trunk/oopsePaper/EmpericalEnergy.tex


\section{\label{sec:empericalEnergy}The Emperical Energy Functions}

\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
representation as possible. They may represent specific atoms of an
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}.

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).

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}) = 
        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
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
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}.

\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} 
Revision:	918
Committed:	Fri Jan 9 20:57:55 2004 UTC (20 years, 5 months ago) by mmeineke
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