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\chapter{\label{chapt:oopse}OOPSE: AN OPEN SOURCE OBJECT-ORIENTED PARALLEL SIMULATION ENGINE FOR MOLECULAR DYNAMICS} |
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%% \begin{abstract} |
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%% We detail the capabilities of a new open-source parallel simulation |
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%% package ({\sc oopse}) that can perform molecular dynamics simulations |
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%% on atom types that are missing from other popular packages. In |
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%% particular, {\sc oopse} is capable of performing orientational |
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%% dynamics on dipolar systems, and it can handle simulations of metallic |
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%% systems using the embedded atom method ({\sc eam}). |
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%% \end{abstract} |
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\lstset{language=C,frame=TB,basicstyle=\small,basicstyle=\ttfamily, % |
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xleftmargin=0.5in, xrightmargin=0.5in,captionpos=b, % |
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abovecaptionskip=0.5cm, belowcaptionskip=0.5cm} |
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\section{\label{sec:intro}Introduction} |
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\begin{itemize} |
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\item Need for package / Niche to fill |
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\item Design Goal |
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\item Open Source |
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\item Discussion of Paper Layout |
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\end{itemize} |
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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 |
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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}[float,caption={[Specifier for molecules and atoms] A sample specification of 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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Atoms can be collected into secondary srtructures such as rigid bodies |
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or molecules. The molecule is a way for {\sc oopse} to keep track of |
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the atoms in a simulation in logical manner. Molecular units store the |
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identities of all the atoms associated with themselves, and are |
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responsible for the evaluation of their own internal interactions |
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(\emph{i.e.}~bonds, bends, and torsions). Scheme \ref{sch:AtmMole} |
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shws how one creates a molecule in the \texttt{.mdl} files. The |
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position of the atoms given in the declaration are relative to the |
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origin of the molecule, and is used when creating a system containing |
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the molecule. |
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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 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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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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\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 $\boldsymbol{\tau}_i$ 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 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 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 numerical instabilities |
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inherent in using the Euler angles, the four parameter ``quaternion'' |
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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 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}. An example |
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definition of a riged body can be seen in Scheme |
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\ref{sch:rigidBody}. The positions in the atom definitions are the |
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placements of the atoms relative to the origin of the rigid body, |
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which itself has a position relative to the origin of the molecule. |
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\begin{lstlisting}[float,caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}] |
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molecule{ |
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name = "TIP3P_water"; |
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nRigidBodies = 1; |
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rigidBody[0]{ |
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nAtoms = 3; |
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atom[0]{ |
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type = "O_TIP3P"; |
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position( 0.0, 0.0, -0.06556 ); |
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} |
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atom[1]{ |
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type = "H_TIP3P"; |
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position( 0.0, 0.75695, 0.52032 ); |
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} |
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atom[2]{ |
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type = "H_TIP3P"; |
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position( 0.0, -0.75695, 0.52032 ); |
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} |
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position( 0.0, 0.0, 0.0 ); |
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orientation( 0.0, 0.0, 1.0 ); |
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} |
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} |
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\end{lstlisting} |
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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The most basic force field implemented in {\sc oopse} is the |
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Lennard-Jones potential, which mimics the van der Waals interaction at |
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long distances, and uses an empirical repulsion at short |
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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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\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 particles $i$ 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. Scheme |
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\ref{sch:LJFF} gives and example partial \texttt{.bass} file that |
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shows a system of 108 Ar particles simulated with the Lennard-Jones |
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force field. |
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\begin{lstlisting}[float,caption={[Invocation of the Lennard-Jones force field] A sample system using the Lennard-Jones force field.},label={sch:LJFF}] |
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/* |
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* The Ar molecule is specified |
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* external to the.bass file |
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*/ |
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#include "argon.mdl" |
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nComponents = 1; |
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component{ |
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type = "Ar"; |
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nMol = 108; |
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} |
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/* |
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* The initial configuration is generated |
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* before the simulation is invoked. |
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*/ |
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initialConfig = "./argon.init"; |
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forceField = "LJ"; |
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\end{lstlisting} |
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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 evaluations to a manageable number, {\sc oopse} employs |
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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 Lennard-Jones |
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length parameter present in the simulation. Truncating the calculation |
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at $r_{\text{cut}}$ 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 potential. This causes the |
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potential to go to zero smoothly 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 dipolar unified-atom force field ({\sc duff}) was developed to |
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simulate lipid bilayers. The simulations require a model capable of |
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forming bilayers, while still being sufficiently computationally |
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efficient to allow large systems ($\approx$100's of phospholipids, |
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$\approx$1000's of waters) to be simulated for long times |
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($\approx$10's of nanoseconds). |
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With this goal in mind, {\sc duff} has no point |
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charges. Charge-neutral distributions were replaced with dipoles, |
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while most atoms and groups of atoms were reduced to Lennard-Jones |
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interaction sites. This simplification cuts the length scale of long |
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range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us |
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to avoid the 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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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 large Lennard-Jones |
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site 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 |
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water model we use to complement the dipoles of the lipids is our |
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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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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{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 |
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is the chain length.} |
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\label{oopseFig:lipidModel} |
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\end{figure} |
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We have used a set of scalable parameters to model the alkyl groups |
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with Lennard-Jones sites. For this, we have borrowed parameters from |
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the TraPPE force field of Siepmann |
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\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
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representation of n-alkanes, which is parametrized against phase |
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equilibria using Gibbs ensemble Monte Carlo simulation |
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techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
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it generalizes the types of atoms in an alkyl chain to keep the number |
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of pseudoatoms to a minimum; the parameters for an atom such as |
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$\text{CH}_2$ do not change depending on what species are bonded to |
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it. |
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TraPPE also constrains all bonds to be of fixed length. Typically, |
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bond vibrations are the fastest motions in a molecular dynamic |
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simulation. Small time steps between force evaluations must be used to |
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ensure adequate sampling of the bond potential to ensure conservation |
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of energy. By constraining the bond lengths, larger time steps may be |
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used when integrating the equations of motion. A simulation using {\sc |
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duff} is illustrated in Scheme \ref{sch:DUFF}. |
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\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
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#include "water.mdl" |
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#include "lipid.mdl" |
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nComponents = 2; |
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component{ |
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type = "simpleLipid_16"; |
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nMol = 60; |
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} |
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component{ |
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type = "SSD_water"; |
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nMol = 1936; |
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} |
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initialConfig = "bilayer.init"; |
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forceField = "DUFF"; |
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\end{lstlisting} |
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\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
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The total potential energy function in {\sc duff} is |
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\begin{equation} |
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V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
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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 molecule $I$: |
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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}}(\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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\biggr] |
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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 the molecule $I$, 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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The bend potential of a molecule is represented by the following function: |
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\begin{equation} |
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V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
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\end{equation} |
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Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
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(see Fig.~\ref{fig:lipidModel}), $\theta_0$ is the equilibrium |
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bond angle, and $k_{\theta}$ is the force constant which determines the |
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strength of the harmonic bend. The parameters for $k_{\theta}$ and |
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$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
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The torsion potential and parameters are also borrowed 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) = 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 $\phi$ is the angle defined by four bonded neighbors $i$, |
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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} in which the angle series is |
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converted to a power series of the form: |
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\begin{equation} |
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V_{\text{torsion}}(\phi) = |
346 |
|
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
347 |
|
|
\label{eq:torsionPot} |
348 |
|
|
\end{equation} |
349 |
|
|
Where: |
350 |
|
|
\begin{align*} |
351 |
|
|
k_0 &= c_1 + c_3 \\ |
352 |
|
|
k_1 &= c_1 - 3c_3 \\ |
353 |
|
|
k_2 &= 2 c_2 \\ |
354 |
|
|
k_3 &= 4c_3 |
355 |
|
|
\end{align*} |
356 |
|
|
By recasting the potential as a power series, repeated trigonometric |
357 |
|
|
evaluations are avoided during the calculation of the potential energy. |
358 |
|
|
|
359 |
|
|
|
360 |
|
|
The cross potential between molecules $I$ and $J$, $V^{IJ}_{\text{Cross}}$, is |
361 |
|
|
as follows: |
362 |
|
|
\begin{equation} |
363 |
|
|
V^{IJ}_{\text{Cross}} = |
364 |
|
|
\sum_{i \in I} \sum_{j \in J} |
365 |
|
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
366 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
367 |
|
|
+ V_{\text{sticky}} |
368 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
369 |
|
|
\biggr] |
370 |
|
|
\label{eq:crossPotentail} |
371 |
|
|
\end{equation} |
372 |
|
|
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
373 |
|
|
$V_{\text{dipole}}$ is the dipole dipole potential, and |
374 |
|
|
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
375 |
|
|
(Sec.~\ref{sec:SSD}). Note that not all atom types include all |
376 |
|
|
interactions. |
377 |
|
|
|
378 |
|
|
The dipole-dipole potential has the following form: |
379 |
|
|
\begin{equation} |
380 |
|
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
381 |
|
|
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
382 |
|
|
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
383 |
|
|
- |
384 |
|
|
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
385 |
|
|
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
386 |
|
|
{r^{2}_{ij}} \biggr] |
387 |
|
|
\label{eq:dipolePot} |
388 |
|
|
\end{equation} |
389 |
|
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
390 |
|
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
391 |
|
|
are the orientational degrees of freedom for atoms $i$ and $j$ |
392 |
|
|
respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom |
393 |
|
|
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
394 |
|
|
vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is |
395 |
|
|
the unit vector pointing along $\mathbf{r}_{ij}$. |
396 |
|
|
|
397 |
|
|
|
398 |
|
|
\subsubsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
399 |
|
|
|
400 |
|
|
In the interest of computational efficiency, the default solvent used |
401 |
|
|
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
402 |
|
|
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
403 |
|
|
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
404 |
|
|
water model proposed by Bratko, Blum, and |
405 |
|
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
406 |
|
|
with a Lennard-Jones core and a sticky potential that directs the |
407 |
|
|
particles to assume the proper hydrogen bond orientation in the first |
408 |
|
|
solvation shell. Thus, the interaction between two SSD water molecules |
409 |
|
|
\emph{i} and \emph{j} is given by the potential |
410 |
|
|
\begin{equation} |
411 |
|
|
V_{ij} = |
412 |
|
|
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
413 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
414 |
|
|
V_{ij}^{sp} |
415 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
416 |
|
|
\label{eq:ssdPot} |
417 |
|
|
\end{equation} |
418 |
|
|
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
419 |
|
|
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
420 |
|
|
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
421 |
|
|
orientations of the respective molecules. The Lennard-Jones and dipole |
422 |
|
|
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
423 |
|
|
and \ref{eq:dipolePot} respectively. The sticky part is described by |
424 |
|
|
the following, |
425 |
|
|
\begin{equation} |
426 |
|
|
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
427 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
428 |
|
|
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
429 |
|
|
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
430 |
|
|
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
431 |
|
|
\label{eq:stickyPot} |
432 |
|
|
\end{equation} |
433 |
|
|
where $\nu_0$ is a strength parameter for the sticky potential, and |
434 |
|
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
435 |
|
|
sticky interaction beyond the first solvation shell. The $w$ function |
436 |
|
|
can be thought of as an attractive potential with tetrahedral |
437 |
|
|
geometry: |
438 |
|
|
\begin{equation} |
439 |
|
|
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
440 |
|
|
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
441 |
|
|
\label{eq:stickyW} |
442 |
|
|
\end{equation} |
443 |
|
|
while the $w^\prime$ function counters the normal aligned and |
444 |
|
|
anti-aligned structures favored by point dipoles: |
445 |
|
|
\begin{equation} |
446 |
|
|
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
447 |
|
|
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
448 |
|
|
\label{eq:stickyWprime} |
449 |
|
|
\end{equation} |
450 |
|
|
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
451 |
|
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
452 |
|
|
enhances the tetrahedral geometry for hydrogen bonded structures), |
453 |
|
|
while $w^\prime$ is a purely empirical function. A more detailed |
454 |
|
|
description of the functional parts and variables in this potential |
455 |
|
|
can be found in the original SSD |
456 |
|
|
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
457 |
|
|
|
458 |
|
|
Since SSD is a single-point {\it dipolar} model, the force |
459 |
|
|
calculations are simplified significantly relative to the standard |
460 |
|
|
{\it charged} multi-point models. In the original Monte Carlo |
461 |
|
|
simulations using this model, Ichiye {\it et al.} reported that using |
462 |
|
|
SSD decreased computer time by a factor of 6-7 compared to other |
463 |
|
|
models.\cite{liu96:new_model} What is most impressive is that these savings |
464 |
|
|
did not come at the expense of accurate depiction of the liquid state |
465 |
|
|
properties. Indeed, SSD maintains reasonable agreement with the Soper |
466 |
|
|
diffraction data for the structural features of liquid |
467 |
|
|
water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
468 |
|
|
exhibited by SSD agree with experiment better than those of more |
469 |
|
|
computationally expensive models (like TIP3P and |
470 |
|
|
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
471 |
|
|
of solvent properties makes SSD a very attractive model for the |
472 |
|
|
simulation of large scale biochemical simulations. |
473 |
|
|
|
474 |
|
|
Recent constant pressure simulations revealed issues in the original |
475 |
|
|
SSD model that led to lower than expected densities at all target |
476 |
|
|
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
477 |
|
|
is therefore SSD/E, a density corrected derivative of SSD that |
478 |
|
|
exhibits improved liquid structure and transport behavior. If the use |
479 |
|
|
of a reaction field long-range interaction correction is desired, it |
480 |
|
|
is recommended that the parameters be modified to those of the SSD/RF |
481 |
|
|
model. Solvent parameters can be easily modified in an accompanying |
482 |
|
|
{\sc BASS} file as illustrated in the scheme below. A table of the |
483 |
|
|
parameter values and the drawbacks and benefits of the different |
484 |
|
|
density corrected SSD models can be found in reference |
485 |
|
|
\ref{Gezelter04}. |
486 |
|
|
|
487 |
mmeineke |
1045 |
\begin{lstlisting}[float,caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
488 |
mmeineke |
1044 |
|
489 |
|
|
#include "water.mdl" |
490 |
|
|
|
491 |
|
|
nComponents = 1; |
492 |
|
|
component{ |
493 |
|
|
type = "SSD_water"; |
494 |
|
|
nMol = 864; |
495 |
|
|
} |
496 |
|
|
|
497 |
|
|
initialConfig = "liquidWater.init"; |
498 |
|
|
|
499 |
|
|
forceField = "DUFF"; |
500 |
|
|
|
501 |
|
|
/* |
502 |
|
|
* The reactionField flag toggles reaction |
503 |
|
|
* field corrections. |
504 |
|
|
*/ |
505 |
|
|
|
506 |
|
|
reactionField = false; // defaults to false |
507 |
|
|
dielectric = 80.0; // dielectric for reaction field |
508 |
|
|
|
509 |
|
|
/* |
510 |
|
|
* The following two flags set the cutoff |
511 |
|
|
* radius for the electrostatic forces |
512 |
|
|
* as well as the skin thickness of the switching |
513 |
|
|
* function. |
514 |
|
|
*/ |
515 |
|
|
|
516 |
|
|
electrostaticCutoffRadius = 9.2; |
517 |
|
|
electrostaticSkinThickness = 1.38; |
518 |
|
|
|
519 |
|
|
\end{lstlisting} |
520 |
|
|
|
521 |
|
|
|
522 |
|
|
\subsection{\label{sec:eam}Embedded Atom Method} |
523 |
|
|
|
524 |
|
|
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
525 |
|
|
capacity to simulate metallic systems, including some that have |
526 |
|
|
parallel computational abilities\cite{plimpton93}. Potentials that |
527 |
|
|
describe bonding transition metal |
528 |
|
|
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
529 |
|
|
attractive interaction which models ``Embedding'' |
530 |
|
|
a positively charged metal ion in the electron density due to the |
531 |
|
|
free valance ``sea'' of electrons created by the surrounding atoms in |
532 |
|
|
the system. A mostly repulsive pairwise part of the potential |
533 |
|
|
describes the interaction of the positively charged metal core ions |
534 |
|
|
with one another. A particular potential description called the |
535 |
|
|
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
536 |
|
|
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
537 |
|
|
good review of {\sc eam} and other metallic potential formulations was done |
538 |
|
|
by Voter.\cite{voter} |
539 |
|
|
|
540 |
|
|
The {\sc eam} potential has the form: |
541 |
|
|
\begin{eqnarray} |
542 |
|
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
543 |
|
|
\phi_{ij}({\bf r}_{ij}) \\ |
544 |
|
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
545 |
|
|
\end{eqnarray}S |
546 |
|
|
|
547 |
|
|
where $F_{i} $ is the embedding function that equates the energy required to embed a |
548 |
|
|
positively-charged core ion $i$ into a linear superposition of |
549 |
|
|
spherically averaged atomic electron densities given by |
550 |
|
|
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
551 |
|
|
between atoms $i$ and $j$. In the original formulation of |
552 |
|
|
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
553 |
|
|
in later refinements to EAM have shown that non-uniqueness between $F$ |
554 |
|
|
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
555 |
|
|
There is a cutoff distance, $r_{cut}$, which limits the |
556 |
|
|
summations in the {\sc eam} equation to the few dozen atoms |
557 |
|
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
558 |
|
|
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}. |
559 |
|
|
|
560 |
|
|
|
561 |
|
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
562 |
|
|
|
563 |
|
|
\newcommand{\roundme}{\operatorname{round}} |
564 |
|
|
|
565 |
|
|
\textit{Periodic boundary conditions} are widely used to simulate truly |
566 |
|
|
macroscopic systems with a relatively small number of particles. The |
567 |
|
|
simulation box is replicated throughout space to form an infinite lattice. |
568 |
|
|
During the simulation, when a particle moves in the primary cell, its image in |
569 |
|
|
other boxes move in exactly the same direction with exactly the same |
570 |
|
|
orientation.Thus, as a particle leaves the primary cell, one of its images |
571 |
|
|
will enter through the opposite face.If the simulation box is large enough to |
572 |
|
|
avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
573 |
|
|
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
574 |
|
|
parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
575 |
|
|
describe the property of the simulation box. Therefore, both the size and |
576 |
|
|
shape of the simulation box can be changed during the simulation. The |
577 |
|
|
transformation from box space vector $\mathbf{s}$ to its corresponding real |
578 |
|
|
space vector $\mathbf{r}$ is defined by |
579 |
|
|
\begin{equation} |
580 |
|
|
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
581 |
|
|
\end{equation} |
582 |
|
|
|
583 |
|
|
|
584 |
|
|
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
585 |
|
|
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
586 |
|
|
simulation box respectively. |
587 |
|
|
|
588 |
|
|
To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
589 |
|
|
to its corresponding vector in box space first, \bigskip% |
590 |
|
|
\begin{equation} |
591 |
|
|
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
592 |
|
|
\end{equation} |
593 |
|
|
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
594 |
|
|
\begin{equation} |
595 |
|
|
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
596 |
|
|
\end{equation} |
597 |
|
|
where |
598 |
|
|
|
599 |
|
|
% |
600 |
|
|
|
601 |
|
|
\begin{equation} |
602 |
|
|
\roundme(x)=\left\{ |
603 |
|
|
\begin{array}{cc}% |
604 |
mmeineke |
1045 |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant 0 \\ |
605 |
mmeineke |
1044 |
\lceil{x-0.5}\rceil & \text{otherwise}% |
606 |
|
|
\end{array} |
607 |
|
|
\right. |
608 |
|
|
\end{equation} |
609 |
|
|
|
610 |
|
|
|
611 |
|
|
For example, $\roundme(3.6)=4$,$\roundme(3.1)=3$, $\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
612 |
|
|
|
613 |
|
|
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
614 |
|
|
transforming back to real space,% |
615 |
|
|
|
616 |
|
|
\begin{equation} |
617 |
|
|
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
618 |
|
|
\end{equation} |
619 |
|
|
|
620 |
|
|
|
621 |
|
|
\section{Input and Output Files} |
622 |
|
|
|
623 |
|
|
\subsection{{\sc bass} and Model Files} |
624 |
|
|
|
625 |
|
|
Every {\sc oopse} simuation begins with a {\sc bass} file. {\sc bass} |
626 |
|
|
(\underline{B}izarre \underline{A}tom \underline{S}imulation |
627 |
|
|
\underline{S}yntax) is a script syntax that is parsed by {\sc oopse} at |
628 |
|
|
runtime. The {\sc bass} file allows for the user to completely describe the |
629 |
|
|
system they are to simulate, as well as tailor {\sc oopse}'s behavior during |
630 |
|
|
the simulation. {\sc bass} files are denoted with the extension |
631 |
|
|
\texttt{.bass}, an example file is shown in |
632 |
|
|
Fig.~\ref{fig:bassExample}. |
633 |
|
|
|
634 |
|
|
\begin{figure} |
635 |
|
|
\centering |
636 |
|
|
\framebox[\linewidth]{\rule{0cm}{0.75\linewidth}I'm a {\sc bass} file!} |
637 |
|
|
\caption{Here is an example \texttt{.bass} file} |
638 |
|
|
\label{fig:bassExample} |
639 |
|
|
\end{figure} |
640 |
|
|
|
641 |
|
|
Within the \texttt{.bass} file it is neccassary to provide a complete |
642 |
|
|
description of the molecule before it is actually placed in the |
643 |
|
|
simulation. The {\sc bass} syntax was originally developed with this goal in |
644 |
|
|
mind, and allows for the specification of all the atoms in a molecular |
645 |
|
|
prototype, as well as any bonds, bends, or torsions. These |
646 |
|
|
descriptions can become lengthy for complex molecules, and it would be |
647 |
|
|
inconvient to duplicate the simulation at the begining of each {\sc bass} |
648 |
|
|
script. Addressing this issue {\sc bass} allows for the inclusion of model |
649 |
|
|
files at the top of a \texttt{.bass} file. These model files, denoted |
650 |
|
|
with the \texttt{.mdl} extension, allow the user to describe a |
651 |
|
|
molecular prototype once, then simply include it into each simulation |
652 |
|
|
containing that molecule. |
653 |
|
|
|
654 |
|
|
\subsection{\label{subSec:coordFiles}Coordinate Files} |
655 |
|
|
|
656 |
|
|
The standard format for storage of a systems coordinates is a modified |
657 |
|
|
xyz-file syntax, the exact details of which can be seen in |
658 |
|
|
App.~\ref{appCoordFormat}. As all bonding and molecular information is |
659 |
|
|
stored in the \texttt{.bass} and \texttt{.mdl} files, the coordinate |
660 |
|
|
files are simply the complete set of coordinates for each atom at a |
661 |
|
|
given simulation time. |
662 |
|
|
|
663 |
|
|
There are three major files used by {\sc oopse} written in the coordinate |
664 |
|
|
format, they are as follows: the initialization file, the simulation |
665 |
|
|
trajectory file, and the final coordinates of the simulation. The |
666 |
|
|
initialization file is neccassary for {\sc oopse} to start the simulation |
667 |
|
|
with the proper coordinates. It is typically denoted with the |
668 |
|
|
extension \texttt{.init}. The trajectory file is created at the |
669 |
|
|
beginning of the simulation, and is used to store snapshots of the |
670 |
|
|
simulation at regular intervals. The first frame is a duplication of |
671 |
|
|
the \texttt{.init} file, and each subsequent frame is appended to the |
672 |
|
|
file at an interval specified in the \texttt{.bass} file. The |
673 |
|
|
trajectory file is given the extension \texttt{.dump}. The final |
674 |
|
|
coordinate file is the end of run or \texttt{.eor} file. The |
675 |
|
|
\texttt{.eor} file stores the final configuration of teh system for a |
676 |
|
|
given simulation. The file is updated at the same time as the |
677 |
|
|
\texttt{.dump} file. However, it only contains the most recent |
678 |
|
|
frame. In this way, an \texttt{.eor} file may be used as the |
679 |
|
|
initialization file to a second simulation in order to continue or |
680 |
|
|
recover the previous simulation. |
681 |
|
|
|
682 |
|
|
\subsection{Generation of Initial Coordinates} |
683 |
|
|
|
684 |
|
|
As was stated in Sec.~\ref{subSec:coordFiles}, an initialization file |
685 |
|
|
is needed to provide the starting coordinates for a simulation. The |
686 |
|
|
{\sc oopse} package provides a program called \texttt{sysBuilder} to aid in |
687 |
|
|
the creation of the \texttt{.init} file. \texttt{sysBuilder} is {\sc bass} |
688 |
|
|
aware, and will recognize arguments and parameters in the |
689 |
|
|
\texttt{.bass} file that would otherwise be ignored by the |
690 |
|
|
simulation. The program itself is under contiunual development, and is |
691 |
|
|
offered here as a helper tool only. |
692 |
|
|
|
693 |
|
|
\subsection{The Statistics File} |
694 |
|
|
|
695 |
|
|
The last output file generated by {\sc oopse} is the statistics file. This |
696 |
|
|
file records such statistical quantities as the instantaneous |
697 |
|
|
temperature, volume, pressure, etc. It is written out with the |
698 |
|
|
frequency specified in the \texttt{.bass} file. The file allows the |
699 |
|
|
user to observe the system variables as a function od simulation time |
700 |
|
|
while the simulation is in progress. One useful function the |
701 |
|
|
statistics file serves is to monitor the conserved quantity of a given |
702 |
|
|
simulation ensemble, this allows the user to observe the stability of |
703 |
|
|
the integrator. The statistics file is denoted with the \texttt{.stat} |
704 |
|
|
file extension. |
705 |
|
|
|
706 |
|
|
\section{\label{sec:mechanics}Mechanics} |
707 |
|
|
|
708 |
|
|
\subsection{\label{integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
709 |
|
|
|
710 |
|
|
Integration of the equations of motion was carried out using the |
711 |
|
|
symplectic splitting method proposed by Dullweber \emph{et |
712 |
|
|
al.}.\cite{Dullweber1997} The reason for this integrator selection |
713 |
|
|
deals with poor energy conservation of rigid body systems using |
714 |
|
|
quaternions. While quaternions work well for orientational motion in |
715 |
|
|
alternate ensembles, the microcanonical ensemble has a constant energy |
716 |
|
|
requirement that is quite sensitive to errors in the equations of |
717 |
|
|
motion. The original implementation of this code utilized quaternions |
718 |
|
|
for rotational motion propagation; however, a detailed investigation |
719 |
|
|
showed that they resulted in a steady drift in the total energy, |
720 |
|
|
something that has been observed by others.\cite{Laird97} |
721 |
|
|
|
722 |
|
|
The key difference in the integration method proposed by Dullweber |
723 |
|
|
\emph{et al.} is that the entire rotation matrix is propagated from |
724 |
|
|
one time step to the next. In the past, this would not have been as |
725 |
|
|
feasible a option, being that the rotation matrix for a single body is |
726 |
|
|
nine elements long as opposed to 3 or 4 elements for Euler angles and |
727 |
|
|
quaternions respectively. System memory has become much less of an |
728 |
|
|
issue in recent times, and this has resulted in substantial benefits |
729 |
|
|
in energy conservation. There is still the issue of 5 or 6 additional |
730 |
|
|
elements for describing the orientation of each particle, which will |
731 |
|
|
increase dump files substantially. Simply translating the rotation |
732 |
|
|
matrix into its component Euler angles or quaternions for storage |
733 |
|
|
purposes relieves this burden. |
734 |
|
|
|
735 |
|
|
The symplectic splitting method allows for Verlet style integration of |
736 |
|
|
both linear and angular motion of rigid bodies. In the integration |
737 |
|
|
method, the orientational propagation involves a sequence of matrix |
738 |
|
|
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
739 |
|
|
matrix rotations end up being more costly computationally than the |
740 |
|
|
simpler arithmetic quaternion propagation. With the same time step, a |
741 |
|
|
1000 SSD particle simulation shows an average 7\% increase in |
742 |
|
|
computation time using the symplectic step method in place of |
743 |
|
|
quaternions. This cost is more than justified when comparing the |
744 |
|
|
energy conservation of the two methods as illustrated in figure |
745 |
|
|
\ref{timestep}. |
746 |
|
|
|
747 |
|
|
\begin{figure} |
748 |
mmeineke |
1045 |
\centering |
749 |
|
|
\includegraphics[width=\linewidth]{timeStep.eps} |
750 |
mmeineke |
1044 |
\caption{Energy conservation using quaternion based integration versus |
751 |
|
|
the symplectic step method proposed by Dullweber \emph{et al.} with |
752 |
|
|
increasing time step. For each time step, the dotted line is total |
753 |
|
|
energy using the symplectic step integrator, and the solid line comes |
754 |
|
|
from the quaternion integrator. The larger time step plots are shifted |
755 |
|
|
up from the true energy baseline for clarity.} |
756 |
|
|
\label{timestep} |
757 |
|
|
\end{figure} |
758 |
|
|
|
759 |
|
|
In figure \ref{timestep}, the resulting energy drift at various time |
760 |
|
|
steps for both the symplectic step and quaternion integration schemes |
761 |
|
|
is compared. All of the 1000 SSD particle simulations started with the |
762 |
|
|
same configuration, and the only difference was the method for |
763 |
|
|
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
764 |
|
|
methods for propagating particle rotation conserve energy fairly well, |
765 |
|
|
with the quaternion method showing a slight energy drift over time in |
766 |
|
|
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
767 |
|
|
energy conservation benefits of the symplectic step method are clearly |
768 |
|
|
demonstrated. Thus, while maintaining the same degree of energy |
769 |
|
|
conservation, one can take considerably longer time steps, leading to |
770 |
|
|
an overall reduction in computation time. |
771 |
|
|
|
772 |
|
|
Energy drift in these SSD particle simulations was unnoticeable for |
773 |
|
|
time steps up to three femtoseconds. A slight energy drift on the |
774 |
|
|
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
775 |
|
|
four femtoseconds, and as expected, this drift increases dramatically |
776 |
|
|
with increasing time step. To insure accuracy in the constant energy |
777 |
|
|
simulations, time steps were set at 2 fs and kept at this value for |
778 |
|
|
constant pressure simulations as well. |
779 |
|
|
|
780 |
|
|
|
781 |
|
|
\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
782 |
|
|
|
783 |
|
|
|
784 |
|
|
{\sc oopse} implements a |
785 |
|
|
|
786 |
|
|
|
787 |
|
|
\subsubsection{\label{sec:noseHooverThermo}Nose-Hoover Thermostatting} |
788 |
|
|
|
789 |
|
|
To mimic the effects of being in a constant temperature ({\sc nvt}) |
790 |
|
|
ensemble, {\sc oopse} uses the Nose-Hoover extended system |
791 |
|
|
approach.\cite{Hoover85} In this method, the equations of motion for |
792 |
|
|
the particle positions and velocities are |
793 |
|
|
\begin{eqnarray} |
794 |
|
|
\dot{{\bf r}} & = & {\bf v} \\ |
795 |
|
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} |
796 |
|
|
\label{eq:nosehoovereom} |
797 |
|
|
\end{eqnarray} |
798 |
|
|
|
799 |
|
|
$\chi$ is an ``extra'' variable included in the extended system, and |
800 |
|
|
it is propagated using the first order equation of motion |
801 |
|
|
\begin{equation} |
802 |
|
|
\dot{\chi} = \frac{1}{\tau_{T}} \left( \frac{T}{T_{target}} - 1 \right) |
803 |
|
|
\label{eq:nosehooverext} |
804 |
|
|
\end{equation} |
805 |
|
|
where $T_{target}$ is the target temperature for the simulation, and |
806 |
|
|
$\tau_{T}$ is a time constant for the thermostat. |
807 |
|
|
|
808 |
|
|
To select the Nose-Hoover {\sc nvt} ensemble, the {\tt ensemble = NVT;} |
809 |
|
|
command would be used in the simulation's {\sc bass} file. There is |
810 |
|
|
some subtlety in choosing values for $\tau_{T}$, and it is usually set |
811 |
|
|
to values of a few ps. Within a {\sc bass} file, $\tau_{T}$ could be |
812 |
|
|
set to 1 ps using the {\tt tauThermostat = 1000; } command. |
813 |
|
|
|
814 |
|
|
|
815 |
|
|
\subsection{\label{Sec:zcons}Z-Constraint Method} |
816 |
|
|
|
817 |
|
|
Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation |
818 |
|
|
method was developed to investigate the dynamics of ions inside the ion |
819 |
|
|
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated |
820 |
|
|
from the deviation of the instaneous force from its mean force. |
821 |
|
|
|
822 |
|
|
% |
823 |
|
|
|
824 |
|
|
\begin{equation} |
825 |
|
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
826 |
|
|
\end{equation} |
827 |
|
|
|
828 |
|
|
|
829 |
|
|
where% |
830 |
|
|
\begin{equation} |
831 |
|
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
832 |
|
|
\end{equation} |
833 |
|
|
|
834 |
|
|
|
835 |
|
|
If the time-dependent friction decay rapidly, static friction coefficient can |
836 |
|
|
be approximated by% |
837 |
|
|
|
838 |
|
|
\begin{equation} |
839 |
|
|
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt |
840 |
|
|
\end{equation} |
841 |
|
|
|
842 |
|
|
|
843 |
|
|
Hence, diffusion constant can be estimated by |
844 |
|
|
\begin{equation} |
845 |
|
|
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
846 |
|
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}% |
847 |
|
|
\end{equation} |
848 |
|
|
|
849 |
|
|
|
850 |
|
|
\bigskip Z-Constraint method, which fixed the z coordinates of the molecules |
851 |
|
|
with respect to the center of the mass of the system, was proposed to obtain |
852 |
|
|
the forces required in force auto-correlation method.\cite{Marrink94} However, |
853 |
|
|
simply resetting the coordinate will move the center of the mass of the whole |
854 |
|
|
system. To avoid this problem, a new method was used at {\sc oopse}. Instead of |
855 |
|
|
resetting the coordinate, we reset the forces of z-constraint molecules as |
856 |
|
|
well as subtract the total constraint forces from the rest of the system after |
857 |
|
|
force calculation at each time step. |
858 |
|
|
\begin{verbatim} |
859 |
|
|
$F_{\alpha i}=0$ |
860 |
|
|
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}$ |
861 |
|
|
$F_{\alpha i}=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}$ |
862 |
|
|
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}$ |
863 |
|
|
\end{verbatim} |
864 |
|
|
|
865 |
|
|
At the very beginning of the simulation, the molecules may not be at its |
866 |
|
|
constraint position. To move the z-constraint molecule to the specified |
867 |
|
|
position, a simple harmonic potential is used% |
868 |
|
|
|
869 |
|
|
\begin{equation} |
870 |
|
|
U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}% |
871 |
|
|
\end{equation} |
872 |
|
|
where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is |
873 |
|
|
current z coordinate of the center of mass of the z-constraint molecule, and |
874 |
|
|
$z_{cons}$ is the restraint position. Therefore, the harmonic force operated |
875 |
|
|
on the z-constraint molecule at time $t$ can be calculated by% |
876 |
|
|
\begin{equation} |
877 |
|
|
F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}% |
878 |
|
|
(z(t)-z_{cons}) |
879 |
|
|
\end{equation} |
880 |
|
|
Worthy of mention, other kinds of potential functions can also be used to |
881 |
|
|
drive the z-constraint molecule. |
882 |
|
|
|
883 |
|
|
\section{\label{sec:analysis}Trajectory Analysis} |
884 |
|
|
|
885 |
|
|
\subsection{\label{subSec:staticProps}Static Property Analysis} |
886 |
|
|
|
887 |
|
|
The static properties of the trajectories are analyzed with the |
888 |
|
|
program \texttt{staticProps}. The code is capable of calculating the following |
889 |
|
|
pair correlations between species A and B: |
890 |
|
|
\begin{itemize} |
891 |
|
|
\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr} |
892 |
|
|
\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta} |
893 |
|
|
\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega} |
894 |
|
|
\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ} |
895 |
|
|
\item $\langle \cos \omega \rangle_{\text{AB}}(r)$: |
896 |
|
|
Eq.~\ref{eq:cosOmegaOfR} |
897 |
|
|
\end{itemize} |
898 |
|
|
|
899 |
|
|
The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
900 |
|
|
\begin{equation} |
901 |
|
|
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %% |
902 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} %% |
903 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr} |
904 |
|
|
\end{equation} |
905 |
|
|
Where $\mathbf{r}_{ij}$ is the vector |
906 |
|
|
\begin{equation*} |
907 |
|
|
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag |
908 |
|
|
\end{equation*} |
909 |
|
|
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over |
910 |
|
|
the expected pair density at a given $r$. |
911 |
|
|
|
912 |
|
|
The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and |
913 |
|
|
$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two |
914 |
|
|
dimensional histograms. Both use $r$ for the primary axis then a |
915 |
|
|
$\cos$ for the secondary axis ($\cos \theta$ for |
916 |
|
|
Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for |
917 |
|
|
Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to |
918 |
|
|
correlate alignment on directional entities. $g_{\text{AB}}(r, \cos |
919 |
|
|
\theta)$ is defined as follows: |
920 |
|
|
\begin{equation} |
921 |
|
|
g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
922 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
923 |
|
|
\delta( \cos \theta - \cos \theta_{ij}) |
924 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle |
925 |
|
|
\label{eq:gofrCosTheta} |
926 |
|
|
\end{equation} |
927 |
|
|
Where |
928 |
|
|
\begin{equation*} |
929 |
|
|
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
930 |
|
|
\end{equation*} |
931 |
|
|
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
932 |
|
|
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector |
933 |
|
|
$\mathbf{r}_{ij}$. |
934 |
|
|
|
935 |
|
|
The second two dimensional histogram is of the form: |
936 |
|
|
\begin{equation} |
937 |
|
|
g_{\text{AB}}(r, \cos \omega) = |
938 |
|
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
939 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
940 |
|
|
\delta( \cos \omega - \cos \omega_{ij}) |
941 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
942 |
|
|
\end{equation} |
943 |
|
|
Here |
944 |
|
|
\begin{equation*} |
945 |
|
|
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
946 |
|
|
\end{equation*} |
947 |
|
|
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit |
948 |
|
|
directional vectors of species $i$ and $j$. |
949 |
|
|
|
950 |
|
|
The static analysis code is also cable of calculating a three |
951 |
|
|
dimensional pair correlation of the form: |
952 |
|
|
\begin{equation}\label{eq:gofrXYZ} |
953 |
|
|
g_{\text{AB}}(x, y, z) = |
954 |
|
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
955 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
956 |
|
|
\delta( x - x_{ij}) |
957 |
|
|
\delta( y - y_{ij}) |
958 |
|
|
\delta( z - z_{ij}) \rangle |
959 |
|
|
\end{equation} |
960 |
|
|
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
961 |
|
|
components respectively of vector $\mathbf{r}_{ij}$. |
962 |
|
|
|
963 |
|
|
The final pair correlation is similar to |
964 |
|
|
Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega |
965 |
|
|
\rangle_{\text{AB}}(r)$ is calculated in the following way: |
966 |
|
|
\begin{equation}\label{eq:cosOmegaOfR} |
967 |
|
|
\langle \cos \omega \rangle_{\text{AB}}(r) = |
968 |
|
|
\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}} |
969 |
|
|
(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle |
970 |
|
|
\end{equation} |
971 |
|
|
Here $\cos \omega_{ij}$ is defined in the same way as in |
972 |
|
|
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair |
973 |
|
|
correlation that gives the average correlation of two directional |
974 |
|
|
entities as a function of their distance from each other. |
975 |
|
|
|
976 |
|
|
All static properties are calculated on a frame by frame basis. The |
977 |
|
|
trajectory is read a single frame at a time, and the appropriate |
978 |
|
|
calculations are done on each frame. Once one frame is finished, the |
979 |
|
|
next frame is read in, and a running average of the property being |
980 |
|
|
calculated is accumulated in each frame. The program allows for the |
981 |
|
|
user to specify more than one property be calculated in single run, |
982 |
|
|
preventing the need to read a file multiple times. |
983 |
|
|
|
984 |
|
|
\subsection{\label{dynamicProps}Dynamic Property Analysis} |
985 |
|
|
|
986 |
|
|
The dynamic properties of a trajectory are calculated with the program |
987 |
|
|
\texttt{dynamicProps}. The program will calculate the following properties: |
988 |
|
|
\begin{gather} |
989 |
|
|
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\ |
990 |
|
|
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\ |
991 |
|
|
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr} |
992 |
|
|
\end{gather} |
993 |
|
|
|
994 |
|
|
Eq.~\ref{eq:rms} is the root mean square displacement |
995 |
|
|
function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the |
996 |
|
|
velocity and angular velocity correlation functions respectively. The |
997 |
|
|
latter is only applicable to directional species in the simulation. |
998 |
|
|
|
999 |
|
|
The \texttt{dynamicProps} program handles he file in a manner different from |
1000 |
|
|
\texttt{staticProps}. As the properties calculated by this program are time |
1001 |
|
|
dependent, multiple frames must be read in simultaneously by the |
1002 |
|
|
program. For small trajectories this is no problem, and the entire |
1003 |
|
|
trajectory is read into memory. However, for long trajectories of |
1004 |
|
|
large systems, the files can be quite large. In order to accommodate |
1005 |
|
|
large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory |
1006 |
|
|
are allocated to read in several frames each. |
1007 |
|
|
|
1008 |
|
|
In this two block scheme, the correlation functions are first |
1009 |
|
|
calculated within each memory block, then the cross correlations |
1010 |
|
|
between the frames contained within the two blocks are |
1011 |
|
|
calculated. Once completed, the memory blocks are incremented, and the |
1012 |
|
|
process is repeated. A diagram illustrating the process is shown in |
1013 |
|
|
Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps}, |
1014 |
|
|
multiple properties may be calculated in a single run to avoid |
1015 |
|
|
multiple reads on the same file. |
1016 |
|
|
|
1017 |
|
|
\begin{figure} |
1018 |
mmeineke |
1045 |
\centering |
1019 |
|
|
\includegraphics[width=\linewidth]{dynamicPropsMem.eps} |
1020 |
mmeineke |
1044 |
\caption{This diagram illustrates the dynamic memory allocation used by \texttt{dynamicProps}, which follows the scheme: $\sum^{N_{\text{memory blocks}}}_{i=1}[ \operatorname{self}(i) + \sum^{N_{\text{memory blocks}}}_{j>i} \operatorname{cross}(i,j)]$. The shaded region represents the self correlation of the memory block, and the open blocks are read one at a time and the cross correlations between blocks are calculated.} |
1021 |
|
|
\label{fig:dynamicPropsMemory} |
1022 |
|
|
\end{figure} |
1023 |
|
|
|
1024 |
|
|
\section{\label{sec:ProgramDesign}Program Design} |
1025 |
|
|
|
1026 |
|
|
\subsection{\label{sec:architecture} OOPSE Architecture} |
1027 |
|
|
|
1028 |
|
|
The core of OOPSE is divided into two main object libraries: {\texttt |
1029 |
|
|
libBASS} and {\texttt libmdtools}. {\texttt libBASS} is the library |
1030 |
|
|
developed around the parseing engine and {\texttt libmdtools} is the |
1031 |
|
|
software library developed around the simulation engine. |
1032 |
|
|
|
1033 |
|
|
|
1034 |
|
|
|
1035 |
|
|
\subsection{\label{sec:programLang} Programming Languages } |
1036 |
|
|
|
1037 |
|
|
\subsection{\label{sec:parallelization} Parallelization of OOPSE} |
1038 |
|
|
|
1039 |
|
|
Although processor power is doubling roughly every 18 months according |
1040 |
|
|
to the famous Moore's Law\cite{moore}, it is still unreasonable to |
1041 |
|
|
simulate systems of more then a 1000 atoms on a single processor. To |
1042 |
|
|
facilitate study of larger system sizes or smaller systems on long |
1043 |
|
|
time scales in a reasonable period of time, parallel methods were |
1044 |
|
|
developed allowing multiple CPU's to share the simulation |
1045 |
|
|
workload. Three general categories of parallel decomposition method's |
1046 |
|
|
have been developed including atomic, spatial and force decomposition |
1047 |
|
|
methods. |
1048 |
|
|
|
1049 |
|
|
Algorithmically simplest of the three method's is atomic decomposition |
1050 |
|
|
where N particles in a simulation are split among P processors for the |
1051 |
|
|
duration of the simulation. Computational cost scales as an optimal |
1052 |
|
|
$O(N/P)$ for atomic decomposition. Unfortunately all processors must |
1053 |
|
|
communicate positions and forces with all other processors leading |
1054 |
|
|
communication to scale as an unfavorable $O(N)$ independent of the |
1055 |
|
|
number of processors. This communication bottleneck led to the |
1056 |
|
|
development of spatial and force decomposition methods in which |
1057 |
|
|
communication among processors scales much more favorably. Spatial or |
1058 |
|
|
domain decomposition divides the physical spatial domain into 3D boxes |
1059 |
|
|
in which each processor is responsible for calculation of forces and |
1060 |
|
|
positions of particles located in its box. Particles are reassigned to |
1061 |
|
|
different processors as they move through simulation space. To |
1062 |
|
|
calculate forces on a given particle, a processor must know the |
1063 |
|
|
positions of particles within some cutoff radius located on nearby |
1064 |
|
|
processors instead of the positions of particles on all |
1065 |
|
|
processors. Both communication between processors and computation |
1066 |
|
|
scale as $O(N/P)$ in the spatial method. However, spatial |
1067 |
|
|
decomposition adds algorithmic complexity to the simulation code and |
1068 |
|
|
is not very efficient for small N since the overall communication |
1069 |
|
|
scales as the surface to volume ratio $(N/P)^{2/3}$ in three |
1070 |
|
|
dimensions. |
1071 |
|
|
|
1072 |
|
|
Force decomposition assigns particles to processors based on a block |
1073 |
|
|
decomposition of the force matrix. Processors are split into a |
1074 |
|
|
optimally square grid forming row and column processor groups. Forces |
1075 |
|
|
are calculated on particles in a given row by particles located in |
1076 |
|
|
that processors column assignment. Force decomposition is less complex |
1077 |
|
|
to implement then the spatial method but still scales computationally |
1078 |
|
|
as $O(N/P)$ and scales as $(N/\sqrt{p})$ in communication |
1079 |
|
|
cost. Plimpton also found that force decompositions scales more |
1080 |
|
|
favorably then spatial decomposition up to 10,000 atoms and favorably |
1081 |
|
|
competes with spatial methods for up to 100,000 atoms. |
1082 |
|
|
|
1083 |
|
|
\subsection{\label{sec:memory}Memory Allocation in Analysis} |
1084 |
|
|
|
1085 |
|
|
\subsection{\label{sec:documentation}Documentation} |
1086 |
|
|
|
1087 |
|
|
\subsection{\label{openSource}Open Source and Distribution License} |
1088 |
|
|
|
1089 |
|
|
|
1090 |
|
|
\section{\label{sec:conclusion}Conclusion} |
1091 |
|
|
|
1092 |
|
|
\begin{itemize} |
1093 |
|
|
|
1094 |
|
|
\item Restate capabilities |
1095 |
|
|
|
1096 |
|
|
\item recap major structure / design choices |
1097 |
|
|
|
1098 |
|
|
\begin{itemize} |
1099 |
|
|
|
1100 |
|
|
\item parallel |
1101 |
|
|
\item symplectic integration |
1102 |
|
|
\item languages |
1103 |
|
|
|
1104 |
|
|
\end{itemize} |
1105 |
|
|
|
1106 |
|
|
\item How well does it meet the primary goal |
1107 |
|
|
|
1108 |
|
|
\end{itemize} |