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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\section{\label{sec:empiricalEnergy}The Empirical Energy Functions} |
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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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representation as possible. They may represent specific atoms of an |
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element, or be used for collections of atoms such as a methyl |
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group. The atoms are also capable of having a directional component |
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associated with them, often in the form of a dipole. Charges on atoms |
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are not currently suporrted by {\sc oopse}. |
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The basic unit of an {\sc oopse} simulation is the atom. The |
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parameters describing the atom are generalized to make the atom as |
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flexible a representation as possible. They may represent specific |
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atoms of an element, or be used for collections of atoms such as |
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methyl and carbonyl groups. The atoms are also capable of having |
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directional components associated with them (\emph{e.g.}~permanent |
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dipoles). Charges on atoms are not currently supported by {\sc oopse}. |
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\begin{lstlisting}[caption={[Specifier for molecules and atoms] An example specifying the simple Ar molecule},label=sch:AtmMole] |
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molecule{ |
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name = "Ar"; |
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nAtoms = 1; |
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atom[0]{ |
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type="Ar"; |
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position( 0.0, 0.0, 0.0 ); |
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} |
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} |
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\end{lstlisting} |
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The second most basic building block of a simulation is the |
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molecule. The molecule is a way for {\sc oopse} to keep track of the |
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atoms in a simulation in logical manner. This particular unit will |
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responsible for the evaluation of its own bonded interaction |
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(i.e.~bonds, bends, and torsions). |
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|
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As stated previously, one of the features that sets {\sc OOPSE} apart |
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As stated previously, one of the features that sets {\sc 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 most 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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included in most simulation packages because of the requirement to |
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propagate the orientational degrees of freedom. Until recently, |
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integrators which propagate orientational motion have been lacking. |
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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 internal 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 more complex than the force in that it is |
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the torque applied on the center of mass that dictates rotational |
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motion. The summation of this torque is given by |
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translational and rotational motion in turn. In order to accumulate |
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the total force on a rigid body, the external forces and torques must |
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first be calculated for all the internal particles. The total force on |
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the rigid body is simply the sum of these external forces. |
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Accumulation of the total torque on the rigid body is more complex |
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than the force in that it is the torque applied on the center of mass |
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that dictates rotational motion. The torque on rigid body {\it i} is |
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\begin{equation} |
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\mathbf{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia}, |
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\boldsymbol{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}, |
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\label{eq:torqueAccumulate} |
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\end{equation} |
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where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$ |
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and $\mathbf{r}_{ia}$ are the force on and position of the component |
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particles of the rigid body.\cite{allen87:csl} |
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where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$, |
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$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on, |
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position of, and torque on the component particles 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 summation 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 must be |
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maintained for each rigid body. At a minimum, the rotation matrix |
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(\textbf{A}) can be described and propagated by the three Euler angles |
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($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of |
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(\textbf{A}) can be described by the three Euler angles ($\phi, |
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\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
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trigonometric operations involving $\phi, \theta,$ and |
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$\psi$.\cite{Goldstein01} In order to avoid rotational limitations |
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$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
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inherent in using the Euler angles, the four parameter ``quaternion'' |
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scheme can be used instead, where \textbf{A} is composed of arithmetic |
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operations involving the four components of a quaternion ($q_0, q_1, |
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q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads |
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to performance enhancements, particularly for very small |
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scheme is often used. The elements of \textbf{A} can be expressed as |
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arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
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and $q_3$).\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} |
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{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine |
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parameter rotation matrix internally. Further discussion on this |
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choice can be found in Sec.~\ref{sec:integrate}. |
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{\sc oopse} utilizes a relatively new scheme that propagates the |
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entire nine parameter rotation matrix internally. Further discussion |
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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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The most basic force field implemented in {\sc oopse} is the Lennard-Jones |
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potential. The Lennard-Jones potential. Which mimics the Van der Waals |
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interaction at long distances, and uses an empirical repulsion at |
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short distances. 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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\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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Where $r_{ij}$ is the distance between particle $i$ and $j$, |
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$\sigma_{ij}$ scales the length of the interaction, and |
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$\epsilon_{ij}$ scales the well depth of the potential. |
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Because the potential is calculated between all pairs, the force |
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Because this 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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keep the pair evaluation to a manageable number, {\sc oopse} employs a |
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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 Lennard-Jones 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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the potential 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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|
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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|
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The \underline{D}ipolar \underline{U}nified-Atom |
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\underline{F}orce \underline{F}ield ({\sc duff}) was developed to |
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simulate lipid bilayers. We needed a model capable of forming |
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The Dipolar Unified-atom Force Field ({\sc duff}) was developed to |
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simulate lipid bilayers. The systems require a model capable of forming |
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bilayers, while still being sufficiently computationally efficient to |
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allow simulations of large systems ($\approx$100's of phospholipids, |
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$\approx$1000's of waters) for long times ($\approx$10's of |
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nanoseconds). |
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|
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With this goal in mind, we have eliminated all point charges. Charge |
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distributions were replaced with dipoles, and charge-neutral |
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distributions were reduced to Lennard-Jones interaction sites. This |
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With this goal in mind, {\sc duff} has no point charges. Charge |
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neutral distributions were replaced with dipoles, while most atoms and |
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groups of atoms were reduced to Lennard-Jones interaction sites. This |
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simplification cuts the length scale of long range interactions from |
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$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
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computationally expensive Ewald-Sum. Instead, we can use |
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neighbor-lists and cutoff radii for the dipolar interactions. |
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computationally expensive Ewald sum. Instead, we can use |
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neighbor-lists, reaction field, and cutoff radii for the dipolar |
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interactions. |
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|
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As an example, lipid head groups in {\sc duff} are represented as point |
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dipole interaction sites.PC and PE Lipid head groups are typically |
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zwitterionic in nature, with charges separated by as much as |
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6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
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center of mass, our model mimics the head group of PC.\cite{Cevc87} |
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Additionally, a Lennard-Jones site is located at the |
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pseudoatom's center of mass. The model is illustrated by the dark grey |
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atom in Fig.~\ref{fig:lipidModel}. |
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As an example, lipid head-groups in {\sc duff} are represented as |
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point dipole interaction sites. By placing a dipole of 20.6~Debye at |
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the head group center of mass, our model mimics the head group of |
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phosphatidylcholine.\cite{Cevc87} Additionally, a Lennard-Jones site |
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is located at the pseudoatom's center of mass. The model is |
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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 |
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reparameterization of the soft sticky dipole (SSD) model of Ichiye |
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\emph{et al.}\cite{liu96:new_model} |
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|
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\begin{figure} |
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\epsfxsize=\linewidth |
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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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is the bend angle, $\mu$ is the dipole moment of the head group, and n |
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is the chain length.} |
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\label{fig:lipidModel} |
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\end{figure} |
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|
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The water model we use to complement the dipoles of the lipids is |
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the soft sticky dipole (SSD) model of Ichiye \emph{et |
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al.}\cite{liu96:new_model} This model is discussed in greater detail |
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in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
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Lennard-Jones interaction site. The site also contains a dipole to |
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mimic the partial charges on the hydrogens and the oxygen. However, |
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what makes the SSD model unique is the inclusion of a tetrahedral |
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short range potential to recover the hydrogen bonding of water, an |
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important factor when modeling bilayers, as it has been shown that |
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hydrogen bond network formation is a leading contribution to the |
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entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
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|
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– |
|
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Turning to the tails of the phospholipids, we have used a set of |
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scalable parameters to model the alkyl groups with Lennard-Jones |
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sites. For this, we have used the TraPPE force field of Siepmann |
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The total energy of function in {\sc duff} is given by the following: |
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\begin{equation} |
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V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
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+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
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+ \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
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\label{eq:totalPotential} |
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\end{equation} |
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Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
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\begin{equation} |
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V^{I}_{\text{Internal}} = |
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\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
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+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl}) |
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+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
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+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\label{eq:internalPotential} |
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\end{equation} |
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Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
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within in the molecule. $V_{\text{torsion}}$ is the torsion The |
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pairwise portions of the internal potential are excluded for pairs |
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that ar closer than three bonds, i.e.~atom pairs farther away than a |
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torsion are included in the pair-wise loop. |
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within the molecule, and $V_{\text{torsion}}$ is the torsion potential |
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for all 1, 4 bonded pairs. The pairwise portions of the internal |
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potential are excluded for pairs that are closer than three bonds, |
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i.e.~atom pairs farther away than a torsion are included in the |
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pair-wise loop. |
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|
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The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
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as follows: |
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\begin{equation} |
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V^{IJ}_{\text{Cross}} = |
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\sum_{i \in I} \sum_{j \in J} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ V_{\text{sticky}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\biggr] |
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\label{eq:crossPotentail} |
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\end{equation} |
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Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
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$V_{\text{dipole}}$ is the dipole dipole potential, and |
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$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
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|
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The bend potential of a molecule is represented by the following function: |
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\begin{equation} |
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The torsion potential and parameters are also taken from TraPPE. It is |
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|
of the form: |
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|
\begin{equation} |
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< |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
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> |
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
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|
+ c_2[1 + \cos(2\phi)] |
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+ c_3[1 + \cos(3\phi)] |
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\label{eq:origTorsionPot} |
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\end{equation} |
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Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
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$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
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< |
for computaional efficency, the torsion potentail has been recast |
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< |
after the method of CHARMM\cite{charmm1983} whereby the angle series |
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< |
is converted to a power series of the form: |
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> |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For |
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> |
computational efficiency, the torsion potential has been recast after |
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> |
the method of CHARMM\cite{charmm1983} whereby the angle series is |
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> |
converted to a power series of the form: |
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|
\begin{equation} |
| 231 |
|
V_{\text{torsion}}(\phi_{ijkl}) = |
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|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
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|
evaluations are avoided during the calculation of the potential. |
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|
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|
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+ |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
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+ |
as follows: |
| 248 |
+ |
\begin{equation} |
| 249 |
+ |
V^{IJ}_{\text{Cross}} = |
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+ |
\sum_{i \in I} \sum_{j \in J} |
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+ |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ |
+ V_{\text{sticky}} |
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+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ |
\biggr] |
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+ |
\label{eq:crossPotentail} |
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+ |
\end{equation} |
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+ |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
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+ |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
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+ |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD |
| 261 |
+ |
model. Note that not all atom types include all interactions. |
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|
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The dipole-dipole potential has the following form: |
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|
\begin{equation} |
| 265 |
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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< |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 267 |
< |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 266 |
> |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 267 |
> |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 268 |
|
- |
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< |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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< |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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< |
{r^{5}_{ij}} \biggr] |
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> |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
| 270 |
> |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
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> |
{r^{2}_{ij}} \biggr] |
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|
\label{eq:dipolePot} |
| 273 |
|
\end{equation} |
| 274 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 275 |
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 276 |
< |
are the Euler angles of atom $i$ and $j$ |
| 277 |
< |
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 278 |
< |
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
| 276 |
> |
are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is |
| 277 |
> |
the magnitude of the dipole moment of atom $i$ and |
| 278 |
> |
$\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of |
| 279 |
> |
$\boldsymbol{\Omega}_i$. |
| 280 |
|
|
| 281 |
|
|
| 282 |
< |
\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
| 282 |
> |
\subsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
| 283 |
|
|
| 284 |
|
In the interest of computational efficiency, the default solvent used |
| 285 |
< |
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
| 286 |
< |
developed by Ichiye \emph{et al.} as a modified form of the |
| 287 |
< |
hard-sphere water model proposed by Bratko, Blum, and |
| 285 |
> |
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
| 286 |
> |
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
| 287 |
> |
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
| 288 |
> |
water model proposed by Bratko, Blum, and |
| 289 |
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 290 |
|
with a Lennard-Jones core and a sticky potential that directs the |
| 291 |
|
particles to assume the proper hydrogen bond orientation in the first |
| 337 |
|
while $w^\prime$ is a purely empirical function. A more detailed |
| 338 |
|
description of the functional parts and variables in this potential |
| 339 |
|
can be found in the original SSD |
| 340 |
< |
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
| 340 |
> |
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
| 341 |
|
|
| 342 |
|
Since SSD is a single-point {\it dipolar} model, the force |
| 343 |
|
calculations are simplified significantly relative to the standard |
| 344 |
|
{\it charged} multi-point models. In the original Monte Carlo |
| 345 |
|
simulations using this model, Ichiye {\it et al.} reported that using |
| 346 |
|
SSD decreased computer time by a factor of 6-7 compared to other |
| 347 |
< |
models.\cite{Ichiye96} What is most impressive is that this savings |
| 347 |
> |
models.\cite{liu96:new_model} What is most impressive is that these savings |
| 348 |
|
did not come at the expense of accurate depiction of the liquid state |
| 349 |
|
properties. Indeed, SSD maintains reasonable agreement with the Soper |
| 350 |
< |
data for the structural features of liquid |
| 351 |
< |
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
| 350 |
> |
diffraction data for the structural features of liquid |
| 351 |
> |
water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
| 352 |
|
exhibited by SSD agree with experiment better than those of more |
| 353 |
|
computationally expensive models (like TIP3P and |
| 354 |
< |
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
| 354 |
> |
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
| 355 |
|
of solvent properties makes SSD a very attractive model for the |
| 356 |
|
simulation of large scale biochemical simulations. |
| 357 |
|
|
| 358 |
|
Recent constant pressure simulations revealed issues in the original |
| 359 |
|
SSD model that led to lower than expected densities at all target |
| 360 |
|
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
| 361 |
< |
is SSD/E, a density corrected derivative of SSD that exhibits improved |
| 362 |
< |
liquid structure and transport behavior. If the use of a reaction |
| 363 |
< |
field long-range interaction correction is desired, it is recommended |
| 364 |
< |
that the parameters be modified to those of the SSD/RF model. Solvent |
| 365 |
< |
parameters can be easily modified in an accompanying {\sc BASS} file |
| 366 |
< |
as illustrated in the scheme below. A table of the parameter values |
| 367 |
< |
and the drawbacks and benefits of the different density corrected SSD |
| 368 |
< |
models can be found in reference \ref{Gezelter04}. |
| 361 |
> |
is therefore SSD/E, a density corrected derivative of SSD that |
| 362 |
> |
exhibits improved liquid structure and transport behavior. If the use |
| 363 |
> |
of a reaction field long-range interaction correction is desired, it |
| 364 |
> |
is recommended that the parameters be modified to those of the SSD/RF |
| 365 |
> |
model. Solvent parameters can be easily modified in an accompanying |
| 366 |
> |
{\sc BASS} file as illustrated in the scheme below. A table of the |
| 367 |
> |
parameter values and the drawbacks and benefits of the different |
| 368 |
> |
density corrected SSD models can be found in reference |
| 369 |
> |
\ref{Gezelter04}. |
| 370 |
|
|
| 371 |
|
!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
| 372 |
|
|
| 373 |
< |
\subsection{\label{sec:eam}Embedded Atom Model} |
| 373 |
> |
\subsection{\label{sec:eam}Embedded Atom Method} |
| 374 |
|
|
| 375 |
< |
Several molecular dynamics codes\cite{dynamo86} exist which have the |
| 375 |
> |
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
| 376 |
|
capacity to simulate metallic systems, including some that have |
| 377 |
|
parallel computational abilities\cite{plimpton93}. Potentials that |
| 378 |
|
describe bonding transition metal |
| 379 |
|
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
| 380 |
< |
attractive interaction which models the stabilization of ``Embedding'' |
| 381 |
< |
a positively charged metal ion in an electron density created by the |
| 380 |
> |
attractive interaction which models ``Embedding'' |
| 381 |
> |
a positively charged metal ion in the electron density due to the |
| 382 |
|
free valance ``sea'' of electrons created by the surrounding atoms in |
| 383 |
|
the system. A mostly repulsive pairwise part of the potential |
| 384 |
|
describes the interaction of the positively charged metal core ions |
| 385 |
|
with one another. A particular potential description called the |
| 386 |
< |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has |
| 387 |
< |
particularly wide adoption has been selected for inclusion in OOPSE. A |
| 388 |
< |
good review of EAM and other metallic potential formulations was done |
| 386 |
> |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
| 387 |
> |
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
| 388 |
> |
good review of {\sc eam} and other metallic potential formulations was done |
| 389 |
|
by Voter.\cite{voter} |
| 390 |
|
|
| 391 |
|
The {\sc eam} potential has the form: |
| 393 |
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 394 |
|
\phi_{ij}({\bf r}_{ij}) \\ |
| 395 |
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 396 |
< |
\end{eqnarray} |
| 396 |
> |
\end{eqnarray}S |
| 397 |
|
|
| 398 |
< |
where $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 399 |
< |
between atoms $i$ and $j$.In the origional formulation of |
| 400 |
< |
EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 401 |
< |
in later refinements to EAM have shown that nonuniqueness between $F$ |
| 402 |
< |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The |
| 403 |
< |
embedding function $F_{i}$ is the energy required to embedded an |
| 404 |
< |
positively-charged core ion $i$ into a linear supeposition of |
| 405 |
< |
sperically averaged atomic electron densities given by |
| 406 |
< |
$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the |
| 398 |
> |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 399 |
> |
positively-charged core ion $i$ into a linear superposition of |
| 400 |
> |
spherically averaged atomic electron densities given by |
| 401 |
> |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 402 |
> |
between atoms $i$ and $j$. In the original formulation of |
| 403 |
> |
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 404 |
> |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 405 |
> |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 406 |
> |
There is a cutoff distance, $r_{cut}$, which limits the |
| 407 |
|
summations in the {\sc eam} equation to the few dozen atoms |
| 408 |
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 409 |
< |
interactions. |
| 409 |
> |
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}. |
| 410 |
|
|
| 411 |
+ |
|
| 412 |
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 413 |
+ |
|
| 414 |
+ |
\newcommand{\roundme}{\operatorname{round}} |
| 415 |
|
|
| 416 |
< |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 417 |
< |
macroscopic systems with a relatively small number of particles. Simulation |
| 418 |
< |
box is replicated throughout space to form an infinite lattice. During the |
| 419 |
< |
simulation, when a particle moves in the primary cell, its periodic image |
| 420 |
< |
particles in other boxes move in exactly the same direction with exactly the |
| 421 |
< |
same orientation.Thus, as a particle leaves the primary cell, one of its |
| 422 |
< |
images will enter through the opposite face.If the simulation box is large |
| 423 |
< |
enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
| 424 |
< |
effect could be ignored. Cubic and parallelepiped are the available periodic |
| 425 |
< |
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
| 426 |
< |
the property of the simulation box. Therefore, not only the size of the |
| 427 |
< |
simulation box could be changed during the simulation, but also the shape of |
| 428 |
< |
it. The transformation from box space vector $\overrightarrow{s}$ to its |
| 429 |
< |
corresponding real space vector $\overrightarrow{r}$ is defined by |
| 430 |
< |
\begin{equation} |
| 431 |
< |
\overrightarrow{r}=H\overrightarrow{s}% |
| 432 |
< |
\end{equation} |
| 433 |
< |
|
| 434 |
< |
|
| 435 |
< |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 436 |
< |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
| 437 |
< |
simulation box respectively. |
| 438 |
< |
|
| 439 |
< |
To find the minimum image, we need to convert the real vector to its |
| 440 |
< |
corresponding vector in box space first, \bigskip% |
| 441 |
< |
\begin{equation} |
| 442 |
< |
\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
| 443 |
< |
\end{equation} |
| 444 |
< |
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
| 445 |
< |
to 0.5, |
| 446 |
< |
\begin{equation} |
| 447 |
< |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 448 |
< |
\end{equation} |
| 449 |
< |
where% |
| 450 |
< |
|
| 451 |
< |
\begin{equation} |
| 452 |
< |
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
| 453 |
< |
}x\geqslant0 |
| 454 |
< |
\end{equation} |
| 455 |
< |
% |
| 456 |
< |
|
| 457 |
< |
\begin{equation} |
| 458 |
< |
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
| 459 |
< |
\end{equation} |
| 460 |
< |
|
| 461 |
< |
|
| 462 |
< |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 463 |
< |
|
| 464 |
< |
Finally, we could get the minimum image by transforming back to real space,% |
| 465 |
< |
|
| 466 |
< |
\begin{equation} |
| 467 |
< |
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
| 468 |
< |
\end{equation} |
| 416 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 417 |
> |
macroscopic systems with a relatively small number of particles. The |
| 418 |
> |
simulation box is replicated throughout space to form an infinite |
| 419 |
> |
lattice. During the simulation, when a particle moves in the primary |
| 420 |
> |
cell, its image in other boxes move in exactly the same direction with |
| 421 |
> |
exactly the same orientation.Thus, as a particle leaves the primary |
| 422 |
> |
cell, one of its images will enter through the opposite face.If the |
| 423 |
> |
simulation box is large enough to avoid "feeling" the symmetries of |
| 424 |
> |
the periodic lattice, surface effects can be ignored. Cubic, |
| 425 |
> |
orthorhombic and parallelepiped are the available periodic cells In |
| 426 |
> |
{\sc oopse}. We use a matrix to describe the property of the simulation |
| 427 |
> |
box. Therefore, both the size and shape of the simulation box can be |
| 428 |
> |
changed during the simulation. The transformation from box space |
| 429 |
> |
vector $\mathbf{s}$ to its corresponding real space vector |
| 430 |
> |
$\mathbf{r}$ is defined by |
| 431 |
> |
\begin{equation} |
| 432 |
> |
\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}% |
| 433 |
> |
\end{equation} |
| 434 |
> |
|
| 435 |
> |
|
| 436 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of |
| 437 |
> |
the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the |
| 438 |
> |
three sides of the simulation box respectively. |
| 439 |
> |
|
| 440 |
> |
To find the minimum image, we convert the real vector to its |
| 441 |
> |
corresponding vector in box space first, \bigskip% |
| 442 |
> |
\begin{equation} |
| 443 |
> |
\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}% |
| 444 |
> |
\end{equation} |
| 445 |
> |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 446 |
> |
\begin{equation} |
| 447 |
> |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 448 |
> |
\end{equation} |
| 449 |
> |
where |
| 450 |
> |
|
| 451 |
> |
% |
| 452 |
> |
|
| 453 |
> |
\begin{equation} |
| 454 |
> |
\roundme(x)=\left\{ |
| 455 |
> |
\begin{array}{cc} |
| 456 |
> |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 457 |
> |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 458 |
> |
\end{array} |
| 459 |
> |
\right. |
| 460 |
> |
\end{equation} |
| 461 |
> |
For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$, |
| 462 |
> |
$\roundme(-3.1)=-3$. |
| 463 |
> |
|
| 464 |
> |
Finally, we obtain the minimum image coordinates by transforming back |
| 465 |
> |
to real space,% |
| 466 |
> |
|
| 467 |
> |
\begin{equation} |
| 468 |
> |
\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 469 |
> |
\end{equation} |
| 470 |
> |
|
