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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\subsection{\label{sec:atomsMolecules}Atoms and Molecules} |
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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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are not currently suporrted by {\sc oopse}. |
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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 atoms |
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in a simulation in logical manner. This particular unit will store the |
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identities of all the atoms associated with itself and is responsible |
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for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
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and torsions). |
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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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store the identities of all the atoms associated with itself and is |
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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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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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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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\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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\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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The application 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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trigonometric operations involving $\phi, \theta,$ and |
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$\psi$.\cite{Goldstein01} In order to avoid rotational limitations |
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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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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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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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The most basic force field implemented in OOPSE is the Lennard-Jones |
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potential. The Lennard-Jones potential mimics the attractive forces |
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two charge neutral particles experience when spontaneous dipoles are |
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induced on each other. This is the predominant intermolecular force in |
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systems of such as noble gases and simple alkanes. |
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The Lennard-Jones potential is given by: |
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\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 particle $i$ and $j$, $\sigma_{ij}$ |
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scales the length of the interaction, and $\epsilon_{ij}$ scales the |
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energy well depth of the potential. |
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Because the potential is calculated between all pairs, the force |
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evaluation can become computationally expensive for large systems. To |
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keep the pair evaluation to a manegable number, OOPSE employs the use |
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of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
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$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length |
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parameter in the system. Truncating the calculation at |
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$r_{\text{cut}}$ introduces a discontinuity into the potential |
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energy. To offset this discontinuity, the energy value at |
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$r_{\text{cut}}$ is subtracted from the entire potential. This causes |
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the equation to go to zero at the cut-off radius. |
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Interactions between dissimilar particles requires the generation of |
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cross term parameters for $\sigma$ and $\epsilon$. These are |
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calculated through the Lorentz-Berthelot mixing |
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rules:\cite{allen87:csl} |
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\begin{equation} |
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\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}] |
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\label{eq:sigmaMix} |
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\end{equation} |
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and |
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\begin{equation} |
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\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}} |
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\label{eq:epsilonMix} |
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\end{equation} |
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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The \underline{D}ipolar \underline{U}nified-Atom |
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atom in Fig.~\ref{fig:lipidModel}. |
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\begin{figure} |
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\epsfxsize=6in |
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\epsfbox{lipidModel.epsi} |
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\epsfbox{lipidModel.eps} |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
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is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
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\label{fig:lipidModel} |
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By recasting the equation to a power series, repeated trigonometric |
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evaluations are avoided during the calculation of the potential. |
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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 atoms $i$ and $j$, $\sigma_{ij}$ |
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scales the length of the interaction, and $\epsilon_{ij}$ scales the |
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energy of the potential. |
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The dipole-dipole potential has the following form: |
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\begin{equation} |
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V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
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\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
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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 native solvent used |
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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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solvation shell. Thus, the interaction between two SSD water molecules |
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\emph{i} and \emph{j} is given by the potential |
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\begin{equation} |
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u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
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V_{ij} = |
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V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
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V_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
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\label{eq:ssdPot} |
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\end{equation} |
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where the $\mathbf{r}_{ij}$ is the position vector between molecules |
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\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
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\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
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$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
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orientations of the respective molecules. The Lennard-Jones, dipole, |
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and sticky parts of the potential are giving by the following |
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equations, |
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orientations of the respective molecules. The Lennard-Jones and dipole |
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parts of the potential are given by equations \ref{eq:lennardJonesPot} |
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and \ref{eq:dipolePot} respectively. The sticky part is described by |
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the following, |
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\begin{equation} |
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u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
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u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
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\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
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s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
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\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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\label{eq:stickyPot} |
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\end{equation} |
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where $\nu_0$ is a strength parameter for the sticky potential, and |
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$s$ and $s^\prime$ are cubic switching functions which turn off the |
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sticky interaction beyond the first solvation shell. The $w$ function |
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can be thought of as an attractive potential with tetrahedral |
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geometry: |
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\begin{equation} |
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u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ , |
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w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
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\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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\label{eq:stickyW} |
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\end{equation} |
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while the $w^\prime$ function counters the normal aligned and |
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anti-aligned structures favored by point dipoles: |
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\begin{equation} |
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\begin{split} |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
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&= |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
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& \quad \ + |
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s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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\end{split} |
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w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
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(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
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\label{eq:stickyWprime} |
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\end{equation} |
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where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
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unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
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$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
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$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
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functions take the following forms, |
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\begin{equation} |
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w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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\end{equation} |
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\begin{equation} |
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w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
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\end{equation} |
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where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
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term that promotes hydrogen bonding orientations within the first |
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solvation shell, and $w^\prime$ is a dipolar repulsion term that |
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repels unrealistic dipolar arrangements within the first solvation |
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shell. A more detailed description of the functional parts and |
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variables in this potential can be found in other |
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articles.\cite{liu96:new_model,chandra99:ssd_md} |
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It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
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and $Y_3^{-2}$ spherical harmonics (a linear combination which |
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enhances the tetrahedral geometry for hydrogen bonded structures), |
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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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Since SSD is a one-site point dipole model, the force calculations are |
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simplified significantly from a computational standpoint, in that the |
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number of long-range interactions is dramatically reduced. In the |
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original Monte Carlo simulations using this model, Ichiye \emph{et |
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al.} reported a calculation speed up of up to an order of magnitude |
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over other comparable models while maintaining the structural behavior |
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of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
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SSD, it was shown that it actually improves upon the prediction of |
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water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
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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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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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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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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} Reparameterizations of the |
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original SSD have resulted in improved density behavior, as well as |
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alterations in the water structure and transport behavior. {\sc oopse} is |
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easily modified to impliment these new potential parameter sets for |
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the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
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variable parameters are listed in the accompanying BASS file, and |
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these parameters simply need to be changed to the updated values. |
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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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|
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!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
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\subsection{\label{sec:eam}Embedded Atom Model} |
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|
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here there be Monsters |
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Several molecular dynamics codes\cite{dynamo86} exist which have the |
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capacity to simulate metallic systems, including some that have |
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parallel computational abilities\cite{plimpton93}. Potentials that |
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describe bonding transition metal |
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systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
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attractive interaction which models the stabilization of ``Embedding'' |
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a positively charged metal ion in an electron density created by the |
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free valance ``sea'' of electrons created by the surrounding atoms in |
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the system. A mostly repulsive pairwise part of the potential |
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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}(EAM) that has |
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particularly wide adoption has been selected for inclusion in OOPSE. A |
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good review of EAM and other metallic potential formulations was done |
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by Voter.\cite{voter} |
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|
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The {\sc eam} potential has the form: |
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\begin{eqnarray} |
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V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
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\phi_{ij}({\bf r}_{ij}) \\ |
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\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
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\end{eqnarray} |
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|
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where $\phi_{ij}$ is a primarily repulsive pairwise interaction |
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between atoms $i$ and $j$.In the origional formulation of |
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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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and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The |
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embedding function $F_{i}$ is the energy required to embedded an |
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positively-charged core ion $i$ into a linear supeposition of |
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sperically averaged atomic electron densities given by |
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$\rho_{i}$. 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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surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
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interactions. |
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|
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\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
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|
| 410 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 411 |
> |
macroscopic systems with a relatively small number of particles. Simulation |
| 412 |
> |
box is replicated throughout space to form an infinite lattice. During the |
| 413 |
> |
simulation, when a particle moves in the primary cell, its periodic image |
| 414 |
> |
particles in other boxes move in exactly the same direction with exactly the |
| 415 |
> |
same orientation.Thus, as a particle leaves the primary cell, one of its |
| 416 |
> |
images will enter through the opposite face.If the simulation box is large |
| 417 |
> |
enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
| 418 |
> |
effect could be ignored. Cubic and parallelepiped are the available periodic |
| 419 |
> |
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
| 420 |
> |
the property of the simulation box. Therefore, not only the size of the |
| 421 |
> |
simulation box could be changed during the simulation, but also the shape of |
| 422 |
> |
it. The transformation from box space vector $\overrightarrow{s}$ to its |
| 423 |
> |
corresponding real space vector $\overrightarrow{r}$ is defined by |
| 424 |
> |
\begin{equation} |
| 425 |
> |
\overrightarrow{r}=H\overrightarrow{s}% |
| 426 |
> |
\end{equation} |
| 427 |
> |
|
| 428 |
> |
|
| 429 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 430 |
> |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
| 431 |
> |
simulation box respectively. |
| 432 |
> |
|
| 433 |
> |
To find the minimum image, we need to convert the real vector to its |
| 434 |
> |
corresponding vector in box space first, \bigskip% |
| 435 |
> |
\begin{equation} |
| 436 |
> |
\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
| 437 |
> |
\end{equation} |
| 438 |
> |
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
| 439 |
> |
to 0.5, |
| 440 |
> |
\begin{equation} |
| 441 |
> |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 442 |
> |
\end{equation} |
| 443 |
> |
where% |
| 444 |
> |
|
| 445 |
> |
\begin{equation} |
| 446 |
> |
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
| 447 |
> |
}x\geqslant0 |
| 448 |
> |
\end{equation} |
| 449 |
> |
% |
| 450 |
> |
|
| 451 |
> |
\begin{equation} |
| 452 |
> |
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
| 453 |
> |
\end{equation} |
| 454 |
> |
|
| 455 |
> |
|
| 456 |
> |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 457 |
> |
|
| 458 |
> |
Finally, we could get the minimum image by transforming back to real space,% |
| 459 |
> |
|
| 460 |
> |
\begin{equation} |
| 461 |
> |
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
| 462 |
> |
\end{equation} |
