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-<
+\section{The Emperical Energy Functions}
->
+\section{\label{sec:empericalEnergy}The Emperical Energy Functions}
-<
+\subsection{Atoms and Molecules}
->
+\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies}
+The basic unit of an {\sc oopse} simulation is the atom. The parameters
+describing the atom are generalized to make the atom as flexible a
+identities of all the atoms associated with itself and is responsible
+for the evaluation of its own bonded interaction (i.e.~bonds, bends,
+and torsions).
-+
-+
+As stated in the previously, one of the features that sets 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 many standard simulation packages because of the need to
-+
+consider orientational degrees of freedom and include an integrator
-+
+that accurately propagates these motions in time.
-+
-+
+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 interal 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 similar to the force in that it is a sum
-+
+of the torque applied on each internal particle, mapped onto the
-+
+center of mass of the rigid body.
-+
-+
+The application 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 be
-+
+maintained for each rigid body. At a minimum, the rotation matrix can
-+
+be described and propagated by the three Euler
-+
+angles.\cite{Goldstein01} In order to avoid rotational limitations
-+
+when using Euler angles, the four parameter ``quaternion'' scheme can
-+
+be used instead.\cite{allen87:csl} Use of quaternions also leads to
-+
+performance enhancements, particularly for very small
-+
+systems.\cite{Evans77} OOPSE utilizes a relatively new scheme that
-+
+propagates the entire nine parameter rotation matrix. Further
-+
+discussion on this choice can be found in Sec.~\ref{sec:integrate}.
-+
-+
+\subsection{\label{sec:LJPot}The Lennard Jones Potential}
-+
-+
+The most basic force field implemented in OOPSE is the Lennard-Jones
-+
+potential. The Lennard-Jones potential mimics the attractive forces
-+
+two charge neutral particles experience when spontaneous dipoles are
-+
+induced on each other. This is the predominant intermolecular force in
-+
+systems of such as noble gases and simple alkanes.
-+
-+
+The Lennard-Jones potential is given by:
-+
+\begin{equation}
-+
+V_{\text{LJ}}(r_{ij}) =
-+
+\epsilon_{ij} \biggl[
-+
+        \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
-+
+        - \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
-+
+        \biggr]
-+
+\label{eq:lennardJonesPot}
-+
+\end{equation}
-+
+Where $r_ij$ is the distance between particle $i$ and $j$, $\sigma_{ij}$
-+
+scales the length of the interaction, and $\epsilon_{ij}$ scales the
-+
+energy well depth of the potential.
-+
-+
+Because the potential is calculated between all pairs, the force
-+
+evaluation can become computationally expensive for large systems. To
-+
+keep the pair evaluation to a manegable number, OOPSE employs the use
-+
+of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
-+
+$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length
-+
+parameter in the system. Truncating the calculation at
-+
+$r_{\text{cut}}$ introduces a discontinuity into the potential
-+
+energy. To offset this discontinuity, the energy value at
-+
+$r_{\text{cut}}$ is subtracted from the entire potential. This causes
-+
+the equation to go to zero at the cut-off radius.
-+
-+
+Interactions between dissimilar particles requires the generation of
-+
+cross term parameters for $\sigma$ and $\epsilon$. These are
-+
+calculated through the Lorentz-Berthelot mixing
-+
+rules:\cite{allen87:csl}
-+
+\begin{equation}
-+
+\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}]
-+
+\label{eq:sigmaMix}
-+
+\end{equation}
-+
+and
-+
+\begin{equation}
-+
+\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}
-+
+\label{eq:epsilonMix}
-+
+\end{equation}
-+
-+
-+
+\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field}
-+
-+
+The \underline{D}ipolar \underline{U}nified-Atom
-+
+\underline{F}orce \underline{F}ield ({\sc duff}) was developed to
-+
+simulate lipid bilayers. We needed a model capable of forming
-+
+bilayers, while still being sufficiently computationally efficient to
-+
+allow simulations of large systems ($\approx$100's of phospholipids,
-+
+$\approx$1000's of waters) for long times ($\approx$10's of
-+
+nanoseconds).
-+
-+
+With this goal in mind, we have eliminated all point charges. Charge
-+
+distributions were replaced with dipoles, and charge-neutral
-+
+distributions were reduced to Lennard-Jones interaction sites. This
-+
+simplification cuts the length scale of long range interactions from
-+
+$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the
-+
+computationally expensive Ewald-Sum. Instead, we can use
-+
+neighbor-lists and cutoff radii for the dipolar interactions.
-+
-+
+As an example, lipid head groups in {\sc duff} are represented as point
-+
+dipole interaction sites.PC and PE Lipid head groups are typically
-+
+zwitterionic in nature, with charges separated by as much as
-+
+~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group
-+
+center of mass, our model mimics the head group of PC.\cite{Cevc87}
-+
+Additionally, a Lennard-Jones site is located at the
-+
+pseudoatom's center of mass. The model is illustrated by the dark grey
-+
+atom in Fig.~\ref{fig:lipidModel}.
-+
-+
+\begin{figure}
-+
+\epsfbox{lipidModel.eps}
-+
+\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ %
-+
+is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.}
-+
+\label{fig:lipidModel}
-+
+\end{figure}
-+
-+
+The water model we use to complement the dipoles of the lipids is
-+
+the soft sticky dipole (SSD) model of Ichiye \emph{et
-+
+al.}\cite{liu96:new_model} This model is discussed in greater detail
-+
+in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single
-+
+Lennard-Jones interaction site. The site also contains a dipole to
-+
+mimic the partial charges on the hydrogens and the oxygen. However,
-+
+what makes the SSD model unique is the inclusion of a tetrahedral
-+
+short range potential to recover the hydrogen bonding of water, an
-+
+important factor when modeling bilayers, as it has been shown that
-+
+hydrogen bond network formation is a leading contribution to the
-+
+entropic driving force towards lipid bilayer formation.\cite{Cevc87}
-+
-+
-+
+Turning to the tails of the phospholipids, we have used a set of
-+
+scalable parameters to model the alkyl groups with Lennard-Jones
-+
+sites. For this, we have used the TraPPE force field of Siepmann
-+
+\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom
-+
+representation of n-alkanes, which is parametrized against phase
-+
+equilibria using Gibbs Monte Carlo simulation
-+
+techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that
-+
+it generalizes the types of atoms in an alkyl chain to keep the number
-+
+of pseudoatoms to a minimum; the parameters for an atom such as
-+
+$\text{CH}_2$ do not change depending on what species are bonded to
-+
+it.
-+
-+
+TraPPE also constrains of all bonds to be of fixed length. Typically,
-+
+bond vibrations are the fastest motions in a molecular dynamic
-+
+simulation. Small time steps between force evaluations must be used to
-+
+ensure adequate sampling of the bond potential conservation of
-+
+energy. By constraining the bond lengths, larger time steps may be
-+
+used when integrating the equations of motion.
-+
-+
-+
+\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions}
-+
-+
+The total energy of function in {\sc duff} is given by the following:
-+
+\begin{equation}
-+
+V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}}
-+
+        + \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}}
-+
+\label{eq:totalPotential}
-+
+\end{equation}
-+
+Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule:
-+
+\begin{equation}
-+
+ V^{I}_{\text{Internal}} =
-+
+        \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk})
-+
+        + \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl})
-+
+        + \sum_{i \in I} \sum_{(j>i+4) \in I}
-+
+        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
-+
+        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
-+
+        \biggr]
-+
+\label{eq:internalPotential}
-+
+\end{equation}
-+
+Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs
-+
+within in the molecule. $V_{\text{torsion}}$ is the torsion The
-+
+pairwise portions of the internal potential are excluded for pairs
-+
+that ar closer than three bonds, i.e.~atom pairs farther away than a
-+
+torsion are included in the pair-wise loop.
-+
-+
+The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
-+
+as follows:
-+
+\begin{equation}
-+
+V^{IJ}_{\text{Cross}} =
-+
+        \sum_{i \in I} \sum_{j \in J}
-+
+        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
-+
+        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
-+
+        + V_{\text{sticky}}
-+
+        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
-+
+        \biggr]
-+
+\label{eq:crossPotentail}
-+
+\end{equation}
-+
+Where $V_{\text{LJ}}$ is the Lennard Jones potential,
-+
+$V_{\text{dipole}}$ is the dipole dipole potential, and
-+
+$V_{\text{sticky}}$ is the sticky potential defined by the SSD model.
-+
-+
+The bend potential of a molecule is represented by the following function:
-+
+\begin{equation}
-+
+V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot}
-+
+\end{equation}
-+
+Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$
-+
+(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium
-+
+bond angle. $k_{\theta}$ is the force constant which determines the
-+
+strength of the harmonic bend. The parameters for $k_{\theta}$ and
-+
+$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998}
-+
-+
+The torsion potential and parameters are also taken from TraPPE. It is
-+
+of the form:
-+
+\begin{equation}
-+
+V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi]
-+
+        + c_2[1 + \cos(2\phi)]
-+
+        + c_3[1 + \cos(3\phi)]
-+
+\label{eq:origTorsionPot}
-+
+\end{equation}
-+
+Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$,
-+
+$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}).  However,
-+
+for computaional efficency, the torsion potentail has been recast
-+
+after the method of CHARMM\cite{charmm1983} whereby the angle series
-+
+is converted to a power series of the form:
-+
+\begin{equation}
-+
+V_{\text{torsion}}(\phi_{ijkl}) =
-+
+        k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0
-+
+\label{eq:torsionPot}
-+
+\end{equation}
-+
+Where:
-+
+\begin{align*}
-+
+k_0 &= c_1 + c_3 \\
-+
+k_1 &= c_1 - 3c_3 \\
-+
+k_2 &= 2 c_2 \\
-+
+k_3 &= 4c_3
-+
+\end{align*}
-+
+By recasting the equation to a power series, repeated trigonometric
-+
+evaluations are avoided during the calculation of the potential.
-+
-+
-+
-+
+The dipole-dipole potential has the following form:
-+
+\begin{equation}
-+
+V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
-+
+        \boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[
-+
+        \frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
-+
+        -
-+
+        \frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
-+
+                (\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
-+
+                {r^{5}_{ij}} \biggr]
-+
+\label{eq:dipolePot}
-+
+\end{equation}
-+
+Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing
-+
+towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$
-+
+are the Euler angles of atom $i$ and $j$
-+
+respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom
-+
+$i$ it takes its orientation from $\boldsymbol{\Omega}_i$.
-+
-+
-+
+\subsection{\label{sec:SSD}Water Model: SSD and Derivatives}
-+
-+
+In the interest of computational efficiency, the native solvent used
-+
+in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was
-+
+developed by Ichiye \emph{et al.} as a modified form of the
-+
+hard-sphere water model proposed by Bratko, Blum, and
-+
+Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole
-+
+with a Lennard-Jones core and a sticky potential that directs the
-+
+particles to assume the proper hydrogen bond orientation in the first
-+
+solvation shell. Thus, the interaction between two SSD water molecules
-+
+\emph{i} and \emph{j} is given by the potential
-+
+\begin{equation}
-+
+u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
-+
+(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
-+
+u_{ij}^{sp}
-+
+(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
-+
+\end{equation}
-+
+where the $\mathbf{r}_{ij}$ is the position vector between molecules
-+
+\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and
-+
+$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
-+
+orientations of the respective molecules. The Lennard-Jones, dipole,
-+
+and sticky parts of the potential are giving by the following
-+
+equations,
-+
+\begin{equation}
-+
+u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
-+
+\end{equation}
-+
+\begin{equation}
-+
+u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ ,
-+
+\end{equation}
-+
+\begin{equation}
-+
+\begin{split}
-+
+u_{ij}^{sp}
-+
+(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)
-+
+&=
-+
+\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\
-+
+& \quad \ +
-+
+s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
-+
+\end{split}
-+
+\end{equation}
-+
+where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
-+
+unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
-+
+$\nu_0$ scales the strength of the overall sticky potential, $s$ and
-+
+$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
-+
+functions take the following forms,
-+
+\begin{equation}
-+
+w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
-+
+\end{equation}
-+
+\begin{equation}
-+
+w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
-+
+\end{equation}
-+
+where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
-+
+term that promotes hydrogen bonding orientations within the first
-+
+solvation shell, and $w^\prime$ is a dipolar repulsion term that
-+
+repels unrealistic dipolar arrangements within the first solvation
-+
+shell. A more detailed description of the functional parts and
-+
+variables in this potential can be found in other
-+
+articles.\cite{liu96:new_model,chandra99:ssd_md}
-+
-+
+Since SSD is a one-site point dipole model, the force calculations are
-+
+simplified significantly from a computational standpoint, in that the
-+
+number of long-range interactions is dramatically reduced. In the
-+
+original Monte Carlo simulations using this model, Ichiye \emph{et
-+
+al.} reported a calculation speed up of up to an order of magnitude
-+
+over other comparable models while maintaining the structural behavior
-+
+of water.\cite{liu96:new_model} In the original molecular dynamics studies of
-+
+SSD, it was shown that it actually improves upon the prediction of
-+
+water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md}
-+
-+
+Recent constant pressure simulations revealed issues in the original
-+
+SSD model that led to lower than expected densities at all target
-+
+pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the
-+
+original SSD have resulted in improved density behavior, as well as
-+
+alterations in the water structure and transport behavior. {\sc oopse} is
-+
+easily modified to impliment these new potential parameter sets for
-+
+the derivative water models: SSD1, SSD/E, and SSD/RF. All of the
-+
+variable parameters are listed in the accompanying BASS file, and
-+
+these parameters simply need to be changed to the updated values.
-+
-+
-+
+\subsection{\label{sec:eam}Embedded Atom Model}
-+
-+
+Several molecular dynamics codes\cite{dynamo86} exist which have the
-+
+capacity to simulate metallic systems, including some that have
-+
+parallel computational abilities\cite{plimpton93}. Potentials that
-+
+describe bonding transition metal
-+
+systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a
-+
+attractive interaction which models the stabilization of ``Embedding''
-+
+a positively charged metal ion in an electron density created by the
-+
+free valance ``sea'' of electrons created by the surrounding atoms in
-+
+the system. A mostly repulsive pairwise part of the potential
-+
+describes the interaction of the positively charged metal core ions
-+
+with one another. A particular potential description called the
-+
+Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has
-+
+particularly wide adoption has been selected for inclusion in OOPSE. A
-+
+good review of EAM and other metallic potential formulations was done
-+
+by Voter.\cite{voter}
-+
-+
+The {\sc eam} potential has the form:
-+
+\begin{eqnarray}
-+
+V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i}
-+
+\phi_{ij}({\bf r}_{ij})  \\
-+
+\rho_{i}  & = & \sum_{j \neq i} f_{j}({\bf r}_{ij})
-+
+\end{eqnarray}
-+
-+
+where $\phi_{ij}$ is a primarily repulsive pairwise interaction
-+
+between atoms $i$ and $j$.In the origional formulation of
-+
+EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however
-+
+in later refinements to EAM have shown that nonuniqueness between $F$
-+
+and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The
-+
+embedding function $F_{i}$ is the energy required to embedded an
-+
+positively-charged core ion $i$ into a linear supeposition of
-+
+sperically averaged atomic electron densities given by
-+
+$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the
-+
+summations in the {\sc eam} equation to the few dozen atoms
-+
+surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$
-+
+interactions.
-+
-+
+\subsection{\label{Sec:pbc}Periodic Boundary Conditions}
-+
-+
+\textit{Periodic boundary conditions} are widely used to simulate truly
-+
+macroscopic systems with a relatively small number of particles. Simulation
-+
+box is replicated throughout space to form an infinite lattice. During the
-+
+simulation, when a particle moves in the primary cell, its periodic image
-+
+particles in other boxes move in exactly the same direction with exactly the
-+
+same orientation.Thus, as a particle leaves the primary cell, one of its
-+
+images will enter through the opposite face.If the simulation box is large
-+
+enough to avoid "feeling" the symmetric of the periodic lattice,the surface
-+
+effect could be ignored. Cubic and parallelepiped are the available periodic
-+
+cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe
-+
+the property of the simulation box. Therefore, not only the size of the
-+
+simulation box could be changed during the simulation, but also the shape of
-+
+it. The transformation from box space vector $\overrightarrow{s}$ to its
-+
+corresponding real space vector $\overrightarrow{r}$ is defined by
-+
+\begin{equation}
-+
+\overrightarrow{r}=H\overrightarrow{s}%
-+
+\end{equation}
-+
-+
-+
+where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
-+
+box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the
-+
+simulation box respectively.
-+
-+
+To find the minimum image, we need to convert the real vector to its
-+
+corresponding vector in box space first, \bigskip%
-+
+\begin{equation}
-+
+\overrightarrow{s}=H^{-1}\overrightarrow{r}%
-+
+\end{equation}
-+
+And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5
-+
+to 0.5,
-+
+\begin{equation}
-+
+s_{i}^{\prime}=s_{i}-round(s_{i})
-+
+\end{equation}
-+
+where%
-+
-+
+\begin{equation}
-+
+round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{
-+
+}x\geqslant0
-+
+\end{equation}
-+
+%
-+
-+
+\begin{equation}
-+
+round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0
-+
+\end{equation}
-+
-+
-+
+For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.
-+
-+
+Finally, we could get the minimum image by transforming back to real space,%
-+
-+
+\begin{equation}
-+
+\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}%
-+
+\end{equation}

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Comparing trunk/oopsePaper/EmpericalEnergy.tex (file contents): Revision 818 by gezelter, Fri Oct 24 21:27:59 2003 UTC vs. Revision 918 by mmeineke, Fri Jan 9 20:57:55 2004 UTC

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Comparing trunk/oopsePaper/EmpericalEnergy.tex (file contents):
Revision 818 by gezelter, Fri Oct 24 21:27:59 2003 UTC vs.
Revision 918 by mmeineke, Fri Jan 9 20:57:55 2004 UTC