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21
22		\begin{document}
23
24	<	\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models}
24	>	\title{On the structural and transport properties of the soft sticky
25	>	dipole (SSD) and related single point water models}
26
27		\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
28		Department of Chemistry and Biochemistry\\ University of Notre Dame\\
#	Line 32 | Line 33 | NVE and NPT molecular dynamics simulations were perfor
33		\maketitle
34
35		\begin{abstract}
36	<	NVE and NPT molecular dynamics simulations were performed in order to
37	<	investigate the density maximum and temperature dependent transport
38	<	for SSD and related water models, both with and without the use of
39	<	reaction field. The constant pressure simulations of the melting of
40	<	both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most
41	<	cases, the calculated densities were significantly lower than the
42	<	densities calculated in simulations of other water models. Analysis of
43	<	particle diffusion showed SSD to capture the transport properties of
36	>	The density maximum and temperature dependence of the self-diffusion
37	>	constant were investigated for the soft sticky dipole (SSD) water
38	>	model and two related re-parameterizations of this single-point model.
39	>	A combination of microcanonical and isobaric-isothermal molecular
40	>	dynamics simulations were used to calculate these properties, both
41	>	with and without the use of reaction field to handle long-range
42	>	electrostatics.  The isobaric-isothermal (NPT) simulations of the
43	>	melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near
44	>	260 K.  In most cases, the use of the reaction field resulted in
45	>	calculated densities which were were significantly lower than
46	>	experimental densities.  Analysis of self-diffusion constants shows
47	>	that the original SSD model captures the transport properties of
48		experimental water very well in both the normal and super-cooled
49	<	liquid regimes. In order to correct the density behavior, SSD was
50	<	reparameterized for use both with and without a long-range interaction
51	<	correction, SSD/RF and SSD/E respectively. Compared to the density
52	<	corrected version of SSD (SSD1), these modified models were shown to
53	<	maintain or improve upon the structural and transport properties.
49	>	liquid regimes.  We also present our re-parameterized versions of SSD
50	>	for use both with the reaction field or without any long-range
51	>	electrostatic corrections.  These are called the SSD/RF and SSD/E
52	>	models respectively.  These modified models were shown to maintain or
53	>	improve upon the experimental agreement with the structural and
54	>	transport properties that can be obtained with either the original SSD
55	>	or the density corrected version of the original model (SSD1).
56	>	Additionally, a novel low-density ice structure is presented
57	>	which appears to be the most stable ice structure for the entire SSD
58	>	family.
59		\end{abstract}
60
61		\newpage
#	Line 60 | Line 70 | systems is the proper depiction of water and water sol
70		\section{Introduction}
71
72		One of the most important tasks in the simulation of biochemical
73	<	systems is the proper depiction of water and water solvation. In fact,
74	<	the bulk of the calculations performed in solvated simulations are of
75	<	interactions with or between solvent molecules. Thus, the outcomes of
76	<	these types of simulations are highly dependent on the physical
77	<	properties of water, both as individual molecules and in clusters or
78	<	bulk. Due to the fact that explicit solvent accounts for a massive
79	<	portion of the calculations, it necessary to simplify the solvent to
70	<	some extent in order to complete simulations in a reasonable amount of
71	<	time. In the case of simulating water in biomolecular studies, the
72	<	balance between accurate properties and computational efficiency is
73	<	especially delicate, and it has resulted in a variety of different
74	<	water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these
75	<	models predict specific properties more accurately than their
76	<	predecessors, but often at the cost of other properties or of computer
77	<	time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves
78	<	upon the structural and transport properties of water relative to the
79	<	previous TIP models, yet this comes at a greater than 50\% increase in
80	<	computational cost.\cite{Jorgensen01,Jorgensen00} One recently
81	<	developed model that succeeds in both retaining the accuracy of system
82	<	properties and simplifying calculations to increase computational
83	<	efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}
73	>	systems is the proper depiction of the aqueous environment of the
74	>	molecules of interest.  In some cases (such as in the simulation of
75	>	phospholipid bilayers), the majority of the calculations that are
76	>	performed involve interactions with or between solvent molecules.
77	>	Thus, the properties one may observe in biochemical simulations are
78	>	going to be highly dependent on the physical properties of the water
79	>	model that is chosen.
80
81	<	The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye
82	<	\emph{et al.} as a modified form of the hard-sphere water model
83	<	proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD
84	<	consists of a single point dipole with a Lennard-Jones core and a
85	<	sticky potential that directs the particles to assume the proper
86	<	hydrogen bond orientation in the first solvation shell. Thus, the
87	<	interaction between two SSD water molecules \emph{i} and \emph{j} is
88	<	given by the potential
81	>	There is an especially delicate balance between computational
82	>	efficiency and the ability of the water model to accurately predict
83	>	the properties of bulk
84	>	water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the
85	>	TIP5P model improves on the structural and transport properties of
86	>	water relative to the previous TIP models, yet this comes at a greater
87	>	than 50\% increase in computational
88	>	cost.\cite{Jorgensen01,Jorgensen00}
89	>
90	>	One recently developed model that largely succeeds in retaining the
91	>	accuracy of bulk properties while greatly reducing the computational
92	>	cost is the Soft Sticky Dipole (SSD) water
93	>	model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model
94	>	was developed by Ichiye \emph{et al.} as a modified form of the
95	>	hard-sphere water model proposed by Bratko, Blum, and
96	>	Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model
97	>	which has an interaction site that is both a point dipole along with a
98	>	Lennard-Jones core.  However, since the normal aligned and
99	>	anti-aligned geometries favored by point dipoles are poor mimics of
100	>	local structure in liquid water, a short ranged ``sticky'' potential
101	>	is also added.  The sticky potential directs the molecules to assume
102	>	the proper hydrogen bond orientation in the first solvation shell.
103	>
104	>	The interaction between two SSD water molecules \emph{i} and \emph{j}
105	>	is given by the potential
106		\begin{equation}
107		u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
108	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
108	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
109		u_{ij}^{sp}
110	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
110	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
111		\end{equation}
112	<	where the $\mathbf{r}_{ij}$ is the position vector between molecules
113	<	\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and
114	<	$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
115	<	orientations of the respective molecules. The Lennard-Jones, dipole,
116	<	and sticky parts of the potential are giving by the following
104	<	equations:
112	>	where the ${\bf r}_{ij}$ is the position vector between molecules
113	>	\emph{i} and \emph{j} with magnitude $r_{ij}$, and
114	>	${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of
115	>	the two molecules. The Lennard-Jones and dipole interactions are given
116	>	by the following familiar forms:
117		\begin{equation}
118	<	u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
118	>	u_{ij}^{LJ}(r_{ij}) = 4\epsilon
119	>	\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]
120	>	\ ,
121		\end{equation}
122	+	and
123		\begin{equation}
124	<	u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ ,
124	>	u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left(
125	>	\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf
126	>	r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ ,
127		\end{equation}
128	+	where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along
129	+	the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and
130	+	$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf
131	+	r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule
132	+	$i$.
133	+
134	+	The sticky potential is somewhat less familiar:
135		\begin{equation}
136		u_{ij}^{sp}
137	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
138	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
137	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
138	>	\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)
139	>	+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf
140	>	\Omega}_j)]\ .
141	>	\label{stickyfunction}
142		\end{equation}
143	<	where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
144	<	unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
145	<	$\nu_0$ scales the strength of the overall sticky potential, and $s$
146	<	and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
147	<	functions take the following forms:
143	>	Here, $\nu_0$ is a strength parameter for the sticky potential, and
144	>	$s$ and $s^\prime$ are cubic switching functions which turn off the
145	>	sticky interaction beyond the first solvation shell. The $w$ function
146	>	can be thought of as an attractive potential with tetrahedral
147	>	geometry:
148		\begin{equation}
149	<	w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
149	>	w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
150		\end{equation}
151	+	while the $w^\prime$ function counters the normal aligned and
152	+	anti-aligned structures favored by point dipoles:
153		\begin{equation}
154	<	w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
154	>	w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ,
155		\end{equation}
156	<	where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
157	<	term that promotes hydrogen bonding orientations within the first
158	<	solvation shell, and $w^\prime$ is a dipolar repulsion term that
159	<	repels unrealistic dipolar arrangements within the first solvation
160	<	shell. A more detailed description of the functional parts and
161	<	variables in this potential can be found in other
162	<	articles.\cite{Ichiye96,Ichiye99}
156	>	It should be noted that $w$ is proportional to the sum of the $Y_3^2$
157	>	and $Y_3^{-2}$ spherical harmonics (a linear combination which
158	>	enhances the tetrahedral geometry for hydrogen bonded structures),
159	>	while $w^\prime$ is a purely empirical function.  A more detailed
160	>	description of the functional parts and variables in this potential
161	>	can be found in the original SSD
162	>	articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
163
164	<	Being that this is a one-site point dipole model, the actual force
165	<	calculations are simplified significantly. In the original Monte Carlo
166	<	simulations using this model, Ichiye \emph{et al.} reported an
167	<	increase in calculation efficiency of up to an order of magnitude over
168	<	other comparable models, while maintaining the structural behavior of
169	<	water.\cite{Ichiye96} In the original molecular dynamics studies, it
170	<	was shown that SSD improves on the prediction of many of water's
171	<	dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This
172	<	attractive combination of speed and accurate depiction of solvent
173	<	properties makes SSD a model of interest for the simulation of large
174	<	scale biological systems, such as membrane phase behavior.
164	>	Since SSD is a single-point {\it dipolar} model, the force
165	>	calculations are simplified significantly relative to the standard
166	>	{\it charged} multi-point models. In the original Monte Carlo
167	>	simulations using this model, Ichiye {\it et al.} reported that using
168	>	SSD decreased computer time by a factor of 6-7 compared to other
169	>	models.\cite{Ichiye96} What is most impressive is that this savings
170	>	did not come at the expense of accurate depiction of the liquid state
171	>	properties.  Indeed, SSD maintains reasonable agreement with the
172	>	Soper data for the structural features of liquid
173	>	water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
174	>	exhibited by SSD agree with experiment better than those of more
175	>	computationally expensive models (like TIP3P and
176	>	SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
177	>	of solvent properties makes SSD a very attractive model for the
178	>	simulation of large scale biochemical simulations.
179
180	<	One of the key limitations of this water model, however, is that it
181	<	has been parameterized for use with the Ewald Sum technique for the
182	<	handling of long-ranged interactions.  When studying very large
183	<	systems, the Ewald summation and even particle-mesh Ewald become
184	<	computational burdens, with their respective ideal $N^\frac{3}{2}$ and
185	<	$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99}
186	<	In applying this water model in these types of systems, it would be
187	<	useful to know its properties and behavior with the more
188	<	computationally efficient reaction field (RF) technique, and even with
189	<	a cutoff that lacks any form of long-range correction. This study
190	<	addresses these issues by looking at the structural and transport
191	<	behavior of SSD over a variety of temperatures with the purpose of
192	<	utilizing the RF correction technique. We then suggest alterations to
193	<	the parameters that result in more water-like behavior. It should be
194	<	noted that in a recent publication, some of the original investigators of
195	<	the SSD water model have put forth adjustments to the SSD water model
196	<	to address abnormal density behavior (also observed here), calling the
197	<	corrected model SSD1.\cite{Ichiye03} This study will make comparisons
198	<	with SSD1's behavior with the goal of improving upon the
199	<	depiction of water under conditions without the Ewald Sum.
180	>	One feature of the SSD model is that it was parameterized for
181	>	use with the Ewald sum to handle long-range interactions.  This would
182	>	normally be the best way of handling long-range interactions in
183	>	systems that contain other point charges.  However, our group has
184	>	recently become interested in systems with point dipoles as mimics for
185	>	neutral, but polarized regions on molecules (e.g. the zwitterionic
186	>	head group regions of phospholipids).  If the system of interest does
187	>	not contain point charges, the Ewald sum and even particle-mesh Ewald
188	>	become computational bottlenecks.  Their respective ideal
189	>	$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$
190	>	particles can become prohibitive when $N$ becomes
191	>	large.\cite{Darden99} In applying this water model in these types of
192	>	systems, it would be useful to know its properties and behavior under
193	>	the more computationally efficient reaction field (RF) technique, or
194	>	even with a simple cutoff. This study addresses these issues by
195	>	looking at the structural and transport behavior of SSD over a
196	>	variety of temperatures with the purpose of utilizing the RF
197	>	correction technique.  We then suggest modifications to the parameters
198	>	that result in more realistic bulk phase behavior.  It should be noted
199	>	that in a recent publication, some of the original investigators of
200	>	the SSD water model have suggested adjustments to the SSD
201	>	water model to address abnormal density behavior (also observed here),
202	>	calling the corrected model SSD1.\cite{Ichiye03} In what
203	>	follows, we compare our reparamaterization of SSD with both the
204	>	original SSD and SSD1 models with the goal of improving
205	>	the bulk phase behavior of an SSD-derived model in simulations
206	>	utilizing the Reaction Field.
207
208		\section{Methods}
209
210	<	As stated previously, the long-range dipole-dipole interactions were
211	<	accounted for in this study by using the reaction field method. The
212	<	magnitude of the reaction field acting on dipole \emph{i} is given by
210	>	Long-range dipole-dipole interactions were accounted for in this study
211	>	by using either the reaction field method or by resorting to a simple
212	>	cubic switching function at a cutoff radius.  The reaction field
213	>	method was actually first used in Monte Carlo simulations of liquid
214	>	water.\cite{Barker73} Under this method, the magnitude of the reaction
215	>	field acting on dipole $i$ is
216		\begin{equation}
217		\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
218	<	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\  ,
218	>	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}),
219		\label{rfequation}
220		\end{equation}
221		where $\mathcal{R}$ is the cavity defined by the cutoff radius
222		($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
223	<	system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment
224	<	vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching
223	>	system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole
224	>	moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching
225		function.\cite{AllenTildesley} The reaction field contribution to the
226	<	total energy by particle \emph{i} is given by
227	<	$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
228	<	on dipole \emph{i} by
229	<	$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use
230	<	of reaction field is known to alter the orientational dynamic
231	<	properties, such as the dielectric relaxation time, based on changes
232	<	in the length of the cutoff radius.\cite{Berendsen98} This variable
233	<	behavior makes reaction field a less attractive method than other
234	<	methods, like the Ewald summation; however, for the simulation of
235	<	large-scale systems, the computational cost benefit of reaction field
236	<	is dramatic. To address some of the dynamical property alterations due
237	<	to the use of reaction field, simulations were also performed without
238	<	a surrounding dielectric and suggestions are presented on how to make
239	<	SSD more accurate both with and without a reaction field.
226	>	total energy by particle $i$ is given by $-\frac{1}{2}{\bf
227	>	\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf
228	>	\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley}  Use of the reaction
229	>	field is known to alter the bulk orientational properties, such as the
230	>	dielectric relaxation time.  There is particular sensitivity of this
231	>	property on changes in the length of the cutoff
232	>	radius.\cite{Berendsen98} This variable behavior makes reaction field
233	>	a less attractive method than the Ewald sum.  However, for very large
234	>	systems, the computational benefit of reaction field is dramatic.
235	>
236	>	We have also performed a companion set of simulations {\it without} a
237	>	surrounding dielectric (i.e. using a simple cubic switching function
238	>	at the cutoff radius), and as a result we have two reparamaterizations
239	>	of SSD which could be used either with or without the reaction
240	>	field turned on.
241
242	<	Simulations were performed in both the isobaric-isothermal and
243	<	microcanonical ensembles. The constant pressure simulations were
242	>	Simulations to obtain the preferred densities of the models were
243	>	performed in the isobaric-isothermal (NPT) ensemble, while all
244	>	dynamical properties were obtained from microcanonical (NVE)
245	>	simulations done at densities matching the NPT density for a
246	>	particular target temperature.  The constant pressure simulations were
247		implemented using an integral thermostat and barostat as outlined by
248	<	Hoover.\cite{Hoover85,Hoover86} All particles were treated as
248	>	Hoover.\cite{Hoover85,Hoover86} All molecules were treated as
249		non-linear rigid bodies. Vibrational constraints are not necessary in
250	<	simulations of SSD, because there are no explicit hydrogen atoms, and
251	<	thus no molecular vibrational modes need to be considered.
250	>	simulations of SSD, because there are no explicit hydrogen
251	>	atoms, and thus no molecular vibrational modes need to be considered.
252
253		Integration of the equations of motion was carried out using the
254	<	symplectic splitting method proposed by Dullweber \emph{et
255	<	al.}\cite{Dullweber1997} The reason for this integrator selection
256	<	deals with poor energy conservation of rigid body systems using
257	<	quaternions. While quaternions work well for orientational motion in
258	<	alternate ensembles, the microcanonical ensemble has a constant energy
259	<	requirement that is quite sensitive to errors in the equations of
260	<	motion. The original implementation of this code utilized quaternions
261	<	for rotational motion propagation; however, a detailed investigation
262	<	showed that they resulted in a steady drift in the total energy,
216	<	something that has been observed by others.\cite{Laird97}
254	>	symplectic splitting method proposed by Dullweber, Leimkuhler, and
255	>	McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting
256	>	this integrator centers on poor energy conservation of rigid body
257	>	dynamics using traditional quaternion
258	>	integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble
259	>	simulations, the energy drift when using quaternions was substantially
260	>	greater than when using the {\sc dlm} method (fig. \ref{timestep}).
261	>	This steady drift in the total energy has also been observed by Kol
262	>	{\it et al.}\cite{Laird97}
263
264		The key difference in the integration method proposed by Dullweber
265		\emph{et al.} is that the entire rotation matrix is propagated from
266	<	one time step to the next. In the past, this would not have been as
267	<	feasible an option, being that the rotation matrix for a single body is
268	<	nine elements long as opposed to 3 or 4 elements for Euler angles and
269	<	quaternions respectively. System memory has become much less of an
224	<	issue in recent times, and this has resulted in substantial benefits
225	<	in energy conservation. There is still the issue of 5 or 6 additional
226	<	elements for describing the orientation of each particle, which will
227	<	increase dump files substantially. Simply translating the rotation
228	<	matrix into its component Euler angles or quaternions for storage
229	<	purposes relieves this burden.
266	>	one time step to the next.  The additional memory required by the
267	>	algorithm is inconsequential on modern computers, and translating the
268	>	rotation matrix into quaternions for storage purposes makes trajectory
269	>	data quite compact.
270
271	<	The symplectic splitting method allows for Verlet style integration of
272	<	both linear and angular motion of rigid bodies. In this integration
273	<	method, the orientational propagation involves a sequence of matrix
274	<	evaluations to update the rotation matrix.\cite{Dullweber1997} These
275	<	matrix rotations are more costly computationally than the simpler
276	<	arithmetic quaternion propagation. With the same time step, a 1000 SSD
277	<	particle simulation shows an average 7\% increase in computation time
278	<	using the symplectic step method in place of quaternions. This cost is
279	<	more than justified when comparing the energy conservation of the two
280	<	methods as illustrated in figure \ref{timestep}.
271	>	The {\sc dlm} method allows for Verlet style integration of both
272	>	translational and orientational motion of rigid bodies. In this
273	>	integration method, the orientational propagation involves a sequence
274	>	of matrix evaluations to update the rotation
275	>	matrix.\cite{Dullweber1997} These matrix rotations are more costly
276	>	than the simpler arithmetic quaternion propagation. With the same time
277	>	step, a 1000 SSD particle simulation shows an average 7\%
278	>	increase in computation time using the {\sc dlm} method in place of
279	>	quaternions. The additional expense per step is justified when one
280	>	considers the ability to use time steps that are nearly twice as large
281	>	under {\sc dlm} than would be usable under quaternion dynamics.  The
282	>	energy conservation of the two methods using a number of different
283	>	time steps is illustrated in figure
284	>	\ref{timestep}.
285
286		\begin{figure}
287		\begin{center}
288		\epsfxsize=6in
289		\epsfbox{timeStep.epsi}
290	<	\caption{Energy conservation using quaternion based integration versus
291	<	the symplectic step method proposed by Dullweber \emph{et al.} with
292	<	increasing time step. The larger time step plots are shifted up from
293	<	the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
290	>	\caption{Energy conservation using both quaternion-based integration and
291	>	the {\sc dlm} method with increasing time step. The larger time step plots
292	>	are shifted from the true energy baseline (that of $\Delta t$ = 0.1
293	>	fs) for clarity.}
294		\label{timestep}
295		\end{center}
296		\end{figure}
297
298		In figure \ref{timestep}, the resulting energy drift at various time
299	<	steps for both the symplectic step and quaternion integration schemes
300	<	is compared. All of the 1000 SSD particle simulations started with the
301	<	same configuration, and the only difference was the method used to
302	<	handle rotational motion. At time steps of 0.1 and 0.5 fs, both
303	<	methods for propagating particle rotation conserve energy fairly well,
304	<	with the quaternion method showing a slight energy drift over time in
305	<	the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
306	<	energy conservation benefits of the symplectic step method are clearly
307	<	demonstrated. Thus, while maintaining the same degree of energy
308	<	conservation, one can take considerably longer time steps, leading to
309	<	an overall reduction in computation time.
299	>	steps for both the {\sc dlm} and quaternion integration schemes is
300	>	compared.  All of the 1000 SSD particle simulations started with
301	>	the same configuration, and the only difference was the method used to
302	>	handle orientational motion. At time steps of 0.1 and 0.5 fs, both
303	>	methods for propagating the orientational degrees of freedom conserve
304	>	energy fairly well, with the quaternion method showing a slight energy
305	>	drift over time in the 0.5 fs time step simulation. At time steps of 1
306	>	and 2 fs, the energy conservation benefits of the {\sc dlm} method are
307	>	clearly demonstrated. Thus, while maintaining the same degree of
308	>	energy conservation, one can take considerably longer time steps,
309	>	leading to an overall reduction in computation time.
310
311	<	Energy drift in the symplectic step simulations was unnoticeable for
312	<	time steps up to three femtoseconds. A slight energy drift on the
311	>	Energy drift in the simulations using {\sc dlm} integration was
312	>	unnoticeable for time steps up to 3 fs. A slight energy drift on the
313		order of 0.012 kcal/mol per nanosecond was observed at a time step of
314	<	four femtoseconds, and as expected, this drift increases dramatically
315	<	with increasing time step. To insure accuracy in the constant energy
314	>	4 fs, and as expected, this drift increases dramatically with
315	>	increasing time step. To insure accuracy in our microcanonical
316		simulations, time steps were set at 2 fs and kept at this value for
317		constant pressure simulations as well.
318
319	<	Ice crystals in both the $I_h$ and $I_c$ lattices were generated as
320	<	starting points for all simulations. The $I_h$ crystals were formed by
321	<	first arranging the centers of mass of the SSD particles into a
322	<	``hexagonal'' ice lattice of 1024 particles. Because of the crystal
323	<	structure of $I_h$ ice, the simulation box assumed a rectangular shape
324	<	with an edge length ratio of approximately
325	<	1.00$\times$1.06$\times$1.23. The particles were then allowed to
326	<	orient freely about fixed positions with angular momenta randomized at
327	<	400 K for varying times. The rotational temperature was then scaled
328	<	down in stages to slowly cool the crystals to 25 K. The particles were
329	<	then allowed to translate with fixed orientations at a constant
330	<	pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were
331	<	removed and the ice crystals were allowed to equilibrate for 50 ps at
332	<	25 K and a constant pressure of 1 atm.  This procedure resulted in
333	<	structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
319	>	Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices
320	>	were generated as starting points for all simulations. The $I_h$
321	>	crystals were formed by first arranging the centers of mass of the
322	>	SSD particles into a ``hexagonal'' ice lattice of 1024
323	>	particles. Because of the crystal structure of $I_h$ ice, the
324	>	simulation box assumed an orthorhombic shape with an edge length ratio
325	>	of approximately 1.00$\times$1.06$\times$1.23. The particles were then
326	>	allowed to orient freely about fixed positions with angular momenta
327	>	randomized at 400 K for varying times. The rotational temperature was
328	>	then scaled down in stages to slowly cool the crystals to 25 K. The
329	>	particles were then allowed to translate with fixed orientations at a
330	>	constant pressure of 1 atm for 50 ps at 25 K. Finally, all constraints
331	>	were removed and the ice crystals were allowed to equilibrate for 50
332	>	ps at 25 K and a constant pressure of 1 atm.  This procedure resulted
333	>	in structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
334		rules.\cite{Bernal33,Rahman72} This method was also utilized in the
335		making of diamond lattice $I_c$ ice crystals, with each cubic
336		simulation box consisting of either 512 or 1000 particles. Only
#	Line 297 | Line 341 | constant pressure and temperature dynamics. During mel
341		\section{Results and discussion}
342
343		Melting studies were performed on the randomized ice crystals using
344	<	constant pressure and temperature dynamics. During melting
345	<	simulations, the melting transition and the density maximum can both
346	<	be observed, provided that the density maximum occurs in the liquid
347	<	and not the supercooled regime. An ensemble average from five separate
348	<	melting simulations was acquired, each starting from different ice
349	<	crystals generated as described previously. All simulations were
350	<	equilibrated for 100 ps prior to a 200 ps data collection run at each
351	<	temperature setting. The temperature range of study spanned from 25 to
352	<	400 K, with a maximum degree increment of 25 K. For regions of
353	<	interest along this stepwise progression, the temperature increment
354	<	was decreased from 25 K to 10 and 5 K. The above equilibration and
355	<	production times were sufficient in that the system volume
356	<	fluctuations dampened out in all but the very cold simulations (below
313	<	225 K).
344	>	isobaric-isothermal (NPT) dynamics. During melting simulations, the
345	>	melting transition and the density maximum can both be observed,
346	>	provided that the density maximum occurs in the liquid and not the
347	>	supercooled regime. An ensemble average from five separate melting
348	>	simulations was acquired, each starting from different ice crystals
349	>	generated as described previously. All simulations were equilibrated
350	>	for 100 ps prior to a 200 ps data collection run at each temperature
351	>	setting. The temperature range of study spanned from 25 to 400 K, with
352	>	a maximum degree increment of 25 K. For regions of interest along this
353	>	stepwise progression, the temperature increment was decreased from 25
354	>	K to 10 and 5 K.  The above equilibration and production times were
355	>	sufficient in that fluctuations in the volume autocorrelation function
356	>	were damped out in all simulations in under 20 ps.
357
358		\subsection{Density Behavior}
316	–	Initial simulations focused on the original SSD water model, and an
317	–	average density versus temperature plot is shown in figure
318	–	\ref{dense1}. Note that the density maximum when using a reaction
319	–	field appears between 255 and 265 K, where the calculated densities
320	–	within this range were nearly indistinguishable. The greater certainty
321	–	of the average value at 260 K makes a good argument for the actual
322	–	density maximum residing at this midpoint value. Figure \ref{dense1}
323	–	was constructed using ice $I_h$ crystals for the initial
324	–	configuration; though not pictured, the simulations starting from ice
325	–	$I_c$ crystal configurations showed similar results, with a
326	–	liquid-phase density maximum in this same region (between 255 and 260
327	–	K). In addition, the $I_c$ crystals are more fragile than the $I_h$
328	–	crystals, leading to deformation into a dense glassy state at lower
329	–	temperatures. This resulted in an overall low temperature density
330	–	maximum at 200 K, while still retaining a liquid state density maximum
331	–	in common with the $I_h$ simulations.
359
360	+	Our initial simulations focused on the original SSD water model,
361	+	and an average density versus temperature plot is shown in figure
362	+	\ref{dense1}. Note that the density maximum when using a reaction
363	+	field appears between 255 and 265 K.  There were smaller fluctuations
364	+	in the density at 260 K than at either 255 or 265, so we report this
365	+	value as the location of the density maximum. Figure \ref{dense1} was
366	+	constructed using ice $I_h$ crystals for the initial configuration;
367	+	though not pictured, the simulations starting from ice $I_c$ crystal
368	+	configurations showed similar results, with a liquid-phase density
369	+	maximum in this same region (between 255 and 260 K).
370	+
371		\begin{figure}
372		\begin{center}
373		\epsfxsize=6in
374	<	\epsfbox{denseSSD.eps}
375	<	\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b}
376	<	TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction
377	<	Field, SSD, and experiment.\cite{CRC80} The arrows indicate the
378	<	change in densities observed when turning off the reaction field. The
379	<	the lower than expected densities for the SSD model were what
380	<	prompted the original reparameterization.\cite{Ichiye03}}
374	>	\epsfbox{denseSSDnew.eps}
375	>	\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}],
376	>	TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD
377	>	without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The
378	>	arrows indicate the change in densities observed when turning off the
379	>	reaction field. The the lower than expected densities for the SSD
380	>	model were what prompted the original reparameterization of SSD1
381	>	[Ref. \citen{Ichiye03}].}
382		\label{dense1}
383		\end{center}
384		\end{figure}
385
386	<	The density maximum for SSD actually compares quite favorably to other
386	>	The density maximum for SSD compares quite favorably to other
387		simple water models. Figure \ref{dense1} also shows calculated
388		densities of several other models and experiment obtained from other
389		sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
390	<	models, SSD has results closest to the experimentally observed water
391	<	density maximum. Of the listed water models, TIP4P has a density
392	<	maximum behavior most like that seen in SSD. Though not included in
393	<	this particular plot, it is useful to note that TIP5P has a water
394	<	density maximum nearly identical to experiment.
390	>	models, SSD has a temperature closest to the experimentally
391	>	observed density maximum. Of the {\it charge-based} models in
392	>	Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that
393	>	seen in SSD. Though not included in this plot, it is useful to
394	>	note that TIP5P has a density maximum nearly identical to the
395	>	experimentally measured temperature.
396
397	<	It has been observed that densities are dependent on the cutoff radius
398	<	used for a variety of water models in simulations both with and
399	<	without the use of reaction field.\cite{Berendsen98} In order to
400	<	address the possible affect of cutoff radius, simulations were
401	<	performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the
402	<	previous SSD simulations, all performed with a cutoff of 9.0 \AA. All
403	<	of the resulting densities overlapped within error and showed no
404	<	significant trend toward lower or higher densities as a function of
405	<	cutoff radius, for simulations both with and without reaction
406	<	field. These results indicate that there is no major benefit in
407	<	choosing a longer cutoff radius in simulations using SSD. This is
408	<	advantageous in that the use of a longer cutoff radius results in
409	<	significant increases in the time required to obtain a single
370	<	trajectory.
397	>	It has been observed that liquid state densities in water are
398	>	dependent on the cutoff radius used both with and without the use of
399	>	reaction field.\cite{Berendsen98} In order to address the possible
400	>	effect of cutoff radius, simulations were performed with a dipolar
401	>	cutoff radius of 12.0 \AA\ to complement the previous SSD
402	>	simulations, all performed with a cutoff of 9.0 \AA. All of the
403	>	resulting densities overlapped within error and showed no significant
404	>	trend toward lower or higher densities as a function of cutoff radius,
405	>	for simulations both with and without reaction field. These results
406	>	indicate that there is no major benefit in choosing a longer cutoff
407	>	radius in simulations using SSD. This is advantageous in that
408	>	the use of a longer cutoff radius results in a significant increase in
409	>	the time required to obtain a single trajectory.
410
411		The key feature to recognize in figure \ref{dense1} is the density
412		scaling of SSD relative to other common models at any given
413	<	temperature. Note that the SSD model assumes a lower density than any
414	<	of the other listed models at the same pressure, behavior which is
415	<	especially apparent at temperatures greater than 300 K. Lower than
416	<	expected densities have been observed for other systems using a
417	<	reaction field for long-range electrostatic interactions, so the most
418	<	likely reason for the significantly lower densities seen in these
419	<	simulations is the presence of the reaction
420	<	field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the
421	<	reaction field on the density of the systems, the simulations were
422	<	repeated without a reaction field present. The results of these
423	<	simulations are also displayed in figure \ref{dense1}. Without
424	<	reaction field, the densities increase considerably to more
425	<	experimentally reasonable values, especially around the freezing point
426	<	of liquid water. The shape of the curve is similar to the curve
427	<	produced from SSD simulations using reaction field, specifically the
428	<	rapidly decreasing densities at higher temperatures; however, a shift
429	<	in the density maximum location, down to 245 K, is observed. This is a
430	<	more accurate comparison to the other listed water models, in that no
431	<	long range corrections were applied in those
393	<	simulations.\cite{Clancy94,Jorgensen98b} However, even without a
413	>	temperature. SSD assumes a lower density than any of the other
414	>	listed models at the same pressure, behavior which is especially
415	>	apparent at temperatures greater than 300 K. Lower than expected
416	>	densities have been observed for other systems using a reaction field
417	>	for long-range electrostatic interactions, so the most likely reason
418	>	for the significantly lower densities seen in these simulations is the
419	>	presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order
420	>	to test the effect of the reaction field on the density of the
421	>	systems, the simulations were repeated without a reaction field
422	>	present. The results of these simulations are also displayed in figure
423	>	\ref{dense1}. Without the reaction field, the densities increase
424	>	to more experimentally reasonable values, especially around the
425	>	freezing point of liquid water. The shape of the curve is similar to
426	>	the curve produced from SSD simulations using reaction field,
427	>	specifically the rapidly decreasing densities at higher temperatures;
428	>	however, a shift in the density maximum location, down to 245 K, is
429	>	observed. This is a more accurate comparison to the other listed water
430	>	models, in that no long range corrections were applied in those
431	>	simulations.\cite{Clancy94,Jorgensen98b} However, even without the
432		reaction field, the density around 300 K is still significantly lower
433		than experiment and comparable water models. This anomalous behavior
434	<	was what lead Ichiye \emph{et al.} to recently reparameterize SSD and
435	<	make SSD1.\cite{Ichiye03} In discussing potential adjustments later in
436	<	this paper, all comparisons were performed with this new model.
434	>	was what lead Tan {\it et al.} to recently reparameterize
435	>	SSD.\cite{Ichiye03} Throughout the remainder of the paper our
436	>	reparamaterizations of SSD will be compared with their newer SSD1
437	>	model.
438
439		\subsection{Transport Behavior}
401	–	Of importance in these types of studies are the transport properties
402	–	of the particles and their change in responce to altering
403	–	environmental conditions. In order to probe transport, constant energy
404	–	simulations were performed about the average density uncovered by the
405	–	constant pressure simulations. Simulations started with randomized
406	–	velocities and underwent 50 ps of temperature scaling and 50 ps of
407	–	constant energy equilibration before obtaining a 200 ps
408	–	trajectory. Diffusion constants were calculated via root-mean square
409	–	deviation analysis. The averaged results from five sets of NVE
410	–	simulations are displayed in figure \ref{diffuse}, alongside
411	–	experimental, SPC/E, and TIP5P
412	–	results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
440
441	+	Accurate dynamical properties of a water model are particularly
442	+	important when using the model to study permeation or transport across
443	+	biological membranes.  In order to probe transport in bulk water,
444	+	constant energy (NVE) simulations were performed at the average
445	+	density obtained by the NPT simulations at an identical target
446	+	temperature. Simulations started with randomized velocities and
447	+	underwent 50 ps of temperature scaling and 50 ps of constant energy
448	+	equilibration before a 200 ps data collection run. Diffusion constants
449	+	were calculated via linear fits to the long-time behavior of the
450	+	mean-square displacement as a function of time. The averaged results
451	+	from five sets of NVE simulations are displayed in figure
452	+	\ref{diffuse}, alongside experimental, SPC/E, and TIP5P
453	+	results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01}
454	+
455		\begin{figure}
456		\begin{center}
457		\epsfxsize=6in
458		\epsfbox{betterDiffuse.epsi}
459	<	\caption{Average diffusion coefficient over increasing temperature for
460	<	SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental
461	<	data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has
462	<	the least deviation from the experimental values. The rapidly
463	<	increasing diffusion constants for TIP5P and SSD correspond to
464	<	significant decrease in density at the higher temperatures.}
459	>	\caption{Average self-diffusion constant as a function of temperature for
460	>	SSD, SPC/E [Ref. \citen{Clancy94}], and TIP5P
461	>	[Ref. \citen{Jorgensen01}] compared with experimental data
462	>	[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models
463	>	shown, SSD has the least deviation from the experimental values. The
464	>	rapidly increasing diffusion constants for TIP5P and SSD correspond to
465	>	significant decreases in density at the higher temperatures.}
466		\label{diffuse}
467		\end{center}
468		\end{figure}
469
470		The observed values for the diffusion constant point out one of the
471	<	strengths of the SSD model. Of the three experimental models shown,
472	<	the SSD model has the most accurate depiction of the diffusion trend
473	<	seen in experiment in both the supercooled and liquid temperature
474	<	regimes. SPC/E does a respectable job by producing values similar to
475	<	SSD and experiment around 290 K; however, it deviates at both higher
476	<	and lower temperatures, failing to predict the experimental
477	<	trend. TIP5P and SSD both start off low at colder temperatures and
478	<	tend to diffuse too rapidly at higher temperatures. This trend at
479	<	higher temperatures is not surprising in that the densities of both
480	<	TIP5P and SSD are lower than experimental water at these higher
481	<	temperatures. When calculating the diffusion coefficients for SSD at
482	<	experimental densities, the resulting values fall more in line with
483	<	experiment at these temperatures, albeit not at standard pressure.
471	>	strengths of the SSD model. Of the three models shown, the SSD model
472	>	has the most accurate depiction of self-diffusion in both the
473	>	supercooled and liquid regimes.  SPC/E does a respectable job by
474	>	reproducing values similar to experiment around 290 K; however, it
475	>	deviates at both higher and lower temperatures, failing to predict the
476	>	correct thermal trend. TIP5P and SSD both start off low at colder
477	>	temperatures and tend to diffuse too rapidly at higher temperatures.
478	>	This behavior at higher temperatures is not particularly surprising
479	>	since the densities of both TIP5P and SSD are lower than experimental
480	>	water densities at higher temperatures.  When calculating the
481	>	diffusion coefficients for SSD at experimental densities
482	>	(instead of the densities from the NPT simulations), the resulting
483	>	values fall more in line with experiment at these temperatures.
484
485		\subsection{Structural Changes and Characterization}
486	+
487		By starting the simulations from the crystalline state, the melting
488	<	transition and the ice structure can be studied along with the liquid
488	>	transition and the ice structure can be obtained along with the liquid
489		phase behavior beyond the melting point. The constant pressure heat
490		capacity (C$_\text{p}$) was monitored to locate the melting transition
491		in each of the simulations. In the melting simulations of the 1024
#	Line 450 | Line 493 | considerably lower than the experimental value, but th
493		at 245 K, indicating a first order phase transition for the melting of
494		these ice crystals. When the reaction field is turned off, the melting
495		transition occurs at 235 K.  These melting transitions are
496	<	considerably lower than the experimental value, but this is not a
454	<	surprise considering the simplicity of the SSD model.
496	>	considerably lower than the experimental value.
497
498		\begin{figure}
499		\begin{center}
500		\epsfxsize=6in
501		\epsfbox{corrDiag.eps}
502	<	\caption{Two dimensional illustration of angles involved in the
461	<	correlations observed in figure \ref{contour}.}
502	>	\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.}
503		\label{corrAngle}
504		\end{center}
505		\end{figure}
#	Line 467 | Line 508 | correlations observed in figure \ref{contour}.}
508		\begin{center}
509		\epsfxsize=6in
510		\epsfbox{fullContours.eps}
511	<	\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at
512	<	100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for
513	<	clarity: dark areas signify peaks while light areas signify
514	<	depressions. White areas have g(\emph{r}) values below 0.5 and black
515	<	areas have values above 1.5.}
511	>	\caption{Contour plots of 2D angular pair correlation functions for
512	>	512 SSD molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas
513	>	signify regions of enhanced density while light areas signify
514	>	depletion relative to the bulk density. White areas have pair
515	>	correlation values below 0.5 and black areas have values above 1.5.}
516		\label{contour}
517		\end{center}
518		\end{figure}
519
520	<	Additional analysis of the melting phase-transition process was
521	<	performed by using two-dimensional structure and dipole angle
522	<	correlations. Expressions for these correlations are as follows:
520	>	Additional analysis of the melting process was performed using
521	>	two-dimensional structure and dipole angle correlations. Expressions
522	>	for these correlations are as follows:
523
524		\begin{equation}
525	<	g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ ,
525	>	g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
526		\end{equation}
527		\begin{equation}
528		g_{\text{AB}}(r,\cos\omega) =
529	<	\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ ,
529	>	\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
530		\end{equation}
531		where $\theta$ and $\omega$ refer to the angles shown in figure
532		\ref{corrAngle}. By binning over both distance and the cosine of the
533	<	desired angle between the two dipoles, the g(\emph{r}) can be
534	<	dissected to determine the common dipole arrangements that constitute
535	<	the peaks and troughs. Frames A and B of figure \ref{contour} show a
536	<	relatively crystalline state of an ice $I_c$ simulation. The first
537	<	peak of the g(\emph{r}) consists primarily of the preferred hydrogen
533	>	desired angle between the two dipoles, the $g(r)$ can be analyzed to
534	>	determine the common dipole arrangements that constitute the peaks and
535	>	troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
536	>	of figure \ref{contour} show results from an ice $I_c$ simulation. The
537	>	first peak in the $g(r)$ consists primarily of the preferred hydrogen
538		bonding arrangements as dictated by the tetrahedral sticky potential -
539	<	one peak for the donating and the other for the accepting hydrogen
540	<	bonds. Due to the high degree of crystallinity of the sample, the
541	<	second and third solvation shells show a repeated peak arrangement
539	>	one peak for the hydrogen bond donor and the other for the hydrogen
540	>	bond acceptor.  Due to the high degree of crystallinity of the sample,
541	>	the second and third solvation shells show a repeated peak arrangement
542		which decays at distances around the fourth solvation shell, near the
543		imposed cutoff for the Lennard-Jones and dipole-dipole interactions.
544		In the higher temperature simulation shown in frames C and D, these
545	<	longer-ranged repeated peak features deteriorate rapidly. The first
546	<	solvation shell still shows the strong effect of the sticky-potential,
547	<	although it covers a larger area, extending to include a fraction of
548	<	aligned dipole peaks within the first solvation shell. The latter
549	<	peaks lose definition as thermal motion and the competing dipole force
550	<	overcomes the sticky potential's tight tetrahedral structuring of the
510	<	fluid.
545	>	long-range features deteriorate rapidly. The first solvation shell
546	>	still shows the strong effect of the sticky-potential, although it
547	>	covers a larger area, extending to include a fraction of aligned
548	>	dipole peaks within the first solvation shell. The latter peaks lose
549	>	due to thermal motion and as the competing dipole force overcomes the
550	>	sticky potential's tight tetrahedral structuring of the crystal.
551
552		This complex interplay between dipole and sticky interactions was
553		remarked upon as a possible reason for the split second peak in the
554	<	oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the
555	<	second solvation shell peak appears to have two distinct components
556	<	that blend together to form one observable peak. At higher
557	<	temperatures, this split character alters to show the leading 4 \AA\
558	<	peak dominated by equatorial anti-parallel dipole orientations. There
559	<	is also a tightly bunched group of axially arranged dipoles that most
560	<	likely consist of the smaller fraction of aligned dipole pairs. The
561	<	trailing component of the split peak at 5 \AA\ is dominated by aligned
562	<	dipoles that assume hydrogen bond arrangements similar to those seen
563	<	in the first solvation shell. This evidence indicates that the dipole
564	<	pair interaction begins to dominate outside of the range of the
565	<	dipolar repulsion term. Primary energetically favorable dipole
566	<	arrangements populate the region immediately outside this repulsion
567	<	region (around 4 \AA), while arrangements that seek to ideally satisfy
568	<	both the sticky and dipole forces locate themselves just beyond this
569	<	initial buildup (around 5 \AA).
554	>	oxygen-oxygen pair correlation function,
555	>	$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second
556	>	solvation shell peak appears to have two distinct components that
557	>	blend together to form one observable peak. At higher temperatures,
558	>	this split character alters to show the leading 4 \AA\ peak dominated
559	>	by equatorial anti-parallel dipole orientations. There is also a
560	>	tightly bunched group of axially arranged dipoles that most likely
561	>	consist of the smaller fraction of aligned dipole pairs. The trailing
562	>	component of the split peak at 5 \AA\ is dominated by aligned dipoles
563	>	that assume hydrogen bond arrangements similar to those seen in the
564	>	first solvation shell. This evidence indicates that the dipole pair
565	>	interaction begins to dominate outside of the range of the dipolar
566	>	repulsion term.  The energetically favorable dipole arrangements
567	>	populate the region immediately outside this repulsion region (around
568	>	4 \AA), while arrangements that seek to satisfy both the sticky and
569	>	dipole forces locate themselves just beyond this initial buildup
570	>	(around 5 \AA).
571
572		From these findings, the split second peak is primarily the product of
573		the dipolar repulsion term of the sticky potential. In fact, the inner
574		peak can be pushed out and merged with the outer split peak just by
575	<	extending the switching function cutoff ($s^\prime(r_{ij})$) from its
576	<	normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of
575	>	extending the switching function ($s^\prime(r_{ij})$) from its normal
576	>	4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of
577		correction is not recommended for improving the liquid structure,
578		since the second solvation shell would still be shifted too far
579		out. In addition, this would have an even more detrimental effect on
580		the system densities, leading to a liquid with a more open structure
581	<	and a density considerably lower than the normal SSD behavior shown
582	<	previously. A better correction would be to include the
581	>	and a density considerably lower than the already low SSD
582	>	density.  A better correction would be to include the
583		quadrupole-quadrupole interactions for the water particles outside of
584	<	the first solvation shell, but this reduces the simplicity and speed
585	<	advantage of SSD.
584	>	the first solvation shell, but this would remove the simplicity and
585	>	speed advantage of SSD.
586
587		\subsection{Adjusted Potentials: SSD/RF and SSD/E}
588	+
589		The propensity of SSD to adopt lower than expected densities under
590		varying conditions is troubling, especially at higher temperatures. In
591		order to correct this model for use with a reaction field, it is
#	Line 551 | Line 593 | densities while maintaining the accurate transport pro
593		intermolecular interactions. In undergoing a reparameterization, it is
594		important not to focus on just one property and neglect the other
595		important properties. In this case, it would be ideal to correct the
596	<	densities while maintaining the accurate transport properties.
596	>	densities while maintaining the accurate transport behavior.
597
598	<	The parameters available for tuning include the $\sigma$ and $\epsilon$
599	<	Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
600	<	attractive and dipole repulsive terms with their respective
601	<	cutoffs. To alter the attractive and repulsive terms of the sticky
602	<	potential independently, it is necessary to separate the terms as
603	<	follows:
604	<	\begin{equation}
605	<	u_{ij}^{sp}
606	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
607	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
566	<	\end{equation}
598	>	The parameters available for tuning include the $\sigma$ and
599	>	$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the
600	>	strength of the sticky potential ($\nu_0$), and the cutoff distances
601	>	for the sticky attractive and dipole repulsive cubic switching
602	>	function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$
603	>	respectively). The results of the reparameterizations are shown in
604	>	table \ref{params}. We are calling these reparameterizations the Soft
605	>	Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction
606	>	field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve
607	>	the liquid structure in simulations without a long-range correction).
608
568	–	where $\nu_0$ scales the strength of the tetrahedral attraction and
569	–	$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
570	–	term. The separation was performed for purposes of the
571	–	reparameterization, but the final parameters were adjusted so that it
572	–	is unnecessary to separate the terms when implementing the adjusted
573	–	water potentials. The results of the reparameterizations are shown in
574	–	table \ref{params}. Note that the tetrahedral attractive and dipolar
575	–	repulsive terms do not share the same lower cutoff ($r_l$) in the
576	–	newly parameterized potentials - soft sticky dipole reaction field
577	–	(SSD/RF - for use with a reaction field) and soft sticky dipole
578	–	enhanced (SSD/E - an attempt to improve the liquid structure in
579	–	simulations without a long-range correction).
580	–
609		\begin{table}
610		\begin{center}
611		\caption{Parameters for the original and adjusted models}
612		\begin{tabular}{ l  c  c  c  c }
613		\hline \\[-3mm]
614	<	\ \ \ Parameters\ \ \  & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \  & \ SSD/E\ \  & \ SSD/RF \\
614	>	\ \ \ Parameters\ \ \  & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \
615	>	& \ SSD1 [Ref. \citen{Ichiye03}]\ \  & \ SSD/E\ \  & \ SSD/RF \\
616		\hline \\[-3mm]
617		\ \ \ $\sigma$ (\AA)  & 3.051 & 3.016 & 3.035 & 3.019\\
618		\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
619		\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\
620		\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
621	+	\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\
622		\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\
623		\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\
594	–	\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
624		\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
625		\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
626		\end{tabular}
#	Line 603 | Line 632 | simulations without a long-range correction).
632		\begin{center}
633		\epsfxsize=5in
634		\epsfbox{GofRCompare.epsi}
635	<	\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E
636	<	and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with
637	<	reaction field turned on (bottom). The insets show the respective
638	<	first peaks in detail. Note how the changes in parameters have lowered
639	<	and broadened the first peak of SSD/E and SSD/RF.}
635	>	\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with
636	>	SSD/E and SSD1 without reaction field (top), as well as
637	>	SSD/RF and SSD1 with reaction field turned on
638	>	(bottom). The insets show the respective first peaks in detail. Note
639	>	how the changes in parameters have lowered and broadened the first
640	>	peak of SSD/E and SSD/RF.}
641		\label{grcompare}
642		\end{center}
643		\end{figure}
#	Line 615 | Line 645 | and broadened the first peak of SSD/E and SSD/RF.}
645		\begin{figure}
646		\begin{center}
647		\epsfxsize=6in
648	<	\epsfbox{dualsticky.ps}
649	<	\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \&
650	<	SSD/RF (right). Light areas correspond to the tetrahedral attractive
651	<	component, and darker areas correspond to the dipolar repulsive
652	<	component.}
648	>	\epsfbox{dualsticky_bw.eps}
649	>	\caption{Positive and negative isosurfaces of the sticky potential for
650	>	SSD1 (left) and SSD/E \& SSD/RF (right). Light areas
651	>	correspond to the tetrahedral attractive component, and darker areas
652	>	correspond to the dipolar repulsive component.}
653		\label{isosurface}
654		\end{center}
655		\end{figure}
656
657	<	In the paper detailing the development of SSD, Liu and Ichiye placed
658	<	particular emphasis on an accurate description of the first solvation
659	<	shell. This resulted in a somewhat tall and narrow first peak in the
660	<	g(\emph{r}) that integrated to give similar coordination numbers to
657	>	In the original paper detailing the development of SSD, Liu and Ichiye
658	>	placed particular emphasis on an accurate description of the first
659	>	solvation shell. This resulted in a somewhat tall and narrow first
660	>	peak in $g(r)$ that integrated to give similar coordination numbers to
661		the experimental data obtained by Soper and
662		Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering
663		data from the Head-Gordon lab indicates a slightly lower and shifted
664	<	first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made
665	<	while taking into consideration the new experimental
664	>	first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were
665	>	made after taking into consideration the new experimental
666		findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the
667	<	relocation of the first peak of the oxygen-oxygen g(\emph{r}) by
668	<	comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new
667	>	relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
668	>	the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
669		experimental results. Both modified water models have shorter peaks
670	<	that are brought in more closely to the experimental peak (as seen in
671	<	the insets of figure \ref{grcompare}).  This structural alteration was
670	>	that match more closely to the experimental peak (as seen in the
671	>	insets of figure \ref{grcompare}).  This structural alteration was
672		accomplished by the combined reduction in the Lennard-Jones $\sigma$
673	<	variable and adjustment of the sticky potential strength and
674	<	cutoffs. As can be seen in table \ref{params}, the cutoffs for the
675	<	tetrahedral attractive and dipolar repulsive terms were nearly swapped
676	<	with each other. Isosurfaces of the original and modified sticky
677	<	potentials are shown in figure \ref{isosurface}. In these isosurfaces,
678	<	it is easy to see how altering the cutoffs changes the repulsive and
679	<	attractive character of the particles. With a reduced repulsive
680	<	surface (darker region), the particles can move closer to one another,
681	<	increasing the density for the overall system. This change in
682	<	interaction cutoff also results in a more gradual orientational motion
683	<	by allowing the particles to maintain preferred dipolar arrangements
684	<	before they begin to feel the pull of the tetrahedral
685	<	restructuring. As the particles move closer together, the dipolar
686	<	repulsion term becomes active and excludes unphysical nearest-neighbor
687	<	arrangements. This compares with how SSD and SSD1 exclude preferred
688	<	dipole alignments before the particles feel the pull of the ``hydrogen
689	<	bonds''. Aside from improving the shape of the first peak in the
690	<	g(\emph{r}), this modification improves the densities considerably by
691	<	allowing the persistence of full dipolar character below the previous
692	<	4.0 \AA\ cutoff.
673	>	variable and adjustment of the sticky potential strength and cutoffs.
674	>	As can be seen in table \ref{params}, the cutoffs for the tetrahedral
675	>	attractive and dipolar repulsive terms were nearly swapped with each
676	>	other.  Isosurfaces of the original and modified sticky potentials are
677	>	shown in figure \ref{isosurface}. In these isosurfaces, it is easy to
678	>	see how altering the cutoffs changes the repulsive and attractive
679	>	character of the particles. With a reduced repulsive surface (darker
680	>	region), the particles can move closer to one another, increasing the
681	>	density for the overall system.  This change in interaction cutoff
682	>	also results in a more gradual orientational motion by allowing the
683	>	particles to maintain preferred dipolar arrangements before they begin
684	>	to feel the pull of the tetrahedral restructuring. As the particles
685	>	move closer together, the dipolar repulsion term becomes active and
686	>	excludes unphysical nearest-neighbor arrangements. This compares with
687	>	how SSD and SSD1 exclude preferred dipole alignments before the
688	>	particles feel the pull of the ``hydrogen bonds''. Aside from
689	>	improving the shape of the first peak in the g(\emph{r}), this
690	>	modification improves the densities considerably by allowing the
691	>	persistence of full dipolar character below the previous 4.0 \AA\
692	>	cutoff.
693
694	<	While adjusting the location and shape of the first peak of
695	<	g(\emph{r}) improves the densities, these changes alone are
696	<	insufficient to bring the system densities up to the values observed
697	<	experimentally. To further increase the densities, the dipole moments
698	<	were increased in both of the adjusted models. Since SSD is a dipole
699	<	based model, the structure and transport are very sensitive to changes
700	<	in the dipole moment. The original SSD simply used the dipole moment
694	>	While adjusting the location and shape of the first peak of $g(r)$
695	>	improves the densities, these changes alone are insufficient to bring
696	>	the system densities up to the values observed experimentally.  To
697	>	further increase the densities, the dipole moments were increased in
698	>	both of our adjusted models. Since SSD is a dipole based model,
699	>	the structure and transport are very sensitive to changes in the
700	>	dipole moment. The original SSD simply used the dipole moment
701		calculated from the TIP3P water model, which at 2.35 D is
702		significantly greater than the experimental gas phase value of 1.84
703		D. The larger dipole moment is a more realistic value and improves the
#	Line 676 | Line 706 | increasing the dipole moments to 2.42 and 2.48 D for S
706		to values as high as 3.11 D, providing a substantial range of
707		reasonable values for a dipole
708		moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
709	<	increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF,
710	<	respectively, leads to significant changes in the density and
711	<	transport of the water models.
709	>	increasing the dipole moments to 2.42 and 2.48 D for SSD/E and
710	>	SSD/RF, respectively, leads to significant changes in the
711	>	density and transport of the water models.
712
713		In order to demonstrate the benefits of these reparameterizations, a
714		series of NPT and NVE simulations were performed to probe the density
#	Line 693 | Line 723 | collection times as stated earlier in this paper.
723		run at each temperature step, and the final configuration from the
724		previous temperature simulation was used as a starting point. All NVE
725		simulations had the same thermalization, equilibration, and data
726	<	collection times as stated earlier in this paper.
726	>	collection times as stated previously.
727
728		\begin{figure}
729		\begin{center}
730		\epsfxsize=6in
731		\epsfbox{ssdeDense.epsi}
732	<	\caption{Comparison of densities calculated with SSD/E to SSD1 without a
733	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
734	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a
735	<	expansion around 300 K with error bars included to clarify this region
736	<	of interest. Note that both SSD1 and SSD/E show good agreement with
732	>	\caption{Comparison of densities calculated with SSD/E to
733	>	SSD1 without a reaction field, TIP3P [Ref. \citen{Jorgensen98b}],
734	>	TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and
735	>	experiment [Ref. \citen{CRC80}]. The window shows a expansion around
736	>	300 K with error bars included to clarify this region of
737	>	interest. Note that both SSD1 and SSD/E show good agreement with
738		experiment when the long-range correction is neglected.}
739		\label{ssdedense}
740		\end{center}
741		\end{figure}
742
743	<	Figure \ref{ssdedense} shows the density profile for the SSD/E model
744	<	in comparison to SSD1 without a reaction field, other common water
745	<	models, and experimental results. The calculated densities for both
746	<	SSD/E and SSD1 have increased significantly over the original SSD
747	<	model (see figure \ref{dense1}) and are in better agreement with the
748	<	experimental values. At 298 K, the densities of SSD/E and SSD1 without
743	>	Fig. \ref{ssdedense} shows the density profile for the SSD/E
744	>	model in comparison to SSD1 without a reaction field, other
745	>	common water models, and experimental results. The calculated
746	>	densities for both SSD/E and SSD1 have increased
747	>	significantly over the original SSD model (see
748	>	fig. \ref{dense1}) and are in better agreement with the experimental
749	>	values. At 298 K, the densities of SSD/E and SSD1 without
750		a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
751		0.999$\pm$0.001 g/cm$^3$ respectively.  These both compare well with
752		the experimental value of 0.997 g/cm$^3$, and they are considerably
753	<	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to
754	<	the dipole moment and sticky switching functions have improved the
755	<	structuring of the liquid (as seen in figure \ref{grcompare}, but they
756	<	have shifted the density maximum to much lower temperatures. This
757	<	comes about via an increase in the liquid disorder through the
758	<	weakening of the sticky potential and strengthening of the dipolar
759	<	character. However, this increasing disorder in the SSD/E model has
760	<	little effect on the melting transition. By monitoring C$\text{p}$
761	<	throughout these simulations, the melting transition for SSD/E was
762	<	shown to occur at 235 K, the same transition temperature observed with
763	<	SSD and SSD1.
753	>	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The
754	>	changes to the dipole moment and sticky switching functions have
755	>	improved the structuring of the liquid (as seen in figure
756	>	\ref{grcompare}, but they have shifted the density maximum to much
757	>	lower temperatures. This comes about via an increase in the liquid
758	>	disorder through the weakening of the sticky potential and
759	>	strengthening of the dipolar character. However, this increasing
760	>	disorder in the SSD/E model has little effect on the melting
761	>	transition. By monitoring $C_p$ throughout these simulations, the
762	>	melting transition for SSD/E was shown to occur at 235 K.  The
763	>	same transition temperature observed with SSD and SSD1.
764
765		\begin{figure}
766		\begin{center}
767		\epsfxsize=6in
768		\epsfbox{ssdrfDense.epsi}
769	<	\caption{Comparison of densities calculated with SSD/RF to SSD1 with a
770	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
771	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the
772	<	necessity of reparameterization when utilizing a reaction field
773	<	long-ranged correction - SSD/RF provides significantly more accurate
774	<	densities than SSD1 when performing room temperature simulations.}
769	>	\caption{Comparison of densities calculated with SSD/RF to
770	>	SSD1 with a reaction field, TIP3P [Ref. \citen{Jorgensen98b}],
771	>	TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and
772	>	experiment [Ref. \citen{CRC80}]. The inset shows the necessity of
773	>	reparameterization when utilizing a reaction field long-ranged
774	>	correction - SSD/RF provides significantly more accurate
775	>	densities than SSD1 when performing room temperature
776	>	simulations.}
777		\label{ssdrfdense}
778		\end{center}
779		\end{figure}
780
781		Including the reaction field long-range correction in the simulations
782	<	results in a more interesting comparison. A density profile including
782	>	results in a more interesting comparison.  A density profile including
783		SSD/RF and SSD1 with an active reaction field is shown in figure
784		\ref{ssdrfdense}.  As observed in the simulations without a reaction
785		field, the densities of SSD/RF and SSD1 show a dramatic increase over
786		normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
787		of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
788	<	considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and
789	<	the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further
790	<	emphasize the importance of reparameterization in order to model the
791	<	density properly under different simulation conditions. Again, these
792	<	changes have only a minor effect on the melting point, which observed
793	<	at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1
794	<	with a reaction field. Additionally, the difference in density maxima
795	<	is not as extreme, with SSD/RF showing a density maximum at 255 K,
796	<	fairly close to the density maxima of 260 K and 265 K, shown by SSD
797	<	and SSD1 respectively.
788	>	considerably better than the original SSD value of 0.941$\pm$0.001
789	>	g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results
790	>	further emphasize the importance of reparameterization in order to
791	>	model the density properly under different simulation conditions.
792	>	Again, these changes have only a minor effect on the melting point,
793	>	which observed at 245 K for SSD/RF, is identical to SSD and only 5 K
794	>	lower than SSD1 with a reaction field. Additionally, the difference in
795	>	density maxima is not as extreme, with SSD/RF showing a density
796	>	maximum at 255 K, fairly close to the density maxima of 260 K and 265
797	>	K, shown by SSD and SSD1 respectively.
798
799		\begin{figure}
800		\begin{center}
801		\epsfxsize=6in
802		\epsfbox{ssdeDiffuse.epsi}
803	<	\caption{Plots of the diffusion constants calculated from SSD/E and SSD1,
804	<	both without a reaction field, along with experimental
805	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
806	<	at the average densities observed in the 1 atm NPT simulations for
807	<	the respective models. SSD/E is slightly more fluid than experiment
808	<	at all of the temperatures, but it is closer than SSD1 without a
809	<	long-range correction.}
803	>	\caption{The diffusion constants calculated from SSD/E and
804	>	SSD1 (both without a reaction field) along with experimental results
805	>	[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were
806	>	performed at the average densities observed in the 1 atm NPT
807	>	simulations for the respective models. SSD/E is slightly more mobile
808	>	than experiment at all of the temperatures, but it is closer to
809	>	experiment at biologically relevant temperatures than SSD1 without a
810	>	long-range correction.}
811		\label{ssdediffuse}
812		\end{center}
813		\end{figure}
#	Line 780 | Line 815 | the densities, it is important that particle transport
815		The reparameterization of the SSD water model, both for use with and
816		without an applied long-range correction, brought the densities up to
817		what is expected for simulating liquid water. In addition to improving
818	<	the densities, it is important that particle transport be maintained
819	<	or improved. Figure \ref{ssdediffuse} compares the temperature
820	<	dependence of the diffusion constant of SSD/E to SSD1 without an
821	<	active reaction field, both at the densities calculated at 1 atm and
822	<	at the experimentally calculated densities for super-cooled and liquid
823	<	water. The diffusion constant for SSD/E is consistently a little
824	<	higher than experiment, while SSD1 remains lower than experiment until
825	<	relatively high temperatures (greater than 330 K). Both models follow
826	<	the shape of the experimental curve well below 300 K but tend to
827	<	diffuse too rapidly at higher temperatures, something that is
828	<	especially apparent with SSD1. This accelerated increasing of
829	<	diffusion is caused by the rapidly decreasing system density with
830	<	increasing temperature. Though it is difficult to see in figure
831	<	\ref{ssdedense}, the densities of SSD1 decay more rapidly with
832	<	temperature than do those of SSD/E, leading to more visible deviation
833	<	from the experimental diffusion trend. Thus, the changes made to
834	<	improve the liquid structure may have had an adverse affect on the
835	<	density maximum, but they improve the transport behavior of SSD/E
836	<	relative to SSD1.
818	>	the densities, it is important that the diffusive behavior of SSD be
819	>	maintained or improved. Figure \ref{ssdediffuse} compares the
820	>	temperature dependence of the diffusion constant of SSD/E to SSD1
821	>	without an active reaction field at the densities calculated from
822	>	their respective NPT simulations at 1 atm. The diffusion constant for
823	>	SSD/E is consistently higher than experiment, while SSD1 remains lower
824	>	than experiment until relatively high temperatures (around 360
825	>	K). Both models follow the shape of the experimental curve well below
826	>	300 K but tend to diffuse too rapidly at higher temperatures, as seen
827	>	in SSD1's crossing above 360 K.  This increasing diffusion relative to
828	>	the experimental values is caused by the rapidly decreasing system
829	>	density with increasing temperature.  Both SSD1 and SSD/E show this
830	>	deviation in particle mobility, but this trend has different
831	>	implications on the diffusive behavior of the models.  While SSD1
832	>	shows more experimentally accurate diffusive behavior in the high
833	>	temperature regimes, SSD/E shows more accurate behavior in the
834	>	supercooled and biologically relevant temperature ranges.  Thus, the
835	>	changes made to improve the liquid structure may have had an adverse
836	>	affect on the density maximum, but they improve the transport behavior
837	>	of SSD/E relative to SSD1 under the most commonly simulated
838	>	conditions.
839
840		\begin{figure}
841		\begin{center}
842		\epsfxsize=6in
843		\epsfbox{ssdrfDiffuse.epsi}
844	<	\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1,
845	<	both with an active reaction field, along with experimental
846	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
847	<	at the average densities observed in the 1 atm NPT simulations for
848	<	both of the models. Note how accurately SSD/RF simulates the
849	<	diffusion of water throughout this temperature range. The more
850	<	rapidly increasing diffusion constants at high temperatures for both
851	<	models is attributed to the significantly lower densities than
852	<	observed in experiment.}
844	>	\caption{The diffusion constants calculated from SSD/RF and
845	>	SSD1 (both with an active reaction field) along with
846	>	experimental results [Refs. \citen{Gillen72} and \citen{Holz00}]. The
847	>	NVE calculations were performed at the average densities observed in
848	>	the 1 atm NPT simulations for both of the models. SSD/RF
849	>	simulates the diffusion of water throughout this temperature range
850	>	very well. The rapidly increasing diffusion constants at high
851	>	temperatures for both models can be attributed to lower calculated
852	>	densities than those observed in experiment.}
853		\label{ssdrfdiffuse}
854		\end{center}
855		\end{figure}
856
857		In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
858		compared to SSD1 with an active reaction field. Note that SSD/RF
859	<	tracks the experimental results incredibly well, identical within
860	<	error throughout the temperature range shown and with only a slight
861	<	increasing trend at higher temperatures. SSD1 tends to diffuse more
862	<	slowly at low temperatures and deviates to diffuse too rapidly at
863	<	temperatures greater than 330 K. As stated in relation to SSD/E, this
864	<	deviation away from the ideal trend is due to a rapid decrease in
865	<	density at higher temperatures. SSD/RF does not suffer from this
866	<	problem as much as SSD1, because the calculated densities are closer
867	<	to the experimental value. These results again emphasize the
868	<	importance of careful reparameterization when using an altered
832	<	long-range correction.
859	>	tracks the experimental results quantitatively, identical within error
860	>	throughout most of the temperature range shown and exhibiting only a
861	>	slight increasing trend at higher temperatures. SSD1 tends to diffuse
862	>	more slowly at low temperatures and deviates to diffuse too rapidly at
863	>	temperatures greater than 330 K.  As stated above, this deviation away
864	>	from the ideal trend is due to a rapid decrease in density at higher
865	>	temperatures. SSD/RF does not suffer from this problem as much as SSD1
866	>	because the calculated densities are closer to the experimental
867	>	values. These results again emphasize the importance of careful
868	>	reparameterization when using an altered long-range correction.
869
870	+	\begin{table}
871	+	\begin{minipage}{\linewidth}
872	+	\renewcommand{\thefootnote}{\thempfootnote}
873	+	\begin{center}
874	+	\caption{Properties of the single-point water models compared with
875	+	experimental data at ambient conditions}
876	+	\begin{tabular}{ l  c  c  c  c  c }
877	+	\hline \\[-3mm]
878	+	\ \ \ \ \ \  & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \
879	+	\ & \ SSD/RF \ \ \ & \ Expt. \\
880	+	\hline \\[-3mm]
881	+	\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\
882	+	\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\
883	+	\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 &
884	+	2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\
885	+	\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 &
886	+	4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in
887	+	Ref. \citen{Head-Gordon00_1}} \\
888	+	\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 &
889	+	3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in
890	+	Ref. \citen{Soper86}}  \\
891	+	\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 &
892	+	7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\
893	+	\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2
894	+	$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in
895	+	Ref. \citen{Krynicki66}}
896	+	\end{tabular}
897	+	\label{liquidproperties}
898	+	\end{center}
899	+	\end{minipage}
900	+	\end{table}
901	+
902	+	Table \ref{liquidproperties} gives a synopsis of the liquid state
903	+	properties of the water models compared in this study along with the
904	+	experimental values for liquid water at ambient conditions. The
905	+	coordination number ($n_C$) and number of hydrogen bonds per particle
906	+	($n_H$) were calculated by integrating the following relations:
907	+	\begin{equation}
908	+	n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr,
909	+	\end{equation}
910	+	\begin{equation}
911	+	n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr,
912	+	\end{equation}
913	+	where $\rho$ is the number density of the specified pair interactions,
914	+	$a$ and $b$ are the radial locations of the minima following the first
915	+	peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number
916	+	of hydrogen bonds stays relatively constant across all of the models,
917	+	but the coordination numbers of SSD/E and SSD/RF show an
918	+	improvement over SSD1.  This improvement is primarily due to
919	+	extension of the first solvation shell in the new parameter sets.
920	+	Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears
921	+	that all molecules in the first solvation shell are involved in
922	+	hydrogen bonds.  Since $n_H$ and $n_C$ differ in the newly
923	+	parameterized models, the orientations in the first solvation shell
924	+	are a bit more ``fluid''.  Therefore SSD1 overstructures the
925	+	first solvation shell and our reparameterizations have returned this
926	+	shell to more realistic liquid-like behavior.
927	+
928	+	The time constants for the orientational autocorrelation functions
929	+	are also displayed in Table \ref{liquidproperties}. The dipolar
930	+	orientational time correlation functions ($C_{l}$) are described
931	+	by:
932	+	\begin{equation}
933	+	C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle,
934	+	\end{equation}
935	+	where $P_l$ are Legendre polynomials of order $l$ and
936	+	$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular
937	+	dipole.\cite{Rahman71} From these correlation functions, the
938	+	orientational relaxation time of the dipole vector can be calculated
939	+	from an exponential fit in the long-time regime ($t >
940	+	\tau_l$).\cite{Rothschild84} Calculation of these time constants were
941	+	averaged over five detailed NVE simulations performed at the ambient
942	+	conditions for each of the respective models. It should be noted that
943	+	the commonly cited value of 1.9 ps for $\tau_2$ was determined from
944	+	the NMR data in Ref. \citen{Krynicki66} at a temperature near
945	+	34$^\circ$C.\cite{Rahman71} Because of the strong temperature
946	+	dependence of $\tau_2$, it is necessary to recalculate it at 298 K to
947	+	make proper comparisons. The value shown in Table
948	+	\ref{liquidproperties} was calculated from the same NMR data in the
949	+	fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was
950	+	recomputed for 298 K from the data in Ref. \citen{Eisenberg69}.
951	+	Again, SSD/E and SSD/RF show improved behavior over SSD1, both with
952	+	and without an active reaction field. Turning on the reaction field
953	+	leads to much improved time constants for SSD1; however, these results
954	+	also include a corresponding decrease in system density.
955	+	Orientational relaxation times published in the original SSD dynamics
956	+	papers are smaller than the values observed here, and this difference
957	+	can be attributed to the use of the Ewald sum.\cite{Ichiye99}
958	+
959		\subsection{Additional Observations}
960
961		\begin{figure}
962		\begin{center}
963		\epsfxsize=6in
964	<	\epsfbox{povIce.ps}
965	<	\caption{A water lattice built from the crystal structure assumed by
966	<	SSD/E when undergoing an extremely restricted temperature NPT
967	<	simulation. This form of ice is referred to as ice \emph{i} to
968	<	emphasize its simulation origins. This image was taken of the (001)
844	<	face of the crystal.}
964	>	\epsfbox{icei_bw.eps}
965	>	\caption{The most stable crystal structure assumed by the SSD family
966	>	of water models.  We refer to this structure as Ice-{\it i} to
967	>	indicate its origins in computer simulation.  This image was taken of
968	>	the (001) face of the crystal.}
969		\label{weirdice}
970		\end{center}
971		\end{figure}
972
973	<	While performing restricted temperature melting sequences of SSD/E not
974	<	previously discussed, some interesting observations were made. After
975	<	melting at 235 K, two of five systems underwent crystallization events
976	<	near 245 K. As the heating process continued, the two systems remained
977	<	crystalline until finally melting between 320 and 330 K. The final
978	<	configurations of these two melting sequences show an expanded
979	<	zeolite-like crystal structure that does not correspond to any known
980	<	form of ice. For convenience, and to help distinguish it from the
981	<	experimentally observed forms of ice, this crystal structure will
982	<	henceforth be referred to as ice $\sqrt{\smash[b]{-\text{I}}}$ (ice
983	<	\emph{i}). The crystallinity was extensive enough that a near ideal
984	<	crystal structure of ice \emph{i} could be obtained. Figure
985	<	\ref{weirdice} shows the repeating crystal structure of a typical
986	<	crystal at 5 K. Each water molecule is hydrogen bonded to four others;
987	<	however, the hydrogen bonds are flexed rather than perfectly
988	<	straight. This results in a skewed tetrahedral geometry about the
989	<	central molecule. Referring to figure \ref{isosurface}, these flexed
990	<	hydrogen bonds are allowed due to the conical shape of the attractive
991	<	regions, with the greatest attraction along the direct hydrogen bond
973	>	While performing a series of melting simulations on an early iteration
974	>	of SSD/E not discussed in this paper, we observed
975	>	recrystallization into a novel structure not previously known for
976	>	water.  After melting at 235 K, two of five systems underwent
977	>	crystallization events near 245 K.  The two systems remained
978	>	crystalline up to 320 and 330 K, respectively.  The crystal exhibits
979	>	an expanded zeolite-like structure that does not correspond to any
980	>	known form of ice.  This appears to be an artifact of the point
981	>	dipolar models, so to distinguish it from the experimentally observed
982	>	forms of ice, we have denoted the structure
983	>	Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}).  A large enough
984	>	portion of the sample crystallized that we have been able to obtain a
985	>	near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice}
986	>	shows the repeating crystal structure of a typical crystal at 5
987	>	K. Each water molecule is hydrogen bonded to four others; however, the
988	>	hydrogen bonds are bent rather than perfectly straight. This results
989	>	in a skewed tetrahedral geometry about the central molecule.  In
990	>	figure \ref{isosurface}, it is apparent that these flexed hydrogen
991	>	bonds are allowed due to the conical shape of the attractive regions,
992	>	with the greatest attraction along the direct hydrogen bond
993		configuration. Though not ideal, these flexed hydrogen bonds are
994	<	favorable enough to stabilize an entire crystal generated around
870	<	them. In fact, the imperfect ice \emph{i} crystals were so stable that
871	<	they melted at temperatures nearly 100 K greater than both ice I$_c$
872	<	and I$_h$.
994	>	favorable enough to stabilize an entire crystal generated around them.
995
996	<	These initial simulations indicated that ice \emph{i} is the preferred
997	<	ice structure for at least the SSD/E model. To verify this, a
998	<	comparison was made between near ideal crystals of ice $I_h$, ice
999	<	$I_c$, and ice 0 at constant pressure with SSD/E, SSD/RF, and
1000	<	SSD1. Near ideal versions of the three types of crystals were cooled
1001	<	to 1 K, and the potential energies of each were compared using all
1002	<	three water models. With every water model, ice \emph{i} turned out to
1003	<	have the lowest potential energy: 5\% lower than $I_h$ with SSD1,
1004	<	6.5\% lower with SSD/E, and 7.5\% lower with SSD/RF.
996	>	Initial simulations indicated that Ice-{\it i} is the preferred ice
997	>	structure for at least the SSD/E model. To verify this, a comparison
998	>	was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and
999	>	Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and
1000	>	SSD1. Near-ideal versions of the three types of crystals were cooled
1001	>	to 1 K, and enthalpies of formation of each were compared using all
1002	>	three water models.  Enthalpies were estimated from the
1003	>	isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where
1004	>	$P_{\text ext}$ is the applied pressure.  A constant value of -60.158
1005	>	kcal / mol has been added to place our zero for the enthalpies of
1006	>	formation for these systems at the traditional state (elemental forms
1007	>	at standard temperature and pressure).  With every model in the SSD
1008	>	family, Ice-{\it i} had the lowest calculated enthalpy of formation.
1009
1010	<	In addition to these low temperature comparisons, melting sequences
1011	<	were performed with ice \emph{i} as the initial configuration using
1012	<	SSD/E, SSD/RF, and SSD1 both with and without a reaction field. The
1013	<	melting transitions for both SSD/E and SSD1 without a reaction field
1014	<	occurred at temperature in excess of 375 K. SSD/RF and SSD1 with a
1015	<	reaction field showed more reasonable melting transitions near 325
1016	<	K. These melting point observations emphasize the preference for this
1017	<	crystal structure over the most common types of ice when using these
1018	<	single point water models.
1010	>	\begin{table}
1011	>	\begin{center}
1012	>	\caption{Enthalpies of Formation (in kcal / mol) of the three crystal
1013	>	structures (at 1 K) exhibited by the SSD family of water models}
1014	>	\begin{tabular}{ l  c  c  c  }
1015	>	\hline \\[-3mm]
1016	>	\ \ \ Water Model \ \ \  & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \  & \
1017	>	Ice-{\it i} \\
1018	>	\hline \\[-3mm]
1019	>	\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\
1020	>	\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\
1021	>	\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\
1022	>	\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\
1023	>	\end{tabular}
1024	>	\label{iceenthalpy}
1025	>	\end{center}
1026	>	\end{table}
1027
1028	<	Recognizing that the above tests show ice \emph{i} to be both the most
1029	<	stable and lowest density crystal structure for these single point
1030	<	water models, it is interesting to speculate on the relative stability
1031	<	of this crystal structure with charge based water models. As a quick
1032	<	test, these 3 crystal types were converted from SSD type particles to
1033	<	TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
1034	<	minimizations were performed on the crystals to compare the system
1035	<	energies. Again, ice \emph{i} was observed to have the lowest total
902	<	system energy. The total energy of ice \emph{i} was ~2\% lower than
903	<	ice $I_h$, which was in turn ~3\% lower than ice $I_c$. Based on these
904	<	initial studies, it would not be surprising if results from the other
905	<	common water models show ice \emph{i} to be the lowest energy crystal
906	<	structure. A continuation of this work studying ice \emph{i} with
907	<	multi-point water models will be published in a coming article.
1028	>	In addition to these energetic comparisons, melting simulations were
1029	>	performed with ice-{\it i} as the initial configuration using SSD/E,
1030	>	SSD/RF, and SSD1 both with and without a reaction field. The melting
1031	>	transitions for both SSD/E and SSD1 without reaction field occurred at
1032	>	temperature in excess of 375~K.  SSD/RF and SSD1 with a reaction field
1033	>	showed more reasonable melting transitions near 325~K.  These melting
1034	>	point observations clearly show that all of the SSD-derived models
1035	>	prefer the ice-{\it i} structure.
1036
1037		\section{Conclusions}
910	–	The density maximum and temperature dependent transport for the SSD
911	–	water model, both with and without the use of reaction field, were
912	–	studied via a series of NPT and NVE simulations. The constant pressure
913	–	simulations of the melting of both $I_h$ and $I_c$ ice showed a
914	–	density maximum near 260 K. In most cases, the calculated densities
915	–	were significantly lower than the densities calculated in simulations
916	–	of other water models. Analysis of particle diffusion showed SSD to
917	–	capture the transport properties of experimental water well in both
918	–	the liquid and super-cooled liquid regimes. In order to correct the
919	–	density behavior, the original SSD model was reparameterized for use
920	–	both with and without a reaction field (SSD/RF and SSD/E), and
921	–	comparison simulations were performed with SSD1, the density corrected
922	–	version of SSD. Both models improve the liquid structure, density
923	–	values, and diffusive properties under their respective conditions,
924	–	indicating the necessity of reparameterization when altering the
925	–	long-range correction specifics. When taking into account the
926	–	appropriate considerations, these simple water models are excellent
927	–	choices for representing explicit water in large scale simulations of
928	–	biochemical systems.
1038
1039	+	The density maximum and temperature dependence of the self-diffusion
1040	+	constant were studied for the SSD water model, both with and
1041	+	without the use of reaction field, via a series of NPT and NVE
1042	+	simulations. The constant pressure simulations showed a density
1043	+	maximum near 260 K. In most cases, the calculated densities were
1044	+	significantly lower than the densities obtained from other water
1045	+	models (and experiment). Analysis of self-diffusion showed SSD
1046	+	to capture the transport properties of water well in both the liquid
1047	+	and supercooled liquid regimes.
1048	+
1049	+	In order to correct the density behavior, the original SSD model was
1050	+	reparameterized for use both with and without a reaction field (SSD/RF
1051	+	and SSD/E), and comparisons were made with SSD1, Ichiye's density
1052	+	corrected version of SSD. Both models improve the liquid structure,
1053	+	densities, and diffusive properties under their respective simulation
1054	+	conditions, indicating the necessity of reparameterization when
1055	+	changing the method of calculating long-range electrostatic
1056	+	interactions.  In general, however, these simple water models are
1057	+	excellent choices for representing explicit water in large scale
1058	+	simulations of biochemical systems.
1059	+
1060	+	The existence of a novel low-density ice structure that is preferred
1061	+	by the SSD family of water models is somewhat troubling, since
1062	+	liquid simulations on this family of water models at room temperature
1063	+	are effectively simulations of supercooled or metastable liquids.  One
1064	+	way to destabilize this unphysical ice structure would be to make the
1065	+	range of angles preferred by the attractive part of the sticky
1066	+	potential much narrower.  This would require extensive
1067	+	reparameterization to maintain the same level of agreement with the
1068	+	experiments.
1069	+
1070	+	Additionally, our initial calculations show that the Ice-{\it i}
1071	+	structure may also be a preferred crystal structure for at least one
1072	+	other popular multi-point water model (TIP3P), and that much of the
1073	+	simulation work being done using this popular model could also be at
1074	+	risk for crystallization into this unphysical structure.  A future
1075	+	publication will detail the relative stability of the known ice
1076	+	structures for a wide range of popular water models.
1077	+
1078		\section{Acknowledgments}
1079		Support for this project was provided by the National Science
1080		Foundation under grant CHE-0134881. Computation time was provided by
1081		the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
1082	<	DMR 00 79647.
1082	>	DMR-0079647.
1083
936	–
1084		\newpage
1085
1086		\bibliographystyle{jcp}

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