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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\section{\label{sec:empiricalEnergy}The Empirical Energy Functions} |
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\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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The basic unit of an {\sc oopse} simulation is the atom. The parameters |
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describing the atom are generalized to make the atom as flexible a |
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representation as possible. They may represent specific atoms of an |
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element, or be used for collections of atoms such as a methyl |
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group. The atoms are also capable of having a directional component |
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associated with them, often in the form of a dipole. Charges on atoms |
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are not currently suported by {\sc oopse}. |
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The basic unit of an {\sc oopse} simulation is the atom. The |
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parameters describing the atom are generalized to make the atom as |
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flexible a representation as possible. They may represent specific |
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atoms of an element, or be used for collections of atoms such as |
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methyl and carbonyl groups. The atoms are also capable of having |
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directional components associated with them (\emph{e.g.}~permanent |
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dipoles). Charges on atoms are not currently supported by {\sc oopse}. |
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\begin{lstlisting}[caption={[Specifier for molecules and atoms] An example specifying the simple Ar molecule},label=sch:AtmMole] |
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molecule{ |
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name = "Ar"; |
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nAtoms = 1; |
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atom[0]{ |
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type="Ar"; |
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position( 0.0, 0.0, 0.0 ); |
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} |
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} |
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\end{lstlisting} |
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The second most basic building block of a simulation is the |
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molecule. The molecule is a way for {\sc oopse} to keep track of the |
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atoms in a simulation in logical manner. This particular unit will |
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responsible for the evaluation of its own bonded interaction |
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(i.e.~bonds, bends, and torsions). |
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As stated previously, one of the features that sets {\sc OOPSE} apart |
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As stated previously, one of the features that sets {\sc oopse} apart |
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from most of the current molecular simulation packages is the ability |
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to handle rigid body dynamics. Rigid bodies are non-spherical |
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particles or collections of particles that have a constant internal |
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potential and move collectively.\cite{Goldstein01} They are not |
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included in most simulation packages because of the need to |
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consider orientational degrees of freedom and include an integrator |
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that accurately propagates these motions in time. |
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included in most simulation packages because of the requirement to |
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propagate the orientational degrees of freedom. Until recently, |
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integrators which propagate orientational motion have been lacking. |
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Moving a rigid body involves determination of both the force and |
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torque applied by the surroundings, which directly affect the |
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translation and rotation in turn. In order to accumulate the total |
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force on a rigid body, the external forces must first be calculated |
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for all the internal particles. The total force on the rigid body is |
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simply the sum of these external forces. Accumulation of the total |
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torque on the rigid body is more complex than the force in that it is |
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the torque applied on the center of mass that dictates rotational |
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motion. The summation of this torque is given by |
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translational and rotational motion in turn. In order to accumulate |
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the total force on a rigid body, the external forces and torques must |
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first be calculated for all the internal particles. The total force on |
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the rigid body is simply the sum of these external forces. |
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Accumulation of the total torque on the rigid body is more complex |
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than the force in that it is the torque applied on the center of mass |
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that dictates rotational motion. The torque on rigid body {\it i} is |
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\begin{equation} |
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\mathbf{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia}, |
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\boldsymbol{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}, |
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\label{eq:torqueAccumulate} |
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\end{equation} |
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where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$ |
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and $\mathbf{r}_{ia}$ are the force on and position of the component |
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particles of the rigid body.\cite{allen87:csl} |
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where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$, |
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$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on, |
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position of, and torque on the component particles of the rigid body. |
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The application of the total torque is done in the body fixed axis of |
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The summation of the total torque is done in the body fixed axis of |
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the rigid body. In order to move between the space fixed and body |
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fixed coordinate axes, parameters describing the orientation must be |
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maintained for each rigid body. At a minimum, the rotation matrix |
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(\textbf{A}) can be described and propagated by the three Euler angles |
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($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of |
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(\textbf{A}) can be described by the three Euler angles ($\phi, |
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\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
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trigonometric operations involving $\phi, \theta,$ and |
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$\psi$.\cite{Goldstein01} In order to avoid rotational limitations |
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$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
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inherent in using the Euler angles, the four parameter ``quaternion'' |
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scheme can be used instead, where \textbf{A} is composed of arithmetic |
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operations involving the four components of a quaternion ($q_0, q_1, |
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q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads |
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to performance enhancements, particularly for very small |
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scheme is often used. The elements of \textbf{A} can be expressed as |
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arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
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and $q_3$).\cite{allen87:csl} Use of quaternions also leads to |
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performance enhancements, particularly for very small |
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systems.\cite{Evans77} |
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{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine |
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parameter rotation matrix internally. Further discussion on this |
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choice can be found in Sec.~\ref{sec:integrate}. |
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{\sc oopse} utilizes a relatively new scheme that propagates the |
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entire nine parameter rotation matrix internally. Further discussion |
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on this choice can be found in Sec.~\ref{sec:integrate}. |
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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The most basic force field implemented in OOPSE is the Lennard-Jones |
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The most basic force field implemented in {\sc oopse} is the Lennard-Jones |
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potential. The Lennard-Jones potential. Which mimics the Van der Waals |
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interaction at long distances, and uses an emperical repulsion at |
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interaction at long distances, and uses an empirical repulsion at |
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short distances. The Lennard-Jones potential is given by: |
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\begin{equation} |
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V_{\text{LJ}}(r_{ij}) = |
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Because this potential is calculated between all pairs, the force |
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evaluation can become computationally expensive for large systems. To |
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keep the pair evaluation to a manegable number, OOPSE employs a |
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keep the pair evaluation to a manageable number, {\sc oopse} employs a |
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cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
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$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones length |
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parameter in the system. Truncating the calculation at |
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phosphatidylcholine.\cite{Cevc87} Additionally, a Lennard-Jones site |
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is located at the pseudoatom's center of mass. The model is |
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illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The water model we use to complement the dipoles of the lipids is out |
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repaarameterization of the soft sticky dipole (SSD) model of Ichiye |
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reparameterization of the soft sticky dipole (SSD) model of Ichiye |
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\emph{et al.}\cite{liu96:new_model} |
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|
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\begin{figure} |
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\end{equation} |
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Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
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$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For |
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computaional efficency, the torsion potential has been recast after |
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computational efficiency, the torsion potential has been recast after |
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the method of CHARMM\cite{charmm1983} whereby the angle series is |
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converted to a power series of the form: |
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\begin{equation} |
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$\boldsymbol{\Omega}_i$. |
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\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
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\subsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
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In the interest of computational efficiency, the default solvent used |
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in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
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developed by Ichiye \emph{et al.} as a modified form of the |
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hard-sphere water model proposed by Bratko, Blum, and |
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by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
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model.\cite{Gezelter04} The original SSD was developed by Ichiye |
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\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
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water model proposed by Bratko, Blum, and |
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Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
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with a Lennard-Jones core and a sticky potential that directs the |
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particles to assume the proper hydrogen bond orientation in the first |
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while $w^\prime$ is a purely empirical function. A more detailed |
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description of the functional parts and variables in this potential |
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can be found in the original SSD |
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articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
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articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
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Since SSD is a single-point {\it dipolar} model, the force |
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calculations are simplified significantly relative to the standard |
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{\it charged} multi-point models. In the original Monte Carlo |
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simulations using this model, Ichiye {\it et al.} reported that using |
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SSD decreased computer time by a factor of 6-7 compared to other |
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models.\cite{Ichiye96} What is most impressive is that this savings |
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models.\cite{liu96:new_model} What is most impressive is that these savings |
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did not come at the expense of accurate depiction of the liquid state |
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properties. Indeed, SSD maintains reasonable agreement with the Soper |
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data for the structural features of liquid |
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water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
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diffraction data for the structural features of liquid |
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water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
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exhibited by SSD agree with experiment better than those of more |
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computationally expensive models (like TIP3P and |
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SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
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SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
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of solvent properties makes SSD a very attractive model for the |
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simulation of large scale biochemical simulations. |
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Recent constant pressure simulations revealed issues in the original |
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SSD model that led to lower than expected densities at all target |
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pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
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is SSD/E, a density corrected derivative of SSD that exhibits improved |
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liquid structure and transport behavior. If the use of a reaction |
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field long-range interaction correction is desired, it is recommended |
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that the parameters be modified to those of the SSD/RF model. Solvent |
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parameters can be easily modified in an accompanying {\sc BASS} file |
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as illustrated in the scheme below. A table of the parameter values |
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and the drawbacks and benefits of the different density corrected SSD |
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models can be found in reference \ref{Gezelter04}. |
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is therefore SSD/E, a density corrected derivative of SSD that |
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exhibits improved liquid structure and transport behavior. If the use |
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of a reaction field long-range interaction correction is desired, it |
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is recommended that the parameters be modified to those of the SSD/RF |
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model. Solvent parameters can be easily modified in an accompanying |
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{\sc BASS} file as illustrated in the scheme below. A table of the |
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parameter values and the drawbacks and benefits of the different |
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density corrected SSD models can be found in reference |
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\ref{Gezelter04}. |
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|
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!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
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|
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describes the interaction of the positively charged metal core ions |
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with one another. A particular potential description called the |
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Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
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particularly wide adoption has been selected for inclusion in OOPSE. A |
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particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
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good review of {\sc eam} and other metallic potential formulations was done |
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by Voter.\cite{voter} |
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|
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|
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where $F_{i} $ is the embedding function that equates the energy required to embed a |
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positively-charged core ion $i$ into a linear superposition of |
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sperically averaged atomic electron densities given by |
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spherically averaged atomic electron densities given by |
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$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
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between atoms $i$ and $j$. In the original formulation of |
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{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
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in later refinements to EAM have shown that nonuniqueness between $F$ |
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in later refinements to EAM have shown that non-uniqueness between $F$ |
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and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
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There is a cutoff distance, $r_{cut}$, which limits the |
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summations in the {\sc eam} equation to the few dozen atoms |
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\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
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\newcommand{\roundme}{\operatorname{round}} |
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\textit{Periodic boundary conditions} are widely used to simulate truly |
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macroscopic systems with a relatively small number of particles. The |
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simulation box is replicated throughout space to form an infinite |
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lattice. During the simulation, when a particle moves in the primary |
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cell, its image in other boxes move in exactly the same direction with |
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exactly the same orientation.Thus, as a particle leaves the primary |
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cell, its images in other boxes move in exactly the same direction with |
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exactly the same orientation. So, as a particle leaves the primary |
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cell, one of its images will enter through the opposite face.If the |
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simulation box is large enough to avoid "feeling" the symmetries of |
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simulation box is large enough to avoid \textquotedblleft feeling\textquotedblright\ the symmetries of |
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the periodic lattice, surface effects can be ignored. Cubic, |
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orthorhombic and parallelepiped are the available periodic cells In |
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OOPSE. We use a matrix to describe the property of the simulation |
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box. Therefore, both the size and shape of the simulation box can be |
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orthorhombic and parallelepiped are the available periodic cells in |
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{\sc oopse}. We use a matrix to describe the property of the simulation |
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box. Both the size and shape of the simulation box can be |
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changed during the simulation. The transformation from box space |
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vector $\mathbf{s}$ to its corresponding real space vector |
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$\mathbf{r}$ is defined by |
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\begin{equation} |
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\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}% |
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\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
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\end{equation} |
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|
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|
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the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the |
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three sides of the simulation box respectively. |
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|
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To find the minimum image, we convert the real vector to its |
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To find the minimum image of a vector $\mathbf{r}$, we convert the real vector to its |
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corresponding vector in box space first, \bigskip% |
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\begin{equation} |
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\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}% |
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\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
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\end{equation} |
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And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
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\begin{equation} |
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s_{i}^{\prime}=s_{i}-round(s_{i}) |
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s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
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\end{equation} |
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where |
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|
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% |
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|
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\begin{equation} |
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round(x)=\left\{ |
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\begin{array}[c]{c}% |
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\roundme(x)=\left\{ |
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\begin{array}{cc} |
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\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
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\lceil{x-0.5}\rceil & \text{otherwise}% |
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\end{array} |
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\right. |
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\end{equation} |
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For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$, |
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$\roundme(-3.1)=-3$. |
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|
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|
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For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, |
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$round(-3.1)=-3$. |
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|
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Finally, we obtain the minimum image coordinates by transforming back |
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Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by transforming back |
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to real space,% |
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|
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\begin{equation} |
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\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
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\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
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\end{equation} |
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|
