| 9 |
|
element, or be used for collections of atoms such as a methyl |
| 10 |
|
group. The atoms are also capable of having a directional component |
| 11 |
|
associated with them, often in the form of a dipole. Charges on atoms |
| 12 |
< |
are not currently suporrted by {\sc oopse}. |
| 12 |
> |
are not currently suported by {\sc oopse}. |
| 13 |
|
|
| 14 |
|
The second most basic building block of a simulation is the |
| 15 |
|
molecule. The molecule is a way for {\sc oopse} to keep track of the |
| 68 |
|
\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
| 69 |
|
|
| 70 |
|
The most basic force field implemented in OOPSE is the Lennard-Jones |
| 71 |
< |
potential. The Lennard-Jones potential mimics the attractive forces |
| 72 |
< |
two charge neutral particles experience when spontaneous dipoles are |
| 73 |
< |
induced on each other. This is the predominant intermolecular force in |
| 74 |
< |
systems of such as noble gases and simple alkanes. |
| 75 |
< |
|
| 76 |
< |
The Lennard-Jones potential is given by: |
| 71 |
> |
potential. The Lennard-Jones potential. Which mimics the Van der Waals |
| 72 |
> |
interaction at long distances, and uses an emperical repulsion at |
| 73 |
> |
short distances. The Lennard-Jones potential is given by: |
| 74 |
|
\begin{equation} |
| 75 |
|
V_{\text{LJ}}(r_{ij}) = |
| 76 |
|
4\epsilon_{ij} \biggl[ |
| 79 |
|
\biggr] |
| 80 |
|
\label{eq:lennardJonesPot} |
| 81 |
|
\end{equation} |
| 82 |
< |
Where $r_ij$ is the distance between particle $i$ and $j$, $\sigma_{ij}$ |
| 83 |
< |
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
| 84 |
< |
energy well depth of the potential. |
| 82 |
> |
Where $r_{ij}$ is the distance between particle $i$ and $j$, |
| 83 |
> |
$\sigma_{ij}$ scales the length of the interaction, and |
| 84 |
> |
$\epsilon_{ij}$ scales the well depth of the potential. |
| 85 |
|
|
| 86 |
< |
Because the potential is calculated between all pairs, the force |
| 86 |
> |
Because this potential is calculated between all pairs, the force |
| 87 |
|
evaluation can become computationally expensive for large systems. To |
| 88 |
< |
keep the pair evaluation to a manegable number, OOPSE employs the use |
| 89 |
< |
of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
| 90 |
< |
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length |
| 88 |
> |
keep the pair evaluation to a manegable number, OOPSE employs a |
| 89 |
> |
cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
| 90 |
> |
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones length |
| 91 |
|
parameter in the system. Truncating the calculation at |
| 92 |
|
$r_{\text{cut}}$ introduces a discontinuity into the potential |
| 93 |
|
energy. To offset this discontinuity, the energy value at |
| 94 |
|
$r_{\text{cut}}$ is subtracted from the entire potential. This causes |
| 95 |
< |
the equation to go to zero at the cut-off radius. |
| 95 |
> |
the potential to go to zero at the cut-off radius. |
| 96 |
|
|
| 97 |
|
Interactions between dissimilar particles requires the generation of |
| 98 |
|
cross term parameters for $\sigma$ and $\epsilon$. These are |
| 111 |
|
|
| 112 |
|
\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
| 113 |
|
|
| 114 |
< |
The \underline{D}ipolar \underline{U}nified-Atom |
| 115 |
< |
\underline{F}orce \underline{F}ield ({\sc duff}) was developed to |
| 119 |
< |
simulate lipid bilayers. We needed a model capable of forming |
| 114 |
> |
The Dipolar Unified-atom Force Field ({\sc duff}) was developed to |
| 115 |
> |
simulate lipid bilayers. The systems require a model capable of forming |
| 116 |
|
bilayers, while still being sufficiently computationally efficient to |
| 117 |
|
allow simulations of large systems ($\approx$100's of phospholipids, |
| 118 |
|
$\approx$1000's of waters) for long times ($\approx$10's of |
| 119 |
|
nanoseconds). |
| 120 |
|
|
| 121 |
< |
With this goal in mind, we have eliminated all point charges. Charge |
| 122 |
< |
distributions were replaced with dipoles, and charge-neutral |
| 123 |
< |
distributions were reduced to Lennard-Jones interaction sites. This |
| 121 |
> |
With this goal in mind, {\sc duff} has no point charges. Charge |
| 122 |
> |
neutral distributions were replaced with dipoles, while most atoms and |
| 123 |
> |
groups of atoms were reduced to Lennard-Jones interaction sites. This |
| 124 |
|
simplification cuts the length scale of long range interactions from |
| 125 |
|
$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
| 126 |
< |
computationally expensive Ewald-Sum. Instead, we can use |
| 127 |
< |
neighbor-lists and cutoff radii for the dipolar interactions. |
| 126 |
> |
computationally expensive Ewald sum. Instead, we can use |
| 127 |
> |
neighbor-lists, reaction field, and cutoff radii for the dipolar |
| 128 |
> |
interactions. |
| 129 |
|
|
| 130 |
< |
As an example, lipid head groups in {\sc duff} are represented as point |
| 131 |
< |
dipole interaction sites.PC and PE Lipid head groups are typically |
| 132 |
< |
zwitterionic in nature, with charges separated by as much as |
| 133 |
< |
6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
| 134 |
< |
center of mass, our model mimics the head group of PC.\cite{Cevc87} |
| 135 |
< |
Additionally, a Lennard-Jones site is located at the |
| 136 |
< |
pseudoatom's center of mass. The model is illustrated by the dark grey |
| 137 |
< |
atom in Fig.~\ref{fig:lipidModel}. |
| 130 |
> |
As an example, lipid head-groups in {\sc duff} are represented as |
| 131 |
> |
point dipole interaction sites. By placing a dipole of 20.6~Debye at |
| 132 |
> |
the head group center of mass, our model mimics the head group of |
| 133 |
> |
phosphatidylcholine.\cite{Cevc87} Additionally, a Lennard-Jones site |
| 134 |
> |
is located at the pseudoatom's center of mass. The model is |
| 135 |
> |
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 |
| 136 |
> |
repaarameterization of the soft sticky dipole (SSD) model of Ichiye |
| 137 |
> |
\emph{et al.}\cite{liu96:new_model} |
| 138 |
|
|
| 139 |
|
\begin{figure} |
| 140 |
+ |
\epsfxsize=\linewidth |
| 141 |
|
\epsfbox{lipidModel.eps} |
| 142 |
|
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 143 |
< |
is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 143 |
> |
is the bend angle, $\mu$ is the dipole moment of the head group, and n |
| 144 |
> |
is the chain length.} |
| 145 |
|
\label{fig:lipidModel} |
| 146 |
|
\end{figure} |
| 148 |
– |
|
| 149 |
– |
The water model we use to complement the dipoles of the lipids is |
| 150 |
– |
the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 151 |
– |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 152 |
– |
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 153 |
– |
Lennard-Jones interaction site. The site also contains a dipole to |
| 154 |
– |
mimic the partial charges on the hydrogens and the oxygen. However, |
| 155 |
– |
what makes the SSD model unique is the inclusion of a tetrahedral |
| 156 |
– |
short range potential to recover the hydrogen bonding of water, an |
| 157 |
– |
important factor when modeling bilayers, as it has been shown that |
| 158 |
– |
hydrogen bond network formation is a leading contribution to the |
| 159 |
– |
entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 147 |
|
|
| 161 |
– |
|
| 148 |
|
Turning to the tails of the phospholipids, we have used a set of |
| 149 |
|
scalable parameters to model the alkyl groups with Lennard-Jones |
| 150 |
|
sites. For this, we have used the TraPPE force field of Siepmann |
| 170 |
|
The total energy of function in {\sc duff} is given by the following: |
| 171 |
|
\begin{equation} |
| 172 |
|
V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 173 |
< |
+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
| 173 |
> |
+ \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
| 174 |
|
\label{eq:totalPotential} |
| 175 |
|
\end{equation} |
| 176 |
|
Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
| 177 |
|
\begin{equation} |
| 178 |
|
V^{I}_{\text{Internal}} = |
| 179 |
|
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 180 |
< |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl}) |
| 180 |
> |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
| 181 |
|
+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 182 |
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 183 |
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 185 |
|
\label{eq:internalPotential} |
| 186 |
|
\end{equation} |
| 187 |
|
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 188 |
< |
within in the molecule. $V_{\text{torsion}}$ is the torsion The |
| 189 |
< |
pairwise portions of the internal potential are excluded for pairs |
| 190 |
< |
that ar closer than three bonds, i.e.~atom pairs farther away than a |
| 191 |
< |
torsion are included in the pair-wise loop. |
| 188 |
> |
within the molecule, and $V_{\text{torsion}}$ is the torsion potential |
| 189 |
> |
for all 1, 4 bonded pairs. The pairwise portions of the internal |
| 190 |
> |
potential are excluded for pairs that are closer than three bonds, |
| 191 |
> |
i.e.~atom pairs farther away than a torsion are included in the |
| 192 |
> |
pair-wise loop. |
| 193 |
|
|
| 207 |
– |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 208 |
– |
as follows: |
| 209 |
– |
\begin{equation} |
| 210 |
– |
V^{IJ}_{\text{Cross}} = |
| 211 |
– |
\sum_{i \in I} \sum_{j \in J} |
| 212 |
– |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 213 |
– |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 214 |
– |
+ V_{\text{sticky}} |
| 215 |
– |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 216 |
– |
\biggr] |
| 217 |
– |
\label{eq:crossPotentail} |
| 218 |
– |
\end{equation} |
| 219 |
– |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 220 |
– |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 221 |
– |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
| 194 |
|
|
| 195 |
|
The bend potential of a molecule is represented by the following function: |
| 196 |
|
\begin{equation} |
| 205 |
|
The torsion potential and parameters are also taken from TraPPE. It is |
| 206 |
|
of the form: |
| 207 |
|
\begin{equation} |
| 208 |
< |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
| 208 |
> |
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
| 209 |
|
+ c_2[1 + \cos(2\phi)] |
| 210 |
|
+ c_3[1 + \cos(3\phi)] |
| 211 |
|
\label{eq:origTorsionPot} |
| 212 |
|
\end{equation} |
| 213 |
|
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
| 214 |
< |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
| 215 |
< |
for computaional efficency, the torsion potentail has been recast |
| 216 |
< |
after the method of CHARMM\cite{charmm1983} whereby the angle series |
| 217 |
< |
is converted to a power series of the form: |
| 214 |
> |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For |
| 215 |
> |
computaional efficency, the torsion potential has been recast after |
| 216 |
> |
the method of CHARMM\cite{charmm1983} whereby the angle series is |
| 217 |
> |
converted to a power series of the form: |
| 218 |
|
\begin{equation} |
| 219 |
|
V_{\text{torsion}}(\phi_{ijkl}) = |
| 220 |
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 231 |
|
evaluations are avoided during the calculation of the potential. |
| 232 |
|
|
| 233 |
|
|
| 234 |
+ |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 235 |
+ |
as follows: |
| 236 |
+ |
\begin{equation} |
| 237 |
+ |
V^{IJ}_{\text{Cross}} = |
| 238 |
+ |
\sum_{i \in I} \sum_{j \in J} |
| 239 |
+ |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 240 |
+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 241 |
+ |
+ V_{\text{sticky}} |
| 242 |
+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 243 |
+ |
\biggr] |
| 244 |
+ |
\label{eq:crossPotentail} |
| 245 |
+ |
\end{equation} |
| 246 |
+ |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 247 |
+ |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 248 |
+ |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD |
| 249 |
+ |
model. Note that not all atom types include all interactions. |
| 250 |
|
|
| 251 |
|
The dipole-dipole potential has the following form: |
| 252 |
|
\begin{equation} |
| 253 |
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 254 |
< |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 255 |
< |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 254 |
> |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 255 |
> |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 256 |
|
- |
| 257 |
< |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 258 |
< |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 259 |
< |
{r^{5}_{ij}} \biggr] |
| 257 |
> |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
| 258 |
> |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
| 259 |
> |
{r^{2}_{ij}} \biggr] |
| 260 |
|
\label{eq:dipolePot} |
| 261 |
|
\end{equation} |
| 262 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 263 |
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 264 |
< |
are the Euler angles of atom $i$ and $j$ |
| 265 |
< |
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 266 |
< |
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
| 264 |
> |
are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is |
| 265 |
> |
the magnitude of the dipole moment of atom $i$ and |
| 266 |
> |
$\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of |
| 267 |
> |
$\boldsymbol{\Omega}_i$. |
| 268 |
|
|
| 269 |
|
|
| 270 |
|
\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
| 356 |
|
|
| 357 |
|
!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
| 358 |
|
|
| 359 |
< |
\subsection{\label{sec:eam}Embedded Atom Model} |
| 359 |
> |
\subsection{\label{sec:eam}Embedded Atom Method} |
| 360 |
|
|
| 361 |
|
Several molecular dynamics codes\cite{dynamo86} exist which have the |
| 362 |
|
capacity to simulate metallic systems, including some that have |
| 396 |
|
|
| 397 |
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 398 |
|
|
| 399 |
< |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 400 |
< |
macroscopic systems with a relatively small number of particles. Simulation |
| 401 |
< |
box is replicated throughout space to form an infinite lattice. During the |
| 402 |
< |
simulation, when a particle moves in the primary cell, its periodic image |
| 403 |
< |
particles in other boxes move in exactly the same direction with exactly the |
| 404 |
< |
same orientation.Thus, as a particle leaves the primary cell, one of its |
| 405 |
< |
images will enter through the opposite face.If the simulation box is large |
| 406 |
< |
enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
| 407 |
< |
effect could be ignored. Cubic and parallelepiped are the available periodic |
| 408 |
< |
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
| 409 |
< |
the property of the simulation box. Therefore, not only the size of the |
| 410 |
< |
simulation box could be changed during the simulation, but also the shape of |
| 411 |
< |
it. The transformation from box space vector $\overrightarrow{s}$ to its |
| 412 |
< |
corresponding real space vector $\overrightarrow{r}$ is defined by |
| 413 |
< |
\begin{equation} |
| 414 |
< |
\overrightarrow{r}=H\overrightarrow{s}% |
| 415 |
< |
\end{equation} |
| 416 |
< |
|
| 417 |
< |
|
| 418 |
< |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 419 |
< |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
| 420 |
< |
simulation box respectively. |
| 421 |
< |
|
| 422 |
< |
To find the minimum image, we need to convert the real vector to its |
| 423 |
< |
corresponding vector in box space first, \bigskip% |
| 424 |
< |
\begin{equation} |
| 425 |
< |
\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
| 426 |
< |
\end{equation} |
| 427 |
< |
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
| 428 |
< |
to 0.5, |
| 429 |
< |
\begin{equation} |
| 430 |
< |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 431 |
< |
\end{equation} |
| 432 |
< |
where% |
| 433 |
< |
|
| 434 |
< |
\begin{equation} |
| 435 |
< |
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
| 436 |
< |
}x\geqslant0 |
| 437 |
< |
\end{equation} |
| 438 |
< |
% |
| 439 |
< |
|
| 440 |
< |
\begin{equation} |
| 441 |
< |
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
| 442 |
< |
\end{equation} |
| 443 |
< |
|
| 444 |
< |
|
| 445 |
< |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 446 |
< |
|
| 447 |
< |
Finally, we could get the minimum image by transforming back to real space,% |
| 448 |
< |
|
| 449 |
< |
\begin{equation} |
| 450 |
< |
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
| 451 |
< |
\end{equation} |
| 399 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 400 |
> |
macroscopic systems with a relatively small number of particles. The |
| 401 |
> |
simulation box is replicated throughout space to form an infinite |
| 402 |
> |
lattice. During the simulation, when a particle moves in the primary |
| 403 |
> |
cell, its image in other boxes move in exactly the same direction with |
| 404 |
> |
exactly the same orientation.Thus, as a particle leaves the primary |
| 405 |
> |
cell, one of its images will enter through the opposite face.If the |
| 406 |
> |
simulation box is large enough to avoid "feeling" the symmetries of |
| 407 |
> |
the periodic lattice, surface effects can be ignored. Cubic, |
| 408 |
> |
orthorhombic and parallelepiped are the available periodic cells In |
| 409 |
> |
OOPSE. We use a matrix to describe the property of the simulation |
| 410 |
> |
box. Therefore, both the size and shape of the simulation box can be |
| 411 |
> |
changed during the simulation. The transformation from box space |
| 412 |
> |
vector $\mathbf{s}$ to its corresponding real space vector |
| 413 |
> |
$\mathbf{r}$ is defined by |
| 414 |
> |
\begin{equation} |
| 415 |
> |
\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}% |
| 416 |
> |
\end{equation} |
| 417 |
> |
|
| 418 |
> |
|
| 419 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of |
| 420 |
> |
the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the |
| 421 |
> |
three sides of the simulation box respectively. |
| 422 |
> |
|
| 423 |
> |
To find the minimum image, we convert the real vector to its |
| 424 |
> |
corresponding vector in box space first, \bigskip% |
| 425 |
> |
\begin{equation} |
| 426 |
> |
\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}% |
| 427 |
> |
\end{equation} |
| 428 |
> |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 429 |
> |
\begin{equation} |
| 430 |
> |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 431 |
> |
\end{equation} |
| 432 |
> |
where |
| 433 |
> |
|
| 434 |
> |
% |
| 435 |
> |
|
| 436 |
> |
\begin{equation} |
| 437 |
> |
round(x)=\left\{ |
| 438 |
> |
\begin{array}[c]{c}% |
| 439 |
> |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 440 |
> |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 441 |
> |
\end{array} |
| 442 |
> |
\right. |
| 443 |
> |
\end{equation} |
| 444 |
> |
|
| 445 |
> |
|
| 446 |
> |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, |
| 447 |
> |
$round(-3.1)=-3$. |
| 448 |
> |
|
| 449 |
> |
Finally, we obtain the minimum image coordinates by transforming back |
| 450 |
> |
to real space,% |
| 451 |
> |
|
| 452 |
> |
\begin{equation} |
| 453 |
> |
\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 454 |
> |
\end{equation} |
| 455 |
> |
|
