215 |
|
field acting on dipole $i$ is |
216 |
|
\begin{equation} |
217 |
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
218 |
< |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
218 |
> |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij}), |
219 |
|
\label{rfequation} |
220 |
|
\end{equation} |
221 |
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
222 |
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
223 |
|
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
224 |
< |
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
224 |
> |
moment vector of particle $j$, and $f(r_{ij})$ is a cubic switching |
225 |
|
function.\cite{AllenTildesley} The reaction field contribution to the |
226 |
|
total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
227 |
|
\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
251 |
|
need to be considered. |
252 |
|
|
253 |
|
Integration of the equations of motion was carried out using the |
254 |
< |
symplectic splitting method proposed by Dullweber {\it et |
255 |
< |
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
256 |
< |
centers on poor energy conservation of rigid body dynamics using |
257 |
< |
traditional quaternion integration.\cite{Evans77,Evans77b} In typical |
258 |
< |
microcanonical ensemble simulations, the energy drift when using |
259 |
< |
quaternions was substantially greater than when using the symplectic |
260 |
< |
splitting method (fig. \ref{timestep}). This steady drift in the |
261 |
< |
total energy has also been observed by Kol {\it et al.}\cite{Laird97} |
254 |
> |
symplectic splitting method proposed by Dullweber, Leimkuhler, and |
255 |
> |
McLachlan (DLM).\cite{Dullweber1997} Our reason for selecting this |
256 |
> |
integrator centers on poor energy conservation of rigid body dynamics |
257 |
> |
using traditional quaternion integration.\cite{Evans77,Evans77b} In |
258 |
> |
typical microcanonical ensemble simulations, the energy drift when |
259 |
> |
using quaternions was substantially greater than when using the DLM |
260 |
> |
method (fig. \ref{timestep}). This steady drift in the total energy |
261 |
> |
has also been observed by Kol {\it et al.}\cite{Laird97} |
262 |
|
|
263 |
|
The key difference in the integration method proposed by Dullweber |
264 |
|
\emph{et al.} is that the entire rotation matrix is propagated from |
267 |
|
rotation matrix into quaternions for storage purposes makes trajectory |
268 |
|
data quite compact. |
269 |
|
|
270 |
< |
The symplectic splitting method allows for Verlet style integration of |
271 |
< |
both translational and orientational motion of rigid bodies. In this |
270 |
> |
The DML method allows for Verlet style integration of both |
271 |
> |
translational and orientational motion of rigid bodies. In this |
272 |
|
integration method, the orientational propagation involves a sequence |
273 |
|
of matrix evaluations to update the rotation |
274 |
|
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
275 |
|
than the simpler arithmetic quaternion propagation. With the same time |
276 |
|
step, a 1000 SSD particle simulation shows an average 7\% increase in |
277 |
< |
computation time using the symplectic step method in place of |
278 |
< |
quaternions. The additional expense per step is justified when one |
279 |
< |
considers the ability to use time steps that are nearly twice as large |
280 |
< |
under symplectic splitting than would be usable under quaternion |
281 |
< |
dynamics. The energy conservation of the two methods using a number |
282 |
< |
of different time steps is illustrated in figure |
277 |
> |
computation time using the DML method in place of quaternions. The |
278 |
> |
additional expense per step is justified when one considers the |
279 |
> |
ability to use time steps that are nearly twice as large under DML |
280 |
> |
than would be usable under quaternion dynamics. The energy |
281 |
> |
conservation of the two methods using a number of different time steps |
282 |
> |
is illustrated in figure |
283 |
|
\ref{timestep}. |
284 |
|
|
285 |
|
\begin{figure} |
287 |
|
\epsfxsize=6in |
288 |
|
\epsfbox{timeStep.epsi} |
289 |
|
\caption{Energy conservation using both quaternion based integration and |
290 |
< |
the symplectic step method proposed by Dullweber \emph{et al.} with |
291 |
< |
increasing time step. The larger time step plots are shifted from the |
292 |
< |
true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
290 |
> |
the symplectic splitting method proposed by Dullweber \emph{et al.} |
291 |
> |
with increasing time step. The larger time step plots are shifted from |
292 |
> |
the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
293 |
|
\label{timestep} |
294 |
|
\end{center} |
295 |
|
\end{figure} |
296 |
|
|
297 |
|
In figure \ref{timestep}, the resulting energy drift at various time |
298 |
< |
steps for both the symplectic step and quaternion integration schemes |
299 |
< |
is compared. All of the 1000 SSD particle simulations started with |
300 |
< |
the same configuration, and the only difference was the method used to |
301 |
< |
handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
302 |
< |
methods for propagating the orientational degrees of freedom conserve |
303 |
< |
energy fairly well, with the quaternion method showing a slight energy |
304 |
< |
drift over time in the 0.5 fs time step simulation. At time steps of 1 |
305 |
< |
and 2 fs, the energy conservation benefits of the symplectic step |
306 |
< |
method are clearly demonstrated. Thus, while maintaining the same |
307 |
< |
degree of energy conservation, one can take considerably longer time |
308 |
< |
steps, leading to an overall reduction in computation time. |
298 |
> |
steps for both the DML and quaternion integration schemes is compared. |
299 |
> |
All of the 1000 SSD particle simulations started with the same |
300 |
> |
configuration, and the only difference was the method used to handle |
301 |
> |
orientational motion. At time steps of 0.1 and 0.5 fs, both methods |
302 |
> |
for propagating the orientational degrees of freedom conserve energy |
303 |
> |
fairly well, with the quaternion method showing a slight energy drift |
304 |
> |
over time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
305 |
> |
fs, the energy conservation benefits of the DML method are clearly |
306 |
> |
demonstrated. Thus, while maintaining the same degree of energy |
307 |
> |
conservation, one can take considerably longer time steps, leading to |
308 |
> |
an overall reduction in computation time. |
309 |
|
|
310 |
< |
Energy drift in the symplectic step simulations was unnoticeable for |
311 |
< |
time steps up to 3 fs. A slight energy drift on the |
312 |
< |
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
313 |
< |
4 fs, and as expected, this drift increases dramatically |
314 |
< |
with increasing time step. To insure accuracy in our microcanonical |
315 |
< |
simulations, time steps were set at 2 fs and kept at this value for |
316 |
< |
constant pressure simulations as well. |
310 |
> |
Energy drift in the simulations using DML integration was unnoticeable |
311 |
> |
for time steps up to 3 fs. A slight energy drift on the order of 0.012 |
312 |
> |
kcal/mol per nanosecond was observed at a time step of 4 fs, and as |
313 |
> |
expected, this drift increases dramatically with increasing time |
314 |
> |
step. To insure accuracy in our microcanonical simulations, time steps |
315 |
> |
were set at 2 fs and kept at this value for constant pressure |
316 |
> |
simulations as well. |
317 |
|
|
318 |
|
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
319 |
|
were generated as starting points for all simulations. The $I_h$ |
430 |
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
431 |
|
reaction field, the density around 300 K is still significantly lower |
432 |
|
than experiment and comparable water models. This anomalous behavior |
433 |
< |
was what lead Ichiye {\it et al.} to recently reparameterize |
433 |
> |
was what lead Tan {\it et al.} to recently reparameterize |
434 |
|
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
435 |
|
reparamaterizations of SSD will be compared with the newer SSD1 model. |
436 |
|
|
642 |
|
\begin{figure} |
643 |
|
\begin{center} |
644 |
|
\epsfxsize=6in |
645 |
< |
\epsfbox{dualsticky.ps} |
645 |
> |
\epsfbox{dualsticky_bw.eps} |
646 |
|
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
647 |
|
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
648 |
|
component, and darker areas correspond to the dipolar repulsive |
812 |
|
the densities, it is important that the excellent diffusive behavior |
813 |
|
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
814 |
|
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
815 |
< |
without an active reaction field at the densities calculated from the |
816 |
< |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
817 |
< |
consistently higher than experiment, while SSD1 remains lower than |
818 |
< |
experiment until relatively high temperatures (around 360 K). Both |
819 |
< |
models follow the shape of the experimental curve well below 300 K but |
820 |
< |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
821 |
< |
crossing above 360 K. This increasing diffusion relative to the |
822 |
< |
experimental values is caused by the rapidly decreasing system density |
823 |
< |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
824 |
< |
in diffusive behavior, but this trend has different implications on |
825 |
< |
the diffusive behavior of the models. While SSD1 shows more |
826 |
< |
experimentally accurate diffusive behavior in the high temperature |
827 |
< |
regimes, SSD/E shows more accurate behavior in the supercooled and |
828 |
< |
biologically relevant temperature ranges. Thus, the changes made to |
829 |
< |
improve the liquid structure may have had an adverse affect on the |
830 |
< |
density maximum, but they improve the transport behavior of SSD/E |
831 |
< |
relative to SSD1 under the most commonly simulated conditions. |
815 |
> |
without an active reaction field at the densities calculated from |
816 |
> |
their respective NPT simulations at 1 atm. The diffusion constant for |
817 |
> |
SSD/E is consistently higher than experiment, while SSD1 remains lower |
818 |
> |
than experiment until relatively high temperatures (around 360 |
819 |
> |
K). Both models follow the shape of the experimental curve well below |
820 |
> |
300 K but tend to diffuse too rapidly at higher temperatures, as seen |
821 |
> |
in SSD1's crossing above 360 K. This increasing diffusion relative to |
822 |
> |
the experimental values is caused by the rapidly decreasing system |
823 |
> |
density with increasing temperature. Both SSD1 and SSD/E show this |
824 |
> |
deviation in particle mobility, but this trend has different |
825 |
> |
implications on the diffusive behavior of the models. While SSD1 |
826 |
> |
shows more experimentally accurate diffusive behavior in the high |
827 |
> |
temperature regimes, SSD/E shows more accurate behavior in the |
828 |
> |
supercooled and biologically relevant temperature ranges. Thus, the |
829 |
> |
changes made to improve the liquid structure may have had an adverse |
830 |
> |
affect on the density maximum, but they improve the transport behavior |
831 |
> |
of SSD/E relative to SSD1 under the most commonly simulated |
832 |
> |
conditions. |
833 |
|
|
834 |
|
\begin{figure} |
835 |
|
\begin{center} |
863 |
|
|
864 |
|
\begin{table} |
865 |
|
\begin{center} |
866 |
< |
\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
866 |
> |
\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Calculated for 298 K from data in ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
867 |
|
\begin{tabular}{ l c c c c c } |
868 |
|
\hline \\[-3mm] |
869 |
|
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
873 |
|
\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\ |
874 |
|
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\ |
875 |
|
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
876 |
< |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
877 |
< |
\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\ |
878 |
< |
\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ |
876 |
> |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.5$^\text{c}$ \\ |
877 |
> |
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 5.7$^\text{d}$ \\ |
878 |
> |
\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ |
879 |
|
\end{tabular} |
880 |
|
\label{liquidproperties} |
881 |
|
\end{center} |
884 |
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
885 |
|
properties of the water models compared in this study along with the |
886 |
|
experimental values for liquid water at ambient conditions. The |
887 |
< |
coordination number and hydrogen bonds per particle were calculated by |
888 |
< |
integrating the following relation: |
887 |
> |
coordination number ($N_C$) and hydrogen bonds per particle ($N_H$) |
888 |
> |
were calculated by integrating the following relations: |
889 |
|
\begin{equation} |
890 |
< |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
890 |
> |
N_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
891 |
|
\end{equation} |
892 |
< |
where $\rho$ is the number density of pair interactions, $a$ is the |
893 |
< |
radial location of the minima following the first solvation shell |
894 |
< |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
895 |
< |
calculation of the coordination number or hydrogen bonds per particle |
892 |
> |
\begin{equation} |
893 |
> |
N_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
894 |
> |
\end{equation} |
895 |
> |
where $\rho$ is the number density of the specified pair interactions, |
896 |
> |
$a$ and $b$ are the radial locations of the minima following the first |
897 |
> |
solvation shell peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ |
898 |
|
respectively. The number of hydrogen bonds stays relatively constant |
899 |
|
across all of the models, but the coordination numbers of SSD/E and |
900 |
|
SSD/RF show an improvement over SSD1. This improvement is primarily |
919 |
|
is the unit vector of the particle dipole.\cite{Rahman71} From these |
920 |
|
correlation functions, the orientational relaxation time of the dipole |
921 |
|
vector can be calculated from an exponential fit in the long-time |
922 |
< |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
922 |
> |
regime ($t > \tau_l$).\cite{Rothschild84} Calculation of these |
923 |
|
time constants were averaged from five detailed NVE simulations |
924 |
|
performed at the STP density for each of the respective models. It |
925 |
|
should be noted that the commonly cited value for $\tau_2$ of 1.9 ps |
926 |
|
was determined from the NMR data in reference \citen{Krynicki66} at a |
927 |
< |
temperature near 34$^\circ$C.\cite{Rahman73} Because of the strong |
927 |
> |
temperature near 34$^\circ$C.\cite{Rahman71} Because of the strong |
928 |
|
temperature dependence of $\tau_2$, it is necessary to recalculate it |
929 |
|
at 298 K to make proper comparisons. The value shown in Table |
930 |
|
\ref{liquidproperties} was calculated from the same NMR data in the |
931 |
< |
fashion described in reference \citen{Krynicki66}. Again, SSD/E and |
932 |
< |
SSD/RF show improved behavior over SSD1, both with and without an |
933 |
< |
active reaction field. Turning on the reaction field leads to much |
934 |
< |
improved time constants for SSD1; however, these results also include |
935 |
< |
a corresponding decrease in system density. Numbers published from the |
936 |
< |
original SSD dynamics studies appear closer to the experimental |
937 |
< |
values, and this difference can be attributed to the use of the Ewald |
938 |
< |
sum technique versus a reaction field.\cite{Ichiye99} |
931 |
> |
fashion described in reference \citen{Krynicki66}. Similarly, $\tau_1$ |
932 |
> |
was recomputed for 298 K from the data in ref \citen{Eisenberg69}. |
933 |
> |
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
934 |
> |
and without an active reaction field. Turning on the reaction field |
935 |
> |
leads to much improved time constants for SSD1; however, these results |
936 |
> |
also include a corresponding decrease in system density. Numbers |
937 |
> |
published from the original SSD dynamics studies are shorter than the |
938 |
> |
values observed here, and this difference can be attributed to the use |
939 |
> |
of the Ewald sum technique versus a reaction field.\cite{Ichiye99} |
940 |
|
|
941 |
|
\subsection{Additional Observations} |
942 |
|
|
943 |
|
\begin{figure} |
944 |
|
\begin{center} |
945 |
|
\epsfxsize=6in |
946 |
< |
\epsfbox{povIce.ps} |
946 |
> |
\epsfbox{icei_bw.eps} |
947 |
|
\caption{A water lattice built from the crystal structure assumed by |
948 |
|
SSD/E when undergoing an extremely restricted temperature NPT |
949 |
|
simulation. This form of ice is referred to as ice-{\it i} to |
1023 |
|
significantly lower than the densities obtained from other water |
1024 |
|
models (and experiment). Analysis of self-diffusion showed SSD to |
1025 |
|
capture the transport properties of water well in both the liquid and |
1026 |
< |
super-cooled liquid regimes. |
1026 |
> |
supercooled liquid regimes. |
1027 |
|
|
1028 |
|
In order to correct the density behavior, the original SSD model was |
1029 |
|
reparameterized for use both with and without a reaction field (SSD/RF |
1039 |
|
The existence of a novel low-density ice structure that is preferred |
1040 |
|
by the SSD family of water models is somewhat troubling, since liquid |
1041 |
|
simulations on this family of water models at room temperature are |
1042 |
< |
effectively simulations of super-cooled or metastable liquids. One |
1043 |
< |
way to de-stabilize this unphysical ice structure would be to make the |
1042 |
> |
effectively simulations of supercooled or metastable liquids. One |
1043 |
> |
way to destabilize this unphysical ice structure would be to make the |
1044 |
|
range of angles preferred by the attractive part of the sticky |
1045 |
|
potential much narrower. This would require extensive |
1046 |
|
reparameterization to maintain the same level of agreement with the |