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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{oopseSec:foreword}Foreword} |
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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 major, |
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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 the long |
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range forces in {\sc oopse} (Sec.~\ref{oopseSec:parallelization}) as |
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well as the inclusion of the embedded-atom potential for transition |
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metals (Sec.~\ref{oopseSec:eam}). Teng Lin's contributions include |
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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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system 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. {\sc |
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oopse}, like many other Molecular Dynamics programs, is a work in |
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progress, and will continue to be so for many graduate student |
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lifetimes. |
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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 simulate. 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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In developing {\sc oopse}, we have adhered to the precepts of Open |
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Source development, and are releasing our source code with a |
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permissive license. It is our intent that by doing so, other |
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researchers might benefit from our work, and add their own |
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contributions to the package. The license under which {\sc oopse} is |
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distributed allows any researcher to download and modify the source |
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code for their own use. In this way further development of {\sc oopse} |
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is not limited to only the models of interest to ourselves, but also |
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those of the community of scientists who contribute back to the |
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project. |
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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}). 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 are presented in Sec.~\ref{oopseSec:design}. And |
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lastly, Sec.~\ref{oopseSec:conclusion} concludes the chapter. |
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\section{\label{oopseSec:empiricalEnergy}The Empirical Energy Functions} |
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\subsection{\label{oopseSec: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, 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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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 ``model'' or |
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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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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 algorithmic |
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complexity involved in propagating orientational degrees of |
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freedom. Until recently, integrators which propagate orientational |
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motion have been much worse than those available for translational |
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motion. |
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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 because the torque is applied to the center of mass of |
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the rigid body. The torque on rigid body $i$ is |
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\begin{equation} |
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\boldsymbol{\tau}_i= |
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\sum_{a}\biggl[(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}\biggr] |
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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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each 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. 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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\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 Force Field} |
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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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\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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#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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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 can either be |
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specified in the \texttt{.bass} file, or left as its default value of |
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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 and the force. To offset this discontinuity in the potential, |
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the energy value at $r_{\text{cut}}$ is subtracted from the |
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potential. This causes the potential to go to zero smoothly at the |
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cut-off radius, and preserves conservation of energy in integrating |
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the equations of motion. |
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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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\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 $\frac{1}{r^3}$, and allows |
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us to avoid the computationally expensive Ewald sum. Instead, we can |
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use neighbor-lists and cutoff radii for the dipolar interactions, or |
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include a reaction field to mimic larger range 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 at the head group |
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center of mass, our model mimics the charge separation found in common |
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phospholipids such as phosphatidylcholine.\cite{Cevc87} Additionally, |
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a large Lennard-Jones site is located at the pseudoatom's center of |
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mass. The model is illustrated by the red atom in |
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Fig.~\ref{oopseFig:lipidModel}. The water model we use to complement |
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the dipoles of the lipids is our reparameterization of the soft sticky |
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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} |
313 |
mmeineke |
1045 |
\centering |
314 |
|
|
\includegraphics[width=\linewidth]{lipidModel.eps} |
315 |
mmeineke |
1083 |
\caption[A representation of a lipid model in {\sc duff}]{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
316 |
mmeineke |
1044 |
is the bend angle, $\mu$ is the dipole moment of the head group, and n |
317 |
|
|
is the chain length.} |
318 |
mmeineke |
1045 |
\label{oopseFig:lipidModel} |
319 |
mmeineke |
1044 |
\end{figure} |
320 |
|
|
|
321 |
|
|
We have used a set of scalable parameters to model the alkyl groups |
322 |
|
|
with Lennard-Jones sites. For this, we have borrowed parameters from |
323 |
|
|
the TraPPE force field of Siepmann |
324 |
|
|
\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
325 |
|
|
representation of n-alkanes, which is parametrized against phase |
326 |
|
|
equilibria using Gibbs ensemble Monte Carlo simulation |
327 |
|
|
techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
328 |
|
|
it generalizes the types of atoms in an alkyl chain to keep the number |
329 |
mmeineke |
1068 |
of pseudoatoms to a minimum; the parameters for a unified atom such as |
330 |
mmeineke |
1044 |
$\text{CH}_2$ do not change depending on what species are bonded to |
331 |
|
|
it. |
332 |
|
|
|
333 |
|
|
TraPPE also constrains all bonds to be of fixed length. Typically, |
334 |
|
|
bond vibrations are the fastest motions in a molecular dynamic |
335 |
|
|
simulation. Small time steps between force evaluations must be used to |
336 |
mmeineke |
1068 |
ensure adequate energy conservation in the bond degrees of freedom. By |
337 |
|
|
constraining the bond lengths, larger time steps may be used when |
338 |
|
|
integrating the equations of motion. A simulation using {\sc duff} is |
339 |
|
|
illustrated in Scheme \ref{sch:DUFF}. |
340 |
mmeineke |
1044 |
|
341 |
mmeineke |
1045 |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
342 |
mmeineke |
1044 |
|
343 |
|
|
#include "water.mdl" |
344 |
|
|
#include "lipid.mdl" |
345 |
|
|
|
346 |
|
|
nComponents = 2; |
347 |
|
|
component{ |
348 |
|
|
type = "simpleLipid_16"; |
349 |
|
|
nMol = 60; |
350 |
|
|
} |
351 |
|
|
|
352 |
|
|
component{ |
353 |
|
|
type = "SSD_water"; |
354 |
|
|
nMol = 1936; |
355 |
|
|
} |
356 |
|
|
|
357 |
|
|
initialConfig = "bilayer.init"; |
358 |
|
|
|
359 |
|
|
forceField = "DUFF"; |
360 |
|
|
|
361 |
|
|
\end{lstlisting} |
362 |
|
|
|
363 |
mmeineke |
1051 |
\subsection{\label{oopseSec:energyFunctions}{\sc duff} Energy Functions} |
364 |
mmeineke |
1044 |
|
365 |
|
|
The total potential energy function in {\sc duff} is |
366 |
|
|
\begin{equation} |
367 |
|
|
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
368 |
mmeineke |
1054 |
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
369 |
mmeineke |
1044 |
\label{eq:totalPotential} |
370 |
|
|
\end{equation} |
371 |
|
|
Where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
372 |
|
|
\begin{equation} |
373 |
|
|
V^{I}_{\text{Internal}} = |
374 |
|
|
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
375 |
|
|
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
376 |
|
|
+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
377 |
|
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
378 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
379 |
|
|
\biggr] |
380 |
|
|
\label{eq:internalPotential} |
381 |
|
|
\end{equation} |
382 |
|
|
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
383 |
|
|
within the molecule $I$, and $V_{\text{torsion}}$ is the torsion potential |
384 |
|
|
for all 1, 4 bonded pairs. The pairwise portions of the internal |
385 |
|
|
potential are excluded for pairs that are closer than three bonds, |
386 |
|
|
i.e.~atom pairs farther away than a torsion are included in the |
387 |
|
|
pair-wise loop. |
388 |
|
|
|
389 |
|
|
|
390 |
|
|
The bend potential of a molecule is represented by the following function: |
391 |
|
|
\begin{equation} |
392 |
|
|
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
393 |
|
|
\end{equation} |
394 |
|
|
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
395 |
mmeineke |
1054 |
(see Fig.~\ref{oopseFig:lipidModel}), $\theta_0$ is the equilibrium |
396 |
mmeineke |
1044 |
bond angle, and $k_{\theta}$ is the force constant which determines the |
397 |
|
|
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
398 |
|
|
$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
399 |
|
|
|
400 |
|
|
The torsion potential and parameters are also borrowed from TraPPE. It is |
401 |
|
|
of the form: |
402 |
|
|
\begin{equation} |
403 |
|
|
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
404 |
|
|
+ c_2[1 + \cos(2\phi)] |
405 |
|
|
+ c_3[1 + \cos(3\phi)] |
406 |
|
|
\label{eq:origTorsionPot} |
407 |
|
|
\end{equation} |
408 |
mmeineke |
1068 |
Where: |
409 |
mmeineke |
1044 |
\begin{equation} |
410 |
mmeineke |
1068 |
\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
411 |
|
|
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}) |
412 |
|
|
\label{eq:torsPhi} |
413 |
|
|
\end{equation} |
414 |
|
|
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
415 |
|
|
vectors between atoms $i$, $j$, $k$, and $l$. For computational |
416 |
|
|
efficiency, the torsion potential has been recast after the method of |
417 |
|
|
{\sc charmm},\cite{Brooks83} in which the angle series is converted to |
418 |
|
|
a power series of the form: |
419 |
|
|
\begin{equation} |
420 |
mmeineke |
1044 |
V_{\text{torsion}}(\phi) = |
421 |
|
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
422 |
|
|
\label{eq:torsionPot} |
423 |
|
|
\end{equation} |
424 |
|
|
Where: |
425 |
|
|
\begin{align*} |
426 |
|
|
k_0 &= c_1 + c_3 \\ |
427 |
|
|
k_1 &= c_1 - 3c_3 \\ |
428 |
|
|
k_2 &= 2 c_2 \\ |
429 |
|
|
k_3 &= 4c_3 |
430 |
|
|
\end{align*} |
431 |
|
|
By recasting the potential as a power series, repeated trigonometric |
432 |
|
|
evaluations are avoided during the calculation of the potential energy. |
433 |
|
|
|
434 |
|
|
|
435 |
|
|
The cross potential between molecules $I$ and $J$, $V^{IJ}_{\text{Cross}}$, is |
436 |
|
|
as follows: |
437 |
|
|
\begin{equation} |
438 |
|
|
V^{IJ}_{\text{Cross}} = |
439 |
|
|
\sum_{i \in I} \sum_{j \in J} |
440 |
|
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
441 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
442 |
|
|
+ V_{\text{sticky}} |
443 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
444 |
|
|
\biggr] |
445 |
|
|
\label{eq:crossPotentail} |
446 |
|
|
\end{equation} |
447 |
|
|
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
448 |
|
|
$V_{\text{dipole}}$ is the dipole dipole potential, and |
449 |
|
|
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
450 |
mmeineke |
1054 |
(Sec.~\ref{oopseSec:SSD}). Note that not all atom types include all |
451 |
mmeineke |
1044 |
interactions. |
452 |
|
|
|
453 |
|
|
The dipole-dipole potential has the following form: |
454 |
|
|
\begin{equation} |
455 |
|
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
456 |
|
|
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
457 |
|
|
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
458 |
|
|
- |
459 |
mmeineke |
1068 |
3(\boldsymbol{\hat{u}}_i \cdot \hat{\mathbf{r}}_{ij}) % |
460 |
|
|
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr] |
461 |
mmeineke |
1044 |
\label{eq:dipolePot} |
462 |
|
|
\end{equation} |
463 |
|
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
464 |
|
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
465 |
|
|
are the orientational degrees of freedom for atoms $i$ and $j$ |
466 |
|
|
respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom |
467 |
mmeineke |
1054 |
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector |
468 |
|
|
of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is the |
469 |
|
|
unit vector pointing along $\mathbf{r}_{ij}$ |
470 |
|
|
($\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). |
471 |
mmeineke |
1044 |
|
472 |
mmeineke |
1068 |
To improve computational efficiency of the dipole-dipole interactions, |
473 |
|
|
{\sc oopse} employs an electrostatic cutoff radius. This parameter can |
474 |
|
|
be set in the \texttt{.bass} file, and controls the length scale over |
475 |
|
|
which dipole interactions are felt. To compensate for the |
476 |
|
|
discontinuity in the potential and the forces at the cutoff radius, we |
477 |
|
|
have implemented a switching function to smoothly scale the |
478 |
|
|
dipole-dipole interaction at the cutoff. |
479 |
|
|
\begin{equation} |
480 |
|
|
S(r_{ij}) = |
481 |
|
|
\begin{cases} |
482 |
|
|
1 & \text{if $r_{ij} \le r_t$},\\ |
483 |
|
|
\frac{(r_{\text{cut}} + 2r_{ij} - 3r_t)(r_{\text{cut}} - r_{ij})^2} |
484 |
|
|
{(r_{\text{cut}} - r_t)^2} |
485 |
|
|
& \text{if $r_t < r_{ij} \le r_{\text{cut}}$}, \\ |
486 |
|
|
0 & \text{if $r_{ij} > r_{\text{cut}}$.} |
487 |
|
|
\end{cases} |
488 |
|
|
\label{eq:dipoleSwitching} |
489 |
|
|
\end{equation} |
490 |
|
|
Here $S(r_{ij})$ scales the potential at a given $r_{ij}$, and $r_t$ |
491 |
|
|
is the taper radius some given thickness less than the electrostatic |
492 |
|
|
cutoff. The switching thickness can be set in the \texttt{.bass} file. |
493 |
mmeineke |
1044 |
|
494 |
mmeineke |
1068 |
\subsection{\label{oopseSec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
495 |
mmeineke |
1044 |
|
496 |
|
|
In the interest of computational efficiency, the default solvent used |
497 |
|
|
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
498 |
|
|
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
499 |
|
|
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
500 |
|
|
water model proposed by Bratko, Blum, and |
501 |
|
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
502 |
|
|
with a Lennard-Jones core and a sticky potential that directs the |
503 |
|
|
particles to assume the proper hydrogen bond orientation in the first |
504 |
|
|
solvation shell. Thus, the interaction between two SSD water molecules |
505 |
|
|
\emph{i} and \emph{j} is given by the potential |
506 |
|
|
\begin{equation} |
507 |
|
|
V_{ij} = |
508 |
|
|
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
509 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
510 |
|
|
V_{ij}^{sp} |
511 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
512 |
|
|
\label{eq:ssdPot} |
513 |
|
|
\end{equation} |
514 |
|
|
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
515 |
|
|
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
516 |
|
|
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
517 |
|
|
orientations of the respective molecules. The Lennard-Jones and dipole |
518 |
|
|
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
519 |
|
|
and \ref{eq:dipolePot} respectively. The sticky part is described by |
520 |
|
|
the following, |
521 |
|
|
\begin{equation} |
522 |
|
|
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
523 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
524 |
|
|
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
525 |
|
|
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
526 |
|
|
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
527 |
|
|
\label{eq:stickyPot} |
528 |
|
|
\end{equation} |
529 |
|
|
where $\nu_0$ is a strength parameter for the sticky potential, and |
530 |
|
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
531 |
|
|
sticky interaction beyond the first solvation shell. The $w$ function |
532 |
|
|
can be thought of as an attractive potential with tetrahedral |
533 |
|
|
geometry: |
534 |
|
|
\begin{equation} |
535 |
|
|
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
536 |
|
|
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
537 |
|
|
\label{eq:stickyW} |
538 |
|
|
\end{equation} |
539 |
|
|
while the $w^\prime$ function counters the normal aligned and |
540 |
|
|
anti-aligned structures favored by point dipoles: |
541 |
|
|
\begin{equation} |
542 |
|
|
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
543 |
|
|
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
544 |
|
|
\label{eq:stickyWprime} |
545 |
|
|
\end{equation} |
546 |
|
|
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
547 |
|
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
548 |
|
|
enhances the tetrahedral geometry for hydrogen bonded structures), |
549 |
|
|
while $w^\prime$ is a purely empirical function. A more detailed |
550 |
|
|
description of the functional parts and variables in this potential |
551 |
|
|
can be found in the original SSD |
552 |
|
|
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
553 |
|
|
|
554 |
mmeineke |
1068 |
Since SSD/E is a single-point {\it dipolar} model, the force |
555 |
mmeineke |
1044 |
calculations are simplified significantly relative to the standard |
556 |
|
|
{\it charged} multi-point models. In the original Monte Carlo |
557 |
|
|
simulations using this model, Ichiye {\it et al.} reported that using |
558 |
|
|
SSD decreased computer time by a factor of 6-7 compared to other |
559 |
|
|
models.\cite{liu96:new_model} What is most impressive is that these savings |
560 |
|
|
did not come at the expense of accurate depiction of the liquid state |
561 |
mmeineke |
1068 |
properties. Indeed, SSD/E maintains reasonable agreement with the Head-Gordon |
562 |
mmeineke |
1044 |
diffraction data for the structural features of liquid |
563 |
mmeineke |
1068 |
water.\cite{hura00,liu96:new_model} Additionally, the dynamical properties |
564 |
|
|
exhibited by SSD/E agree with experiment better than those of more |
565 |
mmeineke |
1044 |
computationally expensive models (like TIP3P and |
566 |
|
|
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
567 |
mmeineke |
1068 |
of solvent properties makes SSD/E a very attractive model for the |
568 |
mmeineke |
1044 |
simulation of large scale biochemical simulations. |
569 |
|
|
|
570 |
|
|
Recent constant pressure simulations revealed issues in the original |
571 |
|
|
SSD model that led to lower than expected densities at all target |
572 |
|
|
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
573 |
|
|
is therefore SSD/E, a density corrected derivative of SSD that |
574 |
|
|
exhibits improved liquid structure and transport behavior. If the use |
575 |
|
|
of a reaction field long-range interaction correction is desired, it |
576 |
|
|
is recommended that the parameters be modified to those of the SSD/RF |
577 |
|
|
model. Solvent parameters can be easily modified in an accompanying |
578 |
mmeineke |
1068 |
\texttt{.bass} file as illustrated in the scheme below. A table of the |
579 |
mmeineke |
1044 |
parameter values and the drawbacks and benefits of the different |
580 |
mmeineke |
1054 |
density corrected SSD models can be found in |
581 |
|
|
reference~\cite{Gezelter04}. |
582 |
mmeineke |
1044 |
|
583 |
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}] |
584 |
mmeineke |
1044 |
|
585 |
|
|
#include "water.mdl" |
586 |
|
|
|
587 |
|
|
nComponents = 1; |
588 |
|
|
component{ |
589 |
|
|
type = "SSD_water"; |
590 |
|
|
nMol = 864; |
591 |
|
|
} |
592 |
|
|
|
593 |
|
|
initialConfig = "liquidWater.init"; |
594 |
|
|
|
595 |
|
|
forceField = "DUFF"; |
596 |
|
|
|
597 |
|
|
/* |
598 |
|
|
* The following two flags set the cutoff |
599 |
|
|
* radius for the electrostatic forces |
600 |
|
|
* as well as the skin thickness of the switching |
601 |
|
|
* function. |
602 |
|
|
*/ |
603 |
|
|
|
604 |
|
|
electrostaticCutoffRadius = 9.2; |
605 |
|
|
electrostaticSkinThickness = 1.38; |
606 |
|
|
|
607 |
|
|
\end{lstlisting} |
608 |
|
|
|
609 |
|
|
|
610 |
mmeineke |
1051 |
\subsection{\label{oopseSec:eam}Embedded Atom Method} |
611 |
mmeineke |
1044 |
|
612 |
mmeineke |
1054 |
There are Molecular Dynamics packages which have the |
613 |
mmeineke |
1044 |
capacity to simulate metallic systems, including some that have |
614 |
|
|
parallel computational abilities\cite{plimpton93}. Potentials that |
615 |
|
|
describe bonding transition metal |
616 |
mmeineke |
1068 |
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have an |
617 |
mmeineke |
1044 |
attractive interaction which models ``Embedding'' |
618 |
|
|
a positively charged metal ion in the electron density due to the |
619 |
|
|
free valance ``sea'' of electrons created by the surrounding atoms in |
620 |
mmeineke |
1068 |
the system. A mostly-repulsive pairwise part of the potential |
621 |
mmeineke |
1044 |
describes the interaction of the positively charged metal core ions |
622 |
|
|
with one another. A particular potential description called the |
623 |
|
|
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
624 |
|
|
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
625 |
mmeineke |
1068 |
good review of {\sc eam} and other metallic potential formulations was written |
626 |
mmeineke |
1044 |
by Voter.\cite{voter} |
627 |
|
|
|
628 |
|
|
The {\sc eam} potential has the form: |
629 |
|
|
\begin{eqnarray} |
630 |
|
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
631 |
|
|
\phi_{ij}({\bf r}_{ij}) \\ |
632 |
|
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
633 |
mmeineke |
1068 |
\end{eqnarray} |
634 |
mmeineke |
1083 |
where $F_{i} $ is the embedding function that equates the energy |
635 |
|
|
required to embed a positively-charged core ion $i$ into a linear |
636 |
|
|
superposition of spherically averaged atomic electron densities given |
637 |
|
|
by $\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise |
638 |
|
|
interaction between atoms $i$ and $j$. In the original formulation of |
639 |
|
|
{\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, |
640 |
|
|
however in later refinements to {\sc eam} have shown that non-uniqueness |
641 |
|
|
between $F$ and $\phi$ allow for more general forms for |
642 |
|
|
$\phi$.\cite{Daw89} There is a cutoff distance, $r_{cut}$, which |
643 |
|
|
limits the summations in the {\sc eam} equation to the few dozen atoms |
644 |
mmeineke |
1044 |
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
645 |
mmeineke |
1083 |
interactions. Foiles \emph{et al}.~fit {\sc eam} potentials for the fcc |
646 |
|
|
metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals.\cite{FBD86} |
647 |
|
|
These fits, are included in {\sc oopse}. |
648 |
mmeineke |
1044 |
|
649 |
mmeineke |
1051 |
\subsection{\label{oopseSec:pbc}Periodic Boundary Conditions} |
650 |
mmeineke |
1044 |
|
651 |
|
|
\newcommand{\roundme}{\operatorname{round}} |
652 |
|
|
|
653 |
mmeineke |
1068 |
\textit{Periodic boundary conditions} are widely used to simulate bulk properties with a relatively small number of particles. The |
654 |
|
|
simulation box is replicated throughout space to form an infinite |
655 |
|
|
lattice. During the simulation, when a particle moves in the primary |
656 |
|
|
cell, its image in other cells move in exactly the same direction with |
657 |
|
|
exactly the same orientation. Thus, as a particle leaves the primary |
658 |
|
|
cell, one of its images will enter through the opposite face. If the |
659 |
|
|
simulation box is large enough to avoid ``feeling'' the symmetries of |
660 |
|
|
the periodic lattice, surface effects can be ignored. The available |
661 |
|
|
periodic cells in OOPSE are cubic, orthorhombic and parallelepiped. We |
662 |
mmeineke |
1083 |
use a $3 \times 3$ matrix, $\mathsf{H}$, to describe the shape and |
663 |
|
|
size of the simulation box. $\mathsf{H}$ is defined: |
664 |
mmeineke |
1044 |
\begin{equation} |
665 |
mmeineke |
1083 |
\mathsf{H} = ( \mathbf{h}_x, \mathbf{h}_y, \mathbf{h}_z ) |
666 |
mmeineke |
1044 |
\end{equation} |
667 |
mmeineke |
1068 |
Where $\mathbf{h}_j$ is the column vector of the $j$th axis of the |
668 |
|
|
box. During the course of the simulation both the size and shape of |
669 |
mmeineke |
1083 |
the box can be changed to allow volume fluctuations when constraining |
670 |
mmeineke |
1068 |
the pressure. |
671 |
mmeineke |
1044 |
|
672 |
mmeineke |
1068 |
A real space vector, $\mathbf{r}$ can be transformed in to a box space |
673 |
|
|
vector, $\mathbf{s}$, and back through the following transformations: |
674 |
|
|
\begin{align} |
675 |
mmeineke |
1083 |
\mathbf{s} &= \mathsf{H}^{-1} \mathbf{r} \\ |
676 |
|
|
\mathbf{r} &= \mathsf{H} \mathbf{s} |
677 |
mmeineke |
1068 |
\end{align} |
678 |
|
|
The vector $\mathbf{s}$ is now a vector expressed as the number of box |
679 |
|
|
lengths in the $\mathbf{h}_x$, $\mathbf{h}_y$, and $\mathbf{h}_z$ |
680 |
|
|
directions. To find the minimum image of a vector $\mathbf{r}$, we |
681 |
|
|
first convert it to its corresponding vector in box space, and then, |
682 |
|
|
cast each element to lie on the in the range $[-0.5,0.5]$: |
683 |
mmeineke |
1044 |
\begin{equation} |
684 |
|
|
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
685 |
|
|
\end{equation} |
686 |
mmeineke |
1068 |
Where $s_i$ is the $i$th element of $\mathbf{s}$, and |
687 |
|
|
$\roundme(s_i)$is given by |
688 |
mmeineke |
1044 |
\begin{equation} |
689 |
mmeineke |
1068 |
\roundme(x) = |
690 |
|
|
\begin{cases} |
691 |
|
|
\lfloor x+0.5 \rfloor & \text{if $x \ge 0$} \\ |
692 |
|
|
\lceil x-0.5 \rceil & \text{if $x < 0$ } |
693 |
|
|
\end{cases} |
694 |
mmeineke |
1044 |
\end{equation} |
695 |
mmeineke |
1068 |
Here $\lfloor x \rfloor$ is the floor operator, and gives the largest |
696 |
|
|
integer value that is not greater than $x$, and $\lceil x \rceil$ is |
697 |
|
|
the ceiling operator, and gives the smallest integer that is not less |
698 |
|
|
than $x$. For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, |
699 |
|
|
$\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
700 |
mmeineke |
1044 |
|
701 |
|
|
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
702 |
mmeineke |
1068 |
transforming back to real space, |
703 |
mmeineke |
1044 |
\begin{equation} |
704 |
mmeineke |
1083 |
\mathbf{r}^{\prime}=\mathsf{H}^{-1}\mathbf{s}^{\prime}% |
705 |
mmeineke |
1044 |
\end{equation} |
706 |
mmeineke |
1068 |
In this way, particles are allowed to diffuse freely in $\mathbf{r}$, |
707 |
|
|
but their minimum images, $\mathbf{r}^{\prime}$ are used to compute |
708 |
mmeineke |
1083 |
the inter-atomic forces. |
709 |
mmeineke |
1044 |
|
710 |
|
|
|
711 |
mmeineke |
1051 |
\section{\label{oopseSec:IOfiles}Input and Output Files} |
712 |
mmeineke |
1044 |
|
713 |
|
|
\subsection{{\sc bass} and Model Files} |
714 |
|
|
|
715 |
mmeineke |
1071 |
Every {\sc oopse} simulation begins with a Bizarre Atom Simulation |
716 |
|
|
Syntax ({\sc bass}) file. {\sc bass} is a script syntax that is parsed |
717 |
|
|
by {\sc oopse} at runtime. The {\sc bass} file allows for the user to |
718 |
|
|
completely describe the system they wish to simulate, as well as tailor |
719 |
|
|
{\sc oopse}'s behavior during the simulation. {\sc bass} files are |
720 |
|
|
denoted with the extension |
721 |
mmeineke |
1044 |
\texttt{.bass}, an example file is shown in |
722 |
mmeineke |
1071 |
Scheme~\ref{sch:bassExample}. |
723 |
mmeineke |
1044 |
|
724 |
mmeineke |
1071 |
\begin{lstlisting}[float,caption={[An example of a complete {\sc bass} file] An example showing a complete {\sc bass} file.},label={sch:bassExample}] |
725 |
mmeineke |
1044 |
|
726 |
mmeineke |
1071 |
molecule{ |
727 |
|
|
name = "Ar"; |
728 |
|
|
nAtoms = 1; |
729 |
|
|
atom[0]{ |
730 |
|
|
type="Ar"; |
731 |
|
|
position( 0.0, 0.0, 0.0 ); |
732 |
|
|
} |
733 |
|
|
} |
734 |
|
|
|
735 |
|
|
nComponents = 1; |
736 |
|
|
component{ |
737 |
|
|
type = "Ar"; |
738 |
|
|
nMol = 108; |
739 |
|
|
} |
740 |
|
|
|
741 |
|
|
initialConfig = "./argon.init"; |
742 |
|
|
|
743 |
|
|
forceField = "LJ"; |
744 |
mmeineke |
1083 |
ensemble = "NVE"; // specify the simulation ensemble |
745 |
mmeineke |
1071 |
dt = 1.0; // the time step for integration |
746 |
|
|
runTime = 1e3; // the total simulation run time |
747 |
|
|
sampleTime = 100; // trajectory file frequency |
748 |
|
|
statusTime = 50; // statistics file frequency |
749 |
|
|
|
750 |
|
|
\end{lstlisting} |
751 |
|
|
|
752 |
mmeineke |
1054 |
Within the \texttt{.bass} file it is necessary to provide a complete |
753 |
mmeineke |
1044 |
description of the molecule before it is actually placed in the |
754 |
mmeineke |
1071 |
simulation. The {\sc bass} syntax was originally developed with this |
755 |
|
|
goal in mind, and allows for the specification of all the atoms in a |
756 |
|
|
molecular prototype, as well as any bonds, bends, or torsions. These |
757 |
mmeineke |
1044 |
descriptions can become lengthy for complex molecules, and it would be |
758 |
mmeineke |
1071 |
inconvenient to duplicate the simulation at the beginning of each {\sc |
759 |
|
|
bass} script. Addressing this issue {\sc bass} allows for the |
760 |
|
|
inclusion of model files at the top of a \texttt{.bass} file. These |
761 |
|
|
model files, denoted with the \texttt{.mdl} extension, allow the user |
762 |
|
|
to describe a molecular prototype once, then simply include it into |
763 |
|
|
each simulation containing that molecule. Returning to the example in |
764 |
|
|
Scheme~\ref{sch:bassExample}, the \texttt{.mdl} file's contents would |
765 |
|
|
be Scheme~\ref{sch:mdlExample}, and the new \texttt{.bass} file would |
766 |
|
|
become Scheme~\ref{sch:bassExPrime}. |
767 |
mmeineke |
1044 |
|
768 |
mmeineke |
1071 |
\begin{lstlisting}[float,caption={An example \texttt{.mdl} file.},label={sch:mdlExample}] |
769 |
|
|
|
770 |
|
|
molecule{ |
771 |
|
|
name = "Ar"; |
772 |
|
|
nAtoms = 1; |
773 |
|
|
atom[0]{ |
774 |
|
|
type="Ar"; |
775 |
|
|
position( 0.0, 0.0, 0.0 ); |
776 |
|
|
} |
777 |
|
|
} |
778 |
|
|
|
779 |
|
|
\end{lstlisting} |
780 |
|
|
|
781 |
|
|
\begin{lstlisting}[float,caption={Revised {\sc bass} example.},label={sch:bassExPrime}] |
782 |
|
|
|
783 |
|
|
#include "argon.mdl" |
784 |
|
|
|
785 |
|
|
nComponents = 1; |
786 |
|
|
component{ |
787 |
|
|
type = "Ar"; |
788 |
|
|
nMol = 108; |
789 |
|
|
} |
790 |
|
|
|
791 |
|
|
initialConfig = "./argon.init"; |
792 |
|
|
|
793 |
|
|
forceField = "LJ"; |
794 |
|
|
ensemble = "NVE"; |
795 |
|
|
dt = 1.0; |
796 |
|
|
runTime = 1e3; |
797 |
|
|
sampleTime = 100; |
798 |
|
|
statusTime = 50; |
799 |
|
|
|
800 |
|
|
\end{lstlisting} |
801 |
|
|
|
802 |
mmeineke |
1051 |
\subsection{\label{oopseSec:coordFiles}Coordinate Files} |
803 |
mmeineke |
1044 |
|
804 |
|
|
The standard format for storage of a systems coordinates is a modified |
805 |
|
|
xyz-file syntax, the exact details of which can be seen in |
806 |
mmeineke |
1071 |
Scheme~\ref{sch:dumpFormat}. As all bonding and molecular information |
807 |
|
|
is stored in the \texttt{.bass} and \texttt{.mdl} files, the |
808 |
|
|
coordinate files are simply the complete set of coordinates for each |
809 |
|
|
atom at a given simulation time. One important note, although the |
810 |
|
|
simulation propagates the complete rotation matrix, directional |
811 |
|
|
entities are written out using quanternions, to save space in the |
812 |
|
|
output files. |
813 |
mmeineke |
1044 |
|
814 |
mmeineke |
1083 |
\begin{lstlisting}[float,caption={[The format of the coordinate files]Shows the format of the coordinate files. The fist line is the number of atoms. The second line begins with the time stamp followed by the three $\mathsf{H}$ column vectors. It is important to note, that for extended system ensembles, additional information pertinent to the integrators may be stored on this line as well.. The next lines are the atomic coordinates for all atoms in the system. First is the name followed by position, velocity, quanternions, and lastly angular velocities.},label=sch:dumpFormat] |
815 |
mmeineke |
1071 |
|
816 |
|
|
nAtoms |
817 |
|
|
time; Hxx Hyx Hzx; Hxy Hyy Hzy; Hxz Hyz Hzz; |
818 |
|
|
Name1 x y z vx vy vz q0 q1 q2 q3 jx jy jz |
819 |
|
|
Name2 x y z vx vy vz q0 q1 q2 q3 jx jy jz |
820 |
|
|
etc... |
821 |
|
|
|
822 |
|
|
\end{lstlisting} |
823 |
|
|
|
824 |
|
|
|
825 |
|
|
There are three major files used by {\sc oopse} written in the |
826 |
|
|
coordinate format, they are as follows: the initialization file |
827 |
|
|
(\texttt{.init}), the simulation trajectory file (\texttt{.dump}), and |
828 |
|
|
the final coordinates of the simulation. The initialization file is |
829 |
|
|
necessary for {\sc oopse} to start the simulation with the proper |
830 |
|
|
coordinates, and is generated before the simulation run. The |
831 |
|
|
trajectory file is created at the beginning of the simulation, and is |
832 |
|
|
used to store snapshots of the simulation at regular intervals. The |
833 |
|
|
first frame is a duplication of the |
834 |
|
|
\texttt{.init} file, and each subsequent frame is appended to the file |
835 |
|
|
at an interval specified in the \texttt{.bass} file with the |
836 |
|
|
\texttt{sampleTime} flag. The final coordinate file is the end of run file. The |
837 |
mmeineke |
1054 |
\texttt{.eor} file stores the final configuration of the system for a |
838 |
mmeineke |
1044 |
given simulation. The file is updated at the same time as the |
839 |
mmeineke |
1071 |
\texttt{.dump} file, however, it only contains the most recent |
840 |
mmeineke |
1044 |
frame. In this way, an \texttt{.eor} file may be used as the |
841 |
mmeineke |
1071 |
initialization file to a second simulation in order to continue a |
842 |
|
|
simulation or recover one from a processor that has crashed during the |
843 |
|
|
course of the run. |
844 |
mmeineke |
1044 |
|
845 |
mmeineke |
1054 |
\subsection{\label{oopseSec:initCoords}Generation of Initial Coordinates} |
846 |
mmeineke |
1044 |
|
847 |
mmeineke |
1071 |
As was stated in Sec.~\ref{oopseSec:coordFiles}, an initialization |
848 |
|
|
file is needed to provide the starting coordinates for a |
849 |
mmeineke |
1083 |
simulation. The {\sc oopse} package provides several system building |
850 |
|
|
programs to aid in the creation of the \texttt{.init} |
851 |
|
|
file. The programs use {\sc bass}, and will recognize |
852 |
mmeineke |
1071 |
arguments and parameters in the \texttt{.bass} file that would |
853 |
|
|
otherwise be ignored by the simulation. |
854 |
mmeineke |
1044 |
|
855 |
|
|
\subsection{The Statistics File} |
856 |
|
|
|
857 |
mmeineke |
1071 |
The last output file generated by {\sc oopse} is the statistics |
858 |
|
|
file. This file records such statistical quantities as the |
859 |
|
|
instantaneous temperature, volume, pressure, etc. It is written out |
860 |
|
|
with the frequency specified in the \texttt{.bass} file with the |
861 |
|
|
\texttt{statusTime} keyword. The file allows the user to observe the |
862 |
|
|
system variables as a function of simulation time while the simulation |
863 |
|
|
is in progress. One useful function the statistics file serves is to |
864 |
|
|
monitor the conserved quantity of a given simulation ensemble, this |
865 |
|
|
allows the user to observe the stability of the integrator. The |
866 |
|
|
statistics file is denoted with the \texttt{.stat} file extension. |
867 |
mmeineke |
1044 |
|
868 |
mmeineke |
1051 |
\section{\label{oopseSec:mechanics}Mechanics} |
869 |
mmeineke |
1044 |
|
870 |
mmeineke |
1054 |
\subsection{\label{oopseSec:integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
871 |
mmeineke |
1044 |
|
872 |
|
|
Integration of the equations of motion was carried out using the |
873 |
|
|
symplectic splitting method proposed by Dullweber \emph{et |
874 |
mmeineke |
1071 |
al.}.\cite{Dullweber1997} The reason for the selection of this |
875 |
|
|
integrator, is the poor energy conservation of rigid body systems |
876 |
|
|
using quaternion dynamics. While quaternions work well for |
877 |
|
|
orientational motion in alternate ensembles, the microcanonical |
878 |
|
|
ensemble has a constant energy requirement that is quite sensitive to |
879 |
|
|
errors in the equations of motion. The original implementation of {\sc |
880 |
|
|
oopse} utilized quaternions for rotational motion propagation; |
881 |
|
|
however, a detailed investigation showed that they resulted in a |
882 |
|
|
steady drift in the total energy, something that has been observed by |
883 |
|
|
others.\cite{Laird97} |
884 |
mmeineke |
1044 |
|
885 |
|
|
The key difference in the integration method proposed by Dullweber |
886 |
mmeineke |
1071 |
\emph{et al}.~({\sc dlm}) is that the entire rotation matrix is propagated from |
887 |
|
|
one time step to the next. In the past, this would not have been a |
888 |
|
|
feasible option, since the rotation matrix for a single body is nine |
889 |
|
|
elements long as opposed to three or four elements for Euler angles |
890 |
|
|
and quaternions respectively. System memory has become much less of an |
891 |
|
|
issue in recent times, and the {\sc dlm} method has used memory in |
892 |
|
|
exchange for substantial benefits in energy conservation. |
893 |
mmeineke |
1044 |
|
894 |
mmeineke |
1071 |
The {\sc dlm} method allows for Verlet style integration of both |
895 |
|
|
linear and angular motion of rigid bodies. In the integration method, |
896 |
|
|
the orientational propagation involves a sequence of matrix |
897 |
mmeineke |
1044 |
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
898 |
mmeineke |
1071 |
matrix rotations are more costly computationally than the simpler |
899 |
|
|
arithmetic quaternion propagation. With the same time step, a 1000 SSD |
900 |
|
|
particle simulation shows an average 7\% increase in computation time |
901 |
|
|
using the {\sc dlm} method in place of quaternions. This cost is more |
902 |
|
|
than justified when comparing the energy conservation of the two |
903 |
|
|
methods as illustrated in Fig.~\ref{timestep}. |
904 |
mmeineke |
1044 |
|
905 |
|
|
\begin{figure} |
906 |
mmeineke |
1045 |
\centering |
907 |
|
|
\includegraphics[width=\linewidth]{timeStep.eps} |
908 |
mmeineke |
1071 |
\caption[Energy conservation for quaternion versus {\sc dlm} dynamics]{Energy conservation using quaternion based integration versus |
909 |
|
|
the {\sc dlm} method with |
910 |
mmeineke |
1044 |
increasing time step. For each time step, the dotted line is total |
911 |
mmeineke |
1071 |
energy using the {\sc dlm} integrator, and the solid line comes |
912 |
mmeineke |
1044 |
from the quaternion integrator. The larger time step plots are shifted |
913 |
|
|
up from the true energy baseline for clarity.} |
914 |
|
|
\label{timestep} |
915 |
|
|
\end{figure} |
916 |
|
|
|
917 |
mmeineke |
1071 |
In Fig.~\ref{timestep}, the resulting energy drift at various time |
918 |
|
|
steps for both the {\sc dlm} and quaternion integration schemes |
919 |
mmeineke |
1044 |
is compared. All of the 1000 SSD particle simulations started with the |
920 |
|
|
same configuration, and the only difference was the method for |
921 |
|
|
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
922 |
|
|
methods for propagating particle rotation conserve energy fairly well, |
923 |
|
|
with the quaternion method showing a slight energy drift over time in |
924 |
|
|
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
925 |
mmeineke |
1071 |
energy conservation benefits of the {\sc dlm} method are clearly |
926 |
mmeineke |
1044 |
demonstrated. Thus, while maintaining the same degree of energy |
927 |
|
|
conservation, one can take considerably longer time steps, leading to |
928 |
|
|
an overall reduction in computation time. |
929 |
|
|
|
930 |
|
|
Energy drift in these SSD particle simulations was unnoticeable for |
931 |
|
|
time steps up to three femtoseconds. A slight energy drift on the |
932 |
|
|
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
933 |
|
|
four femtoseconds, and as expected, this drift increases dramatically |
934 |
mmeineke |
1071 |
with increasing time step. |
935 |
mmeineke |
1044 |
|
936 |
|
|
|
937 |
|
|
\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
938 |
|
|
|
939 |
|
|
|
940 |
|
|
{\sc oopse} implements a |
941 |
|
|
|
942 |
|
|
|
943 |
mmeineke |
1071 |
\subsection{\label{oopseSec:noseHooverThermo}Nose-Hoover Thermostatting} |
944 |
mmeineke |
1044 |
|
945 |
|
|
To mimic the effects of being in a constant temperature ({\sc nvt}) |
946 |
|
|
ensemble, {\sc oopse} uses the Nose-Hoover extended system |
947 |
|
|
approach.\cite{Hoover85} In this method, the equations of motion for |
948 |
|
|
the particle positions and velocities are |
949 |
|
|
\begin{eqnarray} |
950 |
|
|
\dot{{\bf r}} & = & {\bf v} \\ |
951 |
|
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} |
952 |
|
|
\label{eq:nosehoovereom} |
953 |
|
|
\end{eqnarray} |
954 |
|
|
|
955 |
|
|
$\chi$ is an ``extra'' variable included in the extended system, and |
956 |
|
|
it is propagated using the first order equation of motion |
957 |
|
|
\begin{equation} |
958 |
|
|
\dot{\chi} = \frac{1}{\tau_{T}} \left( \frac{T}{T_{target}} - 1 \right) |
959 |
|
|
\label{eq:nosehooverext} |
960 |
|
|
\end{equation} |
961 |
|
|
where $T_{target}$ is the target temperature for the simulation, and |
962 |
|
|
$\tau_{T}$ is a time constant for the thermostat. |
963 |
|
|
|
964 |
|
|
To select the Nose-Hoover {\sc nvt} ensemble, the {\tt ensemble = NVT;} |
965 |
|
|
command would be used in the simulation's {\sc bass} file. There is |
966 |
|
|
some subtlety in choosing values for $\tau_{T}$, and it is usually set |
967 |
|
|
to values of a few ps. Within a {\sc bass} file, $\tau_{T}$ could be |
968 |
|
|
set to 1 ps using the {\tt tauThermostat = 1000; } command. |
969 |
|
|
|
970 |
mmeineke |
1071 |
\subsection{\label{oopseSec:rattle}The {\sc rattle} Method for Bond |
971 |
|
|
Constraints} |
972 |
mmeineke |
1044 |
|
973 |
mmeineke |
1071 |
In order to satisfy the constraints of fixed bond lengths within {\sc |
974 |
|
|
oopse}, we have implemented the {\sc rattle} algorithm of |
975 |
|
|
Andersen.\cite{andersen83} The algorithm is a velocity verlet |
976 |
|
|
formulation of the {\sc shake} method\cite{ryckaert77} of iteratively |
977 |
|
|
solving the Lagrange multipliers of constraint. The system of lagrange |
978 |
|
|
multipliers allows one to reformulate the equations of motion with |
979 |
mmeineke |
1083 |
explicit constraint forces.\cite{fowles99:lagrange} |
980 |
mmeineke |
1071 |
|
981 |
mmeineke |
1083 |
Consider a system described by coordinates $q_1$ and $q_2$ subject to an |
982 |
mmeineke |
1071 |
equation of constraint: |
983 |
|
|
\begin{equation} |
984 |
|
|
\sigma(q_1, q_2,t) = 0 |
985 |
|
|
\label{oopseEq:lm1} |
986 |
|
|
\end{equation} |
987 |
|
|
The Lagrange formulation of the equations of motion can be written: |
988 |
|
|
\begin{equation} |
989 |
|
|
\delta\int_{t_1}^{t_2}L\, dt = |
990 |
|
|
\int_{t_1}^{t_2} \sum_i \biggl [ \frac{\partial L}{\partial q_i} |
991 |
|
|
- \frac{d}{dt}\biggl(\frac{\partial L}{\partial \dot{q}_i} |
992 |
|
|
\biggr ) \biggr] \delta q_i \, dt = 0 |
993 |
|
|
\label{oopseEq:lm2} |
994 |
|
|
\end{equation} |
995 |
|
|
Here, $\delta q_i$ is not independent for each $q$, as $q_1$ and $q_2$ |
996 |
|
|
are linked by $\sigma$. However, $\sigma$ is fixed at any given |
997 |
|
|
instant of time, giving: |
998 |
|
|
\begin{align} |
999 |
|
|
\delta\sigma &= \biggl( \frac{\partial\sigma}{\partial q_1} \delta q_1 % |
1000 |
|
|
+ \frac{\partial\sigma}{\partial q_2} \delta q_2 \biggr) = 0 \\ |
1001 |
|
|
% |
1002 |
|
|
\frac{\partial\sigma}{\partial q_1} \delta q_1 &= % |
1003 |
|
|
- \frac{\partial\sigma}{\partial q_2} \delta q_2 \\ |
1004 |
|
|
% |
1005 |
|
|
\delta q_2 &= - \biggl(\frac{\partial\sigma}{\partial q_1} \bigg / % |
1006 |
|
|
\frac{\partial\sigma}{\partial q_2} \biggr) \delta q_1 |
1007 |
|
|
\end{align} |
1008 |
|
|
Substituted back into Eq.~\ref{oopseEq:lm2}, |
1009 |
|
|
\begin{equation} |
1010 |
|
|
\int_{t_1}^{t_2}\biggl [ \biggl(\frac{\partial L}{\partial q_1} |
1011 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
1012 |
|
|
\biggr) |
1013 |
|
|
- \biggl( \frac{\partial L}{\partial q_1} |
1014 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
1015 |
|
|
\biggr) \biggl(\frac{\partial\sigma}{\partial q_1} \bigg / % |
1016 |
|
|
\frac{\partial\sigma}{\partial q_2} \biggr)\biggr] \delta q_1 \, dt = 0 |
1017 |
|
|
\label{oopseEq:lm3} |
1018 |
|
|
\end{equation} |
1019 |
|
|
Leading to, |
1020 |
|
|
\begin{equation} |
1021 |
|
|
\frac{\biggl(\frac{\partial L}{\partial q_1} |
1022 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
1023 |
|
|
\biggr)}{\frac{\partial\sigma}{\partial q_1}} = |
1024 |
|
|
\frac{\biggl(\frac{\partial L}{\partial q_2} |
1025 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_2} |
1026 |
|
|
\biggr)}{\frac{\partial\sigma}{\partial q_2}} |
1027 |
|
|
\label{oopseEq:lm4} |
1028 |
|
|
\end{equation} |
1029 |
|
|
This relation can only be statisfied, if both are equal to a single |
1030 |
|
|
function $-\lambda(t)$, |
1031 |
|
|
\begin{align} |
1032 |
|
|
\frac{\biggl(\frac{\partial L}{\partial q_1} |
1033 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
1034 |
|
|
\biggr)}{\frac{\partial\sigma}{\partial q_1}} &= -\lambda(t) \\ |
1035 |
|
|
% |
1036 |
|
|
\frac{\partial L}{\partial q_1} |
1037 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} &= |
1038 |
|
|
-\lambda(t)\,\frac{\partial\sigma}{\partial q_1} \\ |
1039 |
|
|
% |
1040 |
|
|
\frac{\partial L}{\partial q_1} |
1041 |
|
|
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
1042 |
|
|
+ \mathcal{G}_i &= 0 |
1043 |
|
|
\end{align} |
1044 |
|
|
Where $\mathcal{G}_i$, the force of constraint on $i$, is: |
1045 |
|
|
\begin{equation} |
1046 |
|
|
\mathcal{G}_i = \lambda(t)\,\frac{\partial\sigma}{\partial q_1} |
1047 |
|
|
\label{oopseEq:lm5} |
1048 |
|
|
\end{equation} |
1049 |
|
|
|
1050 |
|
|
In a simulation, this would involve the solution of a set of $(m + n)$ |
1051 |
|
|
number of equations. Where $m$ is the number of constraints, and $n$ |
1052 |
|
|
is the number of constrained coordinates. In practice, this is not |
1053 |
mmeineke |
1083 |
done, as the matrix inversion necessary to solve the system of |
1054 |
mmeineke |
1071 |
equations would be very time consuming to solve. Additionally, the |
1055 |
|
|
numerical error in the solution of the set of $\lambda$'s would be |
1056 |
|
|
compounded by the error inherent in propagating by the Velocity Verlet |
1057 |
mmeineke |
1083 |
algorithm ($\Delta t^4$). The Verlet propagation error is negligible |
1058 |
|
|
in an unconstrained system, as one is interested in the statistics of |
1059 |
mmeineke |
1071 |
the run, and not that the run be numerically exact to the ``true'' |
1060 |
|
|
integration. This relates back to the ergodic hypothesis that a time |
1061 |
mmeineke |
1083 |
integral of a valid trajectory will still give the correct ensemble |
1062 |
mmeineke |
1071 |
average. However, in the case of constraints, if the equations of |
1063 |
|
|
motion leave the ``true'' trajectory, they are departing from the |
1064 |
|
|
constrained surface. The method that is used, is to iteratively solve |
1065 |
|
|
for $\lambda(t)$ at each time step. |
1066 |
|
|
|
1067 |
|
|
In {\sc rattle} the equations of motion are modified subject to the |
1068 |
|
|
following two constraints: |
1069 |
|
|
\begin{align} |
1070 |
|
|
\sigma_{ij}[\mathbf{r}(t)] \equiv |
1071 |
|
|
[ \mathbf{r}_i(t) - \mathbf{r}_j(t)]^2 - d_{ij}^2 &= 0 % |
1072 |
|
|
\label{oopseEq:c1} \\ |
1073 |
|
|
% |
1074 |
|
|
[\mathbf{\dot{r}}_i(t) - \mathbf{\dot{r}}_j(t)] \cdot |
1075 |
|
|
[\mathbf{r}_i(t) - \mathbf{r}_j(t)] &= 0 \label{oopseEq:c2} |
1076 |
|
|
\end{align} |
1077 |
|
|
Eq.~\ref{oopseEq:c1} is the set of bond constraints, where $d_{ij}$ is |
1078 |
|
|
the constrained distance between atom $i$ and |
1079 |
|
|
$j$. Eq.~\ref{oopseEq:c2} constrains the velocities of $i$ and $j$ to |
1080 |
mmeineke |
1083 |
be perpendicular to the bond vector, so that the bond can neither grow |
1081 |
mmeineke |
1071 |
nor shrink. The constrained dynamics equations become: |
1082 |
|
|
\begin{equation} |
1083 |
|
|
m_i \mathbf{\ddot{r}}_i = \mathbf{F}_i + \mathbf{\mathcal{G}}_i |
1084 |
|
|
\label{oopseEq:r1} |
1085 |
|
|
\end{equation} |
1086 |
mmeineke |
1083 |
Where,$\mathbf{\mathcal{G}}_i$ are the forces of constraint on $i$, |
1087 |
|
|
and are defined: |
1088 |
mmeineke |
1071 |
\begin{equation} |
1089 |
|
|
\mathbf{\mathcal{G}}_i = - \sum_j \lambda_{ij}(t)\,\nabla \sigma_{ij} |
1090 |
|
|
\label{oopseEq:r2} |
1091 |
|
|
\end{equation} |
1092 |
|
|
|
1093 |
|
|
In Velocity Verlet, if $\Delta t = h$, the propagation can be written: |
1094 |
|
|
\begin{align} |
1095 |
|
|
\mathbf{r}_i(t+h) &= |
1096 |
|
|
\mathbf{r}_i(t) + h\mathbf{\dot{r}}(t) + |
1097 |
|
|
\frac{h^2}{2m_i}\,\Bigl[ \mathbf{F}_i(t) + |
1098 |
|
|
\mathbf{\mathcal{G}}_{Ri}(t) \Bigr] \label{oopseEq:vv1} \\ |
1099 |
|
|
% |
1100 |
|
|
\mathbf{\dot{r}}_i(t+h) &= |
1101 |
|
|
\mathbf{\dot{r}}_i(t) + \frac{h}{2m_i} |
1102 |
|
|
\Bigl[ \mathbf{F}_i(t) + \mathbf{\mathcal{G}}_{Ri}(t) + |
1103 |
|
|
\mathbf{F}_i(t+h) + \mathbf{\mathcal{G}}_{Vi}(t+h) \Bigr] % |
1104 |
|
|
\label{oopseEq:vv2} |
1105 |
|
|
\end{align} |
1106 |
mmeineke |
1083 |
Where: |
1107 |
|
|
\begin{align} |
1108 |
|
|
\mathbf{\mathcal{G}}_{Ri}(t) &= |
1109 |
|
|
-2 \sum_j \lambda_{Rij}(t) \mathbf{r}_{ij}(t) \\ |
1110 |
|
|
% |
1111 |
|
|
\mathbf{\mathcal{G}}_{Vi}(t+h) &= |
1112 |
|
|
-2 \sum_j \lambda_{Vij}(t+h) \mathbf{r}(t+h) |
1113 |
|
|
\end{align} |
1114 |
|
|
Next, define: |
1115 |
|
|
\begin{align} |
1116 |
|
|
g_{ij} &= h \lambda_{Rij}(t) \\ |
1117 |
|
|
k_{ij} &= h \lambda_{Vij}(t+h) \\ |
1118 |
|
|
\mathbf{q}_i &= \mathbf{\dot{r}}_i(t) + \frac{h}{2m_i} \mathbf{F}_i(t) |
1119 |
|
|
- \frac{1}{m_i}\sum_j g_{ij}\mathbf{r}_{ij}(t) |
1120 |
|
|
\end{align} |
1121 |
|
|
Using these definitions, Eq.~\ref{oopseEq:vv1} and \ref{oopseEq:vv2} |
1122 |
|
|
can be rewritten as, |
1123 |
|
|
\begin{align} |
1124 |
|
|
\mathbf{r}_i(t+h) &= \mathbf{r}_i(t) + h \mathbf{q}_i \\ |
1125 |
|
|
% |
1126 |
|
|
\mathbf{\dot{r}}(t+h) &= \mathbf{q}_i + \frac{h}{2m_i}\mathbf{F}_i(t+h) |
1127 |
|
|
-\frac{1}{m_i}\sum_j k_{ij} \mathbf{r}_{ij}(t+h) |
1128 |
|
|
\end{align} |
1129 |
mmeineke |
1071 |
|
1130 |
mmeineke |
1083 |
To integrate the equations of motion, the {\sc rattle} algorithm first |
1131 |
|
|
solves for $\mathbf{r}(t+h)$. Let, |
1132 |
|
|
\begin{equation} |
1133 |
|
|
\mathbf{q}_i = \mathbf{\dot{r}}(t) + \frac{h}{2m_i}\mathbf{F}_i(t) |
1134 |
|
|
\end{equation} |
1135 |
|
|
Here $\mathbf{q}_i$ corresponds to an initial unconstrained move. Next |
1136 |
|
|
pick a constraint $j$, and let, |
1137 |
|
|
\begin{equation} |
1138 |
|
|
\mathbf{s} = \mathbf{r}_i(t) + h\mathbf{q}_i(t) |
1139 |
|
|
- \mathbf{r}_j(t) + h\mathbf{q}_j(t) |
1140 |
|
|
\label{oopseEq:ra1} |
1141 |
|
|
\end{equation} |
1142 |
|
|
If |
1143 |
|
|
\begin{equation} |
1144 |
|
|
\Big| |\mathbf{s}|^2 - d_{ij}^2 \Big| > \text{tolerance}, |
1145 |
|
|
\end{equation} |
1146 |
|
|
then the constraint is unsatisfied, and corrections are made to the |
1147 |
|
|
positions. First we define a test corrected configuration as, |
1148 |
|
|
\begin{align} |
1149 |
|
|
\mathbf{r}_i^T(t+h) = \mathbf{r}_i(t) + h\biggl[\mathbf{q}_i - |
1150 |
|
|
g_{ij}\,\frac{\mathbf{r}_{ij}(t)}{m_i} \biggr] \\ |
1151 |
|
|
% |
1152 |
|
|
\mathbf{r}_j^T(t+h) = \mathbf{r}_j(t) + h\biggl[\mathbf{q}_j + |
1153 |
|
|
g_{ij}\,\frac{\mathbf{r}_{ij}(t)}{m_j} \biggr] |
1154 |
|
|
\end{align} |
1155 |
|
|
And we chose $g_{ij}$ such that, $|\mathbf{r}_i^T - \mathbf{r}_j^T|^2 |
1156 |
|
|
= d_{ij}^2$. Solving the quadratic for $g_{ij}$ we obtain the |
1157 |
|
|
approximation, |
1158 |
|
|
\begin{equation} |
1159 |
|
|
g_{ij} = \frac{(s^2 - d^2)}{2h[\mathbf{s}\cdot\mathbf{r}_{ij}(t)] |
1160 |
|
|
(\frac{1}{m_i} + \frac{1}{m_j})} |
1161 |
|
|
\end{equation} |
1162 |
|
|
Although not an exact solution for $g_{ij}$, as this is an iterative |
1163 |
|
|
scheme overall, the eventual solution will converge. With a trial |
1164 |
|
|
$g_{ij}$, the new $\mathbf{q}$'s become, |
1165 |
|
|
\begin{align} |
1166 |
|
|
\mathbf{q}_i &= \mathbf{q}^{\text{old}}_i - g_{ij}\, |
1167 |
|
|
\frac{\mathbf{r}_{ij}(t)}{m_i} \\ |
1168 |
|
|
% |
1169 |
|
|
\mathbf{q}_j &= \mathbf{q}^{\text{old}}_j + g_{ij}\, |
1170 |
|
|
\frac{\mathbf{r}_{ij}(t)}{m_j} |
1171 |
|
|
\end{align} |
1172 |
|
|
The whole algorithm is then repeated from Eq.~\ref{oopseEq:ra1} until |
1173 |
|
|
all constraints are satisfied. |
1174 |
mmeineke |
1071 |
|
1175 |
mmeineke |
1083 |
The second step of {\sc rattle}, is to then update the velocities. The |
1176 |
|
|
step starts with, |
1177 |
|
|
\begin{equation} |
1178 |
|
|
\mathbf{\dot{r}}_i(t+h) = \mathbf{q}_i + \frac{h}{2m_i}\mathbf{F}_i(t+h) |
1179 |
|
|
\end{equation} |
1180 |
|
|
Next we pick a constraint $j$, and calculate the dot product $\ell$. |
1181 |
|
|
\begin{equation} |
1182 |
|
|
\ell = \mathbf{r}_{ij}(t+h) \cdot \mathbf{\dot{r}}_{ij}(t+h) |
1183 |
|
|
\label{oopseEq:rv1} |
1184 |
|
|
\end{equation} |
1185 |
|
|
Here if constraint Eq.~\ref{oopseEq:c2} holds, $\ell$ should be |
1186 |
|
|
zero. Therefore if $\ell$ is greater than some tolerance, then |
1187 |
|
|
corrections are made to the $i$ and $j$ velocities. |
1188 |
|
|
\begin{align} |
1189 |
|
|
\mathbf{\dot{r}}_i^T &= \mathbf{\dot{r}}_i(t+h) - k_{ij} |
1190 |
|
|
\frac{\mathbf{\dot{r}}_{ij}(t+h)}{m_i} \\ |
1191 |
|
|
% |
1192 |
|
|
\mathbf{\dot{r}}_j^T &= \mathbf{\dot{r}}_j(t+h) + k_{ij} |
1193 |
|
|
\frac{\mathbf{\dot{r}}_{ij}(t+h)}{m_j} |
1194 |
|
|
\end{align} |
1195 |
|
|
Like in the previous step, we select a value for $k_{ij}$ such that |
1196 |
|
|
$\ell$ is zero. |
1197 |
|
|
\begin{equation} |
1198 |
|
|
k_{ij} = \frac{\ell}{d^2_{ij}(\frac{1}{m_i} + \frac{1}{m_j})} |
1199 |
|
|
\end{equation} |
1200 |
|
|
The test velocities, $\mathbf{\dot{r}}^T_i$ and |
1201 |
|
|
$\mathbf{\dot{r}}^T_j$, then replace their respective velocities, and |
1202 |
|
|
the algorithm is iterated from Eq.~\ref{oopseEq:rv1} until all |
1203 |
|
|
constraints are satisfied. |
1204 |
mmeineke |
1071 |
|
1205 |
mmeineke |
1083 |
|
1206 |
mmeineke |
1054 |
\subsection{\label{oopseSec:zcons}Z-Constraint Method} |
1207 |
mmeineke |
1044 |
|
1208 |
mmeineke |
1083 |
Based on the fluctuation-dissipation theorem, a force auto-correlation |
1209 |
|
|
method was developed by Roux and Karplus to investigate the dynamics |
1210 |
|
|
of ions inside ion channels.\cite{Roux91} The time-dependent friction |
1211 |
|
|
coefficient can be calculated from the deviation of the instantaneous |
1212 |
|
|
force from its mean force. |
1213 |
mmeineke |
1044 |
\begin{equation} |
1214 |
|
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
1215 |
|
|
\end{equation} |
1216 |
|
|
where% |
1217 |
|
|
\begin{equation} |
1218 |
|
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
1219 |
|
|
\end{equation} |
1220 |
|
|
|
1221 |
|
|
|
1222 |
mmeineke |
1083 |
If the time-dependent friction decays rapidly, the static friction |
1223 |
|
|
coefficient can be approximated by |
1224 |
mmeineke |
1044 |
\begin{equation} |
1225 |
|
|
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt |
1226 |
|
|
\end{equation} |
1227 |
mmeineke |
1083 |
Therefore, the diffusion constant can then be estimated by |
1228 |
mmeineke |
1044 |
\begin{equation} |
1229 |
|
|
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
1230 |
|
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}% |
1231 |
|
|
\end{equation} |
1232 |
|
|
|
1233 |
mmeineke |
1083 |
The Z-Constraint method, which fixes the z coordinates of the |
1234 |
|
|
molecules with respect to the center of the mass of the system, has |
1235 |
|
|
been a method suggested to obtain the forces required for the force |
1236 |
|
|
auto-correlation calculation.\cite{Marrink94} However, simply resetting the |
1237 |
|
|
coordinate will move the center of the mass of the whole system. To |
1238 |
|
|
avoid this problem, a new method was used in {\sc oopse}. Instead of |
1239 |
|
|
resetting the coordinate, we reset the forces of z-constraint |
1240 |
|
|
molecules as well as subtract the total constraint forces from the |
1241 |
|
|
rest of the system after force calculation at each time step. |
1242 |
mmeineke |
1054 |
\begin{align} |
1243 |
|
|
F_{\alpha i}&=0\\ |
1244 |
|
|
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}\\ |
1245 |
|
|
F_{\alpha i}&=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}\\ |
1246 |
|
|
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}}} |
1247 |
|
|
\end{align} |
1248 |
mmeineke |
1044 |
|
1249 |
mmeineke |
1083 |
At the very beginning of the simulation, the molecules may not be at their |
1250 |
|
|
constrained positions. To move a z-constrained molecule to its specified |
1251 |
|
|
position, a simple harmonic potential is used |
1252 |
mmeineke |
1044 |
\begin{equation} |
1253 |
mmeineke |
1083 |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2}% |
1254 |
mmeineke |
1044 |
\end{equation} |
1255 |
mmeineke |
1083 |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ is the |
1256 |
|
|
current $z$ coordinate of the center of mass of the constrained molecule, and |
1257 |
|
|
$z_{\text{cons}}$ is the constrained position. The harmonic force operating |
1258 |
|
|
on the z-constrained molecule at time $t$ can be calculated by |
1259 |
mmeineke |
1044 |
\begin{equation} |
1260 |
mmeineke |
1083 |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
1261 |
|
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}) |
1262 |
mmeineke |
1044 |
\end{equation} |
1263 |
|
|
|
1264 |
mmeineke |
1051 |
\section{\label{oopseSec:props}Trajectory Analysis} |
1265 |
mmeineke |
1044 |
|
1266 |
mmeineke |
1051 |
\subsection{\label{oopseSec:staticProps}Static Property Analysis} |
1267 |
mmeineke |
1044 |
|
1268 |
|
|
The static properties of the trajectories are analyzed with the |
1269 |
mmeineke |
1083 |
program \texttt{staticProps}. The code is capable of calculating a |
1270 |
|
|
number of pair correlations between species A and B. Some of which |
1271 |
|
|
only apply to directional entities. The summary of pair correlations |
1272 |
|
|
can be found in Table~\ref{oopseTb:gofrs} |
1273 |
mmeineke |
1044 |
|
1274 |
mmeineke |
1083 |
\begin{table} |
1275 |
|
|
\caption[The list of pair correlations in \texttt{staticProps}]{The different pair correlations in \texttt{staticProps} along with whether atom A or B must be directional.} |
1276 |
|
|
\label{oopseTb:gofrs} |
1277 |
|
|
\begin{center} |
1278 |
|
|
\begin{tabular}{|l|c|c|} |
1279 |
|
|
\hline |
1280 |
|
|
Name & Equation & Directional Atom \\ \hline |
1281 |
|
|
$g_{\text{AB}}(r)$ & Eq.~\ref{eq:gofr} & neither \\ \hline |
1282 |
|
|
$g_{\text{AB}}(r, \cos \theta)$ & Eq.~\ref{eq:gofrCosTheta} & A \\ \hline |
1283 |
|
|
$g_{\text{AB}}(r, \cos \omega)$ & Eq.~\ref{eq:gofrCosOmega} & both \\ \hline |
1284 |
|
|
$g_{\text{AB}}(x, y, z)$ & Eq.~\ref{eq:gofrXYZ} & neither \\ \hline |
1285 |
|
|
$\langle \cos \omega \rangle_{\text{AB}}(r)$ & Eq.~\ref{eq:cosOmegaOfR} &% |
1286 |
|
|
both \\ \hline |
1287 |
|
|
\end{tabular} |
1288 |
|
|
\end{center} |
1289 |
|
|
\end{table} |
1290 |
|
|
|
1291 |
mmeineke |
1044 |
The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
1292 |
|
|
\begin{equation} |
1293 |
|
|
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %% |
1294 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} %% |
1295 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr} |
1296 |
|
|
\end{equation} |
1297 |
|
|
Where $\mathbf{r}_{ij}$ is the vector |
1298 |
|
|
\begin{equation*} |
1299 |
|
|
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag |
1300 |
|
|
\end{equation*} |
1301 |
|
|
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over |
1302 |
|
|
the expected pair density at a given $r$. |
1303 |
|
|
|
1304 |
|
|
The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and |
1305 |
|
|
$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two |
1306 |
|
|
dimensional histograms. Both use $r$ for the primary axis then a |
1307 |
|
|
$\cos$ for the secondary axis ($\cos \theta$ for |
1308 |
|
|
Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for |
1309 |
|
|
Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to |
1310 |
|
|
correlate alignment on directional entities. $g_{\text{AB}}(r, \cos |
1311 |
|
|
\theta)$ is defined as follows: |
1312 |
|
|
\begin{equation} |
1313 |
|
|
g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
1314 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1315 |
|
|
\delta( \cos \theta - \cos \theta_{ij}) |
1316 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle |
1317 |
|
|
\label{eq:gofrCosTheta} |
1318 |
|
|
\end{equation} |
1319 |
|
|
Where |
1320 |
|
|
\begin{equation*} |
1321 |
|
|
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
1322 |
|
|
\end{equation*} |
1323 |
|
|
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
1324 |
|
|
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector |
1325 |
|
|
$\mathbf{r}_{ij}$. |
1326 |
|
|
|
1327 |
|
|
The second two dimensional histogram is of the form: |
1328 |
|
|
\begin{equation} |
1329 |
|
|
g_{\text{AB}}(r, \cos \omega) = |
1330 |
|
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
1331 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1332 |
|
|
\delta( \cos \omega - \cos \omega_{ij}) |
1333 |
|
|
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
1334 |
|
|
\end{equation} |
1335 |
|
|
Here |
1336 |
|
|
\begin{equation*} |
1337 |
|
|
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
1338 |
|
|
\end{equation*} |
1339 |
|
|
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit |
1340 |
|
|
directional vectors of species $i$ and $j$. |
1341 |
|
|
|
1342 |
|
|
The static analysis code is also cable of calculating a three |
1343 |
|
|
dimensional pair correlation of the form: |
1344 |
|
|
\begin{equation}\label{eq:gofrXYZ} |
1345 |
|
|
g_{\text{AB}}(x, y, z) = |
1346 |
|
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
1347 |
|
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1348 |
|
|
\delta( x - x_{ij}) |
1349 |
|
|
\delta( y - y_{ij}) |
1350 |
|
|
\delta( z - z_{ij}) \rangle |
1351 |
|
|
\end{equation} |
1352 |
|
|
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
1353 |
|
|
components respectively of vector $\mathbf{r}_{ij}$. |
1354 |
|
|
|
1355 |
|
|
The final pair correlation is similar to |
1356 |
|
|
Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega |
1357 |
|
|
\rangle_{\text{AB}}(r)$ is calculated in the following way: |
1358 |
|
|
\begin{equation}\label{eq:cosOmegaOfR} |
1359 |
|
|
\langle \cos \omega \rangle_{\text{AB}}(r) = |
1360 |
|
|
\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}} |
1361 |
|
|
(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle |
1362 |
|
|
\end{equation} |
1363 |
|
|
Here $\cos \omega_{ij}$ is defined in the same way as in |
1364 |
|
|
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair |
1365 |
|
|
correlation that gives the average correlation of two directional |
1366 |
|
|
entities as a function of their distance from each other. |
1367 |
|
|
|
1368 |
|
|
\subsection{\label{dynamicProps}Dynamic Property Analysis} |
1369 |
|
|
|
1370 |
|
|
The dynamic properties of a trajectory are calculated with the program |
1371 |
mmeineke |
1083 |
\texttt{dynamicProps}. The program calculates the following properties: |
1372 |
mmeineke |
1044 |
\begin{gather} |
1373 |
|
|
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\ |
1374 |
|
|
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\ |
1375 |
|
|
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr} |
1376 |
|
|
\end{gather} |
1377 |
|
|
|
1378 |
mmeineke |
1083 |
Eq.~\ref{eq:rms} is the root mean square displacement function. Which |
1379 |
|
|
allows one to observe the average displacement of an atom as a |
1380 |
|
|
function of time. The quantity is useful when calculating diffusion |
1381 |
|
|
coefficients because of the Einstein Relation, which is valid at long |
1382 |
|
|
times.\cite{allen87:csl} |
1383 |
|
|
\begin{equation} |
1384 |
|
|
2tD = \langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle |
1385 |
|
|
\label{oopseEq:einstein} |
1386 |
|
|
\end{equation} |
1387 |
|
|
|
1388 |
|
|
Eq.~\ref{eq:velCorr} and \ref{eq:angularVelCorr} are the translational |
1389 |
mmeineke |
1044 |
velocity and angular velocity correlation functions respectively. The |
1390 |
mmeineke |
1083 |
latter is only applicable to directional species in the |
1391 |
|
|
simulation. The velocity autocorrelation functions are useful when |
1392 |
|
|
determining vibrational information about the system of interest. |
1393 |
mmeineke |
1044 |
|
1394 |
mmeineke |
1051 |
\section{\label{oopseSec:design}Program Design} |
1395 |
mmeineke |
1044 |
|
1396 |
mmeineke |
1054 |
\subsection{\label{sec:architecture} {\sc oopse} Architecture} |
1397 |
mmeineke |
1044 |
|
1398 |
mmeineke |
1054 |
The core of OOPSE is divided into two main object libraries: |
1399 |
|
|
\texttt{libBASS} and \texttt{libmdtools}. \texttt{libBASS} is the |
1400 |
|
|
library developed around the parsing engine and \texttt{libmdtools} |
1401 |
|
|
is the software library developed around the simulation engine. These |
1402 |
|
|
two libraries are designed to encompass all the basic functions and |
1403 |
|
|
tools that {\sc oopse} provides. Utility programs, such as the |
1404 |
|
|
property analyzers, need only link against the software libraries to |
1405 |
|
|
gain access to parsing, force evaluation, and input / output |
1406 |
|
|
routines. |
1407 |
mmeineke |
1044 |
|
1408 |
mmeineke |
1054 |
Contained in \texttt{libBASS} are all the routines associated with |
1409 |
|
|
reading and parsing the \texttt{.bass} input files. Given a |
1410 |
|
|
\texttt{.bass} file, \texttt{libBASS} will open it and any associated |
1411 |
|
|
\texttt{.mdl} files; then create structures in memory that are |
1412 |
|
|
templates of all the molecules specified in the input files. In |
1413 |
|
|
addition, any simulation parameters set in the \texttt{.bass} file |
1414 |
|
|
will be placed in a structure for later query by the controlling |
1415 |
|
|
program. |
1416 |
mmeineke |
1044 |
|
1417 |
mmeineke |
1054 |
Located in \texttt{libmdtools} are all other routines necessary to a |
1418 |
|
|
Molecular Dynamics simulation. The library uses the main data |
1419 |
|
|
structures returned by \texttt{libBASS} to initialize the various |
1420 |
|
|
parts of the simulation: the atom structures and positions, the force |
1421 |
|
|
field, the integrator, \emph{et cetera}. After initialization, the |
1422 |
|
|
library can be used to perform a variety of tasks: integrate a |
1423 |
|
|
Molecular Dynamics trajectory, query phase space information from a |
1424 |
|
|
specific frame of a completed trajectory, or even recalculate force or |
1425 |
|
|
energetic information about specific frames from a completed |
1426 |
|
|
trajectory. |
1427 |
mmeineke |
1044 |
|
1428 |
mmeineke |
1054 |
With these core libraries in place, several programs have been |
1429 |
|
|
developed to utilize the routines provided by \texttt{libBASS} and |
1430 |
|
|
\texttt{libmdtools}. The main program of the package is \texttt{oopse} |
1431 |
|
|
and the corresponding parallel version \texttt{oopse\_MPI}. These two |
1432 |
mmeineke |
1083 |
programs will take the \texttt{.bass} file, and create (and integrate) |
1433 |
mmeineke |
1054 |
the simulation specified in the script. The two analysis programs |
1434 |
|
|
\texttt{staticProps} and \texttt{dynamicProps} utilize the core |
1435 |
|
|
libraries to initialize and read in trajectories from previously |
1436 |
|
|
completed simulations, in addition to the ability to use functionality |
1437 |
|
|
from \texttt{libmdtools} to recalculate forces and energies at key |
1438 |
|
|
frames in the trajectories. Lastly, the family of system building |
1439 |
|
|
programs (Sec.~\ref{oopseSec:initCoords}) also use the libraries to |
1440 |
|
|
store and output the system configurations they create. |
1441 |
|
|
|
1442 |
|
|
\subsection{\label{oopseSec:parallelization} Parallelization of {\sc oopse}} |
1443 |
|
|
|
1444 |
mmeineke |
1083 |
Although processor power is continually growing roughly following |
1445 |
|
|
Moore's Law, it is still unreasonable to simulate systems of more then |
1446 |
|
|
a 1000 atoms on a single processor. To facilitate study of larger |
1447 |
|
|
system sizes or smaller systems on long time scales in a reasonable |
1448 |
|
|
period of time, parallel methods were developed allowing multiple |
1449 |
|
|
CPU's to share the simulation workload. Three general categories of |
1450 |
|
|
parallel decomposition methods have been developed including atomic, |
1451 |
|
|
spatial and force decomposition methods. |
1452 |
mmeineke |
1054 |
|
1453 |
mmeineke |
1083 |
Algorithmically simplest of the three methods is atomic decomposition |
1454 |
mmeineke |
1044 |
where N particles in a simulation are split among P processors for the |
1455 |
|
|
duration of the simulation. Computational cost scales as an optimal |
1456 |
|
|
$O(N/P)$ for atomic decomposition. Unfortunately all processors must |
1457 |
mmeineke |
1083 |
communicate positions and forces with all other processors at every |
1458 |
|
|
force evaluation, leading communication costs to scale as an |
1459 |
|
|
unfavorable $O(N)$, \emph{independent of the number of processors}. This |
1460 |
|
|
communication bottleneck led to the development of spatial and force |
1461 |
|
|
decomposition methods in which communication among processors scales |
1462 |
|
|
much more favorably. Spatial or domain decomposition divides the |
1463 |
|
|
physical spatial domain into 3D boxes in which each processor is |
1464 |
|
|
responsible for calculation of forces and positions of particles |
1465 |
|
|
located in its box. Particles are reassigned to different processors |
1466 |
|
|
as they move through simulation space. To calculate forces on a given |
1467 |
|
|
particle, a processor must know the positions of particles within some |
1468 |
|
|
cutoff radius located on nearby processors instead of the positions of |
1469 |
|
|
particles on all processors. Both communication between processors and |
1470 |
|
|
computation scale as $O(N/P)$ in the spatial method. However, spatial |
1471 |
mmeineke |
1044 |
decomposition adds algorithmic complexity to the simulation code and |
1472 |
|
|
is not very efficient for small N since the overall communication |
1473 |
|
|
scales as the surface to volume ratio $(N/P)^{2/3}$ in three |
1474 |
|
|
dimensions. |
1475 |
|
|
|
1476 |
mmeineke |
1083 |
The parallelization method used in {\sc oopse} is the force |
1477 |
|
|
decomposition method. Force decomposition assigns particles to |
1478 |
|
|
processors based on a block decomposition of the force |
1479 |
|
|
matrix. Processors are split into an optimally square grid forming row |
1480 |
|
|
and column processor groups. Forces are calculated on particles in a |
1481 |
|
|
given row by particles located in that processors column |
1482 |
|
|
assignment. Force decomposition is less complex to implement than the |
1483 |
|
|
spatial method but still scales computationally as $O(N/P)$ and scales |
1484 |
|
|
as $O(N/\sqrt{P})$ in communication cost. Plimpton has also found that |
1485 |
|
|
force decompositions scale more favorably than spatial decompositions |
1486 |
|
|
for systems up to 10,000 atoms and favorably compete with spatial |
1487 |
|
|
methods up to 100,000 atoms.\cite{plimpton95} |
1488 |
mmeineke |
1044 |
|
1489 |
mmeineke |
1054 |
\subsection{\label{oopseSec:memAlloc}Memory Issues in Trajectory Analysis} |
1490 |
mmeineke |
1044 |
|
1491 |
mmeineke |
1054 |
For large simulations, the trajectory files can sometimes reach sizes |
1492 |
|
|
in excess of several gigabytes. In order to effectively analyze that |
1493 |
mmeineke |
1083 |
amount of data, two memory management schemes have been devised for |
1494 |
mmeineke |
1054 |
\texttt{staticProps} and for \texttt{dynamicProps}. The first scheme, |
1495 |
|
|
developed for \texttt{staticProps}, is the simplest. As each frame's |
1496 |
|
|
statistics are calculated independent of each other, memory is |
1497 |
|
|
allocated for each frame, then freed once correlation calculations are |
1498 |
|
|
complete for the snapshot. To prevent multiple passes through a |
1499 |
|
|
potentially large file, \texttt{staticProps} is capable of calculating |
1500 |
|
|
all requested correlations per frame with only a single pair loop in |
1501 |
mmeineke |
1083 |
each frame and a single read of the file. |
1502 |
mmeineke |
1044 |
|
1503 |
mmeineke |
1054 |
The second, more advanced memory scheme, is used by |
1504 |
|
|
\texttt{dynamicProps}. Here, the program must have multiple frames in |
1505 |
|
|
memory to calculate time dependent correlations. In order to prevent a |
1506 |
|
|
situation where the program runs out of memory due to large |
1507 |
|
|
trajectories, the user is able to specify that the trajectory be read |
1508 |
|
|
in blocks. The number of frames in each block is specified by the |
1509 |
|
|
user, and upon reading a block of the trajectory, |
1510 |
|
|
\texttt{dynamicProps} will calculate all of the time correlation frame |
1511 |
mmeineke |
1083 |
pairs within the block. After in-block correlations are complete, a |
1512 |
mmeineke |
1054 |
second block of the trajectory is read, and the cross correlations are |
1513 |
|
|
calculated between the two blocks. this second block is then freed and |
1514 |
|
|
then incremented and the process repeated until the end of the |
1515 |
|
|
trajectory. Once the end is reached, the first block is freed then |
1516 |
|
|
incremented, and the again the internal time correlations are |
1517 |
|
|
calculated. The algorithm with the second block is then repeated with |
1518 |
|
|
the new origin block, until all frame pairs have been correlated in |
1519 |
|
|
time. This process is illustrated in |
1520 |
|
|
Fig.~\ref{oopseFig:dynamicPropsMemory}. |
1521 |
mmeineke |
1044 |
|
1522 |
mmeineke |
1054 |
\begin{figure} |
1523 |
|
|
\centering |
1524 |
|
|
\includegraphics[width=\linewidth]{dynamicPropsMem.eps} |
1525 |
|
|
\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.} |
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\label{oopseFig:dynamicPropsMemory} |
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\end{figure} |
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|
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\section{\label{oopseSec:conclusion}Conclusion} |
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|
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We have presented the design and implementation of our open source |
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simulation package {\sc oopse}. The package offers novel capabilities |
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|
|
to the field of Molecular Dynamics simulation packages in the form of |
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|
dipolar force fields, and symplectic integration of rigid body |
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|
|
dynamics. It is capable of scaling across multiple processors through |
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|
the use of force based decomposition using MPI. It also implements |
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|
several advanced integrators allowing the end user control over |
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|
temperature and pressure. In addition, it is capable of integrating |
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|
constrained dynamics through both the {\sc rattle} algorithm and the |
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|
z-constraint method. |
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|
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These features are all brought together in a single open-source |
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program. Allowing researchers to not only benefit from |
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{\sc oopse}, but also contribute to {\sc oopse}'s development as |
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|
well.Documentation and source code for {\sc oopse} can be downloaded |
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|
from \texttt{http://www.openscience.org/oopse/}. |
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|