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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\subsection{\label{sec:atomsMolecules}Atoms and Molecules} |
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\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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The basic unit of an {\sc oopse} simulation is the atom. The parameters |
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describing the atom are generalized to make the atom as flexible a |
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for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
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and torsions). |
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As stated in the previously, one of the features that sets OOPSE apart |
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from most of the current molecular simulation packages is the ability |
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to handle rigid body dynamics. Rigid bodies are non-spherical |
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particles or collections of particles that have a constant internal |
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potential and move collectively.\cite{Goldstein01} They are not |
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included in many standard simulation packages because of the need to |
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consider orientational degrees of freedom and include an integrator |
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that accurately propagates these motions in time. |
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Moving a rigid body involves determination of both the force and |
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torque applied by the surroundings, which directly affect the |
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translation and rotation in turn. In order to accumulate the total |
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force on a rigid body, the external forces must first be calculated |
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for all the interal particles. The total force on the rigid body is |
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simply the sum of these external forces. Accumulation of the total |
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torque on the rigid body is similar to the force in that it is a sum |
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of the torque applied on each internal particle, mapped onto the |
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center of mass of the rigid body. |
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The application of the total torque is done in the body fixed axis of |
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the rigid body. In order to move between the space fixed and body |
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fixed coordinate axes, parameters describing the orientation be |
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maintained for each rigid body. At a minimum, the rotation matrix can |
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be described and propagated by the three Euler |
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angles.\cite{Goldstein01} In order to avoid rotational limitations |
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when using Euler angles, the four parameter ``quaternion'' scheme can |
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be used instead.\cite{allen87:csl} Use of quaternions also leads to |
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performance enhancements, particularly for very small |
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systems.\cite{Evans77} OOPSE utilizes a relatively new scheme that |
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propagates the entire nine parameter rotation matrix. Further |
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discussion on this choice can be found in Sec.~\ref{sec:integrate}. |
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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The most basic force field implemented in OOPSE is the Lennard-Jones |
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potential. The Lennard-Jones potential mimics the attractive forces |
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two charge neutral particles experience when spontaneous dipoles are |
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induced on each other. This is the predominant intermolecular force in |
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systems of such as noble gases and simple alkanes. |
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The Lennard-Jones potential is given by: |
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\begin{equation} |
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V_{\text{LJ}}(r_{ij}) = |
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4\epsilon_{ij} \biggl[ |
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\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
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\biggr] |
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\label{eq:lennardJonesPot} |
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\end{equation} |
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Where $r_ij$ is the distance between particle $i$ and $j$, $\sigma_{ij}$ |
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scales the length of the interaction, and $\epsilon_{ij}$ scales the |
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energy well depth of the potential. |
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Because the potential is calculated between all pairs, the force |
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evaluation can become computationally expensive for large systems. To |
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keep the pair evaluation to a manegable number, OOPSE employs the use |
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of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
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$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length |
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parameter in the system. Truncating the calculation at |
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$r_{\text{cut}}$ introduces a discontinuity into the potential |
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energy. To offset this discontinuity, the energy value at |
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$r_{\text{cut}}$ is subtracted from the entire potential. This causes |
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the equation to go to zero at the cut-off radius. |
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Interactions between dissimilar particles requires the generation of |
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cross term parameters for $\sigma$ and $\epsilon$. These are |
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calculated through the Lorentz-Berthelot mixing |
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rules:\cite{allen87:csl} |
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\begin{equation} |
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\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}] |
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\label{eq:sigmaMix} |
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\end{equation} |
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and |
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\begin{equation} |
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\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}} |
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\label{eq:epsilonMix} |
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\end{equation} |
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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The \underline{D}ipolar \underline{U}nified-Atom |
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atom in Fig.~\ref{fig:lipidModel}. |
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\begin{figure} |
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\epsfxsize=6in |
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\epsfbox{lipidModel.epsi} |
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\epsfbox{lipidModel.eps} |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
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is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
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\label{fig:lipidModel} |
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By recasting the equation to a power series, repeated trigonometric |
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evaluations are avoided during the calculation of the potential. |
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The Lennard-Jones potential is given by: |
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\begin{equation} |
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V_{\text{LJ}}(r_{ij}) = |
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4\epsilon_{ij} \biggl[ |
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\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
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\biggr] |
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\label{eq:lennardJonesPot} |
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\end{equation} |
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Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$ |
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scales the length of the interaction, and $\epsilon_{ij}$ scales the |
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energy of the potential. |
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The dipole-dipole potential has the following form: |
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\begin{equation} |
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V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\subsection{\label{sec:eam}Embedded Atom Model} |
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here there be Monsters |
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Several molecular dynamics codes\cite{dynamo86} exist which have the |
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capacity to simulate metallic systems, including some that have |
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parallel computational abilities\cite{plimpton93}. Potentials that |
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describe bonding transition metal |
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systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
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attractive interaction which models the stabilization of ``Embedding'' |
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a positively charged metal ion in an electron density created by the |
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free valance ``sea'' of electrons created by the surrounding atoms in |
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the system. A mostly repulsive pairwise part of the potential |
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describes the interaction of the positively charged metal core ions |
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with one another. A particular potential description called the |
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Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has |
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particularly wide adoption has been selected for inclusion in OOPSE. A |
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good review of EAM and other metallic potential formulations was done |
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by Voter.\cite{voter} |
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The {\sc eam} potential has the form: |
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\begin{eqnarray} |
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V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
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\phi_{ij}({\bf r}_{ij}) \\ |
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\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
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\end{eqnarray} |
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where $\phi_{ij}$ is a primarily repulsive pairwise interaction |
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between atoms $i$ and $j$.In the origional formulation of |
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EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
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in later refinements to EAM have shown that nonuniqueness between $F$ |
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and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The |
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embedding function $F_{i}$ is the energy required to embedded an |
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positively-charged core ion $i$ into a linear supeposition of |
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sperically averaged atomic electron densities given by |
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$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the |
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summations in the {\sc eam} equation to the few dozen atoms |
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surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
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interactions. |
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\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
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\textit{Periodic boundary conditions} are widely used to simulate truly |
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macroscopic systems with a relatively small number of particles. Simulation |
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box is replicated throughout space to form an infinite lattice. During the |
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simulation, when a particle moves in the primary cell, its periodic image |
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particles in other boxes move in exactly the same direction with exactly the |
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same orientation.Thus, as a particle leaves the primary cell, one of its |
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images will enter through the opposite face.If the simulation box is large |
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enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
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effect could be ignored. Cubic and parallelepiped are the available periodic |
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cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
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the property of the simulation box. Therefore, not only the size of the |
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simulation box could be changed during the simulation, but also the shape of |
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it. The transformation from box space vector $\overrightarrow{s}$ to its |
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corresponding real space vector $\overrightarrow{r}$ is defined by |
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\begin{equation} |
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\overrightarrow{r}=H\overrightarrow{s}% |
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\end{equation} |
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|
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|
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where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
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box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
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simulation box respectively. |
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To find the minimum image, we need to convert the real vector to its |
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corresponding vector in box space first, \bigskip% |
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\begin{equation} |
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\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
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\end{equation} |
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And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
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to 0.5, |
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\begin{equation} |
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s_{i}^{\prime}=s_{i}-round(s_{i}) |
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\end{equation} |
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where% |
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|
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\begin{equation} |
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round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
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}x\geqslant0 |
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\end{equation} |
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% |
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|
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\begin{equation} |
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round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
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\end{equation} |
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For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
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Finally, we could get the minimum image by transforming back to real space,% |
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\begin{equation} |
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\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
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\end{equation} |
