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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{\thefootnote}
28	<	\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
30	<
31	<	\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
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\\
29		Notre Dame, Indiana 46556}
30
31		\date{\today}
32
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 the SSD water model, both with and without the use of reaction
39	<	field. The constant pressure simulations of the melting of both $I_h$
40	<	and $I_c$ ice showed a density maximum near 260 K. In most cases, the
41	<	calculated densities were significantly lower than the densities
42	<	calculated in simulations of other water models. Analysis of particle
43	<	diffusion showed SSD to capture the transport properties of
44	<	experimental very well in both the normal and super-cooled liquid
45	<	regimes. In order to correct the density behavior, SSD was
46	<	reparameterized for use both with and without a long-range interaction
47	<	correction, SSD/RF and SSD/E respectively. In addition to correcting
48	<	the abnormally low densities, these new versions were show to maintain
49	<	or improve upon the transport and structural features of the original
50	<	water model.
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.  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	<	\maketitle
61	>	\newpage
62
63		%\narrowtext
64
#	Line 62 | Line 69 | One of the most important tasks in simulations of bioc
69
70		\section{Introduction}
71
72	<	One of the most important tasks in simulations of biochemical systems
73	<	is the proper depiction of water and water solvation. In fact, the
74	<	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
78	<	groups/bulk. Due to the fact that explicit solvent accounts for a
79	<	massive portion of the calculations, it necessary to simplify the
73	<	solvent to some extent in order to complete simulations in a
74	<	reasonable amount of time. In the case of simulating water in
75	<	bio-molecular studies, the balance between accurate properties and
76	<	computational efficiency is especially delicate, and it has resulted
77	<	in a variety of different water
78	<	models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these models
79	<	get specific properties correct or better than their predecessors, but
80	<	this is often at a cost of some other properties or of computer
81	<	time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P succeeds
82	<	in improving the structural and transport properties over its
83	<	predecessors, yet this comes at a greater than 50\% increase in
84	<	computational cost.\cite{Jorgensen01,Jorgensen00} One recently
85	<	developed model that succeeds in both retaining accuracy of system
86	<	properties and simplifying calculations to increase computational
87	<	efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}
72	>	One of the most important tasks in the simulation of biochemical
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 was
94	>	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 which
97	>	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
103	>	shell.
104	>
105	>	The interaction between two SSD water molecules \emph{i} and \emph{j}
106	>	is given by the potential
107		\begin{equation}
108		u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
109	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
109	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
110		u_{ij}^{sp}
111	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
111	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
112		\end{equation}
113	<	where the $\mathbf{r}_{ij}$ is the position vector between molecules
114	<	\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and
115	<	$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
116	<	orientations of the respective molecules. The Lennard-Jones, dipole,
117	<	and sticky parts of the potential are giving by the following
108	<	equations,
113	>	where the ${\bf r}_{ij}$ is the position vector between molecules
114	>	\emph{i} and \emph{j} with magnitude $r_{ij}$, and
115	>	${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of
116	>	the two molecules. The Lennard-Jones and dipole interactions are given
117	>	by the following familiar forms:
118		\begin{equation}
119	<	u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
119	>	u_{ij}^{LJ}(r_{ij}) = 4\epsilon
120	>	\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]
121	>	\ ,
122		\end{equation}
123	+	and
124		\begin{equation}
125	<	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}\ ,
125	>	u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left(
126	>	\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf
127	>	r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ ,
128		\end{equation}
129	+	where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along
130	+	the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and
131	+	$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf
132	+	r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule
133	+	$i$.
134	+
135	+	The sticky potential is somewhat less familiar:
136		\begin{equation}
116	–	\begin{split}
137		u_{ij}^{sp}
138	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)
139	<	&=
140	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\
141	<	& \quad \ +
142	<	s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
123	<	\end{split}
138	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
139	>	\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)
140	>	+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf
141	>	\Omega}_j)]\ .
142	>	\label{stickyfunction}
143		\end{equation}
144	<	where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
145	<	unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
146	<	$\nu_0$ scales the strength of the overall sticky potential, $s$ and
147	<	$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
148	<	functions take the following forms,
144	>	Here, $\nu_0$ is a strength parameter for the sticky potential, and
145	>	$s$ and $s^\prime$ are cubic switching functions which turn off the
146	>	sticky interaction beyond the first solvation shell. The $w$ function
147	>	can be thought of as an attractive potential with tetrahedral
148	>	geometry:
149		\begin{equation}
150	<	w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
150	>	w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
151		\end{equation}
152	+	while the $w^\prime$ function counters the normal aligned and
153	+	anti-aligned structures favored by point dipoles:
154		\begin{equation}
155	<	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,
155	>	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,
156		\end{equation}
157	<	where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
158	<	term that promotes hydrogen bonding orientations within the first
159	<	solvation shell, and $w^\prime$ is a dipolar repulsion term that
160	<	repels unrealistic dipolar arrangements within the first solvation
161	<	shell. A more detailed description of the functional parts and
162	<	variables in this potential can be found in other
163	<	articles.\cite{Ichiye96,Ichiye99}
157	>	It should be noted that $w$ is proportional to the sum of the $Y_3^2$
158	>	and $Y_3^{-2}$ spherical harmonics (a linear combination which
159	>	enhances the tetrahedral geometry for hydrogen bonded structures),
160	>	while $w^\prime$ is a purely empirical function.  A more detailed
161	>	description of the functional parts and variables in this potential
162	>	can be found in the original SSD
163	>	articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
164
165	<	Being that this is a one-site point dipole model, the actual force
166	<	calculations are simplified significantly. In the original Monte Carlo
167	<	simulations using this model, Ichiye \emph{et al.} reported a
168	<	calculation speed up of up to an order of magnitude over other
169	<	comparable models while maintaining the structural behavior of
170	<	water.\cite{Ichiye96} In the original molecular dynamics studies of
171	<	SSD, it was shown that it actually improves upon the prediction of
172	<	water's dynamical properties 3 and 4-point models.\cite{Ichiye99} This
173	<	attractive combination of speed and accurate depiction of solvent
174	<	properties makes SSD a model of interest for the simulation of large
175	<	scale biological systems, such as membrane phase behavior, a specific
176	<	interest within our group.
165	>	Since SSD is a single-point {\it dipolar} model, the force
166	>	calculations are simplified significantly relative to the standard
167	>	{\it charged} multi-point models. In the original Monte Carlo
168	>	simulations using this model, Ichiye {\it et al.} reported that using
169	>	SSD decreased computer time by a factor of 6-7 compared to other
170	>	models.\cite{Ichiye96} What is most impressive is that this savings
171	>	did not come at the expense of accurate depiction of the liquid state
172	>	properties.  Indeed, SSD maintains reasonable agreement with the Soper
173	>	data for the structural features of liquid
174	>	water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
175	>	exhibited by SSD agree with experiment better than those of more
176	>	computationally expensive models (like TIP3P and
177	>	SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
178	>	of solvent properties makes SSD a very attractive model for the
179	>	simulation of large scale biochemical simulations.
180
181	<	One of the key limitations of this water model, however, is that it
182	<	has been parameterized for use with the Ewald Sum technique for the
183	<	handling of long-ranged interactions.  When studying very large
184	<	systems, the Ewald summation and even particle-mesh Ewald become
185	<	computational burdens with their respective ideal $N^\frac{3}{2}$ and
186	<	$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99}
187	<	In applying this water model in these types of systems, it would be
188	<	useful to know its properties and behavior with the more
189	<	computationally efficient reaction field (RF) technique, and even with
190	<	a cutoff that lacks any form of long range correction. This study
191	<	addresses these issues by looking at the structural and transport
192	<	behavior of SSD over a variety of temperatures, with the purpose of
193	<	utilizing the RF correction technique. Towards the end, we suggest
194	<	alterations to the parameters that result in more water-like
195	<	behavior. It should be noted that in a recent publication, some the
196	<	original investigators of the SSD water model have put forth
197	<	adjustments to the original SSD water model to address abnormal
198	<	density behavior (also observed here), calling the corrected model
199	<	SSD1.\cite{Ichiye03} This study will consider this new model's
200	<	behavior as well, and hopefully improve upon its depiction of water
201	<	under conditions without the Ewald Sum.
181	>	One feature of the SSD model is that it was parameterized for use with
182	>	the Ewald sum to handle long-range interactions.  This would normally
183	>	be the best way of handling long-range interactions in systems that
184	>	contain other point charges.  However, our group has recently become
185	>	interested in systems with point dipoles as mimics for neutral, but
186	>	polarized regions on molecules (e.g. the zwitterionic head group
187	>	regions of phospholipids).  If the system of interest does not contain
188	>	point charges, the Ewald sum and even particle-mesh Ewald become
189	>	computational bottlenecks.  Their respective ideal $N^\frac{3}{2}$ and
190	>	$N\log N$ calculation scaling orders for $N$ particles can become
191	>	prohibitive when $N$ becomes large.\cite{Darden99} In applying this
192	>	water model in these types of systems, it would be useful to know its
193	>	properties and behavior under the more computationally efficient
194	>	reaction field (RF) technique, or even with a simple cutoff. This
195	>	study addresses these issues by looking at the structural and
196	>	transport behavior of SSD over a variety of temperatures with the
197	>	purpose of utilizing the RF correction technique.  We then suggest
198	>	modifications to the parameters that result in more realistic bulk
199	>	phase behavior.  It should be noted that in a recent publication, some
200	>	of the original investigators of the SSD water model have suggested
201	>	adjustments to the SSD water model to address abnormal density
202	>	behavior (also observed here), calling the corrected model
203	>	SSD1.\cite{Ichiye03} In what follows, we compare our
204	>	reparamaterization of SSD with both the original SSD and SSD1 models
205	>	with the goal of improving the bulk phase behavior of an SSD-derived
206	>	model in simulations utilizing the Reaction Field.
207
208		\section{Methods}
209
210	<	As stated previously, in this study the long-range dipole-dipole
211	<	interactions were accounted for 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.  Under the first method,
213	>	the magnitude of the reaction field acting on dipole $i$ is
214		\begin{equation}
215		\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
216	<	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\  ,
216	>	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\  ,
217		\label{rfequation}
218		\end{equation}
219		where $\mathcal{R}$ is the cavity defined by the cutoff radius
220		($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
221	<	system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment
222	<	vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching
221	>	system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole
222	>	moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching
223		function.\cite{AllenTildesley} The reaction field contribution to the
224	<	total energy by particle \emph{i} is given by
225	<	$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
226	<	on dipole \emph{i} by
227	<	$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use
228	<	of reaction field is known to alter the orientational dynamic
229	<	properties, such as the dielectric relaxation time, based on changes
230	<	in the length of the cutoff radius.\cite{Berendsen98} This variable
231	<	behavior makes reaction field a less attractive method than other
232	<	methods, like the Ewald summation; however, for the simulation of
203	<	large-scale system, the computational cost benefit of reaction field
204	<	is dramatic. To address some of the dynamical property alterations due
205	<	to the use of reaction field, simulations were also performed without
206	<	a surrounding dielectric and suggestions are proposed on how to make
207	<	SSD more compatible with a reaction field.
224	>	total energy by particle $i$ is given by $-\frac{1}{2}{\bf
225	>	\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf
226	>	\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley}  Use of the reaction
227	>	field is known to alter the bulk orientational properties, such as the
228	>	dielectric relaxation time.  There is particular sensitivity of this
229	>	property on changes in the length of the cutoff
230	>	radius.\cite{Berendsen98} This variable behavior makes reaction field
231	>	a less attractive method than the Ewald sum.  However, for very large
232	>	systems, the computational benefit of reaction field is dramatic.
233
234	<	Simulations were performed in both the isobaric-isothermal and
235	<	microcanonical ensembles. The constant pressure simulations were
236	<	implemented using an integral thermostat and barostat as outlined by
237	<	Hoover.\cite{Hoover85,Hoover86} For the constant pressure
238	<	simulations, the \emph{Q} parameter for the was set to 5.0 amu
214	<	\(\cdot\)\AA\(^{2}\), and the relaxation time (\(\tau\))\ was set at
215	<	100 ps.
234	>	We have also performed a companion set of simulations {\it without} a
235	>	surrounding dielectric (i.e. using a simple cubic switching function
236	>	at the cutoff radius) and as a result we have two reparamaterizations
237	>	of SSD which could be used either with or without the Reaction Field
238	>	turned on.
239
240	+	Simulations to obtain the preferred density were performed in the
241	+	isobaric-isothermal (NPT) ensemble, while all dynamical properties
242	+	were obtained from microcanonical (NVE) simulations done at densities
243	+	matching the NPT density for a particular target temperature.  The
244	+	constant pressure simulations were implemented using an integral
245	+	thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86}
246	+	All molecules were treated as non-linear rigid bodies. Vibrational
247	+	constraints are not necessary in simulations of SSD, because there are
248	+	no explicit hydrogen atoms, and thus no molecular vibrational modes
249	+	need to be considered.
250	+
251		Integration of the equations of motion was carried out using the
252	<	symplectic splitting method proposed by Dullweber \emph{et
253	<	al.}.\cite{Dullweber1997} The reason for this integrator selection
254	<	deals with poor energy conservation of rigid body systems using
255	<	quaternions. While quaternions work well for orientational motion in
256	<	alternate ensembles, the microcanonical ensemble has a constant energy
257	<	requirement that is actually quite sensitive to errors in the
258	<	equations of motion. The original implementation of this code utilized
259	<	quaternions for rotational motion propagation; however, a detailed
260	<	investigation showed that they resulted in a steady drift in the total
227	<	energy, something that has been observed by others.\cite{Laird97}
252	>	symplectic splitting method proposed by Dullweber {\it et
253	>	al.}\cite{Dullweber1997} Our reason for selecting this integrator
254	>	centers on poor energy conservation of rigid body dynamics using
255	>	traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions
256	>	may work well for orientational motion under NVT or NPT integrators,
257	>	our limits on energy drift in the microcanonical ensemble were quite
258	>	strict, and the drift under quaternions was substantially greater than
259	>	in the symplectic splitting method.  This steady drift in the total
260	>	energy has also been observed by Kol {\it et al.}\cite{Laird97}
261
262		The key difference in the integration method proposed by Dullweber
263		\emph{et al.} is that the entire rotation matrix is propagated from
264	<	one time step to the next. In the past, this would not have been as
265	<	feasible a option, being that the rotation matrix for a single body is
266	<	nine elements long as opposed to 3 or 4 elements for Euler angles and
267	<	quaternions respectively. System memory has become much less of an
235	<	issue in recent times, and this has resulted in substantial benefits
236	<	in energy conservation. There is still the issue of 5 or 6 additional
237	<	elements for describing the orientation of each particle, which will
238	<	increase dump files substantially. Simply translating the rotation
239	<	matrix into its component Euler angles or quaternions for storage
240	<	purposes relieves this burden.
264	>	one time step to the next.  The additional memory required by the
265	>	algorithm is inconsequential on modern computers, and translating the
266	>	rotation matrix into quaternions for storage purposes makes trajectory
267	>	data quite compact.
268
269		The symplectic splitting method allows for Verlet style integration of
270	<	both linear and angular motion of rigid bodies. In the integration
271	<	method, the orientational propagation involves a sequence of matrix
272	<	evaluations to update the rotation matrix.\cite{Dullweber1997} These
273	<	matrix rotations end up being more costly computationally than the
274	<	simpler arithmetic quaternion propagation. On average, a 1000 SSD
275	<	particle simulation shows a 7\% increase in computation time using the
276	<	symplectic step method in place of quaternions. This cost is more than
277	<	justified when comparing the energy conservation of the two methods as
278	<	illustrated in figure \ref{timestep}.
270	>	both translational and orientational motion of rigid bodies. In this
271	>	integration method, the orientational propagation involves a sequence
272	>	of matrix evaluations to update the rotation
273	>	matrix.\cite{Dullweber1997} These matrix rotations are more costly
274	>	than the simpler arithmetic quaternion propagation. With the same time
275	>	step, a 1000 SSD particle simulation shows an average 7\% increase in
276	>	computation time using the symplectic step method in place of
277	>	quaternions. The additional expense per step is justified when one
278	>	considers the ability to use time steps that are nearly twice as large
279	>	under symplectic splitting than would be usable under quaternion
280	>	dynamics.  The energy conservation of the two methods using a number
281	>	of different time steps is illustrated in figure
282	>	\ref{timestep}.
283
284		\begin{figure}
285	<	\includegraphics[width=61mm, angle=-90]{timeStep.epsi}
286	<	\caption{Energy conservation using quaternion based integration versus
287	<	the symplectic step method proposed by Dullweber \emph{et al.} with
288	<	increasing time step. For each time step, the dotted line is total
289	<	energy using the symplectic step integrator, and the solid line comes
290	<	from the quaternion integrator. The larger time step plots are shifted
291	<	up from the true energy baseline for clarity.}
285	>	\begin{center}
286	>	\epsfxsize=6in
287	>	\epsfbox{timeStep.epsi}
288	>	\caption{Energy conservation using both quaternion based integration and
289	>	the symplectic step method proposed by Dullweber \emph{et al.} with
290	>	increasing time step. The larger time step plots are shifted from the
291	>	true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
292		\label{timestep}
293	+	\end{center}
294		\end{figure}
295
296		In figure \ref{timestep}, the resulting energy drift at various time
297		steps for both the symplectic step and quaternion integration schemes
298	<	is compared. All of the 1000 SSD particle simulations started with the
299	<	same configuration, and the only difference was the method for
300	<	handling rotational motion. At time steps of 0.1 and 0.5 fs, both
301	<	methods for propagating particle rotation conserve energy fairly well,
302	<	with the quaternion method showing a slight energy drift over time in
303	<	the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
304	<	energy conservation benefits of the symplectic step method are clearly
305	<	demonstrated. Thus, while maintaining the same degree of energy
306	<	conservation, one can take considerably longer time steps, leading to
307	<	an overall reduction in computation time.
298	>	is compared.  All of the 1000 SSD particle simulations started with
299	>	the same configuration, and the only difference was the method used to
300	>	handle orientational motion. At time steps of 0.1 and 0.5 fs, both
301	>	methods for propagating the orientational degrees of freedom conserve
302	>	energy fairly well, with the quaternion method showing a slight energy
303	>	drift over time in the 0.5 fs time step simulation. At time steps of 1
304	>	and 2 fs, the energy conservation benefits of the symplectic step
305	>	method are clearly demonstrated. Thus, while maintaining the same
306	>	degree of energy conservation, one can take considerably longer time
307	>	steps, leading to an overall reduction in computation time.
308
309	<	Energy drift in these SSD particle simulations was unnoticeable for
310	<	time steps up to three femtoseconds. A slight energy drift on the
309	>	Energy drift in the symplectic step simulations was unnoticeable for
310	>	time steps up to 3 fs. A slight energy drift on the
311		order of 0.012 kcal/mol per nanosecond was observed at a time step of
312	<	four femtoseconds, and as expected, this drift increases dramatically
313	<	with increasing time step. To insure accuracy in the constant energy
312	>	4 fs, and as expected, this drift increases dramatically
313	>	with increasing time step. To insure accuracy in our microcanonical
314		simulations, time steps were set at 2 fs and kept at this value for
315		constant pressure simulations as well.
316
317	<	Ice crystals in both the $I_h$ and $I_c$ lattices were generated as
318	<	starting points for all the simulations. The $I_h$ crystals were
319	<	formed by first arranging the center of masses of the SSD particles
320	<	into a ``hexagonal'' ice lattice of 1024 particles. Because of the
321	<	crystal structure of $I_h$ ice, the simulation box assumed a
322	<	rectangular shape with a edge length ratio of approximately
317	>	Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices
318	>	were generated as starting points for all simulations. The $I_h$
319	>	crystals were formed by first arranging the centers of mass of the SSD
320	>	particles into a ``hexagonal'' ice lattice of 1024 particles. Because
321	>	of the crystal structure of $I_h$ ice, the simulation box assumed an
322	>	orthorhombic shape with an edge length ratio of approximately
323		1.00$\times$1.06$\times$1.23. The particles were then allowed to
324		orient freely about fixed positions with angular momenta randomized at
325		400 K for varying times. The rotational temperature was then scaled
326	<	down in stages to slowly cool the crystals down to 25 K. The particles
327	<	were then allowed translate with fixed orientations at a constant
326	>	down in stages to slowly cool the crystals to 25 K. The particles were
327	>	then allowed to translate with fixed orientations at a constant
328		pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were
329		removed and the ice crystals were allowed to equilibrate for 50 ps at
330		25 K and a constant pressure of 1 atm.  This procedure resulted in
331		structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
332	<	rules\cite{Bernal33,Rahman72}.  This method was also utilized in the
332	>	rules.\cite{Bernal33,Rahman72} This method was also utilized in the
333		making of diamond lattice $I_c$ ice crystals, with each cubic
334		simulation box consisting of either 512 or 1000 particles. Only
335		isotropic volume fluctuations were performed under constant pressure,
#	Line 307 | Line 339 | constant pressure and temperature dynamics. This invol
339		\section{Results and discussion}
340
341		Melting studies were performed on the randomized ice crystals using
342	<	constant pressure and temperature dynamics. This involved an initial
343	<	randomization of velocities about the starting temperature of 25 K for
344	<	varying amounts of time. The systems were all equilibrated for 100 ps
345	<	prior to a 200 ps data collection run at each temperature setting,
346	<	ranging from 25 to 400 K, with a maximum degree increment of 25 K. For
347	<	regions of interest along this stepwise progression, the temperature
348	<	increment was decreased from 25 K to 10 and then 5 K. The above
349	<	equilibration and production times were sufficient in that the system
350	<	volume fluctuations dampened out in all but the very cold simulations
351	<	(below 225 K). In order to further improve statistics, five separate
352	<	simulation progressions were performed, and the averaged results from
353	<	the $I_h$ melting simulations are shown in figure \ref{dense1}.
342	>	isobaric-isothermal (NPT) dynamics. During melting simulations, the
343	>	melting transition and the density maximum can both be observed,
344	>	provided that the density maximum occurs in the liquid and not the
345	>	supercooled regime. An ensemble average from five separate melting
346	>	simulations was acquired, each starting from different ice crystals
347	>	generated as described previously. All simulations were equilibrated
348	>	for 100 ps prior to a 200 ps data collection run at each temperature
349	>	setting. The temperature range of study spanned from 25 to 400 K, with
350	>	a maximum degree increment of 25 K. For regions of interest along this
351	>	stepwise progression, the temperature increment was decreased from 25
352	>	K to 10 and 5 K.  The above equilibration and production times were
353	>	sufficient in that fluctuations in the volume autocorrelation function
354	>	were damped out in all simulations in under 20 ps.
355
323	–	\begin{figure}
324	–	\includegraphics[width=65mm, angle=-90]{1hdense.epsi}
325	–	\caption{Average density of SSD water at increasing temperatures
326	–	starting from ice $I_h$ lattice.}
327	–	\label{dense1}
328	–	\end{figure}
329	–
356		\subsection{Density Behavior}
331	–	In the initial average density versus temperature plot, the density
332	–	maximum clearly appears between 255 and 265 K. The calculated
333	–	densities within this range were nearly indistinguishable, as can be
334	–	seen in the zoom of this region of interest, shown in figure
335	–	\ref{dense1}. The greater certainty of the average value at 260 K makes
336	–	a good argument for the actual density maximum residing at this
337	–	midpoint value. Figure \ref{dense1} was constructed using ice $I_h$
338	–	crystals for the initial configuration; and though not pictured, the
339	–	simulations starting from ice $I_c$ crystal configurations showed
340	–	similar results, with a liquid-phase density maximum in this same
341	–	region (between 255 and 260 K). In addition, the $I_c$ crystals are
342	–	more fragile than the $I_h$ crystals, leading them to deform into a
343	–	dense glassy state at lower temperatures. This resulted in an overall
344	–	low temperature density maximum at 200 K, but they still retained a
345	–	common liquid state density maximum with the $I_h$ simulations.
357
358	+	Our initial simulations focused on the original SSD water model, and
359	+	an average density versus temperature plot is shown in figure
360	+	\ref{dense1}. Note that the density maximum when using a reaction
361	+	field appears between 255 and 265 K.  There were smaller fluctuations
362	+	in the density at 260 K than at either 255 or 265, so we report this
363	+	value as the location of the density maximum. Figure \ref{dense1} was
364	+	constructed using ice $I_h$ crystals for the initial configuration;
365	+	though not pictured, the simulations starting from ice $I_c$ crystal
366	+	configurations showed similar results, with a liquid-phase density
367	+	maximum in this same region (between 255 and 260 K).
368	+
369		\begin{figure}
370	<	\includegraphics[width=65mm,angle=-90]{dense2.eps}
371	<	\caption{Density versus temperature for TIP4P\cite{Jorgensen98b},
372	<	TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction
373	<	Field, SSD, and Experiment\cite{CRC80}. }
374	<	\label{dense2}
370	>	\begin{center}
371	>	\epsfxsize=6in
372	>	\epsfbox{denseSSD.eps}
373	>	\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}],
374	>	TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD
375	>	without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The
376	>	arrows indicate the change in densities observed when turning off the
377	>	reaction field. The the lower than expected densities for the SSD
378	>	model were what prompted the original reparameterization of SSD1
379	>	[Ref. \citen{Ichiye03}].}
380	>	\label{dense1}
381	>	\end{center}
382		\end{figure}
383
384	<	The density maximum for SSD actually compares quite favorably to other
385	<	simple water models. Figure \ref{dense2} shows a plot of these
386	<	findings with the density progression of several other models and
358	<	experiment obtained from other
384	>	The density maximum for SSD compares quite favorably to other simple
385	>	water models. Figure \ref{dense1} also shows calculated densities of
386	>	several other models and experiment obtained from other
387		sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
388	<	models, SSD has results closest to the experimentally observed water
389	<	density maximum. Of the listed water models, TIP4P has a density
390	<	maximum behavior most like that seen in SSD. Though not shown, it is
391	<	useful to note that TIP5P has a water density maximum nearly identical
392	<	to experiment.
388	>	models, SSD has a temperature closest to the experimentally observed
389	>	density maximum. Of the {\it charge-based} models in
390	>	Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that
391	>	seen in SSD. Though not included in this plot, it is useful
392	>	to note that TIP5P has a density maximum nearly identical to the
393	>	experimentally measured temperature.
394
395	<	Possibly of more importance is the density scaling of SSD relative to
396	<	other common models at any given temperature (Fig. \ref{dense2}). Note
397	<	that the SSD model assumes a lower density than any of the other
398	<	listed models at the same pressure, behavior which is especially
399	<	apparent at temperatures greater than 300 K. Lower than expected
400	<	densities have been observed for other systems with the use of a
401	<	reaction field for long-range electrostatic interactions, so the most
402	<	likely reason for these significantly lower densities in these
403	<	simulations is the presence of the reaction field.\cite{Berendsen98}
404	<	In order to test the effect of the reaction field on the density of
405	<	the systems, the simulations were repeated for the temperature region
406	<	of interest without a reaction field present. The results of these
407	<	simulations are also displayed in figure \ref{dense2}. Without
379	<	reaction field, these densities increase considerably to more
380	<	experimentally reasonable values, especially around the freezing point
381	<	of liquid water. The shape of the curve is similar to the curve
382	<	produced from SSD simulations using reaction field, specifically the
383	<	rapidly decreasing densities at higher temperatures; however, a slight
384	<	shift in the density maximum location, down to 245 K, is
385	<	observed. This is probably a more accurate comparison to the other
386	<	listed water models in that no long range corrections were applied in
387	<	those simulations.\cite{Clancy94,Jorgensen98b}
395	>	It has been observed that liquid state densities in water are
396	>	dependent on the cutoff radius used both with and without the use of
397	>	reaction field.\cite{Berendsen98} In order to address the possible
398	>	effect of cutoff radius, simulations were performed with a dipolar
399	>	cutoff radius of 12.0 \AA\ to complement the previous SSD simulations,
400	>	all performed with a cutoff of 9.0 \AA. All of the resulting densities
401	>	overlapped within error and showed no significant trend toward lower
402	>	or higher densities as a function of cutoff radius, for simulations
403	>	both with and without reaction field. These results indicate that
404	>	there is no major benefit in choosing a longer cutoff radius in
405	>	simulations using SSD. This is advantageous in that the use of a
406	>	longer cutoff radius results in a significant increase in the time
407	>	required to obtain a single trajectory.
408
409	<	It has been observed that densities are dependent on the cutoff radius
410	<	used for a variety of water models in simulations both with and
411	<	without the use of reaction field.\cite{Berendsen98} In order to
412	<	address the possible affect of cutoff radius, simulations were
413	<	performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the
414	<	previous SSD simulations, all performed with a cutoff of 9.0 \AA. All
415	<	the resulting densities overlapped within error and showed no
416	<	significant trend in lower or higher densities as a function of cutoff
417	<	radius, both for simulations with and without reaction field. These
418	<	results indicate that there is no major benefit in choosing a longer
419	<	cutoff radius in simulations using SSD. This is comforting in that the
420	<	use of a longer cutoff radius results in a near doubling of the time
421	<	required to compute a single trajectory.
409	>	The key feature to recognize in figure \ref{dense1} is the density
410	>	scaling of SSD relative to other common models at any given
411	>	temperature. SSD assumes a lower density than any of the other listed
412	>	models at the same pressure, behavior which is especially apparent at
413	>	temperatures greater than 300 K. Lower than expected densities have
414	>	been observed for other systems using a reaction field for long-range
415	>	electrostatic interactions, so the most likely reason for the
416	>	significantly lower densities seen in these simulations is the
417	>	presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order
418	>	to test the effect of the reaction field on the density of the
419	>	systems, the simulations were repeated without a reaction field
420	>	present. The results of these simulations are also displayed in figure
421	>	\ref{dense1}. Without the reaction field, the densities increase
422	>	to more experimentally reasonable values, especially around the
423	>	freezing point of liquid water. The shape of the curve is similar to
424	>	the curve produced from SSD simulations using reaction field,
425	>	specifically the rapidly decreasing densities at higher temperatures;
426	>	however, a shift in the density maximum location, down to 245 K, is
427	>	observed. This is a more accurate comparison to the other listed water
428	>	models, in that no long range corrections were applied in those
429	>	simulations.\cite{Clancy94,Jorgensen98b} However, even without the
430	>	reaction field, the density around 300 K is still significantly lower
431	>	than experiment and comparable water models. This anomalous behavior
432	>	was what lead Ichiye {\it et al.} to recently reparameterize
433	>	SSD.\cite{Ichiye03} Throughout the remainder of the paper our
434	>	reparamaterizations of SSD will be compared with the newer SSD1 model.
435
436		\subsection{Transport Behavior}
437	<	Of importance in these types of studies are the transport properties
438	<	of the particles and how they change when altering the environmental
439	<	conditions. In order to probe transport, constant energy simulations
440	<	were performed about the average density uncovered by the constant
441	<	pressure simulations. Simulations started with randomized velocities
442	<	and underwent 50 ps of temperature scaling and 50 ps of constant
443	<	energy equilibration before obtaining a 200 ps trajectory. Diffusion
444	<	constants were calculated via root-mean square deviation analysis. The
445	<	averaged results from 5 sets of these NVE simulations is displayed in
446	<	figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P
437	>
438	>	Accurate dynamical properties of a water model are particularly
439	>	important when using the model to study permeation or transport across
440	>	biological membranes.  In order to probe transport in bulk water,
441	>	constant energy (NVE) simulations were performed at the average
442	>	density obtained by the NPT simulations at an identical target
443	>	temperature. Simulations started with randomized velocities and
444	>	underwent 50 ps of temperature scaling and 50 ps of constant energy
445	>	equilibration before a 200 ps data collection run. Diffusion constants
446	>	were calculated via linear fits to the long-time behavior of the
447	>	mean-square displacement as a function of time. The averaged results
448	>	from five sets of NVE simulations are displayed in figure
449	>	\ref{diffuse}, alongside experimental, SPC/E, and TIP5P
450		results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
451
452		\begin{figure}
453	<	\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi}
454	<	\caption{Average diffusion coefficient over increasing temperature for
455	<	SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental
456	<	data from Gillen \emph{et al.}\cite{Gillen72}, and from
457	<	Mills\cite{Mills73}.}
453	>	\begin{center}
454	>	\epsfxsize=6in
455	>	\epsfbox{betterDiffuse.epsi}
456	>	\caption{Average self-diffusion constant as a function of temperature for
457	>	SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}],
458	>	and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of
459	>	the three water models shown, SSD has the least deviation from the
460	>	experimental values. The rapidly increasing diffusion constants for
461	>	TIP5P and SSD correspond to significant decrease in density at the
462	>	higher temperatures.}
463		\label{diffuse}
464	+	\end{center}
465		\end{figure}
466
467		The observed values for the diffusion constant point out one of the
468	<	strengths of the SSD model. Of the three experimental models shown,
469	<	the SSD model has the most accurate depiction of the diffusion trend
470	<	seen in experiment in both the supercooled and normal regimes. SPC/E
471	<	does a respectable job by getting similar values as SSD and experiment
472	<	around 290 K; however, it deviates at both higher and lower
473	<	temperatures, failing to predict the experimental trend. TIP5P and SSD
474	<	both start off low at the colder temperatures and tend to diffuse too
475	<	rapidly at the higher temperatures. This type of trend at the higher
476	<	temperatures is not surprising in that the densities of both TIP5P and
477	<	SSD are lower than experimental water at temperatures higher than room
478	<	temperature. When calculating the diffusion coefficients for SSD at
479	<	experimental densities, the resulting values fall more in line with
480	<	experiment at these temperatures, albeit not at standard
439	<	pressure. Results under these conditions can be found later in this
440	<	paper.
468	>	strengths of the SSD model. Of the three models shown, the SSD model
469	>	has the most accurate depiction of self-diffusion in both the
470	>	supercooled and liquid regimes.  SPC/E does a respectable job by
471	>	reproducing values similar to experiment around 290 K; however, it
472	>	deviates at both higher and lower temperatures, failing to predict the
473	>	correct thermal trend. TIP5P and SSD both start off low at colder
474	>	temperatures and tend to diffuse too rapidly at higher temperatures.
475	>	This behavior at higher temperatures is not particularly surprising
476	>	since the densities of both TIP5P and SSD are lower than experimental
477	>	water densities at higher temperatures.  When calculating the
478	>	diffusion coefficients for SSD at experimental densities (instead of
479	>	the densities from the NPT simulations), the resulting values fall
480	>	more in line with experiment at these temperatures.
481
482		\subsection{Structural Changes and Characterization}
483	+
484		By starting the simulations from the crystalline state, the melting
485	<	transition and the ice structure can be studied along with the liquid
486	<	phase behavior beyond the melting point. To locate the melting
487	<	transition, the constant pressure heat capacity (C$_\text{p}$) was
488	<	monitored in each of the simulations. In the melting simulations of
489	<	the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$
490	<	occurs at 245 K, indicating a first order phase transition for the
491	<	melting of these ice crystals. When the reaction field is turned off,
492	<	the melting transition occurs at 235 K.  These melting transitions are
493	<	considerably lower than the experimental value, but this is not
453	<	surprising in that SSD is a simple rigid body model with a fixed
454	<	dipole.
485	>	transition and the ice structure can be obtained along with the liquid
486	>	phase behavior beyond the melting point. The constant pressure heat
487	>	capacity (C$_\text{p}$) was monitored to locate the melting transition
488	>	in each of the simulations. In the melting simulations of the 1024
489	>	particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs
490	>	at 245 K, indicating a first order phase transition for the melting of
491	>	these ice crystals. When the reaction field is turned off, the melting
492	>	transition occurs at 235 K.  These melting transitions are
493	>	considerably lower than the experimental value.
494
495	<	\begin{figure}
496	<	\includegraphics[width=85mm]{fullContours.eps}
495	>	\begin{figure}
496	>	\begin{center}
497	>	\epsfxsize=6in
498	>	\epsfbox{corrDiag.eps}
499	>	\caption{Two dimensional illustration of angles involved in the
500	>	correlations observed in Fig. \ref{contour}.}
501	>	\label{corrAngle}
502	>	\end{center}
503	>	\end{figure}
504	>
505	>	\begin{figure}
506	>	\begin{center}
507	>	\epsfxsize=6in
508	>	\epsfbox{fullContours.eps}
509		\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at
510		100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for
511		clarity: dark areas signify peaks while light areas signify
512	<	depressions. White areas have g(\emph{r}) values below 0.5 and black
512	>	depressions. White areas have $g(r)$ values below 0.5 and black
513		areas have values above 1.5.}
514		\label{contour}
515	+	\end{center}
516		\end{figure}
517
518	<	Additional analyses for understanding the melting phase-transition
519	<	process were performed via two-dimensional structure and dipole angle
520	<	correlations. Expressions for the correlations are as follows:
518	>	Additional analysis of the melting process was performed using
519	>	two-dimensional structure and dipole angle correlations. Expressions
520	>	for these correlations are as follows:
521
522	<	\begin{figure}
523	<	\includegraphics[width=45mm]{corrDiag.eps}
524	<	\caption{Two dimensional illustration of the angles involved in the
525	<	correlations observed in figure \ref{contour}.}
526	<	\label{corrAngle}
527	<	\end{figure}
528	<
529	<	\begin{multline}
530	<	g_{\text{AB}}(r,\cos\theta) = \\
531	<	\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\ ,
532	<	\end{multline}
533	<	\begin{multline}
534	<	g_{\text{AB}}(r,\cos\omega) = \\
535	<	\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\ ,
536	<	\end{multline}
537	<	where $\theta$ and $\omega$ refer to the angles shown in the above
538	<	illustration. By binning over both distance and the cosine of the
539	<	desired angle between the two dipoles, the g(\emph{r}) can be
488	<	dissected to determine the common dipole arrangements that constitute
489	<	the peaks and troughs. Frames A and B of figure \ref{contour} show a
490	<	relatively crystalline state of an ice $I_c$ simulation. The first
491	<	peak of the g(\emph{r}) primarily consists of the preferred hydrogen
492	<	bonding arrangements as dictated by the tetrahedral sticky potential,
493	<	one peak for the donating and the other for the accepting hydrogen
494	<	bonds. Due to the high degree of crystallinity of the sample, the
495	<	second and third solvation shells show a repeated peak arrangement
522	>	\begin{equation}
523	>	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\ ,
524	>	\end{equation}
525	>	\begin{equation}
526	>	g_{\text{AB}}(r,\cos\omega) =
527	>	\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\ ,
528	>	\end{equation}
529	>	where $\theta$ and $\omega$ refer to the angles shown in figure
530	>	\ref{corrAngle}. By binning over both distance and the cosine of the
531	>	desired angle between the two dipoles, the $g(r)$ can be analyzed to
532	>	determine the common dipole arrangements that constitute the peaks and
533	>	troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
534	>	of figure \ref{contour} show results from an ice $I_c$ simulation. The
535	>	first peak in the $g(r)$ consists primarily of the preferred hydrogen
536	>	bonding arrangements as dictated by the tetrahedral sticky potential -
537	>	one peak for the hydrogen bond donor and the other for the hydrogen
538	>	bond acceptor.  Due to the high degree of crystallinity of the sample,
539	>	the second and third solvation shells show a repeated peak arrangement
540		which decays at distances around the fourth solvation shell, near the
541		imposed cutoff for the Lennard-Jones and dipole-dipole interactions.
542	<	In the higher temperature simulation shown in frames C and D, the
543	<	repeated peak features are significantly blurred. The first solvation
544	<	shell still shows the strong effect of the sticky-potential, although
545	<	it covers a larger area, extending to include a fraction of aligned
542	>	In the higher temperature simulation shown in frames C and D, these
543	>	long-range features deteriorate rapidly. The first solvation shell
544	>	still shows the strong effect of the sticky-potential, although it
545	>	covers a larger area, extending to include a fraction of aligned
546		dipole peaks within the first solvation shell. The latter peaks lose
547	<	definition as thermal motion and the competing dipole force overcomes
548	<	the sticky potential's tight tetrahedral structuring of the fluid.
547	>	due to thermal motion and as the competing dipole force overcomes the
548	>	sticky potential's tight tetrahedral structuring of the crystal.
549
550		This complex interplay between dipole and sticky interactions was
551		remarked upon as a possible reason for the split second peak in the
552	<	oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the
553	<	second solvation shell peak appears to have two distinct parts that
554	<	blend together to form one observable peak. At higher temperatures,
555	<	this split character alters to show the leading 4 \AA\ peak dominated
556	<	by equatorial anti-parallel dipole orientations, and there is tightly
557	<	bunched group of axially arranged dipoles that most likely consist of
558	<	the smaller fraction aligned dipole pairs. The trailing part of the
559	<	split peak at 5 \AA\ is dominated by aligned dipoles that range
560	<	primarily within the axial to the chief hydrogen bond arrangements
561	<	similar to those seen in the first solvation shell. This evidence
562	<	indicates that the dipole pair interaction begins to dominate outside
563	<	of the range of the dipolar repulsion term, with the primary
564	<	energetically favorable dipole arrangements populating the region
565	<	immediately outside of it's range (around 4 \AA), and arrangements
566	<	that seek to ideally satisfy both the sticky and dipole forces locate
567	<	themselves just beyond this region (around 5 \AA).
552	>	oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures,
553	>	the second solvation shell peak appears to have two distinct
554	>	components that blend together to form one observable peak. At higher
555	>	temperatures, this split character alters to show the leading 4 \AA\
556	>	peak dominated by equatorial anti-parallel dipole orientations. There
557	>	is also a tightly bunched group of axially arranged dipoles that most
558	>	likely consist of the smaller fraction of aligned dipole pairs. The
559	>	trailing component of the split peak at 5 \AA\ is dominated by aligned
560	>	dipoles that assume hydrogen bond arrangements similar to those seen
561	>	in the first solvation shell. This evidence indicates that the dipole
562	>	pair interaction begins to dominate outside of the range of the
563	>	dipolar repulsion term.  The energetically favorable dipole
564	>	arrangements populate the region immediately outside this repulsion
565	>	region (around 4 \AA), while arrangements that seek to satisfy both
566	>	the sticky and dipole forces locate themselves just beyond this
567	>	initial buildup (around 5 \AA).
568
569		From these findings, the split second peak is primarily the product of
570	<	the dipolar repulsion term of the sticky potential. In fact, the
571	<	leading of the two peaks can be pushed out and merged with the outer
572	<	split peak just by extending the switching function cutoff
573	<	($s^\prime(r_{ij})$) from its normal 4.0 \AA\ to values of 4.5 or even
574	<	5 \AA. This type of correction is not recommended for improving the
575	<	liquid structure, because the second solvation shell will still be
576	<	shifted too far out. In addition, this would have an even more
577	<	detrimental effect on the system densities, leading to a liquid with a
578	<	more open structure and a density considerably lower than the normal
579	<	SSD behavior shown previously. A better correction would be to include
580	<	the quadrupole-quadrupole interactions for the water particles outside
581	<	of the first solvation shell, but this reduces the simplicity and
582	<	speed advantage of SSD, so it is not the most desirable path to take.
570	>	the dipolar repulsion term of the sticky potential. In fact, the inner
571	>	peak can be pushed out and merged with the outer split peak just by
572	>	extending the switching function ($s^\prime(r_{ij})$) from its normal
573	>	4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of
574	>	correction is not recommended for improving the liquid structure,
575	>	since the second solvation shell would still be shifted too far
576	>	out. In addition, this would have an even more detrimental effect on
577	>	the system densities, leading to a liquid with a more open structure
578	>	and a density considerably lower than the already low SSD density.  A
579	>	better correction would be to include the quadrupole-quadrupole
580	>	interactions for the water particles outside of the first solvation
581	>	shell, but this would remove the simplicity and speed advantage of
582	>	SSD.
583
584	<	\subsection{Adjusted Potentials: SSD/E and SSD/RF}
584	>	\subsection{Adjusted Potentials: SSD/RF and SSD/E}
585	>
586		The propensity of SSD to adopt lower than expected densities under
587		varying conditions is troubling, especially at higher temperatures. In
588	<	order to correct this behavior, it's necessary to adjust the force
589	<	field parameters for the primary intermolecular interactions. In
590	<	undergoing a reparameterization, it is important not to focus on just
591	<	one property and neglect the other important properties. In this case,
592	<	it would be ideal to correct the densities while maintaining the
593	<	accurate transport properties.
588	>	order to correct this model for use with a reaction field, it is
589	>	necessary to adjust the force field parameters for the primary
590	>	intermolecular interactions. In undergoing a reparameterization, it is
591	>	important not to focus on just one property and neglect the other
592	>	important properties. In this case, it would be ideal to correct the
593	>	densities while maintaining the accurate transport behavior.
594
595	<	The possible parameters for tuning include the $\sigma$ and $\epsilon$
596	<	Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
597	<	attractive and dipole repulsive terms with their respective
598	<	cutoffs. To alter the attractive and repulsive terms of the sticky
599	<	potential independently, it is necessary to separate the terms as
600	<	follows:
601	<	\begin{equation}
602	<	\begin{split}
603	<	u_{ij}^{sp}
604	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) &=
560	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\\
561	<	& \quad \ + \frac{\nu_0^\prime}{2}
562	<	[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
563	<	\end{split}
564	<	\end{equation}
595	>	The parameters available for tuning include the $\sigma$ and
596	>	$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the
597	>	strength of the sticky potential ($\nu_0$), and the sticky attractive
598	>	and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$
599	>	and $r_l^\prime$, $r_u^\prime$ respectively). The results of the
600	>	reparameterizations are shown in table \ref{params}. We are calling
601	>	these reparameterizations the Soft Sticky Dipole / Reaction Field
602	>	(SSD/RF - for use with a reaction field) and Soft Sticky Dipole
603	>	Extended (SSD/E - an attempt to improve the liquid structure in
604	>	simulations without a long-range correction).
605
566	–	where $\nu_0$ scales the strength of the tetrahedral attraction and
567	–	$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
568	–	term. For purposes of the reparameterization, the separation was
569	–	performed, but the final parameters were adjusted so that it is
570	–	unnecessary to separate the terms when implementing the adjusted water
571	–	potentials. The results of the reparameterizations are shown in table
572	–	\ref{params}. Note that both the tetrahedral attractive and dipolar
573	–	repulsive don't share the same lower cutoff ($r_l$) in the newly
574	–	parameterized potentials - soft sticky dipole enhanced (SSD/E) and
575	–	soft sticky dipole reaction field (SSD/RF).
576	–
606		\begin{table}
607	+	\begin{center}
608		\caption{Parameters for the original and adjusted models}
609	<	\begin{tabular}{ l  c  c  c }
609	>	\begin{tabular}{ l  c  c  c  c }
610		\hline \\[-3mm]
611	<	\ Parameters & \ \ \  SSD$^\dagger$\ \ \ \  & \ SSD/E\ \  & \ SSD/RF\ \ \\
611	>	\ \ \ Parameters\ \ \  & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \
612	>	& \ SSD1 [Ref. \citen{Ichiye03}]\ \  & \ SSD/E\ \  & \ SSD/RF \\
613		\hline \\[-3mm]
614	<	\ \ \ $\sigma$ (\AA)  & 3.051 & 3.035 & 3.019\\
615	<	\ \ \ $\epsilon$ (kcal/mol)\ \ & 0.152 & 0.152 & 0.152\\
616	<	\ \ \ $\mu$ (D) & 2.35 & 2.418 & 2.480\\
617	<	\ \ \ $\nu_0$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\
618	<	\ \ \ $r_l$ (\AA) & 2.75 & 2.40 & 2.40\\
619	<	\ \ \ $r_u$ (\AA) & 3.35 & 3.80 & 3.80\\
620	<	\ \ \ $\nu_0^\prime$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\
621	<	\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75\\
622	<	\ \ \ $r_u^\prime$ (\AA) & 4.00 & 3.35 & 3.35\\
592	<	\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96}
614	>	\ \ \ $\sigma$ (\AA)  & 3.051 & 3.016 & 3.035 & 3.019\\
615	>	\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
616	>	\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\
617	>	\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
618	>	\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\
619	>	\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\
620	>	\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\
621	>	\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
622	>	\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
623		\end{tabular}
624		\label{params}
625	+	\end{center}
626		\end{table}
627
628	<	\begin{figure}
629	<	\includegraphics[width=85mm]{gofrCompare.epsi}
630	<	\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E
631	<	and SSD without reaction field (top), as well as SSD/RF and SSD with
628	>	\begin{figure}
629	>	\begin{center}
630	>	\epsfxsize=5in
631	>	\epsfbox{GofRCompare.epsi}
632	>	\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E
633	>	and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with
634		reaction field turned on (bottom). The insets show the respective
635	<	first peaks in detail. Solid Line - experiment, dashed line - SSD/E
636	<	and SSD/RF, and dotted line - SSD (with and without reaction field).}
635	>	first peaks in detail. Note how the changes in parameters have lowered
636	>	and broadened the first peak of SSD/E and SSD/RF.}
637		\label{grcompare}
638	+	\end{center}
639		\end{figure}
640
641	<	\begin{figure}
642	<	\includegraphics[width=85mm]{dualsticky.ps}
643	<	\caption{Isosurfaces of the sticky potential for SSD (left) and SSD/E \&
641	>	\begin{figure}
642	>	\begin{center}
643	>	\epsfxsize=6in
644	>	\epsfbox{dualsticky.ps}
645	>	\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \&
646		SSD/RF (right). Light areas correspond to the tetrahedral attractive
647	<	part, and the darker areas correspond to the dipolar repulsive part.}
647	>	component, and darker areas correspond to the dipolar repulsive
648	>	component.}
649		\label{isosurface}
650	+	\end{center}
651		\end{figure}
652
653	<	In the paper detailing the development of SSD, Liu and Ichiye placed
654	<	particular emphasis on an accurate description of the first solvation
655	<	shell. This resulted in a somewhat tall and sharp first peak that
656	<	integrated to give similar coordination numbers to the experimental
657	<	data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New
658	<	experimental x-ray scattering data from the Head-Gordon lab indicates
659	<	a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so
660	<	adjustments to SSD were made while taking into consideration the new
661	<	experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare}
662	<	shows the relocation of the first peak of the oxygen-oxygen
663	<	g(\emph{r}) by comparing the original SSD (with and without reaction
664	<	field), SSD-E, and SSD-RF to the new experimental results. Both the
665	<	modified water models have shorter peaks that are brought in more
666	<	closely to the experimental peak (as seen in the insets of figure
667	<	\ref{grcompare}). This structural alteration was accomplished by a
668	<	reduction in the Lennard-Jones $\sigma$ variable as well as adjustment
669	<	of the sticky potential strength and cutoffs. The cutoffs for the
670	<	tetrahedral attractive and dipolar repulsive terms were nearly swapped
671	<	with each other. Isosurfaces of the original and modified sticky
672	<	potentials are shown in figure \cite{isosurface}. In these
673	<	isosurfaces, it is easy to see how altering the cutoffs changes the
674	<	repulsive and attractive character of the particles. With a reduced
675	<	repulsive surface (the darker region), the particles can move closer
676	<	to one another, increasing the density for the overall system. This
677	<	change in interaction cutoff also results in a more gradual
678	<	orientational motion by allowing the particles to maintain preferred
679	<	dipolar arrangements before they begin to feel the pull of the
680	<	tetrahedral restructuring. Upon moving closer together, the dipolar
681	<	repulsion term becomes active and excludes the unphysical
682	<	arrangements. This compares with the original SSD's excluding dipolar
683	<	before the particles feel the pull of the ``hydrogen bonds''. Aside
684	<	from improving the shape of the first peak in the g(\emph{r}), this
685	<	improves the densities considerably by allowing the persistence of
686	<	full dipolar character below the previous 4.0 \AA\ cutoff.
653	>	In the original paper detailing the development of SSD, Liu and Ichiye
654	>	placed particular emphasis on an accurate description of the first
655	>	solvation shell. This resulted in a somewhat tall and narrow first
656	>	peak in $g(r)$ that integrated to give similar coordination numbers to
657	>	the experimental data obtained by Soper and
658	>	Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering
659	>	data from the Head-Gordon lab indicates a slightly lower and shifted
660	>	first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were
661	>	made while taking into consideration the new experimental
662	>	findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the
663	>	relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
664	>	the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
665	>	experimental results. Both modified water models have shorter peaks
666	>	that match more closely to the experimental peak (as seen in the
667	>	insets of figure \ref{grcompare}).  This structural alteration was
668	>	accomplished by the combined reduction in the Lennard-Jones $\sigma$
669	>	variable and adjustment of the sticky potential strength and cutoffs.
670	>	As can be seen in table \ref{params}, the cutoffs for the tetrahedral
671	>	attractive and dipolar repulsive terms were nearly swapped with each
672	>	other.  Isosurfaces of the original and modified sticky potentials are
673	>	shown in figure \ref{isosurface}. In these isosurfaces, it is easy to
674	>	see how altering the cutoffs changes the repulsive and attractive
675	>	character of the particles. With a reduced repulsive surface (darker
676	>	region), the particles can move closer to one another, increasing the
677	>	density for the overall system.  This change in interaction cutoff also
678	>	results in a more gradual orientational motion by allowing the
679	>	particles to maintain preferred dipolar arrangements before they begin
680	>	to feel the pull of the tetrahedral restructuring. As the particles
681	>	move closer together, the dipolar repulsion term becomes active and
682	>	excludes unphysical nearest-neighbor arrangements. This compares with
683	>	how SSD and SSD1 exclude preferred dipole alignments before the
684	>	particles feel the pull of the ``hydrogen bonds''. Aside from
685	>	improving the shape of the first peak in the g(\emph{r}), this
686	>	modification improves the densities considerably by allowing the
687	>	persistence of full dipolar character below the previous 4.0 \AA\
688	>	cutoff.
689
690	<	While adjusting the location and shape of the first peak of
691	<	g(\emph{r}) improves the densities to some degree, these changes alone
692	<	are insufficient to bring the system densities up to the values
693	<	observed experimentally. To finish bringing up the densities, the
694	<	dipole moments were increased in both the adjusted models. Being a
695	<	dipole based model, the structure and transport are very sensitive to
696	<	changes in the dipole moment. The original SSD simply used the dipole
697	<	moment calculated from the TIP3P water model, which at 2.35 D is
698	<	significantly greater than the experimental gas phase value of 1.84
699	<	D. The larger dipole moment is a more realistic value and improve the
700	<	dielectric properties of the fluid. Both theoretical and experimental
701	<	measurements indicate a liquid phase dipole moment ranging from 2.4 D
702	<	to values as high as 3.11 D, so there is quite a range available for
703	<	adjusting the dipole
704	<	moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of
705	<	the dipole moments to 2.418 and 2.48 D for SSD/E and SSD/RF
706	<	respectively is moderate in the range of the experimental values;
667	<	however, it leads to significant changes in the density and transport
668	<	of the water models.
690	>	While adjusting the location and shape of the first peak of $g(r)$
691	>	improves the densities, these changes alone are insufficient to bring
692	>	the system densities up to the values observed experimentally.  To
693	>	further increase the densities, the dipole moments were increased in
694	>	both of our adjusted models. Since SSD is a dipole based model, the
695	>	structure and transport are very sensitive to changes in the dipole
696	>	moment. The original SSD simply used the dipole moment calculated from
697	>	the TIP3P water model, which at 2.35 D is significantly greater than
698	>	the experimental gas phase value of 1.84 D. The larger dipole moment
699	>	is a more realistic value and improves the dielectric properties of
700	>	the fluid. Both theoretical and experimental measurements indicate a
701	>	liquid phase dipole moment ranging from 2.4 D to values as high as
702	>	3.11 D, providing a substantial range of reasonable values for a
703	>	dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
704	>	increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF,
705	>	respectively, leads to significant changes in the density and
706	>	transport of the water models.
707
708	<	In order to demonstrate the benefits of this reparameterization, a
708	>	In order to demonstrate the benefits of these reparameterizations, a
709		series of NPT and NVE simulations were performed to probe the density
710		and transport properties of the adapted models and compare the results
711		to the original SSD model. This comparison involved full NPT melting
712		sequences for both SSD/E and SSD/RF, as well as NVE transport
713	<	calculations at both self-consistent and experimental
714	<	densities. Again, the results come from five separate simulations of
715	<	1024 particle systems, and the melting sequences were started from
716	<	different ice $I_h$ crystals constructed as stated previously. Like
717	<	before, all of the NPT simulations were equilibrated for 100 ps before
718	<	a 200 ps data collection run, and they used the previous temperature's
719	<	final configuration as a starting point. All of the NVE simulations
720	<	had the same thermalization, equilibration, and data collection times
721	<	stated earlier in this paper.
713	>	calculations at the calculated self-consistent densities. Again, the
714	>	results are obtained from five separate simulations of 1024 particle
715	>	systems, and the melting sequences were started from different ice
716	>	$I_h$ crystals constructed as described previously. Each NPT
717	>	simulation was equilibrated for 100 ps before a 200 ps data collection
718	>	run at each temperature step, and the final configuration from the
719	>	previous temperature simulation was used as a starting point. All NVE
720	>	simulations had the same thermalization, equilibration, and data
721	>	collection times as stated previously.
722
723	<	\begin{figure}
724	<	\includegraphics[width=85mm]{ssdecompare.epsi}
725	<	\caption{Comparison of densities calculated with SSD/E to SSD without a
726	<	reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b},
727	<	SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot
728	<	includes error bars, and the calculated results from the other
729	<	references were removed for clarity.}
723	>	\begin{figure}
724	>	\begin{center}
725	>	\epsfxsize=6in
726	>	\epsfbox{ssdeDense.epsi}
727	>	\caption{Comparison of densities calculated with SSD/E to SSD1 without a
728	>	reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P
729	>	[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and
730	>	experiment [Ref. \citen{CRC80}]. The window shows a expansion around
731	>	300 K with error bars included to clarify this region of
732	>	interest. Note that both SSD1 and SSD/E show good agreement with
733	>	experiment when the long-range correction is neglected.}
734		\label{ssdedense}
735	+	\end{center}
736		\end{figure}
737
738	<	Figure \ref{ssdedense} shows the density profile for the SSD/E water
739	<	model in comparison to the original SSD without a reaction field,
740	<	experiment, and the other common water models considered
741	<	previously. The calculated densities have increased significantly over
742	<	the original SSD model and match the experimental value just below 298
743	<	K. At 298 K, the density of SSD/E is 0.995$\pm$0.001 g/cm$^3$, which
744	<	compares well with the experimental value of 0.997 g/cm$^3$ and is
745	<	considerably better than the SSD value of 0.967$\pm$0.003
746	<	g/cm$^3$. The increased dipole moment in SSD/E has helped to flatten
747	<	out the curve at higher temperatures, only the improvement is marginal
748	<	at best. This steep drop in densities is due to the dipolar rather
749	<	than charge based interactions which decay more rapidly at longer
750	<	distances.
751	<
752	<	By monitoring C$\text{p}$ throughout these simulations, the melting
753	<	transition for SSD/E was observed at 230 K, about 5 degrees lower than
754	<	SSD. The resulting density maximum is located at 240 K, again about 5
755	<	degrees lower than the SSD value of 245 K. Though there is a decrease
756	<	in both of these values, the corrected densities near room temperature
757	<	justify the modifications taken.
738	>	Fig. \ref{ssdedense} shows the density profile for the SSD/E model
739	>	in comparison to SSD1 without a reaction field, other common water
740	>	models, and experimental results. The calculated densities for both
741	>	SSD/E and SSD1 have increased significantly over the original SSD
742	>	model (see fig. \ref{dense1}) and are in better agreement with the
743	>	experimental values. At 298 K, the densities of SSD/E and SSD1 without
744	>	a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
745	>	0.999$\pm$0.001 g/cm$^3$ respectively.  These both compare well with
746	>	the experimental value of 0.997 g/cm$^3$, and they are considerably
747	>	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to
748	>	the dipole moment and sticky switching functions have improved the
749	>	structuring of the liquid (as seen in figure \ref{grcompare}, but they
750	>	have shifted the density maximum to much lower temperatures. This
751	>	comes about via an increase in the liquid disorder through the
752	>	weakening of the sticky potential and strengthening of the dipolar
753	>	character. However, this increasing disorder in the SSD/E model has
754	>	little effect on the melting transition. By monitoring $C_p$
755	>	throughout these simulations, the melting transition for SSD/E was
756	>	shown to occur at 235 K.  The same transition temperature observed
757	>	with SSD and SSD1.
758
759	<	\begin{figure}
760	<	\includegraphics[width=85mm]{ssdrfcompare.epsi}
761	<	\caption{Comparison of densities calculated with SSD/RF to SSD with a
762	<	reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b},
763	<	SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot
764	<	includes error bars, and the calculated results from the other
765	<	references were removed for clarity.}
759	>	\begin{figure}
760	>	\begin{center}
761	>	\epsfxsize=6in
762	>	\epsfbox{ssdrfDense.epsi}
763	>	\caption{Comparison of densities calculated with SSD/RF to SSD1 with a
764	>	reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P
765	>	[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and
766	>	experiment [Ref. \citen{CRC80}]. The inset shows the necessity of
767	>	reparameterization when utilizing a reaction field long-ranged
768	>	correction - SSD/RF provides significantly more accurate densities
769	>	than SSD1 when performing room temperature simulations.}
770		\label{ssdrfdense}
771	+	\end{center}
772		\end{figure}
773
774	<	Figure \ref{ssdrfdense} shows a density comparison between SSD/RF and
775	<	SSD with an active reaction field. Like in the simulations of SSD/E,
776	<	the densities show a dramatic increase over normal SSD. At 298 K,
777	<	SSD/RF has a density of 0.997$\pm$0.001 g/cm$^3$, right in line with
778	<	experiment and considerably better than the SSD value of
779	<	0.941$\pm$0.001 g/cm$^3$. The melting point is observed at 240 K,
780	<	which is 5 degrees lower than SSD with a reaction field, and the
781	<	density maximum at 255 K, again 5 degrees lower than SSD. The density
782	<	at higher temperature still drops off more rapidly than the charge
783	<	based models but is in better agreement than SSD/E.
774	>	Including the reaction field long-range correction in the simulations
775	>	results in a more interesting comparison.  A density profile including
776	>	SSD/RF and SSD1 with an active reaction field is shown in figure
777	>	\ref{ssdrfdense}.  As observed in the simulations without a reaction
778	>	field, the densities of SSD/RF and SSD1 show a dramatic increase over
779	>	normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
780	>	of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
781	>	considerably better than the original SSD value of 0.941$\pm$0.001
782	>	g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results
783	>	further emphasize the importance of reparameterization in order to
784	>	model the density properly under different simulation conditions.
785	>	Again, these changes have only a minor effect on the melting point,
786	>	which observed at 245 K for SSD/RF, is identical to SSD and only 5 K
787	>	lower than SSD1 with a reaction field. Additionally, the difference in
788	>	density maxima is not as extreme, with SSD/RF showing a density
789	>	maximum at 255 K, fairly close to the density maxima of 260 K and 265
790	>	K, shown by SSD and SSD1 respectively.
791
792	+	\begin{figure}
793	+	\begin{center}
794	+	\epsfxsize=6in
795	+	\epsfbox{ssdeDiffuse.epsi}
796	+	\caption{The diffusion constants calculated from SSD/E and SSD1,
797	+	both without a reaction field, along with experimental results
798	+	[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations
799	+	were performed at the average densities observed in the 1 atm NPT
800	+	simulations for the respective models. SSD/E is slightly more mobile
801	+	than experiment at all of the temperatures, but it is closer to
802	+	experiment at biologically relavent temperatures than SSD1 without a
803	+	long-range correction.}
804	+	\label{ssdediffuse}
805	+	\end{center}
806	+	\end{figure}
807	+
808		The reparameterization of the SSD water model, both for use with and
809		without an applied long-range correction, brought the densities up to
810		what is expected for simulating liquid water. In addition to improving
811	<	the densities, it is important that particle transport be maintained
812	<	or improved. Figure \ref{ssdediffuse} compares the temperature
813	<	dependence of the diffusion constant of SSD/E to SSD without an active
814	<	reaction field, both at the densities calculated at 1 atm and at the
815	<	experimentally calculated densities for super-cooled and liquid
816	<	water. In the upper plot, the diffusion constant for SSD/E is
817	<	consistently a little faster than experiment, while SSD starts off
818	<	slower than experiment and crosses to merge with SSD/E at high
819	<	temperatures. Both models follow the experimental trend well, but
820	<	diffuse too rapidly at higher temperatures. This abnormally fast
821	<	diffusion is caused by the decreased system density. Since the
822	<	densities of SSD/E don't deviate as much from experiment as those of
823	<	SSD, it follows the experimental trend more closely. This observation
824	<	is backed up by looking at the lower plot. The diffusion constants for
825	<	SSD/E track with the experimental values while SSD deviates on the low
826	<	side of the trend with increasing temperature. This is again a product
827	<	of SSD/E having densities closer to experiment, and not deviating to
828	<	lower densities with increasing temperature as rapidly.
811	>	the densities, it is important that the excellent diffusive behavior
812	>	of SSD be maintained or improved. Figure \ref{ssdediffuse} compares
813	>	the temperature dependence of the diffusion constant of SSD/E to SSD1
814	>	without an active reaction field at the densities calculated from the
815	>	NPT simulations at 1 atm. The diffusion constant for SSD/E is
816	>	consistently higher than experiment, while SSD1 remains lower than
817	>	experiment until relatively high temperatures (around 360 K). Both
818	>	models follow the shape of the experimental curve well below 300 K but
819	>	tend to diffuse too rapidly at higher temperatures, as seen in SSD1's
820	>	crossing above 360 K.  This increasing diffusion relative to the
821	>	experimental values is caused by the rapidly decreasing system density
822	>	with increasing temperature.  Both SSD1 and SSD/E show this deviation
823	>	in diffusive behavior, but this trend has different implications on
824	>	the diffusive behavior of the models.  While SSD1 shows more
825	>	experimentally accurate diffusive behavior in the high temperature
826	>	regimes, SSD/E shows more accurate behavior in the supercooled and
827	>	biologically relavent temperature ranges.  Thus, the changes made to
828	>	improve the liquid structure may have had an adverse affect on the
829	>	density maximum, but they improve the transport behavior of SSD/E
830	>	relative to SSD1 under the most commonly simulated conditions.
831
832	<	\begin{figure}
833	<	\includegraphics[width=85mm]{ssdediffuse.epsi}
834	<	\caption{Plots of the diffusion constants calculated from SSD/E and SSD,
835	<	both without a reaction field along with experimental results from
836	<	Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The
837	<	upper plot is at densities calculated from the NPT simulations at a
838	<	pressure of 1 atm, while the lower plot is at the experimentally
839	<	calculated densities.}
840	<	\label{ssdediffuse}
841	<	\end{figure}
842	<
843	<	\begin{figure}
844	<	\includegraphics[width=85mm]{ssdrfdiffuse.epsi}
772	<	\caption{Plots of the diffusion constants calculated from SSD/RF and SSD,
773	<	both with an active reaction field along with experimental results
774	<	from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The
775	<	upper plot is at densities calculated from the NPT simulations at a
776	<	pressure of 1 atm, while the lower plot is at the experimentally
777	<	calculated densities.}
832	>	\begin{figure}
833	>	\begin{center}
834	>	\epsfxsize=6in
835	>	\epsfbox{ssdrfDiffuse.epsi}
836	>	\caption{The diffusion constants calculated from SSD/RF and SSD1,
837	>	both with an active reaction field, along with experimental results
838	>	[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations
839	>	were performed at the average densities observed in the 1 atm NPT
840	>	simulations for both of the models. Note how accurately SSD/RF
841	>	simulates the diffusion of water throughout this temperature
842	>	range. The more rapidly increasing diffusion constants at high
843	>	temperatures for both models is attributed to lower calculated
844	>	densities than those observed in experiment.}
845		\label{ssdrfdiffuse}
846	+	\end{center}
847		\end{figure}
848
849		In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
850	<	compared with SSD with an active reaction field. In the upper plot,
851	<	SSD/RF tracks with the experimental results incredibly well, identical
852	<	within error throughout the temperature range and only showing a
853	<	slight increasing trend at higher temperatures. SSD also tracks
854	<	experiment well, only it tends to diffuse a little more slowly at low
855	<	temperatures and deviates to diffuse too rapidly at high
856	<	temperatures. As was stated in the SSD/E comparisons, this deviation
857	<	away from the ideal trend is due to a rapid decrease in density at
858	<	higher temperatures. SSD/RF doesn't suffer from this problem as much
859	<	as SSD, because the calculated densities are more true to
860	<	experiment. This is again emphasized in the lower plot, where SSD/RF
793	<	tracks the experimental diffusion exactly while SSD's diffusion
794	<	constants are slightly too low due to its need for a lower density at
795	<	the specified temperature.
850	>	compared to SSD1 with an active reaction field. Note that SSD/RF
851	>	tracks the experimental results quantitatively, identical within error
852	>	throughout most of the temperature range shown and exhibiting only a
853	>	slight increasing trend at higher temperatures. SSD1 tends to diffuse
854	>	more slowly at low temperatures and deviates to diffuse too rapidly at
855	>	temperatures greater than 330 K.  As stated above, this deviation away
856	>	from the ideal trend is due to a rapid decrease in density at higher
857	>	temperatures. SSD/RF does not suffer from this problem as much as SSD1
858	>	because the calculated densities are closer to the experimental
859	>	values. These results again emphasize the importance of careful
860	>	reparameterization when using an altered long-range correction.
861
862	<	\subsection{Additional Observations}
862	>	\begin{table}
863	>	\begin{center}
864	>	\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.}
865	>	\begin{tabular}{ l  c  c  c  c  c }
866	>	\hline \\[-3mm]
867	>	\ \ \ \ \ \  & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \
868	>	\ & \ SSD/RF \ \ \ & \ Expt. \\
869	>	\hline \\[-3mm]
870	>	\ \ \ $\rho$ (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 \\
871	>	\ \ \ $C_p$ (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 \\
872	>	\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\
873	>	\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\
874	>	\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\
875	>	\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\
876	>	\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\
877	>	\end{tabular}
878	>	\label{liquidproperties}
879	>	\end{center}
880	>	\end{table}
881
882	<	While performing the melting sequences of SSD/E, some interesting
883	<	observations were made. After melting at 230 K, two of the systems
884	<	underwent crystallization events near 245 K. As the heating process
885	<	continued, the two systems remained crystalline until finally melting
886	<	between 320 and 330 K. These simulations were excluded from the data
887	<	set shown in figure \ref{ssdedense} and replaced with two additional
888	<	melting sequences that did not undergo this anomalous phase
889	<	transition, while this crystallization event was investigated
890	<	separately.
882	>	Table \ref{liquidproperties} gives a synopsis of the liquid state
883	>	properties of the water models compared in this study along with the
884	>	experimental values for liquid water at ambient conditions. The
885	>	coordination number and hydrogen bonds per particle were calculated by
886	>	integrating the following relation:
887	>	\begin{equation}
888	>	4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr,
889	>	\end{equation}
890	>	where $\rho$ is the number density of pair interactions, $a$ is the
891	>	radial location of the minima following the first solvation shell
892	>	peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for
893	>	calculation of the coordination number or hydrogen bonds per particle
894	>	respectively.
895
896	<	\begin{figure}
897	<	\includegraphics[width=85mm]{povIce.ps}
898	<	\caption{Crystal structure of an ice 0 lattice shown from the (001) face.}
899	<	\label{weirdice}
900	<	\end{figure}
896	>	The time constants for the self orientational autocorrelation function
897	>	are also displayed in Table \ref{liquidproperties}. The dipolar
898	>	orientational time correlation function ($\Gamma_{l}$) is described
899	>	by:
900	>	\begin{equation}
901	>	\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle,
902	>	\end{equation}
903	>	where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$
904	>	is the unit vector of the particle dipole.\cite{Rahman71} From these
905	>	correlation functions, the orientational relaxation time of the dipole
906	>	vector can be calculated from an exponential fit in the long-time
907	>	regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these
908	>	time constants were averaged from five detailed NVE simulations
909	>	performed at the STP density for each of the respective models.
910
911	<	The final configurations of these two melting sequences shows an
816	<	expanded zeolite-like crystal structure that does not correspond to
817	<	any known form of ice. For convenience and to help distinguish it from
818	<	the experimentally observed forms of ice, this crystal structure will
819	<	henceforth be referred to as ice-zero (ice 0). The crystallinity was
820	<	extensive enough than a near ideal crystal structure could be
821	<	obtained. Figure \ref{weirdice} shows the repeating crystal structure
822	<	of a typical crystal at 5 K. The unit cell contains eight molecules,
823	<	and figure \ref{unitcell} shows a unit cell built from the water
824	<	particle center of masses that can be used to construct a repeating
825	<	lattice, similar to figure \ref{weirdice}. Each molecule is hydrogen
826	<	bonded to four other water molecules; however, the hydrogen bonds are
827	<	flexed rather than perfectly straight. This results in a skewed
828	<	tetrahedral geometry about the central molecule. Looking back at
829	<	figure \ref{isosurface}, it is easy to see how these flexed hydrogen
830	<	bonds are allowed in that the attractive regions are conical in shape,
831	<	with the greatest attraction in the central region. Though not ideal,
832	<	these flexed hydrogen bonds are favorable enough to stabilize an
833	<	entire crystal generated around them. In fact, the imperfect ice 0
834	<	crystals were so stable that they melted at greater than room
835	<	temperature.
911	>	\subsection{Additional Observations}
912
913		\begin{figure}
914	<	\includegraphics[width=65mm]{ice0cell.eps}
915	<	\caption{Simple unit cell for constructing ice 0. In this cell, $c$ is
916	<	equal to $0.4714\times a$, and a typical value for $a$ is 8.25 \AA.}
917	<	\label{unitcell}
914	>	\begin{center}
915	>	\epsfxsize=6in
916	>	\epsfbox{povIce.ps}
917	>	\caption{A water lattice built from the crystal structure assumed by
918	>	SSD/E when undergoing an extremely restricted temperature NPT
919	>	simulation. This form of ice is referred to as ice-{\it i} to
920	>	emphasize its simulation origins. This image was taken of the (001)
921	>	face of the crystal.}
922	>	\label{weirdice}
923	>	\end{center}
924		\end{figure}
925
926	<	The initial simulations indicated that ice 0 is the preferred ice
927	<	structure for at least SSD/E. To verify this, a comparison was made
928	<	between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at
929	<	constant pressure with SSD/E, SSD/RF, and SSD. Near ideal versions of
930	<	the three types of crystals were cooled to ~1 K, and the potential
931	<	energies of each were compared using all three water models. With
932	<	every water model, ice 0 turned out to have the lowest potential
933	<	energy: 5\% lower than $I_h$ with SSD, 6.5\% lower with SSD/E, and
934	<	7.5\% lower with SSD/RF. In all three of these water models, ice $I_c$
935	<	was observed to be ~2\% less stable than ice $I_h$. In addition to
936	<	having the lowest potential energy, ice 0 was the most expanded of the
937	<	three ice crystals, ~5\% less dense than ice $I_h$ with all of the
938	<	water models. In all three water models, ice $I_c$ was observed to be
939	<	~2\% more dense than ice $I_h$.
926	>	While performing a series of melting simulations on an early iteration
927	>	of SSD/E not discussed in this paper, we observed recrystallization
928	>	into a novel structure not previously known for water.  After melting
929	>	at 235 K, two of five systems underwent crystallization events near
930	>	245 K.  The two systems remained crystalline up to 320 and 330 K,
931	>	respectively.  The crystal exhibits an expanded zeolite-like structure
932	>	that does not correspond to any known form of ice.  This appears to be
933	>	an artifact of the point dipolar models, so to distinguish it from the
934	>	experimentally observed forms of ice, we have denoted the structure
935	>	Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}).  A large enough
936	>	portion of the sample crystallized that we have been able to obtain a
937	>	near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice}
938	>	shows the repeating crystal structure of a typical crystal at 5
939	>	K. Each water molecule is hydrogen bonded to four others; however, the
940	>	hydrogen bonds are bent rather than perfectly straight. This results
941	>	in a skewed tetrahedral geometry about the central molecule.  In
942	>	figure \ref{isosurface}, it is apparent that these flexed hydrogen
943	>	bonds are allowed due to the conical shape of the attractive regions,
944	>	with the greatest attraction along the direct hydrogen bond
945	>	configuration. Though not ideal, these flexed hydrogen bonds are
946	>	favorable enough to stabilize an entire crystal generated around them.
947
948	<	In addition to the low temperature comparisons, melting sequences were
949	<	performed with ice 0 as the initial configuration using SSD/E, SSD/RF,
950	<	and SSD both with and without a reaction field. The melting
951	<	transitions for both SSD/E and SSD without a reaction field occurred
952	<	at temperature in excess of 375 K. SSD/RF and SSD with a reaction
953	<	field had more reasonable melting transitions, down near 325 K. These
954	<	melting point observations emphasize how preferred this crystal
955	<	structure is over the most common types of ice when using these single
956	<	point water models.
948	>	Initial simulations indicated that ice-{\it i} is the preferred ice
949	>	structure for at least the SSD/E model. To verify this, a comparison
950	>	was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and
951	>	ice-{\it i} at constant pressure with SSD/E, SSD/RF, and
952	>	SSD1. Near-ideal versions of the three types of crystals were cooled
953	>	to 1 K, and the enthalpies of each were compared using all three water
954	>	models. With every model in the SSD family, ice-{\it i} had the lowest
955	>	calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with
956	>	SSD/E, and 7.5\% lower with SSD/RF.  The enthalpy data is summarized
957	>	in Table \ref{iceenthalpy}.
958
959	<	Recognizing that the above tests show ice 0 to be both the most stable
960	<	and lowest density crystal structure for these single point water
961	<	models, it is interesting to speculate on the favorability of this
962	<	crystal structure with the different charge based models. As a quick
963	<	test, these 3 crystal types were converted from SSD type particles to
964	<	TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
965	<	minimizations were performed on all of these crystals to compare the
966	<	system energies. Again, ice 0 was observed to have the lowest total
967	<	system energy. The total energy of ice 0 was ~2\% lower than ice
968	<	$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial
969	<	results, we would not be surprised if results from the other common
970	<	water models show ice 0 to be the lowest energy crystal structure. A
971	<	continuation on work studing ice 0 with multipoint water models will
972	<	be published in a coming article.
959	>	\begin{table}
960	>	\begin{center}
961	>	\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1
962	>	K) exhibited by the SSD family of water models}
963	>	\begin{tabular}{ l  c  c  c  }
964	>	\hline \\[-3mm]
965	>	\ \ \ Water Model \ \ \  & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \  & \
966	>	Ice-{\it i} \\
967	>	\hline \\[-3mm]
968	>	\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\
969	>	\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\
970	>	\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\
971	>	\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\
972	>	\end{tabular}
973	>	\label{iceenthalpy}
974	>	\end{center}
975	>	\end{table}
976
977	+	In addition to these energetic comparisons, melting simulations were
978	+	performed with ice-{\it i} as the initial configuration using SSD/E,
979	+	SSD/RF, and SSD1 both with and without a reaction field. The melting
980	+	transitions for both SSD/E and SSD1 without reaction field occurred at
981	+	temperature in excess of 375~K.  SSD/RF and SSD1 with a reaction field
982	+	showed more reasonable melting transitions near 325~K.  These melting
983	+	point observations clearly show that all of the SSD-derived models
984	+	prefer the ice-{\it i} structure.
985	+
986		\section{Conclusions}
987	<	The density maximum and temperature dependent transport for the SSD
988	<	water model, both with and without the use of reaction field, were
989	<	studied via a series of NPT and NVE simulations. The constant pressure
990	<	simulations of the melting of both $I_h$ and $I_c$ ice showed a
991	<	density maximum near 260 K. In most cases, the calculated densities
992	<	were significantly lower than the densities calculated in simulations
993	<	of other water models. Analysis of particle diffusion showed SSD to
994	<	capture the transport properties of experimental very well in both the
995	<	normal and super-cooled liquid regimes. In order to correct the
996	<	density behavior, SSD was reparameterized for use both with and
997	<	without a long-range interaction correction, SSD/RF and SSD/E
998	<	respectively. In addition to correcting the abnormally low densities,
999	<	these new versions were show to maintain or improve upon the transport
1000	<	and structural features of the original water model, all while
1001	<	maintaining the fast performance of the SSD water model. This work
1002	<	shows these simple water models, and in particular SSD/E and SSD/RF,
1003	<	to be excellent choices to represent explicit water in future
987	>
988	>	The density maximum and temperature dependence of the self-diffusion
989	>	constant were studied for the SSD water model, both with and without
990	>	the use of reaction field, via a series of NPT and NVE
991	>	simulations. The constant pressure simulations showed a density
992	>	maximum near 260 K. In most cases, the calculated densities were
993	>	significantly lower than the densities obtained from other water
994	>	models (and experiment). Analysis of self-diffusion showed SSD to
995	>	capture the transport properties of water well in both the liquid and
996	>	super-cooled liquid regimes.
997	>
998	>	In order to correct the density behavior, the original SSD model was
999	>	reparameterized for use both with and without a reaction field (SSD/RF
1000	>	and SSD/E), and comparisons were made with SSD1, Ichiye's density
1001	>	corrected version of SSD. Both models improve the liquid structure,
1002	>	densities, and diffusive properties under their respective simulation
1003	>	conditions, indicating the necessity of reparameterization when
1004	>	changing the method of calculating long-range electrostatic
1005	>	interactions.  In general, however, these simple water models are
1006	>	excellent choices for representing explicit water in large scale
1007		simulations of biochemical systems.
1008
1009	+	The existence of a novel low-density ice structure that is preferred
1010	+	by the SSD family of water models is somewhat troubling, since liquid
1011	+	simulations on this family of water models at room temperature are
1012	+	effectively simulations of super-cooled or metastable liquids.  One
1013	+	way to de-stabilize this unphysical ice structure would be to make the
1014	+	range of angles preferred by the attractive part of the sticky
1015	+	potential much narrower.  This would require extensive
1016	+	reparameterization to maintain the same level of agreement with the
1017	+	experiments.
1018	+
1019	+	Additionally, our initial calculations show that the ice-{\it i}
1020	+	structure may also be a preferred crystal structure for at least one
1021	+	other popular multi-point water model (TIP3P), and that much of the
1022	+	simulation work being done using this popular model could also be at
1023	+	risk for crystallization into this unphysical structure.  A future
1024	+	publication will detail the relative stability of the known ice
1025	+	structures for a wide range of popular water models.
1026	+
1027		\section{Acknowledgments}
1028	<	The authors would like to thank the National Science Foundation for
1029	<	funding under grant CHE-0134881. Computation time was provided by the
1030	<	Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant DMR
1031	<	00 79647.
1028	>	Support for this project was provided by the National Science
1029	>	Foundation under grant CHE-0134881. Computation time was provided by
1030	>	the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
1031	>	DMR-0079647.
1032
1033	<	\bibliographystyle{jcp}
1033	>	\newpage
1034
1035	+	\bibliographystyle{jcp}
1036		\bibliography{nptSSD}
1037
1038		%\pagebreak

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