--- trunk/oopsePaper/EmpericalEnergy.tex	2004/01/06 18:53:58	899
+++ trunk/oopsePaper/EmpericalEnergy.tex	2004/01/19 17:24:52	961
@@ -1,7 +1,7 @@
 
 \section{\label{sec:empericalEnergy}The Emperical Energy Functions}
 
-\subsection{\label{sec:atomsMolecules}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
@@ -9,63 +9,143 @@ are not currently suporrted by {\sc oopse}.
 element, or be used for collections of atoms such as a methyl
 group. The atoms are also capable of having a directional component
 associated with them, often in the form of a dipole. Charges on atoms
-are not currently suporrted by {\sc oopse}.
+are not currently suported by {\sc oopse}.
 
 The second most basic building block of a simulation is the
-molecule. The molecule is a way for {\sc oopse} to keep track of the atoms
-in a simulation in logical manner. This particular unit will store the
-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).
+molecule. The molecule is a way for {\sc oopse} to keep track of the
+atoms in a simulation in logical manner. This particular unit will
+store the 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 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 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
+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}
+\boldsymbol{\tau}_i=
+	\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} 
+	+ \boldsymbol{\tau}_{ia},
+\label{eq:torqueAccumulate}
+\end{equation}
+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 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 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 numerical instabilities
+inherent in using the Euler angles, the four parameter ``quaternion''
+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 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}
+
+The most basic force field implemented in OOPSE is the Lennard-Jones
+potential. The Lennard-Jones potential. Which mimics the Van der Waals
+interaction at long distances, and uses an emperical repulsion at
+short distances. The Lennard-Jones potential is given by:
+\begin{equation}
+V_{\text{LJ}}(r_{ij}) = 
+	4\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 well depth of the potential.
+
+Because this 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 a
+cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
+$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones 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 potential 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
+The Dipolar Unified-atom Force Field ({\sc duff}) was developed to
+simulate lipid bilayers. The systems require 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
+With this goal in mind, {\sc duff} has no point charges. Charge
+neutral distributions were replaced with dipoles, while most atoms and
+groups of atoms 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.
+computationally expensive Ewald sum. Instead, we can use
+neighbor-lists, reaction field, 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
-6~$\mbox{\AA}$. 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}.
+As an example, lipid head-groups in {\sc duff} are represented as
+point dipole interaction sites. By placing a dipole of 20.6~Debye at
+the head group center of mass, our model mimics the head group of
+phosphatidylcholine.\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}. The water model we use to complement the dipoles of the lipids is out
+repaarameterization of the soft sticky dipole (SSD) model of Ichiye
+\emph{et al.}\cite{liu96:new_model}
 
 \begin{figure}
-\epsfxsize=6in
-\epsfbox{lipidModel.epsi}
+\epsfxsize=\linewidth
+\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.}
+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
@@ -91,14 +171,14 @@ V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Interna
 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}}
+	+ \sum^{N}_{I=1} \sum_{J>I} 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_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{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})
@@ -106,26 +186,12 @@ within in the molecule. $V_{\text{torsion}}$ is the to
 \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. 
+within the molecule, and $V_{\text{torsion}}$ is the torsion potential
+for all 1, 4 bonded pairs. The pairwise portions of the internal
+potential are excluded for pairs that are 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}
@@ -140,16 +206,16 @@ V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] 
 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] 
+V_{\text{torsion}}(\phi) = 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:
+$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For
+computaional efficency, the torsion potential 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
@@ -165,116 +231,229 @@ The Lennard-Jones potential is given by:
 By recasting the equation to a power series, repeated trigonometric
 evaluations are avoided during the calculation of the potential.
 
-The Lennard-Jones potential is given by:
+
+The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
+as follows:
 \begin{equation}
-V_{\text{LJ}}(r_{ij}) = 
-	4\epsilon_{ij} \biggl[
-	\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
-	- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
+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:lennardJonesPot}
+\label{eq:crossPotentail}
 \end{equation}
-Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$
-scales the length of the interaction, and $\epsilon_{ij}$ scales the
-energy of the potential.
+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. Note that not all atom types include all interactions.
 
 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}}
+	\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[
+	\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j}
 	-
-	\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
-		(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
-		{r^{5}_{ij}} \biggr]
+	\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) %
+		(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) }
+		{r^{2}_{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 $\boldsymbol{\Omega}_i$.
+are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is
+the magnitude of the dipole moment of atom $i$ and
+$\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of
+$\boldsymbol{\Omega}_i$.
 
 
-\subsection{\label{sec:SSD}Water Model: SSD and Derivatives}
+\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF}
 
-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
+In the interest of computational efficiency, the default solvent used
+by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water
+model.\cite{Gezelter04} The original SSD was developed by Ichiye
+\emph{et al.}\cite{Ichiye96} 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),
+V_{ij} = 
+	V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp}
+	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
+	V_{ij}^{sp}
+	(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
+\label{eq:ssdPot}
 \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
+\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,
+orientations of the respective molecules. The Lennard-Jones and dipole
+parts of the potential are given by equations \ref{eq:lennardJonesPot}
+and \ref{eq:dipolePot} respectively. The sticky part is described by
+the following,
 \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],
+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) +
+	s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},
+	\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
+\label{eq:stickyPot}
 \end{equation}
+where $\nu_0$ is a strength parameter for the sticky potential, and
+$s$ and $s^\prime$ are cubic switching functions which turn off the
+sticky interaction beyond the first solvation shell. The $w$ function
+can be thought of as an attractive potential with tetrahedral
+geometry:
 \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}\ ,
+w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
+	\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
+\label{eq:stickyW}
 \end{equation}
+while the $w^\prime$ function counters the normal aligned and
+anti-aligned structures favored by point dipoles:
 \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}
+w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
+	(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
+\label{eq:stickyWprime}
 \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,
+It should be noted that $w$ is proportional to the sum of the $Y_3^2$
+and $Y_3^{-2}$ spherical harmonics (a linear combination which
+enhances the tetrahedral geometry for hydrogen bonded structures),
+while $w^\prime$ is a purely empirical function.  A more detailed
+description of the functional parts and variables in this potential
+can be found in the original SSD
+articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
+
+Since SSD is a single-point {\it dipolar} model, the force
+calculations are simplified significantly relative to the standard
+{\it charged} multi-point models. In the original Monte Carlo
+simulations using this model, Ichiye {\it et al.} reported that using
+SSD decreased computer time by a factor of 6-7 compared to other
+models.\cite{Ichiye96} What is most impressive is that these savings
+did not come at the expense of accurate depiction of the liquid state
+properties.  Indeed, SSD maintains reasonable agreement with the Soper
+diffraction data for the structural features of liquid
+water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
+exhibited by SSD agree with experiment better than those of more
+computationally expensive models (like TIP3P and
+SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
+of solvent properties makes SSD a very attractive model for the
+simulation of large scale biochemical simulations.
+
+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} The default model in {\sc oopse}
+is therefore SSD/E, a density corrected derivative of SSD that
+exhibits improved liquid structure and transport behavior. If the use
+of a reaction field long-range interaction correction is desired, it
+is recommended that the parameters be modified to those of the SSD/RF
+model. Solvent parameters can be easily modified in an accompanying
+{\sc BASS} file as illustrated in the scheme below. A table of the
+parameter values and the drawbacks and benefits of the different
+density corrected SSD models can be found in reference
+\ref{Gezelter04}.
+
+!!!Place a {\sc BASS} scheme showing SSD parameters around here!!!
+
+\subsection{\label{sec:eam}Embedded Atom Method}
+
+Several other molecular dynamics packages\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  ``Embedding''
+a positively charged metal ion in the electron density due to 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}({\sc eam}) that has
+particularly wide adoption has been selected for inclusion in OOPSE. A
+good review of {\sc 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}S
+
+where $F_{i} $ is the embedding function that equates the energy required to embed a
+positively-charged core ion $i$ into a linear superposition of
+sperically averaged atomic electron densities given by
+$\rho_{i}$.  $\phi_{ij}$ is a primarily repulsive pairwise interaction
+between atoms $i$ and $j$. In the original formulation of
+{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however
+in later refinements to EAM have shown that non-uniqueness between $F$
+and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} 
+ 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. 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}. 
+
+
+\subsection{\label{Sec:pbc}Periodic Boundary Conditions} 
+
+\newcommand{\roundme}{\operatorname{round}}
+ 
+\textit{Periodic boundary conditions} are widely used to simulate truly
+macroscopic systems with a relatively small number of particles. The
+simulation box is replicated throughout space to form an infinite
+lattice.  During the simulation, when a particle moves in the primary
+cell, its image 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 symmetries of
+the periodic lattice, surface effects can be ignored. Cubic,
+orthorhombic and parallelepiped are the available periodic cells In
+OOPSE. We use a matrix to describe the property of the simulation
+box. Therefore, both the size and shape of the simulation box can be
+changed during the simulation. The transformation from box space
+vector $\mathbf{s}$ to its corresponding real space vector
+$\mathbf{r}$ is defined by
 \begin{equation}
-w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
+\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{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
+three sides of the simulation box respectively.
+
+To find the minimum image, we convert the real vector to its
+corresponding vector in box space first, \bigskip%
 \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,
+\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
 \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}
+And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
+\begin{equation}
+s_{i}^{\prime}=s_{i}-\roundme(s_{i})
+\end{equation}
+where
 
-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.
+\begin{equation}
+\roundme(x)=\left\{
+\begin{array}{cc}
+\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
+\lceil{x-0.5}\rceil & \text{otherwise}%
+\end{array}
+\right.
+\end{equation}
+For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$,
+$\roundme(-3.1)=-3$.
 
+Finally, we obtain the minimum image coordinates by transforming back
+to real space,%
 
-\subsection{\label{sec:eam}Embedded Atom Model}
+\begin{equation}
+\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
+\end{equation}
 
-here there be Monsters