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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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|
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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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|
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\section{\label{oopseSec:foreword}Foreword} |
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|
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In this chapter, I present and detail the capabilities of the open |
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source simulation package {\sc oopse}. It is important to note, that a |
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simulation package of this size and scope would not have been possible |
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without the collaborative efforts of my colleagues: Charles |
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F.~Vardeman II, Teng Lin, Christopher J.~Fennell and J.~Daniel |
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Gezelter. Although my contributions to {\sc oopse} are significant, |
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consideration of my work apart from the others, would not give a |
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complete description to the package's capabilities. As such, all |
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contributions to {\sc oopse} to date are presented in this chapter. |
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|
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Charles Vardeman is responsible for the parallelization of {\sc oopse} |
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(Sec.~\ref{oopseSec:parallelization}) as well as the inclusion of the |
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embedded-atom potential (Sec.~\ref{oopseSec:eam}). Teng Lin's |
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contributions include refinement of the periodic boundary conditions |
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(Sec.~\ref{oopseSec:pbc}), the z-constraint method |
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(Sec.~\ref{oopseSec:zcons}), refinement of the property analysis |
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programs (Sec.~\ref{oopseSec:props}), and development in the extended |
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state integrators (Sec.~\ref{oopseSec:noseHooverThermo}). Christopher |
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Fennell worked on the symplectic integrator |
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(Sec.~\ref{oopseSec:integrate}) and the refinement of the {\sc ssd} |
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water model (Sec.~\ref{oopseSec:SSD}). Daniel Gezelter lent his |
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talents in the development of the extended system integrators |
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(Sec.~\ref{oopseSec:noseHooverThermo}) as well as giving general |
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direction and oversight to the entire project. My responsibilities |
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covered the creation and specification of {\sc bass} |
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(Sec.~\ref{oopseSec:IOfiles}), the original development of the single |
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processor version of {\sc oopse}, contributions to the extended state |
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integrators (Sec.~\ref{oopseSec:noseHooverThermo}), the implementation |
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of the Lennard-Jones (Sec.~\ref{sec:LJPot}) and {\sc duff} |
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(Sec.~\ref{oopseSec:DUFF}) force fields, and initial implementation of |
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the property analysis (Sec.~\ref{oopseSec:props}) and system |
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initialization (Sec.~\ref{oopseSec:initCoords}) utility programs. |
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|
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\section{\label{sec:intro}Introduction} |
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|
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When choosing to simulate a chemical system with molecular dynamics, |
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there are a variety of options available. For simple systems, one |
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might consider writing one's own programming code. However, as systems |
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grow larger and more complex, building and maintaining code for the |
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simulations becomes a time consuming task. In such cases it is usually |
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more convenient for a researcher to turn to pre-existing simulation |
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packages. These packages, such as {\sc amber}\cite{pearlman:1995} and |
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{\sc charmm}\cite{Brooks83}, provide powerful tools for researchers to |
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conduct simulations of their systems without spending their time |
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developing a code base to conduct their research. This then frees them |
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to perhaps explore experimental analogues to their models. |
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|
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Despite their utility, problems with these packages arise when |
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researchers try to develop techniques or energetic models that the |
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code was not originally designed to do. Examples of uncommonly |
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implemented techniques and energetics include; dipole-dipole |
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interactions, rigid body dynamics, and metallic embedded |
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potentials. When faced with these obstacles, a researcher must either |
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develop their own code or license and extend one of the commercial |
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packages. What we have elected to do, is develop a package of |
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simulation code capable of implementing the types of models upon which |
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our research is based. |
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|
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Having written {\sc oopse} we are implementing the concept of Open |
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Source development, and releasing our source code into the public |
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domain. It is our intent that by doing so, other researchers might |
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benefit from our work, and add their own contributions to the |
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package. The license under which {\sc oopse} is distributed allows any |
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researcher to download and modify the source code for their own |
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use. In this way further development of {\sc oopse} is not limited to |
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only the models of interest to ourselves, but also those of the |
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community of scientists who contribute back to the project. |
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|
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We have structured this chapter to first discuss the empirical energy |
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functions that {\sc oopse } implements in |
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Sec.~\ref{oopseSec:empiricalEnergy}. Following that is a discussion of |
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the various input and output files associated with the package |
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(Sec.~\ref{oopseSec:IOfiles}). In Sec.~\ref{oopseSec:mechanics} |
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elucidates the various Molecular Dynamics algorithms {\sc oopse} |
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implements in the integration of the Newtonian equations of |
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motion. Basic analysis of the trajectories obtained from the |
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simulation is discussed in Sec.~\ref{oopseSec:props}. Program design |
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considerations as well as the software distribution license is |
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presented in Sec.~\ref{oopseSec:design}. And lastly, |
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Sec.~\ref{oopseSec:conclusion} concludes the chapter. |
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|
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\section{\label{oopseSec:empiricalEnergy}The Empirical Energy Functions} |
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|
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\subsection{\label{oopseSec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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|
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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, permanent dipoles, and Lennard-Jones parameters for |
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a given atom type are set in the force field parameter files |
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|
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\begin{lstlisting}[float,caption={[Specifier for molecules and atoms] A sample specification of an 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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|
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|
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Atoms can be collected into secondary structures 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 and rigid bodies associated with |
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themselves, and are responsible for the evaluation of their own |
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internal interactions (\emph{i.e.}~bonds, bends, and torsions). Scheme |
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\ref{sch:AtmMole} shows how one creates a molecule in a |
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\texttt{.mdl} file. The position of the atoms given in the |
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declaration are relative to the origin of the molecule, and is used |
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when creating a system containing the molecule. |
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|
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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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|
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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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|
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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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|
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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{oopseSec:integrate}. An example |
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definition of a rigid 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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|
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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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|
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\subsection{\label{sec:LJPot}The Lennard Jones Force Field} |
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|
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The most basic force field implemented in {\sc oopse} is the |
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Lennard-Jones force field, 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 \texttt{.bass} file that |
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sets up a system of 108 Ar particles to be simulated using the |
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Lennard-Jones force field. |
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|
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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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/* |
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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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|
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#include "argon.mdl" |
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|
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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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/* |
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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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|
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initialConfig = "./argon.init"; |
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|
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forceField = "LJ"; |
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\end{lstlisting} |
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|
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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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|
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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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|
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|
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|
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\subsection{\label{oopseSec:DUFF}Dipolar Unified-Atom Force Field} |
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|
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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 ($\sim$100's of phospholipids, |
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$\sim$1000's of waters) to be simulated for long times |
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($\sim$10's of nanoseconds). |
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|
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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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|
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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{oopseFig: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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|
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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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|
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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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|
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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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|
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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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|
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#include "water.mdl" |
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#include "lipid.mdl" |
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|
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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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|
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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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|
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initialConfig = "bilayer.init"; |
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|
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forceField = "DUFF"; |
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|
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\end{lstlisting} |
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|
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\subsection{\label{oopseSec:energyFunctions}{\sc duff} Energy Functions} |
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|
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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-1}_{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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|
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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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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{oopseFig: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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|
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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{oopseFig:lipidModel}). For |
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computational efficiency, the torsion potential has been recast after |
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the method of {\sc charmm},\cite{Brooks83} 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) = |
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k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
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\label{eq:torsionPot} |
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\end{equation} |
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Where: |
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\begin{align*} |
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k_0 &= c_1 + c_3 \\ |
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k_1 &= c_1 - 3c_3 \\ |
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k_2 &= 2 c_2 \\ |
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k_3 &= 4c_3 |
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\end{align*} |
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By recasting the potential as a power series, repeated trigonometric |
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evaluations are avoided during the calculation of the potential energy. |
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|
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|
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The cross potential between molecules $I$ and $J$, $V^{IJ}_{\text{Cross}}$, 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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(Sec.~\ref{oopseSec:SSD}). Note that not all atom types include all |
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interactions. |
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|
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The dipole-dipole potential has the following form: |
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\begin{equation} |
451 |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
452 |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
453 |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
454 |
- |
455 |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
456 |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
457 |
{r^{2}_{ij}} \biggr] |
458 |
\label{eq:dipolePot} |
459 |
\end{equation} |
460 |
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
461 |
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
462 |
are the orientational degrees of freedom for atoms $i$ and $j$ |
463 |
respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom |
464 |
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector |
465 |
of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is the |
466 |
unit vector pointing along $\mathbf{r}_{ij}$ |
467 |
($\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). |
468 |
|
469 |
|
470 |
\subsubsection{\label{oopseSec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
471 |
|
472 |
In the interest of computational efficiency, the default solvent used |
473 |
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
474 |
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
475 |
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
476 |
water model proposed by Bratko, Blum, and |
477 |
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
478 |
with a Lennard-Jones core and a sticky potential that directs the |
479 |
particles to assume the proper hydrogen bond orientation in the first |
480 |
solvation shell. Thus, the interaction between two SSD water molecules |
481 |
\emph{i} and \emph{j} is given by the potential |
482 |
\begin{equation} |
483 |
V_{ij} = |
484 |
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
485 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
486 |
V_{ij}^{sp} |
487 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
488 |
\label{eq:ssdPot} |
489 |
\end{equation} |
490 |
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
491 |
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
492 |
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
493 |
orientations of the respective molecules. The Lennard-Jones and dipole |
494 |
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
495 |
and \ref{eq:dipolePot} respectively. The sticky part is described by |
496 |
the following, |
497 |
\begin{equation} |
498 |
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
499 |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
500 |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
501 |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
502 |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
503 |
\label{eq:stickyPot} |
504 |
\end{equation} |
505 |
where $\nu_0$ is a strength parameter for the sticky potential, and |
506 |
$s$ and $s^\prime$ are cubic switching functions which turn off the |
507 |
sticky interaction beyond the first solvation shell. The $w$ function |
508 |
can be thought of as an attractive potential with tetrahedral |
509 |
geometry: |
510 |
\begin{equation} |
511 |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
512 |
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
513 |
\label{eq:stickyW} |
514 |
\end{equation} |
515 |
while the $w^\prime$ function counters the normal aligned and |
516 |
anti-aligned structures favored by point dipoles: |
517 |
\begin{equation} |
518 |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
519 |
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
520 |
\label{eq:stickyWprime} |
521 |
\end{equation} |
522 |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
523 |
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
524 |
enhances the tetrahedral geometry for hydrogen bonded structures), |
525 |
while $w^\prime$ is a purely empirical function. A more detailed |
526 |
description of the functional parts and variables in this potential |
527 |
can be found in the original SSD |
528 |
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
529 |
|
530 |
Since SSD is a single-point {\it dipolar} model, the force |
531 |
calculations are simplified significantly relative to the standard |
532 |
{\it charged} multi-point models. In the original Monte Carlo |
533 |
simulations using this model, Ichiye {\it et al.} reported that using |
534 |
SSD decreased computer time by a factor of 6-7 compared to other |
535 |
models.\cite{liu96:new_model} What is most impressive is that these savings |
536 |
did not come at the expense of accurate depiction of the liquid state |
537 |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
538 |
diffraction data for the structural features of liquid |
539 |
water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
540 |
exhibited by SSD agree with experiment better than those of more |
541 |
computationally expensive models (like TIP3P and |
542 |
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
543 |
of solvent properties makes SSD a very attractive model for the |
544 |
simulation of large scale biochemical simulations. |
545 |
|
546 |
Recent constant pressure simulations revealed issues in the original |
547 |
SSD model that led to lower than expected densities at all target |
548 |
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
549 |
is therefore SSD/E, a density corrected derivative of SSD that |
550 |
exhibits improved liquid structure and transport behavior. If the use |
551 |
of a reaction field long-range interaction correction is desired, it |
552 |
is recommended that the parameters be modified to those of the SSD/RF |
553 |
model. Solvent parameters can be easily modified in an accompanying |
554 |
{\sc BASS} file as illustrated in the scheme below. A table of the |
555 |
parameter values and the drawbacks and benefits of the different |
556 |
density corrected SSD models can be found in |
557 |
reference~\cite{Gezelter04}. |
558 |
|
559 |
\begin{lstlisting}[float,caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
560 |
|
561 |
#include "water.mdl" |
562 |
|
563 |
nComponents = 1; |
564 |
component{ |
565 |
type = "SSD_water"; |
566 |
nMol = 864; |
567 |
} |
568 |
|
569 |
initialConfig = "liquidWater.init"; |
570 |
|
571 |
forceField = "DUFF"; |
572 |
|
573 |
/* |
574 |
* The reactionField flag toggles reaction |
575 |
* field corrections. |
576 |
*/ |
577 |
|
578 |
reactionField = false; // defaults to false |
579 |
dielectric = 80.0; // dielectric for reaction field |
580 |
|
581 |
/* |
582 |
* The following two flags set the cutoff |
583 |
* radius for the electrostatic forces |
584 |
* as well as the skin thickness of the switching |
585 |
* function. |
586 |
*/ |
587 |
|
588 |
electrostaticCutoffRadius = 9.2; |
589 |
electrostaticSkinThickness = 1.38; |
590 |
|
591 |
\end{lstlisting} |
592 |
|
593 |
|
594 |
\subsection{\label{oopseSec:eam}Embedded Atom Method} |
595 |
|
596 |
There are Molecular Dynamics packages which have the |
597 |
capacity to simulate metallic systems, including some that have |
598 |
parallel computational abilities\cite{plimpton93}. Potentials that |
599 |
describe bonding transition metal |
600 |
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
601 |
attractive interaction which models ``Embedding'' |
602 |
a positively charged metal ion in the electron density due to the |
603 |
free valance ``sea'' of electrons created by the surrounding atoms in |
604 |
the system. A mostly repulsive pairwise part of the potential |
605 |
describes the interaction of the positively charged metal core ions |
606 |
with one another. A particular potential description called the |
607 |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
608 |
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
609 |
good review of {\sc eam} and other metallic potential formulations was done |
610 |
by Voter.\cite{voter} |
611 |
|
612 |
The {\sc eam} potential has the form: |
613 |
\begin{eqnarray} |
614 |
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
615 |
\phi_{ij}({\bf r}_{ij}) \\ |
616 |
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
617 |
\end{eqnarray}S |
618 |
|
619 |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
620 |
positively-charged core ion $i$ into a linear superposition of |
621 |
spherically averaged atomic electron densities given by |
622 |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
623 |
between atoms $i$ and $j$. In the original formulation of |
624 |
{\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
625 |
in later refinements to EAM have shown that non-uniqueness between $F$ |
626 |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
627 |
There is a cutoff distance, $r_{cut}$, which limits the |
628 |
summations in the {\sc eam} equation to the few dozen atoms |
629 |
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
630 |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FBD86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
631 |
|
632 |
|
633 |
\subsection{\label{oopseSec:pbc}Periodic Boundary Conditions} |
634 |
|
635 |
\newcommand{\roundme}{\operatorname{round}} |
636 |
|
637 |
\textit{Periodic boundary conditions} are widely used to simulate truly |
638 |
macroscopic systems with a relatively small number of particles. The |
639 |
simulation box is replicated throughout space to form an infinite lattice. |
640 |
During the simulation, when a particle moves in the primary cell, its image in |
641 |
other boxes move in exactly the same direction with exactly the same |
642 |
orientation.Thus, as a particle leaves the primary cell, one of its images |
643 |
will enter through the opposite face.If the simulation box is large enough to |
644 |
avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
645 |
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
646 |
parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
647 |
describe the property of the simulation box. Therefore, both the size and |
648 |
shape of the simulation box can be changed during the simulation. The |
649 |
transformation from box space vector $\mathbf{s}$ to its corresponding real |
650 |
space vector $\mathbf{r}$ is defined by |
651 |
\begin{equation} |
652 |
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
653 |
\end{equation} |
654 |
|
655 |
|
656 |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
657 |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
658 |
simulation box respectively. |
659 |
|
660 |
To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
661 |
to its corresponding vector in box space first, \bigskip% |
662 |
\begin{equation} |
663 |
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
664 |
\end{equation} |
665 |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
666 |
\begin{equation} |
667 |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
668 |
\end{equation} |
669 |
where |
670 |
|
671 |
% |
672 |
|
673 |
\begin{equation} |
674 |
\roundme(x)=\left\{ |
675 |
\begin{array}{cc}% |
676 |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant 0 \\ |
677 |
\lceil{x-0.5}\rceil & \text{otherwise}% |
678 |
\end{array} |
679 |
\right. |
680 |
\end{equation} |
681 |
|
682 |
|
683 |
For example, $\roundme(3.6)=4$,$\roundme(3.1)=3$, $\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
684 |
|
685 |
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
686 |
transforming back to real space,% |
687 |
|
688 |
\begin{equation} |
689 |
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
690 |
\end{equation} |
691 |
|
692 |
|
693 |
\section{\label{oopseSec:IOfiles}Input and Output Files} |
694 |
|
695 |
\subsection{{\sc bass} and Model Files} |
696 |
|
697 |
Every {\sc oopse} simulation begins with a {\sc bass} file. {\sc bass} |
698 |
(\underline{B}izarre \underline{A}tom \underline{S}imulation |
699 |
\underline{S}yntax) is a script syntax that is parsed by {\sc oopse} at |
700 |
runtime. The {\sc bass} file allows for the user to completely describe the |
701 |
system they are to simulate, as well as tailor {\sc oopse}'s behavior during |
702 |
the simulation. {\sc bass} files are denoted with the extension |
703 |
\texttt{.bass}, an example file is shown in |
704 |
Fig.~\ref{fig:bassExample}. |
705 |
|
706 |
\begin{figure} |
707 |
\centering |
708 |
\framebox[\linewidth]{\rule{0cm}{0.75\linewidth}I'm a {\sc bass} file!} |
709 |
\caption{Here is an example \texttt{.bass} file} |
710 |
\label{fig:bassExample} |
711 |
\end{figure} |
712 |
|
713 |
Within the \texttt{.bass} file it is necessary to provide a complete |
714 |
description of the molecule before it is actually placed in the |
715 |
simulation. The {\sc bass} syntax was originally developed with this goal in |
716 |
mind, and allows for the specification of all the atoms in a molecular |
717 |
prototype, as well as any bonds, bends, or torsions. These |
718 |
descriptions can become lengthy for complex molecules, and it would be |
719 |
inconvenient to duplicate the simulation at the beginning of each {\sc bass} |
720 |
script. Addressing this issue {\sc bass} allows for the inclusion of model |
721 |
files at the top of a \texttt{.bass} file. These model files, denoted |
722 |
with the \texttt{.mdl} extension, allow the user to describe a |
723 |
molecular prototype once, then simply include it into each simulation |
724 |
containing that molecule. |
725 |
|
726 |
\subsection{\label{oopseSec:coordFiles}Coordinate Files} |
727 |
|
728 |
The standard format for storage of a systems coordinates is a modified |
729 |
xyz-file syntax, the exact details of which can be seen in |
730 |
App.~\ref{appCoordFormat}. As all bonding and molecular information is |
731 |
stored in the \texttt{.bass} and \texttt{.mdl} files, the coordinate |
732 |
files are simply the complete set of coordinates for each atom at a |
733 |
given simulation time. |
734 |
|
735 |
There are three major files used by {\sc oopse} written in the coordinate |
736 |
format, they are as follows: the initialization file, the simulation |
737 |
trajectory file, and the final coordinates of the simulation. The |
738 |
initialization file is necessary for {\sc oopse} to start the simulation |
739 |
with the proper coordinates. It is typically denoted with the |
740 |
extension \texttt{.init}. The trajectory file is created at the |
741 |
beginning of the simulation, and is used to store snapshots of the |
742 |
simulation at regular intervals. The first frame is a duplication of |
743 |
the \texttt{.init} file, and each subsequent frame is appended to the |
744 |
file at an interval specified in the \texttt{.bass} file. The |
745 |
trajectory file is given the extension \texttt{.dump}. The final |
746 |
coordinate file is the end of run or \texttt{.eor} file. The |
747 |
\texttt{.eor} file stores the final configuration of the system for a |
748 |
given simulation. The file is updated at the same time as the |
749 |
\texttt{.dump} file. However, it only contains the most recent |
750 |
frame. In this way, an \texttt{.eor} file may be used as the |
751 |
initialization file to a second simulation in order to continue or |
752 |
recover the previous simulation. |
753 |
|
754 |
\subsection{\label{oopseSec:initCoords}Generation of Initial Coordinates} |
755 |
|
756 |
As was stated in Sec.~\ref{oopseSec:coordFiles}, an initialization file |
757 |
is needed to provide the starting coordinates for a simulation. The |
758 |
{\sc oopse} package provides a program called \texttt{sysBuilder} to aid in |
759 |
the creation of the \texttt{.init} file. \texttt{sysBuilder} is {\sc bass} |
760 |
aware, and will recognize arguments and parameters in the |
761 |
\texttt{.bass} file that would otherwise be ignored by the |
762 |
simulation. The program itself is under continual development, and is |
763 |
offered here as a helper tool only. |
764 |
|
765 |
\subsection{The Statistics File} |
766 |
|
767 |
The last output file generated by {\sc oopse} is the statistics file. This |
768 |
file records such statistical quantities as the instantaneous |
769 |
temperature, volume, pressure, etc. It is written out with the |
770 |
frequency specified in the \texttt{.bass} file. The file allows the |
771 |
user to observe the system variables as a function of simulation time |
772 |
while the simulation is in progress. One useful function the |
773 |
statistics file serves is to monitor the conserved quantity of a given |
774 |
simulation ensemble, this allows the user to observe the stability of |
775 |
the integrator. The statistics file is denoted with the \texttt{.stat} |
776 |
file extension. |
777 |
|
778 |
\section{\label{oopseSec:mechanics}Mechanics} |
779 |
|
780 |
\subsection{\label{oopseSec:integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
781 |
|
782 |
Integration of the equations of motion was carried out using the |
783 |
symplectic splitting method proposed by Dullweber \emph{et |
784 |
al.}.\cite{Dullweber1997} The reason for this integrator selection |
785 |
deals with poor energy conservation of rigid body systems using |
786 |
quaternions. While quaternions work well for orientational motion in |
787 |
alternate ensembles, the microcanonical ensemble has a constant energy |
788 |
requirement that is quite sensitive to errors in the equations of |
789 |
motion. The original implementation of this code utilized quaternions |
790 |
for rotational motion propagation; however, a detailed investigation |
791 |
showed that they resulted in a steady drift in the total energy, |
792 |
something that has been observed by others.\cite{Laird97} |
793 |
|
794 |
The key difference in the integration method proposed by Dullweber |
795 |
\emph{et al.} is that the entire rotation matrix is propagated from |
796 |
one time step to the next. In the past, this would not have been as |
797 |
feasible a option, being that the rotation matrix for a single body is |
798 |
nine elements long as opposed to 3 or 4 elements for Euler angles and |
799 |
quaternions respectively. System memory has become much less of an |
800 |
issue in recent times, and this has resulted in substantial benefits |
801 |
in energy conservation. There is still the issue of 5 or 6 additional |
802 |
elements for describing the orientation of each particle, which will |
803 |
increase dump files substantially. Simply translating the rotation |
804 |
matrix into its component Euler angles or quaternions for storage |
805 |
purposes relieves this burden. |
806 |
|
807 |
The symplectic splitting method allows for Verlet style integration of |
808 |
both linear and angular motion of rigid bodies. In the integration |
809 |
method, the orientational propagation involves a sequence of matrix |
810 |
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
811 |
matrix rotations end up being more costly computationally than the |
812 |
simpler arithmetic quaternion propagation. With the same time step, a |
813 |
1000 SSD particle simulation shows an average 7\% increase in |
814 |
computation time using the symplectic step method in place of |
815 |
quaternions. This cost is more than justified when comparing the |
816 |
energy conservation of the two methods as illustrated in figure |
817 |
\ref{timestep}. |
818 |
|
819 |
\begin{figure} |
820 |
\centering |
821 |
\includegraphics[width=\linewidth]{timeStep.eps} |
822 |
\caption{Energy conservation using quaternion based integration versus |
823 |
the symplectic step method proposed by Dullweber \emph{et al.} with |
824 |
increasing time step. For each time step, the dotted line is total |
825 |
energy using the symplectic step integrator, and the solid line comes |
826 |
from the quaternion integrator. The larger time step plots are shifted |
827 |
up from the true energy baseline for clarity.} |
828 |
\label{timestep} |
829 |
\end{figure} |
830 |
|
831 |
In figure \ref{timestep}, the resulting energy drift at various time |
832 |
steps for both the symplectic step and quaternion integration schemes |
833 |
is compared. All of the 1000 SSD particle simulations started with the |
834 |
same configuration, and the only difference was the method for |
835 |
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
836 |
methods for propagating particle rotation conserve energy fairly well, |
837 |
with the quaternion method showing a slight energy drift over time in |
838 |
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
839 |
energy conservation benefits of the symplectic step method are clearly |
840 |
demonstrated. Thus, while maintaining the same degree of energy |
841 |
conservation, one can take considerably longer time steps, leading to |
842 |
an overall reduction in computation time. |
843 |
|
844 |
Energy drift in these SSD particle simulations was unnoticeable for |
845 |
time steps up to three femtoseconds. A slight energy drift on the |
846 |
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
847 |
four femtoseconds, and as expected, this drift increases dramatically |
848 |
with increasing time step. To insure accuracy in the constant energy |
849 |
simulations, time steps were set at 2 fs and kept at this value for |
850 |
constant pressure simulations as well. |
851 |
|
852 |
|
853 |
\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
854 |
|
855 |
|
856 |
{\sc oopse} implements a |
857 |
|
858 |
|
859 |
\subsubsection{\label{oopseSec:noseHooverThermo}Nose-Hoover Thermostatting} |
860 |
|
861 |
To mimic the effects of being in a constant temperature ({\sc nvt}) |
862 |
ensemble, {\sc oopse} uses the Nose-Hoover extended system |
863 |
approach.\cite{Hoover85} In this method, the equations of motion for |
864 |
the particle positions and velocities are |
865 |
\begin{eqnarray} |
866 |
\dot{{\bf r}} & = & {\bf v} \\ |
867 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} |
868 |
\label{eq:nosehoovereom} |
869 |
\end{eqnarray} |
870 |
|
871 |
$\chi$ is an ``extra'' variable included in the extended system, and |
872 |
it is propagated using the first order equation of motion |
873 |
\begin{equation} |
874 |
\dot{\chi} = \frac{1}{\tau_{T}} \left( \frac{T}{T_{target}} - 1 \right) |
875 |
\label{eq:nosehooverext} |
876 |
\end{equation} |
877 |
where $T_{target}$ is the target temperature for the simulation, and |
878 |
$\tau_{T}$ is a time constant for the thermostat. |
879 |
|
880 |
To select the Nose-Hoover {\sc nvt} ensemble, the {\tt ensemble = NVT;} |
881 |
command would be used in the simulation's {\sc bass} file. There is |
882 |
some subtlety in choosing values for $\tau_{T}$, and it is usually set |
883 |
to values of a few ps. Within a {\sc bass} file, $\tau_{T}$ could be |
884 |
set to 1 ps using the {\tt tauThermostat = 1000; } command. |
885 |
|
886 |
|
887 |
\subsection{\label{oopseSec:zcons}Z-Constraint Method} |
888 |
|
889 |
Based on fluctuation-dissipation theorem,\bigskip\ force auto-correlation |
890 |
method was developed to investigate the dynamics of ions inside the ion |
891 |
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated |
892 |
from the deviation of the instantaneous force from its mean force. |
893 |
|
894 |
% |
895 |
|
896 |
\begin{equation} |
897 |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
898 |
\end{equation} |
899 |
where% |
900 |
\begin{equation} |
901 |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
902 |
\end{equation} |
903 |
|
904 |
|
905 |
If the time-dependent friction decay rapidly, static friction coefficient can |
906 |
be approximated by% |
907 |
|
908 |
\begin{equation} |
909 |
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt |
910 |
\end{equation} |
911 |
|
912 |
|
913 |
Hence, diffusion constant can be estimated by |
914 |
\begin{equation} |
915 |
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
916 |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}% |
917 |
\end{equation} |
918 |
|
919 |
|
920 |
\bigskip Z-Constraint method, which fixed the z coordinates of the molecules |
921 |
with respect to the center of the mass of the system, was proposed to obtain |
922 |
the forces required in force auto-correlation method.\cite{Marrink94} However, |
923 |
simply resetting the coordinate will move the center of the mass of the whole |
924 |
system. To avoid this problem, a new method was used at {\sc oopse}. Instead of |
925 |
resetting the coordinate, we reset the forces of z-constraint molecules as |
926 |
well as subtract the total constraint forces from the rest of the system after |
927 |
force calculation at each time step. |
928 |
\begin{align} |
929 |
F_{\alpha i}&=0\\ |
930 |
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}\\ |
931 |
F_{\alpha i}&=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}\\ |
932 |
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}}} |
933 |
\end{align} |
934 |
|
935 |
At the very beginning of the simulation, the molecules may not be at its |
936 |
constraint position. To move the z-constraint molecule to the specified |
937 |
position, a simple harmonic potential is used% |
938 |
|
939 |
\begin{equation} |
940 |
U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}% |
941 |
\end{equation} |
942 |
where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is |
943 |
current z coordinate of the center of mass of the z-constraint molecule, and |
944 |
$z_{cons}$ is the restraint position. Therefore, the harmonic force operated |
945 |
on the z-constraint molecule at time $t$ can be calculated by% |
946 |
\begin{equation} |
947 |
F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}% |
948 |
(z(t)-z_{cons}) |
949 |
\end{equation} |
950 |
Worthy of mention, other kinds of potential functions can also be used to |
951 |
drive the z-constraint molecule. |
952 |
|
953 |
\section{\label{oopseSec:props}Trajectory Analysis} |
954 |
|
955 |
\subsection{\label{oopseSec:staticProps}Static Property Analysis} |
956 |
|
957 |
The static properties of the trajectories are analyzed with the |
958 |
program \texttt{staticProps}. The code is capable of calculating the following |
959 |
pair correlations between species A and B: |
960 |
\begin{itemize} |
961 |
\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr} |
962 |
\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta} |
963 |
\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega} |
964 |
\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ} |
965 |
\item $\langle \cos \omega \rangle_{\text{AB}}(r)$: |
966 |
Eq.~\ref{eq:cosOmegaOfR} |
967 |
\end{itemize} |
968 |
|
969 |
The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
970 |
\begin{equation} |
971 |
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %% |
972 |
\sum_{i \in \text{A}} \sum_{j \in \text{B}} %% |
973 |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr} |
974 |
\end{equation} |
975 |
Where $\mathbf{r}_{ij}$ is the vector |
976 |
\begin{equation*} |
977 |
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag |
978 |
\end{equation*} |
979 |
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over |
980 |
the expected pair density at a given $r$. |
981 |
|
982 |
The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and |
983 |
$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two |
984 |
dimensional histograms. Both use $r$ for the primary axis then a |
985 |
$\cos$ for the secondary axis ($\cos \theta$ for |
986 |
Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for |
987 |
Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to |
988 |
correlate alignment on directional entities. $g_{\text{AB}}(r, \cos |
989 |
\theta)$ is defined as follows: |
990 |
\begin{equation} |
991 |
g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
992 |
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
993 |
\delta( \cos \theta - \cos \theta_{ij}) |
994 |
\delta( r - |\mathbf{r}_{ij}|) \rangle |
995 |
\label{eq:gofrCosTheta} |
996 |
\end{equation} |
997 |
Where |
998 |
\begin{equation*} |
999 |
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
1000 |
\end{equation*} |
1001 |
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
1002 |
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector |
1003 |
$\mathbf{r}_{ij}$. |
1004 |
|
1005 |
The second two dimensional histogram is of the form: |
1006 |
\begin{equation} |
1007 |
g_{\text{AB}}(r, \cos \omega) = |
1008 |
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
1009 |
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1010 |
\delta( \cos \omega - \cos \omega_{ij}) |
1011 |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
1012 |
\end{equation} |
1013 |
Here |
1014 |
\begin{equation*} |
1015 |
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
1016 |
\end{equation*} |
1017 |
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit |
1018 |
directional vectors of species $i$ and $j$. |
1019 |
|
1020 |
The static analysis code is also cable of calculating a three |
1021 |
dimensional pair correlation of the form: |
1022 |
\begin{equation}\label{eq:gofrXYZ} |
1023 |
g_{\text{AB}}(x, y, z) = |
1024 |
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
1025 |
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1026 |
\delta( x - x_{ij}) |
1027 |
\delta( y - y_{ij}) |
1028 |
\delta( z - z_{ij}) \rangle |
1029 |
\end{equation} |
1030 |
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
1031 |
components respectively of vector $\mathbf{r}_{ij}$. |
1032 |
|
1033 |
The final pair correlation is similar to |
1034 |
Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega |
1035 |
\rangle_{\text{AB}}(r)$ is calculated in the following way: |
1036 |
\begin{equation}\label{eq:cosOmegaOfR} |
1037 |
\langle \cos \omega \rangle_{\text{AB}}(r) = |
1038 |
\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1039 |
(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle |
1040 |
\end{equation} |
1041 |
Here $\cos \omega_{ij}$ is defined in the same way as in |
1042 |
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair |
1043 |
correlation that gives the average correlation of two directional |
1044 |
entities as a function of their distance from each other. |
1045 |
|
1046 |
All static properties are calculated on a frame by frame basis. The |
1047 |
trajectory is read a single frame at a time, and the appropriate |
1048 |
calculations are done on each frame. Once one frame is finished, the |
1049 |
next frame is read in, and a running average of the property being |
1050 |
calculated is accumulated in each frame. The program allows for the |
1051 |
user to specify more than one property be calculated in single run, |
1052 |
preventing the need to read a file multiple times. |
1053 |
|
1054 |
\subsection{\label{dynamicProps}Dynamic Property Analysis} |
1055 |
|
1056 |
The dynamic properties of a trajectory are calculated with the program |
1057 |
\texttt{dynamicProps}. The program will calculate the following properties: |
1058 |
\begin{gather} |
1059 |
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\ |
1060 |
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\ |
1061 |
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr} |
1062 |
\end{gather} |
1063 |
|
1064 |
Eq.~\ref{eq:rms} is the root mean square displacement |
1065 |
function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the |
1066 |
velocity and angular velocity correlation functions respectively. The |
1067 |
latter is only applicable to directional species in the simulation. |
1068 |
|
1069 |
The \texttt{dynamicProps} program handles he file in a manner different from |
1070 |
\texttt{staticProps}. As the properties calculated by this program are time |
1071 |
dependent, multiple frames must be read in simultaneously by the |
1072 |
program. For small trajectories this is no problem, and the entire |
1073 |
trajectory is read into memory. However, for long trajectories of |
1074 |
large systems, the files can be quite large. In order to accommodate |
1075 |
large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory |
1076 |
are allocated to read in several frames each. |
1077 |
|
1078 |
In this two block scheme, the correlation functions are first |
1079 |
calculated within each memory block, then the cross correlations |
1080 |
between the frames contained within the two blocks are |
1081 |
calculated. Once completed, the memory blocks are incremented, and the |
1082 |
process is repeated. A diagram illustrating the process is shown in |
1083 |
Fig.~\ref{oopseFig:dynamicPropsMemory}. As was the case with |
1084 |
\texttt{staticProps}, multiple properties may be calculated in a |
1085 |
single run to avoid multiple reads on the same file. |
1086 |
|
1087 |
|
1088 |
|
1089 |
\section{\label{oopseSec:design}Program Design} |
1090 |
|
1091 |
\subsection{\label{sec:architecture} {\sc oopse} Architecture} |
1092 |
|
1093 |
The core of OOPSE is divided into two main object libraries: |
1094 |
\texttt{libBASS} and \texttt{libmdtools}. \texttt{libBASS} is the |
1095 |
library developed around the parsing engine and \texttt{libmdtools} |
1096 |
is the software library developed around the simulation engine. These |
1097 |
two libraries are designed to encompass all the basic functions and |
1098 |
tools that {\sc oopse} provides. Utility programs, such as the |
1099 |
property analyzers, need only link against the software libraries to |
1100 |
gain access to parsing, force evaluation, and input / output |
1101 |
routines. |
1102 |
|
1103 |
Contained in \texttt{libBASS} are all the routines associated with |
1104 |
reading and parsing the \texttt{.bass} input files. Given a |
1105 |
\texttt{.bass} file, \texttt{libBASS} will open it and any associated |
1106 |
\texttt{.mdl} files; then create structures in memory that are |
1107 |
templates of all the molecules specified in the input files. In |
1108 |
addition, any simulation parameters set in the \texttt{.bass} file |
1109 |
will be placed in a structure for later query by the controlling |
1110 |
program. |
1111 |
|
1112 |
Located in \texttt{libmdtools} are all other routines necessary to a |
1113 |
Molecular Dynamics simulation. The library uses the main data |
1114 |
structures returned by \texttt{libBASS} to initialize the various |
1115 |
parts of the simulation: the atom structures and positions, the force |
1116 |
field, the integrator, \emph{et cetera}. After initialization, the |
1117 |
library can be used to perform a variety of tasks: integrate a |
1118 |
Molecular Dynamics trajectory, query phase space information from a |
1119 |
specific frame of a completed trajectory, or even recalculate force or |
1120 |
energetic information about specific frames from a completed |
1121 |
trajectory. |
1122 |
|
1123 |
With these core libraries in place, several programs have been |
1124 |
developed to utilize the routines provided by \texttt{libBASS} and |
1125 |
\texttt{libmdtools}. The main program of the package is \texttt{oopse} |
1126 |
and the corresponding parallel version \texttt{oopse\_MPI}. These two |
1127 |
programs will take the \texttt{.bass} file, and create then integrate |
1128 |
the simulation specified in the script. The two analysis programs |
1129 |
\texttt{staticProps} and \texttt{dynamicProps} utilize the core |
1130 |
libraries to initialize and read in trajectories from previously |
1131 |
completed simulations, in addition to the ability to use functionality |
1132 |
from \texttt{libmdtools} to recalculate forces and energies at key |
1133 |
frames in the trajectories. Lastly, the family of system building |
1134 |
programs (Sec.~\ref{oopseSec:initCoords}) also use the libraries to |
1135 |
store and output the system configurations they create. |
1136 |
|
1137 |
\subsection{\label{oopseSec:parallelization} Parallelization of {\sc oopse}} |
1138 |
|
1139 |
Although processor power is continually growing month by month, it is |
1140 |
still unreasonable to simulate systems of more then a 1000 atoms on a |
1141 |
single processor. To facilitate study of larger system sizes or |
1142 |
smaller systems on long time scales in a reasonable period of time, |
1143 |
parallel methods were developed allowing multiple CPU's to share the |
1144 |
simulation workload. Three general categories of parallel |
1145 |
decomposition method's have been developed including atomic, spatial |
1146 |
and force decomposition methods. |
1147 |
|
1148 |
Algorithmically simplest of the three method's is atomic decomposition |
1149 |
where N particles in a simulation are split among P processors for the |
1150 |
duration of the simulation. Computational cost scales as an optimal |
1151 |
$O(N/P)$ for atomic decomposition. Unfortunately all processors must |
1152 |
communicate positions and forces with all other processors leading |
1153 |
communication to scale as an unfavorable $O(N)$ independent of the |
1154 |
number of processors. This communication bottleneck led to the |
1155 |
development of spatial and force decomposition methods in which |
1156 |
communication among processors scales much more favorably. Spatial or |
1157 |
domain decomposition divides the physical spatial domain into 3D boxes |
1158 |
in which each processor is responsible for calculation of forces and |
1159 |
positions of particles located in its box. Particles are reassigned to |
1160 |
different processors as they move through simulation space. To |
1161 |
calculate forces on a given particle, a processor must know the |
1162 |
positions of particles within some cutoff radius located on nearby |
1163 |
processors instead of the positions of particles on all |
1164 |
processors. Both communication between processors and computation |
1165 |
scale as $O(N/P)$ in the spatial method. However, spatial |
1166 |
decomposition adds algorithmic complexity to the simulation code and |
1167 |
is not very efficient for small N since the overall communication |
1168 |
scales as the surface to volume ratio $(N/P)^{2/3}$ in three |
1169 |
dimensions. |
1170 |
|
1171 |
Force decomposition assigns particles to processors based on a block |
1172 |
decomposition of the force matrix. Processors are split into a |
1173 |
optimally square grid forming row and column processor groups. Forces |
1174 |
are calculated on particles in a given row by particles located in |
1175 |
that processors column assignment. Force decomposition is less complex |
1176 |
to implement then the spatial method but still scales computationally |
1177 |
as $O(N/P)$ and scales as $(N/\sqrt{p})$ in communication |
1178 |
cost. Plimpton also found that force decompositions scales more |
1179 |
favorably then spatial decomposition up to 10,000 atoms and favorably |
1180 |
competes with spatial methods for up to 100,000 atoms. |
1181 |
|
1182 |
\subsection{\label{oopseSec:memAlloc}Memory Issues in Trajectory Analysis} |
1183 |
|
1184 |
For large simulations, the trajectory files can sometimes reach sizes |
1185 |
in excess of several gigabytes. In order to effectively analyze that |
1186 |
amount of data+, two memory management schemes have been devised for |
1187 |
\texttt{staticProps} and for \texttt{dynamicProps}. The first scheme, |
1188 |
developed for \texttt{staticProps}, is the simplest. As each frame's |
1189 |
statistics are calculated independent of each other, memory is |
1190 |
allocated for each frame, then freed once correlation calculations are |
1191 |
complete for the snapshot. To prevent multiple passes through a |
1192 |
potentially large file, \texttt{staticProps} is capable of calculating |
1193 |
all requested correlations per frame with only a single pair loop in |
1194 |
each frame and a single read through of the file. |
1195 |
|
1196 |
The second, more advanced memory scheme, is used by |
1197 |
\texttt{dynamicProps}. Here, the program must have multiple frames in |
1198 |
memory to calculate time dependent correlations. In order to prevent a |
1199 |
situation where the program runs out of memory due to large |
1200 |
trajectories, the user is able to specify that the trajectory be read |
1201 |
in blocks. The number of frames in each block is specified by the |
1202 |
user, and upon reading a block of the trajectory, |
1203 |
\texttt{dynamicProps} will calculate all of the time correlation frame |
1204 |
pairs within the block. After in block correlations are complete, a |
1205 |
second block of the trajectory is read, and the cross correlations are |
1206 |
calculated between the two blocks. this second block is then freed and |
1207 |
then incremented and the process repeated until the end of the |
1208 |
trajectory. Once the end is reached, the first block is freed then |
1209 |
incremented, and the again the internal time correlations are |
1210 |
calculated. The algorithm with the second block is then repeated with |
1211 |
the new origin block, until all frame pairs have been correlated in |
1212 |
time. This process is illustrated in |
1213 |
Fig.~\ref{oopseFig:dynamicPropsMemory}. |
1214 |
|
1215 |
\begin{figure} |
1216 |
\centering |
1217 |
\includegraphics[width=\linewidth]{dynamicPropsMem.eps} |
1218 |
\caption[A representation of the block correlations in \texttt{dynamicProps}]{This diagram illustrates the memory management 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.} |
1219 |
\label{oopseFig:dynamicPropsMemory} |
1220 |
\end{figure} |
1221 |
|
1222 |
\subsection{\label{openSource}Open Source and Distribution License} |
1223 |
|
1224 |
\section{\label{oopseSec:conclusion}Conclusion} |
1225 |
|
1226 |
We have presented the design and implementation of our open source |
1227 |
simulation package {\sc oopse}. The package offers novel |
1228 |
capabilities to the field of Molecular Dynamics simulation packages in |
1229 |
the form of dipolar force fields, and symplectic integration of rigid |
1230 |
body dynamics. It is capable of scaling across multiple processors |
1231 |
through the use of MPI. It also implements several integration |
1232 |
ensembles allowing the end user control over temperature and |
1233 |
pressure. In addition, it is capable of integrating constrained |
1234 |
dynamics through both the {\sc rattle} algorithm and the z-constraint |
1235 |
method. |
1236 |
|
1237 |
These features are all brought together in a single open-source |
1238 |
development package. This allows researchers to not only benefit from |
1239 |
{\sc oopse}, but also contribute to {\sc oopse}'s development as |
1240 |
well.Documentation and source code for {\sc oopse} can be downloaded |
1241 |
from \texttt{http://www.openscience.org/oopse/}. |
1242 |
|