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\section{The Emperical Energy Functions} |
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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\subsection{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 OOPSE simulation is the atom. The parameters |
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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 suporrted by OOPSE. |
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are not currently suported 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 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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|
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As stated previously, one of the features that sets {\sc oopse} apart |
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from most of the current molecular simulation packages is the ability |
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to handle rigid body dynamics. Rigid bodies are non-spherical |
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particles or collections of particles that have a constant internal |
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potential and move collectively.\cite{Goldstein01} They are not |
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included in most simulation packages because of the requirement to |
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propagate the orientational degrees of freedom. Until recently, |
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integrators which propagate orientational motion have been lacking. |
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|
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Moving a rigid body involves determination of both the force and |
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torque applied by the surroundings, which directly affect the |
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translational and rotational motion in turn. In order to accumulate |
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the total force on a rigid body, the external forces and torques must |
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first be calculated for all the internal particles. The total force on |
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the rigid body is simply the sum of these external forces. |
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Accumulation of the total torque on the rigid body is more complex |
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than the force in that it is the torque applied on the center of mass |
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that dictates rotational motion. The torque on rigid body {\it i} is |
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\begin{equation} |
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\boldsymbol{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}, |
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\label{eq:torqueAccumulate} |
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\end{equation} |
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where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$, |
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$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on, |
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position of, and torque on the component particles of the rigid body. |
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The summation of the total torque is done in the body fixed axis of |
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the rigid body. In order to move between the space fixed and body |
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fixed coordinate axes, parameters describing the orientation must be |
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maintained for each rigid body. At a minimum, the rotation matrix |
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(\textbf{A}) can be described by the three Euler angles ($\phi, |
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\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
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trigonometric operations involving $\phi, \theta,$ and |
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$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
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inherent in using the Euler angles, the four parameter ``quaternion'' |
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scheme is often used. The elements of \textbf{A} can be expressed as |
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arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
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and $q_3$).\cite{allen87:csl} Use of quaternions also leads to |
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performance enhancements, particularly for very small |
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systems.\cite{Evans77} |
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|
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{\sc oopse} utilizes a relatively new scheme that propagates the |
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entire nine parameter rotation matrix internally. Further discussion |
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on this choice can be found in Sec.~\ref{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. Which mimics the Van der Waals |
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interaction at long distances, and uses an emperical 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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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 $j$, |
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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. |
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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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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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$r_{\text{cut}}$ 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 potential 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 Dipolar Unified-atom Force Field ({\sc duff}) was developed to |
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simulate lipid bilayers. The systems require a model capable of forming |
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bilayers, while still being sufficiently computationally efficient to |
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allow simulations of large systems ($\approx$100's of phospholipids, |
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$\approx$1000's of waters) for long times ($\approx$10's of |
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nanoseconds). |
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With this goal in mind, {\sc duff} has no point charges. Charge |
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neutral distributions were replaced with dipoles, while most atoms and |
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groups of atoms were reduced to Lennard-Jones interaction sites. This |
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simplification cuts the length scale of long range interactions from |
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$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
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computationally expensive Ewald sum. Instead, we can use |
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neighbor-lists, reaction field, and cutoff radii for the dipolar |
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interactions. |
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|
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As an example, lipid head-groups in {\sc duff} are represented as |
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point dipole interaction sites. By placing a dipole of 20.6~Debye at |
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the head group center of mass, our model mimics the head group of |
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phosphatidylcholine.\cite{Cevc87} Additionally, a 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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\emph{et al.}\cite{liu96:new_model} |
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|
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\begin{figure} |
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\epsfxsize=\linewidth |
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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 |
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is the chain length.} |
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\label{fig:lipidModel} |
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\end{figure} |
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|
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Turning to the tails of the phospholipids, we have used a set of |
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scalable parameters to model the alkyl groups with Lennard-Jones |
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sites. For this, we have used the TraPPE force field of Siepmann |
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\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
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representation of n-alkanes, which is parametrized against phase |
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equilibria using Gibbs Monte Carlo simulation |
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techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
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it generalizes the types of atoms in an alkyl chain to keep the number |
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of pseudoatoms to a minimum; the parameters for an atom such as |
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$\text{CH}_2$ do not change depending on what species are bonded to |
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it. |
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|
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TraPPE also constrains of all bonds to be of fixed length. Typically, |
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bond vibrations are the fastest motions in a molecular dynamic |
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simulation. Small time steps between force evaluations must be used to |
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ensure adequate sampling of the bond potential conservation of |
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energy. By constraining the bond lengths, larger time steps may be |
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used when integrating the equations of motion. |
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\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
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The total energy of function in {\sc duff} is given by the following: |
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\begin{equation} |
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V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
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+ \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
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\label{eq:totalPotential} |
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\end{equation} |
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Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
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\begin{equation} |
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V^{I}_{\text{Internal}} = |
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\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
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+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
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+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\biggr] |
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\label{eq:internalPotential} |
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\end{equation} |
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Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
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within the molecule, and $V_{\text{torsion}}$ is the torsion potential |
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for all 1, 4 bonded pairs. The pairwise portions of the internal |
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potential are excluded for pairs that are closer than three bonds, |
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i.e.~atom pairs farther away than a torsion are included in the |
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pair-wise loop. |
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|
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|
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The bend potential of a molecule is represented by the following function: |
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\begin{equation} |
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V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
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\end{equation} |
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Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
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(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
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bond angle. $k_{\theta}$ is the force constant which determines the |
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strength of the harmonic bend. The parameters for $k_{\theta}$ and |
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$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
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|
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The torsion potential and parameters are also taken from TraPPE. It is |
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of the form: |
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\begin{equation} |
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V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
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+ c_2[1 + \cos(2\phi)] |
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+ c_3[1 + \cos(3\phi)] |
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\label{eq:origTorsionPot} |
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\end{equation} |
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Here $\phi_{ijkl}$ 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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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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V_{\text{torsion}}(\phi_{ijkl}) = |
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k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
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\label{eq:torsionPot} |
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\end{equation} |
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Where: |
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\begin{align*} |
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k_0 &= c_1 + c_3 \\ |
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k_1 &= c_1 - 3c_3 \\ |
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k_2 &= 2 c_2 \\ |
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k_3 &= 4c_3 |
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\end{align*} |
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By recasting the equation to a power series, repeated trigonometric |
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evaluations are avoided during the calculation of the potential. |
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|
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|
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The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
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as follows: |
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\begin{equation} |
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V^{IJ}_{\text{Cross}} = |
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\sum_{i \in I} \sum_{j \in J} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ V_{\text{sticky}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\biggr] |
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\label{eq:crossPotentail} |
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\end{equation} |
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Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
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$V_{\text{dipole}}$ is the dipole dipole potential, and |
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$V_{\text{sticky}}$ is the sticky potential defined by the SSD |
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model. Note that not all atom types include all interactions. |
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|
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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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\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
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\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
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- |
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\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
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{r^{2}_{ij}} \biggr] |
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\label{eq:dipolePot} |
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\end{equation} |
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Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
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towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
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are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is |
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the magnitude of the dipole moment of atom $i$ and |
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$\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of |
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$\boldsymbol{\Omega}_i$. |
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|
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|
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\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
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|
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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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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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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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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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$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
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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}^{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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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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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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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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|
| 330 |
> |
Since SSD is a single-point {\it dipolar} model, the force |
| 331 |
> |
calculations are simplified significantly relative to the standard |
| 332 |
> |
{\it charged} multi-point models. In the original Monte Carlo |
| 333 |
> |
simulations using this model, Ichiye {\it et al.} reported that using |
| 334 |
> |
SSD decreased computer time by a factor of 6-7 compared to other |
| 335 |
> |
models.\cite{Ichiye96} What is most impressive is that this savings |
| 336 |
> |
did not come at the expense of accurate depiction of the liquid state |
| 337 |
> |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
| 338 |
> |
data for the structural features of liquid |
| 339 |
> |
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
| 340 |
> |
exhibited by SSD agree with experiment better than those of more |
| 341 |
> |
computationally expensive models (like TIP3P and |
| 342 |
> |
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
| 343 |
> |
of solvent properties makes SSD a very attractive model for the |
| 344 |
> |
simulation of large scale biochemical simulations. |
| 345 |
> |
|
| 346 |
> |
Recent constant pressure simulations revealed issues in the original |
| 347 |
> |
SSD model that led to lower than expected densities at all target |
| 348 |
> |
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
| 349 |
> |
is SSD/E, a density corrected derivative of SSD that exhibits improved |
| 350 |
> |
liquid structure and transport behavior. If the use of a reaction |
| 351 |
> |
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 |
| 353 |
> |
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 |
| 355 |
> |
and the drawbacks and benefits of the different density corrected SSD |
| 356 |
> |
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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> |
|
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> |
\subsection{\label{sec:eam}Embedded Atom Method} |
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> |
|
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> |
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
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> |
capacity to simulate metallic systems, including some that have |
| 364 |
> |
parallel computational abilities\cite{plimpton93}. Potentials that |
| 365 |
> |
describe bonding transition metal |
| 366 |
> |
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
| 367 |
> |
attractive interaction which models ``Embedding'' |
| 368 |
> |
a positively charged metal ion in the electron density due to the |
| 369 |
> |
free valance ``sea'' of electrons created by the surrounding atoms in |
| 370 |
> |
the system. A mostly repulsive pairwise part of the potential |
| 371 |
> |
describes the interaction of the positively charged metal core ions |
| 372 |
> |
with one another. A particular potential description called the |
| 373 |
> |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
| 374 |
> |
particularly wide adoption has been selected for inclusion in OOPSE. A |
| 375 |
> |
good review of {\sc eam} and other metallic potential formulations was done |
| 376 |
> |
by Voter.\cite{voter} |
| 377 |
> |
|
| 378 |
> |
The {\sc eam} potential has the form: |
| 379 |
> |
\begin{eqnarray} |
| 380 |
> |
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 381 |
> |
\phi_{ij}({\bf r}_{ij}) \\ |
| 382 |
> |
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 383 |
> |
\end{eqnarray}S |
| 384 |
> |
|
| 385 |
> |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 386 |
> |
positively-charged core ion $i$ into a linear superposition of |
| 387 |
> |
sperically averaged atomic electron densities given by |
| 388 |
> |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 389 |
> |
between atoms $i$ and $j$. In the original formulation of |
| 390 |
> |
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 391 |
> |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 392 |
> |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 393 |
> |
There is a cutoff distance, $r_{cut}$, which limits the |
| 394 |
> |
summations in the {\sc eam} equation to the few dozen atoms |
| 395 |
> |
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 396 |
> |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FDB86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 397 |
> |
|
| 398 |
> |
|
| 399 |
> |
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 400 |
> |
|
| 401 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 402 |
> |
macroscopic systems with a relatively small number of particles. The |
| 403 |
> |
simulation box is replicated throughout space to form an infinite |
| 404 |
> |
lattice. During the simulation, when a particle moves in the primary |
| 405 |
> |
cell, its image in other boxes move in exactly the same direction with |
| 406 |
> |
exactly the same orientation.Thus, as a particle leaves the primary |
| 407 |
> |
cell, one of its images will enter through the opposite face.If the |
| 408 |
> |
simulation box is large enough to avoid "feeling" the symmetries of |
| 409 |
> |
the periodic lattice, surface effects can be ignored. Cubic, |
| 410 |
> |
orthorhombic and parallelepiped are the available periodic cells In |
| 411 |
> |
OOPSE. We use a matrix to describe the property of the simulation |
| 412 |
> |
box. Therefore, both the size and shape of the simulation box can be |
| 413 |
> |
changed during the simulation. The transformation from box space |
| 414 |
> |
vector $\mathbf{s}$ to its corresponding real space vector |
| 415 |
> |
$\mathbf{r}$ is defined by |
| 416 |
> |
\begin{equation} |
| 417 |
> |
\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}% |
| 418 |
> |
\end{equation} |
| 419 |
> |
|
| 420 |
> |
|
| 421 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of |
| 422 |
> |
the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the |
| 423 |
> |
three sides of the simulation box respectively. |
| 424 |
> |
|
| 425 |
> |
To find the minimum image, we convert the real vector to its |
| 426 |
> |
corresponding vector in box space first, \bigskip% |
| 427 |
> |
\begin{equation} |
| 428 |
> |
\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}% |
| 429 |
> |
\end{equation} |
| 430 |
> |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 431 |
> |
\begin{equation} |
| 432 |
> |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 433 |
> |
\end{equation} |
| 434 |
> |
where |
| 435 |
> |
|
| 436 |
> |
% |
| 437 |
> |
|
| 438 |
> |
\begin{equation} |
| 439 |
> |
round(x)=\left\{ |
| 440 |
> |
\begin{array}[c]{c}% |
| 441 |
> |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 442 |
> |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 443 |
> |
\end{array} |
| 444 |
> |
\right. |
| 445 |
> |
\end{equation} |
| 446 |
> |
|
| 447 |
> |
|
| 448 |
> |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, |
| 449 |
> |
$round(-3.1)=-3$. |
| 450 |
> |
|
| 451 |
> |
Finally, we obtain the minimum image coordinates by transforming back |
| 452 |
> |
to real space,% |
| 453 |
> |
|
| 454 |
> |
\begin{equation} |
| 455 |
> |
\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 456 |
> |
\end{equation} |
| 457 |
> |
|
