approximations.\cite{Smith82,Smith98} This implementation was tested |
598 |
> |
extensively against the analytic energy constants for the multipolar |
599 |
> |
lattices that are discussed in reference \onlinecite{PaperI}. In all |
600 |
> |
cases discussed below, the quantities being compared are the |
601 |
> |
electrostatic contributions to energies, force, and torques. All |
602 |
> |
other contributions to these quantities (i.e. from Lennard-Jones |
603 |
> |
interactions) are removed prior to the comparisons. |
604 |
|
|
605 |
|
The convergence parameter ($\alpha$) also plays a role in the balance |
606 |
|
of the real-space and reciprocal-space portions of the Ewald |
794 |
|
|
795 |
|
\begin{figure} |
796 |
|
\centering |
797 |
< |
\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
797 |
> |
\includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps} |
798 |
|
\caption{Statistical analysis of the quality of configurational |
799 |
|
energy differences for the real-space electrostatic methods |
800 |
|
compared with the reference Ewald sum. Results with a value equal |
867 |
|
|
868 |
|
\begin{figure} |
869 |
|
\centering |
870 |
< |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
870 |
> |
\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} |
871 |
|
\caption{Statistical analysis of the quality of the force vector |
872 |
|
magnitudes for the real-space electrostatic methods compared with |
873 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
881 |
|
|
882 |
|
\begin{figure} |
883 |
|
\centering |
884 |
< |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
884 |
> |
\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps} |
885 |
|
\caption{Statistical analysis of the quality of the torque vector |
886 |
|
magnitudes for the real-space electrostatic methods compared with |
887 |
|
the reference Ewald sum. Results with a value equal to 1 (dashed |
939 |
|
|
940 |
|
\begin{figure} |
941 |
|
\centering |
942 |
< |
\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
942 |
> |
\includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps} |
943 |
|
\caption{The circular variance of the direction of the force and |
944 |
|
torque vectors obtained from the real-space methods around the |
945 |
|
reference Ewald vectors. A variance equal to 0 (dashed line) |
959 |
|
in this series and provides the most comprehensive test of the new |
960 |
|
methods. A liquid-phase system was created with 2000 water molecules |
961 |
|
and 48 dissolved ions at a density of 0.98 g cm$^{-3}$ and a |
962 |
< |
temperature of 300K. After equilibration, this liquid-phase system |
963 |
< |
was run for 1 ns under the Ewald, Hard, SP, GSF, and TSF methods with |
964 |
< |
a cutoff radius of 12\AA. The value of the damping coefficient was |
965 |
< |
also varied from the undamped case ($\alpha = 0$) to a heavily damped |
966 |
< |
case ($\alpha = 0.3$ \AA$^{-1}$) for all of the real space methods. A |
967 |
< |
sample was also run using the multipolar Ewald sum with the same |
968 |
< |
real-space cutoff. |
962 |
> |
temperature of 300K. After equilibration in the canonical (NVT) |
963 |
> |
ensemble using a Nos\'e-Hoover thermostat, this liquid-phase system |
964 |
> |
was run for 1 ns in the microcanonical (NVE) ensemble under the Ewald, |
965 |
> |
Hard, SP, GSF, and TSF methods with a cutoff radius of 12\AA. The |
966 |
> |
value of the damping coefficient was also varied from the undamped |
967 |
> |
case ($\alpha = 0$) to a heavily damped case ($\alpha = 0.3$ |
968 |
> |
\AA$^{-1}$) for all of the real space methods. A sample was also run |
969 |
> |
using the multipolar Ewald sum with the same real-space cutoff. |
970 |
|
|
971 |
|
In figure~\ref{fig:energyDrift} we show the both the linear drift in |
972 |
|
energy over time, $\delta E_1$, and the standard deviation of energy |
987 |
|
|
988 |
|
\begin{figure} |
989 |
|
\centering |
990 |
< |
\includegraphics[width=\textwidth]{newDrift_12.pdf} |
990 |
> |
\includegraphics[width=\textwidth]{newDrift_12.eps} |
991 |
|
\label{fig:energyDrift} |
992 |
|
\caption{Analysis of the energy conservation of the real-space |
993 |
< |
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
994 |
< |
energy over time (in kcal / mol / particle / ns) and $\delta |
995 |
< |
\mathrm{E}_0$ is the standard deviation of energy fluctuations |
996 |
< |
around this drift (in kcal / mol / particle). All simulations were |
997 |
< |
of a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
993 |
> |
methods. $\delta \mathrm{E}_1$ is the linear drift in energy over |
994 |
> |
time (in kcal / mol / particle / ns) and $\delta \mathrm{E}_0$ is |
995 |
> |
the standard deviation of energy fluctuations around this drift (in |
996 |
> |
kcal / mol / particle). Points that appear below the dashed grey |
997 |
> |
(Ewald) lines exhibit better energy conservation than commonly-used |
998 |
> |
parameters for Ewald-based electrostatics. All simulations were of |
999 |
> |
a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
1000 |
|
300 K starting from the same initial configuration. All runs |
1001 |
|
utilized the same real-space cutoff, $r_c = 12$\AA.} |
1002 |
+ |
\end{figure} |
1003 |
+ |
|
1004 |
+ |
\subsection{Reproduction of Structural \& Dynamical Features\label{sec:structure}} |
1005 |
+ |
The most important test of the modified interaction potentials is the |
1006 |
+ |
fidelity with which they can reproduce structural features and |
1007 |
+ |
dynamical properties in a liquid. One commonly-utilized measure of |
1008 |
+ |
structural ordering is the pair distribution function, $g(r)$, which |
1009 |
+ |
measures local density deviations in relation to the bulk density. In |
1010 |
+ |
the electrostatic approaches studied here, the short-range repulsion |
1011 |
+ |
from the Lennard-Jones potential is identical for the various |
1012 |
+ |
electrostatic methods, and since short range repulsion determines much |
1013 |
+ |
of the local liquid ordering, one would not expect to see many |
1014 |
+ |
differences in $g(r)$. Indeed, the pair distributions are essentially |
1015 |
+ |
identical for all of the electrostatic methods studied (for each of |
1016 |
+ |
the different systems under investigation). An example of this |
1017 |
+ |
agreement for the SSDQ water/ion system is shown in |
1018 |
+ |
Fig. \ref{fig:gofr}. |
1019 |
+ |
|
1020 |
+ |
\begin{figure} |
1021 |
+ |
\centering |
1022 |
+ |
\includegraphics[width=\textwidth]{gofr_ssdqc.eps} |
1023 |
+ |
\label{fig:gofr} |
1024 |
+ |
\caption{The pair distribution functions, $g(r)$, for the SSDQ |
1025 |
+ |
water/ion system obtained using the different real-space methods are |
1026 |
+ |
essentially identical with the result from the Ewald |
1027 |
+ |
treatment.} |
1028 |
|
\end{figure} |
1029 |
|
|
1030 |
+ |
There is a very slight overstructuring of the first solvation shell |
1031 |
+ |
when using when using TSF at lower values of the damping coefficient |
1032 |
+ |
($\alpha \le 0.1$) or when using undamped GSF. With moderate damping, |
1033 |
+ |
GSF and SP produce pair distributions that are identical (within |
1034 |
+ |
numerical noise) to their Ewald counterparts. |
1035 |
|
|
1036 |
+ |
A structural property that is a more demanding test of modified |
1037 |
+ |
electrostatics is the mean value of the electrostatic energy $\langle |
1038 |
+ |
U_\mathrm{elect} \rangle / N$ which is obtained by sampling the |
1039 |
+ |
liquid-state configurations experienced by a liquid evolving entirely |
1040 |
+ |
under the influence of each of the methods. In table \ref{tab:Props} |
1041 |
+ |
we demonstrate how $\langle U_\mathrm{elect} \rangle / N$ varies with |
1042 |
+ |
the damping parameter, $\alpha$, for each of the methods. |
1043 |
+ |
|
1044 |
+ |
As in the crystals studied in the first paper, damping is important |
1045 |
+ |
for converging the mean electrostatic energy values, particularly for |
1046 |
+ |
the two shifted force methods (GSF and TSF). A value of $\alpha |
1047 |
+ |
\approx 0.2$ \AA$^{-1}$ is sufficient to converge the SP and GSF |
1048 |
+ |
energies with a cutoff of 12 \AA, while shorter cutoffs require more |
1049 |
+ |
dramatic damping ($\alpha \approx 0.3$ \AA$^{-1}$ for $r_c = 9$ \AA). |
1050 |
+ |
Overdamping the real-space electrostatic methods occurs with $\alpha > |
1051 |
+ |
0.4$, causing the estimate of the energy to drop below the Ewald |
1052 |
+ |
results. |
1053 |
+ |
|
1054 |
+ |
These ``optimal'' values of the damping coefficient are slightly |
1055 |
+ |
larger than what were observed for DSF electrostatics for purely |
1056 |
+ |
point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for |
1057 |
+ |
$r_c = 12$\AA appears to be an excellent compromise for mixed charge |
1058 |
+ |
multipole systems. |
1059 |
+ |
|
1060 |
+ |
To test the fidelity of the electrostatic methods at reproducing |
1061 |
+ |
dynamics in a multipolar liquid, it is also useful to look at |
1062 |
+ |
transport properties, particularly the diffusion constant, |
1063 |
+ |
\begin{equation} |
1064 |
+ |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left| |
1065 |
+ |
\mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle |
1066 |
+ |
\label{eq:diff} |
1067 |
+ |
\end{equation} |
1068 |
+ |
which measures long-time behavior and is sensitive to the forces on |
1069 |
+ |
the multipoles. For the soft dipolar fluid and the SSDQ liquid |
1070 |
+ |
systems, the self-diffusion constants (D) were calculated from linear |
1071 |
+ |
fits to the long-time portion of the mean square displacement, |
1072 |
+ |
$\langle r^{2}(t) \rangle$.\cite{Allen87} |
1073 |
+ |
|
1074 |
+ |
In addition to translational diffusion, orientational relaxation times |
1075 |
+ |
were calculated for comparisons with the Ewald simulations and with |
1076 |
+ |
experiments. These values were determined from the same 1~ns |
1077 |
+ |
microcanonical trajectories used for translational diffusion by |
1078 |
+ |
calculating the orientational time correlation function, |
1079 |
+ |
\begin{equation} |
1080 |
+ |
C_l^\gamma(t) = \left\langle P_l\left[\hat{\mathbf{A}}_\gamma(t) |
1081 |
+ |
\cdot\hat{\mathbf{A}}_\gamma(0)\right]\right\rangle, |
1082 |
+ |
\label{eq:OrientCorr} |
1083 |
+ |
\end{equation} |
1084 |
+ |
where $P_l$ is the Legendre polynomial of order $l$ and |
1085 |
+ |
$\hat{\mathbf{A}}_\gamma$ is the space-frame unit vector for body axis |
1086 |
+ |
$\gamma$ on a molecule.. Th body-fixed reference frame used for our |
1087 |
+ |
models has the $z$-axis running along the dipoles, and for the SSDQ |
1088 |
+ |
water model, the $y$-axis connects the two implied hydrogen atom |
1089 |
+ |
positions. From the orientation autocorrelation functions, we can |
1090 |
+ |
obtain time constants for rotational relaxation either by fitting an |
1091 |
+ |
exponential function or by integrating the entire correlation |
1092 |
+ |
function. In a good water model, these decay times would be |
1093 |
+ |
comparable to water orientational relaxation times from nuclear |
1094 |
+ |
magnetic resonance (NMR). The relaxation constant obtained from |
1095 |
+ |
$C_2^y(t)$ is normally of experimental interest because it describes |
1096 |
+ |
the relaxation of the principle axis connecting the hydrogen |
1097 |
+ |
atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular portion |
1098 |
+ |
of the dipole-dipole relaxation from a proton NMR signal and should |
1099 |
+ |
provide an estimate of the NMR relaxation time constant.\cite{Impey82} |
1100 |
+ |
|
1101 |
+ |
Results for the diffusion constants and orientational relaxation times |
1102 |
+ |
are shown in figure \ref{tab:Props}. From this data, it is apparent |
1103 |
+ |
that the values for both $D$ and $\tau_2$ using the Ewald sum are |
1104 |
+ |
reproduced with reasonable fidelity by the GSF method. |
1105 |
+ |
|
1106 |
+ |
The $\tau_2$ results in \ref{tab:Props} show a much greater difference |
1107 |
+ |
between the real-space and the Ewald results. |
1108 |
+ |
|
1109 |
+ |
\begin{table} |
1110 |
+ |
\label{tab:Props} |
1111 |
+ |
\caption{Comparison of the structural and dynamic properties for the |
1112 |
+ |
soft dipolar liquid test for all of the real-space methods.} |
1113 |
+ |
\begin{tabular}{l|c|cccc|cccc|cccc} |
1114 |
+ |
& Ewald & \multicolumn{4}{c|}{SP} & \multicolumn{4}{c|}{GSF} & \multicolumn{4}{c|}{TSF} \\ |
1115 |
+ |
$\alpha$ (\AA$^{-1}$) & & |
1116 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
1117 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
1118 |
+ |
0.0 & 0.1 & 0.2 & 0.3 \\ \cline{2-6}\cline{6-10}\cline{10-14} |
1119 |
+ |
|
1120 |
+ |
$\langle U_\mathrm{elect} \rangle /N$ &&&&&&&&&&&&&\\ |
1121 |
+ |
D ($10^{-4}~\mathrm{cm}^2/\mathrm{s}$)& |
1122 |
+ |
470.2(6) & |
1123 |
+ |
416.6(5) & |
1124 |
+ |
379.6(5) & |
1125 |
+ |
438.6(5) & |
1126 |
+ |
476.0(6) & |
1127 |
+ |
412.8(5) & |
1128 |
+ |
421.1(5) & |
1129 |
+ |
400.5(5) & |
1130 |
+ |
437.5(6) & |
1131 |
+ |
434.6(5) & |
1132 |
+ |
411.4(5) & |
1133 |
+ |
545.3(7) & |
1134 |
+ |
459.6(6) \\ |
1135 |
+ |
$\tau_2$ (fs) & |
1136 |
+ |
1.136 & |
1137 |
+ |
1.041 & |
1138 |
+ |
1.064 & |
1139 |
+ |
1.109 & |
1140 |
+ |
1.211 & |
1141 |
+ |
1.119 & |
1142 |
+ |
1.039 & |
1143 |
+ |
1.058 & |
1144 |
+ |
1.21 & |
1145 |
+ |
1.15 & |
1146 |
+ |
1.172 & |
1147 |
+ |
1.153 & |
1148 |
+ |
1.125 \\ |
1149 |
+ |
\end{tabular} |
1150 |
+ |
\end{table} |
1151 |
+ |
|
1152 |
+ |
|
1153 |
|
\section{CONCLUSION} |
1154 |
|
In the first paper in this series, we generalized the |
1155 |
|
charge-neutralized electrostatic energy originally developed by Wolf |