| 69 |
|
\begin{abstract} |
| 70 |
|
We have tested the real-space shifted potential (SP), |
| 71 |
|
gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods |
| 72 |
< |
for multipoles that were developed in the first paper in this series |
| 73 |
< |
against a reference method. The tests were carried out in a variety |
| 74 |
< |
of condensed-phase environments which were designed to test all |
| 75 |
< |
levels of the multipole-multipole interactions. Comparisons of the |
| 76 |
< |
energy differences between configurations, molecular forces, and |
| 77 |
< |
torques were used to analyze how well the real-space models perform |
| 78 |
< |
relative to the more computationally expensive Ewald sum. We have |
| 79 |
< |
also investigated the energy conservation properties of the new |
| 80 |
< |
methods in molecular dynamics simulations using all of these |
| 81 |
< |
methods. The SP method shows excellent agreement with |
| 82 |
< |
configurational energy differences, forces, and torques, and would |
| 83 |
< |
be suitable for use in Monte Carlo calculations. Of the two new |
| 84 |
< |
shifted-force methods, the GSF approach shows the best agreement |
| 85 |
< |
with Ewald-derived energies, forces, and torques and exhibits energy |
| 86 |
< |
conservation properties that make it an excellent choice for |
| 87 |
< |
efficiently computing electrostatic interactions in molecular |
| 88 |
< |
dynamics simulations. |
| 72 |
> |
for multipole interactions that were developed in the first paper in |
| 73 |
> |
this series, using the multipolar Ewald sum as a reference |
| 74 |
> |
method. The tests were carried out in a variety of condensed-phase |
| 75 |
> |
environments which were designed to test all levels of the |
| 76 |
> |
multipole-multipole interactions. Comparisons of the energy |
| 77 |
> |
differences between configurations, molecular forces, and torques |
| 78 |
> |
were used to analyze how well the real-space models perform relative |
| 79 |
> |
to the more computationally expensive Ewald treatment. We have also |
| 80 |
> |
investigated the energy conservation properties of the new methods |
| 81 |
> |
in molecular dynamics simulations. The SP method shows excellent |
| 82 |
> |
agreement with configurational energy differences, forces, and |
| 83 |
> |
torques, and would be suitable for use in Monte Carlo calculations. |
| 84 |
> |
Of the two new shifted-force methods, the GSF approach shows the |
| 85 |
> |
best agreement with Ewald-derived energies, forces, and torques and |
| 86 |
> |
exhibits energy conservation properties that make it an excellent |
| 87 |
> |
choice for efficient computation of electrostatic interactions in |
| 88 |
> |
molecular dynamics simulations. |
| 89 |
|
\end{abstract} |
| 90 |
|
|
| 91 |
|
%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
| 92 |
|
% Classification Scheme. |
| 93 |
< |
\keywords{Electrostatics, Multipoles, Real-space} |
| 93 |
> |
%\keywords{Electrostatics, Multipoles, Real-space} |
| 94 |
|
|
| 95 |
|
\maketitle |
| 96 |
|
|
| 97 |
|
|
| 98 |
|
\section{\label{sec:intro}Introduction} |
| 99 |
|
Computing the interactions between electrostatic sites is one of the |
| 100 |
< |
most expensive aspects of molecular simulations, which is why there |
| 101 |
< |
have been significant efforts to develop practical, efficient and |
| 102 |
< |
convergent methods for handling these interactions. Ewald's method is |
| 103 |
< |
perhaps the best known and most accurate method for evaluating |
| 104 |
< |
energies, forces, and torques in explicitly-periodic simulation |
| 105 |
< |
cells. In this approach, the conditionally convergent electrostatic |
| 106 |
< |
energy is converted into two absolutely convergent contributions, one |
| 107 |
< |
which is carried out in real space with a cutoff radius, and one in |
| 108 |
< |
reciprocal space.\cite{Clarke:1986eu,Woodcock75} |
| 100 |
> |
most expensive aspects of molecular simulations. There have been |
| 101 |
> |
significant efforts to develop practical, efficient and convergent |
| 102 |
> |
methods for handling these interactions. Ewald's method is perhaps the |
| 103 |
> |
best known and most accurate method for evaluating energies, forces, |
| 104 |
> |
and torques in explicitly-periodic simulation cells. In this approach, |
| 105 |
> |
the conditionally convergent electrostatic energy is converted into |
| 106 |
> |
two absolutely convergent contributions, one which is carried out in |
| 107 |
> |
real space with a cutoff radius, and one in reciprocal |
| 108 |
> |
space. BETTER CITATIONS\cite{Clarke:1986eu,Woodcock75} |
| 109 |
|
|
| 110 |
|
When carried out as originally formulated, the reciprocal-space |
| 111 |
|
portion of the Ewald sum exhibits relatively poor computational |
| 116 |
|
the computational cost from $O(N^2)$ down to $O(N \log |
| 117 |
|
N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}. |
| 118 |
|
|
| 119 |
< |
Because of the artificial periodicity required for the Ewald sum, the |
| 120 |
< |
method may require modification to compute interactions for |
| 119 |
> |
Because of the artificial periodicity required for the Ewald sum, |
| 120 |
|
interfacial molecular systems such as membranes and liquid-vapor |
| 121 |
< |
interfaces.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
| 122 |
< |
To simulate interfacial systems, Parry's extension of the 3D Ewald sum |
| 123 |
< |
is appropriate for slab geometries.\cite{Parry:1975if} The inherent |
| 124 |
< |
periodicity in the Ewald’s method can also be problematic for |
| 125 |
< |
interfacial molecular systems.\cite{Fennell:2006lq} Modified Ewald |
| 126 |
< |
methods that were developed to handle two-dimensional (2D) |
| 127 |
< |
electrostatic interactions in interfacial systems have not had similar |
| 128 |
< |
particle-mesh treatments.\cite{Parry:1975if, Parry:1976fq, Clarke77, |
| 129 |
< |
Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} |
| 121 |
> |
interfaces require modifications to the |
| 122 |
> |
method.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
| 123 |
> |
Parry's extension of the three dimensional Ewald sum is appropriate |
| 124 |
> |
for slab geometries.\cite{Parry:1975if} Modified Ewald methods that |
| 125 |
> |
were developed to handle two-dimensional (2D) electrostatic |
| 126 |
> |
interactions in interfacial systems have not seen similar |
| 127 |
> |
particle-mesh treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77, |
| 128 |
> |
Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly |
| 129 |
> |
with system size. The inherent periodicity in the Ewald’s method can |
| 130 |
> |
also be problematic for interfacial molecular |
| 131 |
> |
systems.\cite{Fennell:2006lq} |
| 132 |
|
|
| 133 |
|
\subsection{Real-space methods} |
| 134 |
|
Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
| 135 |
|
method for calculating electrostatic interactions between point |
| 136 |
< |
charges. They argued that the effective Coulomb interaction in |
| 137 |
< |
condensed systems is actually short ranged.\cite{Wolf92,Wolf95} For an |
| 138 |
< |
ordered lattice (e.g., when computing the Madelung constant of an |
| 139 |
< |
ionic solid), the material can be considered as a set of ions |
| 140 |
< |
interacting with neutral dipolar or quadrupolar ``molecules'' giving |
| 141 |
< |
an effective distance dependence for the electrostatic interactions of |
| 142 |
< |
$r^{-5}$ (see figure \ref{fig:schematic}). For this reason, careful |
| 143 |
< |
applications of Wolf's method are able to obtain accurate estimates of |
| 144 |
< |
Madelung constants using relatively short cutoff radii. Recently, |
| 145 |
< |
Fukuda used neutralization of the higher order moments for the |
| 146 |
< |
calculation of the electrostatic interaction of the point charges |
| 147 |
< |
system.\cite{Fukuda:2013sf} |
| 136 |
> |
charges. They argued that the effective Coulomb interaction in most |
| 137 |
> |
condensed phase systems is effectively short |
| 138 |
> |
ranged.\cite{Wolf92,Wolf95} For an ordered lattice (e.g., when |
| 139 |
> |
computing the Madelung constant of an ionic solid), the material can |
| 140 |
> |
be considered as a set of ions interacting with neutral dipolar or |
| 141 |
> |
quadrupolar ``molecules'' giving an effective distance dependence for |
| 142 |
> |
the electrostatic interactions of $r^{-5}$ (see figure |
| 143 |
> |
\ref{fig:schematic}). If one views the \ce{NaCl} crystal as a simple |
| 144 |
> |
cubic (SC) structure with an octupolar \ce{(NaCl)4} basis, the |
| 145 |
> |
electrostatic energy per ion converges more rapidly to the Madelung |
| 146 |
> |
energy than the dipolar approximation.\cite{Wolf92} To find the |
| 147 |
> |
correct Madelung constant, Lacman suggested that the NaCl structure |
| 148 |
> |
could be constructed in a way that the finite crystal terminates with |
| 149 |
> |
complete \ce{(NaCl)4} molecules.\cite{Lacman65} The central ion sees |
| 150 |
> |
what is effectively a set of octupoles at large distances. These facts |
| 151 |
> |
suggest that the Madelung constants are relatively short ranged for |
| 152 |
> |
perfect ionic crystals.\cite{Wolf:1999dn} For this reason, careful |
| 153 |
> |
application of Wolf's method are able to obtain accurate estimates of |
| 154 |
> |
Madelung constants using relatively short cutoff radii. |
| 155 |
|
|
| 156 |
+ |
Direct truncation of interactions at a cutoff radius creates numerical |
| 157 |
+ |
errors. Wolf \textit{et al.} argued that truncation errors are due |
| 158 |
+ |
to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To |
| 159 |
+ |
neutralize this charge they proposed placing an image charge on the |
| 160 |
+ |
surface of the cutoff sphere for every real charge inside the cutoff. |
| 161 |
+ |
These charges are present for the evaluation of both the pair |
| 162 |
+ |
interaction energy and the force, although the force expression |
| 163 |
+ |
maintained a discontinuity at the cutoff sphere. In the original Wolf |
| 164 |
+ |
formulation, the total energy for the charge and image were not equal |
| 165 |
+ |
to the integral of their force expression, and as a result, the total |
| 166 |
+ |
energy would not be conserved in molecular dynamics (MD) |
| 167 |
+ |
simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and |
| 168 |
+ |
Gezelter later proposed shifted force variants of the Wolf method with |
| 169 |
+ |
commensurate force and energy expressions that do not exhibit this |
| 170 |
+ |
problem.\cite{Fennell:2006lq} Related real-space methods were also |
| 171 |
+ |
proposed by Chen \textit{et |
| 172 |
+ |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
| 173 |
+ |
and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has used |
| 174 |
+ |
neutralization of the higher order moments for the calculation of the |
| 175 |
+ |
electrostatic interaction of the point charge |
| 176 |
+ |
systems.\cite{Fukuda:2013sf} |
| 177 |
+ |
|
| 178 |
|
\begin{figure} |
| 179 |
|
\centering |
| 180 |
|
\includegraphics[width=\linewidth]{schematic.pdf} |
| 181 |
|
\caption{Top: Ionic systems exhibit local clustering of dissimilar |
| 182 |
|
charges (in the smaller grey circle), so interactions are |
| 183 |
< |
effectively charge-multipole in order at longer distances. With |
| 184 |
< |
hard cutoffs, motion of individual charges in and out of the |
| 185 |
< |
cutoff sphere can break the effective multipolar ordering. |
| 186 |
< |
Bottom: dipolar crystals and fluids have a similar effective |
| 183 |
> |
effectively charge-multipole at longer distances. With hard |
| 184 |
> |
cutoffs, motion of individual charges in and out of the cutoff |
| 185 |
> |
sphere can break the effective multipolar ordering. Bottom: |
| 186 |
> |
dipolar crystals and fluids have a similar effective |
| 187 |
|
\textit{quadrupolar} ordering (in the smaller grey circles), and |
| 188 |
|
orientational averaging helps to reduce the effective range of the |
| 189 |
|
interactions in the fluid. Placement of reversed image multipoles |
| 192 |
|
\label{fig:schematic} |
| 193 |
|
\end{figure} |
| 194 |
|
|
| 195 |
< |
The direct truncation of interactions at a cutoff radius creates |
| 196 |
< |
truncation defects. Wolf \textit{et al.} further argued that |
| 197 |
< |
truncation errors are due to net charge remaining inside the cutoff |
| 198 |
< |
sphere.\cite{Wolf:1999dn} To neutralize this charge they proposed |
| 199 |
< |
placing an image charge on the surface of the cutoff sphere for every |
| 200 |
< |
real charge inside the cutoff. These charges are present for the |
| 201 |
< |
evaluation of both the pair interaction energy and the force, although |
| 202 |
< |
the force expression maintained a discontinuity at the cutoff sphere. |
| 203 |
< |
In the original Wolf formulation, the total energy for the charge and |
| 204 |
< |
image were not equal to the integral of their force expression, and as |
| 175 |
< |
a result, the total energy would not be conserved in molecular |
| 176 |
< |
dynamics (MD) simulations.\cite{Zahn:2002hc} Zahn \textit{et al.} and |
| 177 |
< |
Fennel and Gezelter later proposed shifted force variants of the Wolf |
| 178 |
< |
method with commensurate force and energy expressions that do not |
| 179 |
< |
exhibit this problem.\cite{Fennell:2006lq} Related real-space |
| 180 |
< |
methods were also proposed by Chen \textit{et |
| 181 |
< |
al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw} |
| 182 |
< |
and by Wu and Brooks.\cite{Wu:044107} |
| 183 |
< |
|
| 184 |
< |
Considering the interaction of one central ion in an ionic crystal |
| 185 |
< |
with a portion of the crystal at some distance, the effective Columbic |
| 186 |
< |
potential is found to decrease as $r^{-5}$. If one views the \ce{NaCl} |
| 187 |
< |
crystal as a simple cubic (SC) structure with an octupolar |
| 188 |
< |
\ce{(NaCl)4} basis, the electrostatic energy per ion converges more |
| 189 |
< |
rapidly to the Madelung energy than the dipolar |
| 190 |
< |
approximation.\cite{Wolf92} To find the correct Madelung constant, |
| 191 |
< |
Lacman suggested that the NaCl structure could be constructed in a way |
| 192 |
< |
that the finite crystal terminates with complete \ce{(NaCl)4} |
| 193 |
< |
molecules.\cite{Lacman65} The central ion sees what is effectively a |
| 194 |
< |
set of octupoles at large distances. These facts suggest that the |
| 195 |
< |
Madelung constants are relatively short ranged for perfect ionic |
| 196 |
< |
crystals.\cite{Wolf:1999dn} |
| 197 |
< |
|
| 198 |
< |
One can make a similar argument for crystals of point multipoles. The |
| 199 |
< |
Luttinger and Tisza treatment of energy constants for dipolar lattices |
| 200 |
< |
utilizes 24 basis vectors that contain dipoles at the eight corners of |
| 201 |
< |
a unit cube. Only three of these basis vectors, $X_1, Y_1, |
| 202 |
< |
\mathrm{~and~} Z_1,$ retain net dipole moments, while the rest have |
| 203 |
< |
zero net dipole and retain contributions only from higher order |
| 204 |
< |
multipoles. The effective interaction between a dipole at the center |
| 195 |
> |
One can make a similar effective range argument for crystals of point |
| 196 |
> |
\textit{multipoles}. The Luttinger and Tisza treatment of energy |
| 197 |
> |
constants for dipolar lattices utilizes 24 basis vectors that contain |
| 198 |
> |
dipoles at the eight corners of a unit cube. Only three of these |
| 199 |
> |
basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole |
| 200 |
> |
moments, while the rest have zero net dipole and retain contributions |
| 201 |
> |
only from higher order multipoles. The lowest energy crystalline |
| 202 |
> |
structures are built out of basis vectors that have only residual |
| 203 |
> |
quadrupolar moments (e.g. the $Z_5$ array). In these low energy |
| 204 |
> |
structures, the effective interaction between a dipole at the center |
| 205 |
|
of a crystal and a group of eight dipoles farther away is |
| 206 |
|
significantly shorter ranged than the $r^{-3}$ that one would expect |
| 207 |
|
for raw dipole-dipole interactions. Only in crystals which retain a |
| 258 |
|
densities.\cite{Shi:2013ij,Kannam:2012rr,Acevedo13,Space12,English08,Lawrence13,Vergne13} |
| 259 |
|
|
| 260 |
|
\subsection{The damping function} |
| 261 |
< |
The damping function used in our research has been discussed in detail |
| 262 |
< |
in the first paper of this series.\cite{PaperI} The radial kernel |
| 263 |
< |
$1/r$ for the interactions between point charges can be replaced by |
| 264 |
< |
the complementary error function $\mathrm{erfc}(\alpha r)/r$ to |
| 265 |
< |
accelerate the rate of convergence, where $\alpha$ is a damping |
| 266 |
< |
parameter with units of inverse distance. Altering the value of |
| 267 |
< |
$\alpha$ is equivalent to changing the width of Gaussian charge |
| 268 |
< |
distributions that replace each point charge -- Gaussian overlap |
| 269 |
< |
integrals yield complementary error functions when truncated at a |
| 270 |
< |
finite distance. |
| 261 |
> |
The damping function has been discussed in detail in the first paper |
| 262 |
> |
of this series.\cite{PaperI} The $1/r$ Coulombic kernel for the |
| 263 |
> |
interactions between point charges can be replaced by the |
| 264 |
> |
complementary error function $\mathrm{erfc}(\alpha r)/r$ to accelerate |
| 265 |
> |
convergence, where $\alpha$ is a damping parameter with units of |
| 266 |
> |
inverse distance. Altering the value of $\alpha$ is equivalent to |
| 267 |
> |
changing the width of Gaussian charge distributions that replace each |
| 268 |
> |
point charge, as Coulomb integrals with Gaussian charge distributions |
| 269 |
> |
produce complementary error functions when truncated at a finite |
| 270 |
> |
distance. |
| 271 |
|
|
| 272 |
< |
By using suitable value of damping alpha ($\alpha \sim 0.2$) for a |
| 273 |
< |
cutoff radius ($r_{c}=9 A$), Fennel and Gezelter produced very good |
| 274 |
< |
agreement with SPME for the interaction energies, forces and torques |
| 275 |
< |
for charge-charge interactions.\cite{Fennell:2006lq} |
| 272 |
> |
With moderate damping coefficients, $\alpha \sim 0.2$, the DSF method |
| 273 |
> |
produced very good agreement with SPME for interaction energies, |
| 274 |
> |
forces and torques for charge-charge |
| 275 |
> |
interactions.\cite{Fennell:2006lq} |
| 276 |
|
|
| 277 |
|
\subsection{Point multipoles in molecular modeling} |
| 278 |
|
Coarse-graining approaches which treat entire molecular subsystems as |
| 279 |
|
a single rigid body are now widely used. A common feature of many |
| 280 |
|
coarse-graining approaches is simplification of the electrostatic |
| 281 |
|
interactions between bodies so that fewer site-site interactions are |
| 282 |
< |
required to compute configurational energies. Many coarse-grained |
| 283 |
< |
molecular structures would normally consist of equal positive and |
| 284 |
< |
negative charges, and rather than use multiple site-site interactions, |
| 285 |
< |
the interaction between higher order multipoles can also be used to |
| 286 |
< |
evaluate a single molecule-molecule |
| 287 |
< |
interaction.\cite{Ren06,Essex10,Essex11} |
| 282 |
> |
required to compute configurational |
| 283 |
> |
energies.\cite{Ren06,Essex10,Essex11} |
| 284 |
|
|
| 285 |
|
Because electrons in a molecule are not localized at specific points, |
| 286 |
< |
the assignment of partial charges to atomic centers is a relatively |
| 287 |
< |
rough approximation. Atomic sites can also be assigned point |
| 288 |
< |
multipoles and polarizabilities to increase the accuracy of the |
| 289 |
< |
molecular model. Recently, water has been modeled with point |
| 290 |
< |
multipoles up to octupolar order using the soft sticky |
| 295 |
< |
dipole-quadrupole-octupole (SSDQO) |
| 286 |
> |
the assignment of partial charges to atomic centers is always an |
| 287 |
> |
approximation. Atomic sites can also be assigned point multipoles and |
| 288 |
> |
polarizabilities to increase the accuracy of the molecular model. |
| 289 |
> |
Recently, water has been modeled with point multipoles up to octupolar |
| 290 |
> |
order using the soft sticky dipole-quadrupole-octupole (SSDQO) |
| 291 |
|
model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point |
| 292 |
|
multipoles up to quadrupolar order have also been coupled with point |
| 293 |
|
polarizabilities in the high-quality AMOEBA and iAMOEBA water |
| 294 |
< |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} But |
| 295 |
< |
using point multipole with the real space truncation without |
| 296 |
< |
accounting for multipolar neutrality will create energy conservation |
| 302 |
< |
issues in molecular dynamics (MD) simulations. |
| 294 |
> |
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However, |
| 295 |
> |
truncating point multipoles without smoothing the forces and torques |
| 296 |
> |
will create energy conservation issues in molecular dynamics simulations. |
| 297 |
|
|
| 298 |
|
In this paper we test a set of real-space methods that were developed |
| 299 |
|
for point multipolar interactions. These methods extend the damped |
| 333 |
|
where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is |
| 334 |
|
expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf |
| 335 |
|
a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object |
| 336 |
< |
$\bf a$. |
| 336 |
> |
$\bf a$, etc. |
| 337 |
|
|
| 338 |
|
% Interactions between multipoles can be expressed as higher derivatives |
| 339 |
|
% of the bare Coulomb potential, so one way of ensuring that the forces |
| 406 |
|
connection to unmodified electrostatics as well as the smooth |
| 407 |
|
transition to zero in both these functions as $r\rightarrow r_c$. The |
| 408 |
|
electrostatic forces and torques acting on the central multipole due |
| 409 |
< |
to another site within cutoff sphere are derived from |
| 409 |
> |
to another site within the cutoff sphere are derived from |
| 410 |
|
Eq.~\ref{generic}, accounting for the appropriate number of |
| 411 |
|
derivatives. Complete energy, force, and torque expressions are |
| 412 |
|
presented in the first paper in this series (Reference |
| 424 |
|
U_{D_{\bf a}D_{\bf b}}(r) = U_{D_{\bf a}D_{\bf b}}(r) - |
| 425 |
|
U_{D_{\bf a} D_{\bf b}}(r_c) |
| 426 |
|
- (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot |
| 427 |
< |
\vec{\nabla} U_{D_{\bf a}D_{\bf b}}(r) \Big \lvert _{r_c} |
| 427 |
> |
\nabla U_{D_{\bf a}D_{\bf b}}(r_c). |
| 428 |
|
\end{equation} |
| 429 |
|
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf |
| 430 |
|
a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance |
| 448 |
|
In general, the gradient shifted potential between a central multipole |
| 449 |
|
and any multipolar site inside the cutoff radius is given by, |
| 450 |
|
\begin{equation} |
| 451 |
< |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 452 |
< |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
| 453 |
< |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} \right] |
| 451 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 452 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}} |
| 453 |
> |
\cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 454 |
|
\label{generic2} |
| 455 |
|
\end{equation} |
| 456 |
|
where the sum describes a separate force-shifting that is applied to |
| 457 |
< |
each orientational contribution to the energy. |
| 457 |
> |
each orientational contribution to the energy. In this expression, |
| 458 |
> |
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles |
| 459 |
> |
($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ |
| 460 |
> |
represent the orientations the multipoles. |
| 461 |
|
|
| 462 |
|
The third term converges more rapidly than the first two terms as a |
| 463 |
|
function of radius, hence the contribution of the third term is very |
| 464 |
|
small for large cutoff radii. The force and torque derived from |
| 465 |
< |
equation \ref{generic2} are consistent with the energy expression and |
| 465 |
> |
Eq. \ref{generic2} are consistent with the energy expression and |
| 466 |
|
approach zero as $r \rightarrow r_c$. Both the GSF and TSF methods |
| 467 |
|
can be considered generalizations of the original DSF method for |
| 468 |
|
higher order multipole interactions. GSF and TSF are also identical up |
| 470 |
|
the energy, force and torque for higher order multipole-multipole |
| 471 |
|
interactions. Complete energy, force, and torque expressions for the |
| 472 |
|
GSF potential are presented in the first paper in this series |
| 473 |
< |
(Reference~\onlinecite{PaperI}) |
| 473 |
> |
(Reference~\onlinecite{PaperI}). |
| 474 |
|
|
| 475 |
|
|
| 476 |
|
\subsection{Shifted potential (SP) } |
| 484 |
|
effectively shifts the total potential to zero at the cutoff radius, |
| 485 |
|
\begin{equation} |
| 486 |
|
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 487 |
< |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 487 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 488 |
|
\label{eq:SP} |
| 489 |
|
\end{equation} |
| 490 |
|
where the sum describes separate potential shifting that is done for |
