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\usepackage{graphicx}% Include figure files |
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\usepackage{dcolumn}% Align table columns on decimal point |
| 33 |
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%\usepackage{bm}% bold math |
| 34 |
+ |
\usepackage{times} |
| 35 |
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\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
| 36 |
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\usepackage{url} |
| 37 |
|
|
| 83 |
|
\maketitle |
| 84 |
|
|
| 85 |
|
\section{Introduction} |
| 86 |
< |
|
| 87 |
< |
There is increasing interest in real-space methods for calculating |
| 88 |
< |
electrostatic interactions in computer simulations of condensed |
| 88 |
< |
molecular |
| 86 |
> |
There has been increasing interest in real-space methods for |
| 87 |
> |
calculating electrostatic interactions in computer simulations of |
| 88 |
> |
condensed molecular |
| 89 |
|
systems.\cite{Wolf99,Zahn02,Kast03,BeckD.A.C._bi0486381,Ma05,Fennell:2006zl,Chen:2004du,Chen:2006ii,Rodgers:2006nw,Denesyuk:2008ez,Izvekov:2008wo} |
| 90 |
|
The simplest of these techniques was developed by Wolf {\it et al.} |
| 91 |
|
in their work towards an $\mathcal{O}(N)$ Coulombic sum.\cite{Wolf99} |
| 92 |
< |
Fennell and Gezelter showed that a simple damped shifted force (DSF) |
| 93 |
< |
modification to Wolf's method could give nearly quantitative agreement |
| 94 |
< |
with smooth particle mesh Ewald (SPME)\cite{Essmann95} for quantities |
| 95 |
< |
of interest in Monte Carlo (i.e., configurational energy differences) |
| 96 |
< |
and Molecular Dynamics (i.e., atomic force and molecular torque |
| 97 |
< |
vectors).\cite{Fennell:2006zl} |
| 92 |
> |
For systems of point charges, Fennell and Gezelter showed that a |
| 93 |
> |
simple damped shifted force (DSF) modification to Wolf's method could |
| 94 |
> |
give nearly quantitative agreement with smooth particle mesh Ewald |
| 95 |
> |
(SPME)\cite{Essmann95} configurational energy differences as well as |
| 96 |
> |
atomic force and molecular torque vectors.\cite{Fennell:2006zl} |
| 97 |
|
|
| 98 |
|
The computational efficiency and the accuracy of the DSF method are |
| 99 |
|
surprisingly good, particularly for systems with uniform charge |
| 111 |
|
compromise between the computational speed of real-space cutoffs and |
| 112 |
|
the accuracy of fully-periodic Ewald treatments. |
| 113 |
|
|
| 114 |
+ |
\subsection{Coarse Graining using Point Multipoles} |
| 115 |
|
One common feature of many coarse-graining approaches, which treat |
| 116 |
|
entire molecular subsystems as a single rigid body, is simplification |
| 117 |
|
of the electrostatic interactions between these bodies so that fewer |
| 154 |
|
a different orientation can cause energy discontinuities. |
| 155 |
|
|
| 156 |
|
This paper outlines an extension of the original DSF electrostatic |
| 157 |
< |
kernel to point multipoles. We present in this paper two distinct |
| 158 |
< |
real-space interaction models for higher-order multipoles based on two |
| 159 |
< |
truncated Taylor expansions that are carried out at the cutoff radius. |
| 160 |
< |
We are calling these models {\bf Taylor-shifted} and {\bf |
| 161 |
< |
Gradient-shifted} electrostatics. Because of differences in the |
| 162 |
< |
initial assumptions, the two methods yield related, but different |
| 163 |
< |
expressions for energies, forces, and torques. |
| 157 |
> |
kernel to point multipoles. We have developed two distinct real-space |
| 158 |
> |
interaction models for higher-order multipoles based on two truncated |
| 159 |
> |
Taylor expansions that are carried out at the cutoff radius. We are |
| 160 |
> |
calling these models {\bf Taylor-shifted} and {\bf Gradient-shifted} |
| 161 |
> |
electrostatics. Because of differences in the initial assumptions, |
| 162 |
> |
the two methods yield related, but different expressions for energies, |
| 163 |
> |
forces, and torques. |
| 164 |
|
|
| 165 |
+ |
In this paper we outline the new methodology and give functional forms |
| 166 |
+ |
for the energies, forces, and torques up to quadrupole-quadrupole |
| 167 |
+ |
order. We also compare the new methods to analytic energy constants |
| 168 |
+ |
for periodic arrays of point multipoles. In the following paper, we |
| 169 |
+ |
provide extensive numerical comparisons to Ewald-based electrostatics |
| 170 |
+ |
in common simulation enviornments. |
| 171 |
+ |
|
| 172 |
|
\section{Methodology} |
| 173 |
|
|
| 174 |
|
\subsection{Self-neutralization, damping, and force-shifting} |
| 168 |
– |
|
| 175 |
|
The DSF and Wolf methods operate by neutralizing the total charge |
| 176 |
|
contained within the cutoff sphere surrounding each particle. This is |
| 177 |
|
accomplished by shifting the potential functions to generate image |
| 178 |
|
charges on the surface of the cutoff sphere for each pair interaction |
| 179 |
< |
computed within $R_\textrm{c}$. An Ewald-like complementary error |
| 180 |
< |
function damping is applied to the potential to accelerate |
| 181 |
< |
convergence. The potential for the DSF method, where $\alpha$ is the |
| 182 |
< |
adjustable damping parameter, is given by |
| 179 |
> |
computed within $R_\textrm{c}$. Damping using a complementary error |
| 180 |
> |
function is applied to the potential to accelerate convergence. The |
| 181 |
> |
potential for the DSF method, where $\alpha$ is the adjustable damping |
| 182 |
> |
parameter, is given by |
| 183 |
|
\begin{equation*} |
| 184 |
|
V_\mathrm{DSF}(r) = C_a C_b \Biggr{[} |
| 185 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 201 |
|
utilization of the complimentary error function for electrostatic |
| 202 |
|
damping.\cite{deLeeuw80,Wolf99} |
| 203 |
|
|
| 204 |
+ |
There have been recent efforts to extend the Wolf self-neutralization |
| 205 |
+ |
method to zero out the dipole and higher order multipoles contained |
| 206 |
+ |
within the cutoff |
| 207 |
+ |
sphere.\cite{Fukuda:2011jk,Fukuda:2012yu,Fukuda:2013qv} |
| 208 |
+ |
|
| 209 |
+ |
In this work, we will be extending the idea of self-neutralization for |
| 210 |
+ |
the point multipoles in a similar way. In Figure |
| 211 |
+ |
\ref{fig:shiftedMultipoles}, the central dipolar site $\mathbf{D}_i$ |
| 212 |
+ |
is interacting with point dipole $\mathbf{D}_j$ and point quadrupole, |
| 213 |
+ |
$\mathbf{Q}_k$. The self-neutralization scheme for point multipoles |
| 214 |
+ |
involves projecting opposing multipoles for sites $j$ and $k$ on the |
| 215 |
+ |
surface of the cutoff sphere. |
| 216 |
+ |
|
| 217 |
|
\begin{figure} |
| 218 |
< |
\includegraphics[width=\linewidth]{SM} |
| 218 |
> |
\includegraphics[width=3in]{SM} |
| 219 |
|
\caption{Reversed multipoles are projected onto the surface of the |
| 220 |
|
cutoff sphere. The forces, torques, and potential are then smoothly |
| 221 |
|
shifted to zero as the sites leave the cutoff region.} |
| 222 |
|
\label{fig:shiftedMultipoles} |
| 223 |
|
\end{figure} |
| 224 |
|
|
| 225 |
< |
\subsection{Multipole Expansion} |
| 225 |
> |
As in the point-charge approach, there is a contribution from |
| 226 |
> |
self-neutralization of site $i$. The self term for multipoles is |
| 227 |
> |
described in section \ref{sec:selfTerm}. |
| 228 |
|
|
| 229 |
+ |
\subsection{The multipole expansion} |
| 230 |
+ |
|
| 231 |
|
Consider two discrete rigid collections of point charges, denoted as |
| 232 |
< |
objects $\bf a$ and $\bf b$. In the following, we assume that the two |
| 233 |
< |
objects only interact via electrostatics and describe those |
| 234 |
< |
interactions in terms of a multipole expansion. Putting the origin of |
| 235 |
< |
the coordinate system at the center of mass of $\bf a$, we use vectors |
| 232 |
> |
$\bf a$ and $\bf b$. In the following, we assume that the two objects |
| 233 |
> |
interact via electrostatics only and describe those interactions in |
| 234 |
> |
terms of a standard multipole expansion. Putting the origin of the |
| 235 |
> |
coordinate system at the center of mass of $\bf a$, we use vectors |
| 236 |
|
$\mathbf{r}_k$ to denote the positions of all charges $q_k$ in $\bf |
| 237 |
|
a$. Then the electrostatic potential of object $\bf a$ at |
| 238 |
|
$\mathbf{r}$ is given by |
| 240 |
|
V_a(\mathbf r) = \frac{1}{4\pi\epsilon_0} |
| 241 |
|
\sum_{k \, \text{in \bf a}} \frac{q_k}{\lvert \mathbf{r} - \mathbf{r}_k \rvert}. |
| 242 |
|
\end{equation} |
| 243 |
< |
We write the Taylor series expansion in $r$ using an implied summation |
| 244 |
< |
notation, Greek indices are used to indicate space coordinates ($x$, |
| 245 |
< |
$y$, $z$) and the subscripts $k$ and $j$ are reserved for labelling |
| 246 |
< |
specific charges in $\bf a$ and $\bf b$ respectively, and find: |
| 243 |
> |
The Taylor expansion in $r$ can be written using an implied summation |
| 244 |
> |
notation. Here Greek indices are used to indicate space coordinates |
| 245 |
> |
($x$, $y$, $z$) and the subscripts $k$ and $j$ are reserved for |
| 246 |
> |
labelling specific charges in $\bf a$ and $\bf b$ respectively. The |
| 247 |
> |
Taylor expansion, |
| 248 |
|
\begin{equation} |
| 249 |
|
\frac{1}{\lvert \mathbf{r} - \mathbf{r}_k \rvert} = |
| 250 |
|
\left( 1 |
| 251 |
|
- r_{k\alpha} \frac{\partial}{\partial r_{\alpha}} |
| 252 |
|
+ \frac{1}{2} r_{k\alpha} r_{k\beta} \frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} +\dots |
| 253 |
|
\right) |
| 254 |
< |
\frac{1}{r} . |
| 254 |
> |
\frac{1}{r} , |
| 255 |
|
\end{equation} |
| 256 |
< |
The electrostatic potential on $\bf a$ can then be expressed in terms |
| 257 |
< |
of multipole operators, |
| 256 |
> |
can then be used to express the electrostatic potential on $\bf a$ in |
| 257 |
> |
terms of multipole operators, |
| 258 |
|
\begin{equation} |
| 259 |
|
V_{\bf a}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\hat{M}_{\bf a} \frac{1}{r} |
| 260 |
|
\end{equation} |
| 266 |
|
\end{equation} |
| 267 |
|
Here, the point charge, dipole, and quadrupole for object $\bf a$ are |
| 268 |
|
given by $C_{\bf a}$, $D_{{\bf a}\alpha}$, and $Q_{{\bf |
| 269 |
< |
a}\alpha\beta}$, respectively. These can be tied to distribution |
| 270 |
< |
of charges |
| 271 |
< |
\begin{equation} |
| 272 |
< |
C_{\bf a}=\sum_{k \, \text{in \bf a}} q_k , |
| 273 |
< |
\end{equation} |
| 274 |
< |
\begin{equation} |
| 275 |
< |
D_{{\bf a}\alpha} = \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} , |
| 276 |
< |
\end{equation} |
| 277 |
< |
\begin{equation} |
| 278 |
< |
Q_{{\bf a}\alpha\beta} = \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
| 255 |
< |
\end{equation} |
| 269 |
> |
a}\alpha\beta}$, respectively. These are the primitive multipoles |
| 270 |
> |
which can be expressed as a distribution of charges, |
| 271 |
> |
\begin{align} |
| 272 |
> |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \\ |
| 273 |
> |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha} ,\\ |
| 274 |
> |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
| 275 |
> |
\end{align} |
| 276 |
> |
Note that the definition of the primitive quadrupole here differs from |
| 277 |
> |
the standard traceless form, and contains an additional Taylor-series |
| 278 |
> |
based factor of $1/2$. |
| 279 |
|
|
| 280 |
|
It is convenient to locate charges $q_j$ relative to the center of mass of $\bf b$. Then with $\bf{r}$ pointing from |
| 281 |
|
$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
| 259 |
– |
\begin{eqnarray} |
| 260 |
– |
U_{\bf{ab}}(r) =&& \frac{1}{4\pi \epsilon_0} |
| 261 |
– |
\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
| 262 |
– |
\frac{q_k q_j}{\vert \bf{r}_k - (\bf{r}+\bf{r}_j) \vert} \nonumber\\ |
| 263 |
– |
=&& \frac{1}{4\pi \epsilon_0} |
| 264 |
– |
\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
| 265 |
– |
\frac{q_k q_j}{\vert \bf{r}+ (\bf{r}_j-\bf{r}_k) \vert} \nonumber\\ |
| 266 |
– |
=&&\frac{1}{4\pi \epsilon_0} \sum_{j \, \text{in \bf b}} q_j V_a(\bf{r}+\bf{r}_j) \nonumber\\ |
| 267 |
– |
=&&\frac{1}{4\pi \epsilon_0} \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
| 268 |
– |
\end{eqnarray} |
| 269 |
– |
The last expression can also be expanded as a Taylor series in $r$. Using a notation similar to before, we define |
| 282 |
|
\begin{equation} |
| 283 |
+ |
U_{\bf{ab}}(r) |
| 284 |
+ |
= \frac{1}{4\pi \epsilon_0} \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
| 285 |
+ |
\end{equation} |
| 286 |
+ |
This can also be expanded as a Taylor series in $r$. Using a notation |
| 287 |
+ |
similar to before to define the multipoles on object {\bf b}, |
| 288 |
+ |
\begin{equation} |
| 289 |
|
\hat{M}_{\bf b} = C_{\bf b} + D_{{\bf b}\alpha} \frac{\partial}{\partial r_{\alpha}} |
| 290 |
|
+ Q_{{\bf b}\alpha\beta} |
| 291 |
|
\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
| 292 |
|
\end{equation} |
| 293 |
< |
and |
| 293 |
> |
we arrive at the multipole expression for the total interaction energy. |
| 294 |
|
\begin{equation} |
| 295 |
|
U_{\bf{ab}}(r)=\frac{\hat{M}_{\bf a} \hat{M}_{\bf b}}{4\pi \epsilon_0} \frac{1}{r} \label{kernel}. |
| 296 |
|
\end{equation} |
| 297 |
< |
Note the ease of separting out the respective energies of interaction of the charge, dipole, and quadrupole of $\bf a$ from those of $\bf b$. |
| 297 |
> |
This form has the benefit of separating out the energies of |
| 298 |
> |
interaction into contributions from the charge, dipole, and quadrupole |
| 299 |
> |
of $\bf a$ interacting with the same multipoles $\bf b$. |
| 300 |
|
|
| 301 |
< |
\subsection{Bare Coulomb versus smeared charge} |
| 302 |
< |
|
| 303 |
< |
With the four types of methods being considered here, it is desirable to list the approximations in as transparent a form |
| 304 |
< |
as possible. First, one may use the bare Coulomb potential, with radial dependence $1/r$, |
| 305 |
< |
as shown in Eq.~(\ref{kernel}). Alternatively, one may use |
| 306 |
< |
a smeared charge distribution, then the``kernel'' $1/r$ of the expansion is replaced with a function: |
| 301 |
> |
\subsection{Damped Coulomb interactions} |
| 302 |
> |
In the standard multipole expansion, one typically uses the bare |
| 303 |
> |
Coulomb potential, with radial dependence $1/r$, as shown in |
| 304 |
> |
Eq.~(\ref{kernel}). It is also quite common to use a damped Coulomb |
| 305 |
> |
interaction, which results from replacing point charges with Gaussian |
| 306 |
> |
distributions of charge with width $\alpha$. In damped multipole |
| 307 |
> |
electrostatics, the kernel ($1/r$) of the expansion is replaced with |
| 308 |
> |
the function: |
| 309 |
|
\begin{equation} |
| 310 |
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r} |
| 311 |
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
| 312 |
|
\end{equation} |
| 313 |
< |
We develop equations using a function $f(r)$ to represent either $1/r$ or $B_0(r)$, dependent on the the type of approach being considered. |
| 314 |
< |
Smith's convenient functions $B_l(r)$ are summarized in Appendix A. |
| 313 |
> |
We develop equations below using the function $f(r)$ to represent |
| 314 |
> |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied |
| 315 |
> |
either to bare or damped Coulomb kernels as long as derivatives of |
| 316 |
> |
these functions are known. Smith's convenient functions $B_l(r)$ are |
| 317 |
> |
summarized in Appendix A. |
| 318 |
|
|
| 319 |
< |
\subsection{Shifting the force, method 1} |
| 319 |
> |
\subsection{Taylor-shifted force (TSF) electrostatics} |
| 320 |
|
|
| 321 |
< |
As discussed in the introduction, it is desirable to cutoff the electrostatic energy at a radius |
| 322 |
< |
$r_c$. For consistency in approximation, we want the interaction energy as well as the force and |
| 323 |
< |
torque to go to zero at $r=r_c$. |
| 324 |
< |
To describe how this goal may be met using a radial approximation, we use two examples, charge-charge |
| 325 |
< |
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain the idea. |
| 321 |
> |
The main goal of this work is to smoothly cut off the interaction |
| 322 |
> |
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 323 |
> |
describe how this goal may be met, we use two examples, charge-charge |
| 324 |
> |
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain |
| 325 |
> |
the idea. |
| 326 |
|
|
| 327 |
< |
In the shifted-force approximation, the interaction energy $U_{\bf{ab}}(r_c)=0$ |
| 328 |
< |
for two charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is written: |
| 327 |
> |
In the shifted-force approximation, the interaction energy for two |
| 328 |
> |
charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is |
| 329 |
> |
written: |
| 330 |
|
\begin{equation} |
| 331 |
|
U_{C_{\bf a}C_{\bf b}}(r)=\frac{1}{4\pi \epsilon_0} C_{\bf a} C_{\bf b} |
| 332 |
|
\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
| 333 |
|
\right) . |
| 334 |
|
\end{equation} |
| 335 |
< |
Two shifting terms appear in this equations because we want the force to |
| 336 |
< |
also be shifted due to an image charge located at a distance $r_c$. |
| 337 |
< |
Since one derivative of the interaction energy is needed for the force, we want a term |
| 338 |
< |
linear in $(r-r_c)$ in the interaction energy, that is: |
| 335 |
> |
Two shifting terms appear in this equations, one from the |
| 336 |
> |
neutralization procedure ($-1/r_c$), and one that will make the first |
| 337 |
> |
derivative also vanish at the cutoff radius. |
| 338 |
> |
|
| 339 |
> |
Since one derivative of the interaction energy is needed for the |
| 340 |
> |
force, the minimal perturbation is a term linear in $(r-r_c)$ in the |
| 341 |
> |
interaction energy, that is: |
| 342 |
|
\begin{equation} |
| 343 |
|
\frac{d\,}{dr} |
| 344 |
|
\left( {\frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
| 345 |
|
\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
| 346 |
|
\right) . |
| 347 |
|
\end{equation} |
| 348 |
< |
This demonstrates the need of the third term in the brackets of the energy expression, but leads to the question, how does this idea generalize for higher-order multipole energies? |
| 348 |
> |
There are a number of ways to generalize this derivative shift for |
| 349 |
> |
higher-order multipoles. |
| 350 |
|
|
| 351 |
< |
In method 1, the procedure that we follow is based on the number of derivatives need for each energy, force, or torque. That is, |
| 352 |
< |
a quadrupole-quadrupole interaction energy will have four derivatives, |
| 353 |
< |
$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, |
| 354 |
< |
and the force or torque will bring in yet another derivative. |
| 355 |
< |
We thus want shifted energy expressions to include terms so that all energies, forces, and torques |
| 356 |
< |
are zero at $r=r_c$. In each case, we will subtract off a function $f_n^{\text{shift}}(r)$ from the |
| 357 |
< |
kernel $f(r)=1/r$. The index $n$ indicates the number of derivatives to be taken when |
| 358 |
< |
deriving a given multipole energy. |
| 359 |
< |
We choose a function with guaranteed smooth derivatives --- a truncated Taylor series of the function |
| 360 |
< |
$f(r)$, e.g., |
| 351 |
> |
In the Taylor-shifted force (TSF) method, the procedure that we follow |
| 352 |
> |
is based on a Taylor expansion containing the same number of |
| 353 |
> |
derivatives required for each force term to vanish at the cutoff. For |
| 354 |
> |
example, the quadrupole-quadrupole interaction energy requires four |
| 355 |
> |
derivatives of the kernel, and the force requires one additional |
| 356 |
> |
derivative. We therefore require shifted energy expressions that |
| 357 |
> |
include enough terms so that all energies, forces, and torques are |
| 358 |
> |
zero as $r \rightarrow r_c$. In each case, we will subtract off a |
| 359 |
> |
function $f_n^{\text{shift}}(r)$ from the kernel $f(r)=1/r$. The |
| 360 |
> |
index $n$ indicates the number of derivatives to be taken when |
| 361 |
> |
deriving a given multipole energy. We choose a function with |
| 362 |
> |
guaranteed smooth derivatives --- a truncated Taylor series of the |
| 363 |
> |
function $f(r)$, e.g., |
| 364 |
|
% |
| 365 |
|
\begin{equation} |
| 366 |
|
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
| 373 |
|
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} . |
| 374 |
|
\end{equation} |
| 375 |
|
% |
| 376 |
< |
Continuing with the example of a charge $\bf a$ interacting with a dipole $\bf b$, we write |
| 376 |
> |
Continuing with the example of a charge $\bf a$ interacting with a |
| 377 |
> |
dipole $\bf b$, we write |
| 378 |
|
% |
| 379 |
|
\begin{equation} |
| 380 |
|
U_{C_{\bf a}D_{\bf b}}(r)= |
| 425 |
|
of the many derivatives that are taken, the algebra is tedious and are summarized |
| 426 |
|
in Appendices A and B. |
| 427 |
|
|
| 428 |
< |
\subsection{Shifting the force, method 2} |
| 428 |
> |
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 429 |
|
|
| 430 |
|
Note the method used in the previous subsection to shift a force is basically that of using |
| 431 |
|
a truncated Taylor Series in the radius $r$. An alternate method exists, best explained by |
| 528 |
|
be discussed below. |
| 529 |
|
|
| 530 |
|
|
| 531 |
< |
\section{The Self-Interaction} |
| 531 |
> |
\subsection{The Self-Interaction \label{sec:selfTerm}} |
| 532 |
> |
|
| 533 |
|
The Wolf summation~\cite{Wolf99} and the later damped shifted force |
| 534 |
|
(DSF) extension~\cite{Fennell06} included self-interactions that are |
| 535 |
|
handled separately from the pairwise interactions between sites. The |
