--- trunk/oopsePaper/EmpericalEnergy.tex	2004/01/13 20:03:21	933
+++ trunk/oopsePaper/EmpericalEnergy.tex	2004/01/13 20:10:23	935
@@ -18,52 +18,53 @@ As stated previously, one of the features that sets {\
 responsible for the evaluation of its own bonded interaction
 (i.e.~bonds, bends, and torsions).
 
-As stated previously, one of the features that sets {\sc OOPSE} apart
+As stated previously, one of the features that sets {\sc oopse} apart
 from most of the current molecular simulation packages is the ability
 to handle rigid body dynamics. Rigid bodies are non-spherical
 particles or collections of particles that have a constant internal
 potential and move collectively.\cite{Goldstein01} They are not
-included in most simulation packages because of the need to
-consider orientational degrees of freedom and include an integrator
-that accurately propagates these motions in time.
+included in most simulation packages because of the requirement to
+propagate the orientational degrees of freedom. Until recently,
+integrators which propagate orientational motion have been lacking.
 
 Moving a rigid body involves determination of both the force and
 torque applied by the surroundings, which directly affect the
-translation and rotation in turn. In order to accumulate the total
-force on a rigid body, the external forces must first be calculated
-for all the internal particles. The total force on the rigid body is
-simply the sum of these external forces.  Accumulation of the total
-torque on the rigid body is more complex than the force in that it is
-the torque applied on the center of mass that dictates rotational
-motion. The summation of this torque is given by
+translational and rotational motion in turn. In order to accumulate
+the total force on a rigid body, the external forces and torques must
+first be calculated for all the internal particles. The total force on
+the rigid body is simply the sum of these external forces.
+Accumulation of the total torque on the rigid body is more complex
+than the force in that it is the torque applied on the center of mass
+that dictates rotational motion. The torque on rigid body {\it i} is
 \begin{equation}
-\mathbf{\tau}_i=
-	\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia},
+\boldsymbol{\tau}_i=
+	\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} 
+	+ \boldsymbol{\tau}_{ia},
 \label{eq:torqueAccumulate}
 \end{equation}
-where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and
-position of the center of mass respectively, while $\mathbf{f}_{ia}$
-and $\mathbf{r}_{ia}$ are the force on and position of the component
-particles of the rigid body.\cite{allen87:csl}
+where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and
+position of the center of mass respectively, while $\mathbf{f}_{ia}$,
+$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on,
+position of, and torque on the component particles of the rigid body.
 
-The application of the total torque is done in the body fixed axis of
+The summation of the total torque is done in the body fixed axis of
 the rigid body. In order to move between the space fixed and body
 fixed coordinate axes, parameters describing the orientation must be
 maintained for each rigid body. At a minimum, the rotation matrix
-(\textbf{A}) can be described and propagated by the three Euler angles
-($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of
+(\textbf{A}) can be described by the three Euler angles ($\phi,
+\theta,$ and $\psi$), where the elements of \textbf{A} are composed of
 trigonometric operations involving $\phi, \theta,$ and
-$\psi$.\cite{Goldstein01} In order to avoid rotational limitations
+$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities
 inherent in using the Euler angles, the four parameter ``quaternion''
-scheme can be used instead, where \textbf{A} is composed of arithmetic
-operations involving the four components of a quaternion ($q_0, q_1,
-q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads
-to performance enhancements, particularly for very small
+scheme is often used. The elements of \textbf{A} can be expressed as
+arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$
+and $q_3$).\cite{allen87:csl} Use of quaternions also leads to
+performance enhancements, particularly for very small
 systems.\cite{Evans77}
 
-{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine
-parameter rotation matrix internally. Further discussion on this
-choice can be found in Sec.~\ref{sec:integrate}.
+{\sc oopse} utilizes a relatively new scheme that propagates the
+entire nine parameter rotation matrix internally. Further discussion
+on this choice can be found in Sec.~\ref{sec:integrate}.
 
 \subsection{\label{sec:LJPot}The Lennard Jones Potential}