multipole/multipole_2/multipole2.tex

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\begin{document}

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\title{Real space electrostatics for multipoles. II. Comparisons with
  the Ewald Sum}

\author{Madan Lamichhane}
 \affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}

\author{Kathie E. Newman}
\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}

\author{J. Daniel Gezelter}%
 \email{gezelter@nd.edu.}
\affiliation{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556
}

\date{\today}

\begin{abstract}
  We report on tests of the shifted potential (SP), gradient shifted
  force (GSF), and Taylor shifted force (TSF) real-space methods for
  multipole interactions developed in the first paper in this series,
  using the multipolar Ewald sum as a reference method. The tests were
  carried out in a variety of condensed-phase environments designed to
  test up to quadrupole-quadrupole interactions.  Comparisons of the
  energy differences between configurations, molecular forces, and
  torques were used to analyze how well the real-space models perform
  relative to the more computationally expensive Ewald treatment.  We
  have also investigated the energy conservation properties of the new
  methods in molecular dynamics simulations. The SP method shows
  excellent agreement with configurational energy differences, forces,
  and torques, and would be suitable for use in Monte Carlo
  calculations.  Of the two new shifted-force methods, the GSF
  approach shows the best agreement with Ewald-derived energies,
  forces, and torques and also exhibits energy conservation properties
  that make it an excellent choice for efficient computation of
  electrostatic interactions in molecular dynamics simulations.
\end{abstract}

%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Electrostatics, Multipoles, Real-space}

\maketitle

\section{\label{sec:intro}Introduction}
Computing the interactions between electrostatic sites is one of the
most expensive aspects of molecular simulations. There have been
significant efforts to develop practical, efficient and convergent
methods for handling these interactions. Ewald's method is perhaps the
best known and most accurate method for evaluating energies, forces,
and torques in explicitly-periodic simulation cells. In this approach,
the conditionally convergent electrostatic energy is converted into
two absolutely convergent contributions, one which is carried out in
real space with a cutoff radius, and one in reciprocal
space.\cite{Ewald21,deLeeuw80,Smith81,Allen87} 

When carried out as originally formulated, the reciprocal-space
portion of the Ewald sum exhibits relatively poor computational
scaling, making it prohibitive for large systems. By utilizing a
particle mesh and three dimensional fast Fourier transforms (FFT), the
particle-mesh Ewald (PME), particle-particle particle-mesh Ewald
(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME)
methods can decrease the computational cost from $O(N^2)$ down to $O(N
\log
N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}

Because of the artificial periodicity required for the Ewald sum,
interfacial molecular systems such as membranes and liquid-vapor
interfaces require modifications to the method.  Parry's extension of
the three dimensional Ewald sum is appropriate for slab
geometries.\cite{Parry:1975if} Modified Ewald methods that were
developed to handle two-dimensional (2-D) electrostatic
interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl}
These methods were originally quite computationally
expensive.\cite{Spohr97,Yeh99} There have been several successful
efforts that reduced the computational cost of 2-D lattice summations,
bringing them more in line with the scaling for the full 3-D
treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The
inherent periodicity required by the Ewald method can also be
problematic in a number of protein/solvent and ionic solution
environments.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq}

\subsection{Real-space methods}
Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$
method for calculating electrostatic interactions between point
charges. They argued that the effective Coulomb interaction in most
condensed phase systems is effectively short
ranged.\cite{Wolf92,Wolf95} For an ordered lattice (e.g., when
computing the Madelung constant of an ionic solid), the material can
be considered as a set of ions interacting with neutral dipolar or
quadrupolar ``molecules'' giving an effective distance dependence for
the electrostatic interactions of $r^{-5}$ (see figure
\ref{fig:schematic}). If one views the \ce{NaCl} crystal as a simple
cubic (SC) structure with an octupolar \ce{(NaCl)4} basis, the
electrostatic energy per ion converges more rapidly to the Madelung
energy than the dipolar approximation.\cite{Wolf92} To find the
correct Madelung constant, Lacman suggested that the NaCl structure
could be constructed in a way that the finite crystal terminates with
complete \ce{(NaCl)4} molecules.\cite{Lacman65} The central ion sees
what is effectively a set of octupoles at large distances. These facts
suggest that the Madelung constants are relatively short ranged for
perfect ionic crystals.\cite{Wolf:1999dn} For this reason, careful
application of Wolf's method can provide accurate estimates of
Madelung constants using relatively short cutoff radii.

Direct truncation of interactions at a cutoff radius creates numerical
errors.  Wolf \textit{et al.} suggest that truncation errors are due
to net charge remaining inside the cutoff sphere.\cite{Wolf:1999dn} To
neutralize this charge they proposed placing an image charge on the
surface of the cutoff sphere for every real charge inside the cutoff.
These charges are present for the evaluation of both the pair
interaction energy and the force, although the force expression
maintains a discontinuity at the cutoff sphere.  In the original Wolf
formulation, the total energy for the charge and image were not equal
to the integral of the force expression, and as a result, the total
energy would not be conserved in molecular dynamics (MD)
simulations.\cite{Zahn:2002hc} Zahn \textit{et al.}, and Fennel and
Gezelter later proposed shifted force variants of the Wolf method with
commensurate force and energy expressions that do not exhibit this
problem.\cite{Zahn:2002hc,Fennell:2006lq} Related real-space methods
were also proposed by Chen \textit{et
  al.}\cite{Chen:2004du,Chen:2006ii,Denesyuk:2008ez,Rodgers:2006nw}
and by Wu and Brooks.\cite{Wu:044107} Recently, Fukuda has successfuly
used additional neutralization of higher order moments for systems of
point charges.\cite{Fukuda:2013sf}

\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{schematic.eps}
  \caption{Top: Ionic systems exhibit local clustering of dissimilar
    charges (in the smaller grey circle), so interactions are
    effectively charge-multipole at longer distances.  With hard
    cutoffs, motion of individual charges in and out of the cutoff
    sphere can break the effective multipolar ordering.  Bottom:
    dipolar crystals and fluids have a similar effective
    \textit{quadrupolar} ordering (in the smaller grey circles), and
    orientational averaging helps to reduce the effective range of the
    interactions in the fluid.  Placement of reversed image multipoles
    on the surface of the cutoff sphere recovers the effective
    higher-order multipole behavior.}
  \label{fig:schematic}
\end{figure}

One can make a similar effective range argument for crystals of point
\textit{multipoles}. The Luttinger and Tisza treatment of energy
constants for dipolar lattices utilizes 24 basis vectors that contain
dipoles at the eight corners of a unit cube.\cite{LT} Only three of
these basis vectors, $X_1, Y_1, \mathrm{~and~} Z_1,$ retain net dipole
moments, while the rest have zero net dipole and retain contributions
only from higher order multipoles.  The lowest-energy crystalline
structures are built out of basis vectors that have only residual
quadrupolar moments (e.g. the $Z_5$ array). In these low energy
structures, the effective interaction between a dipole at the center
of a crystal and a group of eight dipoles farther away is
significantly shorter ranged than the $r^{-3}$ that one would expect
for raw dipole-dipole interactions.  Only in crystals which retain a
bulk dipole moment (e.g. ferroelectrics) does the analogy with the
ionic crystal break down -- ferroelectric dipolar crystals can exist,
while ionic crystals with net charge in each unit cell would be
unstable.

In ionic crystals, real-space truncation can break the effective
multipolar arrangements (see Fig. \ref{fig:schematic}), causing
significant swings in the electrostatic energy as individual ions move
back and forth across the boundary.  This is why the image charges are
necessary for the Wolf sum to exhibit rapid convergence.  Similarly,
the real-space truncation of point multipole interactions breaks
higher order multipole arrangements, and image multipoles are required
for real-space treatments of electrostatic energies.

The shorter effective range of electrostatic interactions is not
limited to perfect crystals, but can also apply in disordered fluids.
Even at elevated temperatures, there is local charge balance in an
ionic liquid, where each positive ion has surroundings dominated by
negaitve ions and vice versa.  The reversed-charge images on the
cutoff sphere that are integral to the Wolf and DSF approaches retain
the effective multipolar interactions as the charges traverse the
cutoff boundary.

In multipolar fluids (see Fig. \ref{fig:schematic}) there is
significant orientational averaging that additionally reduces the
effect of long-range multipolar interactions.  The image multipoles
that are introduced in the TSF, GSF, and SP methods mimic this effect
and reduce the effective range of the multipolar interactions as
interacting molecules traverse each other's cutoff boundaries.

% Because of this reason, although the nature of electrostatic
% interaction short ranged, the hard cutoff sphere creates very large
% fluctuation in the electrostatic energy for the perfect crystal. In
% addition, the charge neutralized potential proposed by Wolf et
% al. converged to correct Madelung constant but still holds oscillation
% in the energy about correct Madelung energy.\cite{Wolf:1999dn}.  This
% oscillation in the energy around its fully converged value can be due
% to the non-neutralized value of the higher order moments within the
% cutoff sphere.

Forces and torques acting on atomic sites are fundamental in driving
dynamics in molecular simulations, and the damped shifted force (DSF)
energy kernel provides consistent energies and forces on charged atoms
within the cutoff sphere. Both the energy and the force go smoothly to
zero as an atom aproaches the cutoff radius. The comparisons of the
accuracy these quantities between the DSF kernel and SPME was
surprisingly good.\cite{Fennell:2006lq} As a result, the DSF method
has seen increasing use in molecular systems with relatively uniform
charge
densities.\cite{English08,Kannam:2012rr,Space12,Lawrence13,Acevedo13,Shi:2013ij,Vergne13}

\subsection{The damping function}
The damping function has been discussed in detail in the first paper
of this series.\cite{PaperI} The $1/r$ Coulombic kernel for the
interactions between point charges can be replaced by the
complementary error function $\mathrm{erfc}(\alpha r)/r$ to accelerate
convergence, where $\alpha$ is a damping parameter with units of
inverse distance.  Altering the value of $\alpha$ is equivalent to
changing the width of Gaussian charge distributions that replace each
point charge, as Coulomb integrals with Gaussian charge distributions
produce complementary error functions when truncated at a finite
distance.

With moderate damping coefficients, $\alpha \sim 0.2$, the DSF method
produced very good agreement with SPME for interaction energies,
forces and torques for charge-charge
interactions.\cite{Fennell:2006lq}

\subsection{Point multipoles in molecular modeling}
Coarse-graining approaches which treat entire molecular subsystems as
a single rigid body are now widely used. A common feature of many
coarse-graining approaches is simplification of the electrostatic
interactions between bodies so that fewer site-site interactions are
required to compute configurational
energies.\cite{Ren06,Essex10,Essex11}

Additionally, because electrons in a molecule are not localized at
specific points, the assignment of partial charges to atomic centers
is always an approximation.  For increased accuracy, atomic sites can
also be assigned point multipoles and polarizabilities.  Recently,
water has been modeled with point multipoles up to octupolar order
using the soft sticky dipole-quadrupole-octupole (SSDQO)
model.\cite{Ichiye10_1,Ichiye10_2,Ichiye10_3} Atom-centered point
multipoles up to quadrupolar order have also been coupled with point
polarizabilities in the high-quality AMOEBA and iAMOEBA water
models.\cite{Ren:2003uq,Ren:2004kx,Ponder:2010vl,Wang:2013fk} However,
truncating point multipoles without smoothing the forces and torques
can create energy conservation issues in molecular dynamics
simulations.

In this paper we test a set of real-space methods that were developed
for point multipolar interactions.  These methods extend the damped
shifted force (DSF) and Wolf methods originally developed for
charge-charge interactions and generalize them for higher order
multipoles.  The detailed mathematical development of these methods
has been presented in the first paper in this series, while this work
covers the testing of energies, forces, torques, and energy
conservation properties of the methods in realistic simulation
environments.  In all cases, the methods are compared with the
reference method, a full multipolar Ewald treatment.\cite{Smith82,Smith98}


%\subsection{Conservation of total energy } 
%To conserve the total energy in MD simulations, the energy, force, and torque on a central molecule due to another molecule should smoothly tends to zero as second molecule approaches to cutoff radius. In addition, the force should be derivable from the energy and vice versa. If only the first condition holds but not the second, the total energy does not conserve.\cite{Fennell:2006lq}. The hard cutoff method does not ensure the smooth transition of the energy, force, and torque at the cutoff radius.\cite{Wolf:1999dn} By placing image charge on the surface of the cutoff sphere, the smooth transition of the energy can be ensured but the force and torque remains discontinuous. Therefore the purposed methods should have smooth transition of the energy, force and torque to ensure the total energy conservation and the expression should be close to idea of placing image multipole on the surface of the cutoff sphere.

\section{\label{sec:method}Review of Methods}
Any real-space electrostatic method that is suitable for MD
simulations should have the electrostatic energy, forces and torques
between two sites go smoothly to zero as the distance between the
sites, $r_{\bf ab}$ approaches the cutoff radius, $r_c$.  Requiring
this continuity at the cutoff is essential for energy conservation in
MD simulations.  The mathematical details of the shifted potential
(SP), gradient-shifted-force (GSF) and Taylor shifted-force (TSF)
methods have been discussed in detail in the previous paper in this
series.\cite{PaperI} Here we briefly review the new methods and
describe their essential features.

\subsection{Taylor-shifted force (TSF)}

The electrostatic potential energy between point multipoles can be
expressed as the product of two multipole operators and a Coulombic
kernel,
\begin{equation}
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r}  \label{kernel}.
\end{equation}
where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is
expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf
    a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object
$\bf a$, etc.

% Interactions between multipoles can be expressed as higher derivatives
% of the bare Coulomb potential, so one way of ensuring that the forces
% and torques vanish at the cutoff distance is to include a larger
% number of terms in the truncated Taylor expansion, e.g.,
% %
% \begin{equation}
% f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert  _{r_c}  .
% \end{equation}
% %
% The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$.
% Thus, for $f(r)=1/r$, we find
% %
% \begin{equation}
% 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} .
% \end{equation}
% This function is an approximate electrostatic potential that has
% vanishing second derivatives at the cutoff radius, making it suitable
% for shifting the forces and torques of charge-dipole interactions.

The TSF potential for any multipole-multipole interaction can be
written
\begin{equation}
U^{\text{TSF}}= (\text{prefactor}) (\text{derivatives}) f_n(r)
\label{generic}
\end{equation}
where $f_n(r)$ is a shifted kernel that is appropriate for the order
of the interaction (see Ref. \onlinecite{PaperI}), with $n=0$ for
charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole
and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for
quadrupole-quadrupole.  To ensure smooth convergence of the energy,
force, and torques, a Taylor expansion with $n$ terms must be
performed at cutoff radius ($r_c$) to obtain $f_n(r)$.

% To carry out the same procedure for a damped electrostatic kernel, we
% replace $1/r$ in the Coulomb potential with $\text{erfc}(\alpha r)/r$.
% Many of the derivatives of the damped kernel are well known from
% Smith's early work on multipoles for the Ewald
% summation.\cite{Smith82,Smith98} 

% Note that increasing the value of $n$ will add additional terms to the
% electrostatic potential, e.g., $f_2(r)$ includes orders up to
% $(r-r_c)^3/r_c^4$, and so on.  Successive derivatives of the $f_n(r)$
% functions are denoted $g_2(r) = f^\prime_2(r)$, $h_2(r) =
% f^{\prime\prime}_2(r)$, etc.  These higher derivatives are required
% for computing multipole energies, forces, and torques, and smooth
% cutoffs of these quantities can be guaranteed as long as the number of
% terms in the Taylor series exceeds the derivative order required.

For multipole-multipole interactions, following this procedure results
in separate radial functions for each of the distinct orientational
contributions to the potential, and ensures that the forces and
torques from each of these contributions will vanish at the cutoff
radius.  For example, the direct dipole dot product
($\mathbf{D}_{\bf a}
\cdot \mathbf{D}_{\bf b}$) is treated differently than the dipole-distance
dot products:
\begin{equation}
U_{D_{\bf a}D_{\bf b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left(
  \mathbf{D}_{\bf a} \cdot
\mathbf{D}_{\bf b} \right) v_{21}(r) +
\left( \mathbf{D}_{\bf a} \cdot \hat{r} \right)
\left( \mathbf{D}_{\bf b} \cdot \hat{r} \right) v_{22}(r) \right]
\end{equation}

For the Taylor shifted (TSF) method with the undamped kernel,
$v_{21}(r) = -\frac{1}{r^3} + \frac{3 r}{r_c^4} - \frac{8}{r_c^3} +
\frac{6}{r r_c^2}$ and $v_{22}(r) = \frac{3}{r^3} + \frac{3 r}{r_c^4}
- \frac{6}{r r_c^2}$.  In these functions, one can easily see the
connection to unmodified electrostatics as well as the smooth
transition to zero in both these functions as $r\rightarrow r_c$.  The
electrostatic forces and torques acting on the central multipole due
to another site within the cutoff sphere are derived from
Eq.~\ref{generic}, accounting for the appropriate number of
derivatives. Complete energy, force, and torque expressions are
presented in the first paper in this series (Reference
\onlinecite{PaperI}).

\subsection{Gradient-shifted force (GSF)}

A second (and conceptually simpler) method involves shifting the
gradient of the raw Coulomb potential for each particular multipole
order.  For example, the raw dipole-dipole potential energy may be
shifted smoothly by finding the gradient for two interacting dipoles
which have been projected onto the surface of the cutoff sphere
without changing their relative orientation,
\begin{equation}
U_{D_{\bf a}D_{\bf b}}(r)  = U_{D_{\bf a}D_{\bf b}}(r) -
U_{D_{\bf a} D_{\bf b}}(r_c)
   - (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot
  \nabla U_{D_{\bf a}D_{\bf b}}(r_c).
\end{equation}
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf
  a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance
(although the signs are reversed for the dipole that has been
projected onto the cutoff sphere).  In many ways, this simpler
approach is closer in spirit to the original shifted force method, in
that it projects a neutralizing multipole (and the resulting forces
from this multipole) onto a cutoff sphere. The resulting functional
forms for the potentials, forces, and torques turn out to be quite
similar in form to the Taylor-shifted approach, although the radial
contributions are significantly less perturbed by the gradient-shifted
approach than they are in the Taylor-shifted method.

For the gradient shifted (GSF) method with the undamped kernel,
$v_{21}(r) = -\frac{1}{r^3} + \frac{3 r}{r_c^4} + \frac{4}{r_c^3}$ and
$v_{22}(r) = \frac{3}{r^3} + \frac{9 r}{r_c^4} - \frac{12}{r_c^3}$.
Again, these functions go smoothly to zero as $r\rightarrow r_c$, and
because the Taylor expansion retains only one term, they are
significantly less perturbed than the TSF functions.

In general, the gradient shifted potential between a central multipole
and any multipolar site inside the cutoff radius is given by,
\begin{equation}
  U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
    U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}}
    \cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
\label{generic2}
\end{equation}
where the sum describes a separate force-shifting that is applied to
each orientational contribution to the energy.  In this expression,
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles
($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$
represent the orientations the multipoles.

The third term converges more rapidly than the first two terms as a
function of radius, hence the contribution of the third term is very
small for large cutoff radii.  The force and torque derived from
Eq. \ref{generic2} are consistent with the energy expression and
approach zero as $r \rightarrow r_c$.  Both the GSF and TSF methods
can be considered generalizations of the original DSF method for
higher order multipole interactions. GSF and TSF are also identical up
to the charge-dipole interaction but generate different expressions in
the energy, force and torque for higher order multipole-multipole
interactions. Complete energy, force, and torque expressions for the
GSF potential are presented in the first paper in this series
(Reference~\onlinecite{PaperI}).


\subsection{Shifted potential (SP) }
A discontinuous truncation of the electrostatic potential at the
cutoff sphere introduces a severe artifact (oscillation in the
electrostatic energy) even for molecules with the higher-order
multipoles.\cite{PaperI} We have also formulated an extension of the
Wolf approach for point multipoles by simply projecting the image
multipole onto the surface of the cutoff sphere, and including the
interactions with the central multipole and the image. This
effectively shifts the total potential to zero at the cutoff radius,
\begin{equation}
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
\label{eq:SP}
\end{equation}          
where the sum describes separate potential shifting that is done for
each orientational contribution to the energy (e.g. the direct dipole
product contribution is shifted {\it separately} from the
dipole-distance terms in dipole-dipole interactions).  Note that this
is not a simple shifting of the total potential at $r_c$. Each radial
contribution is shifted separately.  One consequence of this is that
multipoles that reorient after leaving the cutoff sphere can re-enter
the cutoff sphere without perturbing the total energy.

For the shifted potential (SP) method with the undamped kernel,
$v_{21}(r) = -\frac{1}{r^3} + \frac{1}{r_c^3}$ and $v_{22}(r) =
\frac{3}{r^3} - \frac{3}{r_c^3}$.  The potential energy between a
central multipole and other multipolar sites goes smoothly to zero as
$r \rightarrow r_c$.  However, the force and torque obtained from the
shifted potential (SP) are discontinuous at $r_c$.  MD simulations
will still experience energy drift while operating under the SP
potential, but it may be suitable for Monte Carlo approaches where the
configurational energy differences are the primary quantity of
interest.

\subsection{The Self Term}
In the TSF, GSF, and SP methods, a self-interaction is retained for
the central multipole interacting with its own image on the surface of
the cutoff sphere.  This self interaction is nearly identical with the
self-terms that arise in the Ewald sum for multipoles.  Complete
expressions for the self terms are presented in the first paper in
this series (Reference \onlinecite{PaperI}). 


\section{\label{sec:methodology}Methodology}

To understand how the real-space multipole methods behave in computer
simulations, it is vital to test against established methods for
computing electrostatic interactions in periodic systems, and to
evaluate the size and sources of any errors that arise from the
real-space cutoffs.  In the first paper of this series, we compared
the dipolar and quadrupolar energy expressions against analytic
expressions for ordered dipolar and quadrupolar
arrays.\cite{Sauer,LT,Nagai01081960,Nagai01091963} In this work, we
used the multipolar Ewald sum as a reference method for comparing
energies, forces, and torques for molecular models that mimic
disordered and ordered condensed-phase systems.  The parameters used
in the test cases are given in table~\ref{tab:pars}.

\begin{table}
\label{tab:pars}
\caption{The parameters used in the systems used to evaluate the new
  real-space methods.  The most comprehensive test was a liquid
  composed of 2000 SSDQ molecules with 48 dissolved ions (24 \ce{Na+} and 24 \ce{Cl-}
  ions).  This test excercises all orders of the multipolar
  interactions developed in the first paper.}
\begin{tabularx}{\textwidth}{r|cc|YYccc|Yccc} \hline
             & \multicolumn{2}{c|}{LJ parameters} &
             \multicolumn{5}{c|}{Electrostatic moments} & & & & \\
 Test system & $\sigma$& $\epsilon$ & $C$ & $D$  &
 $Q_{xx}$ & $Q_{yy}$ & $Q_{zz}$ & mass  & $I_{xx}$ & $I_{yy}$ &
 $I_{zz}$ \\ \cline{6-8}\cline{10-12}
 & (\AA) & (kcal/mol) & (e) & (debye) & \multicolumn{3}{c|}{(debye \AA)} & (amu) & \multicolumn{3}{c}{(amu
 \AA\textsuperscript{2})} \\ \hline
    Soft Dipolar fluid & 3.051 & 0.152 &  & 2.35 & & & & 18.0153 & 1.77&0.6145& 1.155 \\
    Soft Dipolar solid & 2.837 & 1.0   &  & 2.35 & & & & $10^4$  & 17.6 &17.6 & 0 \\
Soft Quadrupolar fluid & 3.051 & 0.152 &  &  & -1&-1&-2.5 & 18.0153 & 1.77&0.6145&1.155  \\
Soft Quadrupolar solid & 2.837 & 1.0   &  &  & -1&-1&-2.5 & $10^4$  & 17.6&17.6&0 \\
      SSDQ water  & 3.051 & 0.152 &  & 2.35 & -1.35&0&-0.68 & 18.0153 & 1.77&0.6145&1.155 \\
              \ce{Na+} & 2.579 & 0.118 & +1& & & & & 22.99 & & &\\
              \ce{Cl-} & 4.445 & 0.1   & -1& & & & & 35.4527& & & \\ \hline
\end{tabularx}
\end{table}
The systems consist of pure multipolar solids (both dipole and
quadrupole), pure multipolar liquids (both dipole and quadrupole), a
fluid composed of sites containing both dipoles and quadrupoles
simultaneously, and a final test case that includes ions with point
charges in addition to the multipolar fluid.  The solid-phase
parameters were chosen so that the systems can explore some
orientational freedom for the multipolar sites, while maintaining
relatively strict translational order.  The SSDQ model used here is
not a particularly accurate water model, but it does test
dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole
interactions at roughly the same magnitudes. The last test case, SSDQ
water with dissolved ions, exercises \textit{all} levels of the
multipole-multipole interactions we have derived so far and represents
the most complete test of the new methods.

In the following section, we present results for the total
electrostatic energy, as well as the electrostatic contributions to
the force and torque on each molecule.  These quantities have been
computed using the SP, TSF, and GSF methods, as well as a hard cutoff,
and have been compared with the values obtained from the multipolar
Ewald sum.  In Monte Carlo (MC) simulations, the energy differences
between two configurations is the primary quantity that governs how
the simulation proceeds. These differences are the most important
indicators of the reliability of a method even if the absolute
energies are not exact.  For each of the multipolar systems listed
above, we have compared the change in electrostatic potential energy
($\Delta E$) between 250 statistically-independent configurations.  In
molecular dynamics (MD) simulations, the forces and torques govern the
behavior of the simulation, so we also compute the electrostatic
contributions to the forces and torques.

\subsection{Implementation}
The real-space methods developed in the first paper in this series
have been implemented in our group's open source molecular simulation
program, OpenMD,\cite{Meineke05,openmd} which was used for all calculations in
this work.  The complementary error function can be a relatively slow
function on some processors, so all of the radial functions are
precomputed on a fine grid and are spline-interpolated to provide
values when required.  

Using the same simulation code, we compare to a multipolar Ewald sum
with a reciprocal space cutoff, $k_\mathrm{max} = 7$.  Our version of
the Ewald sum is a re-implementation of the algorithm originally
proposed by Smith that does not use the particle mesh or smoothing
approximations.\cite{Smith82,Smith98} In all cases, the quantities
being compared are the electrostatic contributions to energies, force,
and torques.  All other contributions to these quantities (i.e. from
Lennard-Jones interactions) are removed prior to the comparisons.

The convergence parameter ($\alpha$) also plays a role in the balance
of the real-space and reciprocal-space portions of the Ewald
calculation.  Typical molecular mechanics packages set this to a value
that depends on the cutoff radius and a tolerance (typically less than
$1 \times 10^{-4}$ kcal/mol).  Smaller tolerances are typically
associated with increasing accuracy at the expense of computational
time spent on the reciprocal-space portion of the
summation.\cite{Perram88,Essmann:1995pb} A default tolerance of $1 \times
10^{-8}$ kcal/mol was used in all Ewald calculations, resulting in
Ewald coefficient 0.3119 \AA$^{-1}$ for a cutoff radius of 12 \AA.

The real-space models have self-interactions that provide
contributions to the energies only.  Although the self interaction is
a rapid calculation, we note that in systems with fluctuating charges
or point polarizabilities, the self-term is not static and must be
recomputed at each time step.

\subsection{Model systems}
To sample independent configurations of the multipolar crystals, body
centered cubic (bcc) crystals, which exhibit the minimum energy
structures for point dipoles, were generated using 3,456 molecules.
The multipoles were translationally locked in their respective crystal
sites for equilibration at a relatively low temperature (50K) so that
dipoles or quadrupoles could freely explore all accessible
orientations.  The translational constraints were then removed, the
systems were re-equilibrated, and the crystals were simulated for an
additional 10 ps in the microcanonical (NVE) ensemble with an average
temperature of 50 K.  The balance between moments of inertia and
particle mass were chosen to allow orientational sampling without
significant translational motion.  Configurations were sampled at
equal time intervals in order to compare configurational energy
differences.  The crystals were simulated far from the melting point
in order to avoid translational deformation away of the ideal lattice
geometry.

For dipolar, quadrupolar, and mixed-multipole \textit{liquid}
simulations, each system was created with 2,048 randomly-oriented
molecules.  These were equilibrated at a temperature of 300K for 1 ns.
Each system was then simulated for 1 ns in the microcanonical (NVE)
ensemble.  We collected 250 different configurations at equal time
intervals. For the liquid system that included ionic species, we
converted 48 randomly-distributed molecules into 24 \ce{Na+} and 24
\ce{Cl-} ions and re-equilibrated. After equilibration, the system was
run under the same conditions for 1 ns. A total of 250 configurations
were collected. In the following comparisons of energies, forces, and
torques, the Lennard-Jones potentials were turned off and only the
purely electrostatic quantities were compared with the same values
obtained via the Ewald sum.

\subsection{Accuracy of Energy Differences, Forces and Torques}
The pairwise summation techniques (outlined above) were evaluated for
use in MC simulations by studying the energy differences between
different configurations.  We took the Ewald-computed energy
difference between two conformations to be the correct behavior. An
ideal performance by one of the new methods would reproduce these
energy differences exactly. The configurational energies being used
here contain only contributions from electrostatic interactions.
Lennard-Jones interactions were omitted from the comparison as they
should be identical for all methods.

Since none of the real-space methods provide exact energy differences,
we used least square regressions analysis for the six different
molecular systems to compare $\Delta E$ from Hard, SP, GSF, and TSF
with the multipolar Ewald reference method.  Unitary results for both
the correlation (slope) and correlation coefficient for these
regressions indicate perfect agreement between the real-space method
and the multipolar Ewald sum.

Molecular systems were run long enough to explore independent
configurations and 250 configurations were recorded for comparison.
Each system provided 31,125 energy differences for a total of 186,750
data points.  Similarly, the magnitudes of the forces and torques have
also been compared using least squares regression analysis. In the
forces and torques comparison, the magnitudes of the forces acting in
each molecule for each configuration were evaluated. For example, our
dipolar liquid simulation contains 2048 molecules and there are 250
different configurations for each system resulting in 3,072,000 data
points for comparison of forces and torques.

\subsection{Analysis of vector quantities}
Getting the magnitudes of the force and torque vectors correct is only
part of the issue for carrying out accurate molecular dynamics
simulations.  Because the real space methods reweight the different
orientational contributions to the energies, it is also important to
understand how the methods impact the \textit{directionality} of the
force and torque vectors. Fisher developed a probablity density
function to analyse directional data sets,
\begin{equation}
p_f(\theta) = \frac{\kappa}{2 \sinh\kappa}\sin\theta e^{\kappa \cos\theta}
\label{eq:pdf}
\end{equation}
where $\kappa$ measures directional dispersion of the data around the
mean direction.\cite{fisher53} This quantity $(\kappa)$ can be
estimated as a reciprocal of the circular variance.\cite{Allen91} To
quantify the directional error, forces obtained from the Ewald sum
were taken as the mean (or correct) direction and the angle between
the forces obtained via the Ewald sum and the real-space methods were
evaluated,
\begin{equation}
\cos\theta_i =  \frac{\vec{f}_i^\mathrm{~Ewald} \cdot
  \vec{f}_i^\mathrm{~GSF}}{\left|\vec{f}_i^\mathrm{~Ewald}\right| \left|\vec{f}_i^\mathrm{~GSF}\right|}
\end{equation}
The total angular displacement of the vectors was calculated as,
\begin{equation}
R = \sqrt{\left(\sum\limits_{i=1}^N \cos\theta_i\right)^2 + \left(\sum\limits_{i=1}^N \sin\theta_i\right)^2}
\label{eq:displacement}
\end{equation} 
where $N$ is number of force vectors.  The circular variance is
defined as
\begin{equation}
\mathrm{Var}(\theta) \approx 1/\kappa = 1 - R/N
\end{equation}
The circular variance takes on values between from 0 to 1, with 0
indicating a perfect directional match between the Ewald force vectors
and the real-space forces. Lower values of $\mathrm{Var}(\theta)$
correspond to higher values of $\kappa$, which indicates tighter
clustering of the real-space force vectors around the Ewald forces.

A similar analysis was carried out for the electrostatic contribution
to the molecular torques as well as forces.  

\subsection{Energy conservation}
To test conservation the energy for the methods, the mixed molecular
system of 2000 SSDQ water molecules with 24 \ce{Na+} and 24 \ce{Cl-}
ions was run for 1 ns in the microcanonical ensemble at an average
temperature of 300K.  Each of the different electrostatic methods
(Ewald, Hard, SP, GSF, and TSF) was tested for a range of different
damping values. The molecular system was started with same initial
positions and velocities for all cutoff methods. The energy drift
($\delta E_1$) and standard deviation of the energy about the slope
($\delta E_0$) were evaluated from the total energy of the system as a
function of time.  Although both measures are valuable at
investigating new methods for molecular dynamics, a useful interaction
model must allow for long simulation times with minimal energy drift.

\section{\label{sec:result}RESULTS}
\subsection{Configurational energy differences}
%The magnitude of the fluctuation in the total electrostatic energy per molecule for a dipolar crystal is very high as shown in (Fig … paper I).\cite{PaperI}As soon as, the net dipole moment within a cutoff radius is neutralized in the SP method, the magnitude of the fluctuation in the total electrostatic energy per molecule reduced significantly and rapidly converged to the correct energy constant (Refer figure … Paper I).\cite{PaperI} The GSF potential energy also converged to the correct energy constant for the cutoff radius rc = 6a for the undamped case. The potential energy from the TSF method converges towards the correct value for a very large cutoff radius. The speed of convergence for the all the cutoff methods can be increased by using damping function as shown in figure … Paper I\cite{PaperI}. For the quadrupolar crystals, the fluctuation in the total electrostatic energy for the hard cutoff method is small and short ranged as compared to the dipolar crystals (figure … in the paper I).\cite{PaperI}  Similar to the dipolar crystals, the net quadrupolar neutralization of the cutoff sphere in the SP method reduces oscillation rapidly and converge electrostatic energy to the correct energy constant.
%The oscillation in the the electrostatic energy for the hard cutoff method is even true for the dipolar liquids as shown in Figure ~\ref{fig:rcutConvergence_dipolarLiquid}. As we placed image on the surface of the cutoff sphere in SP method, the oscillation in the energy is reduced. The fluctuation in the energy in liquid is much smaller as compared to the crystal (This result is similar to the results observed by \textit{Wolf et al.} in the case of ionic crystal and Mgo melt). The large magnitude in the fluctuation of the electrostatic energy in the crystal is because of large range of multipole ordering in the crystal. When the energy is evaluated by the direct truncation, it breaks up large number of multipolar ordering leaving behind net multipole moment. But in the case of liquid, there is only local ordering of the multipoles and their ordering disappears in the long range. Therefore, the direct truncation results a small oscillation in the electrostatic energy (which is smaller than deviation SP energy from the Ewald Figure ~\ref{fig:rcutConvergence_dipolarLiquid}) in the case of liquid. Although, the oscillation in the energy is very small for the case of liquid, this affects the change in potential energy ($\triangle E$), which is observed when $\triangle E$ evaluated from the SP method compared with Ewald as shown in figure 4a and 4b.
%\begin{figure}[h!]
%        \centering
%        \includegraphics[width=0.50 \textwidth]{rcutConvergence_dipolarLiquid-crop.pdf}
%        \caption{The energy per molecule plotted against cutoff radius, rc for i) Hard ii) SP iii) GSF, and iv) TSF method. The hard cutoff method shows fluctuation in the electorstatic energy and it disappers in all other methods.  }
%        \label{fig:rcutConvergence_dipolarLiquid}
%    \end{figure}
%In MC simulations, the electrostatic differences between the molecules are important parameter for sampling. We have compared $\triangle E$ from the different methods (Hard, SP, GSF, and TSF) with the Ewald using linear regression analysis. The correlation coefficient ($R^2$) of the regression line measures the deviation of the evaluated quantities from the mean slope. We know that Ewald method evaluates accurate value of the electrostatic energy. Hence, if the proposed methods can quantify the electrostatic energy as good as Ewald then the correlation coefficient is 1.The correlation coefficient is 0 for the completely random result for any physical quantity measured by the proposed method. The slope is a measure of the accuracy of the average of a physical quantity obtained from the proposed methods. If the slope is 1 then we can conclude that the average of the physical quantity measured by the method is as good as Ewald. The deviation of the slope from 1 state that the method used in quantifying physical quantities is statistically biased as compared to Ewald.
%\begin{figure}
%        \centering
%        \includegraphics[width=0.45 \textwidth]{slopeComparision_undamped.pdf}
%        \label{fig:barGraph1}
%        \end{figure}
%        \begin{figure}
%        \centering
 %       \includegraphics[width=0.45 \textwidth]{slopeComparision_undamped.pdf}
%        \caption{}
       
%        \label{fig:barGraph2}
%      \end{figure} 
%The correlation coefficient ($R^2$) and slope of the linear
%regression plots for the energy differences for all six different
%molecular systems is shown in figure 4a and 4b.The plot shows that
%the correlation coefficient improves for the SP cutoff method as
%compared to the undamped hard cutoff method in the case of SSDQC,
%SSDQ, dipolar crystal, and dipolar liquid. For the quadrupolar
%crystal and liquid, the correlation coefficient is almost unchanged
%and close to 1.  The correlation coefficient is smallest (0.696276
%for $r_c$ = 9 $A^\circ$) for the SSDQC liquid because of the presence of
%charge-charge and charge-multipole interactions. Since the
%charge-charge and charge-multipole interaction is long ranged, there
%is huge deviation of correlation coefficient from 1. Similarly, the
%quarupole–quadrupole interaction is short ranged ($\sim 1/r^6$) with
%compared to interactions in the other multipolar systems, thus the
%correlation coefficient very close to 1 even for hard cutoff
%method. The idea of placing image multipole on the surface of the
%cutoff sphere improves the correlation coefficient and makes it close
%to 1 for all types of multipolar systems. Similarly the slope is
%hugely deviated from the correct value for the lower order
%multipole-multipole interaction and slightly deviated for higher
%order multipole – multipole interaction. The SP method improves both
%correlation coefficient ($R^2$) and slope significantly in SSDQC and
%dipolar systems.  The Slope is found to be deviated more in dipolar
%crystal as compared to liquid which is associated with the large
%fluctuation in the electrostatic energy in crystal. The GSF also
%produced better values of correlation coefficient and slope with the
%proper selection of the damping alpha (Interested reader can consult
%accompanying supporting material). The TSF method gives good value of
%correlation coefficient for the dipolar crystal, dipolar liquid,
%SSDQ, and SSDQC (not for the quadrupolar crystal and liquid) but the
%regression slopes are significantly deviated.
 
\begin{figure}
  \centering
  \includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps}
  \caption{Statistical analysis of the quality of configurational
    energy differences for the real-space electrostatic methods
    compared with the reference Ewald sum.  Results with a value equal
    to 1 (dashed line) indicate $\Delta E$ values indistinguishable
    from those obtained using the multipolar Ewald sum.  Different
    values of the cutoff radius are indicated with different symbols
    (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = inverted
    triangles).}
  \label{fig:slopeCorr_energy}
\end{figure} 

The combined correlation coefficient and slope for all six systems is
shown in Figure ~\ref{fig:slopeCorr_energy}.  Most of the methods
reproduce the Ewald configurational energy differences with remarkable
fidelity.  Undamped hard cutoffs introduce a significant amount of
random scatter in the energy differences which is apparent in the
reduced value of the correlation coefficient for this method.  This
can be easily understood as configurations which exhibit small
traversals of a few dipoles or quadrupoles out of the cutoff sphere
will see large energy jumps when hard cutoffs are used.  The
orientations of the multipoles (particularly in the ordered crystals)
mean that these energy jumps can go in either direction, producing a
significant amount of random scatter, but no systematic error.

The TSF method produces energy differences that are highly correlated
with the Ewald results, but it also introduces a significant
systematic bias in the values of the energies, particularly for
smaller cutoff values. The TSF method alters the distance dependence
of different orientational contributions to the energy in a
non-uniform way, so the size of the cutoff sphere can have a large
effect, particularly for the crystalline systems.

Both the SP and GSF methods appear to reproduce the Ewald results with
excellent fidelity, particularly for moderate damping ($\alpha =
0.1-0.2$\AA$^{-1}$) and with a commonly-used cutoff value ($r_c =
12$\AA).  With the exception of the undamped hard cutoff, and the TSF
method with short cutoffs, all of the methods would be appropriate for
use in Monte Carlo simulations.
 
\subsection{Magnitude of the force and torque vectors}

The comparisons of the magnitudes of the forces and torques for the
data accumulated from all six systems are shown in Figures
~\ref{fig:slopeCorr_force} and \ref{fig:slopeCorr_torque}. The
correlation and slope for the forces agree well with the Ewald sum
even for the hard cutoffs.

For systems of molecules with only multipolar interactions, the pair
energy contributions are quite short ranged.  Moreover, the force
decays more rapidly than the electrostatic energy, hence the hard
cutoff method can also produce reasonable agreement for this quantity.
Although the pure cutoff gives reasonably good electrostatic forces
for pairs of molecules included within each other's cutoff spheres,
the discontinuity in the force at the cutoff radius can potentially
cause energy conservation problems as molecules enter and leave the
cutoff spheres.  This is discussed in detail in section
\ref{sec:conservation}.

The two shifted-force methods (GSF and TSF) exhibit a small amount of
systematic variation and scatter compared with the Ewald forces.  The
shifted-force models intentionally perturb the forces between pairs of
molecules inside each other's cutoff spheres in order to correct the
energy conservation issues, and this perturbation is evident in the
statistics accumulated for the molecular forces.  The GSF
perturbations are minimal, particularly for moderate damping and
commonly-used cutoff values ($r_c = 12$\AA).  The TSF method shows
reasonable agreement in the correlation coefficient but again the
systematic error in the forces is concerning if replication of Ewald
forces is desired.

\begin{figure}
  \centering
  \includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} 
  \caption{Statistical analysis of the quality of the force vector
    magnitudes for the real-space electrostatic methods compared with
    the reference Ewald sum. Results with a value equal to 1 (dashed
    line) indicate force magnitude values indistinguishable from those
    obtained using the multipolar Ewald sum.  Different values of the
    cutoff radius are indicated with different symbols (9\AA\ =
    circles, 12\AA\ = squares, and 15\AA\ = inverted triangles). }
  \label{fig:slopeCorr_force}
\end{figure} 


\begin{figure}
  \centering
  \includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps}
  \caption{Statistical analysis of the quality of the torque vector
    magnitudes for the real-space electrostatic methods compared with
    the reference Ewald sum. Results with a value equal to 1 (dashed
    line) indicate force magnitude values indistinguishable from those
    obtained using the multipolar Ewald sum.  Different values of the
    cutoff radius are indicated with different symbols (9\AA\ =
    circles, 12\AA\ = squares, and 15\AA\ = inverted triangles).}
  \label{fig:slopeCorr_torque}
\end{figure}

The torques (Fig. \ref{fig:slopeCorr_torque}) appear to be
significantly influenced by the choice of real-space method.  The
torque expressions have the same distance dependence as the energies,
which are naturally longer-ranged expressions than the inter-site
forces.  Torques are also quite sensitive to orientations of
neighboring molecules, even those that are near the cutoff distance.

The results shows that the torque from the hard cutoff method
reproduces the torques in quite good agreement with the Ewald sum.
The other real-space methods can cause some deviations, but excellent
agreement with the Ewald sum torques is recovered at moderate values
of the damping coefficient ($\alpha = 0.1-0.2$\AA$^{-1}$) and cutoff
radius ($r_c \ge 12$\AA).  The TSF method exhibits only fair agreement
in the slope when compared with the Ewald torques even for larger
cutoff radii.  It appears that the severity of the perturbations in
the TSF method are most in evidence for the torques.

\subsection{Directionality of the force and torque vectors}   

The accurate evaluation of force and torque directions is just as
important for molecular dynamics simulations as the magnitudes of
these quantities. Force and torque vectors for all six systems were
analyzed using Fisher statistics, and the quality of the vector
directionality is shown in terms of circular variance
($\mathrm{Var}(\theta)$) in figure
\ref{fig:slopeCorr_circularVariance}. The force and torque vectors
from the new real-space methods exhibit nearly-ideal Fisher probability
distributions (Eq.~\ref{eq:pdf}). Both the hard and SP cutoff methods
exhibit the best vectorial agreement with the Ewald sum. The force and
torque vectors from GSF method also show good agreement with the Ewald
method, which can also be systematically improved by using moderate
damping and a reasonable cutoff radius. For $\alpha = 0.2$ and $r_c =
12$\AA, we observe $\mathrm{Var}(\theta) = 0.00206$, which corresponds
to a distribution with 95\% of force vectors within $6.37^\circ$ of
the corresponding Ewald forces. The TSF method produces the poorest
agreement with the Ewald force directions.

Torques are again more perturbed than the forces by the new real-space
methods, but even here the variance is reasonably small.  For the same
method (GSF) with the same parameters ($\alpha = 0.2, r_c = 12$\AA),
the circular variance was 0.01415, corresponds to a distribution which
has 95\% of torque vectors are within $16.75^\circ$ of the Ewald
results. Again, the direction of the force and torque vectors can be
systematically improved by varying $\alpha$ and $r_c$.

\begin{figure}
  \centering
  \includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps}
  \caption{The circular variance of the direction of the force and
    torque vectors obtained from the real-space methods around the
    reference Ewald vectors. A variance equal to 0 (dashed line)
    indicates direction of the force or torque vectors are
    indistinguishable from those obtained from the Ewald sum. Here
    different symbols represent different values of the cutoff radius
    (9 \AA\ = circle, 12 \AA\ = square, 15 \AA\ = inverted triangle)}
  \label{fig:slopeCorr_circularVariance}
\end{figure}

\subsection{Energy conservation\label{sec:conservation}}

We have tested the conservation of energy one can expect to see with
the new real-space methods using the SSDQ water model with a small
fraction of solvated ions. This is a test system which exercises all
orders of multipole-multipole interactions derived in the first paper
in this series and provides the most comprehensive test of the new
methods.  A liquid-phase system was created with 2000 water molecules
and 48 dissolved ions at a density of 0.98 g cm$^{-3}$ and a
temperature of 300K.  After equilibration, this liquid-phase system
was run for 1 ns under the Ewald, Hard, SP, GSF, and TSF methods with
a cutoff radius of 12\AA.  The value of the damping coefficient was
also varied from the undamped case ($\alpha = 0$) to a heavily damped
case ($\alpha = 0.3$ \AA$^{-1}$) for all of the real space methods.  A
sample was also run using the multipolar Ewald sum with the same
real-space cutoff.

In figure~\ref{fig:energyDrift} we show the both the linear drift in
energy over time, $\delta E_1$, and the standard deviation of energy
fluctuations around this drift $\delta E_0$.  Both of the
shifted-force methods (GSF and TSF) provide excellent energy
conservation (drift less than $10^{-5}$ kcal / mol / ns / particle),
while the hard cutoff is essentially unusable for molecular dynamics.
SP provides some benefit over the hard cutoff because the energetic
jumps that happen as particles leave and enter the cutoff sphere are
somewhat reduced, but like the Wolf method for charges, the SP method
would not be as useful for molecular dynamics as either of the
shifted-force methods.

We note that for all tested values of the cutoff radius, the new
real-space methods can provide better energy conservation behavior
than the multipolar Ewald sum, even when utilizing a relatively large
$k$-space cutoff values.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{newDrift_12.eps}
\label{fig:energyDrift}        
\caption{Analysis of the energy conservation of the real-space
  electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in
  energy over time (in kcal / mol / particle / ns) and $\delta
  \mathrm{E}_0$ is the standard deviation of energy fluctuations
  around this drift (in kcal / mol / particle).  All simulations were
  of a 2000-molecule simulation of SSDQ water with 48 ionic charges at
  300 K starting from the same initial configuration. All runs
  utilized the same real-space cutoff, $r_c = 12$\AA.}
\end{figure} 

\subsection{Reproduction of Structural Features\label{sec:structure}}
One of the best tests of modified interaction potentials is the
fidelity with which they can reproduce structural features in a
liquid.  One commonly-utilized measure of structural ordering is the
pair distribution function, $g(r)$, which measures local density
deviations in relation to the bulk density.  In the electrostatic
approaches studied here, the short-range repulsion from the
Lennard-Jones potential is identical for the various electrostatic
methods, and since short range repulsion determines much of the local
liquid ordering, one would not expect to see any differences in
$g(r)$.  Indeed, the pair distributions are essentially identical for
all of the electrostatic methods studied (for each of the different
systems under investigation).  Interested readers may consult the
supplementary information for plots of these pair distribution
functions.

A direct measure of the structural features that is a more
enlightening test of the modified electrostatic methods is the average
value of the electrostatic energy $\langle U_\mathrm{elect} \rangle$
which is obtained by sampling the liquid-state configurations
experienced by a liquid evolving entirely under the influence of the
methods being investigated.  In figure \ref{fig:Uelect} we show how
$\langle U_\mathrm{elect} \rangle$ for varies with the damping parameter,
$\alpha$, for each of the methods.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{averagePotentialEnergy_r9_12.eps}
\label{fig:Uelect}        
\caption{The average electrostatic potential energy, 
  $\langle U_\mathrm{elect} \rangle$ for the SSDQ water with ions as a function
  of the damping parameter, $\alpha$, for each of the real-space
  electrostatic methods. Top panel: simulations run with a real-space
  cutoff, $r_c = 9$\AA.  Bottom panel: the same quantity, but with a
  larger cutoff, $r_c = 12$\AA.}
\end{figure} 

It is clear that moderate damping is important for converging the mean
potential energy values, particularly for the two shifted force
methods (GSF and TSF).  A value of $\alpha \approx 0.18$ \AA$^{-1}$ is
sufficient to converge the SP and GSF energies with a cutoff of 12
\AA, while shorter cutoffs require more dramatic damping ($\alpha
\approx 0.36$ \AA$^{-1}$ for $r_c = 9$ \AA).  It is also clear from
fig. \ref{fig:Uelect} that it is possible to overdamp the real-space
electrostatic methods, causing the estimate of the energy to drop
below the Ewald results.

These ``optimal'' values of the damping coefficient are slightly
larger than what were observed for DSF electrostatics for purely
point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for
$r_c = 12$\AA appears to be an excellent compromise for mixed charge
multipole systems.

\subsection{Reproduction of Dynamic Properties\label{sec:structure}}
To test the fidelity of the electrostatic methods at reproducing
dynamics in a multipolar liquid, it is also useful to look at
transport properties, particularly the diffusion constant,
\begin{equation}
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left|
  \mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle
\label{eq:diff}
\end{equation}
which measures long-time behavior and is sensitive to the forces on
the multipoles.  For the soft dipolar fluid, and the SSDQ liquid
systems, the self-diffusion constants (D) were calculated from linear
fits to the long-time portion of the mean square displacement
($\langle r^{2}(t) \rangle$).\cite{Allen87}

In addition to translational diffusion, orientational relaxation times
were calculated for comparisons with the Ewald simulations and with
experiments. These values were determined from the same 1~ns $NVE$
trajectories used for translational diffusion by calculating the
orientational time correlation function,
\begin{equation}
C_l^\alpha(t) = \left\langle P_l\left[\hat{\mathbf{u}}_i^\gamma(t)
                \cdot\hat{\mathbf{u}}_i^\gamma(0)\right]\right\rangle,
\label{eq:OrientCorr}
\end{equation}
where $P_l$ is the Legendre polynomial of order $l$ and
$\hat{\mathbf{u}}_i^\alpha$ is the unit vector of molecule $i$ along
axis $\gamma$.  The body-fixed reference frame used for our
orientational correlation functions has the $z$-axis running along the
dipoles, and for the SSDQ water model, the $y$-axis connects the two
implied hydrogen atoms.

From the orientation autocorrelation functions, we can obtain time
constants for rotational relaxation either by fitting an exponential
function or by integrating the entire correlation function.  These
decay times are directly comparable to water orientational relaxation
times from nuclear magnetic resonance (NMR). The relaxation constant
obtained from $C_2^y(t)$ is normally of experimental interest because
it describes the relaxation of the principle axis connecting the
hydrogen atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular
portion of the dipole-dipole relaxation from a proton NMR signal and
should provide an estimate of the NMR relaxation time
constant.\cite{Impey82}

Results for the diffusion constants and orientational relaxation times
are shown in figure \ref{fig:dynamics}. From this data, it is apparent
that the values for both $D$ and $\tau_2$ using the Ewald sum are
reproduced with high fidelity by the GSF method. 

The $\tau_2$ results in \ref{fig:dynamics} show a much greater
difference between the real-space and the Ewald results.


\section{CONCLUSION}
In the first paper in this series, we generalized the
charge-neutralized electrostatic energy originally developed by Wolf
\textit{et al.}\cite{Wolf:1999dn} to multipole-multipole interactions
up to quadrupolar order.  The SP method is essentially a
multipole-capable version of the Wolf model.  The SP method for
multipoles provides excellent agreement with Ewald-derived energies,
forces and torques, and is suitable for Monte Carlo simulations,
although the forces and torques retain discontinuities at the cutoff
distance that prevents its use in molecular dynamics.

We also developed two natural extensions of the damped shifted-force
(DSF) model originally proposed by Fennel and
Gezelter.\cite{Fennell:2006lq} The GSF and TSF approaches provide
smooth truncation of energies, forces, and torques at the real-space
cutoff, and both converge to DSF electrostatics for point-charge
interactions.  The TSF model is based on a high-order truncated Taylor
expansion which can be relatively perturbative inside the cutoff
sphere.  The GSF model takes the gradient from an images of the
interacting multipole that has been projected onto the cutoff sphere
to derive shifted force and torque expressions, and is a significantly
more gentle approach.

Of the two newly-developed shifted force models, the GSF method
produced quantitative agreement with Ewald energy, force, and torques.
It also performs well in conserving energy in MD simulations.  The
Taylor-shifted (TSF) model provides smooth dynamics, but these take
place on a potential energy surface that is significantly perturbed
from Ewald-based electrostatics.  

% The direct truncation of any electrostatic potential energy without
% multipole neutralization creates large fluctuations in molecular
% simulations.  This fluctuation in the energy is very large for the case
% of crystal because of long range of multipole ordering (Refer paper
% I).\cite{PaperI} This is also significant in the case of the liquid
% because of the local multipole ordering in the molecules. If the net
% multipole within cutoff radius neutralized within cutoff sphere by
% placing image multiples on the surface of the sphere, this fluctuation
% in the energy reduced significantly. Also, the multipole
% neutralization in the generalized SP method showed very good agreement
% with the Ewald as compared to direct truncation for the evaluation of
% the $\triangle E$ between the configurations.  In MD simulations, the
% energy conservation is very important. The conservation of the total
% energy can be ensured by i) enforcing the smooth truncation of the
% energy, force and torque in the cutoff radius and ii) making the
% energy, force and torque consistent with each other. The GSF and TSF
% methods ensure the consistency and smooth truncation of the energy,
% force and torque at the cutoff radius, as a result show very good
% total energy conservation. But the TSF method does not show good
% agreement in the absolute value of the electrostatic energy, force and
% torque with the Ewald.  The GSF method has mimicked Ewald’s force,
% energy and torque accurately and also conserved energy. 

The only cases we have found where the new GSF and SP real-space
methods can be problematic are those which retain a bulk dipole moment
at large distances (e.g. the $Z_1$ dipolar lattice).  In ferroelectric
materials, uniform weighting of the orientational contributions can be
important for converging the total energy.  In these cases, the
damping function which causes the non-uniform weighting can be
replaced by the bare electrostatic kernel, and the energies return to
the expected converged values.

Based on the results of this work, the GSF method is a suitable and
efficient replacement for the Ewald sum for evaluating electrostatic
interactions in MD simulations.  Both methods retain excellent
fidelity to the Ewald energies, forces and torques.  Additionally, the
energy drift and fluctuations from the GSF electrostatics are better
than a multipolar Ewald sum for finite-sized reciprocal spaces.
Because they use real-space cutoffs with moderate cutoff radii, the
GSF and SP models approach $\mathcal{O}(N)$ scaling as the system size
increases.  Additionally, they can be made extremely efficient using
spline interpolations of the radial functions.  They require no
Fourier transforms or $k$-space sums, and guarantee the smooth
handling of energies, forces, and torques as multipoles cross the
real-space cutoff boundary.

\begin{acknowledgments}
  JDG acknowledges helpful discussions with Christopher
  Fennell. Support for this project was provided by the National
  Science Foundation under grant CHE-1362211. Computational time was
  provided by the Center for Research Computing (CRC) at the
  University of Notre Dame.
\end{acknowledgments}

%\bibliographystyle{aip}
\newpage
\bibliography{references}
\end{document}

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