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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 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	<	\maketitle
61	>	\newpage
62
63		%\narrowtext
64
#	Line 61 | 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
72	<	solvent to some extent in order to complete simulations in a
73	<	reasonable amount of time. In the case of simulating water in
74	<	bio-molecular studies, the balance between accurate properties and
75	<	computational efficiency is especially delicate, and it has resulted
76	<	in a variety of different water
77	<	models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these models
78	<	get specific properties correct or better than their predecessors, but
79	<	this is often at a cost of some other properties or of computer
80	<	time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P succeeds
81	<	in improving the structural and transport properties over its
82	<	predecessors, yet this comes at a greater than 50\% increase in
83	<	computational cost.\cite{Jorgensen01,Jorgensen00} One recently
84	<	developed model that succeeds in both retaining accuracy of system
85	<	properties and simplifying calculations to increase computational
86	<	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
107	<	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}
115	–	\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 \ +
121	<	s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
122	<	\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		\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, $s$ and
146	<	$s^\prime$ 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 a
167	<	calculation speed up of up to an order of magnitude over other
168	<	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. Toward the end, we suggest
193	<	alterations to the parameters that result in more water-like
194	<	behavior. It should be noted that in a recent publication, some the
195	<	original investigators of the SSD water model have put forth
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} This study will make comparisons with this new
203	<	model's behavior with the goal of improving upon the depiction of
204	<	water under conditions without the Ewald Sum.
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, in this study the long-range dipole-dipole
210	<	interactions were accounted for 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 system, 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 proposed 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,
225	<	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 a 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
233	<	issue in recent times, and this has resulted in substantial benefits
234	<	in energy conservation. There is still the issue of 5 or 6 additional
235	<	elements for describing the orientation of each particle, which will
236	<	increase dump files substantially. Simply translating the rotation
237	<	matrix into its component Euler angles or quaternions for storage
238	<	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 the integration
270	<	method, the orientational propagation involves a sequence of matrix
271	<	evaluations to update the rotation matrix.\cite{Dullweber1997} These
272	<	matrix rotations end up being more costly computationally than the
273	<	simpler arithmetic quaternion propagation. With the same time step, a
274	<	1000 SSD particle simulation shows an average 7\% increase in
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. This cost is more than justified when comparing the
277	<	energy conservation of the two methods as illustrated in figure
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	<	\includegraphics[width=61mm, angle=-90]{timeStep.epsi}
285	<	\caption{Energy conservation using quaternion based integration versus
284	>	\begin{center}
285	>	\epsfxsize=6in
286	>	\epsfbox{timeStep.epsi}
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. For each time step, the dotted line is total
290	<	energy using the symplectic step integrator, and the solid line comes
258	<	from the quaternion integrator. The larger time step plots are shifted
259	<	up from the true energy baseline 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 for
299	<	handling 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 these SSD particle simulations was unnoticeable for
309	<	time steps up to three femtoseconds. A slight energy drift on the
308	>	Energy drift in the symplectic step simulations was unnoticeable for
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 the simulations. The $I_h$ crystals were
318	<	formed by first arranging the center of masses of the SSD particles
319	<	into a ``hexagonal'' ice lattice of 1024 particles. Because of the
320	<	crystal structure of $I_h$ ice, the simulation box assumed a
321	<	rectangular shape with a 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
325	<	down in stages to slowly cool the crystals down to 25 K. The particles
326	<	were then allowed translate with fixed orientations at a constant
325	>	down in stages to slowly cool the crystals to 25 K. The particles were
326	>	then allowed to translate with fixed orientations at a constant
327		pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were
328		removed and the ice crystals were allowed to equilibrate for 50 ps at
329		25 K and a constant pressure of 1 atm.  This procedure resulted in
330		structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
331	<	rules\cite{Bernal33,Rahman72}.  This method was also utilized in the
331	>	rules.\cite{Bernal33,Rahman72} This method was also utilized in the
332		making of diamond lattice $I_c$ ice crystals, with each cubic
333		simulation box consisting of either 512 or 1000 particles. Only
334		isotropic volume fluctuations were performed under constant pressure,
#	Line 306 | Line 338 | constant pressure and temperature dynamics. By perform
338		\section{Results and discussion}
339
340		Melting studies were performed on the randomized ice crystals using
341	<	constant pressure and temperature dynamics. By performing melting
342	<	simulations, the melting transition can be determined by monitoring
311	<	the heat capacity, in addition to determining the density maximum -
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
#	Line 317 | Line 348 | K to 10 and 5 K. The above equilibration and productio
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 the system volume fluctuations dampened out in all
353	<	but the very cold simulations (below 225 K).
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}
325	–	Initial simulations focused on the original SSD water model, and an
326	–	average density versus temperature plot is shown in figure
327	–	\ref{dense1}. Note that the density maximum when using a reaction
328	–	field appears between 255 and 265 K, where the calculated densities
329	–	within this range were nearly indistinguishable. The greater certainty
330	–	of the average value at 260 K makes a good argument for the actual
331	–	density maximum residing at this midpoint value. Figure \ref{dense1}
332	–	was constructed using ice $I_h$ crystals for the initial
333	–	configuration; and though not pictured, the simulations starting from
334	–	ice $I_c$ crystal configurations showed similar results, with a
335	–	liquid-phase density maximum in this same region (between 255 and 260
336	–	K). In addition, the $I_c$ crystals are more fragile than the $I_h$
337	–	crystals, leading them to deform into a dense glassy state at lower
338	–	temperatures. This resulted in an overall low temperature density
339	–	maximum at 200 K, but they still retained a common liquid state
340	–	density maximum 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	<	\includegraphics[width=65mm,angle=-90]{dense2.eps}
370	<	\caption{Density versus temperature for TIP4P\cite{Jorgensen98b},
371	<	TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction
372	<	Field, SSD, and Experiment\cite{CRC80}. The arrows indicate the
373	<	change in densities observed when turning off the reaction field. The
374	<	the lower than expected densities for the SSD model were what
375	<	prompted the original reparameterization.\cite{Ichiye03}}
369	>	\begin{center}
370	>	\epsfxsize=6in
371	>	\epsfbox{denseSSD.eps}
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	<	the resulting densities overlapped within error and showed no
401	<	significant trend in lower or higher densities as a function of cutoff
402	<	radius, both for simulations with and without reaction field. These
403	<	results indicate that there is no major benefit in choosing a longer
404	<	cutoff radius in simulations using SSD. This is comforting in that the
405	<	use of a longer cutoff radius results in significant increases in the
406	<	time required to obtain a single 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 most important thing to recognize in figure \ref{dense1} is the
409	<	density 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 with the use
414	<	of a reaction field for long-range electrostatic interactions, so the
415	<	most likely reason for these significantly lower densities 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, these 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
427	<	probably a more accurate comparison to the other listed water models,
428	<	in that no long range corrections were applied in those
398	<	simulations.\cite{Clancy94,Jorgensen98b} However, even without a
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. 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}
436	<	Of importance in these types of studies are the transport properties
437	<	of the particles and how they change when altering the environmental
438	<	conditions. In order to probe transport, constant energy simulations
439	<	were performed about the average density uncovered by the constant
440	<	pressure simulations. Simulations started with randomized velocities
441	<	and underwent 50 ps of temperature scaling and 50 ps of constant
442	<	energy equilibration before obtaining a 200 ps trajectory. Diffusion
443	<	constants were calculated via root-mean square deviation analysis. The
444	<	averaged results from 5 sets of these NVE simulations is displayed in
445	<	figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P
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	<	\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi}
453	<	\caption{Average diffusion coefficient over increasing temperature for
454	<	SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental
455	<	data from Gillen \emph{et al.}\cite{Gillen72}, and from
456	<	Mills\cite{Mills73}.}
452	>	\begin{center}
453	>	\epsfxsize=6in
454	>	\epsfbox{betterDiffuse.epsi}
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 normal regimes. SPC/E
470	<	does a respectable job by getting similar values as SSD and experiment
471	<	around 290 K; however, it deviates at both higher and lower
472	<	temperatures, failing to predict the experimental trend. TIP5P and SSD
473	<	both start off low at the colder temperatures and tend to diffuse too
474	<	rapidly at the higher temperatures. This type of trend at the higher
475	<	temperatures is not surprising in that the densities of both TIP5P and
476	<	SSD are lower than experimental water at temperatures higher than room
477	<	temperature. 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
485	<	phase behavior beyond the melting point. To locate the melting
486	<	transition, the constant pressure heat capacity (C$_\text{p}$) was
487	<	monitored in each of the simulations. In the melting simulations of
488	<	the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$
489	<	occurs at 245 K, indicating a first order phase transition for the
490	<	melting of these ice crystals. When the reaction field is turned off,
491	<	the melting transition occurs at 235 K.  These melting transitions are
492	<	considerably lower than the experimental value, but this is not
453	<	surprising when considering the simplicity of the SSD model.
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
488	>	particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs
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.
493
494	<	\begin{figure}
495	<	\includegraphics[width=85mm]{fullContours.eps}
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 Fig. \ref{contour}.}
500	>	\label{corrAngle}
501	>	\end{center}
502	>	\end{figure}
503	>
504	>	\begin{figure}
505	>	\begin{center}
506	>	\epsfxsize=6in
507	>	\epsfbox{fullContours.eps}
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	<	\begin{figure}
518	<	\includegraphics[width=45mm]{corrDiag.eps}
519	<	\caption{Two dimensional illustration of the angles involved in the
468	<	correlations observed in figure \ref{contour}.}
469	<	\label{corrAngle}
470	<	\end{figure}
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	<	Additional analysis of the melting phase-transition process was
522	<	performed by using two-dimensional structure and dipole angle
523	<	correlations. Expressions for these correlations are as follows:
524	<
525	<	\begin{multline}
526	<	g_{\text{AB}}(r,\cos\theta) = \\
527	<	\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\ ,
479	<	\end{multline}
480	<	\begin{multline}
481	<	g_{\text{AB}}(r,\cos\omega) = \\
482	<	\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\ ,
483	<	\end{multline}
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|{\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|{\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
504	<	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 parts that
553	<	blend together to form one observable peak. At higher temperatures,
554	<	this split character alters to show the leading 4 \AA\ peak dominated
555	<	by equatorial anti-parallel dipole orientations, and there is tightly
556	<	bunched group of axially arranged dipoles that most likely consist of
557	<	the smaller fraction aligned dipole pairs. The trailing part of the
558	<	split peak at 5 \AA\ is dominated by aligned dipoles that range
559	<	primarily within the axial to the chief hydrogen bond arrangements
560	<	similar to those seen in the first solvation shell. This evidence
561	<	indicates that the dipole pair interaction begins to dominate outside
562	<	of the range of the dipolar repulsion term, with the primary
563	<	energetically favorable dipole arrangements populating the region
564	<	immediately outside this repulsion region (around 4 \AA), and
565	<	arrangements that seek to ideally satisfy both the sticky and dipole
566	<	forces locate themselves just beyond this initial buildup (around 5
524	<	\AA).
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
557	>	likely consist of the smaller fraction of aligned dipole pairs. The
558	>	trailing component of the split peak at 5 \AA\ is dominated by aligned
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.  The energetically favorable dipole
563	>	arrangements populate the region immediately outside this repulsion
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	<	because the second solvation shell will still be shifted too far
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 546 | Line 589 | densities while maintaining the accurate transport pro
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 possible parameters for tuning include the $\sigma$ and $\epsilon$
594	>	The parameters available for tuning include the $\sigma$ and $\epsilon$
595		Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
596		attractive and dipole repulsive terms with their respective
597		cutoffs. To alter the attractive and repulsive terms of the sticky
598		potential independently, it is necessary to separate the terms as
599		follows:
600		\begin{equation}
558	–	\begin{split}
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)]\\
562	<	& \quad \ + \frac{\nu_0^\prime}{2}
563	<	[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
564	<	\end{split}
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}
566	–
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. For purposes of the reparameterization, the separation was
608	<	performed, but the final parameters were adjusted so that it is
609	<	unnecessary to separate the terms when implementing the adjusted water
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 both the tetrahedral attractive and dipolar
612	<	repulsive don't share the same lower cutoff ($r_l$) in the newly
613	<	parameterized potentials - soft sticky dipole reaction field (SSD/RF -
614	<	for use with a reaction field) and soft sticky dipole enhanced (SSD/E
615	<	- an attempt to improve the liquid structure in simulations without a
616	<	long-range correction).
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.  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}
620	+	\begin{center}
621		\caption{Parameters for the original and adjusted models}
622		\begin{tabular}{ l  c  c  c  c }
623		\hline \\[-3mm]
624	<	\ \ \ Parameters\ \ \  & \ \ \ SSD$^\dagger$ \ \ \ & \ SSD1$^\ddagger$\ \  & \ 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 592 | Line 633 | long-range correction).
633		\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
634		\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
635		\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
595	–	\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96}
596	–	\\$^\ddagger$ ref. \onlinecite{Ichiye03}
636		\end{tabular}
637		\label{params}
638	+	\end{center}
639		\end{table}
640
641	<	\begin{figure}
642	<	\includegraphics[width=85mm]{GofRCompare.epsi}
643	<	\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E
641	>	\begin{figure}
642	>	\begin{center}
643	>	\epsfxsize=5in
644	>	\epsfbox{GofRCompare.epsi}
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. Solid Line - experiment, dashed line - SSD/E
649	<	and SSD/RF, and dotted line - SSD1 (with and without reaction field).}
648	>	first peaks in detail. Note how the changes in parameters have lowered
649	>	and broadened the first peak of SSD/E and SSD/RF.}
650		\label{grcompare}
651	+	\end{center}
652		\end{figure}
653
654	<	\begin{figure}
655	<	\includegraphics[width=85mm]{dualsticky.ps}
654	>	\begin{figure}
655	>	\begin{center}
656	>	\epsfxsize=6in
657	>	\epsfbox{dualsticky.ps}
658		\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \&
659		SSD/RF (right). Light areas correspond to the tetrahedral attractive
660	<	part, and the darker areas correspond to the dipolar repulsive part.}
660	>	component, and darker areas correspond to the dipolar repulsive
661	>	component.}
662		\label{isosurface}
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 sharp first peak that
669	<	integrated to give similar coordination numbers to the experimental
670	<	data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New
671	<	experimental x-ray scattering data from the Head-Gordon lab indicates
672	<	a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so
673	<	adjustments to SSD were made while taking into consideration the new
674	<	experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare}
675	<	shows the relocation of the first peak of the oxygen-oxygen
676	<	g(\emph{r}) by comparing the revised SSD model (SSD1), SSD-E, and
677	<	SSD-RF to the new experimental results. Both the modified water models
678	<	have shorter peaks that are brought in more closely to the
679	<	experimental peak (as seen in the insets of figure \ref{grcompare}).
680	<	This structural alteration was accomplished by the combined reduction
681	<	in the Lennard-Jones $\sigma$ variable and adjustment of the sticky
682	<	potential strength and cutoffs. As can be seen in table \ref{params},
683	<	the cutoffs for the tetrahedral attractive and dipolar repulsive terms
684	<	were nearly swapped with each other. Isosurfaces of the original and
685	<	modified sticky potentials are shown in figure \cite{isosurface}. In
686	<	these isosurfaces, it is easy to see how altering the cutoffs changes
687	<	the repulsive and attractive character of the particles. With a
688	<	reduced repulsive surface (the darker region), the particles can move
689	<	closer to one another, increasing the density for the overall
690	<	system. This change in interaction cutoff also results in a more
691	<	gradual orientational motion by allowing the particles to maintain
692	<	preferred dipolar arrangements before they begin to feel the pull of
693	<	the tetrahedral restructuring. Upon moving closer together, the
694	<	dipolar repulsion term becomes active and excludes unphysical
695	<	nearest-neighbor arrangements. This compares with how SSD and SSD1
696	<	exclude preferred dipole alignments before the particles feel the pull
697	<	of the ``hydrogen bonds''. Aside from improving the shape of the first
698	<	peak in the g(\emph{r}), this improves the densities considerably by
699	<	allowing the persistence of full dipolar character below the previous
700	<	4.0 \AA\ cutoff.
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 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 $g(r)$ 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 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 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 finish bringing up the densities, the dipole
707	<	moments were increased in both the adjusted models. Being 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, so there is quite a range of available
716	<	values for a reasonable dipole
717	<	moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of
718	<	the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF
719	<	respectively is moderate in this range; however, it leads to
672	<	significant changes in the density and transport of the water models.
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.
720
721		In order to demonstrate the benefits of these reparameterizations, a
722		series of NPT and NVE simulations were performed to probe the density
#	Line 677 | Line 724 | results come from five separate simulations of 1024 pa
724		to the original SSD model. This comparison involved full NPT melting
725		sequences for both SSD/E and SSD/RF, as well as NVE transport
726		calculations at the calculated self-consistent densities. Again, the
727	<	results come from five separate simulations of 1024 particle systems,
728	<	and the melting sequences were started from different ice $I_h$
729	<	crystals constructed as stated earlier. Like before, each NPT
727	>	results are obtained from five separate simulations of 1024 particle
728	>	systems, and the melting sequences were started from different ice
729	>	$I_h$ crystals constructed as described previously. Each NPT
730		simulation was equilibrated for 100 ps before a 200 ps data collection
731	<	run at each temperature step, and they used the final configuration
732	<	from the previous temperature simulation as a starting point. All of
733	<	the NVE simulations had the same thermalization, equilibration, and
734	<	data collection times stated earlier in this paper.
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 previously.
735
736	<	\begin{figure}
737	<	\includegraphics[width=62mm, angle=-90]{ssdeDense.epsi}
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
752	<	in comparison to SSD1 without a reaction field, experiment, and other
753	<	common water models. The calculated densities for both SSD/E and SSD1
754	<	have increased significantly over the original SSD model (see figure
755	<	\ref{dense1} and are in significantly better agreement with the
756	<	experimental values. At 298 K, the density of SSD/E and SSD1 without a
757	<	long-range correction are 0.996$\pm$0.001 g/cm$^3$ and 0.999$\pm$0.001
758	<	g/cm$^3$ respectively.  These both compare well with the experimental
759	<	value of 0.997 g/cm$^3$, and they are considerably better than the SSD
760	<	value of 0.967$\pm$0.003 g/cm$^3$. The changes to the dipole moment
761	<	and sticky switching functions have improved the structuring of the
762	<	liquid (as seen in figure \ref{grcompare}, but they have shifted the
763	<	density maximum to much lower temperatures. This comes about via an
764	<	increase of the liquid disorder through the weakening of the sticky
765	<	potential and strengthening of the dipolar character. However, this
766	<	increasing disorder in the SSD/E model has little affect on the
767	<	melting transition. By monitoring C$\text{p}$ throughout these
768	<	simulations, the melting transition for SSD/E occurred at 235 K, the
769	<	same transition temperature observed with SSD and SSD1.
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 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
759	>	the experimental value of 0.997 g/cm$^3$, and they are considerably
760	>	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to
761	>	the dipole moment and sticky switching functions have improved the
762	>	structuring of the liquid (as seen in figure \ref{grcompare}, but they
763	>	have shifted the density maximum to much lower temperatures. This
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 $C_p$
768	>	throughout these simulations, the melting transition for SSD/E was
769	>	shown to occur at 235 K.  The same transition temperature observed
770	>	with SSD and SSD1.
771
772	<	\begin{figure}
773	<	\includegraphics[width=62mm, angle=-90]{ssdrfDense.epsi}
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 results in a more
788	<	interesting comparison. A density profile including SSD/RF and SSD1
789	<	with an active reaction field is shown in figure \ref{ssdrfdense}.  As
790	<	observed in the simulations without a reaction field, the densities of
791	<	SSD/RF and SSD1 show a dramatic increase over normal SSD (see figure
792	<	\ref{dense1}). At 298 K, SSD/RF has a density of 0.997$\pm$0.001
793	<	g/cm$^3$, right in line with experiment and considerably better than
794	<	the SSD value of 0.941$\pm$0.001 g/cm$^3$ and the SSD1 value of
795	<	0.972$\pm$0.002 g/cm$^3$. These results further emphasize the
796	<	importance of reparameterization in order to model the density
797	<	properly under different simulation conditions. Again, these changes
798	<	don't have that profound an effect on the melting point which is
799	<	observed at 245 K for SSD/RF, identical to SSD and only 5 K lower than
800	<	SSD1 with a reaction field. However, the difference in density maxima
801	<	is not quite as extreme with SSD/RF showing a density maximum at 255
802	<	K, fairly close to 260 and 265 K, the density maxima for SSD and SSD1
803	<	respectively.
787	>	Including the reaction field long-range correction in the simulations
788	>	results in a more interesting comparison.  A density profile including
789	>	SSD/RF and SSD1 with an active reaction field is shown in figure
790	>	\ref{ssdrfdense}.  As observed in the simulations without a reaction
791	>	field, the densities of SSD/RF and SSD1 show a dramatic increase over
792	>	normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
793	>	of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
794	>	considerably better than the original SSD value of 0.941$\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	<	\includegraphics[width=65mm, angle=-90]{ssdeDiffuse.epsi}
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 results are
811	<	from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The
812	<	NVE calculations were performed at the average densities observed in
813	<	the 1 atm NPT simulations for the respective models. SSD/E is
814	<	slightly more fluid than experiment at all of the temperatures, but
815	<	it is closer than SSD1 without a 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}
819
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. In the upper plot, the diffusion constant for SSD/E is
829	<	consistently a little faster than experiment, while SSD1 remains
830	<	slower than experiment until relatively high temperatures (greater
831	<	than 330 K). Both models follow the shape of the experimental trend
832	<	well below 300 K, but the trend leans toward diffusing too rapidly at
833	<	higher temperatures, something that is especially apparent with
834	<	SSD1. This accelerated increasing of diffusion is caused by the
835	<	rapidly decreasing system density with increasing temperature. Though
836	<	it is difficult to see in figure \ref{ssdedense}, the densities of SSD1
837	<	decay more rapidly with temperature than do those of SSD/E, leading to
838	<	more visible deviation from the experimental diffusion trend. Thus,
839	<	the changes made to improve the liquid structure may have had an
840	<	adverse affect on the density maximum, but they improve the transport
782	<	behavior of the water model.
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 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	<	\includegraphics[width=65mm, angle=-90]{ssdrfDiffuse.epsi}
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 results
848	<	from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The
849	<	NVE calculations were performed at the average densities observed in
850	<	the 1 atm NPT simulations for both of the models. Note how accurately
851	<	SSD/RF simulates the diffusion of water throughout this temperature
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 with 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 only showing a slight
860	>	compared to SSD1 with an active reaction field. Note that SSD/RF
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 was stated in the SSD/E
866	<	comparisons, this deviation away from the ideal trend is due to a
867	<	rapid decrease in density at higher temperatures. SSD/RF doesn't
868	<	suffer from this problem as much as SSD1, because the calculated
869	<	densities are more true to experiment. These results again emphasize
870	<	the importance of careful reparameterization when using an altered
810	<	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
874		\begin{figure}
875	<	\includegraphics[width=85mm]{povIce.ps}
876	<	\caption{A water lattice built from the crystal structure that SSD/E
877	<	assumed when undergoing an extremely restricted temperature NPT
878	<	simulation. This form of ice is referred to as ice 0 to emphasize its
879	<	simulation origins. This image was taken of the (001) face of the
880	<	crystal.}
875	>	\begin{center}
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-{\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	<	discussed earlier in this paper, some interesting observations were
889	<	made. After melting at 235 K, two of five systems underwent
890	<	crystallization events near 245 K. As the heating process continued,
891	<	the two systems remained crystalline until finally melting between 320
892	<	and 330 K. The final configurations of these two melting sequences
893	<	show an expanded zeolite-like crystal structure that does not
894	<	correspond to any known form of ice. For convenience and to help
895	<	distinguish it from the experimentally observed forms of ice, this
896	<	crystal structure will henceforth be referred to as ice-zero (ice
897	<	0). The crystallinity was extensive enough that a near ideal crystal
898	<	structure could be obtained. Figure \ref{weirdice} shows the repeating
899	<	crystal structure of a typical crystal at 5 K. Each water molecule is
900	<	hydrogen bonded to four others; however, the hydrogen bonds are flexed
901	<	rather than perfectly straight. This results in a skewed tetrahedral
902	<	geometry about the central molecule. Looking back at figure
903	<	\ref{isosurface}, it is easy to see how these flexed hydrogen bonds
904	<	are allowed in that the attractive regions are conical in shape, with
905	<	the greatest attraction in the central region. Though not ideal, these
906	<	flexed hydrogen bonds are favorable enough to stabilize an entire
907	<	crystal generated around them. In fact, the imperfect ice 0 crystals
845	<	were so stable that they melted at temperatures nearly 100 K greater
846	<	than both ice I$_c$ and I$_h$.
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 them.
908
909	<	These initial simulations indicated that ice 0 is the preferred ice
910	<	structure for at least SSD/E. To verify this, a comparison was made
911	<	between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at
912	<	constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal versions of
913	<	the three types of crystals were cooled to 1 K, and the potential
914	<	energies of each were compared using all three water models. With
915	<	every water model, ice 0 turned out to have the lowest potential
916	<	energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with SSD/E, and
917	<	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 0 as the initial configuration using SSD/E,
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	>	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 a reaction field occurred
942	<	at temperature in excess of 375 K. SSD/RF and SSD1 with a reaction
943	<	field had more reasonable melting transitions, down near 325 K. These
944	<	melting point observations emphasize how preferred this crystal
945	<	structure is over the most common types of ice when using these single
866	<	point water models.
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
868	–	Recognizing that the above tests show ice 0 to be both the most stable
869	–	and lowest density crystal structure for these single point water
870	–	models, it is interesting to speculate on the relative stability of
871	–	this crystal structure with charge based water models. As a quick
872	–	test, these 3 crystal types were converted from SSD type particles to
873	–	TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
874	–	minimizations were performed on all of these crystals to compare the
875	–	system energies. Again, ice 0 was observed to have the lowest total
876	–	system energy. The total energy of ice 0 was ~2\% lower than ice
877	–	$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial
878	–	results, we would not be surprised if results from the other common
879	–	water models show ice 0 to be the lowest energy crystal structure. A
880	–	continuation on work studying ice 0 with multi-point water models will
881	–	be published in a coming article.
882	–
947		\section{Conclusions}
884	–	The density maximum and temperature dependent transport for the SSD
885	–	water model, both with and without the use of reaction field, were
886	–	studied via a series of NPT and NVE simulations. The constant pressure
887	–	simulations of the melting of both $I_h$ and $I_c$ ice showed a
888	–	density maximum near 260 K. In most cases, the calculated densities
889	–	were significantly lower than the densities calculated in simulations
890	–	of other water models. Analysis of particle diffusion showed SSD to
891	–	capture the transport properties of experimental very well in both the
892	–	normal and super-cooled liquid regimes. In order to correct the
893	–	density behavior, the original SSD model was reparameterized for use
894	–	both with and without a reaction field (SSD/RF and SSD/E), and
895	–	comparison simulations were performed with SSD1, the density corrected
896	–	version of SSD. Both models improve the liquid structure, density
897	–	values, and diffusive properties under their respective conditions,
898	–	indicating the necessity of reparameterization when altering the
899	–	long-range correction specifics. When taking the appropriate
900	–	considerations, these simple water models are excellent choices for
901	–	representing explicit water in large scale simulations of biochemical
902	–	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
994	<	\bibliographystyle{jcp}
994	>	\newpage
995
996	+	\bibliographystyle{jcp}
997		\bibliography{nptSSD}
998
999		%\pagebreak

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