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\frontmatter

\title{APPLICATION AND DEVELOPMENT OF MOLECULAR DYNAMICS TECHNIQUES FOR THE 
STUDY OF WATER}     
\author{Christopher Joseph Fennell} 
\work{Dissertation}
\degprior{B.Sc.}
\degaward{Doctor of Philosophy}
\advisor{J. Daniel Gezelter}
\department{Chemistry and Biochemistry}

\maketitle

\begin{abstract}
\end{abstract}

\begin{dedication}
\end{dedication}

\tableofcontents
\listoffigures
\listoftables

\begin{acknowledge}
\end{acknowledge}

\mainmatter

\chapter{\label{chap:intro}INTRODUCTION AND BACKGROUND}

\chapter{\label{chap:electrostatics}ELECTROSTATIC INTERACTION CORRECTION 
TECHNIQUES}

In molecular simulations, proper accumulation of the electrostatic
interactions is essential and is one of the most
computationally-demanding tasks.  The common molecular mechanics force
fields represent atomic sites with full or partial charges protected
by Lennard-Jones (short range) interactions.  This means that nearly
every pair interaction involves a calculation of charge-charge forces.
Coupled with relatively long-ranged $r^{-1}$ decay, the monopole
interactions quickly become the most expensive part of molecular
simulations.  Historically, the electrostatic pair interaction would
not have decayed appreciably within the typical box lengths that could
be feasibly simulated.  In the larger systems that are more typical of
modern simulations, large cutoffs should be used to incorporate
electrostatics correctly.

There have been many efforts to address the proper and practical
handling of electrostatic interactions, and these have resulted in a
variety of techniques.\cite{Roux99,Sagui99,Tobias01} These are
typically classified as implicit methods (i.e., continuum dielectrics,
static dipolar fields),\cite{Born20,Grossfield00} explicit methods
(i.e., Ewald summations, interaction shifting or
truncation),\cite{Ewald21,Steinbach94} or a mixture of the two (i.e.,
reaction field type methods, fast multipole
methods).\cite{Onsager36,Rokhlin85} The explicit or mixed methods are
often preferred because they physically incorporate solvent molecules
in the system of interest, but these methods are sometimes difficult
to utilize because of their high computational cost.\cite{Roux99} In
addition to the computational cost, there have been some questions
regarding possible artifacts caused by the inherent periodicity of the
explicit Ewald summation.\cite{Tobias01}

In this chapter, we focus on a new set of pairwise methods devised by
Wolf {\it et al.},\cite{Wolf99} which we further extend.  These
methods along with a few other mixed methods (i.e. reaction field) are
compared with the smooth particle mesh Ewald
sum,\cite{Onsager36,Essmann99} which is our reference method for
handling long-range electrostatic interactions. The new methods for
handling electrostatics have the potential to scale linearly with
increasing system size since they involve only a simple modification
to the direct pairwise sum.  They also lack the added periodicity of
the Ewald sum, so they can be used for systems which are non-periodic
or which have one- or two-dimensional periodicity.  Below, these
methods are evaluated using a variety of model systems to
establish their usability in molecular simulations.

\section{The Ewald Sum}

The complete accumulation of the electrostatic interactions in a system with
periodic boundary conditions (PBC) requires the consideration of the
effect of all charges within a (cubic) simulation box as well as those
in the periodic replicas,
\begin{equation}
V_\textrm{elec} = \frac{1}{2} {\sum_{\mathbf{n}}}^\prime 
\left[ \sum_{i=1}^N\sum_{j=1}^N \phi
\left( \mathbf{r}_{ij} + L\mathbf{n},\bm{\Omega}_i,\bm{\Omega}_j\right) 
\right],
\label{eq:PBCSum}
\end{equation}
where the sum over $\mathbf{n}$ is a sum over all periodic box
replicas with integer coordinates $\mathbf{n} = (l,m,n)$, and the
prime indicates $i = j$ are neglected for $\mathbf{n} =
0$.\cite{deLeeuw80} Within the sum, $N$ is the number of electrostatic
particles, $\mathbf{r}_{ij}$ is $\mathbf{r}_j - \mathbf{r}_i$, $L$ is
the cell length, $\bm{\Omega}_{i,j}$ are the Euler angles for $i$ and
$j$, and $\phi$ is the solution to Poisson's equation
($\phi(\mathbf{r}_{ij}) = q_i q_j |\mathbf{r}_{ij}|^{-1}$ for
charge-charge interactions). In the case of monopole electrostatics,
eq. (\ref{eq:PBCSum}) is conditionally convergent and is divergent for
non-neutral systems.

The electrostatic summation problem was originally studied by Ewald
for the case of an infinite crystal.\cite{Ewald21}. The approach he
took was to convert this conditionally convergent sum into two
absolutely convergent summations: a short-ranged real-space summation
and a long-ranged reciprocal-space summation,
\begin{equation}
\begin{split}
V_\textrm{elec} = \frac{1}{2}& 
\sum_{i=1}^N\sum_{j=1}^N \Biggr[ q_i q_j\Biggr( {\sum_{\mathbf{n}}}^\prime 
\frac{\textrm{erfc}\left( \alpha|\mathbf{r}_{ij}+\mathbf{n}|\right)}
{|\mathbf{r}_{ij}+\mathbf{n}|} \\ 
&+ \frac{1}{\pi L^3}\sum_{\mathbf{k}\ne 0}\frac{4\pi^2}{|\mathbf{k}|^2} 
\exp{\left(-\frac{\pi^2|\mathbf{k}|^2}{\alpha^2}\right) 
\cos\left(\mathbf{k}\cdot\mathbf{r}_{ij}\right)}\Biggr)\Biggr] \\ 
&- \frac{\alpha}{\pi^{1/2}}\sum_{i=1}^N q_i^2 
+ \frac{2\pi}{(2\epsilon_\textrm{S}+1)L^3}
\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2,
\end{split}
\label{eq:EwaldSum}
\end{equation}
where $\alpha$ is the damping or convergence parameter with units of
\AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and are equal to
$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric
constant of the surrounding medium. The final two terms of
eq. (\ref{eq:EwaldSum}) are a particle-self term and a dipolar term
for interacting with a surrounding dielectric.\cite{Allen87} This
dipolar term was neglected in early applications in molecular
simulations,\cite{Brush66,Woodcock71} until it was introduced by de
Leeuw {\it et al.} to address situations where the unit cell has a
dipole moment which is magnified through replication of the periodic
images.\cite{deLeeuw80,Smith81} If this term is taken to be zero, the
system is said to be using conducting (or ``tin-foil'') boundary
conditions, $\epsilon_{\rm S} = \infty$. Figure
\ref{fig:ewaldTime} shows how the Ewald sum has been applied over
time.  Initially, due to the small system sizes that could be
simulated feasibly, the entire simulation box was replicated to
convergence.  In more modern simulations, the systems have grown large
enough that a real-space cutoff could potentially give convergent
behavior.  Indeed, it has been observed that with the choice of a
small $\alpha$, the reciprocal-space portion of the Ewald sum can be
rapidly convergent and small relative to the real-space
portion.\cite{Karasawa89,Kolafa92}

\begin{figure}
\includegraphics[width=\linewidth]{./figures/ewaldProgression.pdf}
\caption{The change in the need for the Ewald sum with
increasing computational power.  A:~Initially, only small systems
could be studied, and the Ewald sum replicated the simulation box to
convergence.  B:~Now, radial cutoff methods should be able to reach
convergence for the larger systems of charges that are common today.}
\label{fig:ewaldTime}
\end{figure}

The original Ewald summation is an $\mathscr{O}(N^2)$ algorithm.  The
convergence parameter $(\alpha)$ plays an important role in balancing
the computational cost between the direct and reciprocal-space
portions of the summation.  The choice of this value allows one to
select whether the real-space or reciprocal space portion of the
summation is an $\mathscr{O}(N^2)$ calculation (with the other being
$\mathscr{O}(N)$).\cite{Sagui99} With the appropriate choice of
$\alpha$ and thoughtful algorithm development, this cost can be
reduced to $\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route
taken to reduce the cost of the Ewald summation even further is to set
$\alpha$ such that the real-space interactions decay rapidly, allowing
for a short spherical cutoff. Then the reciprocal space summation is
optimized.  These optimizations usually involve utilization of the
fast Fourier transform (FFT),\cite{Hockney81} leading to the
particle-particle particle-mesh (P3M) and particle mesh Ewald (PME)
methods.\cite{Shimada93,Luty94,Luty95,Darden93,Essmann95} In these
methods, the cost of the reciprocal-space portion of the Ewald
summation is reduced from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N
\log N)$.

These developments and optimizations have made the use of the Ewald
summation routine in simulations with periodic boundary
conditions. However, in certain systems, such as vapor-liquid
interfaces and membranes, the intrinsic three-dimensional periodicity
can prove problematic.  The Ewald sum has been reformulated to handle
2-D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} but these
methods are computationally expensive.\cite{Spohr97,Yeh99} More
recently, there have been several successful efforts toward reducing
the computational cost of 2-D lattice
summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04}
bringing them more in line with the cost of the full 3-D summation.

Several studies have recognized that the inherent periodicity in the
Ewald sum can also have an effect on three-dimensional
systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00}
Solvated proteins are essentially kept at high concentration due to
the periodicity of the electrostatic summation method.  In these
systems, the more compact folded states of a protein can be
artificially stabilized by the periodic replicas introduced by the
Ewald summation.\cite{Weber00} Thus, care must be taken when
considering the use of the Ewald summation where the assumed
periodicity would introduce spurious effects in the system dynamics.


\section{The Wolf and Zahn Methods}

In a recent paper by Wolf \textit{et al.}, a procedure was outlined
for the accurate accumulation of electrostatic interactions in an
efficient pairwise fashion.  This procedure lacks the inherent
periodicity of the Ewald summation.\cite{Wolf99} Wolf \textit{et al.}
observed that the electrostatic interaction is effectively
short-ranged in condensed phase systems and that neutralization of the
charge contained within the cutoff radius is crucial for potential
stability. They devised a pairwise summation method that ensures
charge neutrality and gives results similar to those obtained with the
Ewald summation.  The resulting shifted Coulomb potential includes
image-charges subtracted out through placement on the cutoff sphere
and a distance-dependent damping function (identical to that seen in
the real-space portion of the Ewald sum) to aid convergence
\begin{equation}
V_{\textrm{Wolf}}(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}
- \lim_{r_{ij}\rightarrow R_\textrm{c}}
\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}.
\label{eq:WolfPot}
\end{equation}
Eq. (\ref{eq:WolfPot}) is essentially the common form of a shifted
potential.  However, neutralizing the charge contained within each
cutoff sphere requires the placement of a self-image charge on the
surface of the cutoff sphere.  This additional self-term in the total
potential enabled Wolf {\it et al.}  to obtain excellent estimates of
Madelung energies for many crystals.

In order to use their charge-neutralized potential in molecular
dynamics simulations, Wolf \textit{et al.} suggested taking the
derivative of this potential prior to evaluation of the limit.  This
procedure gives an expression for the forces,
\begin{equation}
\begin{split}
F_{\textrm{Wolf}}(r_{ij}) = q_i q_j \Biggr\{& 
\Biggr[\frac{\textrm{erfc}\left(\alpha r_{ij}\right)}{r^2_{ij}} 
+ \frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r_{ij}^2\right)}}{r_{ij}}
\Biggr]\\ 
&-\Biggr[
\frac{\textrm{erfc}\left(\alpha R_{\textrm{c}}\right)}{R_{\textrm{c}}^2} 
+ \frac{2\alpha}{\pi^{1/2}}
\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}
\Biggr]\Biggr\},
\end{split}
\label{eq:WolfForces}
\end{equation}
that incorporates both image charges and damping of the electrostatic
interaction. 

More recently, Zahn \textit{et al.} investigated these potential and
force expressions for use in simulations involving water.\cite{Zahn02}
In their work, they pointed out that the forces and derivative of
the potential are not commensurate.  Attempts to use both
eqs. (\ref{eq:WolfPot}) and (\ref{eq:WolfForces}) together will lead
to poor energy conservation.  They correctly observed that taking the
limit shown in equation (\ref{eq:WolfPot}) {\it after} calculating the
derivatives gives forces for a different potential energy function
than the one shown in eq. (\ref{eq:WolfPot}).

Zahn \textit{et al.} introduced a modified form of this summation
method as a way to use the technique in Molecular Dynamics
simulations.  They proposed a new damped Coulomb potential,
\begin{equation}
\begin{split}
V_{\textrm{Zahn}}(r_{ij}) = q_iq_j\Biggr\{&
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} \\
&- \left[\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2}
+ \frac{2\alpha}{\pi^{1/2}}
\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}}
\right]\left(r_{ij}-R_\mathrm{c}\right)\Biggr\},
\end{split}
\label{eq:ZahnPot}
\end{equation}
and showed that this potential does fairly well at capturing the
structural and dynamic properties of water compared the same
properties obtained using the Ewald sum.

\section{Simple Forms for Pairwise Electrostatics}

The potentials proposed by Wolf \textit{et al.} and Zahn \textit{et
al.} are constructed using two different (and separable) computational
tricks: 

\begin{enumerate}
\item shifting through the use of image charges, and 
\item damping the electrostatic interaction.
\end{enumerate}  
Wolf \textit{et al.} treated the
development of their summation method as a progressive application of
these techniques,\cite{Wolf99} while Zahn \textit{et al.} founded
their damped Coulomb modification (Eq. (\ref{eq:ZahnPot})) on the
post-limit forces (Eq. (\ref{eq:WolfForces})) which were derived using
both techniques.  It is possible, however, to separate these
tricks and study their effects independently. 

Starting with the original observation that the effective range of the
electrostatic interaction in condensed phases is considerably less
than $r^{-1}$, either the cutoff sphere neutralization or the
distance-dependent damping technique could be used as a foundation for
a new pairwise summation method.  Wolf \textit{et al.} made the
observation that charge neutralization within the cutoff sphere plays
a significant role in energy convergence; therefore we will begin our
analysis with the various shifted forms that maintain this charge
neutralization.  We can evaluate the methods of Wolf
\textit{et al.}  and Zahn \textit{et al.} by considering the standard
shifted potential,
\begin{equation}
V_\textrm{SP}(r) =      \begin{cases}
v(r)-v_\textrm{c} &\quad r\leqslant R_\textrm{c} \\ 0 &\quad r >
R_\textrm{c}  
\end{cases},
\label{eq:shiftingPotForm}
\end{equation}
and shifted force,
\begin{equation}
V_\textrm{SF}(r) = \begin{cases}
v(r) - v_\textrm{c}
- \left(\frac{d v(r)}{d r}\right)_{r=R_\textrm{c}}(r-R_\textrm{c})
&\quad r\leqslant R_\textrm{c} \\ 0 &\quad r > R_\textrm{c} 
\end{cases},
\label{eq:shiftingForm}
\end{equation}
functions where $v(r)$ is the unshifted form of the potential, and
$v_c$ is $v(R_\textrm{c})$.  The Shifted Force ({\sc sf}) form ensures
that both the potential and the forces goes to zero at the cutoff
radius, while the Shifted Potential ({\sc sp}) form only ensures the
potential is smooth at the cutoff radius
($R_\textrm{c}$).\cite{Allen87}

The forces associated with the shifted potential are simply the forces
of the unshifted potential itself (when inside the cutoff sphere),
\begin{equation}
F_{\textrm{SP}} = -\left( \frac{d v(r)}{dr} \right),
\end{equation}
and are zero outside.  Inside the cutoff sphere, the forces associated
with the shifted force form can be written,
\begin{equation}
F_{\textrm{SF}} = -\left( \frac{d v(r)}{dr} \right) + \left(\frac{d
v(r)}{dr} \right)_{r=R_\textrm{c}}.
\end{equation}
 
If the potential, $v(r)$, is taken to be the normal Coulomb potential,
\begin{equation}
v(r) = \frac{q_i q_j}{r},
\label{eq:Coulomb}
\end{equation}
then the Shifted Potential ({\sc sp}) forms will give Wolf {\it et
al.}'s undamped prescription:
\begin{equation}
V_\textrm{SP}(r) =
q_iq_j\left(\frac{1}{r}-\frac{1}{R_\textrm{c}}\right) \quad
r\leqslant R_\textrm{c},
\label{eq:SPPot}
\end{equation}
with associated forces,
\begin{equation}
F_\textrm{SP}(r) = q_iq_j\left(\frac{1}{r^2}\right) 
\quad r\leqslant R_\textrm{c}.
\label{eq:SPForces}
\end{equation}
These forces are identical to the forces of the standard Coulomb
interaction, and cutting these off at $R_c$ was addressed by Wolf
\textit{et al.} as undesirable.  They pointed out that the effect of
the image charges is neglected in the forces when this form is
used,\cite{Wolf99} thereby eliminating any benefit from the method in
molecular dynamics.  Additionally, there is a discontinuity in the
forces at the cutoff radius which results in energy drift during MD
simulations.

The shifted force ({\sc sf}) form using the normal Coulomb potential
will give,
\begin{equation}
V_\textrm{SF}(r) = q_iq_j\left[\frac{1}{r}-\frac{1}{R_\textrm{c}} 
+ \left(\frac{1}{R_\textrm{c}^2}\right)(r-R_\textrm{c})\right] 
\quad r\leqslant R_\textrm{c}.
\label{eq:SFPot}
\end{equation}
with associated forces,
\begin{equation}
F_\textrm{SF}(r) =  q_iq_j\left(\frac{1}{r^2}-\frac{1}{R_\textrm{c}^2}\right) 
\quad r\leqslant R_\textrm{c}.
\label{eq:SFForces}
\end{equation}
This formulation has the benefits that there are no discontinuities at
the cutoff radius, while the neutralizing image charges are present in
both the energy and force expressions.  It would be simple to add the
self-neutralizing term back when computing the total energy of the
system, thereby maintaining the agreement with the Madelung energies.
A side effect of this treatment is the alteration in the shape of the
potential that comes from the derivative term.  Thus, a degree of
clarity about agreement with the empirical potential is lost in order
to gain functionality in dynamics simulations.

Wolf \textit{et al.} originally discussed the energetics of the
shifted Coulomb potential (Eq. \ref{eq:SPPot}) and found that it was
insufficient for accurate determination of the energy with reasonable
cutoff distances.  The calculated Madelung energies fluctuated around
the expected value as the cutoff radius was increased, but the
oscillations converged toward the correct value.\cite{Wolf99} A
damping function was incorporated to accelerate the convergence; and
though alternative forms for the damping function could be
used,\cite{Jones56,Heyes81} the complimentary error function was
chosen to mirror the effective screening used in the Ewald summation.
Incorporating this error function damping into the simple Coulomb
potential,
\begin{equation}
v(r) = \frac{\mathrm{erfc}\left(\alpha r\right)}{r},
\label{eq:dampCoulomb}
\end{equation}
the shifted potential (eq. (\ref{eq:SPPot})) becomes
\begin{equation}
V_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r}
- \frac{\textrm{erfc}\left(\alpha R_\textrm{c}\right)}{R_\textrm{c}}\right) 
\quad r\leqslant R_\textrm{c},
\label{eq:DSPPot}
\end{equation}
with associated forces,
\begin{equation}
F_{\textrm{DSP}}(r) = q_iq_j
\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r^2} 
+ \frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r^2\right)}}{r}\right) 
\quad r\leqslant R_\textrm{c}.
\label{eq:DSPForces}
\end{equation}
Again, this damped shifted potential suffers from a
force-discontinuity at the cutoff radius, and the image charges play
no role in the forces.  To remedy these concerns, one may derive a
{\sc sf} variant by including the derivative term in
eq. (\ref{eq:shiftingForm}),
\begin{equation}
\begin{split}
V_\mathrm{DSF}(r) = q_iq_j\Biggr{[}&
\frac{\mathrm{erfc}\left(\alpha r\right)}{r} 
- \frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}} \\ 
&+ \left(\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2} 
+ \frac{2\alpha}{\pi^{1/2}}
\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}}
\right)\left(r-R_\mathrm{c}\right)\ \Biggr{]} 
\quad r\leqslant R_\textrm{c}.
\label{eq:DSFPot}
\end{split}
\end{equation}
The derivative of the above potential will lead to the following forces,
\begin{equation}
\begin{split}
F_\mathrm{DSF}(r) = q_iq_j\Biggr{[}&
\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r^2}
+ \frac{2\alpha}{\pi^{1/2}}\frac{\exp{\left(-\alpha^2r^2\right)}}{r}\right) \\ 
&- \left(\frac{\textrm{erfc}\left(\alpha R_{\textrm{c}}\right)}
{R_{\textrm{c}}^2}
+ \frac{2\alpha}{\pi^{1/2}}
\frac{\exp{\left(-\alpha^2R_{\textrm{c}}^2\right)}}{R_{\textrm{c}}}
\right)\Biggr{]} 
\quad r\leqslant R_\textrm{c}.
\label{eq:DSFForces}
\end{split}
\end{equation}
If the damping parameter $(\alpha)$ is set to zero, the undamped case,
eqs. (\ref{eq:SPPot} through \ref{eq:SFForces}) are correctly
recovered from eqs. (\ref{eq:DSPPot} through \ref{eq:DSFForces}).

This new {\sc sf} potential is similar to equation \ref{eq:ZahnPot}
derived by Zahn \textit{et al.}; however, there are two important
differences.\cite{Zahn02} First, the $v_\textrm{c}$ term from
eq. (\ref{eq:shiftingForm}) is equal to eq. (\ref{eq:dampCoulomb})
with $r$ replaced by $R_\textrm{c}$.  This term is {\it not} present
in the Zahn potential, resulting in a potential discontinuity as
particles cross $R_\textrm{c}$.  Second, the sign of the derivative
portion is different.  The missing $v_\textrm{c}$ term would not
affect molecular dynamics simulations (although the computed energy
would be expected to have sudden jumps as particle distances crossed
$R_c$).  The sign problem is a potential source of errors, however.
In fact, it introduces a discontinuity in the forces at the cutoff,
because the force function is shifted in the wrong direction and
doesn't cross zero at $R_\textrm{c}$.

Eqs. (\ref{eq:DSFPot}) and (\ref{eq:DSFForces}) result in an
electrostatic summation method in which the potential and forces are
continuous at the cutoff radius and which incorporates the damping
function proposed by Wolf \textit{et al.}\cite{Wolf99} In the rest of
this paper, we will evaluate exactly how good these methods ({\sc sp},
{\sc sf}, damping) are at reproducing the correct electrostatic
summation performed by the Ewald sum.


\section{Evaluating Pairwise Summation Techniques}

In classical molecular mechanics simulations, there are two primary
techniques utilized to obtain information about the system of
interest: Monte Carlo (MC) and Molecular Dynamics (MD).  Both of these
techniques utilize pairwise summations of interactions between
particle sites, but they use these summations in different ways.

In MC, the potential energy difference between configurations dictates
the progression of MC sampling.  Going back to the origins of this
method, the acceptance criterion for the canonical ensemble laid out
by Metropolis \textit{et al.} states that a subsequent configuration
is accepted if $\Delta E < 0$ or if $\xi < \exp(-\Delta E/kT)$, where
$\xi$ is a random number between 0 and 1.\cite{Metropolis53}
Maintaining the correct $\Delta E$ when using an alternate method for
handling the long-range electrostatics will ensure proper sampling
from the ensemble.

In MD, the derivative of the potential governs how the system will
progress in time.  Consequently, the force and torque vectors on each
body in the system dictate how the system evolves.  If the magnitude
and direction of these vectors are similar when using alternate
electrostatic summation techniques, the dynamics in the short term
will be indistinguishable.  Because error in MD calculations is
cumulative, one should expect greater deviation at longer times,
although methods which have large differences in the force and torque
vectors will diverge from each other more rapidly.

\subsection{Monte Carlo and the Energy Gap}\label{sec:MCMethods}

The pairwise summation techniques (outlined in section
\ref{sec:ESMethods}) were evaluated for use in MC simulations by 
studying the energy differences between conformations.  We took the
{\sc spme}-computed energy difference between two conformations to be the
correct behavior. An ideal performance by an alternative method would
reproduce these energy differences exactly (even if the absolute
energies calculated by the methods are different).  Since none of the
methods provide exact energy differences, we used linear least squares
regressions of energy gap data to evaluate how closely the methods
mimicked the Ewald energy gaps.  Unitary results for both the
correlation (slope) and correlation coefficient for these regressions
indicate perfect agreement between the alternative method and {\sc spme}.
Sample correlation plots for two alternate methods are shown in
Fig. \ref{fig:linearFit}.

\begin{figure}
\centering
\includegraphics[width = \linewidth]{./figures/dualLinear.pdf}
\caption{Example least squares regressions of the configuration energy
differences for SPC/E water systems. The upper plot shows a data set
with a poor correlation coefficient ($R^2$), while the lower plot
shows a data set with a good correlation coefficient.} 
\label{fig:linearFit} 
\end{figure}

Each of the seven system types (detailed in section \ref{sec:RepSims})
were represented using 500 independent configurations.  Thus, each of
the alternative (non-Ewald) electrostatic summation methods was
evaluated using an accumulated 873,250 configurational energy
differences.

Results and discussion for the individual analysis of each of the
system types appear in sections \ref{sec:SystemResults}, while the
cumulative results over all the investigated systems appear below in
sections \ref{sec:EnergyResults}.

\subsection{Molecular Dynamics and the Force and Torque 
Vectors}\label{sec:MDMethods} We evaluated the pairwise methods
(outlined in section \ref{sec:ESMethods}) for use in MD simulations by
comparing the force and torque vectors with those obtained using the
reference Ewald summation ({\sc spme}).  Both the magnitude and the
direction of these vectors on each of the bodies in the system were
analyzed.  For the magnitude of these vectors, linear least squares
regression analyses were performed as described previously for
comparing $\Delta E$ values.  Instead of a single energy difference
between two system configurations, we compared the magnitudes of the
forces (and torques) on each molecule in each configuration.  For a
system of 1000 water molecules and 40 ions, there are 1040 force
vectors and 1000 torque vectors.  With 500 configurations, this
results in 520,000 force and 500,000 torque vector comparisons.
Additionally, data from seven different system types was aggregated
before the comparison was made.

The {\it directionality} of the force and torque vectors was
investigated through measurement of the angle ($\theta$) formed
between those computed from the particular method and those from {\sc spme},
\begin{equation}
\theta_f = \cos^{-1} \left(\hat{F}_\textrm{SPME} 
\cdot \hat{F}_\textrm{M}\right),
\end{equation}
where $\hat{F}_\textrm{M}$ is the unit vector pointing along the force
vector computed using method M.  Each of these $\theta$ values was
accumulated in a distribution function and weighted by the area on the
unit sphere.  Since this distribution is a measure of angular error
between two different electrostatic summation methods, there is no
{\it a priori} reason for the profile to adhere to any specific
shape. Thus, gaussian fits were used to measure the width of the
resulting distributions. The variance ($\sigma^2$) was extracted from
each of these fits and was used to compare distribution widths.
Values of $\sigma^2$ near zero indicate vector directions
indistinguishable from those calculated when using the reference
method ({\sc spme}).

\subsection{Short-time Dynamics}

The effects of the alternative electrostatic summation methods on the
short-time dynamics of charged systems were evaluated by considering a
NaCl crystal at a temperature of 1000 K.  A subset of the best
performing pairwise methods was used in this comparison.  The NaCl
crystal was chosen to avoid possible complications from the treatment
of orientational motion in molecular systems.  All systems were
started with the same initial positions and velocities.  Simulations
were performed under the microcanonical ensemble, and velocity
autocorrelation functions (Eq. \ref{eq:vCorr}) were computed for each
of the trajectories,
\begin{equation}
C_v(t) = \frac{\langle v(0)\cdot v(t)\rangle}{\langle v^2\rangle}.
\label{eq:vCorr}
\end{equation}
Velocity autocorrelation functions require detailed short time data,
thus velocity information was saved every 2 fs over 10 ps
trajectories. Because the NaCl crystal is composed of two different
atom types, the average of the two resulting velocity autocorrelation
functions was used for comparisons.

\subsection{Long-Time and Collective Motion}\label{sec:LongTimeMethods}

The effects of the same subset of alternative electrostatic methods on
the {\it long-time} dynamics of charged systems were evaluated using
the same model system (NaCl crystals at 1000K).  The power spectrum
($I(\omega)$) was obtained via Fourier transform of the velocity
autocorrelation function, \begin{equation} I(\omega) =
\frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt,
\label{eq:powerSpec}
\end{equation}
where the frequency, $\omega=0,\ 1,\ ...,\ N-1$. Again, because the
NaCl crystal is composed of two different atom types, the average of
the two resulting power spectra was used for comparisons. Simulations
were performed under the microcanonical ensemble, and velocity
information was saved every 5~fs over 100~ps trajectories. 

\subsection{Representative Simulations}\label{sec:RepSims}
A variety of representative molecular simulations were analyzed to
determine the relative effectiveness of the pairwise summation
techniques in reproducing the energetics and dynamics exhibited by
{\sc spme}.  We wanted to span the space of typical molecular
simulations (i.e. from liquids of neutral molecules to ionic
crystals), so the systems studied were:

\begin{enumerate}
\item liquid water (SPC/E),\cite{Berendsen87}
\item crystalline water (Ice I$_\textrm{c}$ crystals of SPC/E),
\item NaCl crystals,
\item NaCl melts,
\item a low ionic strength solution of NaCl in water (0.11 M),
\item a high ionic strength solution of NaCl in water (1.1 M), and
\item a 6\AA\  radius sphere of Argon in water.
\end{enumerate}

By utilizing the pairwise techniques (outlined in section
\ref{sec:ESMethods}) in systems composed entirely of neutral groups,
charged particles, and mixtures of the two, we hope to discern under
which conditions it will be possible to use one of the alternative
summation methodologies instead of the Ewald sum.

For the solid and liquid water configurations, configurations were
taken at regular intervals from high temperature trajectories of 1000
SPC/E water molecules.  Each configuration was equilibrated
independently at a lower temperature (300K for the liquid, 200K for
the crystal).  The solid and liquid NaCl systems consisted of 500
$\textrm{Na}^{+}$ and 500 $\textrm{Cl}^{-}$ ions.  Configurations for
these systems were selected and equilibrated in the same manner as the
water systems. In order to introduce measurable fluctuations in the
configuration energy differences, the crystalline simulations were
equilibrated at 1000K, near the $T_\textrm{m}$ for NaCl. The liquid
NaCl configurations needed to represent a fully disordered array of
point charges, so the high temperature of 7000K was selected for
equilibration. The ionic solutions were made by solvating 4 (or 40)
ions in a periodic box containing 1000 SPC/E water molecules.  Ion and
water positions were then randomly swapped, and the resulting
configurations were again equilibrated individually.  Finally, for the
Argon / Water ``charge void'' systems, the identities of all the SPC/E
waters within 6\AA\ of the center of the equilibrated water
configurations were converted to argon.

These procedures guaranteed us a set of representative configurations
from chemically-relevant systems sampled from appropriate
ensembles. Force field parameters for the ions and Argon were taken
from the force field utilized by {\sc oopse}.\cite{Meineke05}

\subsection{Comparison of Summation Methods}\label{sec:ESMethods}
We compared the following alternative summation methods with results
from the reference method ({\sc spme}):

\begin{enumerate}
\item {\sc sp} with damping parameters ($\alpha$) of 0.0, 0.1, 0.2,
and 0.3\AA$^{-1}$,
\item {\sc sf} with damping parameters ($\alpha$) of 0.0, 0.1, 0.2,
and 0.3\AA$^{-1}$,
\item reaction field with an infinite dielectric constant, and 
\item an unmodified cutoff.
\end{enumerate}

Group-based cutoffs with a fifth-order polynomial switching function
were utilized for the reaction field simulations.  Additionally, we
investigated the use of these cutoffs with the {\sc sp}, {\sc sf}, and pure
cutoff.  The {\sc spme} electrostatics were performed using the {\sc tinker}
implementation of {\sc spme},\cite{Ponder87} while all other calculations
were performed using the {\sc oopse} molecular mechanics
package.\cite{Meineke05} All other portions of the energy calculation
(i.e. Lennard-Jones interactions) were handled in exactly the same
manner across all systems and configurations.

The alternative methods were also evaluated with three different
cutoff radii (9, 12, and 15\AA).  As noted previously, the
convergence parameter ($\alpha$) 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 dependent 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,Essmann95}
The default {\sc tinker} tolerance of $1 \times 10^{-8}$ kcal/mol was used
in all {\sc spme} calculations, resulting in Ewald coefficients of 0.4200,
0.3119, and 0.2476\AA$^{-1}$ for cutoff radii of 9, 12, and 15\AA\
respectively.


\section{Discussion on the Pairwise Technique Evaluation}

\subsection{Configuration Energy Differences}\label{sec:EnergyResults}
In order to evaluate the performance of the pairwise electrostatic
summation methods for Monte Carlo simulations, the energy differences
between configurations were compared to the values obtained when using
{\sc spme}.  The results for the combined regression analysis of all
of the systems are shown in figure \ref{fig:delE}.

\begin{figure}
\centering
\includegraphics[width=4.75in]{./figures/delEplot.pdf}
\caption{Statistical analysis of the quality of configurational energy
differences for a given electrostatic method compared with the
reference Ewald sum.  Results with a value equal to 1 (dashed line)
indicate $\Delta E$ values indistinguishable from those obtained using
{\sc spme}.  Different values of the cutoff radius are indicated with
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
inverted triangles).} 
\label{fig:delE}
\end{figure}

The most striking feature of this plot is how well the Shifted Force
({\sc sf}) and Shifted Potential ({\sc sp}) methods capture the energy
differences.  For the undamped {\sc sf} method, and the
moderately-damped {\sc sp} methods, the results are nearly
indistinguishable from the Ewald results.  The other common methods do
significantly less well.

The unmodified cutoff method is essentially unusable.  This is not
surprising since hard cutoffs give large energy fluctuations as atoms
or molecules move in and out of the cutoff
radius.\cite{Rahman71,Adams79} These fluctuations can be alleviated to
some degree by using group based cutoffs with a switching
function.\cite{Adams79,Steinbach94,Leach01} However, we do not see
significant improvement using the group-switched cutoff because the
salt and salt solution systems contain non-neutral groups.  Section
\ref{sec:SystemResults} includes results for systems comprised entirely 
of neutral groups.

For the {\sc sp} method, inclusion of electrostatic damping improves
the agreement with Ewald, and using an $\alpha$ of 0.2\AA $^{-1}$
shows an excellent correlation and quality of fit with the {\sc spme}
results, particularly with a cutoff radius greater than 12
\AA .  Use of a larger damping parameter is more helpful for the
shortest cutoff shown, but it has a detrimental effect on simulations
with larger cutoffs.  

In the {\sc sf} sets, increasing damping results in progressively {\it
worse} correlation with Ewald.  Overall, the undamped case is the best
performing set, as the correlation and quality of fits are
consistently superior regardless of the cutoff distance.  The undamped
case is also less computationally demanding (because no evaluation of
the complementary error function is required).

The reaction field results illustrates some of that method's
limitations, primarily that it was developed for use in homogenous
systems; although it does provide results that are an improvement over
those from an unmodified cutoff.

\sub

\subsection{Magnitudes of the Force and Torque Vectors}

Evaluation of pairwise methods for use in Molecular Dynamics
simulations requires consideration of effects on the forces and
torques.  Figures \ref{fig:frcMag} and \ref{fig:trqMag} show the
regression results for the force and torque vector magnitudes,
respectively.  The data in these figures was generated from an
accumulation of the statistics from all of the system types.

\begin{figure}
\centering
\includegraphics[width=4.75in]{./figures/frcMagplot.pdf}
\caption{Statistical analysis of the quality of the force vector 
magnitudes for a given electrostatic method compared with the
reference Ewald sum.  Results with a value equal to 1 (dashed line)
indicate force magnitude values indistinguishable from those obtained
using {\sc spme}.  Different values of the cutoff radius are indicated with
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
inverted triangles).}
\label{fig:frcMag}
\end{figure}

Again, it is striking how well the Shifted Potential and Shifted Force
methods are doing at reproducing the {\sc spme} forces.  The undamped and
weakly-damped {\sc sf} method gives the best agreement with Ewald.
This is perhaps expected because this method explicitly incorporates a
smooth transition in the forces at the cutoff radius as well as the
neutralizing image charges.

Figure \ref{fig:frcMag}, for the most part, parallels the results seen
in the previous $\Delta E$ section.  The unmodified cutoff results are
poor, but using group based cutoffs and a switching function provides
an improvement much more significant than what was seen with $\Delta
E$.

With moderate damping and a large enough cutoff radius, the {\sc sp}
method is generating usable forces.  Further increases in damping,
while beneficial for simulations with a cutoff radius of 9\AA\ , is
detrimental to simulations with larger cutoff radii.

The reaction field results are surprisingly good, considering the poor
quality of the fits for the $\Delta E$ results.  There is still a
considerable degree of scatter in the data, but the forces correlate
well with the Ewald forces in general.  We note that the reaction
field calculations do not include the pure NaCl systems, so these
results are partly biased towards conditions in which the method
performs more favorably.

\begin{figure}
\centering
\includegraphics[width=4.75in]{./figures/trqMagplot.pdf}
\caption{Statistical analysis of the quality of the torque vector 
magnitudes for a given electrostatic method compared with the
reference Ewald sum.  Results with a value equal to 1 (dashed line)
indicate torque magnitude values indistinguishable from those obtained
using {\sc spme}.  Different values of the cutoff radius are indicated with
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ =
inverted triangles).}
\label{fig:trqMag}
\end{figure}

Molecular torques were only available from the systems which contained
rigid molecules (i.e. the systems containing water).  The data in
fig. \ref{fig:trqMag} is taken from this smaller sampling pool.

Torques appear to be much more sensitive to charges at a longer
distance.   The striking feature in comparing the new electrostatic
methods with {\sc spme} is how much the agreement improves with increasing
cutoff radius.  Again, the weakly damped and undamped {\sc sf} method
appears to be reproducing the {\sc spme} torques most accurately.  

Water molecules are dipolar, and the reaction field method reproduces
the effect of the surrounding polarized medium on each of the
molecular bodies. Therefore it is not surprising that reaction field
performs best of all of the methods on molecular torques.

\subsection{Directionality of the Force and Torque Vectors}

It is clearly important that a new electrostatic method can reproduce
the magnitudes of the force and torque vectors obtained via the Ewald
sum. However, the {\it directionality} of these vectors will also be
vital in calculating dynamical quantities accurately.  Force and
torque directionalities were investigated by measuring the angles
formed between these vectors and the same vectors calculated using
{\sc spme}.  The results (Fig. \ref{fig:frcTrqAng}) are compared through the
variance ($\sigma^2$) of the Gaussian fits of the angle error
distributions of the combined set over all system types.

\begin{figure}
\centering
\includegraphics[width=4.75in]{./figures/frcTrqAngplot.pdf}
\caption{Statistical analysis of the width of the angular distribution
that the force and torque vectors from a given electrostatic method
make with their counterparts obtained using the reference Ewald sum.
Results with a variance ($\sigma^2$) equal to zero (dashed line)
indicate force and torque directions indistinguishable from those
obtained using {\sc spme}.  Different values of the cutoff radius are
indicated with different symbols (9\AA\ = circles, 12\AA\ = squares,
and 15\AA\ = inverted triangles).}
\label{fig:frcTrqAng}
\end{figure}

Both the force and torque $\sigma^2$ results from the analysis of the
total accumulated system data are tabulated in figure
\ref{fig:frcTrqAng}. Here it is clear that the Shifted Potential ({\sc
sp}) method would be essentially unusable for molecular dynamics
unless the damping function is added.  The Shifted Force ({\sc sf})
method, however, is generating force and torque vectors which are
within a few degrees of the Ewald results even with weak (or no)
damping.

All of the sets (aside from the over-damped case) show the improvement
afforded by choosing a larger cutoff radius.  Increasing the cutoff
from 9 to 12\AA\ typically results in a halving of the width of the
distribution, with a similar improvement when going from 12 to 15
\AA . 

The undamped {\sc sf}, group-based cutoff, and reaction field methods
all do equivalently well at capturing the direction of both the force
and torque vectors.  Using the electrostatic damping improves the
angular behavior significantly for the {\sc sp} and moderately for the
{\sc sf} methods.  Overdamping is detrimental to both methods.  Again
it is important to recognize that the force vectors cover all
particles in all seven systems, while torque vectors are only
available for neutral molecular groups.  Damping is more beneficial to
charged bodies, and this observation is investigated further in
section \ref{SystemResults}.

Although not discussed previously, group based cutoffs can be applied
to both the {\sc sp} and {\sc sf} methods.  The group-based cutoffs
will reintroduce small discontinuities at the cutoff radius, but the
effects of these can be minimized by utilizing a switching function.
Though there are no significant benefits or drawbacks observed in
$\Delta E$ and the force and torque magnitudes when doing this, there
is a measurable improvement in the directionality of the forces and
torques. Table \ref{tab:groupAngle} shows the angular variances
obtained both without (N) and with (Y) group based cutoffs and a
switching function.  Note that the $\alpha$ values have units of
\AA$^{-1}$ and the variance values have units of degrees$^2$.  The
{\sc sp} (with an $\alpha$ of 0.2\AA$^{-1}$ or smaller) shows much
narrower angular distributions when using group-based cutoffs. The
{\sc sf} method likewise shows improvement in the undamped and lightly
damped cases.

\begin{table}
\caption{STATISTICAL ANALYSIS OF THE ANGULAR DISTRIBUTIONS ($\sigma^2$) 
THAT THE FORCE ({\it upper}) AND TORQUE ({\it lower}) VECTORS FROM A
GIVEN ELECTROSTATIC METHOD MAKE WITH THEIR COUNTERPARTS OBTAINED USING
THE REFERENCE EWALD SUMMATION}

\footnotesize
\begin{center}
\begin{tabular}{@{} ccrrrrrrrr @{}} \\ 
\toprule 
\toprule 

& &\multicolumn{4}{c}{Shifted Potential} & \multicolumn{4}{c}{Shifted
Force} \\
\cmidrule(lr){3-6} \cmidrule(l){7-10} $R_\textrm{c}$ & Groups & 
$\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$ &
$\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$\\

\midrule
     
9\AA   & N & 29.545 & 12.003 & 5.489 & 0.610 & 2.323 & 2.321 & 0.429 & 0.603 \\
       & \textbf{Y} & \textbf{2.486} & \textbf{2.160} & \textbf{0.667} & \textbf{0.608} & \textbf{1.768} & \textbf{1.766} & \textbf{0.676} & \textbf{0.609} \\
12\AA  & N & 19.381 & 3.097 & 0.190 & 0.608 & 0.920 & 0.736 & 0.133 & 0.612 \\
       & \textbf{Y} & \textbf{0.515} & \textbf{0.288} & \textbf{0.127} & \textbf{0.586} & \textbf{0.308} & \textbf{0.249} & \textbf{0.127} & \textbf{0.586} \\
15\AA  & N & 12.700 & 1.196 & 0.123 & 0.601 & 0.339 & 0.160 & 0.123 & 0.601 \\
       & \textbf{Y} & \textbf{0.228} & \textbf{0.099} & \textbf{0.121} & \textbf{0.598} & \textbf{0.144} & \textbf{0.090} & \textbf{0.121} & \textbf{0.598} \\

\midrule
      
9\AA   & N & 262.716 & 116.585 & 5.234 & 5.103 & 2.392 & 2.350 & 1.770 & 5.122 \\
       & \textbf{Y} & \textbf{2.115} & \textbf{1.914} & \textbf{1.878} & \textbf{5.142} & \textbf{2.076} & \textbf{2.039} & \textbf{1.972} & \textbf{5.146} \\
12\AA  & N & 129.576 & 25.560 & 1.369 & 5.080 & 0.913 & 0.790 & 1.362 & 5.124 \\
       & \textbf{Y} & \textbf{0.810} & \textbf{0.685} & \textbf{1.352} & \textbf{5.082} & \textbf{0.765} & \textbf{0.714} & \textbf{1.360} & \textbf{5.082} \\
15\AA  & N & 87.275 & 4.473 & 1.271 & 5.000 & 0.372 & 0.312 & 1.271 & 5.000 \\
       & \textbf{Y} & \textbf{0.282} & \textbf{0.294} & \textbf{1.272} & \textbf{4.999} & \textbf{0.324} & \textbf{0.318} & \textbf{1.272} & \textbf{4.999} \\

\bottomrule 
\end{tabular} 
\end{center}
\label{tab:groupAngle}
\end{table}

One additional trend in table \ref{tab:groupAngle} is that the
$\sigma^2$ values for both {\sc sp} and {\sc sf} converge as $\alpha$
increases, something that is more obvious with group-based cutoffs.
The complimentary error function inserted into the potential weakens
the electrostatic interaction as the value of $\alpha$ is increased.
However, at larger values of $\alpha$, it is possible to overdamp the
electrostatic interaction and to remove it completely.  Kast
\textit{et al.}  developed a method for choosing appropriate $\alpha$
values for these types of electrostatic summation methods by fitting
to $g(r)$ data, and their methods indicate optimal values of 0.34,
0.25, and 0.16\AA$^{-1}$ for cutoff values of 9, 12, and 15\AA\
respectively.\cite{Kast03} These appear to be reasonable choices to
obtain proper MC behavior (Fig. \ref{fig:delE}); however, based on
these findings, choices this high would introduce error in the
molecular torques, particularly for the shorter cutoffs.  Based on our
observations, empirical damping up to 0.2\AA$^{-1}$ is beneficial,
but damping may be unnecessary when using the {\sc sf} method.

\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals}

Zahn {\it et al.} investigated the structure and dynamics of water
using eqs. (\ref{eq:ZahnPot}) and
(\ref{eq:WolfForces}).\cite{Zahn02,Kast03} Their results indicated
that a method similar (but not identical with) the damped {\sc sf}
method resulted in properties very similar to those obtained when
using the Ewald summation.  The properties they studied (pair
distribution functions, diffusion constants, and velocity and
orientational correlation functions) may not be particularly sensitive
to the long-range and collective behavior that governs the
low-frequency behavior in crystalline systems.  Additionally, the
ionic crystals are the worst case scenario for the pairwise methods
because they lack the reciprocal space contribution contained in the
Ewald summation.  

We are using two separate measures to probe the effects of these
alternative electrostatic methods on the dynamics in crystalline
materials.  For short- and intermediate-time dynamics, we are
computing the velocity autocorrelation function, and for long-time
and large length-scale collective motions, we are looking at the
low-frequency portion of the power spectrum.

\begin{figure}
\centering
\includegraphics[width = \linewidth]{./figures/vCorrPlot.pdf}
\caption{Velocity autocorrelation functions of NaCl crystals at 
1000K using {\sc spme}, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc
sp} ($\alpha$ = 0.2). The inset is a magnification of the area around
the first minimum.  The times to first collision are nearly identical,
but differences can be seen in the peaks and troughs, where the
undamped and weakly damped methods are stiffer than the moderately
damped and {\sc spme} methods.}
\label{fig:vCorrPlot}
\end{figure}

The short-time decay of the velocity autocorrelation function through
the first collision are nearly identical in figure
\ref{fig:vCorrPlot}, but the peaks and troughs of the functions show
how the methods differ.  The undamped {\sc sf} method has deeper
troughs (see inset in Fig. \ref{fig:vCorrPlot}) and higher peaks than
any of the other methods.  As the damping parameter ($\alpha$) is
increased, these peaks are smoothed out, and the {\sc sf} method
approaches the {\sc spme} results.  With $\alpha$ values of 0.2\AA$^{-1}$,
the {\sc sf} and {\sc sp} functions are nearly identical and track the
{\sc spme} features quite well.  This is not surprising because the {\sc sf}
and {\sc sp} potentials become nearly identical with increased
damping.  However, this appears to indicate that once damping is
utilized, the details of the form of the potential (and forces)
constructed out of the damped electrostatic interaction are less
important.

\subsection{Collective Motion: Power Spectra of NaCl Crystals}

To evaluate how the differences between the methods affect the
collective long-time motion, we computed power spectra from long-time
traces of the velocity autocorrelation function. The power spectra for
the best-performing alternative methods are shown in
fig. \ref{fig:methodPS}.  Apodization of the correlation functions via
a cubic switching function between 40 and 50 ps was used to reduce the
ringing resulting from data truncation.  This procedure had no
noticeable effect on peak location or magnitude.

\begin{figure}
\centering
\includegraphics[width = \linewidth]{./figures/spectraSquare.pdf}
\caption{Power spectra obtained from the velocity auto-correlation 
functions of NaCl crystals at 1000K while using {\sc spme}, {\sc sf}
($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2).  The inset
shows the frequency region below 100 cm$^{-1}$ to highlight where the
spectra differ.}
\label{fig:methodPS}
\end{figure}

While the high frequency regions of the power spectra for the
alternative methods are quantitatively identical with Ewald spectrum,
the low frequency region shows how the summation methods differ.
Considering the low-frequency inset (expanded in the upper frame of
figure \ref{fig:dampInc}), at frequencies below 100 cm$^{-1}$, the
correlated motions are blue-shifted when using undamped or weakly
damped {\sc sf}.  When using moderate damping ($\alpha = 0.2$
\AA$^{-1}$) both the {\sc sf} and {\sc sp} methods give nearly identical
correlated motion to the Ewald method (which has a convergence
parameter of 0.3119\AA$^{-1}$).  This weakening of the electrostatic
interaction with increased damping explains why the long-ranged
correlated motions are at lower frequencies for the moderately damped
methods than for undamped or weakly damped methods. 

To isolate the role of the damping constant, we have computed the
spectra for a single method ({\sc sf}) with a range of damping
constants and compared this with the {\sc spme} spectrum.
Fig. \ref{fig:dampInc} shows more clearly that increasing the
electrostatic damping red-shifts the lowest frequency phonon modes.
However, even without any electrostatic damping, the {\sc sf} method
has at most a 10 cm$^{-1}$ error in the lowest frequency phonon mode.
Without the {\sc sf} modifications, an undamped (pure cutoff) method
would predict the lowest frequency peak near 325 cm$^{-1}$.  {\it
Most} of the collective behavior in the crystal is accurately captured
using the {\sc sf} method.  Quantitative agreement with Ewald can be
obtained using moderate damping in addition to the shifting at the
cutoff distance.

\begin{figure}
\centering
\includegraphics[width = \linewidth]{./figures/increasedDamping.pdf}
\caption{Effect of damping on the two lowest-frequency phonon modes in
the NaCl crystal at 1000K.  The undamped shifted force ({\sc sf})
method is off by less than 10 cm$^{-1}$, and increasing the
electrostatic damping to 0.25\AA$^{-1}$ gives quantitative agreement
with the power spectrum obtained using the Ewald sum.  Overdamping can
result in underestimates of frequencies of the long-wavelength
motions.}
\label{fig:dampInc}
\end{figure}


\chapter{\label{chap:water}SIMPLE MODELS FOR WATER}

\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS}

\chapter{\label{chap:shapes}SPHERICAL HARMONIC APPROXIMATIONS FOR MOLECULAR 
SIMULATIONS}

\chapter{\label{chap:conclusion}CONCLUSION}

\backmatter
 
\bibliographystyle{ndthesis}
\bibliography{dissertation}           

\end{document}


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