| 220 |
|
parameter. This method has been shown to be reversible and provide |
| 221 |
|
results in excellent agreement with other established |
| 222 |
|
methods.\cite{Baez95b} |
| 223 |
+ |
|
| 224 |
+ |
The Helmholtz free energy error was determined in the same manner in |
| 225 |
+ |
both the solid and the liquid free energy calculations . At each point |
| 226 |
+ |
along the integration path, we calculated the standard deviation of |
| 227 |
+ |
the potential energy difference. Addition or subtraction of these |
| 228 |
+ |
values to each of their respective points and integrating the curve |
| 229 |
+ |
again provides the upper and lower bounds of the uncertainty in the |
| 230 |
+ |
Helmholtz free energy. |
| 231 |
|
|
| 232 |
|
Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
| 233 |
|
Lennard-Jones interactions were gradually reduced by a cubic switching |
| 435 |
|
\cmidrule(lr){2-6} |
| 436 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 437 |
|
\midrule |
| 438 |
< |
TIP5P-E & & & & - & \\ |
| 439 |
< |
TIP4P-Ew & & -13.09(3) & & - & -12.98(3) \\ |
| 440 |
< |
SPC/E & -12.99(3) & -13.00(3) & & - & -12.99(3) \\ |
| 438 |
> |
TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(3) \\ |
| 439 |
> |
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
| 440 |
> |
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
| 441 |
|
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
| 442 |
|
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
| 443 |
|
\end{tabular} |
| 450 |
|
the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
| 451 |
|
free energy (nearly isoenergetic). The free energy differences between |
| 452 |
|
ice polymorphs for TRED water parallel SSD/RF, with the exception that |
| 453 |
< |
ice B is destabilized such that it is no longer nearly isoenergetic |
| 454 |
< |
with Ice-$i$. |
| 453 |
> |
ice B is destabilized such that it is not very close to Ice-$i$. The |
| 454 |
> |
SPC/E results really show the near isoenergetic behavior when using |
| 455 |
> |
the electrostatics correction. Ice B has the lowest Helmholtz free |
| 456 |
> |
energy; however, all the polymorph results overlap within error. |
| 457 |
|
|
| 458 |
+ |
The most interesting results from these calculations come from the |
| 459 |
+ |
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 460 |
+ |
optimized for use with an electrostatic correction and are |
| 461 |
+ |
geometrically arranged to mimic water following two different |
| 462 |
+ |
ideas. In TIP5P-E, the primary location for the negative charge in the |
| 463 |
+ |
molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
| 464 |
+ |
places the negative charge near the center-of-mass along the H-O-H |
| 465 |
+ |
bisector. There is some debate as to which is the proper choice for |
| 466 |
+ |
the negative charge location, and this has in part led to a six-site |
| 467 |
+ |
water model that balances both of these options.\cite{Vega05,Nada03} |
| 468 |
+ |
The limited results in table \ref{tab:dampedFreeEnergy} support the |
| 469 |
+ |
results of Vega {\it et al.}, which indicate the TIP4P charge location |
| 470 |
+ |
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
| 471 |
+ |
water model, the experimentally observed polymorph (ice |
| 472 |
+ |
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 473 |
+ |
higher in energy, though overlapping within error, and the less |
| 474 |
+ |
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
| 475 |
+ |
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
| 476 |
+ |
real free energy distinction between the various polymorphs and lend |
| 477 |
+ |
credence to other results indicating the preferred form of TIP5P at |
| 478 |
+ |
1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} |
| 479 |
+ |
These results indicate that TIP4P-Ew is a better mimic of real water |
| 480 |
+ |
than these other models when studying crystallization and solid forms |
| 481 |
+ |
of water. |
| 482 |
+ |
|
| 483 |
|
\section{Conclusions} |
| 484 |
|
|
| 485 |
|
In this work, thermodynamic integration was used to determine the |
