| 1 |
|
| 2 |
\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
| 3 |
|
| 4 |
\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
| 5 |
|
| 6 |
The basic unit of an {\sc oopse} simulation is the atom. The parameters |
| 7 |
describing the atom are generalized to make the atom as flexible a |
| 8 |
representation as possible. They may represent specific atoms of an |
| 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}. |
| 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 atoms |
| 16 |
in a simulation in logical manner. This particular unit will store the |
| 17 |
identities of all the atoms associated with itself and is responsible |
| 18 |
for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
| 19 |
and torsions). |
| 20 |
|
| 21 |
As stated in the previously, one of the features that sets OOPSE apart |
| 22 |
from most of the current molecular simulation packages is the ability |
| 23 |
to handle rigid body dynamics. Rigid bodies are non-spherical |
| 24 |
particles or collections of particles that have a constant internal |
| 25 |
potential and move collectively.\cite{Goldstein01} They are not |
| 26 |
included in many standard simulation packages because of the need to |
| 27 |
consider orientational degrees of freedom and include an integrator |
| 28 |
that accurately propagates these motions in time. |
| 29 |
|
| 30 |
Moving a rigid body involves determination of both the force and |
| 31 |
torque applied by the surroundings, which directly affect the |
| 32 |
translation and rotation in turn. In order to accumulate the total |
| 33 |
force on a rigid body, the external forces must first be calculated |
| 34 |
for all the interal particles. The total force on the rigid body is |
| 35 |
simply the sum of these external forces. Accumulation of the total |
| 36 |
torque on the rigid body is similar to the force in that it is a sum |
| 37 |
of the torque applied on each internal particle, mapped onto the |
| 38 |
center of mass of the rigid body. |
| 39 |
|
| 40 |
The application of the total torque is done in the body fixed axis of |
| 41 |
the rigid body. In order to move between the space fixed and body |
| 42 |
fixed coordinate axes, parameters describing the orientation be |
| 43 |
maintained for each rigid body. At a minimum, the rotation matrix can |
| 44 |
be described and propagated by the three Euler |
| 45 |
angles.\cite{Goldstein01} In order to avoid rotational limitations |
| 46 |
when using Euler angles, the four parameter ``quaternion'' scheme can |
| 47 |
be used instead.\cite{allen87:csl} Use of quaternions also leads to |
| 48 |
performance enhancements, particularly for very small |
| 49 |
systems.\cite{Evans77} OOPSE utilizes a relatively new scheme that |
| 50 |
propagates the entire nine parameter rotation matrix. Further |
| 51 |
discussion on this choice can be found in Sec.~\ref{sec:integrate}. |
| 52 |
|
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
| 54 |
|
| 55 |
The most basic force field implemented in OOPSE is the Lennard-Jones |
| 56 |
potential. The Lennard-Jones potential mimics the attractive forces |
| 57 |
two charge neutral particles experience when spontaneous dipoles are |
| 58 |
induced on each other. This is the predominant intermolecular force in |
| 59 |
systems of such as noble gases and simple alkanes. |
| 60 |
|
| 61 |
The Lennard-Jones potential is given by: |
| 62 |
\begin{equation} |
| 63 |
V_{\text{LJ}}(r_{ij}) = |
| 64 |
4\epsilon_{ij} \biggl[ |
| 65 |
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
| 67 |
\biggr] |
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\label{eq:lennardJonesPot} |
| 69 |
\end{equation} |
| 70 |
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 |
| 72 |
energy well depth of the potential. |
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|
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Because the potential is calculated between all pairs, the force |
| 75 |
evaluation can become computationally expensive for large systems. To |
| 76 |
keep the pair evaluation to a manegable number, OOPSE employs the use |
| 77 |
of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
| 78 |
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length |
| 79 |
parameter in the system. Truncating the calculation at |
| 80 |
$r_{\text{cut}}$ introduces a discontinuity into the potential |
| 81 |
energy. To offset this discontinuity, the energy value at |
| 82 |
$r_{\text{cut}}$ is subtracted from the entire potential. This causes |
| 83 |
the equation to go to zero at the cut-off radius. |
| 84 |
|
| 85 |
Interactions between dissimilar particles requires the generation of |
| 86 |
cross term parameters for $\sigma$ and $\epsilon$. These are |
| 87 |
calculated through the Lorentz-Berthelot mixing |
| 88 |
rules:\cite{allen87:csl} |
| 89 |
\begin{equation} |
| 90 |
\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}] |
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\label{eq:sigmaMix} |
| 92 |
\end{equation} |
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and |
| 94 |
\begin{equation} |
| 95 |
\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}} |
| 96 |
\label{eq:epsilonMix} |
| 97 |
\end{equation} |
| 98 |
|
| 99 |
|
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
| 101 |
|
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The \underline{D}ipolar \underline{U}nified-Atom |
| 103 |
\underline{F}orce \underline{F}ield ({\sc duff}) was developed to |
| 104 |
simulate lipid bilayers. We needed a model capable of forming |
| 105 |
bilayers, while still being sufficiently computationally efficient to |
| 106 |
allow simulations of large systems ($\approx$100's of phospholipids, |
| 107 |
$\approx$1000's of waters) for long times ($\approx$10's of |
| 108 |
nanoseconds). |
| 109 |
|
| 110 |
With this goal in mind, we have eliminated all point charges. Charge |
| 111 |
distributions were replaced with dipoles, and charge-neutral |
| 112 |
distributions were reduced to Lennard-Jones interaction sites. This |
| 113 |
simplification cuts the length scale of long range interactions from |
| 114 |
$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
| 115 |
computationally expensive Ewald-Sum. Instead, we can use |
| 116 |
neighbor-lists and cutoff radii for the dipolar interactions. |
| 117 |
|
| 118 |
As an example, lipid head groups in {\sc duff} are represented as point |
| 119 |
dipole interaction sites.PC and PE Lipid head groups are typically |
| 120 |
zwitterionic in nature, with charges separated by as much as |
| 121 |
6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
| 122 |
center of mass, our model mimics the head group of PC.\cite{Cevc87} |
| 123 |
Additionally, a Lennard-Jones site is located at the |
| 124 |
pseudoatom's center of mass. The model is illustrated by the dark grey |
| 125 |
atom in Fig.~\ref{fig:lipidModel}. |
| 126 |
|
| 127 |
\begin{figure} |
| 128 |
\epsfxsize=6in |
| 129 |
\epsfbox{lipidModel.epsi} |
| 130 |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 131 |
is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 132 |
\label{fig:lipidModel} |
| 133 |
\end{figure} |
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|
| 135 |
The water model we use to complement the dipoles of the lipids is |
| 136 |
the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 137 |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 138 |
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 139 |
Lennard-Jones interaction site. The site also contains a dipole to |
| 140 |
mimic the partial charges on the hydrogens and the oxygen. However, |
| 141 |
what makes the SSD model unique is the inclusion of a tetrahedral |
| 142 |
short range potential to recover the hydrogen bonding of water, an |
| 143 |
important factor when modeling bilayers, as it has been shown that |
| 144 |
hydrogen bond network formation is a leading contribution to the |
| 145 |
entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 146 |
|
| 147 |
|
| 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 |
| 151 |
\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
| 152 |
representation of n-alkanes, which is parametrized against phase |
| 153 |
equilibria using Gibbs Monte Carlo simulation |
| 154 |
techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
| 155 |
it generalizes the types of atoms in an alkyl chain to keep the number |
| 156 |
of pseudoatoms to a minimum; the parameters for an atom such as |
| 157 |
$\text{CH}_2$ do not change depending on what species are bonded to |
| 158 |
it. |
| 159 |
|
| 160 |
TraPPE also constrains of all bonds to be of fixed length. Typically, |
| 161 |
bond vibrations are the fastest motions in a molecular dynamic |
| 162 |
simulation. Small time steps between force evaluations must be used to |
| 163 |
ensure adequate sampling of the bond potential conservation of |
| 164 |
energy. By constraining the bond lengths, larger time steps may be |
| 165 |
used when integrating the equations of motion. |
| 166 |
|
| 167 |
|
| 168 |
\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
| 169 |
|
| 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}} |
| 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}) |
| 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}) |
| 184 |
\biggr] |
| 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. |
| 192 |
|
| 193 |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 194 |
as follows: |
| 195 |
\begin{equation} |
| 196 |
V^{IJ}_{\text{Cross}} = |
| 197 |
\sum_{i \in I} \sum_{j \in J} |
| 198 |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 199 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 200 |
+ V_{\text{sticky}} |
| 201 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 202 |
\biggr] |
| 203 |
\label{eq:crossPotentail} |
| 204 |
\end{equation} |
| 205 |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 206 |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 207 |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
| 208 |
|
| 209 |
The bend potential of a molecule is represented by the following function: |
| 210 |
\begin{equation} |
| 211 |
V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 212 |
\end{equation} |
| 213 |
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 214 |
(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
| 215 |
bond angle. $k_{\theta}$ is the force constant which determines the |
| 216 |
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 217 |
$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
| 218 |
|
| 219 |
The torsion potential and parameters are also taken from TraPPE. It is |
| 220 |
of the form: |
| 221 |
\begin{equation} |
| 222 |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
| 223 |
+ c_2[1 + \cos(2\phi)] |
| 224 |
+ c_3[1 + \cos(3\phi)] |
| 225 |
\label{eq:origTorsionPot} |
| 226 |
\end{equation} |
| 227 |
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
| 228 |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
| 229 |
for computaional efficency, the torsion potentail has been recast |
| 230 |
after the method of CHARMM\cite{charmm1983} whereby the angle series |
| 231 |
is converted to a power series of the form: |
| 232 |
\begin{equation} |
| 233 |
V_{\text{torsion}}(\phi_{ijkl}) = |
| 234 |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 235 |
\label{eq:torsionPot} |
| 236 |
\end{equation} |
| 237 |
Where: |
| 238 |
\begin{align*} |
| 239 |
k_0 &= c_1 + c_3 \\ |
| 240 |
k_1 &= c_1 - 3c_3 \\ |
| 241 |
k_2 &= 2 c_2 \\ |
| 242 |
k_3 &= 4c_3 |
| 243 |
\end{align*} |
| 244 |
By recasting the equation to a power series, repeated trigonometric |
| 245 |
evaluations are avoided during the calculation of the potential. |
| 246 |
|
| 247 |
|
| 248 |
|
| 249 |
The dipole-dipole potential has the following form: |
| 250 |
\begin{equation} |
| 251 |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 252 |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 253 |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 254 |
- |
| 255 |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 256 |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 257 |
{r^{5}_{ij}} \biggr] |
| 258 |
\label{eq:dipolePot} |
| 259 |
\end{equation} |
| 260 |
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 261 |
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 262 |
are the Euler angles of atom $i$ and $j$ |
| 263 |
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 264 |
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
| 265 |
|
| 266 |
|
| 267 |
\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
| 268 |
|
| 269 |
In the interest of computational efficiency, the native solvent used |
| 270 |
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
| 271 |
developed by Ichiye \emph{et al.} as a modified form of the |
| 272 |
hard-sphere water model proposed by Bratko, Blum, and |
| 273 |
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 274 |
with a Lennard-Jones core and a sticky potential that directs the |
| 275 |
particles to assume the proper hydrogen bond orientation in the first |
| 276 |
solvation shell. Thus, the interaction between two SSD water molecules |
| 277 |
\emph{i} and \emph{j} is given by the potential |
| 278 |
\begin{equation} |
| 279 |
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
| 280 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 281 |
u_{ij}^{sp} |
| 282 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 283 |
\end{equation} |
| 284 |
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
| 285 |
\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
| 286 |
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
| 287 |
orientations of the respective molecules. The Lennard-Jones, dipole, |
| 288 |
and sticky parts of the potential are giving by the following |
| 289 |
equations, |
| 290 |
\begin{equation} |
| 291 |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
| 292 |
\end{equation} |
| 293 |
\begin{equation} |
| 294 |
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}\ , |
| 295 |
\end{equation} |
| 296 |
\begin{equation} |
| 297 |
\begin{split} |
| 298 |
u_{ij}^{sp} |
| 299 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
| 300 |
&= |
| 301 |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
| 302 |
& \quad \ + |
| 303 |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 304 |
\end{split} |
| 305 |
\end{equation} |
| 306 |
where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
| 307 |
unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
| 308 |
$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
| 309 |
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
| 310 |
functions take the following forms, |
| 311 |
\begin{equation} |
| 312 |
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 313 |
\end{equation} |
| 314 |
\begin{equation} |
| 315 |
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, |
| 316 |
\end{equation} |
| 317 |
where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
| 318 |
term that promotes hydrogen bonding orientations within the first |
| 319 |
solvation shell, and $w^\prime$ is a dipolar repulsion term that |
| 320 |
repels unrealistic dipolar arrangements within the first solvation |
| 321 |
shell. A more detailed description of the functional parts and |
| 322 |
variables in this potential can be found in other |
| 323 |
articles.\cite{liu96:new_model,chandra99:ssd_md} |
| 324 |
|
| 325 |
Since SSD is a one-site point dipole model, the force calculations are |
| 326 |
simplified significantly from a computational standpoint, in that the |
| 327 |
number of long-range interactions is dramatically reduced. In the |
| 328 |
original Monte Carlo simulations using this model, Ichiye \emph{et |
| 329 |
al.} reported a calculation speed up of up to an order of magnitude |
| 330 |
over other comparable models while maintaining the structural behavior |
| 331 |
of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
| 332 |
SSD, it was shown that it actually improves upon the prediction of |
| 333 |
water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
| 334 |
|
| 335 |
Recent constant pressure simulations revealed issues in the original |
| 336 |
SSD model that led to lower than expected densities at all target |
| 337 |
pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the |
| 338 |
original SSD have resulted in improved density behavior, as well as |
| 339 |
alterations in the water structure and transport behavior. {\sc oopse} is |
| 340 |
easily modified to impliment these new potential parameter sets for |
| 341 |
the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
| 342 |
variable parameters are listed in the accompanying BASS file, and |
| 343 |
these parameters simply need to be changed to the updated values. |
| 344 |
|
| 345 |
|
| 346 |
\subsection{\label{sec:eam}Embedded Atom Model} |
| 347 |
|
| 348 |
here there be Monsters |
| 349 |
|
| 350 |
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 351 |
|
| 352 |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 353 |
macroscopic systems with a relatively small number of particles. Simulation |
| 354 |
box is replicated throughout space to form an infinite lattice. During the |
| 355 |
simulation, when a particle moves in the primary cell, its periodic image |
| 356 |
particles in other boxes move in exactly the same direction with exactly the |
| 357 |
same orientation.Thus, as a particle leaves the primary cell, one of its |
| 358 |
images will enter through the opposite face.If the simulation box is large |
| 359 |
enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
| 360 |
effect could be ignored. Cubic and parallelepiped are the available periodic |
| 361 |
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
| 362 |
the property of the simulation box. Therefore, not only the size of the |
| 363 |
simulation box could be changed during the simulation, but also the shape of |
| 364 |
it. The transformation from box space vector $\overrightarrow{s}$ to its |
| 365 |
corresponding real space vector $\overrightarrow{r}$ is defined by |
| 366 |
\begin{equation} |
| 367 |
\overrightarrow{r}=H\overrightarrow{s}% |
| 368 |
\end{equation} |
| 369 |
|
| 370 |
|
| 371 |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 372 |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
| 373 |
simulation box respectively. |
| 374 |
|
| 375 |
To find the minimum image, we need to convert the real vector to its |
| 376 |
corresponding vector in box space first, \bigskip% |
| 377 |
\begin{equation} |
| 378 |
\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
| 379 |
\end{equation} |
| 380 |
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
| 381 |
to 0.5, |
| 382 |
\begin{equation} |
| 383 |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 384 |
\end{equation} |
| 385 |
where% |
| 386 |
|
| 387 |
\begin{equation} |
| 388 |
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
| 389 |
}x\geqslant0 |
| 390 |
\end{equation} |
| 391 |
% |
| 392 |
|
| 393 |
\begin{equation} |
| 394 |
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
| 395 |
\end{equation} |
| 396 |
|
| 397 |
|
| 398 |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 399 |
|
| 400 |
Finally, we could get the minimum image by transforming back to real space,% |
| 401 |
|
| 402 |
\begin{equation} |
| 403 |
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
| 404 |
\end{equation} |
