--- trunk/oopsePaper/EmpericalEnergy.tex	2004/01/12 18:43:56	925
+++ trunk/oopsePaper/EmpericalEnergy.tex	2004/01/13 20:03:21	933
@@ -9,7 +9,7 @@ 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
@@ -68,12 +68,9 @@ potential. The Lennard-Jones potential mimics the attr
 \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:
+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[
@@ -82,20 +79,20 @@ Where $r_ij$ is the distance between particle $i$ and 
 	\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.
+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 the potential is calculated between all pairs, the force
+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 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
+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 equation to go to zero at the cut-off radius.
+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
@@ -114,51 +111,40 @@ The \underline{D}ipolar \underline{U}nified-Atom
 
 \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=\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
@@ -184,14 +170,14 @@ 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})
@@ -199,26 +185,12 @@ 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}
@@ -233,16 +205,16 @@ 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
@@ -259,23 +231,40 @@ evaluations are avoided during the calculation of the 
 evaluations are avoided during the calculation of the potential.
 
 
+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. 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}The {\sc DUFF} Water Models: SSD/E and SSD/RF}
@@ -367,22 +356,22 @@ models can be found in reference \ref{Gezelter04}.
 
 !!!Place a {\sc BASS} scheme showing SSD parameters around here!!!
 
-\subsection{\label{sec:eam}Embedded Atom Model}
+\subsection{\label{sec:eam}Embedded Atom Method}
 
-Several molecular dynamics codes\cite{dynamo86} exist which have the
+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 the stabilization of ``Embedding''
-a positively charged metal ion in an electron density created by the
+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}(EAM) that has
+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 EAM and other metallic potential formulations was done
+good review of {\sc eam} and other metallic potential formulations was done
 by Voter.\cite{voter}
 
 The {\sc eam} potential has the form:
@@ -390,73 +379,78 @@ V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i}
 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}
+\end{eqnarray}S
 
-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
+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
-$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the
+$\rho_{i}$.  $\phi_{ij}$ 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.
+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} 
  
-\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} 
+\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}
+\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}
+\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
+\end{equation}
+And then, each element of $\mathbf{s}$ is wrapped 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)=\left\{
+\begin{array}[c]{c}%
+\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
+\lceil{x-0.5}\rceil & \text{otherwise}%
+\end{array}
+\right.
+\end{equation}
+
+
+For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$,
+$round(-3.1)=-3$.
+
+Finally, we obtain the minimum image coordinates by transforming back
+to real space,%
+
+\begin{equation}
+\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
+\end{equation}
+
