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 |
333 |
< |
$\bf a$, 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, $\mathbf{D}_{\bf |
427 |
< |
a}$ 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 |
956 |
|
in this series and provides the most comprehensive test of the new |
957 |
|
methods. A liquid-phase system was created with 2000 water molecules |
958 |
|
and 48 dissolved ions at a density of 0.98 g cm$^{-3}$ and a |
959 |
< |
temperature of 300K. After equilibration, this liquid-phase system |
960 |
< |
was run for 1 ns under the Ewald, Hard, SP, GSF, and TSF methods with |
961 |
< |
a cutoff radius of 12\AA. The value of the damping coefficient was |
962 |
< |
also varied from the undamped case ($\alpha = 0$) to a heavily damped |
963 |
< |
case ($\alpha = 0.3$ \AA$^{-1}$) for all of the real space methods. A |
964 |
< |
sample was also run using the multipolar Ewald sum with the same |
965 |
< |
real-space cutoff. |
959 |
> |
temperature of 300K. After equilibration in the canonical (NVT) |
960 |
> |
ensemble using a Nos\'e-Hoover thermostat, this liquid-phase system |
961 |
> |
was run for 1 ns in the microcanonical (NVE) ensemble under the Ewald, |
962 |
> |
Hard, SP, GSF, and TSF methods with a cutoff radius of 12\AA. The |
963 |
> |
value of the damping coefficient was also varied from the undamped |
964 |
> |
case ($\alpha = 0$) to a heavily damped case ($\alpha = 0.3$ |
965 |
> |
\AA$^{-1}$) for all of the real space methods. A sample was also run |
966 |
> |
using the multipolar Ewald sum with the same real-space cutoff. |
967 |
|
|
968 |
|
In figure~\ref{fig:energyDrift} we show the both the linear drift in |
969 |
|
energy over time, $\delta E_1$, and the standard deviation of energy |
987 |
|
\includegraphics[width=\textwidth]{newDrift_12.eps} |
988 |
|
\label{fig:energyDrift} |
989 |
|
\caption{Analysis of the energy conservation of the real-space |
990 |
< |
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
991 |
< |
energy over time (in kcal / mol / particle / ns) and $\delta |
992 |
< |
\mathrm{E}_0$ is the standard deviation of energy fluctuations |
993 |
< |
around this drift (in kcal / mol / particle). All simulations were |
994 |
< |
of a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
990 |
> |
methods. $\delta \mathrm{E}_1$ is the linear drift in energy over |
991 |
> |
time (in kcal / mol / particle / ns) and $\delta \mathrm{E}_0$ is |
992 |
> |
the standard deviation of energy fluctuations around this drift (in |
993 |
> |
kcal / mol / particle). Points that appear below the dashed grey |
994 |
> |
(Ewald) lines exhibit better energy conservation than commonly-used |
995 |
> |
parameters for Ewald-based electrostatics. All simulations were of |
996 |
> |
a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
997 |
|
300 K starting from the same initial configuration. All runs |
998 |
|
utilized the same real-space cutoff, $r_c = 12$\AA.} |
999 |
|
\end{figure} |
1000 |
|
|
1001 |
< |
\subsection{Reproduction of Structural Features\label{sec:structure}} |
1002 |
< |
One of the best tests of modified interaction potentials is the |
1003 |
< |
fidelity with which they can reproduce structural features in a |
1004 |
< |
liquid. One commonly-utilized measure of structural ordering is the |
1005 |
< |
pair distribution function, $g(r)$, which measures local density |
1006 |
< |
deviations in relation to the bulk density. In the electrostatic |
1007 |
< |
approaches studied here, the short-range repulsion from the |
1008 |
< |
Lennard-Jones potential is identical for the various electrostatic |
1009 |
< |
methods, and since short range repulsion determines much of the local |
1010 |
< |
liquid ordering, one would not expect to see any differences in |
1011 |
< |
$g(r)$. Indeed, the pair distributions are essentially identical for |
1012 |
< |
all of the electrostatic methods studied (for each of the different |
1013 |
< |
systems under investigation). Interested readers may consult the |
1014 |
< |
supplementary information for plots of these pair distribution |
1015 |
< |
functions. |
1001 |
> |
\subsection{Reproduction of Structural \& Dynamical Features\label{sec:structure}} |
1002 |
> |
The most important test of the modified interaction potentials is the |
1003 |
> |
fidelity with which they can reproduce structural features and |
1004 |
> |
dynamical properties in a liquid. One commonly-utilized measure of |
1005 |
> |
structural ordering is the pair distribution function, $g(r)$, which |
1006 |
> |
measures local density deviations in relation to the bulk density. In |
1007 |
> |
the electrostatic approaches studied here, the short-range repulsion |
1008 |
> |
from the Lennard-Jones potential is identical for the various |
1009 |
> |
electrostatic methods, and since short range repulsion determines much |
1010 |
> |
of the local liquid ordering, one would not expect to see many |
1011 |
> |
differences in $g(r)$. Indeed, the pair distributions are essentially |
1012 |
> |
identical for all of the electrostatic methods studied (for each of |
1013 |
> |
the different systems under investigation). An example of this |
1014 |
> |
agreement for the SSDQ water/ion system is shown in |
1015 |
> |
Fig. \ref{fig:gofr}. |
1016 |
|
|
1016 |
– |
A direct measure of the structural features that is a more |
1017 |
– |
enlightening test of the modified electrostatic methods is the average |
1018 |
– |
value of the electrostatic energy $\langle U_\mathrm{elect} \rangle$ |
1019 |
– |
which is obtained by sampling the liquid-state configurations |
1020 |
– |
experienced by a liquid evolving entirely under the influence of the |
1021 |
– |
methods being investigated. In figure \ref{fig:Uelect} we show how |
1022 |
– |
$\langle U_\mathrm{elect} \rangle$ for varies with the damping parameter, |
1023 |
– |
$\alpha$, for each of the methods. |
1024 |
– |
|
1017 |
|
\begin{figure} |
1018 |
|
\centering |
1019 |
< |
\includegraphics[width=\textwidth]{averagePotentialEnergy_r9_12.eps} |
1020 |
< |
\label{fig:Uelect} |
1021 |
< |
\caption{The average electrostatic potential energy, |
1022 |
< |
$\langle U_\mathrm{elect} \rangle$ for the SSDQ water with ions as a function |
1023 |
< |
of the damping parameter, $\alpha$, for each of the real-space |
1024 |
< |
electrostatic methods. Top panel: simulations run with a real-space |
1033 |
< |
cutoff, $r_c = 9$\AA. Bottom panel: the same quantity, but with a |
1034 |
< |
larger cutoff, $r_c = 12$\AA.} |
1019 |
> |
\includegraphics[width=\textwidth]{gofr_ssdqc.eps} |
1020 |
> |
\label{fig:gofr} |
1021 |
> |
\caption{The pair distribution functions, $g(r)$, for the SSDQ |
1022 |
> |
water/ion system obtained using the different real-space methods are |
1023 |
> |
essentially identical with the result from the Ewald |
1024 |
> |
treatment.} |
1025 |
|
\end{figure} |
1026 |
|
|
1027 |
< |
It is clear that moderate damping is important for converging the mean |
1028 |
< |
potential energy values, particularly for the two shifted force |
1029 |
< |
methods (GSF and TSF). A value of $\alpha \approx 0.18$ \AA$^{-1}$ is |
1030 |
< |
sufficient to converge the SP and GSF energies with a cutoff of 12 |
1031 |
< |
\AA, while shorter cutoffs require more dramatic damping ($\alpha |
1042 |
< |
\approx 0.36$ \AA$^{-1}$ for $r_c = 9$ \AA). It is also clear from |
1043 |
< |
fig. \ref{fig:Uelect} that it is possible to overdamp the real-space |
1044 |
< |
electrostatic methods, causing the estimate of the energy to drop |
1045 |
< |
below the Ewald results. |
1027 |
> |
There is a very slight overstructuring of the first solvation shell |
1028 |
> |
when using when using TSF at lower values of the damping coefficient |
1029 |
> |
($\alpha \le 0.1$) or when using undamped GSF. With moderate damping, |
1030 |
> |
GSF and SP produce pair distributions that are identical (within |
1031 |
> |
numerical noise) to their Ewald counterparts. |
1032 |
|
|
1033 |
+ |
A structural property that is a more demanding test of modified |
1034 |
+ |
electrostatics is the mean value of the electrostatic energy $\langle |
1035 |
+ |
U_\mathrm{elect} \rangle / N$ which is obtained by sampling the |
1036 |
+ |
liquid-state configurations experienced by a liquid evolving entirely |
1037 |
+ |
under the influence of each of the methods. In table \ref{tab:Props} |
1038 |
+ |
we demonstrate how $\langle U_\mathrm{elect} \rangle / N$ varies with |
1039 |
+ |
the damping parameter, $\alpha$, for each of the methods. |
1040 |
+ |
|
1041 |
+ |
As in the crystals studied in the first paper, damping is important |
1042 |
+ |
for converging the mean electrostatic energy values, particularly for |
1043 |
+ |
the two shifted force methods (GSF and TSF). A value of $\alpha |
1044 |
+ |
\approx 0.2$ \AA$^{-1}$ is sufficient to converge the SP and GSF |
1045 |
+ |
energies with a cutoff of 12 \AA, while shorter cutoffs require more |
1046 |
+ |
dramatic damping ($\alpha \approx 0.3$ \AA$^{-1}$ for $r_c = 9$ \AA). |
1047 |
+ |
Overdamping the real-space electrostatic methods occurs with $\alpha > |
1048 |
+ |
0.4$, causing the estimate of the energy to drop below the Ewald |
1049 |
+ |
results. |
1050 |
+ |
|
1051 |
|
These ``optimal'' values of the damping coefficient are slightly |
1052 |
|
larger than what were observed for DSF electrostatics for purely |
1053 |
|
point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for |
1054 |
|
$r_c = 12$\AA appears to be an excellent compromise for mixed charge |
1055 |
|
multipole systems. |
1056 |
|
|
1053 |
– |
\subsection{Reproduction of Dynamic Properties\label{sec:structure}} |
1057 |
|
To test the fidelity of the electrostatic methods at reproducing |
1058 |
|
dynamics in a multipolar liquid, it is also useful to look at |
1059 |
|
transport properties, particularly the diffusion constant, |
1063 |
|
\label{eq:diff} |
1064 |
|
\end{equation} |
1065 |
|
which measures long-time behavior and is sensitive to the forces on |
1066 |
< |
the multipoles. For the soft dipolar fluid, and the SSDQ liquid |
1066 |
> |
the multipoles. For the soft dipolar fluid and the SSDQ liquid |
1067 |
|
systems, the self-diffusion constants (D) were calculated from linear |
1068 |
< |
fits to the long-time portion of the mean square displacement |
1069 |
< |
($\langle r^{2}(t) \rangle$).\cite{Allen87} |
1068 |
> |
fits to the long-time portion of the mean square displacement, |
1069 |
> |
$\langle r^{2}(t) \rangle$.\cite{Allen87} |
1070 |
|
|
1071 |
|
In addition to translational diffusion, orientational relaxation times |
1072 |
|
were calculated for comparisons with the Ewald simulations and with |
1073 |
< |
experiments. These values were determined from the same 1~ns $NVE$ |
1074 |
< |
trajectories used for translational diffusion by calculating the |
1075 |
< |
orientational time correlation function, |
1073 |
> |
experiments. These values were determined from the same 1~ns |
1074 |
> |
microcanonical trajectories used for translational diffusion by |
1075 |
> |
calculating the orientational time correlation function, |
1076 |
|
\begin{equation} |
1077 |
< |
C_l^\alpha(t) = \left\langle P_l\left[\hat{\mathbf{u}}_i^\gamma(t) |
1078 |
< |
\cdot\hat{\mathbf{u}}_i^\gamma(0)\right]\right\rangle, |
1077 |
> |
C_l^\gamma(t) = \left\langle P_l\left[\hat{\mathbf{A}}_\gamma(t) |
1078 |
> |
\cdot\hat{\mathbf{A}}_\gamma(0)\right]\right\rangle, |
1079 |
|
\label{eq:OrientCorr} |
1080 |
|
\end{equation} |
1081 |
|
where $P_l$ is the Legendre polynomial of order $l$ and |
1082 |
< |
$\hat{\mathbf{u}}_i^\alpha$ is the unit vector of molecule $i$ along |
1083 |
< |
axis $\gamma$. The body-fixed reference frame used for our |
1084 |
< |
orientational correlation functions has the $z$-axis running along the |
1085 |
< |
dipoles, and for the SSDQ water model, the $y$-axis connects the two |
1086 |
< |
implied hydrogen atoms. |
1082 |
> |
$\hat{\mathbf{A}}_\gamma$ is the space-frame unit vector for body axis |
1083 |
> |
$\gamma$ on a molecule.. Th body-fixed reference frame used for our |
1084 |
> |
models has the $z$-axis running along the dipoles, and for the SSDQ |
1085 |
> |
water model, the $y$-axis connects the two implied hydrogen atom |
1086 |
> |
positions. From the orientation autocorrelation functions, we can |
1087 |
> |
obtain time constants for rotational relaxation either by fitting an |
1088 |
> |
exponential function or by integrating the entire correlation |
1089 |
> |
function. In a good water model, these decay times would be |
1090 |
> |
comparable to water orientational relaxation times from nuclear |
1091 |
> |
magnetic resonance (NMR). The relaxation constant obtained from |
1092 |
> |
$C_2^y(t)$ is normally of experimental interest because it describes |
1093 |
> |
the relaxation of the principle axis connecting the hydrogen |
1094 |
> |
atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular portion |
1095 |
> |
of the dipole-dipole relaxation from a proton NMR signal and should |
1096 |
> |
provide an estimate of the NMR relaxation time constant.\cite{Impey82} |
1097 |
|
|
1085 |
– |
From the orientation autocorrelation functions, we can obtain time |
1086 |
– |
constants for rotational relaxation either by fitting an exponential |
1087 |
– |
function or by integrating the entire correlation function. These |
1088 |
– |
decay times are directly comparable to water orientational relaxation |
1089 |
– |
times from nuclear magnetic resonance (NMR). The relaxation constant |
1090 |
– |
obtained from $C_2^y(t)$ is normally of experimental interest because |
1091 |
– |
it describes the relaxation of the principle axis connecting the |
1092 |
– |
hydrogen atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular |
1093 |
– |
portion of the dipole-dipole relaxation from a proton NMR signal and |
1094 |
– |
should provide an estimate of the NMR relaxation time |
1095 |
– |
constant.\cite{Impey82} |
1096 |
– |
|
1098 |
|
Results for the diffusion constants and orientational relaxation times |
1099 |
< |
are shown in figure \ref{fig:dynamics}. From this data, it is apparent |
1099 |
> |
are shown in figure \ref{tab:Props}. From this data, it is apparent |
1100 |
|
that the values for both $D$ and $\tau_2$ using the Ewald sum are |
1101 |
< |
reproduced with high fidelity by the GSF method. |
1101 |
> |
reproduced with reasonable fidelity by the GSF method. |
1102 |
|
|
1103 |
< |
The $\tau_2$ results in \ref{fig:dynamics} show a much greater |
1104 |
< |
difference between the real-space and the Ewald results. |
1103 |
> |
The $\tau_2$ results in \ref{tab:Props} show a much greater difference |
1104 |
> |
between the real-space and the Ewald results. |
1105 |
|
|
1106 |
+ |
\begin{table} |
1107 |
+ |
\label{tab:Props} |
1108 |
+ |
\caption{Comparison of the structural and dynamic properties for the |
1109 |
+ |
soft dipolar liquid test for all of the real-space methods.} |
1110 |
+ |
\begin{tabular}{l|c|cccc|cccc|cccc} |
1111 |
+ |
& Ewald & \multicolumn{4}{c|}{SP} & \multicolumn{4}{c|}{GSF} & \multicolumn{4}{c|}{TSF} \\ |
1112 |
+ |
$\alpha$ (\AA$^{-1}$) & & |
1113 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
1114 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
1115 |
+ |
0.0 & 0.1 & 0.2 & 0.3 \\ \cline{2-6}\cline{6-10}\cline{10-14} |
1116 |
|
|
1117 |
+ |
$\langle U_\mathrm{elect} \rangle /N$ &&&&&&&&&&&&&\\ |
1118 |
+ |
D ($10^{-4}~\mathrm{cm}^2/\mathrm{s}$)& |
1119 |
+ |
470.2(6) & |
1120 |
+ |
416.6(5) & |
1121 |
+ |
379.6(5) & |
1122 |
+ |
438.6(5) & |
1123 |
+ |
476.0(6) & |
1124 |
+ |
412.8(5) & |
1125 |
+ |
421.1(5) & |
1126 |
+ |
400.5(5) & |
1127 |
+ |
437.5(6) & |
1128 |
+ |
434.6(5) & |
1129 |
+ |
411.4(5) & |
1130 |
+ |
545.3(7) & |
1131 |
+ |
459.6(6) \\ |
1132 |
+ |
$\tau_2$ (fs) & |
1133 |
+ |
1.136 & |
1134 |
+ |
1.041 & |
1135 |
+ |
1.064 & |
1136 |
+ |
1.109 & |
1137 |
+ |
1.211 & |
1138 |
+ |
1.119 & |
1139 |
+ |
1.039 & |
1140 |
+ |
1.058 & |
1141 |
+ |
1.21 & |
1142 |
+ |
1.15 & |
1143 |
+ |
1.172 & |
1144 |
+ |
1.153 & |
1145 |
+ |
1.125 \\ |
1146 |
+ |
\end{tabular} |
1147 |
+ |
\end{table} |
1148 |
+ |
|
1149 |
+ |
|
1150 |
|
\section{CONCLUSION} |
1151 |
|
In the first paper in this series, we generalized the |
1152 |
|
charge-neutralized electrostatic energy originally developed by Wolf |