| 1 |
– |
|
| 1 |
|
%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
| 2 |
< |
\documentclass[preprint,aps,endfloats]{revtex4} |
| 2 |
> |
\documentclass[11pt]{article} |
| 3 |
|
%\documentclass[11pt]{article} |
| 4 |
< |
%\usepackage{endfloat} |
| 4 |
> |
\usepackage{endfloat} |
| 5 |
|
\usepackage{amsmath} |
| 6 |
|
\usepackage{epsf} |
| 7 |
|
\usepackage{berkeley} |
| 8 |
< |
%\usepackage{setspace} |
| 9 |
< |
%\usepackage{tabularx} |
| 8 |
> |
\usepackage{setspace} |
| 9 |
> |
\usepackage{tabularx} |
| 10 |
|
\usepackage{graphicx} |
| 11 |
< |
%\usepackage[ref]{overcite} |
| 12 |
< |
%\pagestyle{plain} |
| 13 |
< |
%\pagenumbering{arabic} |
| 14 |
< |
%\oddsidemargin 0.0cm \evensidemargin 0.0cm |
| 15 |
< |
%\topmargin -21pt \headsep 10pt |
| 16 |
< |
%\textheight 9.0in \textwidth 6.5in |
| 17 |
< |
%\brokenpenalty=10000 |
| 11 |
> |
\usepackage[ref]{overcite} |
| 12 |
> |
\pagestyle{plain} |
| 13 |
> |
\pagenumbering{arabic} |
| 14 |
> |
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
| 15 |
> |
\topmargin -21pt \headsep 10pt |
| 16 |
> |
\textheight 9.0in \textwidth 6.5in |
| 17 |
> |
\brokenpenalty=10000 |
| 18 |
> |
\renewcommand{\baselinestretch}{1.2} |
| 19 |
> |
\renewcommand\citemid{\ } % no comma in optional reference note |
| 20 |
|
|
| 20 |
– |
%\renewcommand\citemid{\ } % no comma in optional reference note |
| 21 |
– |
|
| 21 |
|
\begin{document} |
| 22 |
|
|
| 23 |
< |
\title{A Free Energy Study of Low Temperature and Anomalous Ice} |
| 23 |
> |
\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more |
| 24 |
> |
stable than Ice $I_h$ for point-charge and point-dipole water models} |
| 25 |
|
|
| 26 |
< |
\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
| 27 |
< |
\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
| 28 |
< |
|
| 29 |
< |
\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
| 26 |
> |
\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
| 27 |
> |
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
| 28 |
|
Notre Dame, Indiana 46556} |
| 29 |
|
|
| 30 |
|
\date{\today} |
| 31 |
|
|
| 32 |
< |
%\maketitle |
| 32 |
> |
\maketitle |
| 33 |
|
%\doublespacing |
| 34 |
|
|
| 35 |
|
\begin{abstract} |
| 36 |
|
The free energies of several ice polymorphs in the low pressure regime |
| 37 |
< |
were calculated using thermodynamic integration of systems consisting |
| 38 |
< |
of a variety of common water models. Ice-{\it i}, a recent |
| 39 |
< |
computationally observed solid structure, was determined to be the |
| 40 |
< |
stable state with the lowest free energy for all the water models |
| 41 |
< |
investigated. Phase diagrams were generated, and melting and boiling |
| 42 |
< |
points for all the models were determined and show relatively good |
| 43 |
< |
agreement with experiment, although the solid phase is different |
| 44 |
< |
between simulation and experiment. In addition, potential truncation |
| 45 |
< |
was shown to have an effect on the calculated free energies, and may |
| 46 |
< |
result in altered free energy landscapes. |
| 37 |
> |
were calculated using thermodynamic integration. These integrations |
| 38 |
> |
were done for most of the common water models. Ice-{\it i}, a |
| 39 |
> |
structure we recently observed to be stable in one of the single-point |
| 40 |
> |
water models, was determined to be the stable crystalline state (at 1 |
| 41 |
> |
atm) for {\it all} the water models investigated. Phase diagrams were |
| 42 |
> |
generated, and phase coexistence lines were determined for all of the |
| 43 |
> |
known low-pressure ice structures under all of the common water |
| 44 |
> |
models. Additionally, potential truncation was shown to have an |
| 45 |
> |
effect on the calculated free energies, and can result in altered free |
| 46 |
> |
energy landscapes. |
| 47 |
|
\end{abstract} |
| 48 |
|
|
| 51 |
– |
\maketitle |
| 52 |
– |
|
| 53 |
– |
\newpage |
| 54 |
– |
|
| 49 |
|
%\narrowtext |
| 50 |
|
|
| 51 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 54 |
|
|
| 55 |
|
\section{Introduction} |
| 56 |
|
|
| 57 |
< |
Molecular dynamics has developed into a valuable tool for studying the |
| 58 |
< |
phase behavior of systems ranging from small or simple |
| 59 |
< |
molecules\cite{smallStuff} to complex biological |
| 60 |
< |
species.\cite{bigStuff} Many techniques have been developed in order |
| 61 |
< |
to investigate the thermodynamic properites of model substances, |
| 62 |
< |
providing both qualitative and quantitative comparisons between |
| 63 |
< |
simulations and experiment.\cite{thermMethods} Investigation of these |
| 64 |
< |
properties leads to the development of new and more accurate models, |
| 65 |
< |
leading to better understanding and depiction of physical processes |
| 66 |
< |
and intricate molecular systems. |
| 57 |
> |
Computer simulations are a valuable tool for studying the phase |
| 58 |
> |
behavior of systems ranging from small or simple molecules to complex |
| 59 |
> |
biological species.\cite{Matsumoto02,Sanz04,Marrink01} Useful techniques |
| 60 |
> |
have been developed to investigate the thermodynamic properites of |
| 61 |
> |
model substances, providing both qualitative and quantitative |
| 62 |
> |
comparisons between simulations and |
| 63 |
> |
experiment.\cite{Widom63,Frenkel84} Investigation of these properties |
| 64 |
> |
leads to the development of new and more accurate models, leading to |
| 65 |
> |
better understanding and depiction of physical processes and intricate |
| 66 |
> |
molecular systems. |
| 67 |
|
|
| 68 |
|
Water has proven to be a challenging substance to depict in |
| 69 |
< |
simulations, and has resulted in a variety of models that attempt to |
| 70 |
< |
describe its behavior under a varying simulation |
| 71 |
< |
conditions.\cite{lotsOfWaterPapers} Many of these models have been |
| 72 |
< |
used to investigate important physical phenomena like phase |
| 73 |
< |
transitions and the hydrophobic effect.\cite{evenMorePapers} With the |
| 74 |
< |
advent of numerous differing models, it is only natural that attention |
| 75 |
< |
is placed on the properties of the models themselves in an attempt to |
| 76 |
< |
clarify their benefits and limitations when applied to a system of |
| 77 |
< |
interest.\cite{modelProps} One important but challenging property to |
| 78 |
< |
quantify is the free energy, particularly of the solid forms of |
| 79 |
< |
water. Difficulty in these types of studies typically arises from the |
| 80 |
< |
assortment of possible crystalline polymorphs that water that water |
| 81 |
< |
adopts over a wide range of pressures and temperatures. There are |
| 82 |
< |
currently 13 recognized forms of ice, and it is a challenging task to |
| 83 |
< |
investigate the entire free energy landscape.\cite{Sanz04} Ideally, |
| 84 |
< |
research is focused on the phases having the lowest free energy, |
| 85 |
< |
because these phases will dictate the true transition temperatures and |
| 86 |
< |
pressures for their respective model. |
| 69 |
> |
simulations, and a variety of models have been developed to describe |
| 70 |
> |
its behavior under varying simulation |
| 71 |
> |
conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} |
| 72 |
> |
These models have been used to investigate important physical |
| 73 |
> |
phenomena like phase transitions, molecule transport, and the |
| 74 |
> |
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
| 75 |
> |
choice of models available, it is only natural to compare the models |
| 76 |
> |
under interesting thermodynamic conditions in an attempt to clarify |
| 77 |
> |
the limitations of each of the |
| 78 |
> |
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
| 79 |
> |
important property to quantify are the Gibbs and Helmholtz free |
| 80 |
> |
energies, particularly for the solid forms of water. Difficulty in |
| 81 |
> |
these types of studies typically arises from the assortment of |
| 82 |
> |
possible crystalline polymorphs that water adopts over a wide range of |
| 83 |
> |
pressures and temperatures. There are currently 13 recognized forms |
| 84 |
> |
of ice, and it is a challenging task to investigate the entire free |
| 85 |
> |
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
| 86 |
> |
phases having the lowest free energy at a given state point, because |
| 87 |
> |
these phases will dictate the true transition temperatures and |
| 88 |
> |
pressures for the model. |
| 89 |
|
|
| 90 |
< |
In this paper, standard reference state methods were applied to the |
| 91 |
< |
study of crystalline water polymorphs in the low pressure regime. This |
| 92 |
< |
work is unique in the fact that one of the crystal lattices was |
| 93 |
< |
arrived at through crystallization of a computationally efficient |
| 94 |
< |
water model under constant pressure and temperature |
| 95 |
< |
conditions. Crystallization events are interesting in and of |
| 96 |
< |
themselves\cite{nucleationStudies}; however, the crystal structure |
| 97 |
< |
obtained in this case was different from any previously observed ice |
| 98 |
< |
polymorphs, in experiment or simulation.\cite{Fennell04} This crystal |
| 99 |
< |
was termed Ice-{\it i} in homage to its origin in computational |
| 90 |
> |
In this paper, standard reference state methods were applied to known |
| 91 |
> |
crystalline water polymorphs in the low pressure regime. This work is |
| 92 |
> |
unique in the fact that one of the crystal lattices was arrived at |
| 93 |
> |
through crystallization of a computationally efficient water model |
| 94 |
> |
under constant pressure and temperature conditions. Crystallization |
| 95 |
> |
events are interesting in and of |
| 96 |
> |
themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure |
| 97 |
> |
obtained in this case is different from any previously observed ice |
| 98 |
> |
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
| 99 |
> |
this structure Ice-{\it i} to indicate its origin in computational |
| 100 |
|
simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight |
| 101 |
|
water molecules that stack in rows of interlocking water |
| 102 |
|
tetramers. Proton ordering can be accomplished by orienting two of the |
| 103 |
< |
waters so that both of their donated hydrogen bonds are internal to |
| 103 |
> |
molecules so that both of their donated hydrogen bonds are internal to |
| 104 |
|
their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
| 105 |
|
constructed of water tetramers, the hydrogen bonds are not as linear |
| 106 |
|
as those observed in ice $I_h$, however the interlocking of these |
| 109 |
|
open octagonal cavities that are typically greater than 6.3 \AA\ in |
| 110 |
|
diameter. This relatively open overall structure leads to crystals |
| 111 |
|
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
| 112 |
+ |
|
| 113 |
|
\begin{figure} |
| 114 |
< |
\includegraphics[scale=1.0]{unitCell.eps} |
| 115 |
< |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-2{\it i}, the elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ relation is given by $a = 2.1214c$, while for Ice-2{\it i}, $a = 1.7850c$.} |
| 114 |
> |
\includegraphics[width=\linewidth]{unitCell.eps} |
| 115 |
> |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
| 116 |
> |
elongated variant of Ice-{\it i}. The spheres represent the |
| 117 |
> |
center-of-mass locations of the water molecules. The $a$ to $c$ |
| 118 |
> |
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
| 119 |
> |
$a:2.1214c$ and $a:1.7850c$ respectively.} |
| 120 |
|
\label{iceiCell} |
| 121 |
|
\end{figure} |
| 122 |
+ |
|
| 123 |
|
\begin{figure} |
| 124 |
< |
\includegraphics[scale=1.0]{orderedIcei.eps} |
| 124 |
> |
\includegraphics[width=\linewidth]{orderedIcei.eps} |
| 125 |
|
\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
| 126 |
|
down the (001) crystal face. The rows of water tetramers surrounded by |
| 127 |
|
octagonal pores leads to a crystal structure that is significantly |
| 129 |
|
\label{protOrder} |
| 130 |
|
\end{figure} |
| 131 |
|
|
| 132 |
< |
Results in the previous study indicated that Ice-{\it i} is the |
| 133 |
< |
minimum energy crystal structure for the single point water models |
| 134 |
< |
being studied (for discussions on these single point dipole models, |
| 135 |
< |
see the previous work and related |
| 136 |
< |
articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only |
| 137 |
< |
consider energetic stabilization and neglect entropic contributions to |
| 138 |
< |
the overall free energy. To address this issue, the absolute free |
| 139 |
< |
energy of this crystal was calculated using thermodynamic integration |
| 140 |
< |
and compared to the free energies of cubic and hexagonal ice $I$ (the |
| 141 |
< |
experimental low density ice polymorphs) and ice B (a higher density, |
| 142 |
< |
but very stable crystal structure observed by B\`{a}ez and Clancy in |
| 143 |
< |
free energy studies of SPC/E).\cite{Baez95b} This work includes |
| 144 |
< |
results for the water model from which Ice-{\it i} was crystallized |
| 145 |
< |
(soft sticky dipole extended, SSD/E) in addition to several common |
| 146 |
< |
water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field |
| 147 |
< |
parametrized single point dipole water model (soft sticky dipole |
| 148 |
< |
reaction field, SSD/RF). In should be noted that a second version of |
| 149 |
< |
Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E, |
| 150 |
< |
TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B) |
| 151 |
< |
is similar to the Ice-{\it i} unit it is extended in the direction of |
| 152 |
< |
the (001) face and compressed along the other two faces. |
| 132 |
> |
Results from our previous study indicated that Ice-{\it i} is the |
| 133 |
> |
minimum energy crystal structure for the single point water models we |
| 134 |
> |
investigated (for discussions on these single point dipole models, see |
| 135 |
> |
our previous work and related |
| 136 |
> |
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
| 137 |
> |
considered energetic stabilization and neglected entropic |
| 138 |
> |
contributions to the overall free energy. To address this issue, the |
| 139 |
> |
absolute free energy of this crystal was calculated using |
| 140 |
> |
thermodynamic integration and compared to the free energies of cubic |
| 141 |
> |
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
| 142 |
> |
and ice B (a higher density, but very stable crystal structure |
| 143 |
> |
observed by B\`{a}ez and Clancy in free energy studies of |
| 144 |
> |
SPC/E).\cite{Baez95b} This work includes results for the water model |
| 145 |
> |
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
| 146 |
> |
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
| 147 |
> |
field parametrized single point dipole water model (SSD/RF). It should |
| 148 |
> |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used |
| 149 |
> |
in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
| 150 |
> |
this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
| 151 |
> |
it is extended in the direction of the (001) face and compressed along |
| 152 |
> |
the other two faces. |
| 153 |
|
|
| 154 |
|
\section{Methods} |
| 155 |
|
|
| 156 |
|
Canonical ensemble (NVT) molecular dynamics calculations were |
| 157 |
< |
performed using the OOPSE (Object-Oriented Parallel Simulation Engine) |
| 158 |
< |
molecular mechanics package. All molecules were treated as rigid |
| 159 |
< |
bodies, with orientational motion propagated using the symplectic DLM |
| 160 |
< |
integration method. Details about the implementation of these |
| 161 |
< |
techniques can be found in a recent publication.\cite{Meineke05} |
| 157 |
> |
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
| 158 |
> |
All molecules were treated as rigid bodies, with orientational motion |
| 159 |
> |
propagated using the symplectic DLM integration method. Details about |
| 160 |
> |
the implementation of this technique can be found in a recent |
| 161 |
> |
publication.\cite{Dullweber1997} |
| 162 |
|
|
| 163 |
< |
Thermodynamic integration was utilized to calculate the free energy of |
| 164 |
< |
several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
| 165 |
< |
SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
| 166 |
< |
400 K for all of these water models were also determined using this |
| 167 |
< |
same technique, in order to determine melting points and generate |
| 163 |
> |
Thermodynamic integration is an established technique for |
| 164 |
> |
determination of free energies of condensed phases of |
| 165 |
> |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
| 166 |
> |
method, implemented in the same manner illustrated by B\`{a}ez and |
| 167 |
> |
Clancy, was utilized to calculate the free energy of several ice |
| 168 |
> |
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
| 169 |
> |
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
| 170 |
> |
and 400 K for all of these water models were also determined using |
| 171 |
> |
this same technique in order to determine melting points and generate |
| 172 |
|
phase diagrams. All simulations were carried out at densities |
| 173 |
|
resulting in a pressure of approximately 1 atm at their respective |
| 174 |
|
temperatures. |
| 175 |
|
|
| 176 |
|
A single thermodynamic integration involves a sequence of simulations |
| 177 |
|
over which the system of interest is converted into a reference system |
| 178 |
< |
for which the free energy is known. This transformation path is then |
| 179 |
< |
integrated in order to determine the free energy difference between |
| 180 |
< |
the two states: |
| 178 |
> |
for which the free energy is known analytically. This transformation |
| 179 |
> |
path is then integrated in order to determine the free energy |
| 180 |
> |
difference between the two states: |
| 181 |
|
\begin{equation} |
| 182 |
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
| 183 |
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
| 184 |
|
\end{equation} |
| 185 |
|
where $V$ is the interaction potential and $\lambda$ is the |
| 186 |
|
transformation parameter that scales the overall |
| 187 |
< |
potential. Simulations are distributed unevenly along this path in |
| 188 |
< |
order to sufficiently sample the regions of greatest change in the |
| 187 |
> |
potential. Simulations are distributed strategically along this path |
| 188 |
> |
in order to sufficiently sample the regions of greatest change in the |
| 189 |
|
potential. Typical integrations in this study consisted of $\sim$25 |
| 190 |
|
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
| 191 |
|
(near the reference state) in length. |
| 192 |
|
|
| 193 |
|
For the thermodynamic integration of molecular crystals, the Einstein |
| 194 |
< |
Crystal is chosen as the reference state that the system is converted |
| 195 |
< |
to over the course of the simulation. In an Einstein Crystal, the |
| 196 |
< |
molecules are harmonically restrained at their ideal lattice locations |
| 197 |
< |
and orientations. The partition function for a molecular crystal |
| 198 |
< |
restrained in this fashion has been evaluated, and the Helmholtz Free |
| 199 |
< |
Energy ({\it A}) is given by |
| 194 |
> |
crystal was chosen as the reference system. In an Einstein crystal, |
| 195 |
> |
the molecules are restrained at their ideal lattice locations and |
| 196 |
> |
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
| 197 |
> |
Clancy, the total potential for this reference crystal |
| 198 |
> |
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
| 199 |
> |
\begin{equation} |
| 200 |
> |
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
| 201 |
> |
\frac{K_\omega\omega^2}{2}, |
| 202 |
> |
\end{equation} |
| 203 |
> |
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
| 204 |
> |
the spring constants restraining translational motion and deflection |
| 205 |
> |
of and rotation around the principle axis of the molecule |
| 206 |
> |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
| 207 |
> |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
| 208 |
> |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
| 209 |
> |
restrained in this fashion can be evaluated analytically, and the |
| 210 |
> |
Helmholtz Free Energy ({\it A}) is given by |
| 211 |
|
\begin{eqnarray} |
| 212 |
|
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
| 213 |
|
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
| 219 |
|
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
| 220 |
|
\label{ecFreeEnergy} |
| 221 |
|
\end{eqnarray} |
| 222 |
< |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
| 223 |
< |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
| 224 |
< |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
| 208 |
< |
motion and deflection of and rotation around the principle axis of the |
| 209 |
< |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
| 210 |
< |
minimum potential energy of the ideal crystal. In the case of |
| 211 |
< |
molecular liquids, the ideal vapor is chosen as the target reference |
| 212 |
< |
state. |
| 222 |
> |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
| 223 |
> |
potential energy of the ideal crystal.\cite{Baez95a} |
| 224 |
> |
|
| 225 |
|
\begin{figure} |
| 226 |
< |
\includegraphics[scale=1.0]{rotSpring.eps} |
| 226 |
> |
\includegraphics[width=\linewidth]{rotSpring.eps} |
| 227 |
|
\caption{Possible orientational motions for a restrained molecule. |
| 228 |
|
$\theta$ angles correspond to displacement from the body-frame {\it |
| 229 |
|
z}-axis, while $\omega$ angles correspond to rotation about the |
| 233 |
|
\label{waterSpring} |
| 234 |
|
\end{figure} |
| 235 |
|
|
| 236 |
+ |
In the case of molecular liquids, the ideal vapor is chosen as the |
| 237 |
+ |
target reference state. There are several examples of liquid state |
| 238 |
+ |
free energy calculations of water models present in the |
| 239 |
+ |
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
| 240 |
+ |
typically differ in regard to the path taken for switching off the |
| 241 |
+ |
interaction potential to convert the system to an ideal gas of water |
| 242 |
+ |
molecules. In this study, we apply of one of the most convenient |
| 243 |
+ |
methods and integrate over the $\lambda^4$ path, where all interaction |
| 244 |
+ |
parameters are scaled equally by this transformation parameter. This |
| 245 |
+ |
method has been shown to be reversible and provide results in |
| 246 |
+ |
excellent agreement with other established methods.\cite{Baez95b} |
| 247 |
+ |
|
| 248 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
| 249 |
< |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By |
| 250 |
< |
applying this function, these interactions are smoothly truncated, |
| 251 |
< |
thereby avoiding poor energy conserving dynamics resulting from |
| 252 |
< |
harsher truncation schemes. The effect of a long-range correction was |
| 253 |
< |
also investigated on select model systems in a variety of manners. For |
| 254 |
< |
the SSD/RF model, a reaction field with a fixed dielectric constant of |
| 255 |
< |
80 was applied in all simulations.\cite{Onsager36} For a series of the |
| 256 |
< |
least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
| 257 |
< |
simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
| 258 |
< |
compare with the 9 \AA\ cutoff results. Finally, results from the use |
| 259 |
< |
of an Ewald summation were estimated for TIP3P and SPC/E by performing |
| 249 |
> |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
| 250 |
> |
). By applying this function, these interactions are smoothly |
| 251 |
> |
truncated, thereby avoiding the poor energy conservation which results |
| 252 |
> |
from harsher truncation schemes. The effect of a long-range correction |
| 253 |
> |
was also investigated on select model systems in a variety of |
| 254 |
> |
manners. For the SSD/RF model, a reaction field with a fixed |
| 255 |
> |
dielectric constant of 80 was applied in all |
| 256 |
> |
simulations.\cite{Onsager36} For a series of the least computationally |
| 257 |
> |
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 258 |
> |
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 259 |
> |
\AA\ cutoff results. Finally, results from the use of an Ewald |
| 260 |
> |
summation were estimated for TIP3P and SPC/E by performing |
| 261 |
|
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
| 262 |
< |
mechanics software package. TINKER was chosen because it can also |
| 263 |
< |
propagate the motion of rigid-bodies, and provides the most direct |
| 264 |
< |
comparison to the results from OOPSE. The calculated energy difference |
| 265 |
< |
in the presence and absence of PME was applied to the previous results |
| 241 |
< |
in order to predict changes in the free energy landscape. |
| 262 |
> |
mechanics software package.\cite{Tinker} The calculated energy |
| 263 |
> |
difference in the presence and absence of PME was applied to the |
| 264 |
> |
previous results in order to predict changes to the free energy |
| 265 |
> |
landscape. |
| 266 |
|
|
| 267 |
|
\section{Results and discussion} |
| 268 |
|
|
| 272 |
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 273 |
|
and thought to be the minimum free energy structure for the SPC/E |
| 274 |
|
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 275 |
< |
Ice XI, the experimentally observed proton ordered variant of ice |
| 276 |
< |
$I_h$, was investigated initially, but it was found not to be as |
| 277 |
< |
stable as antiferroelectric variants of proton ordered or even proton |
| 278 |
< |
disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
| 279 |
< |
ice $I_h$ used here is a simple antiferroelectric version that has an |
| 280 |
< |
8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
| 281 |
< |
ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
| 282 |
< |
$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
| 275 |
> |
Ice XI, the experimentally-observed proton-ordered variant of ice |
| 276 |
> |
$I_h$, was investigated initially, but was found to be not as stable |
| 277 |
> |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 278 |
> |
proton ordered variant of ice $I_h$ used here is a simple |
| 279 |
> |
antiferroelectric version that has an 8 molecule unit |
| 280 |
> |
cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
| 281 |
> |
for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
| 282 |
> |
ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
| 283 |
|
were necessary for simulations involving larger cutoff values. |
| 284 |
|
|
| 285 |
|
\begin{table*} |
| 289 |
|
\caption{Calculated free energies for several ice polymorphs with a |
| 290 |
|
variety of common water models. All calculations used a cutoff radius |
| 291 |
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 292 |
< |
kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
| 292 |
> |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
| 293 |
> |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
| 294 |
|
\begin{tabular}{ l c c c c } |
| 295 |
< |
\hline \\[-7mm] |
| 296 |
< |
\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
| 297 |
< |
\hline \\[-3mm] |
| 298 |
< |
\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
| 299 |
< |
\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
| 300 |
< |
\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
| 301 |
< |
\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
| 302 |
< |
\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
| 303 |
< |
\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
| 295 |
> |
\hline |
| 296 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 297 |
> |
\hline |
| 298 |
> |
TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ |
| 299 |
> |
TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ |
| 300 |
> |
TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ |
| 301 |
> |
SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ |
| 302 |
> |
SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ |
| 303 |
> |
SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ |
| 304 |
|
\end{tabular} |
| 305 |
|
\label{freeEnergy} |
| 306 |
|
\end{center} |
| 310 |
|
The free energy values computed for the studied polymorphs indicate |
| 311 |
|
that Ice-{\it i} is the most stable state for all of the common water |
| 312 |
|
models studied. With the free energy at these state points, the |
| 313 |
< |
temperature and pressure dependence of the free energy was used to |
| 314 |
< |
project to other state points and build phase diagrams. Figures |
| 313 |
> |
Gibbs-Helmholtz equation was used to project to other state points and |
| 314 |
> |
to build phase diagrams. Figures |
| 315 |
|
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 316 |
|
from the free energy results. All other models have similar structure, |
| 317 |
< |
only the crossing points between these phases exist at different |
| 318 |
< |
temperatures and pressures. It is interesting to note that ice $I$ |
| 319 |
< |
does not exist in either cubic or hexagonal form in any of the phase |
| 320 |
< |
diagrams for any of the models. For purposes of this study, ice B is |
| 321 |
< |
representative of the dense ice polymorphs. A recent study by Sanz |
| 322 |
< |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 323 |
< |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 317 |
> |
although the crossing points between the phases exist at slightly |
| 318 |
> |
different temperatures and pressures. It is interesting to note that |
| 319 |
> |
ice $I$ does not exist in either cubic or hexagonal form in any of the |
| 320 |
> |
phase diagrams for any of the models. For purposes of this study, ice |
| 321 |
> |
B is representative of the dense ice polymorphs. A recent study by |
| 322 |
> |
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 323 |
> |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 324 |
> |
|
| 325 |
|
\begin{figure} |
| 326 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 327 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 331 |
|
higher in energy and don't appear in the phase diagram.} |
| 332 |
|
\label{tp3phasedia} |
| 333 |
|
\end{figure} |
| 334 |
+ |
|
| 335 |
|
\begin{figure} |
| 336 |
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 337 |
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 348 |
|
\renewcommand{\thefootnote}{\thempfootnote} |
| 349 |
|
\begin{center} |
| 350 |
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 351 |
< |
temperatures of several common water models compared with experiment.} |
| 351 |
> |
temperatures at 1 atm for several common water models compared with |
| 352 |
> |
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
| 353 |
> |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 354 |
> |
liquid or gas state.} |
| 355 |
|
\begin{tabular}{ l c c c c c c c } |
| 356 |
< |
\hline \\[-7mm] |
| 357 |
< |
\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
| 358 |
< |
\hline \\[-3mm] |
| 359 |
< |
\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
| 360 |
< |
\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
| 361 |
< |
\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
| 356 |
> |
\hline |
| 357 |
> |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 358 |
> |
\hline |
| 359 |
> |
$T_m$ (K) & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ |
| 360 |
> |
$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ |
| 361 |
> |
$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ |
| 362 |
|
\end{tabular} |
| 363 |
|
\label{meltandboil} |
| 364 |
|
\end{center} |
| 374 |
|
studies in the literature. Earlier free energy studies of ice $I$ |
| 375 |
|
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 376 |
|
being attributed to choice of interaction truncation and different |
| 377 |
< |
ordered and disordered molecular arrangements). If the presence of ice |
| 378 |
< |
B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 377 |
> |
ordered and disordered molecular |
| 378 |
> |
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 379 |
> |
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 380 |
|
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 381 |
|
calculated at 265 K, significantly higher in temperature than the |
| 382 |
|
previous studies. Also of interest in these results is that SSD/E does |
| 395 |
|
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
| 396 |
|
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
| 397 |
|
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
| 398 |
< |
\AA\. These crystals are unstable at 200 K and rapidly convert into a |
| 399 |
< |
liquid. The connecting lines are qualitative visual aid.} |
| 398 |
> |
\AA . These crystals are unstable at 200 K and rapidly convert into |
| 399 |
> |
liquids. The connecting lines are qualitative visual aid.} |
| 400 |
|
\label{incCutoff} |
| 401 |
|
\end{figure} |
| 402 |
|
|
| 411 |
|
in the SSD/E model that the liquid state is preferred under standard |
| 412 |
|
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 413 |
|
simulations using this model choose interaction truncation radii |
| 414 |
< |
greater than 9 \AA\. This narrowing trend is much more subtle in the |
| 414 |
> |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 415 |
|
case of SSD/RF, indicating that the free energies calculated with a |
| 416 |
|
reaction field present provide a more accurate picture of the free |
| 417 |
|
energy landscape in the absence of potential truncation. |
| 423 |
|
correction. This was accomplished by calculation of the potential |
| 424 |
|
energy of identical crystals with and without PME using TINKER. The |
| 425 |
|
free energies for the investigated polymorphs using the TIP3P and |
| 426 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
| 427 |
< |
are not fully supported in TINKER, so the results for these models |
| 428 |
< |
could not be estimated. The same trend pointed out through increase of |
| 429 |
< |
cutoff radius is observed in these PME results. Ice-{\it i} is the |
| 430 |
< |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
| 431 |
< |
water models; however, there is a narrowing of the free energy |
| 432 |
< |
differences between the various solid forms. In the case of SPC/E this |
| 433 |
< |
narrowing is significant enough that it becomes less clear cut that |
| 434 |
< |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
| 435 |
< |
with respect to ice B and possibly ice $I_c$. However, these results |
| 436 |
< |
do not significantly alter the finding that the Ice-{\it i} polymorph |
| 437 |
< |
is a stable crystal structure that should be considered when studying |
| 438 |
< |
the phase behavior of water models. |
| 426 |
> |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
| 427 |
> |
pointed out through increase of cutoff radius is observed in these PME |
| 428 |
> |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 429 |
> |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 430 |
> |
the free energy differences between the various solid forms is |
| 431 |
> |
significant enough that it becomes less clear that it is the most |
| 432 |
> |
stable polymorph. The free energies of Ice-{\it i} and ice B overlap |
| 433 |
> |
within error, with ice $I_c$ just outside, indicating that Ice-{\it i} |
| 434 |
> |
might be metastable with respect to ice B and possibly ice $I_c$ in |
| 435 |
> |
the SPC/E water model. However, these results do not significantly |
| 436 |
> |
alter the finding that the Ice-{\it i} polymorph is a stable crystal |
| 437 |
> |
structure that should be considered when studying the phase behavior |
| 438 |
> |
of water models. |
| 439 |
|
|
| 440 |
|
\begin{table*} |
| 441 |
|
\begin{minipage}{\linewidth} |
| 445 |
|
the energy difference attributed to the inclusion of the PME |
| 446 |
|
long-range interaction correction. Units are kcal/mol.} |
| 447 |
|
\begin{tabular}{ l c c c c } |
| 448 |
< |
\hline \\[-7mm] |
| 448 |
> |
\hline |
| 449 |
|
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 450 |
< |
\hline \\[-3mm] |
| 451 |
< |
\ \ TIP3P & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\ |
| 452 |
< |
\ \ SPC/E & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\ |
| 450 |
> |
\hline |
| 451 |
> |
TIP3P & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ |
| 452 |
> |
SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ |
| 453 |
|
\end{tabular} |
| 454 |
|
\label{pmeShift} |
| 455 |
|
\end{center} |
| 459 |
|
\section{Conclusions} |
| 460 |
|
|
| 461 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
| 462 |
< |
$I$, ice B, and recently discovered Ice-{\it i} where calculated under |
| 462 |
> |
$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
| 463 |
|
standard conditions for several common water models via thermodynamic |
| 464 |
|
integration. All the water models studied show Ice-{\it i} to be the |
| 465 |
|
minimum free energy crystal structure in the with a 9 \AA\ switching |
| 468 |
|
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
| 469 |
|
interaction truncation was investigated through variation of the |
| 470 |
|
cutoff radius, use of a reaction field parameterized model, and |
| 471 |
< |
estimation of the results in the presence of the Ewald summation |
| 472 |
< |
correction. Interaction truncation has a significant effect on the |
| 471 |
> |
estimation of the results in the presence of the Ewald |
| 472 |
> |
summation. Interaction truncation has a significant effect on the |
| 473 |
|
computed free energy values, and may significantly alter the free |
| 474 |
|
energy landscape for the more complex multipoint water models. Despite |
| 475 |
|
these effects, these results show Ice-{\it i} to be an important ice |
| 477 |
|
|
| 478 |
|
Due to this relative stability of Ice-{\it i} in all manner of |
| 479 |
|
investigated simulation examples, the question arises as to possible |
| 480 |
< |
experimental observation of this polymorph. The rather extensive past |
| 480 |
> |
experimental observation of this polymorph. The rather extensive past |
| 481 |
|
and current experimental investigation of water in the low pressure |
| 482 |
< |
regime leads the authors to be hesitant in ascribing relevance outside |
| 483 |
< |
of computational models, hence the descriptive name presented. That |
| 484 |
< |
being said, there are certain experimental conditions that would |
| 485 |
< |
provide the most ideal situation for possible observation. These |
| 486 |
< |
include the negative pressure or stretched solid regime, small |
| 487 |
< |
clusters in vacuum deposition environments, and in clathrate |
| 488 |
< |
structures involving small non-polar molecules. |
| 482 |
> |
regime makes us hesitant to ascribe any relevance of this work outside |
| 483 |
> |
of the simulation community. It is for this reason that we chose a |
| 484 |
> |
name for this polymorph which involves an imaginary quantity. That |
| 485 |
> |
said, there are certain experimental conditions that would provide the |
| 486 |
> |
most ideal situation for possible observation. These include the |
| 487 |
> |
negative pressure or stretched solid regime, small clusters in vacuum |
| 488 |
> |
deposition environments, and in clathrate structures involving small |
| 489 |
> |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
| 490 |
> |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
| 491 |
> |
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
| 492 |
> |
i} at a temperature of 77K. In a quick comparison of the predicted |
| 493 |
> |
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
| 494 |
> |
water, it is possible that some of the ``spurious'' peaks that could |
| 495 |
> |
not be assigned in HDA could correspond to peaks labeled in this |
| 496 |
> |
S(q).\cite{Bizid87} It should be noted that there is typically poor |
| 497 |
> |
agreement on crystal densities between simulation and experiment, so |
| 498 |
> |
such peak comparisons should be made with caution. We will leave it |
| 499 |
> |
to our experimental colleagues to determine whether this ice polymorph |
| 500 |
> |
is named appropriately or if it should be promoted to Ice-0. |
| 501 |
|
|
| 502 |
+ |
\begin{figure} |
| 503 |
+ |
\includegraphics[width=\linewidth]{iceGofr.eps} |
| 504 |
+ |
\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ |
| 505 |
+ |
calculated from from simulations of the SSD/RF water model at 77 K.} |
| 506 |
+ |
\label{fig:gofr} |
| 507 |
+ |
\end{figure} |
| 508 |
+ |
|
| 509 |
+ |
\begin{figure} |
| 510 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
| 511 |
+ |
\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at |
| 512 |
+ |
77 K. The raw structure factors have been convoluted with a gaussian |
| 513 |
+ |
instrument function (0.075 \AA$^{-1}$ width) to compensate for the |
| 514 |
+ |
trunction effects in our finite size simulations. The labeled peaks |
| 515 |
+ |
compared favorably with ``spurious'' peaks observed in experimental |
| 516 |
+ |
studies of amorphous solid water.\cite{Bizid87}} |
| 517 |
+ |
\label{fig:sofq} |
| 518 |
+ |
\end{figure} |
| 519 |
+ |
|
| 520 |
|
\section{Acknowledgments} |
| 521 |
|
Support for this project was provided by the National Science |
| 522 |
|
Foundation under grant CHE-0134881. Computation time was provided by |
