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21
22		\begin{document}
23
24	<	\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models}
24	>	\title{On the structural and transport properties of the soft sticky
25	>	dipole (SSD) and related single point water models}
26
27		\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
28		Department of Chemistry and Biochemistry\\ University of Notre Dame\\
#	Line 32 \| Line 33 \| Notre Dame, Indiana 46556}
33		\maketitle
34
35		\begin{abstract}
36	<	NVE and NPT molecular dynamics simulations were performed in order to
37	<	investigate the density maximum and temperature dependent transport
38	<	for SSD and related water models, both with and without the use of
39	<	reaction field. The constant pressure simulations of the melting of
40	<	both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most
41	<	cases, the calculated densities were significantly lower than the
42	<	densities calculated in simulations of other water models. Analysis of
43	<	particle diffusion showed SSD to capture the transport properties of
36	>	The density maximum and temperature dependence of the self-diffusion
37	>	constant were investigated for the soft sticky dipole (SSD) water
38	>	model and two related re-parameterizations of this single-point model.
39	>	A combination of microcanonical and isobaric-isothermal molecular
40	>	dynamics simulations were used to calculate these properties, both
41	>	with and without the use of reaction field to handle long-range
42	>	electrostatics. The isobaric-isothermal (NPT) simulations of the
43	>	melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near
44	>	260 K. In most cases, the use of the reaction field resulted in
45	>	calculated densities which were were significantly lower than
46	>	experimental densities. Analysis of self-diffusion constants shows
47	>	that the original SSD model captures the transport properties of
48		experimental water very well in both the normal and super-cooled
49	<	liquid regimes. In order to correct the density behavior, SSD was
50	<	reparameterized for use both with and without a long-range interaction
51	<	correction, SSD/RF and SSD/E respectively. Compared to the density
52	<	corrected version of SSD (SSD1), these modified models were shown to
53	<	maintain or improve upon the structural and transport properties.
49	>	liquid regimes. We also present our re-parameterized versions of SSD
50	>	for use both with the reaction field or without any long-range
51	>	electrostatic corrections. These are called the SSD/RF and SSD/E
52	>	models respectively. These modified models were shown to maintain or
53	>	improve upon the experimental agreement with the structural and
54	>	transport properties that can be obtained with either the original SSD
55	>	or the density corrected version of the original model (SSD1).
56	>	Additionally, a novel low-density ice structure is presented
57	>	which appears to be the most stable ice structure for the entire SSD
58	>	family.
59		\end{abstract}
60
61		\newpage
#	Line 60 \| Line 70 \| One of the most important tasks in the simulation of b
70		\section{Introduction}
71
72		One of the most important tasks in the simulation of biochemical
73	<	systems is the proper depiction of water and water solvation. In fact,
74	<	the bulk of the calculations performed in solvated simulations are of
75	<	interactions with or between solvent molecules. Thus, the outcomes of
76	<	these types of simulations are highly dependent on the physical
77	<	properties of water, both as individual molecules and in clusters or
78	<	bulk. Due to the fact that explicit solvent accounts for a massive
79	<	portion of the calculations, it necessary to simplify the solvent to
70	<	some extent in order to complete simulations in a reasonable amount of
71	<	time. In the case of simulating water in biomolecular studies, the
72	<	balance between accurate properties and computational efficiency is
73	<	especially delicate, and it has resulted in a variety of different
74	<	water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these
75	<	models predict specific properties more accurately than their
76	<	predecessors, but often at the cost of other properties or of computer
77	<	time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves
78	<	upon the structural and transport properties of water relative to the
79	<	previous TIP models, yet this comes at a greater than 50\% increase in
80	<	computational cost.\cite{Jorgensen01,Jorgensen00} One recently
81	<	developed model that succeeds in both retaining the accuracy of system
82	<	properties and simplifying calculations to increase computational
83	<	efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}
73	>	systems is the proper depiction of the aqueous environment of the
74	>	molecules of interest. In some cases (such as in the simulation of
75	>	phospholipid bilayers), the majority of the calculations that are
76	>	performed involve interactions with or between solvent molecules.
77	>	Thus, the properties one may observe in biochemical simulations are
78	>	going to be highly dependent on the physical properties of the water
79	>	model that is chosen.
80
81	<	The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye
82	<	\emph{et al.} as a modified form of the hard-sphere water model
83	<	proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD
84	<	consists of a single point dipole with a Lennard-Jones core and a
85	<	sticky potential that directs the particles to assume the proper
86	<	hydrogen bond orientation in the first solvation shell. Thus, the
87	<	interaction between two SSD water molecules \emph{i} and \emph{j} is
88	<	given by the potential
81	>	There is an especially delicate balance between computational
82	>	efficiency and the ability of the water model to accurately predict
83	>	the properties of bulk
84	>	water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the
85	>	TIP5P model improves on the structural and transport properties of
86	>	water relative to the previous TIP models, yet this comes at a greater
87	>	than 50\% increase in computational
88	>	cost.\cite{Jorgensen01,Jorgensen00}
89	>
90	>	One recently developed model that largely succeeds in retaining the
91	>	accuracy of bulk properties while greatly reducing the computational
92	>	cost is the Soft Sticky Dipole (SSD) water
93	>	model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model 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
104	<	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}
137		u_{ij}^{sp}
138	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
139	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
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		\end{equation}
143	<	where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
144	<	unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
145	<	$\nu_0$ scales the strength of the overall sticky potential, and $s$
146	<	and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
147	<	functions take the following forms:
143	>	Here, $\nu_0$ is a strength parameter for the sticky potential, and
144	>	$s$ and $s^\prime$ are cubic switching functions which turn off the
145	>	sticky interaction beyond the first solvation shell. The $w$ function
146	>	can be thought of as an attractive potential with tetrahedral
147	>	geometry:
148		\begin{equation}
149	<	w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
149	>	w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
150		\end{equation}
151	+	while the $w^\prime$ function counters the normal aligned and
152	+	anti-aligned structures favored by point dipoles:
153		\begin{equation}
154	<	w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
154	>	w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
155		\end{equation}
156	<	where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
157	<	term that promotes hydrogen bonding orientations within the first
158	<	solvation shell, and $w^\prime$ is a dipolar repulsion term that
159	<	repels unrealistic dipolar arrangements within the first solvation
160	<	shell. A more detailed description of the functional parts and
161	<	variables in this potential can be found in other
162	<	articles.\cite{Ichiye96,Ichiye99}
156	>	It should be noted that $w$ is proportional to the sum of the $Y_3^2$
157	>	and $Y_3^{-2}$ spherical harmonics (a linear combination which
158	>	enhances the tetrahedral geometry for hydrogen bonded structures),
159	>	while $w^\prime$ is a purely empirical function. A more detailed
160	>	description of the functional parts and variables in this potential
161	>	can be found in the original SSD
162	>	articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
163
164	<	Being that this is a one-site point dipole model, the actual force
165	<	calculations are simplified significantly. In the original Monte Carlo
166	<	simulations using this model, Ichiye \emph{et al.} reported an
167	<	increase in calculation efficiency of up to an order of magnitude over
168	<	other comparable models, while maintaining the structural behavior of
169	<	water.\cite{Ichiye96} In the original molecular dynamics studies, it
170	<	was shown that SSD improves on the prediction of many of water's
171	<	dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This
172	<	attractive combination of speed and accurate depiction of solvent
173	<	properties makes SSD a model of interest for the simulation of large
174	<	scale biological systems, such as membrane phase behavior.
164	>	Since SSD is a single-point {\it dipolar} model, the force
165	>	calculations are simplified significantly relative to the standard
166	>	{\it charged} multi-point models. In the original Monte Carlo
167	>	simulations using this model, Ichiye {\it et al.} reported that using
168	>	SSD decreased computer time by a factor of 6-7 compared to other
169	>	models.\cite{Ichiye96} What is most impressive is that this savings
170	>	did not come at the expense of accurate depiction of the liquid state
171	>	properties. Indeed, SSD maintains reasonable agreement with the Soper
172	>	data for the structural features of liquid
173	>	water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
174	>	exhibited by SSD agree with experiment better than those of more
175	>	computationally expensive models (like TIP3P and
176	>	SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
177	>	of solvent properties makes SSD a very attractive model for the
178	>	simulation of large scale biochemical simulations.
179
180	<	One of the key limitations of this water model, however, is that it
181	<	has been parameterized for use with the Ewald Sum technique for the
182	<	handling of long-ranged interactions. When studying very large
183	<	systems, the Ewald summation and even particle-mesh Ewald become
184	<	computational burdens, with their respective ideal $N^\frac{3}{2}$ and
185	<	$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99}
186	<	In applying this water model in these types of systems, it would be
187	<	useful to know its properties and behavior with the more
188	<	computationally efficient reaction field (RF) technique, and even with
189	<	a cutoff that lacks any form of long-range correction. This study
190	<	addresses these issues by looking at the structural and transport
191	<	behavior of SSD over a variety of temperatures with the purpose of
192	<	utilizing the RF correction technique. We then suggest alterations to
193	<	the parameters that result in more water-like behavior. It should be
194	<	noted that in a recent publication, some of the original investigators of
195	<	the SSD water model have put forth adjustments to the SSD water model
196	<	to address abnormal density behavior (also observed here), calling the
197	<	corrected model SSD1.\cite{Ichiye03} This study will make comparisons
198	<	with SSD1's behavior with the goal of improving upon the
199	<	depiction of water under conditions without the Ewald Sum.
180	>	One feature of the SSD model is that it was parameterized for use with
181	>	the Ewald sum to handle long-range interactions. This would normally
182	>	be the best way of handling long-range interactions in systems that
183	>	contain other point charges. However, our group has recently become
184	>	interested in systems with point dipoles as mimics for neutral, but
185	>	polarized regions on molecules (e.g. the zwitterionic head group
186	>	regions of phospholipids). If the system of interest does not contain
187	>	point charges, the Ewald sum and even particle-mesh Ewald become
188	>	computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and
189	>	$N\log N$ calculation scaling orders for $N$ particles can become
190	>	prohibitive when $N$ becomes large.\cite{Darden99} In applying this
191	>	water model in these types of systems, it would be useful to know its
192	>	properties and behavior under the more computationally efficient
193	>	reaction field (RF) technique, or even with a simple cutoff. This
194	>	study addresses these issues by looking at the structural and
195	>	transport behavior of SSD over a variety of temperatures with the
196	>	purpose of utilizing the RF correction technique. We then suggest
197	>	modifications to the parameters that result in more realistic bulk
198	>	phase behavior. It should be noted that in a recent publication, some
199	>	of the original investigators of the SSD water model have suggested
200	>	adjustments to the SSD water model to address abnormal density
201	>	behavior (also observed here), calling the corrected model
202	>	SSD1.\cite{Ichiye03} In what follows, we compare our
203	>	reparamaterization of SSD with both the original SSD and SSD1 models
204	>	with the goal of improving the bulk phase behavior of an SSD-derived
205	>	model in simulations utilizing the Reaction Field.
206
207		\section{Methods}
208
209	<	As stated previously, the long-range dipole-dipole interactions were
210	<	accounted for in this study by using the reaction field method. The
211	<	magnitude of the reaction field acting on dipole \emph{i} is given by
209	>	Long-range dipole-dipole interactions were accounted for in this study
210	>	by using either the reaction field method or by resorting to a simple
211	>	cubic switching function at a cutoff radius. Under the first method,
212	>	the magnitude of the reaction field acting on dipole $i$ is
213		\begin{equation}
214		\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
215	<	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\ ,
215	>	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ ,
216		\label{rfequation}
217		\end{equation}
218		where $\mathcal{R}$ is the cavity defined by the cutoff radius
219		($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
220	<	system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment
221	<	vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching
220	>	system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole
221	>	moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching
222		function.\cite{AllenTildesley} The reaction field contribution to the
223	<	total energy by particle \emph{i} is given by
224	<	$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
225	<	on dipole \emph{i} by
226	<	$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use
227	<	of reaction field is known to alter the orientational dynamic
228	<	properties, such as the dielectric relaxation time, based on changes
229	<	in the length of the cutoff radius.\cite{Berendsen98} This variable
230	<	behavior makes reaction field a less attractive method than other
231	<	methods, like the Ewald summation; however, for the simulation of
232	<	large-scale systems, the computational cost benefit of reaction field
233	<	is dramatic. To address some of the dynamical property alterations due
234	<	to the use of reaction field, simulations were also performed without
235	<	a surrounding dielectric and suggestions are presented on how to make
236	<	SSD more accurate both with and without a reaction field.
223	>	total energy by particle $i$ is given by $-\frac{1}{2}{\bf
224	>	\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf
225	>	\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction
226	>	field is known to alter the bulk orientational properties, such as the
227	>	dielectric relaxation time. There is particular sensitivity of this
228	>	property on changes in the length of the cutoff
229	>	radius.\cite{Berendsen98} This variable behavior makes reaction field
230	>	a less attractive method than the Ewald sum. However, for very large
231	>	systems, the computational benefit of reaction field is dramatic.
232	>
233	>	We have also performed a companion set of simulations {\it without} a
234	>	surrounding dielectric (i.e. using a simple cubic switching function
235	>	at the cutoff radius) and as a result we have two reparamaterizations
236	>	of SSD which could be used either with or without the Reaction Field
237	>	turned on.
238
239	<	Simulations were performed in both the isobaric-isothermal and
240	<	microcanonical ensembles. The constant pressure simulations were
241	<	implemented using an integral thermostat and barostat as outlined by
242	<	Hoover.\cite{Hoover85,Hoover86} All particles were treated as
243	<	non-linear rigid bodies. Vibrational constraints are not necessary in
244	<	simulations of SSD, because there are no explicit hydrogen atoms, and
245	<	thus no molecular vibrational modes need to be considered.
239	>	Simulations to obtain the preferred density were performed in the
240	>	isobaric-isothermal (NPT) ensemble, while all dynamical properties
241	>	were obtained from microcanonical (NVE) simulations done at densities
242	>	matching the NPT density for a particular target temperature. The
243	>	constant pressure simulations were implemented using an integral
244	>	thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86}
245	>	All molecules were treated as non-linear rigid bodies. Vibrational
246	>	constraints are not necessary in simulations of SSD, because there are
247	>	no explicit hydrogen atoms, and thus no molecular vibrational modes
248	>	need to be considered.
249
250		Integration of the equations of motion was carried out using the
251	<	symplectic splitting method proposed by Dullweber \emph{et
252	<	al.}\cite{Dullweber1997} The reason for this integrator selection
253	<	deals with poor energy conservation of rigid body systems using
254	<	quaternions. While quaternions work well for orientational motion in
255	<	alternate ensembles, the microcanonical ensemble has a constant energy
256	<	requirement that is quite sensitive to errors in the equations of
257	<	motion. The original implementation of this code utilized quaternions
258	<	for rotational motion propagation; however, a detailed investigation
259	<	showed that they resulted in a steady drift in the total energy,
216	<	something that has been observed by others.\cite{Laird97}
251	>	symplectic splitting method proposed by Dullweber {\it et
252	>	al.}\cite{Dullweber1997} Our reason for selecting this integrator
253	>	centers on poor energy conservation of rigid body dynamics using
254	>	traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions
255	>	may work well for orientational motion under NVT or NPT integrators,
256	>	our limits on energy drift in the microcanonical ensemble were quite
257	>	strict, and the drift under quaternions was substantially greater than
258	>	in the symplectic splitting method. This steady drift in the total
259	>	energy has also been observed by Kol {\it et al.}\cite{Laird97}
260
261		The key difference in the integration method proposed by Dullweber
262		\emph{et al.} is that the entire rotation matrix is propagated from
263	<	one time step to the next. In the past, this would not have been as
264	<	feasible an option, being that the rotation matrix for a single body is
265	<	nine elements long as opposed to 3 or 4 elements for Euler angles and
266	<	quaternions respectively. System memory has become much less of an
224	<	issue in recent times, and this has resulted in substantial benefits
225	<	in energy conservation. There is still the issue of 5 or 6 additional
226	<	elements for describing the orientation of each particle, which will
227	<	increase dump files substantially. Simply translating the rotation
228	<	matrix into its component Euler angles or quaternions for storage
229	<	purposes relieves this burden.
263	>	one time step to the next. The additional memory required by the
264	>	algorithm is inconsequential on modern computers, and translating the
265	>	rotation matrix into quaternions for storage purposes makes trajectory
266	>	data quite compact.
267
268		The symplectic splitting method allows for Verlet style integration of
269	<	both linear and angular motion of rigid bodies. In this integration
270	<	method, the orientational propagation involves a sequence of matrix
271	<	evaluations to update the rotation matrix.\cite{Dullweber1997} These
272	<	matrix rotations are more costly computationally than the simpler
273	<	arithmetic quaternion propagation. With the same time step, a 1000 SSD
274	<	particle simulation shows an average 7\% increase in computation time
275	<	using the symplectic step method in place of quaternions. This cost is
276	<	more than justified when comparing the energy conservation of the two
277	<	methods as illustrated in figure \ref{timestep}.
269	>	both translational and orientational motion of rigid bodies. In this
270	>	integration method, the orientational propagation involves a sequence
271	>	of matrix evaluations to update the rotation
272	>	matrix.\cite{Dullweber1997} These matrix rotations are more costly
273	>	than the simpler arithmetic quaternion propagation. With the same time
274	>	step, a 1000 SSD particle simulation shows an average 7\% increase in
275	>	computation time using the symplectic step method in place of
276	>	quaternions. The additional expense per step is justified when one
277	>	considers the ability to use time steps that are nearly twice as large
278	>	under symplectic splitting than would be usable under quaternion
279	>	dynamics. The energy conservation of the two methods using a number
280	>	of different time steps is illustrated in figure
281	>	\ref{timestep}.
282
283		\begin{figure}
284		\begin{center}
285		\epsfxsize=6in
286		\epsfbox{timeStep.epsi}
287	<	\caption{Energy conservation using quaternion based integration versus
287	>	\caption{Energy conservation using both quaternion based integration and
288		the symplectic step method proposed by Dullweber \emph{et al.} with
289	<	increasing time step. The larger time step plots are shifted up from
290	<	the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
289	>	increasing time step. The larger time step plots are shifted from the
290	>	true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
291		\label{timestep}
292		\end{center}
293		\end{figure}
294
295		In figure \ref{timestep}, the resulting energy drift at various time
296		steps for both the symplectic step and quaternion integration schemes
297	<	is compared. All of the 1000 SSD particle simulations started with the
298	<	same configuration, and the only difference was the method used to
299	<	handle rotational motion. At time steps of 0.1 and 0.5 fs, both
300	<	methods for propagating particle rotation conserve energy fairly well,
301	<	with the quaternion method showing a slight energy drift over time in
302	<	the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
303	<	energy conservation benefits of the symplectic step method are clearly
304	<	demonstrated. Thus, while maintaining the same degree of energy
305	<	conservation, one can take considerably longer time steps, leading to
306	<	an overall reduction in computation time.
297	>	is compared. All of the 1000 SSD particle simulations started with
298	>	the same configuration, and the only difference was the method used to
299	>	handle orientational motion. At time steps of 0.1 and 0.5 fs, both
300	>	methods for propagating the orientational degrees of freedom conserve
301	>	energy fairly well, with the quaternion method showing a slight energy
302	>	drift over time in the 0.5 fs time step simulation. At time steps of 1
303	>	and 2 fs, the energy conservation benefits of the symplectic step
304	>	method are clearly demonstrated. Thus, while maintaining the same
305	>	degree of energy conservation, one can take considerably longer time
306	>	steps, leading to an overall reduction in computation time.
307
308		Energy drift in the symplectic step simulations was unnoticeable for
309	<	time steps up to three femtoseconds. A slight energy drift on the
309	>	time steps up to 3 fs. A slight energy drift on the
310		order of 0.012 kcal/mol per nanosecond was observed at a time step of
311	<	four femtoseconds, and as expected, this drift increases dramatically
312	<	with increasing time step. To insure accuracy in the constant energy
311	>	4 fs, and as expected, this drift increases dramatically
312	>	with increasing time step. To insure accuracy in our microcanonical
313		simulations, time steps were set at 2 fs and kept at this value for
314		constant pressure simulations as well.
315
316	<	Ice crystals in both the $I_h$ and $I_c$ lattices were generated as
317	<	starting points for all simulations. The $I_h$ crystals were formed by
318	<	first arranging the centers of mass of the SSD particles into a
319	<	``hexagonal'' ice lattice of 1024 particles. Because of the crystal
320	<	structure of $I_h$ ice, the simulation box assumed a rectangular shape
321	<	with an edge length ratio of approximately
316	>	Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices
317	>	were generated as starting points for all simulations. The $I_h$
318	>	crystals were formed by first arranging the centers of mass of the SSD
319	>	particles into a ``hexagonal'' ice lattice of 1024 particles. Because
320	>	of the crystal structure of $I_h$ ice, the simulation box assumed an
321	>	orthorhombic shape with an edge length ratio of approximately
322		1.00$\times$1.06$\times$1.23. The particles were then allowed to
323		orient freely about fixed positions with angular momenta randomized at
324		400 K for varying times. The rotational temperature was then scaled
#	Line 297 \| Line 338 \| Melting studies were performed on the randomized ice c
338		\section{Results and discussion}
339
340		Melting studies were performed on the randomized ice crystals using
341	<	constant pressure and temperature dynamics. During melting
342	<	simulations, the melting transition and the density maximum can both
343	<	be observed, provided that the density maximum occurs in the liquid
344	<	and not the supercooled regime. An ensemble average from five separate
345	<	melting simulations was acquired, each starting from different ice
346	<	crystals generated as described previously. All simulations were
347	<	equilibrated for 100 ps prior to a 200 ps data collection run at each
348	<	temperature setting. The temperature range of study spanned from 25 to
349	<	400 K, with a maximum degree increment of 25 K. For regions of
350	<	interest along this stepwise progression, the temperature increment
351	<	was decreased from 25 K to 10 and 5 K. The above equilibration and
352	<	production times were sufficient in that the system volume
353	<	fluctuations dampened out in all but the very cold simulations (below
313	<	225 K).
341	>	isobaric-isothermal (NPT) dynamics. During melting simulations, the
342	>	melting transition and the density maximum can both be observed,
343	>	provided that the density maximum occurs in the liquid and not the
344	>	supercooled regime. An ensemble average from five separate melting
345	>	simulations was acquired, each starting from different ice crystals
346	>	generated as described previously. All simulations were equilibrated
347	>	for 100 ps prior to a 200 ps data collection run at each temperature
348	>	setting. The temperature range of study spanned from 25 to 400 K, with
349	>	a maximum degree increment of 25 K. For regions of interest along this
350	>	stepwise progression, the temperature increment was decreased from 25
351	>	K to 10 and 5 K. The above equilibration and production times were
352	>	sufficient in that fluctuations in the volume autocorrelation function
353	>	were damped out in all simulations in under 20 ps.
354
355		\subsection{Density Behavior}
316	–	Initial simulations focused on the original SSD water model, and an
317	–	average density versus temperature plot is shown in figure
318	–	\ref{dense1}. Note that the density maximum when using a reaction
319	–	field appears between 255 and 265 K, where the calculated densities
320	–	within this range were nearly indistinguishable. The greater certainty
321	–	of the average value at 260 K makes a good argument for the actual
322	–	density maximum residing at this midpoint value. Figure \ref{dense1}
323	–	was constructed using ice $I_h$ crystals for the initial
324	–	configuration; though not pictured, the simulations starting from ice
325	–	$I_c$ crystal configurations showed similar results, with a
326	–	liquid-phase density maximum in this same region (between 255 and 260
327	–	K). In addition, the $I_c$ crystals are more fragile than the $I_h$
328	–	crystals, leading to deformation into a dense glassy state at lower
329	–	temperatures. This resulted in an overall low temperature density
330	–	maximum at 200 K, while still retaining a liquid state density maximum
331	–	in common with the $I_h$ simulations.
356
357	+	Our initial simulations focused on the original SSD water model, and
358	+	an average density versus temperature plot is shown in figure
359	+	\ref{dense1}. Note that the density maximum when using a reaction
360	+	field appears between 255 and 265 K. There were smaller fluctuations
361	+	in the density at 260 K than at either 255 or 265, so we report this
362	+	value as the location of the density maximum. Figure \ref{dense1} was
363	+	constructed using ice $I_h$ crystals for the initial configuration;
364	+	though not pictured, the simulations starting from ice $I_c$ crystal
365	+	configurations showed similar results, with a liquid-phase density
366	+	maximum in this same region (between 255 and 260 K).
367	+
368		\begin{figure}
369		\begin{center}
370		\epsfxsize=6in
371		\epsfbox{denseSSD.eps}
372	<	\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b}
373	<	TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction
374	<	Field, SSD, and experiment.\cite{CRC80} The arrows indicate the
375	<	change in densities observed when turning off the reaction field. The
376	<	the lower than expected densities for the SSD model were what
377	<	prompted the original reparameterization.\cite{Ichiye03}}
372	>	\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}],
373	>	TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD
374	>	without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The
375	>	arrows indicate the change in densities observed when turning off the
376	>	reaction field. The the lower than expected densities for the SSD
377	>	model were what prompted the original reparameterization of SSD1
378	>	[Ref. \citen{Ichiye03}].}
379		\label{dense1}
380		\end{center}
381		\end{figure}
382
383	<	The density maximum for SSD actually compares quite favorably to other
384	<	simple water models. Figure \ref{dense1} also shows calculated
385	<	densities of several other models and experiment obtained from other
383	>	The density maximum for SSD compares quite favorably to other simple
384	>	water models. Figure \ref{dense1} also shows calculated densities of
385	>	several other models and experiment obtained from other
386		sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
387	<	models, SSD has results closest to the experimentally observed water
388	<	density maximum. Of the listed water models, TIP4P has a density
389	<	maximum behavior most like that seen in SSD. Though not included in
390	<	this particular plot, it is useful to note that TIP5P has a water
391	<	density maximum nearly identical to experiment.
387	>	models, SSD has a temperature closest to the experimentally observed
388	>	density maximum. Of the {\it charge-based} models in
389	>	Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that
390	>	seen in SSD. Though not included in this plot, it is useful
391	>	to note that TIP5P has a density maximum nearly identical to the
392	>	experimentally measured temperature.
393
394	<	It has been observed that densities are dependent on the cutoff radius
395	<	used for a variety of water models in simulations both with and
396	<	without the use of reaction field.\cite{Berendsen98} In order to
397	<	address the possible affect of cutoff radius, simulations were
398	<	performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the
399	<	previous SSD simulations, all performed with a cutoff of 9.0 \AA. All
400	<	of the resulting densities overlapped within error and showed no
401	<	significant trend toward lower or higher densities as a function of
402	<	cutoff radius, for simulations both with and without reaction
403	<	field. These results indicate that there is no major benefit in
404	<	choosing a longer cutoff radius in simulations using SSD. This is
405	<	advantageous in that the use of a longer cutoff radius results in
406	<	significant increases in the time required to obtain a single
370	<	trajectory.
394	>	It has been observed that liquid state densities in water are
395	>	dependent on the cutoff radius used both with and without the use of
396	>	reaction field.\cite{Berendsen98} In order to address the possible
397	>	effect of cutoff radius, simulations were performed with a dipolar
398	>	cutoff radius of 12.0 \AA\ to complement the previous SSD simulations,
399	>	all performed with a cutoff of 9.0 \AA. All of the resulting densities
400	>	overlapped within error and showed no significant trend toward lower
401	>	or higher densities as a function of cutoff radius, for simulations
402	>	both with and without reaction field. These results indicate that
403	>	there is no major benefit in choosing a longer cutoff radius in
404	>	simulations using SSD. This is advantageous in that the use of a
405	>	longer cutoff radius results in a significant increase in the time
406	>	required to obtain a single trajectory.
407
408		The key feature to recognize in figure \ref{dense1} is the density
409		scaling of SSD relative to other common models at any given
410	<	temperature. Note that the SSD model assumes a lower density than any
411	<	of the other listed models at the same pressure, behavior which is
412	<	especially apparent at temperatures greater than 300 K. Lower than
413	<	expected densities have been observed for other systems using a
414	<	reaction field for long-range electrostatic interactions, so the most
415	<	likely reason for the significantly lower densities seen in these
416	<	simulations is the presence of the reaction
417	<	field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the
418	<	reaction field on the density of the systems, the simulations were
419	<	repeated without a reaction field present. The results of these
420	<	simulations are also displayed in figure \ref{dense1}. Without
421	<	reaction field, the densities increase considerably to more
422	<	experimentally reasonable values, especially around the freezing point
423	<	of liquid water. The shape of the curve is similar to the curve
424	<	produced from SSD simulations using reaction field, specifically the
425	<	rapidly decreasing densities at higher temperatures; however, a shift
426	<	in the density maximum location, down to 245 K, is observed. This is a
427	<	more accurate comparison to the other listed water models, in that no
428	<	long range corrections were applied in those
393	<	simulations.\cite{Clancy94,Jorgensen98b} However, even without a
410	>	temperature. SSD assumes a lower density than any of the other listed
411	>	models at the same pressure, behavior which is especially apparent at
412	>	temperatures greater than 300 K. Lower than expected densities have
413	>	been observed for other systems using a reaction field for long-range
414	>	electrostatic interactions, so the most likely reason for the
415	>	significantly lower densities seen in these simulations is the
416	>	presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order
417	>	to test the effect of the reaction field on the density of the
418	>	systems, the simulations were repeated without a reaction field
419	>	present. The results of these simulations are also displayed in figure
420	>	\ref{dense1}. Without the reaction field, the densities increase
421	>	to more experimentally reasonable values, especially around the
422	>	freezing point of liquid water. The shape of the curve is similar to
423	>	the curve produced from SSD simulations using reaction field,
424	>	specifically the rapidly decreasing densities at higher temperatures;
425	>	however, a shift in the density maximum location, down to 245 K, is
426	>	observed. This is a more accurate comparison to the other listed water
427	>	models, in that no long range corrections were applied in those
428	>	simulations.\cite{Clancy94,Jorgensen98b} However, even without the
429		reaction field, the density around 300 K is still significantly lower
430		than experiment and comparable water models. This anomalous behavior
431	<	was what lead Ichiye \emph{et al.} to recently reparameterize SSD and
432	<	make SSD1.\cite{Ichiye03} In discussing potential adjustments later in
433	<	this paper, all comparisons were performed with this new model.
431	>	was what lead Ichiye {\it et al.} to recently reparameterize
432	>	SSD.\cite{Ichiye03} Throughout the remainder of the paper our
433	>	reparamaterizations of SSD will be compared with the newer SSD1 model.
434
435		\subsection{Transport Behavior}
401	–	Of importance in these types of studies are the transport properties
402	–	of the particles and their change in responce to altering
403	–	environmental conditions. In order to probe transport, constant energy
404	–	simulations were performed about the average density uncovered by the
405	–	constant pressure simulations. Simulations started with randomized
406	–	velocities and underwent 50 ps of temperature scaling and 50 ps of
407	–	constant energy equilibration before obtaining a 200 ps
408	–	trajectory. Diffusion constants were calculated via root-mean square
409	–	deviation analysis. The averaged results from five sets of NVE
410	–	simulations are displayed in figure \ref{diffuse}, alongside
411	–	experimental, SPC/E, and TIP5P
412	–	results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
436
437	+	Accurate dynamical properties of a water model are particularly
438	+	important when using the model to study permeation or transport across
439	+	biological membranes. In order to probe transport in bulk water,
440	+	constant energy (NVE) simulations were performed at the average
441	+	density obtained by the NPT simulations at an identical target
442	+	temperature. Simulations started with randomized velocities and
443	+	underwent 50 ps of temperature scaling and 50 ps of constant energy
444	+	equilibration before a 200 ps data collection run. Diffusion constants
445	+	were calculated via linear fits to the long-time behavior of the
446	+	mean-square displacement as a function of time. The averaged results
447	+	from five sets of NVE simulations are displayed in figure
448	+	\ref{diffuse}, alongside experimental, SPC/E, and TIP5P
449	+	results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
450	+
451		\begin{figure}
452		\begin{center}
453		\epsfxsize=6in
454		\epsfbox{betterDiffuse.epsi}
455	<	\caption{Average diffusion coefficient over increasing temperature for
456	<	SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental
457	<	data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has
458	<	the least deviation from the experimental values. The rapidly
459	<	increasing diffusion constants for TIP5P and SSD correspond to
460	<	significant decrease in density at the higher temperatures.}
455	>	\caption{Average self-diffusion constant as a function of temperature for
456	>	SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}],
457	>	and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of
458	>	the three water models shown, SSD has the least deviation from the
459	>	experimental values. The rapidly increasing diffusion constants for
460	>	TIP5P and SSD correspond to significant decrease in density at the
461	>	higher temperatures.}
462		\label{diffuse}
463		\end{center}
464		\end{figure}
465
466		The observed values for the diffusion constant point out one of the
467	<	strengths of the SSD model. Of the three experimental models shown,
468	<	the SSD model has the most accurate depiction of the diffusion trend
469	<	seen in experiment in both the supercooled and liquid temperature
470	<	regimes. SPC/E does a respectable job by producing values similar to
471	<	SSD and experiment around 290 K; however, it deviates at both higher
472	<	and lower temperatures, failing to predict the experimental
473	<	trend. TIP5P and SSD both start off low at colder temperatures and
474	<	tend to diffuse too rapidly at higher temperatures. This trend at
475	<	higher temperatures is not surprising in that the densities of both
476	<	TIP5P and SSD are lower than experimental water at these higher
477	<	temperatures. When calculating the diffusion coefficients for SSD at
478	<	experimental densities, the resulting values fall more in line with
479	<	experiment at these temperatures, albeit not at standard pressure.
467	>	strengths of the SSD model. Of the three models shown, the SSD model
468	>	has the most accurate depiction of self-diffusion in both the
469	>	supercooled and liquid regimes. SPC/E does a respectable job by
470	>	reproducing values similar to experiment around 290 K; however, it
471	>	deviates at both higher and lower temperatures, failing to predict the
472	>	correct thermal trend. TIP5P and SSD both start off low at colder
473	>	temperatures and tend to diffuse too rapidly at higher temperatures.
474	>	This behavior at higher temperatures is not particularly surprising
475	>	since the densities of both TIP5P and SSD are lower than experimental
476	>	water densities at higher temperatures. When calculating the
477	>	diffusion coefficients for SSD at experimental densities (instead of
478	>	the densities from the NPT simulations), the resulting values fall
479	>	more in line with experiment at these temperatures.
480
481		\subsection{Structural Changes and Characterization}
482	+
483		By starting the simulations from the crystalline state, the melting
484	<	transition and the ice structure can be studied along with the liquid
484	>	transition and the ice structure can be obtained along with the liquid
485		phase behavior beyond the melting point. The constant pressure heat
486		capacity (C$_\text{p}$) was monitored to locate the melting transition
487		in each of the simulations. In the melting simulations of the 1024
#	Line 450 \| Line 489 \| transition occurs at 235 K. These melting transitions
489		at 245 K, indicating a first order phase transition for the melting of
490		these ice crystals. When the reaction field is turned off, the melting
491		transition occurs at 235 K. These melting transitions are
492	<	considerably lower than the experimental value, but this is not a
454	<	surprise considering the simplicity of the SSD model.
492	>	considerably lower than the experimental value.
493
494		\begin{figure}
495		\begin{center}
496		\epsfxsize=6in
497		\epsfbox{corrDiag.eps}
498		\caption{Two dimensional illustration of angles involved in the
499	<	correlations observed in figure \ref{contour}.}
499	>	correlations observed in Fig. \ref{contour}.}
500		\label{corrAngle}
501		\end{center}
502		\end{figure}
#	Line 470 \| Line 508 \| clarity: dark areas signify peaks while light areas si
508		\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at
509		100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for
510		clarity: dark areas signify peaks while light areas signify
511	<	depressions. White areas have g(\emph{r}) values below 0.5 and black
511	>	depressions. White areas have $g(r)$ values below 0.5 and black
512		areas have values above 1.5.}
513		\label{contour}
514		\end{center}
515		\end{figure}
516
517	<	Additional analysis of the melting phase-transition process was
518	<	performed by using two-dimensional structure and dipole angle
519	<	correlations. Expressions for these correlations are as follows:
517	>	Additional analysis of the melting process was performed using
518	>	two-dimensional structure and dipole angle correlations. Expressions
519	>	for these correlations are as follows:
520
521		\begin{equation}
522	<	g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left\|\mathbf{r}_{ij}\right\|)\rangle\ ,
522	>	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\ ,
523		\end{equation}
524		\begin{equation}
525		g_{\text{AB}}(r,\cos\omega) =
526	<	\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\ ,
526	>	\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\ ,
527		\end{equation}
528		where $\theta$ and $\omega$ refer to the angles shown in figure
529		\ref{corrAngle}. By binning over both distance and the cosine of the
530	<	desired angle between the two dipoles, the g(\emph{r}) can be
531	<	dissected to determine the common dipole arrangements that constitute
532	<	the peaks and troughs. Frames A and B of figure \ref{contour} show a
533	<	relatively crystalline state of an ice $I_c$ simulation. The first
534	<	peak of the g(\emph{r}) consists primarily of the preferred hydrogen
530	>	desired angle between the two dipoles, the $g(r)$ can be analyzed to
531	>	determine the common dipole arrangements that constitute the peaks and
532	>	troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
533	>	of figure \ref{contour} show results from an ice $I_c$ simulation. The
534	>	first peak in the $g(r)$ consists primarily of the preferred hydrogen
535		bonding arrangements as dictated by the tetrahedral sticky potential -
536	<	one peak for the donating and the other for the accepting hydrogen
537	<	bonds. Due to the high degree of crystallinity of the sample, the
538	<	second and third solvation shells show a repeated peak arrangement
536	>	one peak for the hydrogen bond donor and the other for the hydrogen
537	>	bond acceptor. Due to the high degree of crystallinity of the sample,
538	>	the second and third solvation shells show a repeated peak arrangement
539		which decays at distances around the fourth solvation shell, near the
540		imposed cutoff for the Lennard-Jones and dipole-dipole interactions.
541		In the higher temperature simulation shown in frames C and D, these
542	<	longer-ranged repeated peak features deteriorate rapidly. The first
543	<	solvation shell still shows the strong effect of the sticky-potential,
544	<	although it covers a larger area, extending to include a fraction of
545	<	aligned dipole peaks within the first solvation shell. The latter
546	<	peaks lose definition as thermal motion and the competing dipole force
547	<	overcomes the sticky potential's tight tetrahedral structuring of the
510	<	fluid.
542	>	long-range features deteriorate rapidly. The first solvation shell
543	>	still shows the strong effect of the sticky-potential, although it
544	>	covers a larger area, extending to include a fraction of aligned
545	>	dipole peaks within the first solvation shell. The latter peaks lose
546	>	due to thermal motion and as the competing dipole force overcomes the
547	>	sticky potential's tight tetrahedral structuring of the crystal.
548
549		This complex interplay between dipole and sticky interactions was
550		remarked upon as a possible reason for the split second peak in the
551	<	oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the
552	<	second solvation shell peak appears to have two distinct components
553	<	that blend together to form one observable peak. At higher
551	>	oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures,
552	>	the second solvation shell peak appears to have two distinct
553	>	components that blend together to form one observable peak. At higher
554		temperatures, this split character alters to show the leading 4 \AA\
555		peak dominated by equatorial anti-parallel dipole orientations. There
556		is also a tightly bunched group of axially arranged dipoles that most
#	Line 522 \| Line 559 \| pair interaction begins to dominate outside of the ran
559		dipoles that assume hydrogen bond arrangements similar to those seen
560		in the first solvation shell. This evidence indicates that the dipole
561		pair interaction begins to dominate outside of the range of the
562	<	dipolar repulsion term. Primary energetically favorable dipole
562	>	dipolar repulsion term. The energetically favorable dipole
563		arrangements populate the region immediately outside this repulsion
564	<	region (around 4 \AA), while arrangements that seek to ideally satisfy
565	<	both the sticky and dipole forces locate themselves just beyond this
564	>	region (around 4 \AA), while arrangements that seek to satisfy both
565	>	the sticky and dipole forces locate themselves just beyond this
566		initial buildup (around 5 \AA).
567
568		From these findings, the split second peak is primarily the product of
569		the dipolar repulsion term of the sticky potential. In fact, the inner
570		peak can be pushed out and merged with the outer split peak just by
571	<	extending the switching function cutoff ($s^\prime(r_{ij})$) from its
572	<	normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of
571	>	extending the switching function ($s^\prime(r_{ij})$) from its normal
572	>	4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of
573		correction is not recommended for improving the liquid structure,
574		since the second solvation shell would still be shifted too far
575		out. In addition, this would have an even more detrimental effect on
576		the system densities, leading to a liquid with a more open structure
577	<	and a density considerably lower than the normal SSD behavior shown
578	<	previously. A better correction would be to include the
579	<	quadrupole-quadrupole interactions for the water particles outside of
580	<	the first solvation shell, but this reduces the simplicity and speed
581	<	advantage of SSD.
577	>	and a density considerably lower than the already low SSD density. A
578	>	better correction would be to include the quadrupole-quadrupole
579	>	interactions for the water particles outside of the first solvation
580	>	shell, but this would remove the simplicity and speed advantage of
581	>	SSD.
582
583		\subsection{Adjusted Potentials: SSD/RF and SSD/E}
584	+
585		The propensity of SSD to adopt lower than expected densities under
586		varying conditions is troubling, especially at higher temperatures. In
587		order to correct this model for use with a reaction field, it is
#	Line 551 \| Line 589 \| important properties. In this case, it would be ideal
589		intermolecular interactions. In undergoing a reparameterization, it is
590		important not to focus on just one property and neglect the other
591		important properties. In this case, it would be ideal to correct the
592	<	densities while maintaining the accurate transport properties.
592	>	densities while maintaining the accurate transport behavior.
593
594		The parameters available for tuning include the $\sigma$ and $\epsilon$
595		Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
#	Line 561 \| Line 599 \| u_{ij}^{sp}
599		follows:
600		\begin{equation}
601		u_{ij}^{sp}
602	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
603	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
602	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
603	>	\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)],
604		\end{equation}
567	–
605		where $\nu_0$ scales the strength of the tetrahedral attraction and
606	<	$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
607	<	term. The separation was performed for purposes of the
608	<	reparameterization, but the final parameters were adjusted so that it
609	<	is unnecessary to separate the terms when implementing the adjusted
610	<	water potentials. The results of the reparameterizations are shown in
611	<	table \ref{params}. Note that the tetrahedral attractive and dipolar
606	>	$\nu_0^\prime$ scales the dipole repulsion term independently. The
607	>	separation was performed for purposes of the reparameterization, but
608	>	the final parameters were adjusted so that it is not necessary to
609	>	separate the terms when implementing the adjusted water
610	>	potentials. The results of the reparameterizations are shown in table
611	>	\ref{params}. Note that the tetrahedral attractive and dipolar
612		repulsive terms do not share the same lower cutoff ($r_l$) in the
613	<	newly parameterized potentials - soft sticky dipole reaction field
614	<	(SSD/RF - for use with a reaction field) and soft sticky dipole
615	<	enhanced (SSD/E - an attempt to improve the liquid structure in
613	>	newly parameterized potentials. We are calling these
614	>	reparameterizations the Soft Sticky Dipole / Reaction Field
615	>	(SSD/RF - for use with a reaction field) and Soft Sticky Dipole
616	>	Enhanced (SSD/E - an attempt to improve the liquid structure in
617		simulations without a long-range correction).
618
619		\begin{table}
#	Line 583 \| Line 621 \| simulations without a long-range correction).
621		\caption{Parameters for the original and adjusted models}
622		\begin{tabular}{ l c c c c }
623		\hline \\[-3mm]
624	<	\ \ \ Parameters\ \ \ & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \ & \ SSD/E\ \ & \ SSD/RF \\
624	>	\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \
625	>	& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\
626		\hline \\[-3mm]
627		\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\
628		\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
#	Line 603 \| Line 642 \| simulations without a long-range correction).
642		\begin{center}
643		\epsfxsize=5in
644		\epsfbox{GofRCompare.epsi}
645	<	\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E
645	>	\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E
646		and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with
647		reaction field turned on (bottom). The insets show the respective
648		first peaks in detail. Note how the changes in parameters have lowered
#	Line 624 \| Line 663 \| component.}
663		\end{center}
664		\end{figure}
665
666	<	In the paper detailing the development of SSD, Liu and Ichiye placed
667	<	particular emphasis on an accurate description of the first solvation
668	<	shell. This resulted in a somewhat tall and narrow first peak in the
669	<	g(\emph{r}) that integrated to give similar coordination numbers to
666	>	In the original paper detailing the development of SSD, Liu and Ichiye
667	>	placed particular emphasis on an accurate description of the first
668	>	solvation shell. This resulted in a somewhat tall and narrow first
669	>	peak in $g(r)$ that integrated to give similar coordination numbers to
670		the experimental data obtained by Soper and
671		Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering
672		data from the Head-Gordon lab indicates a slightly lower and shifted
673	<	first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made
674	<	while taking into consideration the new experimental
673	>	first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were
674	>	made while taking into consideration the new experimental
675		findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the
676	<	relocation of the first peak of the oxygen-oxygen g(\emph{r}) by
677	<	comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new
676	>	relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
677	>	the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
678		experimental results. Both modified water models have shorter peaks
679	<	that are brought in more closely to the experimental peak (as seen in
680	<	the insets of figure \ref{grcompare}). This structural alteration was
679	>	that match more closely to the experimental peak (as seen in the
680	>	insets of figure \ref{grcompare}). This structural alteration was
681		accomplished by the combined reduction in the Lennard-Jones $\sigma$
682	<	variable and adjustment of the sticky potential strength and
683	<	cutoffs. As can be seen in table \ref{params}, the cutoffs for the
684	<	tetrahedral attractive and dipolar repulsive terms were nearly swapped
685	<	with each other. Isosurfaces of the original and modified sticky
686	<	potentials are shown in figure \ref{isosurface}. In these isosurfaces,
687	<	it is easy to see how altering the cutoffs changes the repulsive and
688	<	attractive character of the particles. With a reduced repulsive
689	<	surface (darker region), the particles can move closer to one another,
690	<	increasing the density for the overall system. This change in
691	<	interaction cutoff also results in a more gradual orientational motion
692	<	by allowing the particles to maintain preferred dipolar arrangements
693	<	before they begin to feel the pull of the tetrahedral
694	<	restructuring. As the particles move closer together, the dipolar
695	<	repulsion term becomes active and excludes unphysical nearest-neighbor
696	<	arrangements. This compares with how SSD and SSD1 exclude preferred
697	<	dipole alignments before the particles feel the pull of the ``hydrogen
698	<	bonds''. Aside from improving the shape of the first peak in the
699	<	g(\emph{r}), this modification improves the densities considerably by
700	<	allowing the persistence of full dipolar character below the previous
701	<	4.0 \AA\ cutoff.
682	>	variable and adjustment of the sticky potential strength and cutoffs.
683	>	As can be seen in table \ref{params}, the cutoffs for the tetrahedral
684	>	attractive and dipolar repulsive terms were nearly swapped with each
685	>	other. Isosurfaces of the original and modified sticky potentials are
686	>	shown in figure \ref{isosurface}. In these isosurfaces, it is easy to
687	>	see how altering the cutoffs changes the repulsive and attractive
688	>	character of the particles. With a reduced repulsive surface (darker
689	>	region), the particles can move closer to one another, increasing the
690	>	density for the overall system. This change in interaction cutoff also
691	>	results in a more gradual orientational motion by allowing the
692	>	particles to maintain preferred dipolar arrangements before they begin
693	>	to feel the pull of the tetrahedral restructuring. As the particles
694	>	move closer together, the dipolar repulsion term becomes active and
695	>	excludes unphysical nearest-neighbor arrangements. This compares with
696	>	how SSD and SSD1 exclude preferred dipole alignments before the
697	>	particles feel the pull of the ``hydrogen bonds''. Aside from
698	>	improving the shape of the first peak in the g(\emph{r}), this
699	>	modification improves the densities considerably by allowing the
700	>	persistence of full dipolar character below the previous 4.0 \AA\
701	>	cutoff.
702
703	<	While adjusting the location and shape of the first peak of
704	<	g(\emph{r}) improves the densities, these changes alone are
705	<	insufficient to bring the system densities up to the values observed
706	<	experimentally. To further increase the densities, the dipole moments
707	<	were increased in both of the adjusted models. Since SSD is a dipole
708	<	based model, the structure and transport are very sensitive to changes
709	<	in the dipole moment. The original SSD simply used the dipole moment
710	<	calculated from the TIP3P water model, which at 2.35 D is
711	<	significantly greater than the experimental gas phase value of 1.84
712	<	D. The larger dipole moment is a more realistic value and improves the
713	<	dielectric properties of the fluid. Both theoretical and experimental
714	<	measurements indicate a liquid phase dipole moment ranging from 2.4 D
715	<	to values as high as 3.11 D, providing a substantial range of
716	<	reasonable values for a dipole
678	<	moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
703	>	While adjusting the location and shape of the first peak of $g(r)$
704	>	improves the densities, these changes alone are insufficient to bring
705	>	the system densities up to the values observed experimentally. To
706	>	further increase the densities, the dipole moments were increased in
707	>	both of our adjusted models. Since SSD is a dipole based model, the
708	>	structure and transport are very sensitive to changes in the dipole
709	>	moment. The original SSD simply used the dipole moment calculated from
710	>	the TIP3P water model, which at 2.35 D is significantly greater than
711	>	the experimental gas phase value of 1.84 D. The larger dipole moment
712	>	is a more realistic value and improves the dielectric properties of
713	>	the fluid. Both theoretical and experimental measurements indicate a
714	>	liquid phase dipole moment ranging from 2.4 D to values as high as
715	>	3.11 D, providing a substantial range of reasonable values for a
716	>	dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
717		increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF,
718		respectively, leads to significant changes in the density and
719		transport of the water models.
#	Line 693 \| Line 731 \| simulations had the same thermalization, equilibration
731		run at each temperature step, and the final configuration from the
732		previous temperature simulation was used as a starting point. All NVE
733		simulations had the same thermalization, equilibration, and data
734	<	collection times as stated earlier in this paper.
734	>	collection times as stated previously.
735
736		\begin{figure}
737		\begin{center}
738		\epsfxsize=6in
739		\epsfbox{ssdeDense.epsi}
740		\caption{Comparison of densities calculated with SSD/E to SSD1 without a
741	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
742	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a
743	<	expansion around 300 K with error bars included to clarify this region
744	<	of interest. Note that both SSD1 and SSD/E show good agreement with
741	>	reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P
742	>	[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and
743	>	experiment [Ref. \citen{CRC80}]. The window shows a expansion around
744	>	300 K with error bars included to clarify this region of
745	>	interest. Note that both SSD1 and SSD/E show good agreement with
746		experiment when the long-range correction is neglected.}
747		\label{ssdedense}
748		\end{center}
749		\end{figure}
750
751	<	Figure \ref{ssdedense} shows the density profile for the SSD/E model
751	>	Fig. \ref{ssdedense} shows the density profile for the SSD/E model
752		in comparison to SSD1 without a reaction field, other common water
753		models, and experimental results. The calculated densities for both
754		SSD/E and SSD1 have increased significantly over the original SSD
755	<	model (see figure \ref{dense1}) and are in better agreement with the
755	>	model (see fig. \ref{dense1}) and are in better agreement with the
756		experimental values. At 298 K, the densities of SSD/E and SSD1 without
757		a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
758		0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with
#	Line 725 \| Line 764 \| character. However, this increasing disorder in the SS
764		comes about via an increase in the liquid disorder through the
765		weakening of the sticky potential and strengthening of the dipolar
766		character. However, this increasing disorder in the SSD/E model has
767	<	little effect on the melting transition. By monitoring C$\text{p}$
767	>	little effect on the melting transition. By monitoring $C_p$
768		throughout these simulations, the melting transition for SSD/E was
769	<	shown to occur at 235 K, the same transition temperature observed with
770	<	SSD and SSD1.
769	>	shown to occur at 235 K. The same transition temperature observed
770	>	with SSD and SSD1.
771
772		\begin{figure}
773		\begin{center}
774		\epsfxsize=6in
775		\epsfbox{ssdrfDense.epsi}
776		\caption{Comparison of densities calculated with SSD/RF to SSD1 with a
777	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
778	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the
779	<	necessity of reparameterization when utilizing a reaction field
780	<	long-ranged correction - SSD/RF provides significantly more accurate
781	<	densities than SSD1 when performing room temperature simulations.}
777	>	reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P
778	>	[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and
779	>	experiment [Ref. \citen{CRC80}]. The inset shows the necessity of
780	>	reparameterization when utilizing a reaction field long-ranged
781	>	correction - SSD/RF provides significantly more accurate densities
782	>	than SSD1 when performing room temperature simulations.}
783		\label{ssdrfdense}
784		\end{center}
785		\end{figure}
786
787		Including the reaction field long-range correction in the simulations
788	<	results in a more interesting comparison. A density profile including
788	>	results in a more interesting comparison. A density profile including
789		SSD/RF and SSD1 with an active reaction field is shown in figure
790		\ref{ssdrfdense}. As observed in the simulations without a reaction
791		field, the densities of SSD/RF and SSD1 show a dramatic increase over
792		normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
793		of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
794	<	considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and
795	<	the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further
796	<	emphasize the importance of reparameterization in order to model the
797	<	density properly under different simulation conditions. Again, these
798	<	changes have only a minor effect on the melting point, which observed
799	<	at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1
800	<	with a reaction field. Additionally, the difference in density maxima
801	<	is not as extreme, with SSD/RF showing a density maximum at 255 K,
802	<	fairly close to the density maxima of 260 K and 265 K, shown by SSD
803	<	and SSD1 respectively.
794	>	considerably better than the original SSD value of 0.941$\pm$0.001
795	>	g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results
796	>	further emphasize the importance of reparameterization in order to
797	>	model the density properly under different simulation conditions.
798	>	Again, these changes have only a minor effect on the melting point,
799	>	which observed at 245 K for SSD/RF, is identical to SSD and only 5 K
800	>	lower than SSD1 with a reaction field. Additionally, the difference in
801	>	density maxima is not as extreme, with SSD/RF showing a density
802	>	maximum at 255 K, fairly close to the density maxima of 260 K and 265
803	>	K, shown by SSD and SSD1 respectively.
804
805		\begin{figure}
806		\begin{center}
807		\epsfxsize=6in
808		\epsfbox{ssdeDiffuse.epsi}
809		\caption{Plots of the diffusion constants calculated from SSD/E and SSD1,
810	<	both without a reaction field, along with experimental
811	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
812	<	at the average densities observed in the 1 atm NPT simulations for
813	<	the respective models. SSD/E is slightly more fluid than experiment
814	<	at all of the temperatures, but it is closer than SSD1 without a
815	<	long-range correction.}
810	>	both without a reaction field, along with experimental results
811	>	[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations were
812	>	performed at the average densities observed in the 1 atm NPT
813	>	simulations for the respective models. SSD/E is slightly more fluid
814	>	than experiment at all of the temperatures, but it is closer than SSD1
815	>	without a long-range correction.}
816		\label{ssdediffuse}
817		\end{center}
818		\end{figure}
#	Line 780 \| Line 820 \| what is expected for simulating liquid water. In addit
820		The reparameterization of the SSD water model, both for use with and
821		without an applied long-range correction, brought the densities up to
822		what is expected for simulating liquid water. In addition to improving
823	<	the densities, it is important that particle transport be maintained
824	<	or improved. Figure \ref{ssdediffuse} compares the temperature
825	<	dependence of the diffusion constant of SSD/E to SSD1 without an
826	<	active reaction field, both at the densities calculated at 1 atm and
827	<	at the experimentally calculated densities for super-cooled and liquid
828	<	water. The diffusion constant for SSD/E is consistently a little
823	>	the densities, it is important that the excellent diffusive behavior
824	>	of SSD be maintained or improved. Figure \ref{ssdediffuse} compares
825	>	the temperature dependence of the diffusion constant of SSD/E to SSD1
826	>	without an active reaction field, both at the densities calculated at
827	>	1 atm and at the experimentally calculated densities for super-cooled
828	>	and liquid water. The diffusion constant for SSD/E is consistently
829		higher than experiment, while SSD1 remains lower than experiment until
830		relatively high temperatures (greater than 330 K). Both models follow
831		the shape of the experimental curve well below 300 K but tend to
832		diffuse too rapidly at higher temperatures, something that is
833	<	especially apparent with SSD1. This accelerated increasing of
834	<	diffusion is caused by the rapidly decreasing system density with
835	<	increasing temperature. Though it is difficult to see in figure
836	<	\ref{ssdedense}, the densities of SSD1 decay more rapidly with
837	<	temperature than do those of SSD/E, leading to more visible deviation
838	<	from the experimental diffusion trend. Thus, the changes made to
839	<	improve the liquid structure may have had an adverse affect on the
840	<	density maximum, but they improve the transport behavior of SSD/E
801	<	relative to SSD1.
833	>	especially apparent with SSD1. This increasing diffusion relative to
834	>	the experimental values is caused by the rapidly decreasing system
835	>	density with increasing temperature. The densities of SSD1 decay more
836	>	rapidly with temperature than do those of SSD/E, leading to more
837	>	visible deviation from the experimental diffusion trend. Thus, the
838	>	changes made to improve the liquid structure may have had an adverse
839	>	affect on the density maximum, but they improve the transport behavior
840	>	of SSD/E relative to SSD1.
841
842		\begin{figure}
843		\begin{center}
844		\epsfxsize=6in
845		\epsfbox{ssdrfDiffuse.epsi}
846		\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1,
847	<	both with an active reaction field, along with experimental
848	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
849	<	at the average densities observed in the 1 atm NPT simulations for
850	<	both of the models. Note how accurately SSD/RF simulates the
851	<	diffusion of water throughout this temperature range. The more
852	<	rapidly increasing diffusion constants at high temperatures for both
853	<	models is attributed to the significantly lower densities than
854	<	observed in experiment.}
847	>	both with an active reaction field, along with experimental results
848	>	[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations
849	>	were performed at the average densities observed in the 1 atm NPT
850	>	simulations for both of the models. Note how accurately SSD/RF
851	>	simulates the diffusion of water throughout this temperature
852	>	range. The more rapidly increasing diffusion constants at high
853	>	temperatures for both models is attributed to the significantly lower
854	>	densities than observed in experiment.}
855		\label{ssdrfdiffuse}
856		\end{center}
857		\end{figure}
858
859		In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
860		compared to SSD1 with an active reaction field. Note that SSD/RF
861	<	tracks the experimental results incredibly well, identical within
862	<	error throughout the temperature range shown and with only a slight
861	>	tracks the experimental results quantitatively, identical within error
862	>	throughout the temperature range shown and with only a slight
863		increasing trend at higher temperatures. SSD1 tends to diffuse more
864		slowly at low temperatures and deviates to diffuse too rapidly at
865	<	temperatures greater than 330 K. As stated in relation to SSD/E, this
866	<	deviation away from the ideal trend is due to a rapid decrease in
867	<	density at higher temperatures. SSD/RF does not suffer from this
868	<	problem as much as SSD1, because the calculated densities are closer
869	<	to the experimental value. These results again emphasize the
870	<	importance of careful reparameterization when using an altered
832	<	long-range correction.
865	>	temperatures greater than 330 K. As stated above, this deviation away
866	>	from the ideal trend is due to a rapid decrease in density at higher
867	>	temperatures. SSD/RF does not suffer from this problem as much as SSD1
868	>	because the calculated densities are closer to the experimental
869	>	values. These results again emphasize the importance of careful
870	>	reparameterization when using an altered long-range correction.
871
872		\subsection{Additional Observations}
873
#	Line 838 \| Line 876 \| long-range correction.
876		\epsfxsize=6in
877		\epsfbox{povIce.ps}
878		\caption{A water lattice built from the crystal structure assumed by
879	<	SSD/E when undergoing an extremely restricted temperature NPT
880	<	simulation. This form of ice is referred to as ice \emph{i} to
881	<	emphasize its simulation origins. This image was taken of the (001)
882	<	face of the crystal.}
879	>	SSD/E when undergoing an extremely restricted temperature NPT
880	>	simulation. This form of ice is referred to as ice-{\it i} to
881	>	emphasize its simulation origins. This image was taken of the (001)
882	>	face of the crystal.}
883		\label{weirdice}
884		\end{center}
885		\end{figure}
886
887	<	While performing restricted temperature melting sequences of SSD/E not
888	<	previously discussed, some interesting observations were made. After
889	<	melting at 235 K, two of five systems underwent crystallization events
890	<	near 245 K. As the heating process continued, the two systems remained
891	<	crystalline until finally melting between 320 and 330 K. The final
892	<	configurations of these two melting sequences show an expanded
893	<	zeolite-like crystal structure that does not correspond to any known
894	<	form of ice. For convenience, and to help distinguish it from the
895	<	experimentally observed forms of ice, this crystal structure will
896	<	henceforth be referred to as ice $\sqrt{\smash[b]{-\text{I}}}$ (ice
897	<	\emph{i}). The crystallinity was extensive enough that a near ideal
898	<	crystal structure of ice \emph{i} could be obtained. Figure
899	<	\ref{weirdice} shows the repeating crystal structure of a typical
900	<	crystal at 5 K. Each water molecule is hydrogen bonded to four others;
901	<	however, the hydrogen bonds are flexed rather than perfectly
902	<	straight. This results in a skewed tetrahedral geometry about the
903	<	central molecule. Referring to figure \ref{isosurface}, these flexed
904	<	hydrogen bonds are allowed due to the conical shape of the attractive
905	<	regions, with the greatest attraction along the direct hydrogen bond
887	>	While performing a series of melting simulations on an early iteration
888	>	of SSD/E not discussed in this paper, we observed recrystallization
889	>	into a novel structure not previously known for water. After melting
890	>	at 235 K, two of five systems underwent crystallization events near
891	>	245 K. The two systems remained crystalline up to 320 and 330 K,
892	>	respectively. The crystal exhibits an expanded zeolite-like structure
893	>	that does not correspond to any known form of ice. This appears to be
894	>	an artifact of the point dipolar models, so to distinguish it from the
895	>	experimentally observed forms of ice, we have denoted the structure
896	>	Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough
897	>	portion of the sample crystallized that we have been able to obtain a
898	>	near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice}
899	>	shows the repeating crystal structure of a typical crystal at 5
900	>	K. Each water molecule is hydrogen bonded to four others; however, the
901	>	hydrogen bonds are bent rather than perfectly straight. This results
902	>	in a skewed tetrahedral geometry about the central molecule. In
903	>	figure \ref{isosurface}, it is apparent that these flexed hydrogen
904	>	bonds are allowed due to the conical shape of the attractive regions,
905	>	with the greatest attraction along the direct hydrogen bond
906		configuration. Though not ideal, these flexed hydrogen bonds are
907	<	favorable enough to stabilize an entire crystal generated around
870	<	them. In fact, the imperfect ice \emph{i} crystals were so stable that
871	<	they melted at temperatures nearly 100 K greater than both ice I$_c$
872	<	and I$_h$.
907	>	favorable enough to stabilize an entire crystal generated around them.
908
909	<	These initial simulations indicated that ice \emph{i} is the preferred
910	<	ice structure for at least the SSD/E model. To verify this, a
911	<	comparison was made between near ideal crystals of ice $I_h$, ice
912	<	$I_c$, and ice 0 at constant pressure with SSD/E, SSD/RF, and
913	<	SSD1. Near ideal versions of the three types of crystals were cooled
914	<	to 1 K, and the potential energies of each were compared using all
915	<	three water models. With every water model, ice \emph{i} turned out to
916	<	have the lowest potential energy: 5\% lower than $I_h$ with SSD1,
917	<	6.5\% lower with SSD/E, and 7.5\% lower with SSD/RF.
909	>	Initial simulations indicated that ice-{\it i} is the preferred ice
910	>	structure for at least the SSD/E model. To verify this, a comparison
911	>	was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and
912	>	ice-{\it i} at constant pressure with SSD/E, SSD/RF, and
913	>	SSD1. Near-ideal versions of the three types of crystals were cooled
914	>	to 1 K, and the enthalpies of each were compared using all three water
915	>	models. With every model in the SSD family, ice-{\it i} had the lowest
916	>	calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with
917	>	SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized
918	>	in Table \ref{iceenthalpy}.
919
920	<	In addition to these low temperature comparisons, melting sequences
921	<	were performed with ice \emph{i} as the initial configuration using
922	<	SSD/E, SSD/RF, and SSD1 both with and without a reaction field. The
923	<	melting transitions for both SSD/E and SSD1 without a reaction field
924	<	occurred at temperature in excess of 375 K. SSD/RF and SSD1 with a
925	<	reaction field showed more reasonable melting transitions near 325
926	<	K. These melting point observations emphasize the preference for this
927	<	crystal structure over the most common types of ice when using these
928	<	single point water models.
920	>	\begin{table}
921	>	\begin{center}
922	>	\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1
923	>	K) exhibited by the SSD family of water models}
924	>	\begin{tabular}{ l c c c }
925	>	\hline \\[-3mm]
926	>	\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \
927	>	Ice-{\it i} \\
928	>	\hline \\[-3mm]
929	>	\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\
930	>	\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\
931	>	\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\
932	>	\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\
933	>	\end{tabular}
934	>	\label{iceenthalpy}
935	>	\end{center}
936	>	\end{table}
937
938	<	Recognizing that the above tests show ice \emph{i} to be both the most
939	<	stable and lowest density crystal structure for these single point
940	<	water models, it is interesting to speculate on the relative stability
941	<	of this crystal structure with charge based water models. As a quick
942	<	test, these 3 crystal types were converted from SSD type particles to
943	<	TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
944	<	minimizations were performed on the crystals to compare the system
945	<	energies. Again, ice \emph{i} was observed to have the lowest total
902	<	system energy. The total energy of ice \emph{i} was ~2\% lower than
903	<	ice $I_h$, which was in turn ~3\% lower than ice $I_c$. Based on these
904	<	initial studies, it would not be surprising if results from the other
905	<	common water models show ice \emph{i} to be the lowest energy crystal
906	<	structure. A continuation of this work studying ice \emph{i} with
907	<	multi-point water models will be published in a coming article.
938	>	In addition to these energetic comparisons, melting simulations were
939	>	performed with ice-{\it i} as the initial configuration using SSD/E,
940	>	SSD/RF, and SSD1 both with and without a reaction field. The melting
941	>	transitions for both SSD/E and SSD1 without reaction field occurred at
942	>	temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field
943	>	showed more reasonable melting transitions near 325~K. These melting
944	>	point observations clearly show that all of the SSD-derived models
945	>	prefer the ice-{\it i} structure.
946
947		\section{Conclusions}
910	–	The density maximum and temperature dependent transport for the SSD
911	–	water model, both with and without the use of reaction field, were
912	–	studied via a series of NPT and NVE simulations. The constant pressure
913	–	simulations of the melting of both $I_h$ and $I_c$ ice showed a
914	–	density maximum near 260 K. In most cases, the calculated densities
915	–	were significantly lower than the densities calculated in simulations
916	–	of other water models. Analysis of particle diffusion showed SSD to
917	–	capture the transport properties of experimental water well in both
918	–	the liquid and super-cooled liquid regimes. In order to correct the
919	–	density behavior, the original SSD model was reparameterized for use
920	–	both with and without a reaction field (SSD/RF and SSD/E), and
921	–	comparison simulations were performed with SSD1, the density corrected
922	–	version of SSD. Both models improve the liquid structure, density
923	–	values, and diffusive properties under their respective conditions,
924	–	indicating the necessity of reparameterization when altering the
925	–	long-range correction specifics. When taking into account the
926	–	appropriate considerations, these simple water models are excellent
927	–	choices for representing explicit water in large scale simulations of
928	–	biochemical systems.
948
949	+	The density maximum and temperature dependence of the self-diffusion
950	+	constant were studied for the SSD water model, both with and without
951	+	the use of reaction field, via a series of NPT and NVE
952	+	simulations. The constant pressure simulations showed a density
953	+	maximum near 260 K. In most cases, the calculated densities were
954	+	significantly lower than the densities obtained from other water
955	+	models (and experiment). Analysis of self-diffusion showed SSD to
956	+	capture the transport properties of water well in both the liquid and
957	+	super-cooled liquid regimes.
958	+
959	+	In order to correct the density behavior, the original SSD model was
960	+	reparameterized for use both with and without a reaction field (SSD/RF
961	+	and SSD/E), and comparisons were made with SSD1, Ichiye's density
962	+	corrected version of SSD. Both models improve the liquid structure,
963	+	densities, and diffusive properties under their respective simulation
964	+	conditions, indicating the necessity of reparameterization when
965	+	changing the method of calculating long-range electrostatic
966	+	interactions. In general, however, these simple water models are
967	+	excellent choices for representing explicit water in large scale
968	+	simulations of biochemical systems.
969	+
970	+	The existence of a novel low-density ice structure that is preferred
971	+	by the SSD family of water models is somewhat troubling, since liquid
972	+	simulations on this family of water models at room temperature are
973	+	effectively simulations of super-cooled or metastable liquids. One
974	+	way to de-stabilize this unphysical ice structure would be to make the
975	+	range of angles preferred by the attractive part of the sticky
976	+	potential much narrower. This would require extensive
977	+	reparameterization to maintain the same level of agreement with the
978	+	experiments.
979	+
980	+	Additionally, our initial calculations show that the ice-{\it i}
981	+	structure may also be a preferred crystal structure for at least one
982	+	other popular multi-point water model (TIP3P), and that much of the
983	+	simulation work being done using this popular model could also be at
984	+	risk for crystallization into this unphysical structure. A future
985	+	publication will detail the relative stability of the known ice
986	+	structures for a wide range of popular water models.
987	+
988		\section{Acknowledgments}
989		Support for this project was provided by the National Science
990		Foundation under grant CHE-0134881. Computation time was provided by
991		the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
992	<	DMR 00 79647.
992	>	DMR-0079647.
993
936	–
994		\newpage
995
996		\bibliographystyle{jcp}

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Revision 863 by chrisfen, Thu Nov 13 15:55:20 2003 UTC vs.
Revision 921 by gezelter, Mon Jan 12 16:20:53 2004 UTC