| 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 |
|
|
| 359 |
|
|
| 360 |
|
\subsection{\label{sec:eam}Embedded Atom Method} |
| 361 |
|
|
| 362 |
< |
Several molecular dynamics codes\cite{dynamo86} exist which have the |
| 362 |
> |
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
| 363 |
|
capacity to simulate metallic systems, including some that have |
| 364 |
|
parallel computational abilities\cite{plimpton93}. Potentials that |
| 365 |
|
describe bonding transition metal |
| 366 |
|
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
| 367 |
< |
attractive interaction which models the stabilization of ``Embedding'' |
| 368 |
< |
a positively charged metal ion in an electron density created by the |
| 367 |
> |
attractive interaction which models ``Embedding'' |
| 368 |
> |
a positively charged metal ion in the electron density due to the |
| 369 |
|
free valance ``sea'' of electrons created by the surrounding atoms in |
| 370 |
|
the system. A mostly repulsive pairwise part of the potential |
| 371 |
|
describes the interaction of the positively charged metal core ions |
| 372 |
|
with one another. A particular potential description called the |
| 373 |
< |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has |
| 373 |
> |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
| 374 |
|
particularly wide adoption has been selected for inclusion in OOPSE. A |
| 375 |
< |
good review of EAM and other metallic potential formulations was done |
| 375 |
> |
good review of {\sc eam} and other metallic potential formulations was done |
| 376 |
|
by Voter.\cite{voter} |
| 377 |
|
|
| 378 |
|
The {\sc eam} potential has the form: |
| 380 |
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 381 |
|
\phi_{ij}({\bf r}_{ij}) \\ |
| 382 |
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 383 |
< |
\end{eqnarray} |
| 383 |
> |
\end{eqnarray}S |
| 384 |
|
|
| 385 |
< |
where $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 386 |
< |
between atoms $i$ and $j$.In the origional formulation of |
| 386 |
< |
EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 387 |
< |
in later refinements to EAM have shown that nonuniqueness between $F$ |
| 388 |
< |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The |
| 389 |
< |
embedding function $F_{i}$ is the energy required to embedded an |
| 390 |
< |
positively-charged core ion $i$ into a linear supeposition of |
| 385 |
> |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 386 |
> |
positively-charged core ion $i$ into a linear superposition of |
| 387 |
|
sperically averaged atomic electron densities given by |
| 388 |
< |
$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the |
| 388 |
> |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 389 |
> |
between atoms $i$ and $j$. In the original formulation of |
| 390 |
> |
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 391 |
> |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 392 |
> |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 393 |
> |
There is a cutoff distance, $r_{cut}$, which limits the |
| 394 |
|
summations in the {\sc eam} equation to the few dozen atoms |
| 395 |
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 396 |
< |
interactions. |
| 396 |
> |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FDB86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 397 |
|
|
| 398 |
+ |
|
| 399 |
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 400 |
+ |
|
| 401 |
+ |
\newcommand{\roundme}{\operatorname{round}} |
| 402 |
|
|
| 403 |
|
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 404 |
|
macroscopic systems with a relatively small number of particles. The |
| 431 |
|
\end{equation} |
| 432 |
|
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 433 |
|
\begin{equation} |
| 434 |
< |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 434 |
> |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 435 |
|
\end{equation} |
| 436 |
|
where |
| 437 |
|
|
| 438 |
|
% |
| 439 |
|
|
| 440 |
|
\begin{equation} |
| 441 |
< |
round(x)=\left\{ |
| 442 |
< |
\begin{array}[c]{c}% |
| 441 |
> |
\roundme(x)=\left\{ |
| 442 |
> |
\begin{array}{cc} |
| 443 |
|
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 444 |
|
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 445 |
|
\end{array} |
| 446 |
|
\right. |
| 447 |
|
\end{equation} |
| 448 |
+ |
For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$, |
| 449 |
+ |
$\roundme(-3.1)=-3$. |
| 450 |
|
|
| 445 |
– |
|
| 446 |
– |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, |
| 447 |
– |
$round(-3.1)=-3$. |
| 448 |
– |
|
| 451 |
|
Finally, we obtain the minimum image coordinates by transforming back |
| 452 |
|
to real space,% |
| 453 |
|
|
