| 18 |
|
responsible for the evaluation of its own bonded interaction |
| 19 |
|
(i.e.~bonds, bends, and torsions). |
| 20 |
|
|
| 21 |
< |
As stated previously, one of the features that sets {\sc OOPSE} apart |
| 21 |
> |
As stated previously, one of the features that sets {\sc oopse} apart |
| 22 |
|
from most of the current molecular simulation packages is the ability |
| 23 |
|
to handle rigid body dynamics. Rigid bodies are non-spherical |
| 24 |
|
particles or collections of particles that have a constant internal |
| 25 |
|
potential and move collectively.\cite{Goldstein01} They are not |
| 26 |
< |
included in most simulation packages because of the need to |
| 27 |
< |
consider orientational degrees of freedom and include an integrator |
| 28 |
< |
that accurately propagates these motions in time. |
| 26 |
> |
included in most simulation packages because of the requirement to |
| 27 |
> |
propagate the orientational degrees of freedom. Until recently, |
| 28 |
> |
integrators which propagate orientational motion have been lacking. |
| 29 |
|
|
| 30 |
|
Moving a rigid body involves determination of both the force and |
| 31 |
|
torque applied by the surroundings, which directly affect the |
| 32 |
< |
translation and rotation in turn. In order to accumulate the total |
| 33 |
< |
force on a rigid body, the external forces must first be calculated |
| 34 |
< |
for all the internal particles. The total force on the rigid body is |
| 35 |
< |
simply the sum of these external forces. Accumulation of the total |
| 36 |
< |
torque on the rigid body is more complex than the force in that it is |
| 37 |
< |
the torque applied on the center of mass that dictates rotational |
| 38 |
< |
motion. The summation of this torque is given by |
| 32 |
> |
translational and rotational motion in turn. In order to accumulate |
| 33 |
> |
the total force on a rigid body, the external forces and torques must |
| 34 |
> |
first be calculated for all the internal particles. The total force on |
| 35 |
> |
the rigid body is simply the sum of these external forces. |
| 36 |
> |
Accumulation of the total torque on the rigid body is more complex |
| 37 |
> |
than the force in that it is the torque applied on the center of mass |
| 38 |
> |
that dictates rotational motion. The torque on rigid body {\it i} is |
| 39 |
|
\begin{equation} |
| 40 |
< |
\mathbf{\tau}_i= |
| 41 |
< |
\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia}, |
| 40 |
> |
\boldsymbol{\tau}_i= |
| 41 |
> |
\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
| 42 |
> |
+ \boldsymbol{\tau}_{ia}, |
| 43 |
|
\label{eq:torqueAccumulate} |
| 44 |
|
\end{equation} |
| 45 |
< |
where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and |
| 46 |
< |
position of the center of mass respectively, while $\mathbf{f}_{ia}$ |
| 47 |
< |
and $\mathbf{r}_{ia}$ are the force on and position of the component |
| 48 |
< |
particles of the rigid body.\cite{allen87:csl} |
| 45 |
> |
where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
| 46 |
> |
position of the center of mass respectively, while $\mathbf{f}_{ia}$, |
| 47 |
> |
$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on, |
| 48 |
> |
position of, and torque on the component particles of the rigid body. |
| 49 |
|
|
| 50 |
< |
The application of the total torque is done in the body fixed axis of |
| 50 |
> |
The summation of the total torque is done in the body fixed axis of |
| 51 |
|
the rigid body. In order to move between the space fixed and body |
| 52 |
|
fixed coordinate axes, parameters describing the orientation must be |
| 53 |
|
maintained for each rigid body. At a minimum, the rotation matrix |
| 54 |
< |
(\textbf{A}) can be described and propagated by the three Euler angles |
| 55 |
< |
($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of |
| 54 |
> |
(\textbf{A}) can be described by the three Euler angles ($\phi, |
| 55 |
> |
\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
| 56 |
|
trigonometric operations involving $\phi, \theta,$ and |
| 57 |
< |
$\psi$.\cite{Goldstein01} In order to avoid rotational limitations |
| 57 |
> |
$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
| 58 |
|
inherent in using the Euler angles, the four parameter ``quaternion'' |
| 59 |
< |
scheme can be used instead, where \textbf{A} is composed of arithmetic |
| 60 |
< |
operations involving the four components of a quaternion ($q_0, q_1, |
| 61 |
< |
q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads |
| 62 |
< |
to performance enhancements, particularly for very small |
| 59 |
> |
scheme is often used. The elements of \textbf{A} can be expressed as |
| 60 |
> |
arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
| 61 |
> |
and $q_3$).\cite{allen87:csl} Use of quaternions also leads to |
| 62 |
> |
performance enhancements, particularly for very small |
| 63 |
|
systems.\cite{Evans77} |
| 64 |
|
|
| 65 |
< |
{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine |
| 66 |
< |
parameter rotation matrix internally. Further discussion on this |
| 67 |
< |
choice can be found in Sec.~\ref{sec:integrate}. |
| 65 |
> |
{\sc oopse} utilizes a relatively new scheme that propagates the |
| 66 |
> |
entire nine parameter rotation matrix internally. Further discussion |
| 67 |
> |
on this choice can be found in Sec.~\ref{sec:integrate}. |
| 68 |
|
|
| 69 |
|
\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
| 70 |
|
|
| 271 |
|
\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
| 272 |
|
|
| 273 |
|
In the interest of computational efficiency, the default solvent used |
| 274 |
< |
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
| 275 |
< |
developed by Ichiye \emph{et al.} as a modified form of the |
| 276 |
< |
hard-sphere water model proposed by Bratko, Blum, and |
| 274 |
> |
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
| 275 |
> |
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
| 276 |
> |
\emph{et al.}\cite{Ichiye96} as a modified form of the hard-sphere |
| 277 |
> |
water model proposed by Bratko, Blum, and |
| 278 |
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 279 |
|
with a Lennard-Jones core and a sticky potential that directs the |
| 280 |
|
particles to assume the proper hydrogen bond orientation in the first |
| 333 |
|
{\it charged} multi-point models. In the original Monte Carlo |
| 334 |
|
simulations using this model, Ichiye {\it et al.} reported that using |
| 335 |
|
SSD decreased computer time by a factor of 6-7 compared to other |
| 336 |
< |
models.\cite{Ichiye96} What is most impressive is that this savings |
| 336 |
> |
models.\cite{Ichiye96} What is most impressive is that these savings |
| 337 |
|
did not come at the expense of accurate depiction of the liquid state |
| 338 |
|
properties. Indeed, SSD maintains reasonable agreement with the Soper |
| 339 |
< |
data for the structural features of liquid |
| 339 |
> |
diffraction data for the structural features of liquid |
| 340 |
|
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
| 341 |
|
exhibited by SSD agree with experiment better than those of more |
| 342 |
|
computationally expensive models (like TIP3P and |
| 347 |
|
Recent constant pressure simulations revealed issues in the original |
| 348 |
|
SSD model that led to lower than expected densities at all target |
| 349 |
|
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
| 350 |
< |
is SSD/E, a density corrected derivative of SSD that exhibits improved |
| 351 |
< |
liquid structure and transport behavior. If the use of a reaction |
| 352 |
< |
field long-range interaction correction is desired, it is recommended |
| 353 |
< |
that the parameters be modified to those of the SSD/RF model. Solvent |
| 354 |
< |
parameters can be easily modified in an accompanying {\sc BASS} file |
| 355 |
< |
as illustrated in the scheme below. A table of the parameter values |
| 356 |
< |
and the drawbacks and benefits of the different density corrected SSD |
| 357 |
< |
models can be found in reference \ref{Gezelter04}. |
| 350 |
> |
is therefore SSD/E, a density corrected derivative of SSD that |
| 351 |
> |
exhibits improved liquid structure and transport behavior. If the use |
| 352 |
> |
of a reaction field long-range interaction correction is desired, it |
| 353 |
> |
is recommended that the parameters be modified to those of the SSD/RF |
| 354 |
> |
model. Solvent parameters can be easily modified in an accompanying |
| 355 |
> |
{\sc BASS} file as illustrated in the scheme below. A table of the |
| 356 |
> |
parameter values and the drawbacks and benefits of the different |
| 357 |
> |
density corrected SSD models can be found in reference |
| 358 |
> |
\ref{Gezelter04}. |
| 359 |
|
|
| 360 |
|
!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
| 361 |
|
|
| 390 |
|
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 391 |
|
between atoms $i$ and $j$. In the original formulation of |
| 392 |
|
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 393 |
< |
in later refinements to EAM have shown that nonuniqueness between $F$ |
| 393 |
> |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 394 |
|
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 395 |
|
There is a cutoff distance, $r_{cut}$, which limits the |
| 396 |
|
summations in the {\sc eam} equation to the few dozen atoms |
| 399 |
|
|
| 400 |
|
|
| 401 |
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 402 |
+ |
|
| 403 |
+ |
\newcommand{\roundme}{\operatorname{round}} |
| 404 |
|
|
| 405 |
|
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 406 |
|
macroscopic systems with a relatively small number of particles. The |
| 433 |
|
\end{equation} |
| 434 |
|
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 435 |
|
\begin{equation} |
| 436 |
< |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 436 |
> |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 437 |
|
\end{equation} |
| 438 |
|
where |
| 439 |
|
|
| 440 |
|
% |
| 441 |
|
|
| 442 |
|
\begin{equation} |
| 443 |
< |
round(x)=\left\{ |
| 444 |
< |
\begin{array}[c]{c}% |
| 443 |
> |
\roundme(x)=\left\{ |
| 444 |
> |
\begin{array}{cc} |
| 445 |
|
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 446 |
|
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 447 |
|
\end{array} |
| 448 |
|
\right. |
| 449 |
|
\end{equation} |
| 450 |
+ |
For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$, |
| 451 |
+ |
$\roundme(-3.1)=-3$. |
| 452 |
|
|
| 446 |
– |
|
| 447 |
– |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, |
| 448 |
– |
$round(-3.1)=-3$. |
| 449 |
– |
|
| 453 |
|
Finally, we obtain the minimum image coordinates by transforming back |
| 454 |
|
to real space,% |
| 455 |
|
|
