| 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 |
|
|
