ViewVC Help

View File | Revision Log | Show Annotations | View Changeset | Root Listing

root/group/trunk/ssdePaper/nptSSD.tex

Comparing trunk/ssdePaper/nptSSD.tex (file contents):
Revision 863 by chrisfen, Thu Nov 13 15:55:20 2003 UTC vs.
Revision 1033 by chrisfen, Fri Feb 6 19:15:44 2004 UTC

#	Line 1 | Line 1
1		%\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4}
2	<	\documentclass[11pt]{article}
3	<	\usepackage{endfloat}
2	>	\documentclass[preprint,aps]{revtex4}
3	>	%\documentclass[11pt]{article}
4	>	%\usepackage{endfloat}
5		\usepackage{amsmath}
6		\usepackage{epsf}
7		\usepackage{berkeley}
8		\usepackage{setspace}
9		\usepackage{tabularx}
10		\usepackage{graphicx}
11	<	\usepackage[ref]{overcite}
11	<	%\usepackage{berkeley}
11	>	%\usepackage[ref]{overcite}
12		%\usepackage{curves}
13	<	\pagestyle{plain}
14	<	\pagenumbering{arabic}
15	<	\oddsidemargin 0.0cm \evensidemargin 0.0cm
16	<	\topmargin -21pt \headsep 10pt
17	<	\textheight 9.0in \textwidth 6.5in
18	<	\brokenpenalty=10000
19	<	\renewcommand{\baselinestretch}{1.2}
20	<	\renewcommand\citemid{\ } % no comma in optional reference note
13	>	%\pagestyle{plain}
14	>	%\pagenumbering{arabic}
15	>	%\oddsidemargin 0.0cm \evensidemargin 0.0cm
16	>	%\topmargin -21pt \headsep 10pt
17	>	%\textheight 9.0in \textwidth 6.5in
18	>	%\brokenpenalty=10000
19	>	%\renewcommand{\baselinestretch}{1.2}
20	>	%\renewcommand\citemid{\ } % no comma in optional reference note
21	>	\newcounter{captions}
22	>	\newcounter{figs}
23
24		\begin{document}
25
26	<	\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models}
26	>	\title{On the structural and transport properties of the soft sticky
27	>	dipole (SSD) and related single point water models}
28
29	<	\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
30	<	Department of Chemistry and Biochemistry\\ University of Notre Dame\\
29	>	\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
30	>
31	>	\affiliation{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
32		Notre Dame, Indiana 46556}
33
34		\date{\today}
35
32	–	\maketitle
36
37		\begin{abstract}
38	<	NVE and NPT molecular dynamics simulations were performed in order to
39	<	investigate the density maximum and temperature dependent transport
40	<	for SSD and related water models, both with and without the use of
41	<	reaction field. The constant pressure simulations of the melting of
42	<	both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most
43	<	cases, the calculated densities were significantly lower than the
44	<	densities calculated in simulations of other water models. Analysis of
45	<	particle diffusion showed SSD to capture the transport properties of
38	>	The density maximum and temperature dependence of the self-diffusion
39	>	constant were investigated for the soft sticky dipole (SSD) water
40	>	model and two related reparameterizations of this single-point model.
41	>	A combination of microcanonical and isobaric-isothermal molecular
42	>	dynamics simulations were used to calculate these properties, both
43	>	with and without the use of reaction field to handle long-range
44	>	electrostatics.  The isobaric-isothermal (NPT) simulations of the
45	>	melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near
46	>	260~K.  In most cases, the use of the reaction field resulted in
47	>	calculated densities which were were significantly lower than
48	>	experimental densities.  Analysis of self-diffusion constants shows
49	>	that the original SSD model captures the transport properties of
50		experimental water very well in both the normal and super-cooled
51	<	liquid regimes. In order to correct the density behavior, SSD was
52	<	reparameterized for use both with and without a long-range interaction
53	<	correction, SSD/RF and SSD/E respectively. Compared to the density
54	<	corrected version of SSD (SSD1), these modified models were shown to
55	<	maintain or improve upon the structural and transport properties.
51	>	liquid regimes.  We also present our reparameterized versions of SSD
52	>	for use both with the reaction field or without any long-range
53	>	electrostatic corrections.  These are called the SSD/RF and SSD/E
54	>	models respectively.  These modified models were shown to maintain or
55	>	improve upon the experimental agreement with the structural and
56	>	transport properties that can be obtained with either the original SSD
57	>	or the density corrected version of the original model (SSD1).
58	>	Additionally, a novel low-density ice structure is presented
59	>	which appears to be the most stable ice structure for the entire SSD
60	>	family.
61		\end{abstract}
62
63	+	\maketitle
64	+
65		\newpage
66
67		%\narrowtext
#	Line 60 | Line 74 | systems is the proper depiction of water and water sol
74		\section{Introduction}
75
76		One of the most important tasks in the simulation of biochemical
77	<	systems is the proper depiction of water and water solvation. In fact,
78	<	the bulk of the calculations performed in solvated simulations are of
79	<	interactions with or between solvent molecules. Thus, the outcomes of
80	<	these types of simulations are highly dependent on the physical
81	<	properties of water, both as individual molecules and in clusters or
82	<	bulk. Due to the fact that explicit solvent accounts for a massive
83	<	portion of the calculations, it necessary to simplify the solvent to
70	<	some extent in order to complete simulations in a reasonable amount of
71	<	time. In the case of simulating water in biomolecular studies, the
72	<	balance between accurate properties and computational efficiency is
73	<	especially delicate, and it has resulted in a variety of different
74	<	water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these
75	<	models predict specific properties more accurately than their
76	<	predecessors, but often at the cost of other properties or of computer
77	<	time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves
78	<	upon the structural and transport properties of water relative to the
79	<	previous TIP models, yet this comes at a greater than 50\% increase in
80	<	computational cost.\cite{Jorgensen01,Jorgensen00} One recently
81	<	developed model that succeeds in both retaining the accuracy of system
82	<	properties and simplifying calculations to increase computational
83	<	efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96}
77	>	systems is the proper depiction of the aqueous environment of the
78	>	molecules of interest.  In some cases (such as in the simulation of
79	>	phospholipid bilayers), the majority of the calculations that are
80	>	performed involve interactions with or between solvent molecules.
81	>	Thus, the properties one may observe in biochemical simulations are
82	>	going to be highly dependent on the physical properties of the water
83	>	model that is chosen.
84
85	<	The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye
86	<	\emph{et al.} as a modified form of the hard-sphere water model
87	<	proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD
88	<	consists of a single point dipole with a Lennard-Jones core and a
89	<	sticky potential that directs the particles to assume the proper
90	<	hydrogen bond orientation in the first solvation shell. Thus, the
91	<	interaction between two SSD water molecules \emph{i} and \emph{j} is
92	<	given by the potential
85	>	There is an especially delicate balance between computational
86	>	efficiency and the ability of the water model to accurately predict
87	>	the properties of bulk
88	>	water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the
89	>	TIP5P model improves on the structural and transport properties of
90	>	water relative to the previous TIP models, yet this comes at a greater
91	>	than 50\% increase in computational
92	>	cost.\cite{Jorgensen01,Jorgensen00}
93	>
94	>	One recently developed model that largely succeeds in retaining the
95	>	accuracy of bulk properties while greatly reducing the computational
96	>	cost is the Soft Sticky Dipole (SSD) water
97	>	model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model
98	>	was developed by Ichiye \emph{et al.} as a modified form of the
99	>	hard-sphere water model proposed by Bratko, Blum, and
100	>	Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model
101	>	which has an interaction site that is both a point dipole and a
102	>	Lennard-Jones core.  However, since the normal aligned and
103	>	anti-aligned geometries favored by point dipoles are poor mimics of
104	>	local structure in liquid water, a short ranged ``sticky'' potential
105	>	is also added.  The sticky potential directs the molecules to assume
106	>	the proper hydrogen bond orientation in the first solvation shell.
107	>
108	>	The interaction between two SSD water molecules \emph{i} and \emph{j}
109	>	is given by the potential
110		\begin{equation}
111		u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
112	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
112	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
113		u_{ij}^{sp}
114	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
114	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
115		\end{equation}
116	<	where the $\mathbf{r}_{ij}$ is the position vector between molecules
117	<	\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and
118	<	$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
119	<	orientations of the respective molecules. The Lennard-Jones, dipole,
120	<	and sticky parts of the potential are giving by the following
104	<	equations:
116	>	where the ${\bf r}_{ij}$ is the position vector between molecules
117	>	\emph{i} and \emph{j} with magnitude $r_{ij}$, and
118	>	${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of
119	>	the two molecules. The Lennard-Jones and dipole interactions are given
120	>	by the following familiar forms:
121		\begin{equation}
122	<	u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
122	>	u_{ij}^{LJ}(r_{ij}) = 4\epsilon
123	>	\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]
124	>	\ ,
125		\end{equation}
126	+	and
127		\begin{equation}
128	<	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}\ ,
128	>	u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left(
129	>	\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf
130	>	r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ ,
131		\end{equation}
132	+	where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along
133	+	the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and
134	+	$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf
135	+	r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule
136	+	$i$.
137	+
138	+	The sticky potential is somewhat less familiar:
139		\begin{equation}
140		u_{ij}^{sp}
141	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
142	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
141	>	({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
142	>	\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)
143	>	+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf
144	>	\Omega}_j)]\ .
145	>	\label{stickyfunction}
146		\end{equation}
147	<	where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
148	<	unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
149	<	$\nu_0$ scales the strength of the overall sticky potential, and $s$
150	<	and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
151	<	functions take the following forms:
147	>	Here, $\nu_0$ is a strength parameter for the sticky potential, and
148	>	$s$ and $s^\prime$ are cubic switching functions which turn off the
149	>	sticky interaction beyond the first solvation shell. The $w$ function
150	>	can be thought of as an attractive potential with tetrahedral
151	>	geometry:
152		\begin{equation}
153	<	w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
153	>	w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
154		\end{equation}
155	+	while the $w^\prime$ function counters the normal aligned and
156	+	anti-aligned structures favored by point dipoles:
157		\begin{equation}
158	<	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,
158	>	w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ,
159		\end{equation}
160	<	where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
161	<	term that promotes hydrogen bonding orientations within the first
162	<	solvation shell, and $w^\prime$ is a dipolar repulsion term that
163	<	repels unrealistic dipolar arrangements within the first solvation
164	<	shell. A more detailed description of the functional parts and
165	<	variables in this potential can be found in other
166	<	articles.\cite{Ichiye96,Ichiye99}
160	>	It should be noted that $w$ is proportional to the sum of the $Y_3^2$
161	>	and $Y_3^{-2}$ spherical harmonics (a linear combination which
162	>	enhances the tetrahedral geometry for hydrogen bonded structures),
163	>	while $w^\prime$ is a purely empirical function.  A more detailed
164	>	description of the functional parts and variables in this potential
165	>	can be found in the original SSD
166	>	articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
167
168	<	Being that this is a one-site point dipole model, the actual force
169	<	calculations are simplified significantly. In the original Monte Carlo
170	<	simulations using this model, Ichiye \emph{et al.} reported an
171	<	increase in calculation efficiency of up to an order of magnitude over
172	<	other comparable models, while maintaining the structural behavior of
173	<	water.\cite{Ichiye96} In the original molecular dynamics studies, it
174	<	was shown that SSD improves on the prediction of many of water's
175	<	dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This
176	<	attractive combination of speed and accurate depiction of solvent
177	<	properties makes SSD a model of interest for the simulation of large
178	<	scale biological systems, such as membrane phase behavior.
168	>	Since SSD is a single-point {\it dipolar} model, the force
169	>	calculations are simplified significantly relative to the standard
170	>	{\it charged} multi-point models. In the original Monte Carlo
171	>	simulations using this model, Liu and Ichiye reported that using SSD
172	>	decreased computer time by a factor of 6-7 compared to other
173	>	models.\cite{Ichiye96} What is most impressive is that this savings
174	>	did not come at the expense of accurate depiction of the liquid state
175	>	properties.  Indeed, SSD maintains reasonable agreement with the Soper
176	>	data for the structural features of liquid
177	>	water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
178	>	exhibited by SSD agree with experiment better than those of more
179	>	computationally expensive models (like TIP3P and
180	>	SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
181	>	of solvent properties makes SSD a very attractive model for the
182	>	simulation of large scale biochemical simulations.
183
184	<	One of the key limitations of this water model, however, is that it
185	<	has been parameterized for use with the Ewald Sum technique for the
186	<	handling of long-ranged interactions.  When studying very large
187	<	systems, the Ewald summation and even particle-mesh Ewald become
188	<	computational burdens, with their respective ideal $N^\frac{3}{2}$ and
189	<	$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99}
190	<	In applying this water model in these types of systems, it would be
191	<	useful to know its properties and behavior with the more
192	<	computationally efficient reaction field (RF) technique, and even with
193	<	a cutoff that lacks any form of long-range correction. This study
194	<	addresses these issues by looking at the structural and transport
195	<	behavior of SSD over a variety of temperatures with the purpose of
196	<	utilizing the RF correction technique. We then suggest alterations to
197	<	the parameters that result in more water-like behavior. It should be
198	<	noted that in a recent publication, some of the original investigators of
199	<	the SSD water model have put forth adjustments to the SSD water model
200	<	to address abnormal density behavior (also observed here), calling the
201	<	corrected model SSD1.\cite{Ichiye03} This study will make comparisons
202	<	with SSD1's behavior with the goal of improving upon the
203	<	depiction of water under conditions without the Ewald Sum.
184	>	One feature of the SSD model is that it was parameterized for
185	>	use with the Ewald sum to handle long-range interactions.  This would
186	>	normally be the best way of handling long-range interactions in
187	>	systems that contain other point charges.  However, our group has
188	>	recently become interested in systems with point dipoles as mimics for
189	>	neutral, but polarized regions on molecules (e.g. the zwitterionic
190	>	head group regions of phospholipids).  If the system of interest does
191	>	not contain point charges, the Ewald sum and even particle-mesh Ewald
192	>	become computational bottlenecks.  Their respective ideal
193	>	$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$
194	>	particles can become prohibitive when $N$ becomes
195	>	large.\cite{Darden99} In applying this water model in these types of
196	>	systems, it would be useful to know its properties and behavior under
197	>	the more computationally efficient reaction field (RF) technique, or
198	>	even with a simple cutoff. This study addresses these issues by
199	>	looking at the structural and transport behavior of SSD over a
200	>	variety of temperatures with the purpose of utilizing the RF
201	>	correction technique.  We then suggest modifications to the parameters
202	>	that result in more realistic bulk phase behavior.  It should be noted
203	>	that in a recent publication, some of the original investigators of
204	>	the SSD water model have suggested adjustments to the SSD
205	>	water model to address abnormal density behavior (also observed here),
206	>	calling the corrected model SSD1.\cite{Ichiye03} In what
207	>	follows, we compare our reparamaterization of SSD with both the
208	>	original SSD and SSD1 models with the goal of improving
209	>	the bulk phase behavior of an SSD-derived model in simulations
210	>	utilizing the reaction field.
211
212		\section{Methods}
213
214	<	As stated previously, the long-range dipole-dipole interactions were
215	<	accounted for in this study by using the reaction field method. The
216	<	magnitude of the reaction field acting on dipole \emph{i} is given by
214	>	Long-range dipole-dipole interactions were accounted for in this study
215	>	by using either the reaction field technique or by resorting to a
216	>	simple cubic switching function at a cutoff radius.  One of the early
217	>	applications of a reaction field was actually in Monte Carlo
218	>	simulations of liquid water.\cite{Barker73} Under this method, the
219	>	magnitude of the reaction field acting on dipole $i$ is
220		\begin{equation}
221		\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
222	<	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\  ,
222	>	\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}),
223		\label{rfequation}
224		\end{equation}
225		where $\mathcal{R}$ is the cavity defined by the cutoff radius
226		($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
227	<	system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment
228	<	vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching
227	>	system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole
228	>	moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching
229		function.\cite{AllenTildesley} The reaction field contribution to the
230	<	total energy by particle \emph{i} is given by
231	<	$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
232	<	on dipole \emph{i} by
233	<	$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use
234	<	of reaction field is known to alter the orientational dynamic
235	<	properties, such as the dielectric relaxation time, based on changes
236	<	in the length of the cutoff radius.\cite{Berendsen98} This variable
237	<	behavior makes reaction field a less attractive method than other
238	<	methods, like the Ewald summation; however, for the simulation of
239	<	large-scale systems, the computational cost benefit of reaction field
240	<	is dramatic. To address some of the dynamical property alterations due
241	<	to the use of reaction field, simulations were also performed without
242	<	a surrounding dielectric and suggestions are presented on how to make
243	<	SSD more accurate both with and without a reaction field.
230	>	total energy by particle $i$ is given by $-\frac{1}{2}{\bf
231	>	\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf
232	>	\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley}  Use of the reaction
233	>	field is known to alter the bulk orientational properties of simulated
234	>	water, and there is particular sensitivity of these properties on
235	>	changes in the length of the cutoff radius.\cite{Berendsen98} This
236	>	variable behavior makes reaction field a less attractive method than
237	>	the Ewald sum.  However, for very large systems, the computational
238	>	benefit of reaction field is dramatic.
239	>
240	>	We have also performed a companion set of simulations {\it without} a
241	>	surrounding dielectric (i.e. using a simple cubic switching function
242	>	at the cutoff radius), and as a result we have two reparamaterizations
243	>	of SSD which could be used either with or without the reaction
244	>	field turned on.
245
246	<	Simulations were performed in both the isobaric-isothermal and
247	<	microcanonical ensembles. The constant pressure simulations were
246	>	Simulations to obtain the preferred densities of the models were
247	>	performed in the isobaric-isothermal (NPT) ensemble, while all
248	>	dynamical properties were obtained from microcanonical (NVE)
249	>	simulations done at densities matching the NPT density for a
250	>	particular target temperature.  The constant pressure simulations were
251		implemented using an integral thermostat and barostat as outlined by
252	<	Hoover.\cite{Hoover85,Hoover86} All particles were treated as
252	>	Hoover.\cite{Hoover85,Hoover86} All molecules were treated as
253		non-linear rigid bodies. Vibrational constraints are not necessary in
254	<	simulations of SSD, because there are no explicit hydrogen atoms, and
255	<	thus no molecular vibrational modes need to be considered.
254	>	simulations of SSD, because there are no explicit hydrogen
255	>	atoms, and thus no molecular vibrational modes need to be considered.
256
257		Integration of the equations of motion was carried out using the
258	<	symplectic splitting method proposed by Dullweber \emph{et
259	<	al.}\cite{Dullweber1997} The reason for this integrator selection
260	<	deals with poor energy conservation of rigid body systems using
261	<	quaternions. While quaternions work well for orientational motion in
262	<	alternate ensembles, the microcanonical ensemble has a constant energy
263	<	requirement that is quite sensitive to errors in the equations of
264	<	motion. The original implementation of this code utilized quaternions
265	<	for rotational motion propagation; however, a detailed investigation
266	<	showed that they resulted in a steady drift in the total energy,
216	<	something that has been observed by others.\cite{Laird97}
258	>	symplectic splitting method proposed by Dullweber, Leimkuhler, and
259	>	McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting
260	>	this integrator centers on poor energy conservation of rigid body
261	>	dynamics using traditional quaternion
262	>	integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble
263	>	simulations, the energy drift when using quaternions was substantially
264	>	greater than when using the {\sc dlm} method (fig. \ref{timestep}).
265	>	This steady drift in the total energy has also been observed by Kol
266	>	{\it et al.}\cite{Laird97}
267
268		The key difference in the integration method proposed by Dullweber
269		\emph{et al.} is that the entire rotation matrix is propagated from
270	<	one time step to the next. In the past, this would not have been as
271	<	feasible an option, being that the rotation matrix for a single body is
272	<	nine elements long as opposed to 3 or 4 elements for Euler angles and
273	<	quaternions respectively. System memory has become much less of an
224	<	issue in recent times, and this has resulted in substantial benefits
225	<	in energy conservation. There is still the issue of 5 or 6 additional
226	<	elements for describing the orientation of each particle, which will
227	<	increase dump files substantially. Simply translating the rotation
228	<	matrix into its component Euler angles or quaternions for storage
229	<	purposes relieves this burden.
270	>	one time step to the next.  The additional memory required by the
271	>	algorithm is inconsequential on modern computers, and translating the
272	>	rotation matrix into quaternions for storage purposes makes trajectory
273	>	data quite compact.
274
275	<	The symplectic splitting method allows for Verlet style integration of
276	<	both linear and angular motion of rigid bodies. In this integration
277	<	method, the orientational propagation involves a sequence of matrix
278	<	evaluations to update the rotation matrix.\cite{Dullweber1997} These
279	<	matrix rotations are more costly computationally than the simpler
280	<	arithmetic quaternion propagation. With the same time step, a 1000 SSD
281	<	particle simulation shows an average 7\% increase in computation time
282	<	using the symplectic step method in place of quaternions. This cost is
283	<	more than justified when comparing the energy conservation of the two
284	<	methods as illustrated in figure \ref{timestep}.
275	>	The {\sc dlm} method allows for Verlet style integration of both
276	>	translational and orientational motion of rigid bodies. In this
277	>	integration method, the orientational propagation involves a sequence
278	>	of matrix evaluations to update the rotation
279	>	matrix.\cite{Dullweber1997} These matrix rotations are more costly
280	>	than the simpler arithmetic quaternion propagation. With the same time
281	>	step, a 1000 SSD particle simulation shows an average 7\%
282	>	increase in computation time using the {\sc dlm} method in place of
283	>	quaternions. The additional expense per step is justified when one
284	>	considers the ability to use time steps that are nearly twice as large
285	>	under {\sc dlm} than would be usable under quaternion dynamics.  The
286	>	energy conservation of the two methods using a number of different
287	>	time steps is illustrated in figure
288	>	\ref{timestep}.
289
290	<	\begin{figure}
291	<	\begin{center}
292	<	\epsfxsize=6in
293	<	\epsfbox{timeStep.epsi}
294	<	\caption{Energy conservation using quaternion based integration versus
295	<	the symplectic step method proposed by Dullweber \emph{et al.} with
296	<	increasing time step. The larger time step plots are shifted up from
297	<	the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.}
298	<	\label{timestep}
299	<	\end{center}
300	<	\end{figure}
290	>	%\begin{figure}
291	>	%\begin{center}
292	>	%\epsfxsize=6in
293	>	%\epsfbox{timeStep.epsi}
294	>	%\caption{Energy conservation using both quaternion-based integration and
295	>	%the {\sc dlm} method with increasing time step. The larger time step
296	>	%plots are shifted from the true energy baseline (that of $\Delta t$ =
297	>	%0.1~fs) for clarity.}
298	>	%\label{timestep}
299	>	%\end{center}
300	>	%\end{figure}
301
302		In figure \ref{timestep}, the resulting energy drift at various time
303	<	steps for both the symplectic step and quaternion integration schemes
304	<	is compared. All of the 1000 SSD particle simulations started with the
305	<	same configuration, and the only difference was the method used to
306	<	handle rotational motion. At time steps of 0.1 and 0.5 fs, both
307	<	methods for propagating particle rotation conserve energy fairly well,
308	<	with the quaternion method showing a slight energy drift over time in
309	<	the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
310	<	energy conservation benefits of the symplectic step method are clearly
311	<	demonstrated. Thus, while maintaining the same degree of energy
312	<	conservation, one can take considerably longer time steps, leading to
313	<	an overall reduction in computation time.
303	>	steps for both the {\sc dlm} and quaternion integration schemes is
304	>	compared.  All of the 1000 SSD particle simulations started with
305	>	the same configuration, and the only difference was the method used to
306	>	handle orientational motion. At time steps of 0.1 and 0.5~fs, both
307	>	methods for propagating the orientational degrees of freedom conserve
308	>	energy fairly well, with the quaternion method showing a slight energy
309	>	drift over time in the 0.5~fs time step simulation. At time steps of 1
310	>	and 2~fs, the energy conservation benefits of the {\sc dlm} method are
311	>	clearly demonstrated. Thus, while maintaining the same degree of
312	>	energy conservation, one can take considerably longer time steps,
313	>	leading to an overall reduction in computation time.
314
315	<	Energy drift in the symplectic step simulations was unnoticeable for
316	<	time steps up to three femtoseconds. A slight energy drift on the
317	<	order of 0.012 kcal/mol per nanosecond was observed at a time step of
318	<	four femtoseconds, and as expected, this drift increases dramatically
319	<	with increasing time step. To insure accuracy in the constant energy
320	<	simulations, time steps were set at 2 fs and kept at this value for
315	>	Energy drift in the simulations using {\sc dlm} integration was
316	>	unnoticeable for time steps up to 3~fs. A slight energy drift on the
317	>	order of 0.012~kcal/mol per nanosecond was observed at a time step of
318	>	4~fs, and as expected, this drift increases dramatically with
319	>	increasing time step. To insure accuracy in our microcanonical
320	>	simulations, time steps were set at 2~fs and kept at this value for
321		constant pressure simulations as well.
322
323	<	Ice crystals in both the $I_h$ and $I_c$ lattices were generated as
324	<	starting points for all simulations. The $I_h$ crystals were formed by
325	<	first arranging the centers of mass of the SSD particles into a
326	<	``hexagonal'' ice lattice of 1024 particles. Because of the crystal
327	<	structure of $I_h$ ice, the simulation box assumed a rectangular shape
328	<	with an edge length ratio of approximately
323	>	Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices
324	>	were generated as starting points for all simulations. The $I_h$
325	>	crystals were formed by first arranging the centers of mass of the SSD
326	>	particles into a ``hexagonal'' ice lattice of 1024 particles. Because
327	>	of the crystal structure of $I_h$ ice, the simulation box assumed an
328	>	orthorhombic shape with an edge length ratio of approximately
329		1.00$\times$1.06$\times$1.23. The particles were then allowed to
330		orient freely about fixed positions with angular momenta randomized at
331	<	400 K for varying times. The rotational temperature was then scaled
332	<	down in stages to slowly cool the crystals to 25 K. The particles were
331	>	400~K for varying times. The rotational temperature was then scaled
332	>	down in stages to slowly cool the crystals to 25~K. The particles were
333		then allowed to translate with fixed orientations at a constant
334	<	pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were
335	<	removed and the ice crystals were allowed to equilibrate for 50 ps at
336	<	25 K and a constant pressure of 1 atm.  This procedure resulted in
334	>	pressure of 1 atm for 50~ps at 25~K. Finally, all constraints were
335	>	removed and the ice crystals were allowed to equilibrate for 50~ps at
336	>	25~K and a constant pressure of 1~atm.  This procedure resulted in
337		structurally stable $I_h$ ice crystals that obey the Bernal-Fowler
338		rules.\cite{Bernal33,Rahman72} This method was also utilized in the
339		making of diamond lattice $I_c$ ice crystals, with each cubic
#	Line 297 | Line 345 | constant pressure and temperature dynamics. During mel
345		\section{Results and discussion}
346
347		Melting studies were performed on the randomized ice crystals using
348	<	constant pressure and temperature dynamics. During melting
349	<	simulations, the melting transition and the density maximum can both
350	<	be observed, provided that the density maximum occurs in the liquid
351	<	and not the supercooled regime. An ensemble average from five separate
352	<	melting simulations was acquired, each starting from different ice
353	<	crystals generated as described previously. All simulations were
354	<	equilibrated for 100 ps prior to a 200 ps data collection run at each
355	<	temperature setting. The temperature range of study spanned from 25 to
356	<	400 K, with a maximum degree increment of 25 K. For regions of
357	<	interest along this stepwise progression, the temperature increment
358	<	was decreased from 25 K to 10 and 5 K. The above equilibration and
359	<	production times were sufficient in that the system volume
360	<	fluctuations dampened out in all but the very cold simulations (below
313	<	225 K).
348	>	isobaric-isothermal (NPT) dynamics. During melting simulations, the
349	>	melting transition and the density maximum can both be observed,
350	>	provided that the density maximum occurs in the liquid and not the
351	>	supercooled regime. An ensemble average from five separate melting
352	>	simulations was acquired, each starting from different ice crystals
353	>	generated as described previously. All simulations were equilibrated
354	>	for 100~ps prior to a 200~ps data collection run at each temperature
355	>	setting. The temperature range of study spanned from 25 to 400~K, with
356	>	a maximum degree increment of 25~K. For regions of interest along this
357	>	stepwise progression, the temperature increment was decreased from
358	>	25~K to 10 and 5~K.  The above equilibration and production times were
359	>	sufficient in that fluctuations in the volume autocorrelation function
360	>	were damped out in all simulations in under 20~ps.
361
362		\subsection{Density Behavior}
316	–	Initial simulations focused on the original SSD water model, and an
317	–	average density versus temperature plot is shown in figure
318	–	\ref{dense1}. Note that the density maximum when using a reaction
319	–	field appears between 255 and 265 K, where the calculated densities
320	–	within this range were nearly indistinguishable. The greater certainty
321	–	of the average value at 260 K makes a good argument for the actual
322	–	density maximum residing at this midpoint value. Figure \ref{dense1}
323	–	was constructed using ice $I_h$ crystals for the initial
324	–	configuration; though not pictured, the simulations starting from ice
325	–	$I_c$ crystal configurations showed similar results, with a
326	–	liquid-phase density maximum in this same region (between 255 and 260
327	–	K). In addition, the $I_c$ crystals are more fragile than the $I_h$
328	–	crystals, leading to deformation into a dense glassy state at lower
329	–	temperatures. This resulted in an overall low temperature density
330	–	maximum at 200 K, while still retaining a liquid state density maximum
331	–	in common with the $I_h$ simulations.
363
364	<	\begin{figure}
365	<	\begin{center}
366	<	\epsfxsize=6in
367	<	\epsfbox{denseSSD.eps}
368	<	\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b}
369	<	TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction
370	<	Field, SSD, and experiment.\cite{CRC80} The arrows indicate the
371	<	change in densities observed when turning off the reaction field. The
372	<	the lower than expected densities for the SSD model were what
373	<	prompted the original reparameterization.\cite{Ichiye03}}
343	<	\label{dense1}
344	<	\end{center}
345	<	\end{figure}
364	>	Our initial simulations focused on the original SSD water model,
365	>	and an average density versus temperature plot is shown in figure
366	>	\ref{dense1}. Note that the density maximum when using a reaction
367	>	field appears between 255 and 265~K.  There were smaller fluctuations
368	>	in the density at 260~K than at either 255 or 265~K, so we report this
369	>	value as the location of the density maximum. Figure \ref{dense1} was
370	>	constructed using ice $I_h$ crystals for the initial configuration;
371	>	though not pictured, the simulations starting from ice $I_c$ crystal
372	>	configurations showed similar results, with a liquid-phase density
373	>	maximum in this same region (between 255 and 260~K).
374
375	<	The density maximum for SSD actually compares quite favorably to other
375	>	%\begin{figure}
376	>	%\begin{center}
377	>	%\epsfxsize=6in
378	>	%\epsfbox{denseSSDnew.eps}
379	>	%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}],
380	>	% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD
381	>	% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The
382	>	% arrows indicate the change in densities observed when turning off the
383	>	% reaction field. The the lower than expected densities for the SSD
384	>	% model were what prompted the original reparameterization of SSD1
385	>	% [Ref. \onlinecite{Ichiye03}].}
386	>	%\label{dense1}
387	>	%\end{center}
388	>	%\end{figure}
389	>
390	>	The density maximum for SSD compares quite favorably to other
391		simple water models. Figure \ref{dense1} also shows calculated
392		densities of several other models and experiment obtained from other
393		sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
394	<	models, SSD has results closest to the experimentally observed water
395	<	density maximum. Of the listed water models, TIP4P has a density
396	<	maximum behavior most like that seen in SSD. Though not included in
397	<	this particular plot, it is useful to note that TIP5P has a water
398	<	density maximum nearly identical to experiment.
394	>	models, SSD has a temperature closest to the experimentally
395	>	observed density maximum. Of the {\it charge-based} models in
396	>	Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that
397	>	seen in SSD. Though not included in this plot, it is useful to
398	>	note that TIP5P has a density maximum nearly identical to the
399	>	experimentally measured temperature.
400
401	<	It has been observed that densities are dependent on the cutoff radius
402	<	used for a variety of water models in simulations both with and
403	<	without the use of reaction field.\cite{Berendsen98} In order to
404	<	address the possible affect of cutoff radius, simulations were
405	<	performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the
406	<	previous SSD simulations, all performed with a cutoff of 9.0 \AA. All
407	<	of the resulting densities overlapped within error and showed no
408	<	significant trend toward lower or higher densities as a function of
409	<	cutoff radius, for simulations both with and without reaction
410	<	field. These results indicate that there is no major benefit in
411	<	choosing a longer cutoff radius in simulations using SSD. This is
412	<	advantageous in that the use of a longer cutoff radius results in
413	<	significant increases in the time required to obtain a single
370	<	trajectory.
401	>	It has been observed that liquid state densities in water are
402	>	dependent on the cutoff radius used both with and without the use of
403	>	reaction field.\cite{Berendsen98} In order to address the possible
404	>	effect of cutoff radius, simulations were performed with a dipolar
405	>	cutoff radius of 12.0~\AA\ to complement the previous SSD
406	>	simulations, all performed with a cutoff of 9.0~\AA. All of the
407	>	resulting densities overlapped within error and showed no significant
408	>	trend toward lower or higher densities as a function of cutoff radius,
409	>	for simulations both with and without reaction field. These results
410	>	indicate that there is no major benefit in choosing a longer cutoff
411	>	radius in simulations using SSD. This is advantageous in that
412	>	the use of a longer cutoff radius results in a significant increase in
413	>	the time required to obtain a single trajectory.
414
415		The key feature to recognize in figure \ref{dense1} is the density
416		scaling of SSD relative to other common models at any given
417	<	temperature. Note that the SSD model assumes a lower density than any
418	<	of the other listed models at the same pressure, behavior which is
419	<	especially apparent at temperatures greater than 300 K. Lower than
420	<	expected densities have been observed for other systems using a
421	<	reaction field for long-range electrostatic interactions, so the most
422	<	likely reason for the significantly lower densities seen in these
423	<	simulations is the presence of the reaction
424	<	field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the
425	<	reaction field on the density of the systems, the simulations were
426	<	repeated without a reaction field present. The results of these
427	<	simulations are also displayed in figure \ref{dense1}. Without
428	<	reaction field, the densities increase considerably to more
429	<	experimentally reasonable values, especially around the freezing point
430	<	of liquid water. The shape of the curve is similar to the curve
431	<	produced from SSD simulations using reaction field, specifically the
432	<	rapidly decreasing densities at higher temperatures; however, a shift
433	<	in the density maximum location, down to 245 K, is observed. This is a
434	<	more accurate comparison to the other listed water models, in that no
435	<	long range corrections were applied in those
436	<	simulations.\cite{Clancy94,Jorgensen98b} However, even without a
394	<	reaction field, the density around 300 K is still significantly lower
417	>	temperature. SSD assumes a lower density than any of the other
418	>	listed models at the same pressure, behavior which is especially
419	>	apparent at temperatures greater than 300~K. Lower than expected
420	>	densities have been observed for other systems using a reaction field
421	>	for long-range electrostatic interactions, so the most likely reason
422	>	for the significantly lower densities seen in these simulations is the
423	>	presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order
424	>	to test the effect of the reaction field on the density of the
425	>	systems, the simulations were repeated without a reaction field
426	>	present. The results of these simulations are also displayed in figure
427	>	\ref{dense1}. Without the reaction field, the densities increase
428	>	to more experimentally reasonable values, especially around the
429	>	freezing point of liquid water. The shape of the curve is similar to
430	>	the curve produced from SSD simulations using reaction field,
431	>	specifically the rapidly decreasing densities at higher temperatures;
432	>	however, a shift in the density maximum location, down to 245~K, is
433	>	observed. This is a more accurate comparison to the other listed water
434	>	models, in that no long range corrections were applied in those
435	>	simulations.\cite{Clancy94,Jorgensen98b} However, even without the
436	>	reaction field, the density around 300~K is still significantly lower
437		than experiment and comparable water models. This anomalous behavior
438	<	was what lead Ichiye \emph{et al.} to recently reparameterize SSD and
439	<	make SSD1.\cite{Ichiye03} In discussing potential adjustments later in
440	<	this paper, all comparisons were performed with this new model.
438	>	was what lead Tan {\it et al.} to recently reparameterize
439	>	SSD.\cite{Ichiye03} Throughout the remainder of the paper our
440	>	reparamaterizations of SSD will be compared with their newer SSD1
441	>	model.
442
443		\subsection{Transport Behavior}
401	–	Of importance in these types of studies are the transport properties
402	–	of the particles and their change in responce to altering
403	–	environmental conditions. In order to probe transport, constant energy
404	–	simulations were performed about the average density uncovered by the
405	–	constant pressure simulations. Simulations started with randomized
406	–	velocities and underwent 50 ps of temperature scaling and 50 ps of
407	–	constant energy equilibration before obtaining a 200 ps
408	–	trajectory. Diffusion constants were calculated via root-mean square
409	–	deviation analysis. The averaged results from five sets of NVE
410	–	simulations are displayed in figure \ref{diffuse}, alongside
411	–	experimental, SPC/E, and TIP5P
412	–	results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01}
444
445	<	\begin{figure}
446	<	\begin{center}
447	<	\epsfxsize=6in
448	<	\epsfbox{betterDiffuse.epsi}
449	<	\caption{Average diffusion coefficient over increasing temperature for
450	<	SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental
451	<	data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has
452	<	the least deviation from the experimental values. The rapidly
453	<	increasing diffusion constants for TIP5P and SSD correspond to
454	<	significant decrease in density at the higher temperatures.}
455	<	\label{diffuse}
456	<	\end{center}
457	<	\end{figure}
445	>	Accurate dynamical properties of a water model are particularly
446	>	important when using the model to study permeation or transport across
447	>	biological membranes.  In order to probe transport in bulk water,
448	>	constant energy (NVE) simulations were performed at the average
449	>	density obtained by the NPT simulations at an identical target
450	>	temperature. Simulations started with randomized velocities and
451	>	underwent 50~ps of temperature scaling and 50~ps of constant energy
452	>	equilibration before a 200~ps data collection run. Diffusion constants
453	>	were calculated via linear fits to the long-time behavior of the
454	>	mean-square displacement as a function of time. The averaged results
455	>	from five sets of NVE simulations are displayed in figure
456	>	\ref{diffuse}, alongside experimental, SPC/E, and TIP5P
457	>	results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01}
458
459	+	%\begin{figure}
460	+	%\begin{center}
461	+	%\epsfxsize=6in
462	+	%\epsfbox{betterDiffuse.epsi}
463	+	%\caption{Average self-diffusion constant as a function of temperature for
464	+	%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P
465	+	%[Ref. \onlinecite{Jorgensen01}] compared with experimental data
466	+	%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models
467	+	%shown, SSD has the least deviation from the experimental values. The
468	+	%rapidly increasing diffusion constants for TIP5P and SSD correspond to
469	+	%significant decreases in density at the higher temperatures.}
470	+	%\label{diffuse}
471	+	%\end{center}
472	+	%\end{figure}
473	+
474		The observed values for the diffusion constant point out one of the
475	<	strengths of the SSD model. Of the three experimental models shown,
476	<	the SSD model has the most accurate depiction of the diffusion trend
477	<	seen in experiment in both the supercooled and liquid temperature
478	<	regimes. SPC/E does a respectable job by producing values similar to
479	<	SSD and experiment around 290 K; however, it deviates at both higher
480	<	and lower temperatures, failing to predict the experimental
481	<	trend. TIP5P and SSD both start off low at colder temperatures and
482	<	tend to diffuse too rapidly at higher temperatures. This trend at
483	<	higher temperatures is not surprising in that the densities of both
484	<	TIP5P and SSD are lower than experimental water at these higher
485	<	temperatures. When calculating the diffusion coefficients for SSD at
486	<	experimental densities, the resulting values fall more in line with
487	<	experiment at these temperatures, albeit not at standard pressure.
475	>	strengths of the SSD model. Of the three models shown, the SSD model
476	>	has the most accurate depiction of self-diffusion in both the
477	>	supercooled and liquid regimes.  SPC/E does a respectable job by
478	>	reproducing values similar to experiment around 290~K; however, it
479	>	deviates at both higher and lower temperatures, failing to predict the
480	>	correct thermal trend. TIP5P and SSD both start off low at colder
481	>	temperatures and tend to diffuse too rapidly at higher temperatures.
482	>	This behavior at higher temperatures is not particularly surprising
483	>	since the densities of both TIP5P and SSD are lower than experimental
484	>	water densities at higher temperatures.  When calculating the
485	>	diffusion coefficients for SSD at experimental densities
486	>	(instead of the densities from the NPT simulations), the resulting
487	>	values fall more in line with experiment at these temperatures.
488
489		\subsection{Structural Changes and Characterization}
490	+
491		By starting the simulations from the crystalline state, the melting
492	<	transition and the ice structure can be studied along with the liquid
492	>	transition and the ice structure can be obtained along with the liquid
493		phase behavior beyond the melting point. The constant pressure heat
494		capacity (C$_\text{p}$) was monitored to locate the melting transition
495		in each of the simulations. In the melting simulations of the 1024
496		particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs
497	<	at 245 K, indicating a first order phase transition for the melting of
497	>	at 245~K, indicating a first order phase transition for the melting of
498		these ice crystals. When the reaction field is turned off, the melting
499	<	transition occurs at 235 K.  These melting transitions are
500	<	considerably lower than the experimental value, but this is not a
454	<	surprise considering the simplicity of the SSD model.
499	>	transition occurs at 235~K.  These melting transitions are
500	>	considerably lower than the experimental value.
501
502	<	\begin{figure}
503	<	\begin{center}
504	<	\epsfxsize=6in
505	<	\epsfbox{corrDiag.eps}
506	<	\caption{Two dimensional illustration of angles involved in the
507	<	correlations observed in figure \ref{contour}.}
508	<	\label{corrAngle}
509	<	\end{center}
464	<	\end{figure}
502	>	%\begin{figure}
503	>	%\begin{center}
504	>	%\epsfxsize=6in
505	>	%\epsfbox{corrDiag.eps}
506	>	%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.}
507	>	%\label{corrAngle}
508	>	%\end{center}
509	>	%\end{figure}
510
511	<	\begin{figure}
512	<	\begin{center}
513	<	\epsfxsize=6in
514	<	\epsfbox{fullContours.eps}
515	<	\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at
516	<	100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for
517	<	clarity: dark areas signify peaks while light areas signify
518	<	depressions. White areas have g(\emph{r}) values below 0.5 and black
519	<	areas have values above 1.5.}
520	<	\label{contour}
521	<	\end{center}
522	<	\end{figure}
511	>	%\begin{figure}
512	>	%\begin{center}
513	>	%\epsfxsize=6in
514	>	%\epsfbox{fullContours.eps}
515	>	%\caption{Contour plots of 2D angular pair correlation functions for
516	>	%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas
517	>	%signify regions of enhanced density while light areas signify
518	>	%depletion relative to the bulk density. White areas have pair
519	>	%correlation values below 0.5 and black areas have values above 1.5.}
520	>	%\label{contour}
521	>	%\end{center}
522	>	%\end{figure}
523
524	<	Additional analysis of the melting phase-transition process was
525	<	performed by using two-dimensional structure and dipole angle
526	<	correlations. Expressions for these correlations are as follows:
524	>	Additional analysis of the melting process was performed using
525	>	two-dimensional structure and dipole angle correlations. Expressions
526	>	for these correlations are as follows:
527
528		\begin{equation}
529	<	g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ ,
529	>	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\ ,
530		\end{equation}
531		\begin{equation}
532		g_{\text{AB}}(r,\cos\omega) =
533	<	\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\ ,
533	>	\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\ ,
534		\end{equation}
535		where $\theta$ and $\omega$ refer to the angles shown in figure
536		\ref{corrAngle}. By binning over both distance and the cosine of the
537	<	desired angle between the two dipoles, the g(\emph{r}) can be
538	<	dissected to determine the common dipole arrangements that constitute
539	<	the peaks and troughs. Frames A and B of figure \ref{contour} show a
540	<	relatively crystalline state of an ice $I_c$ simulation. The first
541	<	peak of the g(\emph{r}) consists primarily of the preferred hydrogen
537	>	desired angle between the two dipoles, the $g(r)$ can be analyzed to
538	>	determine the common dipole arrangements that constitute the peaks and
539	>	troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
540	>	of figure \ref{contour} show results from an ice $I_c$ simulation. The
541	>	first peak in the $g(r)$ consists primarily of the preferred hydrogen
542		bonding arrangements as dictated by the tetrahedral sticky potential -
543	<	one peak for the donating and the other for the accepting hydrogen
544	<	bonds. Due to the high degree of crystallinity of the sample, the
545	<	second and third solvation shells show a repeated peak arrangement
543	>	one peak for the hydrogen bond donor and the other for the hydrogen
544	>	bond acceptor.  Due to the high degree of crystallinity of the sample,
545	>	the second and third solvation shells show a repeated peak arrangement
546		which decays at distances around the fourth solvation shell, near the
547		imposed cutoff for the Lennard-Jones and dipole-dipole interactions.
548		In the higher temperature simulation shown in frames C and D, these
549	<	longer-ranged repeated peak features deteriorate rapidly. The first
550	<	solvation shell still shows the strong effect of the sticky-potential,
551	<	although it covers a larger area, extending to include a fraction of
552	<	aligned dipole peaks within the first solvation shell. The latter
553	<	peaks lose definition as thermal motion and the competing dipole force
554	<	overcomes the sticky potential's tight tetrahedral structuring of the
510	<	fluid.
549	>	long-range features deteriorate rapidly. The first solvation shell
550	>	still shows the strong effect of the sticky-potential, although it
551	>	covers a larger area, extending to include a fraction of aligned
552	>	dipole peaks within the first solvation shell. The latter peaks lose
553	>	due to thermal motion and as the competing dipole force overcomes the
554	>	sticky potential's tight tetrahedral structuring of the crystal.
555
556		This complex interplay between dipole and sticky interactions was
557		remarked upon as a possible reason for the split second peak in the
558	<	oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the
559	<	second solvation shell peak appears to have two distinct components
560	<	that blend together to form one observable peak. At higher
561	<	temperatures, this split character alters to show the leading 4 \AA\
562	<	peak dominated by equatorial anti-parallel dipole orientations. There
563	<	is also a tightly bunched group of axially arranged dipoles that most
564	<	likely consist of the smaller fraction of aligned dipole pairs. The
565	<	trailing component of the split peak at 5 \AA\ is dominated by aligned
566	<	dipoles that assume hydrogen bond arrangements similar to those seen
567	<	in the first solvation shell. This evidence indicates that the dipole
568	<	pair interaction begins to dominate outside of the range of the
569	<	dipolar repulsion term. Primary energetically favorable dipole
570	<	arrangements populate the region immediately outside this repulsion
571	<	region (around 4 \AA), while arrangements that seek to ideally satisfy
572	<	both the sticky and dipole forces locate themselves just beyond this
573	<	initial buildup (around 5 \AA).
558	>	oxygen-oxygen pair correlation function,
559	>	$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second
560	>	solvation shell peak appears to have two distinct components that
561	>	blend together to form one observable peak. At higher temperatures,
562	>	this split character alters to show the leading 4~\AA\ peak dominated
563	>	by equatorial anti-parallel dipole orientations. There is also a
564	>	tightly bunched group of axially arranged dipoles that most likely
565	>	consist of the smaller fraction of aligned dipole pairs. The trailing
566	>	component of the split peak at 5~\AA\ is dominated by aligned dipoles
567	>	that assume hydrogen bond arrangements similar to those seen in the
568	>	first solvation shell. This evidence indicates that the dipole pair
569	>	interaction begins to dominate outside of the range of the dipolar
570	>	repulsion term.  The energetically favorable dipole arrangements
571	>	populate the region immediately outside this repulsion region (around
572	>	4~\AA), while arrangements that seek to satisfy both the sticky and
573	>	dipole forces locate themselves just beyond this initial buildup
574	>	(around 5~\AA).
575
576		From these findings, the split second peak is primarily the product of
577		the dipolar repulsion term of the sticky potential. In fact, the inner
578		peak can be pushed out and merged with the outer split peak just by
579	<	extending the switching function cutoff ($s^\prime(r_{ij})$) from its
580	<	normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of
579	>	extending the switching function ($s^\prime(r_{ij})$) from its normal
580	>	4.0~\AA\ cutoff to values of 4.5 or even 5~\AA. This type of
581		correction is not recommended for improving the liquid structure,
582		since the second solvation shell would still be shifted too far
583		out. In addition, this would have an even more detrimental effect on
584		the system densities, leading to a liquid with a more open structure
585	<	and a density considerably lower than the normal SSD behavior shown
586	<	previously. A better correction would be to include the
585	>	and a density considerably lower than the already low SSD
586	>	density.  A better correction would be to include the
587		quadrupole-quadrupole interactions for the water particles outside of
588	<	the first solvation shell, but this reduces the simplicity and speed
589	<	advantage of SSD.
588	>	the first solvation shell, but this would remove the simplicity and
589	>	speed advantage of SSD.
590
591		\subsection{Adjusted Potentials: SSD/RF and SSD/E}
592	+
593		The propensity of SSD to adopt lower than expected densities under
594		varying conditions is troubling, especially at higher temperatures. In
595		order to correct this model for use with a reaction field, it is
#	Line 551 | Line 597 | densities while maintaining the accurate transport pro
597		intermolecular interactions. In undergoing a reparameterization, it is
598		important not to focus on just one property and neglect the other
599		important properties. In this case, it would be ideal to correct the
600	<	densities while maintaining the accurate transport properties.
600	>	densities while maintaining the accurate transport behavior.
601
602	<	The parameters available for tuning include the $\sigma$ and $\epsilon$
603	<	Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky
604	<	attractive and dipole repulsive terms with their respective
605	<	cutoffs. To alter the attractive and repulsive terms of the sticky
606	<	potential independently, it is necessary to separate the terms as
607	<	follows:
608	<	\begin{equation}
609	<	u_{ij}^{sp}
610	<	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) =
611	<	\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)],
566	<	\end{equation}
602	>	The parameters available for tuning include the $\sigma$ and
603	>	$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the
604	>	strength of the sticky potential ($\nu_0$), and the cutoff distances
605	>	for the sticky attractive and dipole repulsive cubic switching
606	>	function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$
607	>	respectively). The results of the reparameterizations are shown in
608	>	table \ref{params}. We are calling these reparameterizations the Soft
609	>	Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction
610	>	field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve
611	>	the liquid structure in simulations without a long-range correction).
612
568	–	where $\nu_0$ scales the strength of the tetrahedral attraction and
569	–	$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion
570	–	term. The separation was performed for purposes of the
571	–	reparameterization, but the final parameters were adjusted so that it
572	–	is unnecessary to separate the terms when implementing the adjusted
573	–	water potentials. The results of the reparameterizations are shown in
574	–	table \ref{params}. Note that the tetrahedral attractive and dipolar
575	–	repulsive terms do not share the same lower cutoff ($r_l$) in the
576	–	newly parameterized potentials - soft sticky dipole reaction field
577	–	(SSD/RF - for use with a reaction field) and soft sticky dipole
578	–	enhanced (SSD/E - an attempt to improve the liquid structure in
579	–	simulations without a long-range correction).
580	–
613		\begin{table}
614		\begin{center}
615		\caption{Parameters for the original and adjusted models}
616		\begin{tabular}{ l  c  c  c  c }
617		\hline \\[-3mm]
618	<	\ \ \ Parameters\ \ \  & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \  & \ SSD/E\ \  & \ SSD/RF \\
618	>	\ \ \ Parameters\ \ \  & \ \ \ SSD [Ref. \onlinecite{Ichiye96}] \ \ \
619	>	& \ SSD1 [Ref. \onlinecite{Ichiye03}]\ \  & \ SSD/E\ \  & \ \ SSD/RF \\
620		\hline \\[-3mm]
621		\ \ \ $\sigma$ (\AA)  & 3.051 & 3.016 & 3.035 & 3.019\\
622		\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
623		\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\
624		\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
625	+	\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\
626		\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\
627		\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\
594	–	\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
628		\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
629		\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
630		\end{tabular}
#	Line 599 | Line 632 | simulations without a long-range correction).
632		\end{center}
633		\end{table}
634
635	<	\begin{figure}
636	<	\begin{center}
637	<	\epsfxsize=5in
638	<	\epsfbox{GofRCompare.epsi}
639	<	\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E
640	<	and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with
641	<	reaction field turned on (bottom). The insets show the respective
642	<	first peaks in detail. Note how the changes in parameters have lowered
643	<	and broadened the first peak of SSD/E and SSD/RF.}
644	<	\label{grcompare}
645	<	\end{center}
646	<	\end{figure}
647	<
615	<	\begin{figure}
616	<	\begin{center}
617	<	\epsfxsize=6in
618	<	\epsfbox{dualsticky.ps}
619	<	\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \&
620	<	SSD/RF (right). Light areas correspond to the tetrahedral attractive
621	<	component, and darker areas correspond to the dipolar repulsive
622	<	component.}
623	<	\label{isosurface}
624	<	\end{center}
625	<	\end{figure}
635	>	%\begin{figure}
636	>	%\begin{center}
637	>	%\epsfxsize=5in
638	>	%\epsfbox{GofRCompare.epsi}
639	>	%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with
640	>	%SSD/E and SSD1 without reaction field (top), as well as
641	>	%SSD/RF and SSD1 with reaction field turned on
642	>	%(bottom). The insets show the respective first peaks in detail. Note
643	>	%how the changes in parameters have lowered and broadened the first
644	>	%peak of SSD/E and SSD/RF.}
645	>	%\label{grcompare}
646	>	%\end{center}
647	>	%\end{figure}
648
649	<	In the paper detailing the development of SSD, Liu and Ichiye placed
650	<	particular emphasis on an accurate description of the first solvation
651	<	shell. This resulted in a somewhat tall and narrow first peak in the
652	<	g(\emph{r}) that integrated to give similar coordination numbers to
649	>	%\begin{figure}
650	>	%\begin{center}
651	>	%\epsfxsize=6in
652	>	%\epsfbox{dualsticky_bw.eps}
653	>	%\caption{Positive and negative isosurfaces of the sticky potential for
654	>	%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas
655	>	%correspond to the tetrahedral attractive component, and darker areas
656	>	%correspond to the dipolar repulsive component.}
657	>	%\label{isosurface}
658	>	%\end{center}
659	>	%\end{figure}
660	>
661	>	In the original paper detailing the development of SSD, Liu and Ichiye
662	>	placed particular emphasis on an accurate description of the first
663	>	solvation shell. This resulted in a somewhat tall and narrow first
664	>	peak in $g(r)$ that integrated to give similar coordination numbers to
665		the experimental data obtained by Soper and
666		Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering
667		data from the Head-Gordon lab indicates a slightly lower and shifted
668	<	first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made
669	<	while taking into consideration the new experimental
668	>	first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were
669	>	made after taking into consideration the new experimental
670		findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the
671	<	relocation of the first peak of the oxygen-oxygen g(\emph{r}) by
672	<	comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new
671	>	relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
672	>	the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
673		experimental results. Both modified water models have shorter peaks
674	<	that are brought in more closely to the experimental peak (as seen in
675	<	the insets of figure \ref{grcompare}).  This structural alteration was
674	>	that match more closely to the experimental peak (as seen in the
675	>	insets of figure \ref{grcompare}).  This structural alteration was
676		accomplished by the combined reduction in the Lennard-Jones $\sigma$
677	<	variable and adjustment of the sticky potential strength and
678	<	cutoffs. As can be seen in table \ref{params}, the cutoffs for the
679	<	tetrahedral attractive and dipolar repulsive terms were nearly swapped
680	<	with each other. Isosurfaces of the original and modified sticky
681	<	potentials are shown in figure \ref{isosurface}. In these isosurfaces,
682	<	it is easy to see how altering the cutoffs changes the repulsive and
683	<	attractive character of the particles. With a reduced repulsive
684	<	surface (darker region), the particles can move closer to one another,
685	<	increasing the density for the overall system. This change in
686	<	interaction cutoff also results in a more gradual orientational motion
687	<	by allowing the particles to maintain preferred dipolar arrangements
688	<	before they begin to feel the pull of the tetrahedral
689	<	restructuring. As the particles move closer together, the dipolar
690	<	repulsion term becomes active and excludes unphysical nearest-neighbor
691	<	arrangements. This compares with how SSD and SSD1 exclude preferred
692	<	dipole alignments before the particles feel the pull of the ``hydrogen
693	<	bonds''. Aside from improving the shape of the first peak in the
694	<	g(\emph{r}), this modification improves the densities considerably by
695	<	allowing the persistence of full dipolar character below the previous
696	<	4.0 \AA\ cutoff.
677	>	variable and adjustment of the sticky potential strength and cutoffs.
678	>	As can be seen in table \ref{params}, the cutoffs for the tetrahedral
679	>	attractive and dipolar repulsive terms were nearly swapped with each
680	>	other.  Isosurfaces of the original and modified sticky potentials are
681	>	shown in figure \ref{isosurface}. In these isosurfaces, it is easy to
682	>	see how altering the cutoffs changes the repulsive and attractive
683	>	character of the particles. With a reduced repulsive surface (darker
684	>	region), the particles can move closer to one another, increasing the
685	>	density for the overall system.  This change in interaction cutoff
686	>	also results in a more gradual orientational motion by allowing the
687	>	particles to maintain preferred dipolar arrangements before they begin
688	>	to feel the pull of the tetrahedral restructuring. As the particles
689	>	move closer together, the dipolar repulsion term becomes active and
690	>	excludes unphysical nearest-neighbor arrangements. This compares with
691	>	how SSD and SSD1 exclude preferred dipole alignments before the
692	>	particles feel the pull of the ``hydrogen bonds''. Aside from
693	>	improving the shape of the first peak in the g(\emph{r}), this
694	>	modification improves the densities considerably by allowing the
695	>	persistence of full dipolar character below the previous 4.0~\AA\
696	>	cutoff.
697
698	<	While adjusting the location and shape of the first peak of
699	<	g(\emph{r}) improves the densities, these changes alone are
700	<	insufficient to bring the system densities up to the values observed
701	<	experimentally. To further increase the densities, the dipole moments
702	<	were increased in both of the adjusted models. Since SSD is a dipole
703	<	based model, the structure and transport are very sensitive to changes
704	<	in the dipole moment. The original SSD simply used the dipole moment
705	<	calculated from the TIP3P water model, which at 2.35 D is
706	<	significantly greater than the experimental gas phase value of 1.84
707	<	D. The larger dipole moment is a more realistic value and improves the
708	<	dielectric properties of the fluid. Both theoretical and experimental
709	<	measurements indicate a liquid phase dipole moment ranging from 2.4 D
710	<	to values as high as 3.11 D, providing a substantial range of
711	<	reasonable values for a dipole
712	<	moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
679	<	increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF,
698	>	While adjusting the location and shape of the first peak of $g(r)$
699	>	improves the densities, these changes alone are insufficient to bring
700	>	the system densities up to the values observed experimentally.  To
701	>	further increase the densities, the dipole moments were increased in
702	>	both of our adjusted models. Since SSD is a dipole based model, the
703	>	structure and transport are very sensitive to changes in the dipole
704	>	moment. The original SSD simply used the dipole moment calculated from
705	>	the TIP3P water model, which at 2.35~D is significantly greater than
706	>	the experimental gas phase value of 1.84~D. The larger dipole moment
707	>	is a more realistic value and improves the dielectric properties of
708	>	the fluid. Both theoretical and experimental measurements indicate a
709	>	liquid phase dipole moment ranging from 2.4~D to values as high as
710	>	3.11~D, providing a substantial range of reasonable values for a
711	>	dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately
712	>	increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF,
713		respectively, leads to significant changes in the density and
714		transport of the water models.
715
#	Line 689 | Line 722 | simulation was equilibrated for 100 ps before a 200 ps
722		results are obtained from five separate simulations of 1024 particle
723		systems, and the melting sequences were started from different ice
724		$I_h$ crystals constructed as described previously. Each NPT
725	<	simulation was equilibrated for 100 ps before a 200 ps data collection
725	>	simulation was equilibrated for 100~ps before a 200~ps data collection
726		run at each temperature step, and the final configuration from the
727		previous temperature simulation was used as a starting point. All NVE
728		simulations had the same thermalization, equilibration, and data
729	<	collection times as stated earlier in this paper.
729	>	collection times as stated previously.
730
731	+	%\begin{figure}
732	+	%\begin{center}
733	+	%\epsfxsize=6in
734	+	%\epsfbox{ssdeDense.epsi}
735	+	%\caption{Comparison of densities calculated with SSD/E to
736	+	%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
737	+	%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and
738	+	%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around
739	+	%300 K with error bars included to clarify this region of
740	+	%interest. Note that both SSD1 and SSD/E show good agreement with
741	+	%experiment when the long-range correction is neglected.}
742	+	%\label{ssdedense}
743	+	%\end{center}
744	+	%\end{figure}
745	+
746	+	Fig. \ref{ssdedense} shows the density profile for the SSD/E
747	+	model in comparison to SSD1 without a reaction field, other
748	+	common water models, and experimental results. The calculated
749	+	densities for both SSD/E and SSD1 have increased
750	+	significantly over the original SSD model (see
751	+	fig. \ref{dense1}) and are in better agreement with the experimental
752	+	values. At 298 K, the densities of SSD/E and SSD1 without
753	+	a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
754	+	0.999$\pm$0.001 g/cm$^3$ respectively.  These both compare well with
755	+	the experimental value of 0.997 g/cm$^3$, and they are considerably
756	+	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The
757	+	changes to the dipole moment and sticky switching functions have
758	+	improved the structuring of the liquid (as seen in figure
759	+	\ref{grcompare}, but they have shifted the density maximum to much
760	+	lower temperatures. This comes about via an increase in the liquid
761	+	disorder through the weakening of the sticky potential and
762	+	strengthening of the dipolar character. However, this increasing
763	+	disorder in the SSD/E model has little effect on the melting
764	+	transition. By monitoring $C_p$ throughout these simulations, the
765	+	melting transition for SSD/E was shown to occur at 235~K.  The
766	+	same transition temperature observed with SSD and SSD1.
767	+
768	+	%\begin{figure}
769	+	%\begin{center}
770	+	%\epsfxsize=6in
771	+	%\epsfbox{ssdrfDense.epsi}
772	+	%\caption{Comparison of densities calculated with SSD/RF to
773	+	%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
774	+	%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and
775	+	%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of
776	+	%reparameterization when utilizing a reaction field long-ranged
777	+	%correction - SSD/RF provides significantly more accurate
778	+	%densities than SSD1 when performing room temperature
779	+	%simulations.}
780	+	%\label{ssdrfdense}
781	+	%\end{center}
782	+	%\end{figure}
783	+
784	+	Including the reaction field long-range correction in the simulations
785	+	results in a more interesting comparison.  A density profile including
786	+	SSD/RF and SSD1 with an active reaction field is shown in figure
787	+	\ref{ssdrfdense}.  As observed in the simulations without a reaction
788	+	field, the densities of SSD/RF and SSD1 show a dramatic increase over
789	+	normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
790	+	of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
791	+	considerably better than the original SSD value of 0.941$\pm$0.001
792	+	g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results
793	+	further emphasize the importance of reparameterization in order to
794	+	model the density properly under different simulation conditions.
795	+	Again, these changes have only a minor effect on the melting point,
796	+	which observed at 245~K for SSD/RF, is identical to SSD and only 5~K
797	+	lower than SSD1 with a reaction field. Additionally, the difference in
798	+	density maxima is not as extreme, with SSD/RF showing a density
799	+	maximum at 255~K, fairly close to the density maxima of 260~K and
800	+	265~K, shown by SSD and SSD1 respectively.
801	+
802	+	%\begin{figure}
803	+	%\begin{center}
804	+	%\epsfxsize=6in
805	+	%\epsfbox{ssdeDiffuse.epsi}
806	+	%\caption{The diffusion constants calculated from SSD/E and
807	+	%SSD1 (both without a reaction field) along with experimental results
808	+	%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were
809	+	%performed at the average densities observed in the 1 atm NPT
810	+	%simulations for the respective models. SSD/E is slightly more mobile
811	+	%than experiment at all of the temperatures, but it is closer to
812	+	%experiment at biologically relevant temperatures than SSD1 without a
813	+	%long-range correction.}
814	+	%\label{ssdediffuse}
815	+	%\end{center}
816	+	%\end{figure}
817	+
818	+	The reparameterization of the SSD water model, both for use with and
819	+	without an applied long-range correction, brought the densities up to
820	+	what is expected for simulating liquid water. In addition to improving
821	+	the densities, it is important that the diffusive behavior of SSD be
822	+	maintained or improved. Figure \ref{ssdediffuse} compares the
823	+	temperature dependence of the diffusion constant of SSD/E to SSD1
824	+	without an active reaction field at the densities calculated from
825	+	their respective NPT simulations at 1 atm. The diffusion constant for
826	+	SSD/E is consistently higher than experiment, while SSD1 remains lower
827	+	than experiment until relatively high temperatures (around 360
828	+	K). Both models follow the shape of the experimental curve well below
829	+	300~K but tend to diffuse too rapidly at higher temperatures, as seen
830	+	in SSD1's crossing above 360~K.  This increasing diffusion relative to
831	+	the experimental values is caused by the rapidly decreasing system
832	+	density with increasing temperature.  Both SSD1 and SSD/E show this
833	+	deviation in particle mobility, but this trend has different
834	+	implications on the diffusive behavior of the models.  While SSD1
835	+	shows more experimentally accurate diffusive behavior in the high
836	+	temperature regimes, SSD/E shows more accurate behavior in the
837	+	supercooled and biologically relevant temperature ranges.  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 under the most commonly simulated
841	+	conditions.
842	+
843	+	%\begin{figure}
844	+	%\begin{center}
845	+	%\epsfxsize=6in
846	+	%\epsfbox{ssdrfDiffuse.epsi}
847	+	%\caption{The diffusion constants calculated from SSD/RF and
848	+	%SSD1 (both with an active reaction field) along with
849	+	%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The
850	+	%NVE calculations were performed at the average densities observed in
851	+	%the 1 atm NPT simulations for both of the models. SSD/RF
852	+	%simulates the diffusion of water throughout this temperature range
853	+	%very well. The rapidly increasing diffusion constants at high
854	+	%temperatures for both models can be attributed to lower calculated
855	+	%densities than those observed in experiment.}
856	+	%\label{ssdrfdiffuse}
857	+	%\end{center}
858	+	%\end{figure}
859	+
860	+	In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
861	+	compared to SSD1 with an active reaction field. Note that SSD/RF
862	+	tracks the experimental results quantitatively, identical within error
863	+	throughout most of the temperature range shown and exhibiting only a
864	+	slight increasing trend at higher temperatures. SSD1 tends to diffuse
865	+	more slowly at low temperatures and deviates to diffuse too rapidly at
866	+	temperatures greater than 330~K.  As stated above, this deviation away
867	+	from the ideal trend is due to a rapid decrease in density at higher
868	+	temperatures. SSD/RF does not suffer from this problem as much as SSD1
869	+	because the calculated densities are closer to the experimental
870	+	values. These results again emphasize the importance of careful
871	+	reparameterization when using an altered long-range correction.
872	+
873	+	\begin{table}
874	+	\begin{minipage}{\linewidth}
875	+	\renewcommand{\thefootnote}{\thempfootnote}
876	+	\begin{center}
877	+	\caption{Properties of the single-point water models compared with
878	+	experimental data at ambient conditions. Deviations of the of the
879	+	averages are given in parentheses.}
880	+	\begin{tabular}{ l  c  c  c  c  c }
881	+	\hline \\[-3mm]
882	+	\ \ \ \ \ \  & \ \ \ SSD1 \ \ \ & \ \ SSD/E \ \ \ & \ \ SSD1 (RF) \ \
883	+	\ & \ \ SSD/RF \ \ \ & \ \ Expt. \\
884	+	\hline \\[-3mm]
885	+	\ \ $\rho$ (g/cm$^3$) & 0.999(0.001) & 0.996(0.001) & 0.972(0.002) & 0.997(0.001) & 0.997 \\
886	+	\ \ $C_p$ (cal/mol K) & 28.80(0.11) & 25.45(0.09) & 28.28(0.06) & 23.83(0.16) & 17.98 \\
887	+	\ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78(0.7) & 2.51(0.18) & 2.00(0.17) & 2.32(0.06) & 2.299\cite{Mills73} \\
888	+	\ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 &
889	+	4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in
890	+	Ref. \onlinecite{Head-Gordon00_1}} \\
891	+	\ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 &
892	+	3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in
893	+	Ref. \onlinecite{Soper86}}  \\
894	+	\ \ $\tau_1$ (ps) & 10.9(0.6) & 7.3(0.4) & 7.5(0.7) & 7.2(0.4) & 5.7\footnote{Calculated for 298 K from data in Ref. \onlinecite{Eisenberg69}} \\
895	+	\ \ $\tau_2$ (ps) & 4.7(0.4) & 3.1(0.2) & 3.5(0.3) & 3.2(0.2) & 2.3\footnote{Calculated for 298 K from data in
896	+	Ref. \onlinecite{Krynicki66}}
897	+	\end{tabular}
898	+	\label{liquidproperties}
899	+	\end{center}
900	+	\end{minipage}
901	+	\end{table}
902	+
903	+	Table \ref{liquidproperties} gives a synopsis of the liquid state
904	+	properties of the water models compared in this study along with the
905	+	experimental values for liquid water at ambient conditions. The
906	+	coordination number ($n_C$) and number of hydrogen bonds per particle
907	+	($n_H$) were calculated by integrating the following relations:
908	+	\begin{equation}
909	+	n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr,
910	+	\end{equation}
911	+	\begin{equation}
912	+	n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr,
913	+	\end{equation}
914	+	where $\rho$ is the number density of the specified pair interactions,
915	+	$a$ and $b$ are the radial locations of the minima following the first
916	+	peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number
917	+	of hydrogen bonds stays relatively constant across all of the models,
918	+	but the coordination numbers of SSD/E and SSD/RF show an
919	+	improvement over SSD1.  This improvement is primarily due to
920	+	extension of the first solvation shell in the new parameter sets.
921	+	Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears
922	+	that all molecules in the first solvation shell are involved in
923	+	hydrogen bonds.  Since $n_H$ and $n_C$ differ in the newly
924	+	parameterized models, the orientations in the first solvation shell
925	+	are a bit more ``fluid''.  Therefore SSD1 overstructures the
926	+	first solvation shell and our reparameterizations have returned this
927	+	shell to more realistic liquid-like behavior.
928	+
929	+	The time constants for the orientational autocorrelation functions
930	+	are also displayed in Table \ref{liquidproperties}. The dipolar
931	+	orientational time correlation functions ($C_{l}$) are described
932	+	by:
933	+	\begin{equation}
934	+	C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle,
935	+	\end{equation}
936	+	where $P_l$ are Legendre polynomials of order $l$ and
937	+	$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular
938	+	dipole.\cite{Rahman71} From these correlation functions, the
939	+	orientational relaxation time of the dipole vector can be calculated
940	+	from an exponential fit in the long-time regime ($t >
941	+	\tau_l$).\cite{Rothschild84} Calculation of these time constants were
942	+	averaged over five detailed NVE simulations performed at the ambient
943	+	conditions for each of the respective models. It should be noted that
944	+	the commonly cited value of 1.9 ps for $\tau_2$ was determined from
945	+	the NMR data in Ref. \onlinecite{Krynicki66} at a temperature near
946	+	34$^\circ$C.\cite{Rahman71} Because of the strong temperature
947	+	dependence of $\tau_2$, it is necessary to recalculate it at 298~K to
948	+	make proper comparisons. The value shown in Table
949	+	\ref{liquidproperties} was calculated from the same NMR data in the
950	+	fashion described in Ref. \onlinecite{Krynicki66}. Similarly, $\tau_1$ was
951	+	recomputed for 298~K from the data in Ref. \onlinecite{Eisenberg69}.
952	+	Again, SSD/E and SSD/RF show improved behavior over SSD1, both with
953	+	and without an active reaction field. Turning on the reaction field
954	+	leads to much improved time constants for SSD1; however, these results
955	+	also include a corresponding decrease in system density.
956	+	Orientational relaxation times published in the original SSD dynamics
957	+	papers are smaller than the values observed here, and this difference
958	+	can be attributed to the use of the Ewald sum.\cite{Ichiye99}
959	+
960	+	\subsection{Additional Observations}
961	+
962	+	%\begin{figure}
963	+	%\begin{center}
964	+	%\epsfxsize=6in
965	+	%\epsfbox{icei_bw.eps}
966	+	%\caption{The most stable crystal structure assumed by the SSD family
967	+	%of water models.  We refer to this structure as Ice-{\it i} to
968	+	%indicate its origins in computer simulation.  This image was taken of
969	+	%the (001) face of the crystal.}
970	+	%\label{weirdice}
971	+	%\end{center}
972	+	%\end{figure}
973	+
974	+	While performing a series of melting simulations on an early iteration
975	+	of SSD/E not discussed in this paper, we observed
976	+	recrystallization into a novel structure not previously known for
977	+	water.  After melting at 235~K, two of five systems underwent
978	+	crystallization events near 245~K.  The two systems remained
979	+	crystalline up to 320 and 330~K, respectively.  The crystal exhibits
980	+	an expanded zeolite-like structure that does not correspond to any
981	+	known form of ice.  This appears to be an artifact of the point
982	+	dipolar models, so to distinguish it from the experimentally observed
983	+	forms of ice, we have denoted the structure
984	+	Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}).  A large enough
985	+	portion of the sample crystallized that we have been able to obtain a
986	+	near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice}
987	+	shows the repeating crystal structure of a typical crystal at 5
988	+	K. Each water molecule is hydrogen bonded to four others; however, the
989	+	hydrogen bonds are bent rather than perfectly straight. This results
990	+	in a skewed tetrahedral geometry about the central molecule.  In
991	+	figure \ref{isosurface}, it is apparent that these flexed hydrogen
992	+	bonds are allowed due to the conical shape of the attractive regions,
993	+	with the greatest attraction along the direct hydrogen bond
994	+	configuration. Though not ideal, these flexed hydrogen bonds are
995	+	favorable enough to stabilize an entire crystal generated around them.
996	+
997	+	Initial simulations indicated that Ice-{\it i} is the preferred ice
998	+	structure for at least the SSD/E model. To verify this, a comparison
999	+	was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and
1000	+	Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and
1001	+	SSD1. Near-ideal versions of the three types of crystals were cooled
1002	+	to 1 K, and enthalpies of formation of each were compared using all
1003	+	three water models.  Enthalpies were estimated from the
1004	+	isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where
1005	+	$P_{\text ext}$ is the applied pressure.  A constant value of -60.158
1006	+	kcal / mol has been added to place our zero for the enthalpies of
1007	+	formation for these systems at the traditional state (elemental forms
1008	+	at standard temperature and pressure).  With every model in the SSD
1009	+	family, Ice-{\it i} had the lowest calculated enthalpy of formation.
1010	+
1011	+	\begin{table}
1012	+	\begin{center}
1013	+	\caption{Enthalpies of Formation (in kcal / mol) of the three crystal
1014	+	structures (at 1 K) exhibited by the SSD family of water models}
1015	+	\begin{tabular}{ l  c  c  c  }
1016	+	\hline \\[-3mm]
1017	+	\ \ \ Water Model \ \ \  & \ \ \ Ice-$I_h$ \ \ \ & \ \ \ Ice-$I_c$ \ \ \  &
1018	+	\ \ \ \ Ice-{\it i} \\
1019	+	\hline \\[-3mm]
1020	+	\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\
1021	+	\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\
1022	+	\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\
1023	+	\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\
1024	+	\end{tabular}
1025	+	\label{iceenthalpy}
1026	+	\end{center}
1027	+	\end{table}
1028	+
1029	+	In addition to these energetic comparisons, melting simulations were
1030	+	performed with Ice-{\it i} as the initial configuration using SSD/E,
1031	+	SSD/RF, and SSD1 both with and without a reaction field. The melting
1032	+	transitions for both SSD/E and SSD1 without reaction field occurred at
1033	+	temperature in excess of 375~K.  SSD/RF and SSD1 with a reaction field
1034	+	showed more reasonable melting transitions near 325~K.  These melting
1035	+	point observations clearly show that all of the SSD-derived models
1036	+	prefer the ice-{\it i} structure.
1037	+
1038	+	\section{Conclusions}
1039	+
1040	+	The density maximum and temperature dependence of the self-diffusion
1041	+	constant were studied for the SSD water model, both with and
1042	+	without the use of reaction field, via a series of NPT and NVE
1043	+	simulations. The constant pressure simulations showed a density
1044	+	maximum near 260 K. In most cases, the calculated densities were
1045	+	significantly lower than the densities obtained from other water
1046	+	models (and experiment). Analysis of self-diffusion showed SSD
1047	+	to capture the transport properties of water well in both the liquid
1048	+	and supercooled liquid regimes.
1049	+
1050	+	In order to correct the density behavior, the original SSD model was
1051	+	reparameterized for use both with and without a reaction field (SSD/RF
1052	+	and SSD/E), and comparisons were made with SSD1, Ichiye's density
1053	+	corrected version of SSD. Both models improve the liquid structure,
1054	+	densities, and diffusive properties under their respective simulation
1055	+	conditions, indicating the necessity of reparameterization when
1056	+	changing the method of calculating long-range electrostatic
1057	+	interactions.  In general, however, these simple water models are
1058	+	excellent choices for representing explicit water in large scale
1059	+	simulations of biochemical systems.
1060	+
1061	+	The existence of a novel low-density ice structure that is preferred
1062	+	by the SSD family of water models is somewhat troubling, since
1063	+	liquid simulations on this family of water models at room temperature
1064	+	are effectively simulations of supercooled or metastable liquids.  One
1065	+	way to destabilize this unphysical ice structure would be to make the
1066	+	range of angles preferred by the attractive part of the sticky
1067	+	potential much narrower.  This would require extensive
1068	+	reparameterization to maintain the same level of agreement with the
1069	+	experiments.
1070	+
1071	+	Additionally, our initial calculations show that the Ice-{\it i}
1072	+	structure may also be a preferred crystal structure for at least one
1073	+	other popular multi-point water model (TIP3P), and that much of the
1074	+	simulation work being done using this popular model could also be at
1075	+	risk for crystallization into this unphysical structure.  A future
1076	+	publication will detail the relative stability of the known ice
1077	+	structures for a wide range of popular water models.
1078	+
1079	+	\section{Acknowledgments}
1080	+	Support for this project was provided by the National Science
1081	+	Foundation under grant CHE-0134881. Computation time was provided by
1082	+	the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
1083	+	DMR-0079647.
1084	+
1085	+	\newpage
1086	+
1087	+	\bibliographystyle{jcp}
1088	+	\bibliography{nptSSD}
1089	+
1090	+	\newpage
1091	+
1092	+	\begin{list}
1093	+	{Figure \arabic{captions}: }{\usecounter{captions}
1094	+	\setlength{\rightmargin}{\leftmargin}}
1095	+
1096	+	\item Energy conservation using both quaternion-based integration and
1097	+	the {\sc dlm} method with increasing time step. The larger time step
1098	+	plots are shifted from the true energy baseline (that of $\Delta t$ =
1099	+	0.1~fs) for clarity.
1100	+
1101	+	\item Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}],
1102	+	TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E
1103	+	[Ref. \onlinecite{Clancy94}], SSD without Reaction Field, SSD, and
1104	+	experiment [Ref. \onlinecite{CRC80}]. The arrows indicate the change
1105	+	in densities observed when turning off the reaction field. The the
1106	+	lower than expected densities for the SSD model were what prompted the
1107	+	original reparameterization of SSD1 [Ref. \onlinecite{Ichiye03}].
1108	+
1109	+	\item Average self-diffusion constant as a function of temperature for
1110	+	SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P
1111	+	[Ref. \onlinecite{Jorgensen01}] compared with experimental data
1112	+	[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three
1113	+	water models shown, SSD has the least deviation from the experimental
1114	+	values. The rapidly increasing diffusion constants for TIP5P and SSD
1115	+	correspond to significant decreases in density at the higher
1116	+	temperatures.
1117	+
1118	+	\item An illustration of angles involved in the correlations observed in
1119	+	Fig. \ref{contour}.
1120	+
1121	+	\item Contour plots of 2D angular pair correlation functions for
1122	+	512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas
1123	+	signify regions of enhanced density while light areas signify
1124	+	depletion relative to the bulk density. White areas have pair
1125	+	correlation values below 0.5 and black areas have values above 1.5.
1126	+
1127	+	\item Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with
1128	+	SSD/E and SSD1 without reaction field (top), as well as SSD/RF and
1129	+	SSD1 with reaction field turned on (bottom). The insets show the
1130	+	respective first peaks in detail. Note how the changes in parameters
1131	+	have lowered and broadened the first peak of SSD/E and SSD/RF.
1132	+
1133	+	\item Positive and negative isosurfaces of the sticky potential for
1134	+	SSD1 (left) and SSD/E \& SSD/RF (right). Light areas
1135	+	correspond to the tetrahedral attractive component, and darker areas
1136	+	correspond to the dipolar repulsive component.
1137	+
1138	+	\item Comparison of densities calculated with SSD/E to
1139	+	SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
1140	+	TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and
1141	+	experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around
1142	+	300 K with error bars included to clarify this region of
1143	+	interest. Note that both SSD1 and SSD/E show good agreement with
1144	+	experiment when the long-range correction is neglected.
1145	+
1146	+	\item Comparison of densities calculated with SSD/RF to
1147	+	SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
1148	+	TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and
1149	+	experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of
1150	+	reparameterization when utilizing a reaction field long-ranged
1151	+	correction - SSD/RF provides significantly more accurate
1152	+	densities than SSD1 when performing room temperature
1153	+	simulations.
1154	+
1155	+	\item The diffusion constants calculated from SSD/E and
1156	+	SSD1 (both without a reaction field) along with experimental results
1157	+	[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were
1158	+	performed at the average densities observed in the 1 atm NPT
1159	+	simulations for the respective models. SSD/E is slightly more mobile
1160	+	than experiment at all of the temperatures, but it is closer to
1161	+	experiment at biologically relevant temperatures than SSD1 without a
1162	+	long-range correction.
1163	+
1164	+	\item The diffusion constants calculated from SSD/RF and
1165	+	SSD1 (both with an active reaction field) along with
1166	+	experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The
1167	+	NVE calculations were performed at the average densities observed in
1168	+	the 1 atm NPT simulations for both of the models. SSD/RF
1169	+	simulates the diffusion of water throughout this temperature range
1170	+	very well. The rapidly increasing diffusion constants at high
1171	+	temperatures for both models can be attributed to lower calculated
1172	+	densities than those observed in experiment.
1173	+
1174	+	\item The most stable crystal structure assumed by the SSD family
1175	+	of water models.  We refer to this structure as Ice-{\it i} to
1176	+	indicate its origins in computer simulation.  This image was taken of
1177	+	the (001) face of the crystal.
1178	+	\end{list}
1179	+
1180	+	\newpage
1181	+
1182		\begin{figure}
1183	+	\begin{center}
1184	+	\epsfxsize=6in
1185	+	\epsfbox{timeStep.epsi}
1186	+	%\caption{Energy conservation using both quaternion-based integration and
1187	+	%the {\sc dlm} method with increasing time step. The larger time step
1188	+	%plots are shifted from the true energy baseline (that of $\Delta t$ =
1189	+	%0.1~fs) for clarity.}
1190	+	\label{timestep}
1191	+	\end{center}
1192	+	\end{figure}
1193	+
1194	+	\newpage
1195	+
1196	+	\begin{figure}
1197	+	\begin{center}
1198	+	\epsfxsize=6in
1199	+	\epsfbox{denseSSDnew.eps}
1200	+	%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}],
1201	+	% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD
1202	+	% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The
1203	+	% arrows indicate the change in densities observed when turning off the
1204	+	% reaction field. The the lower than expected densities for the SSD
1205	+	% model were what prompted the original reparameterization of SSD1
1206	+	% [Ref. \onlinecite{Ichiye03}].}
1207	+	\label{dense1}
1208	+	\end{center}
1209	+	\end{figure}
1210	+
1211	+	\newpage
1212	+
1213	+	\begin{figure}
1214	+	\begin{center}
1215	+	\epsfxsize=6in
1216	+	\epsfbox{betterDiffuse.epsi}
1217	+	%\caption{Average self-diffusion constant as a function of temperature for
1218	+	%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P
1219	+	%[Ref. \onlinecite{Jorgensen01}] compared with experimental data
1220	+	%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models
1221	+	%shown, SSD has the least deviation from the experimental values. The
1222	+	%rapidly increasing diffusion constants for TIP5P and SSD correspond to
1223	+	%significant decreases in density at the higher temperatures.}
1224	+	\label{diffuse}
1225	+	\end{center}
1226	+	\end{figure}
1227	+
1228	+	\newpage
1229	+
1230	+	\begin{figure}
1231	+	\begin{center}
1232	+	\epsfxsize=6in
1233	+	\epsfbox{corrDiag.eps}
1234	+	%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.}
1235	+	\label{corrAngle}
1236	+	\end{center}
1237	+	\end{figure}
1238	+
1239	+	\newpage
1240	+
1241	+	\begin{figure}
1242		\begin{center}
1243		\epsfxsize=6in
1244	+	\epsfbox{fullContours.eps}
1245	+	%\caption{Contour plots of 2D angular pair correlation functions for
1246	+	%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas
1247	+	%signify regions of enhanced density while light areas signify
1248	+	%depletion relative to the bulk density. White areas have pair
1249	+	%correlation values below 0.5 and black areas have values above 1.5.}
1250	+	\label{contour}
1251	+	\end{center}
1252	+	\end{figure}
1253	+
1254	+	\newpage
1255	+
1256	+	\begin{figure}
1257	+	\begin{center}
1258	+	\epsfxsize=6in
1259	+	\epsfbox{GofRCompare.epsi}
1260	+	%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with
1261	+	%SSD/E and SSD1 without reaction field (top), as well as
1262	+	%SSD/RF and SSD1 with reaction field turned on
1263	+	%(bottom). The insets show the respective first peaks in detail. Note
1264	+	%how the changes in parameters have lowered and broadened the first
1265	+	%peak of SSD/E and SSD/RF.}
1266	+	\label{grcompare}
1267	+	\end{center}
1268	+	\end{figure}
1269	+
1270	+	\newpage
1271	+
1272	+	\begin{figure}
1273	+	\begin{center}
1274	+	\epsfxsize=7in
1275	+	\epsfbox{dualsticky_bw.eps}
1276	+	%\caption{Positive and negative isosurfaces of the sticky potential for
1277	+	%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas
1278	+	%correspond to the tetrahedral attractive component, and darker areas
1279	+	%correspond to the dipolar repulsive component.}
1280	+	\label{isosurface}
1281	+	\end{center}
1282	+	\end{figure}
1283	+
1284	+	\newpage
1285	+
1286	+	\begin{figure}
1287	+	\begin{center}
1288	+	\epsfxsize=6in
1289		\epsfbox{ssdeDense.epsi}
1290	<	\caption{Comparison of densities calculated with SSD/E to SSD1 without a
1291	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
1292	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a
1293	<	expansion around 300 K with error bars included to clarify this region
1294	<	of interest. Note that both SSD1 and SSD/E show good agreement with
1295	<	experiment when the long-range correction is neglected.}
1290	>	%\caption{Comparison of densities calculated with SSD/E to
1291	>	%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
1292	>	%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and
1293	>	%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around
1294	>	%300 K with error bars included to clarify this region of
1295	>	%interest. Note that both SSD1 and SSD/E show good agreement with
1296	>	%experiment when the long-range correction is neglected.}
1297		\label{ssdedense}
1298		\end{center}
1299		\end{figure}
1300
1301	<	Figure \ref{ssdedense} shows the density profile for the SSD/E model
713	<	in comparison to SSD1 without a reaction field, other common water
714	<	models, and experimental results. The calculated densities for both
715	<	SSD/E and SSD1 have increased significantly over the original SSD
716	<	model (see figure \ref{dense1}) and are in better agreement with the
717	<	experimental values. At 298 K, the densities of SSD/E and SSD1 without
718	<	a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and
719	<	0.999$\pm$0.001 g/cm$^3$ respectively.  These both compare well with
720	<	the experimental value of 0.997 g/cm$^3$, and they are considerably
721	<	better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to
722	<	the dipole moment and sticky switching functions have improved the
723	<	structuring of the liquid (as seen in figure \ref{grcompare}, but they
724	<	have shifted the density maximum to much lower temperatures. This
725	<	comes about via an increase in the liquid disorder through the
726	<	weakening of the sticky potential and strengthening of the dipolar
727	<	character. However, this increasing disorder in the SSD/E model has
728	<	little effect on the melting transition. By monitoring C$\text{p}$
729	<	throughout these simulations, the melting transition for SSD/E was
730	<	shown to occur at 235 K, the same transition temperature observed with
731	<	SSD and SSD1.
1301	>	\newpage
1302
1303		\begin{figure}
1304		\begin{center}
1305		\epsfxsize=6in
1306		\epsfbox{ssdrfDense.epsi}
1307	<	\caption{Comparison of densities calculated with SSD/RF to SSD1 with a
1308	<	reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00}
1309	<	SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the
1310	<	necessity of reparameterization when utilizing a reaction field
1311	<	long-ranged correction - SSD/RF provides significantly more accurate
1312	<	densities than SSD1 when performing room temperature simulations.}
1307	>	%\caption{Comparison of densities calculated with SSD/RF to
1308	>	%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}],
1309	>	%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and
1310	>	%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of
1311	>	%reparameterization when utilizing a reaction field long-ranged
1312	>	%correction - SSD/RF provides significantly more accurate
1313	>	%densities than SSD1 when performing room temperature
1314	>	%simulations.}
1315		\label{ssdrfdense}
1316		\end{center}
1317		\end{figure}
1318
1319	<	Including the reaction field long-range correction in the simulations
748	<	results in a more interesting comparison. A density profile including
749	<	SSD/RF and SSD1 with an active reaction field is shown in figure
750	<	\ref{ssdrfdense}.  As observed in the simulations without a reaction
751	<	field, the densities of SSD/RF and SSD1 show a dramatic increase over
752	<	normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density
753	<	of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and
754	<	considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and
755	<	the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further
756	<	emphasize the importance of reparameterization in order to model the
757	<	density properly under different simulation conditions. Again, these
758	<	changes have only a minor effect on the melting point, which observed
759	<	at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1
760	<	with a reaction field. Additionally, the difference in density maxima
761	<	is not as extreme, with SSD/RF showing a density maximum at 255 K,
762	<	fairly close to the density maxima of 260 K and 265 K, shown by SSD
763	<	and SSD1 respectively.
1319	>	\newpage
1320
1321	<	\begin{figure}
1321	>	\begin{figure}
1322		\begin{center}
1323		\epsfxsize=6in
1324		\epsfbox{ssdeDiffuse.epsi}
1325	<	\caption{Plots of the diffusion constants calculated from SSD/E and SSD1,
1326	<	both without a reaction field, along with experimental
1327	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
1328	<	at the average densities observed in the 1 atm NPT simulations for
1329	<	the respective models. SSD/E is slightly more fluid than experiment
1330	<	at all of the temperatures, but it is closer than SSD1 without a
1331	<	long-range correction.}
1325	>	%\caption{The diffusion constants calculated from SSD/E and
1326	>	%SSD1 (both without a reaction field) along with experimental results
1327	>	%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were
1328	>	%performed at the average densities observed in the 1 atm NPT
1329	>	%simulations for the respective models. SSD/E is slightly more mobile
1330	>	%than experiment at all of the temperatures, but it is closer to
1331	>	%experiment at biologically relevant temperatures than SSD1 without a
1332	>	%long-range correction.}
1333		\label{ssdediffuse}
1334		\end{center}
1335		\end{figure}
1336
1337	<	The reparameterization of the SSD water model, both for use with and
781	<	without an applied long-range correction, brought the densities up to
782	<	what is expected for simulating liquid water. In addition to improving
783	<	the densities, it is important that particle transport be maintained
784	<	or improved. Figure \ref{ssdediffuse} compares the temperature
785	<	dependence of the diffusion constant of SSD/E to SSD1 without an
786	<	active reaction field, both at the densities calculated at 1 atm and
787	<	at the experimentally calculated densities for super-cooled and liquid
788	<	water. The diffusion constant for SSD/E is consistently a little
789	<	higher than experiment, while SSD1 remains lower than experiment until
790	<	relatively high temperatures (greater than 330 K). Both models follow
791	<	the shape of the experimental curve well below 300 K but tend to
792	<	diffuse too rapidly at higher temperatures, something that is
793	<	especially apparent with SSD1. This accelerated increasing of
794	<	diffusion is caused by the rapidly decreasing system density with
795	<	increasing temperature. Though it is difficult to see in figure
796	<	\ref{ssdedense}, the densities of SSD1 decay more rapidly with
797	<	temperature than do those of SSD/E, leading to more visible deviation
798	<	from the experimental diffusion trend. Thus, the changes made to
799	<	improve the liquid structure may have had an adverse affect on the
800	<	density maximum, but they improve the transport behavior of SSD/E
801	<	relative to SSD1.
1337	>	\newpage
1338
1339		\begin{figure}
1340		\begin{center}
1341		\epsfxsize=6in
1342		\epsfbox{ssdrfDiffuse.epsi}
1343	<	\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1,
1344	<	both with an active reaction field, along with experimental
1345	<	results.\cite{Gillen72,Mills73} The NVE calculations were performed
1346	<	at the average densities observed in the 1 atm NPT simulations for
1347	<	both of the models. Note how accurately SSD/RF simulates the
1348	<	diffusion of water throughout this temperature range. The more
1349	<	rapidly increasing diffusion constants at high temperatures for both
1350	<	models is attributed to the significantly lower densities than
1351	<	observed in experiment.}
1343	>	%\caption{The diffusion constants calculated from SSD/RF and
1344	>	%SSD1 (both with an active reaction field) along with
1345	>	%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The
1346	>	%NVE calculations were performed at the average densities observed in
1347	>	%the 1 atm NPT simulations for both of the models. SSD/RF
1348	>	%simulates the diffusion of water throughout this temperature range
1349	>	%very well. The rapidly increasing diffusion constants at high
1350	>	%temperatures for both models can be attributed to lower calculated
1351	>	%densities than those observed in experiment.}
1352		\label{ssdrfdiffuse}
1353		\end{center}
1354		\end{figure}
1355
1356	<	In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are
821	<	compared to SSD1 with an active reaction field. Note that SSD/RF
822	<	tracks the experimental results incredibly well, identical within
823	<	error throughout the temperature range shown and with only a slight
824	<	increasing trend at higher temperatures. SSD1 tends to diffuse more
825	<	slowly at low temperatures and deviates to diffuse too rapidly at
826	<	temperatures greater than 330 K. As stated in relation to SSD/E, this
827	<	deviation away from the ideal trend is due to a rapid decrease in
828	<	density at higher temperatures. SSD/RF does not suffer from this
829	<	problem as much as SSD1, because the calculated densities are closer
830	<	to the experimental value. These results again emphasize the
831	<	importance of careful reparameterization when using an altered
832	<	long-range correction.
1356	>	\newpage
1357
834	–	\subsection{Additional Observations}
835	–
1358		\begin{figure}
1359		\begin{center}
1360		\epsfxsize=6in
1361	<	\epsfbox{povIce.ps}
1362	<	\caption{A water lattice built from the crystal structure assumed by
1363	<	SSD/E when undergoing an extremely restricted temperature NPT
1364	<	simulation. This form of ice is referred to as ice \emph{i} to
1365	<	emphasize its simulation origins. This image was taken of the (001)
844	<	face of the crystal.}
1361	>	\epsfbox{icei_bw.eps}
1362	>	%\caption{The most stable crystal structure assumed by the SSD family
1363	>	%of water models.  We refer to this structure as Ice-{\it i} to
1364	>	%indicate its origins in computer simulation.  This image was taken of
1365	>	%the (001) face of the crystal.}
1366		\label{weirdice}
1367		\end{center}
1368	<	\end{figure}
1368	>	\end{figure}
1369
849	–	While performing restricted temperature melting sequences of SSD/E not
850	–	previously discussed, some interesting observations were made. After
851	–	melting at 235 K, two of five systems underwent crystallization events
852	–	near 245 K. As the heating process continued, the two systems remained
853	–	crystalline until finally melting between 320 and 330 K. The final
854	–	configurations of these two melting sequences show an expanded
855	–	zeolite-like crystal structure that does not correspond to any known
856	–	form of ice. For convenience, and to help distinguish it from the
857	–	experimentally observed forms of ice, this crystal structure will
858	–	henceforth be referred to as ice $\sqrt{\smash[b]{-\text{I}}}$ (ice
859	–	\emph{i}). The crystallinity was extensive enough that a near ideal
860	–	crystal structure of ice \emph{i} could be obtained. Figure
861	–	\ref{weirdice} shows the repeating crystal structure of a typical
862	–	crystal at 5 K. Each water molecule is hydrogen bonded to four others;
863	–	however, the hydrogen bonds are flexed rather than perfectly
864	–	straight. This results in a skewed tetrahedral geometry about the
865	–	central molecule. Referring to figure \ref{isosurface}, these flexed
866	–	hydrogen bonds are allowed due to the conical shape of the attractive
867	–	regions, with the greatest attraction along the direct hydrogen bond
868	–	configuration. Though not ideal, these flexed hydrogen bonds are
869	–	favorable enough to stabilize an entire crystal generated around
870	–	them. In fact, the imperfect ice \emph{i} crystals were so stable that
871	–	they melted at temperatures nearly 100 K greater than both ice I$_c$
872	–	and I$_h$.
873	–
874	–	These initial simulations indicated that ice \emph{i} is the preferred
875	–	ice structure for at least the SSD/E model. To verify this, a
876	–	comparison was made between near ideal crystals of ice $I_h$, ice
877	–	$I_c$, and ice 0 at constant pressure with SSD/E, SSD/RF, and
878	–	SSD1. Near ideal versions of the three types of crystals were cooled
879	–	to 1 K, and the potential energies of each were compared using all
880	–	three water models. With every water model, ice \emph{i} turned out to
881	–	have the lowest potential energy: 5\% lower than $I_h$ with SSD1,
882	–	6.5\% lower with SSD/E, and 7.5\% lower with SSD/RF.
883	–
884	–	In addition to these low temperature comparisons, melting sequences
885	–	were performed with ice \emph{i} as the initial configuration using
886	–	SSD/E, SSD/RF, and SSD1 both with and without a reaction field. The
887	–	melting transitions for both SSD/E and SSD1 without a reaction field
888	–	occurred at temperature in excess of 375 K. SSD/RF and SSD1 with a
889	–	reaction field showed more reasonable melting transitions near 325
890	–	K. These melting point observations emphasize the preference for this
891	–	crystal structure over the most common types of ice when using these
892	–	single point water models.
893	–
894	–	Recognizing that the above tests show ice \emph{i} to be both the most
895	–	stable and lowest density crystal structure for these single point
896	–	water models, it is interesting to speculate on the relative stability
897	–	of this crystal structure with charge based water models. As a quick
898	–	test, these 3 crystal types were converted from SSD type particles to
899	–	TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy
900	–	minimizations were performed on the crystals to compare the system
901	–	energies. Again, ice \emph{i} was observed to have the lowest total
902	–	system energy. The total energy of ice \emph{i} was ~2\% lower than
903	–	ice $I_h$, which was in turn ~3\% lower than ice $I_c$. Based on these
904	–	initial studies, it would not be surprising if results from the other
905	–	common water models show ice \emph{i} to be the lowest energy crystal
906	–	structure. A continuation of this work studying ice \emph{i} with
907	–	multi-point water models will be published in a coming article.
908	–
909	–	\section{Conclusions}
910	–	The density maximum and temperature dependent transport for the SSD
911	–	water model, both with and without the use of reaction field, were
912	–	studied via a series of NPT and NVE simulations. The constant pressure
913	–	simulations of the melting of both $I_h$ and $I_c$ ice showed a
914	–	density maximum near 260 K. In most cases, the calculated densities
915	–	were significantly lower than the densities calculated in simulations
916	–	of other water models. Analysis of particle diffusion showed SSD to
917	–	capture the transport properties of experimental water well in both
918	–	the liquid and super-cooled liquid regimes. In order to correct the
919	–	density behavior, the original SSD model was reparameterized for use
920	–	both with and without a reaction field (SSD/RF and SSD/E), and
921	–	comparison simulations were performed with SSD1, the density corrected
922	–	version of SSD. Both models improve the liquid structure, density
923	–	values, and diffusive properties under their respective conditions,
924	–	indicating the necessity of reparameterization when altering the
925	–	long-range correction specifics. When taking into account the
926	–	appropriate considerations, these simple water models are excellent
927	–	choices for representing explicit water in large scale simulations of
928	–	biochemical systems.
929	–
930	–	\section{Acknowledgments}
931	–	Support for this project was provided by the National Science
932	–	Foundation under grant CHE-0134881. Computation time was provided by
933	–	the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
934	–	DMR 00 79647.
935	–
936	–
937	–	\newpage
938	–
939	–	\bibliographystyle{jcp}
940	–	\bibliography{nptSSD}
941	–
942	–	%\pagebreak
943	–
1370		\end{document}

Diff Legend

–	Removed lines
+	Added lines
<	Changed lines
>	Changed lines