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1  
2 < \section{\label{sec:empericalEnergy}The Emperical Energy Functions}
2 > \section{\label{sec:empiricalEnergy}The Empirical Energy Functions}
3  
4   \subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies}
5  
6 < The basic unit of an {\sc oopse} simulation is the atom. The parameters
7 < describing the atom are generalized to make the atom as flexible a
8 < representation as possible. They may represent specific atoms of an
9 < element, or be used for collections of atoms such as a methyl
10 < group. The atoms are also capable of having a directional component
11 < associated with them, often in the form of a dipole. Charges on atoms
12 < are not currently suported by {\sc oopse}.
6 > The basic unit of an {\sc oopse} simulation is the atom. The
7 > parameters describing the atom are generalized to make the atom as
8 > flexible a representation as possible. They may represent specific
9 > atoms of an element, or be used for collections of atoms such as
10 > methyl and carbonyl groups. The atoms are also capable of having
11 > directional components associated with them (\emph{e.g.}~permanent
12 > dipoles). Charges on atoms are not currently supported by {\sc oopse}.
13  
14 < \begin{lstlisting}[caption={[Specifier for molecules and atoms] An example specifing the simple Ar molecule},label=sch:AtmMole]
14 > \begin{lstlisting}[caption={[Specifier for molecules and atoms] A sample specification of the simple Ar molecule},label=sch:AtmMole]
15   molecule{
16    name = "Ar";
17    nAtoms = 1;
18    atom[0]{
19 <     type="Ar";
20 <     position( 0.0, 0.0, 0.0 );
19 >    type="Ar";
20 >    position( 0.0, 0.0, 0.0 );
21    }
22   }
23   \end{lstlisting}
24  
25 < The second most basic building block of a simulation is the
26 < molecule. The molecule is a way for {\sc oopse} to keep track of the
27 < atoms in a simulation in logical manner. This particular unit will
28 < store the identities of all the atoms associated with itself and is
29 < responsible for the evaluation of its own bonded interaction
30 < (i.e.~bonds, bends, and torsions).
25 > Atoms can be collected into secondary srtructures such as rigid bodies
26 > or molecules. The molecule is a way for {\sc oopse} to keep track of
27 > the atoms in a simulation in logical manner. Molecular units store the
28 > identities of all the atoms associated with themselves, and are
29 > responsible for the evaluation of their own internal interactions
30 > (\emph{i.e.}~bonds, bends, and torsions). Scheme \ref{sch:AtmMole}
31 > shws how one creates a molecule in the \texttt{.mdl} files. The
32 > position of the atoms given in the declaration are relative to the
33 > origin of the molecule, and is used when creating a system containing
34 > the molecule.
35  
36   As stated previously, one of the features that sets {\sc oopse} apart
37   from most of the current molecular simulation packages is the ability
# Line 75 | Line 79 | entire nine parameter rotation matrix internally. Furt
79  
80   {\sc oopse} utilizes a relatively new scheme that propagates the
81   entire nine parameter rotation matrix internally. Further discussion
82 < on this choice can be found in Sec.~\ref{sec:integrate}.
82 > on this choice can be found in Sec.~\ref{sec:integrate}. An example
83 > definition of a riged body can be seen in Scheme
84 > \ref{sch:rigidBody}. The positions in the atom definitions are the
85 > placements of the atoms relative to the origin of the rigid body,
86 > which itself has a position relative to the origin of the molecule.
87  
88 + \begin{lstlisting}[caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}]
89 + molecule{
90 +  name = "TIP3P_water";
91 +  nRigidBodies = 1;
92 +  rigidBody[0]{
93 +    nAtoms = 3;
94 +    atom[0]{
95 +      type = "O_TIP3P";
96 +      position( 0.0, 0.0, -0.06556 );    
97 +    }                                    
98 +    atom[1]{
99 +      type = "H_TIP3P";
100 +      position( 0.0, 0.75695, 0.52032 );
101 +    }
102 +    atom[2]{
103 +      type = "H_TIP3P";
104 +      position( 0.0, -0.75695, 0.52032 );
105 +    }
106 +    position( 0.0, 0.0, 0.0 );
107 +    orientation( 0.0, 0.0, 1.0 );
108 +  }
109 + }
110 + \end{lstlisting}
111 +
112   \subsection{\label{sec:LJPot}The Lennard Jones Potential}
113  
114 < The most basic force field implemented in OOPSE is the Lennard-Jones
115 < potential. The Lennard-Jones potential. Which mimics the Van der Waals
116 < interaction at long distances, and uses an emperical repulsion at
117 < short distances. The Lennard-Jones potential is given by:
114 > The most basic force field implemented in {\sc oopse} is the
115 > Lennard-Jones potential, which mimics the van der Waals interaction at
116 > long distances, and uses an empirical repulsion at short
117 > distances. The Lennard-Jones potential is given by:
118   \begin{equation}
119   V_{\text{LJ}}(r_{ij}) =
120          4\epsilon_{ij} \biggl[
# Line 91 | Line 123 | V_{\text{LJ}}(r_{ij}) =
123          \biggr]
124   \label{eq:lennardJonesPot}
125   \end{equation}
126 < Where $r_{ij}$ is the distance between particle $i$ and $j$,
126 > Where $r_{ij}$ is the distance between particles $i$ and $j$,
127   $\sigma_{ij}$ scales the length of the interaction, and
128 < $\epsilon_{ij}$ scales the well depth of the potential.
128 > $\epsilon_{ij}$ scales the well depth of the potential. Scheme
129 > \ref{sch:LJFF} gives and example partial \texttt{.bass} file that
130 > shows a system of 108 Ar particles simulated with the Lennard-Jones
131 > force field.
132  
133 + \begin{lstlisting}[caption={[Invocation of the Lennard-Jones force field] A sample system using the Lennard-Jones force field.},label={sch:LJFF}]
134 +
135 + /*
136 + * The Ar molecule is specified
137 + * external to the.bass file
138 + */
139 +
140 + #include "argon.mdl"
141 +
142 + nComponents = 1;
143 + component{
144 +  type = "Ar";
145 +  nMol = 108;
146 + }
147 +
148 + /*
149 + * The initial configuration is generated
150 + * before the simulation is invoked.
151 + */
152 +
153 + initialConfig = "./argon.init";
154 +
155 + forceField = "LJ";
156 + \end{lstlisting}
157 +
158   Because this potential is calculated between all pairs, the force
159   evaluation can become computationally expensive for large systems. To
160 < keep the pair evaluation to a manegable number, OOPSE employs a
161 < cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
162 < $2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones length
163 < parameter in the system. Truncating the calculation at
164 < $r_{\text{cut}}$ introduces a discontinuity into the potential
160 > keep the pair evaluations to a manageable number, {\sc oopse} employs
161 > a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
162 > $2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones
163 > length parameter present in the simulation. Truncating the calculation
164 > at $r_{\text{cut}}$ introduces a discontinuity into the potential
165   energy. To offset this discontinuity, the energy value at
166 < $r_{\text{cut}}$ is subtracted from the entire potential. This causes
167 < the potential to go to zero at the cut-off radius.
166 > $r_{\text{cut}}$ is subtracted from the potential. This causes the
167 > potential to go to zero smoothly at the cut-off radius.
168  
169   Interactions between dissimilar particles requires the generation of
170   cross term parameters for $\sigma$ and $\epsilon$. These are
# Line 121 | Line 181 | and
181   \end{equation}
182  
183  
184 +
185   \subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field}
186  
187 < The Dipolar Unified-atom Force Field ({\sc duff}) was developed to
188 < simulate lipid bilayers. The systems require a model capable of forming
189 < bilayers, while still being sufficiently computationally efficient to
190 < allow simulations of large systems ($\approx$100's of phospholipids,
191 < $\approx$1000's of waters) for long times ($\approx$10's of
192 < nanoseconds).
187 > The dipolar unified-atom force field ({\sc duff}) was developed to
188 > simulate lipid bilayers. The simulations require a model capable of
189 > forming bilayers, while still being sufficiently computationally
190 > efficient to allow large systems ($\approx$100's of phospholipids,
191 > $\approx$1000's of waters) to be simulated for long times
192 > ($\approx$10's of nanoseconds).
193  
194 < With this goal in mind, {\sc duff} has no point charges. Charge
195 < neutral distributions were replaced with dipoles, while most atoms and
196 < groups of atoms were reduced to Lennard-Jones interaction sites. This
197 < simplification cuts the length scale of long range interactions from
198 < $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the
199 < computationally expensive Ewald sum. Instead, we can use
194 > With this goal in mind, {\sc duff} has no point
195 > charges. Charge-neutral distributions were replaced with dipoles,
196 > while most atoms and groups of atoms were reduced to Lennard-Jones
197 > interaction sites. This simplification cuts the length scale of long
198 > range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us
199 > to avoid the computationally expensive Ewald sum. Instead, we can use
200   neighbor-lists, reaction field, and cutoff radii for the dipolar
201   interactions.
202  
203   As an example, lipid head-groups in {\sc duff} are represented as
204   point dipole interaction sites. By placing a dipole of 20.6~Debye at
205   the head group center of mass, our model mimics the head group of
206 < phosphatidylcholine.\cite{Cevc87} Additionally, a Lennard-Jones site
207 < is located at the pseudoatom's center of mass. The model is
208 < illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The water model we use to complement the dipoles of the lipids is out
209 < repaarameterization of the soft sticky dipole (SSD) model of Ichiye
206 > phosphatidylcholine.\cite{Cevc87} Additionally, a large Lennard-Jones
207 > site is located at the pseudoatom's center of mass. The model is
208 > illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The
209 > water model we use to complement the dipoles of the lipids is our
210 > reparameterization of the soft sticky dipole (SSD) model of Ichiye
211   \emph{et al.}\cite{liu96:new_model}
212  
213   \begin{figure}
# Line 157 | Line 219 | is the chain length.}
219   \label{fig:lipidModel}
220   \end{figure}
221  
222 < Turning to the tails of the phospholipids, we have used a set of
223 < scalable parameters to model the alkyl groups with Lennard-Jones
224 < sites. For this, we have used the TraPPE force field of Siepmann
222 > We have used a set of scalable parameters to model the alkyl groups
223 > with Lennard-Jones sites. For this, we have borrowed parameters from
224 > the TraPPE force field of Siepmann
225   \emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom
226   representation of n-alkanes, which is parametrized against phase
227 < equilibria using Gibbs Monte Carlo simulation
227 > equilibria using Gibbs ensemble Monte Carlo simulation
228   techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that
229   it generalizes the types of atoms in an alkyl chain to keep the number
230   of pseudoatoms to a minimum; the parameters for an atom such as
231   $\text{CH}_2$ do not change depending on what species are bonded to
232   it.
233  
234 < TraPPE also constrains of all bonds to be of fixed length. Typically,
234 > TraPPE also constrains all bonds to be of fixed length. Typically,
235   bond vibrations are the fastest motions in a molecular dynamic
236   simulation. Small time steps between force evaluations must be used to
237 < ensure adequate sampling of the bond potential conservation of
238 < energy. By constraining the bond lengths, larger time steps may be
239 < used when integrating the equations of motion.
237 > ensure adequate sampling of the bond potential to ensure conservation
238 > of energy. By constraining the bond lengths, larger time steps may be
239 > used when integrating the equations of motion. A simulation using {\sc
240 > duff} is illustrated in Scheme \ref{sch:DUFF}.
241  
242 + \begin{lstlisting}[caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}]
243  
244 < \subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions}
244 > #include "water.mdl"
245 > #include "lipid.mdl"
246  
247 < The total energy of function in {\sc duff} is given by the following:
247 > nComponents = 2;
248 > component{
249 >  type = "simpleLipid_16";
250 >  nMol = 60;
251 > }
252 >
253 > component{
254 >  type = "SSD_water";
255 >  nMol = 1936;
256 > }
257 >
258 > initialConfig = "bilayer.init";
259 >
260 > forceField = "DUFF";
261 >
262 > \end{lstlisting}
263 >
264 > \subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions}
265 >
266 > The total potential energy function in {\sc duff} is
267   \begin{equation}
268 < V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}}
268 > V = \sum^{N}_{I=1} V^{I}_{\text{Internal}}
269          + \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}}
270   \label{eq:totalPotential}
271   \end{equation}
272 < Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule:
272 > Where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$:
273   \begin{equation}
274   V^{I}_{\text{Internal}} =
275          \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk})
# Line 197 | Line 281 | Here $V_{\text{bend}}$ is the bend potential for all 1
281   \label{eq:internalPotential}
282   \end{equation}
283   Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs
284 < within the molecule, and $V_{\text{torsion}}$ is the torsion potential
284 > within the molecule $I$, and $V_{\text{torsion}}$ is the torsion potential
285   for all 1, 4 bonded pairs. The pairwise portions of the internal
286   potential are excluded for pairs that are closer than three bonds,
287   i.e.~atom pairs farther away than a torsion are included in the
# Line 206 | Line 290 | The bend potential of a molecule is represented by the
290  
291   The bend potential of a molecule is represented by the following function:
292   \begin{equation}
293 < V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot}
293 > V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot}
294   \end{equation}
295   Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$
296 < (see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium
297 < bond angle. $k_{\theta}$ is the force constant which determines the
296 > (see Fig.~\ref{fig:lipidModel}), $\theta_0$ is the equilibrium
297 > bond angle, and $k_{\theta}$ is the force constant which determines the
298   strength of the harmonic bend. The parameters for $k_{\theta}$ and
299 < $\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998}
299 > $\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998}
300  
301 < The torsion potential and parameters are also taken from TraPPE. It is
301 > The torsion potential and parameters are also borrowed from TraPPE. It is
302   of the form:
303   \begin{equation}
304   V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi]
# Line 222 | Line 306 | V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi]
306          + c_3[1 + \cos(3\phi)]
307   \label{eq:origTorsionPot}
308   \end{equation}
309 < Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$,
309 > Here $\phi$ is the angle defined by four bonded neighbors $i$,
310   $j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For
311 < computaional efficency, the torsion potential has been recast after
312 < the method of CHARMM\cite{charmm1983} whereby the angle series is
311 > computational efficiency, the torsion potential has been recast after
312 > the method of CHARMM,\cite{charmm1983} in which the angle series is
313   converted to a power series of the form:
314   \begin{equation}
315 < V_{\text{torsion}}(\phi_{ijkl}) =  
315 > V_{\text{torsion}}(\phi) =  
316          k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0
317   \label{eq:torsionPot}
318   \end{equation}
# Line 239 | Line 323 | k_3 &= 4c_3
323   k_2 &= 2 c_2 \\
324   k_3 &= 4c_3
325   \end{align*}
326 < By recasting the equation to a power series, repeated trigonometric
327 < evaluations are avoided during the calculation of the potential.
326 > By recasting the potential as a power series, repeated trigonometric
327 > evaluations are avoided during the calculation of the potential energy.
328  
329  
330 < The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
330 > The cross potential between molecules $I$ and $J$, $V^{IJ}_{\text{Cross}}$, is
331   as follows:
332   \begin{equation}
333   V^{IJ}_{\text{Cross}} =
# Line 257 | Line 341 | $V_{\text{dipole}}$ is the dipole dipole potential, an
341   \end{equation}
342   Where $V_{\text{LJ}}$ is the Lennard Jones potential,
343   $V_{\text{dipole}}$ is the dipole dipole potential, and
344 < $V_{\text{sticky}}$ is the sticky potential defined by the SSD
345 < model. Note that not all atom types include all interactions.
344 > $V_{\text{sticky}}$ is the sticky potential defined by the SSD model
345 > (Sec.~\ref{sec:SSD}). Note that not all atom types include all
346 > interactions.
347  
348   The dipole-dipole potential has the following form:
349   \begin{equation}
# Line 273 | Line 358 | towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsym
358   \end{equation}
359   Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing
360   towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$
361 < are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is
362 < the magnitude of the dipole moment of atom $i$ and
363 < $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of
364 < $\boldsymbol{\Omega}_i$.
361 > are the orientational degrees of freedom for atoms $i$ and $j$
362 > respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom
363 > $i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation
364 > vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is
365 > the unit vector pointing along $\mathbf{r}_{ij}$.
366  
367  
368 < \subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF}
368 > \subsubsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF}
369  
370   In the interest of computational efficiency, the default solvent used
371   by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water
372   model.\cite{Gezelter04} The original SSD was developed by Ichiye
373 < \emph{et al.}\cite{Ichiye96} as a modified form of the hard-sphere
373 > \emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere
374   water model proposed by Bratko, Blum, and
375   Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole
376   with a Lennard-Jones core and a sticky potential that directs the
# Line 337 | Line 423 | can be found in the original SSD
423   while $w^\prime$ is a purely empirical function.  A more detailed
424   description of the functional parts and variables in this potential
425   can be found in the original SSD
426 < articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
426 > articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03}
427  
428   Since SSD is a single-point {\it dipolar} model, the force
429   calculations are simplified significantly relative to the standard
430   {\it charged} multi-point models. In the original Monte Carlo
431   simulations using this model, Ichiye {\it et al.} reported that using
432   SSD decreased computer time by a factor of 6-7 compared to other
433 < models.\cite{Ichiye96} What is most impressive is that these savings
433 > models.\cite{liu96:new_model} What is most impressive is that these savings
434   did not come at the expense of accurate depiction of the liquid state
435   properties.  Indeed, SSD maintains reasonable agreement with the Soper
436   diffraction data for the structural features of liquid
437 < water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
437 > water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties
438   exhibited by SSD agree with experiment better than those of more
439   computationally expensive models (like TIP3P and
440 < SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
440 > SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction
441   of solvent properties makes SSD a very attractive model for the
442   simulation of large scale biochemical simulations.
443  
# Line 368 | Line 454 | density corrected SSD models can be found in reference
454   density corrected SSD models can be found in reference
455   \ref{Gezelter04}.
456  
457 < !!!Place a {\sc BASS} scheme showing SSD parameters around here!!!
457 > \begin{lstlisting}[caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}]
458 >
459 > #include "water.mdl"
460 >
461 > nComponents = 1;
462 > component{
463 >  type = "SSD_water";
464 >  nMol = 864;
465 > }
466 >
467 > initialConfig = "liquidWater.init";
468 >
469 > forceField = "DUFF";
470 >
471 > /*
472 > * The reactionField flag toggles reaction
473 > * field corrections.
474 > */
475  
476 + reactionField = false; // defaults to false
477 + dielectric = 80.0; // dielectric for reaction field
478 +
479 + /*
480 + * The following two flags set the cutoff
481 + * radius for the electrostatic forces
482 + * as well as the skin thickness of the switching
483 + * function.
484 + */
485 +
486 + electrostaticCutoffRadius  = 9.2;
487 + electrostaticSkinThickness = 1.38;
488 +
489 + \end{lstlisting}
490 +
491 +
492   \subsection{\label{sec:eam}Embedded Atom Method}
493  
494   Several other molecular dynamics packages\cite{dynamo86} exist which have the
# Line 384 | Line 503 | Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(
503   describes the interaction of the positively charged metal core ions
504   with one another. A particular potential description called the
505   Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has
506 < particularly wide adoption has been selected for inclusion in OOPSE. A
506 > particularly wide adoption has been selected for inclusion in {\sc oopse}. A
507   good review of {\sc eam} and other metallic potential formulations was done
508   by Voter.\cite{voter}
509  
# Line 397 | Line 516 | positively-charged core ion $i$ into a linear superpos
516  
517   where $F_{i} $ is the embedding function that equates the energy required to embed a
518   positively-charged core ion $i$ into a linear superposition of
519 < sperically averaged atomic electron densities given by
519 > spherically averaged atomic electron densities given by
520   $\rho_{i}$.  $\phi_{ij}$ is a primarily repulsive pairwise interaction
521   between atoms $i$ and $j$. In the original formulation of
522   {\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however
# Line 412 | Line 531 | interactions. Foiles et al. fit EAM potentials for fcc
531   \subsection{\label{Sec:pbc}Periodic Boundary Conditions}
532  
533   \newcommand{\roundme}{\operatorname{round}}
534 <
534 >
535   \textit{Periodic boundary conditions} are widely used to simulate truly
536   macroscopic systems with a relatively small number of particles. The
537 < simulation box is replicated throughout space to form an infinite
538 < lattice.  During the simulation, when a particle moves in the primary
539 < cell, its image in other boxes move in exactly the same direction with
540 < exactly the same orientation.Thus, as a particle leaves the primary
541 < cell, one of its images will enter through the opposite face.If the
542 < simulation box is large enough to avoid "feeling" the symmetries of
543 < the periodic lattice, surface effects can be ignored. Cubic,
544 < orthorhombic and parallelepiped are the available periodic cells In
545 < OOPSE. We use a matrix to describe the property of the simulation
546 < box. Therefore, both the size and shape of the simulation box can be
547 < changed during the simulation. The transformation from box space
548 < vector $\mathbf{s}$ to its corresponding real space vector
430 < $\mathbf{r}$ is defined by
537 > simulation box is replicated throughout space to form an infinite lattice.
538 > During the simulation, when a particle moves in the primary cell, its image in
539 > other boxes move in exactly the same direction with exactly the same
540 > orientation.Thus, as a particle leaves the primary cell, one of its images
541 > will enter through the opposite face.If the simulation box is large enough to
542 > avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the
543 > periodic lattice, surface effects can be ignored. Cubic, orthorhombic and
544 > parallelepiped are the available periodic cells In OOPSE. We use a matrix to
545 > describe the property of the simulation box. Therefore, both the size and
546 > shape of the simulation box can be changed during the simulation. The
547 > transformation from box space vector $\mathbf{s}$ to its corresponding real
548 > space vector $\mathbf{r}$ is defined by
549   \begin{equation}
550 < \mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}%
550 > \mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}%
551   \end{equation}
552  
553  
554 < where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of
555 < the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the
556 < three sides of the simulation box respectively.
554 > where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
555 > box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the
556 > simulation box respectively.
557  
558 < To find the minimum image, we convert the real vector to its
559 < corresponding vector in box space first, \bigskip%
558 > To find the minimum image of a vector $\mathbf{r}$, we convert the real vector
559 > to its corresponding vector in box space first, \bigskip%
560   \begin{equation}
561 < \mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
561 > \mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}%
562   \end{equation}
563   And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
564   \begin{equation}
# Line 452 | Line 570 | where
570  
571   \begin{equation}
572   \roundme(x)=\left\{
573 < \begin{array}{cc}
573 > \begin{array}{cc}%
574   \lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
575   \lceil{x-0.5}\rceil & \text{otherwise}%
576   \end{array}
577   \right.
578   \end{equation}
461 – For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$,
462 – $\roundme(-3.1)=-3$.
579  
464 – Finally, we obtain the minimum image coordinates by transforming back
465 – to real space,%
580  
581 + For example, $\roundme(3.6)=4$,$\roundme(3.1)=3$, $\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$.
582 +
583 + Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by
584 + transforming back to real space,%
585 +
586   \begin{equation}
587 < \mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
587 > \mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}%
588   \end{equation}
589  

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