| 18 |
|
\section{\label{oopseSec:foreword}Foreword} |
| 19 |
|
|
| 20 |
|
In this chapter, I present and detail the capabilities of the open |
| 21 |
< |
source simulation package {\sc oopse}. It is important to note, that a |
| 22 |
< |
simulation package of this size and scope would not have been possible |
| 21 |
> |
source simulation program {\sc oopse}. It is important to note that a |
| 22 |
> |
simulation program of this size and scope would not have been possible |
| 23 |
|
without the collaborative efforts of my colleagues: Charles |
| 24 |
|
F.~Vardeman II, Teng Lin, Christopher J.~Fennell and J.~Daniel |
| 25 |
|
Gezelter. Although my contributions to {\sc oopse} are major, |
| 26 |
|
consideration of my work apart from the others would not give a |
| 27 |
< |
complete description to the package's capabilities. As such, all |
| 27 |
> |
complete description to the program's capabilities. As such, all |
| 28 |
|
contributions to {\sc oopse} to date are presented in this chapter. |
| 29 |
|
|
| 30 |
|
Charles Vardeman is responsible for the parallelization of the long |
| 70 |
|
|
| 71 |
|
Despite their utility, problems with these packages arise when |
| 72 |
|
researchers try to develop techniques or energetic models that the |
| 73 |
< |
code was not originally designed to simulate. Examples of uncommonly |
| 74 |
< |
implemented techniques and energetics include; dipole-dipole |
| 75 |
< |
interactions, rigid body dynamics, and metallic embedded |
| 76 |
< |
potentials. When faced with these obstacles, a researcher must either |
| 77 |
< |
develop their own code or license and extend one of the commercial |
| 78 |
< |
packages. What we have elected to do, is develop a package of |
| 79 |
< |
simulation code capable of implementing the types of models upon which |
| 80 |
< |
our research is based. |
| 73 |
> |
code was not originally designed to simulate. Examples of techniques |
| 74 |
> |
and energetics not commonly implemented include; dipole-dipole |
| 75 |
> |
interactions, rigid body dynamics, and metallic potentials. When faced |
| 76 |
> |
with these obstacles, a researcher must either develop their own code |
| 77 |
> |
or license and extend one of the commercial packages. What we have |
| 78 |
> |
elected to do is develop a body of simulation code capable of |
| 79 |
> |
implementing the types of models upon which our research is based. |
| 80 |
|
|
| 81 |
|
In developing {\sc oopse}, we have adhered to the precepts of Open |
| 82 |
|
Source development, and are releasing our source code with a |
| 160 |
|
\begin{equation} |
| 161 |
|
\boldsymbol{\tau}_i= |
| 162 |
|
\sum_{a}\biggl[(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
| 163 |
< |
+ \boldsymbol{\tau}_{ia}\biggr] |
| 163 |
> |
+ \boldsymbol{\tau}_{ia}\biggr], |
| 164 |
|
\label{eq:torqueAccumulate} |
| 165 |
|
\end{equation} |
| 166 |
|
where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
| 172 |
|
each rigid body. In order to move between the space fixed and body |
| 173 |
|
fixed coordinate axes, parameters describing the orientation must be |
| 174 |
|
maintained for each rigid body. At a minimum, the rotation matrix |
| 175 |
< |
(\textbf{A}) can be described by the three Euler angles ($\phi, |
| 176 |
< |
\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
| 175 |
> |
($\mathsf{A}$) can be described by the three Euler angles ($\phi, |
| 176 |
> |
\theta,$ and $\psi$), where the elements of $\mathsf{A}$ are composed of |
| 177 |
|
trigonometric operations involving $\phi, \theta,$ and |
| 178 |
|
$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
| 179 |
|
inherent in using the Euler angles, the four parameter ``quaternion'' |
| 180 |
< |
scheme is often used. The elements of \textbf{A} can be expressed as |
| 180 |
> |
scheme is often used. The elements of $\mathsf{A}$ can be expressed as |
| 181 |
|
arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
| 182 |
|
and $q_3$).\cite{allen87:csl} Use of quaternions also leads to |
| 183 |
|
performance enhancements, particularly for very small |
| 193 |
|
|
| 194 |
|
\begin{lstlisting}[float,caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}] |
| 195 |
|
molecule{ |
| 196 |
< |
name = "TIP3P_water"; |
| 196 |
> |
name = "TIP3P"; |
| 197 |
> |
nAtoms = 3; |
| 198 |
> |
atom[0]{ |
| 199 |
> |
type = "O_TIP3P"; |
| 200 |
> |
position( 0.0, 0.0, -0.06556 ); |
| 201 |
> |
} |
| 202 |
> |
atom[1]{ |
| 203 |
> |
type = "H_TIP3P"; |
| 204 |
> |
position( 0.0, 0.75695, 0.52032 ); |
| 205 |
> |
} |
| 206 |
> |
atom[2]{ |
| 207 |
> |
type = "H_TIP3P"; |
| 208 |
> |
position( 0.0, -0.75695, 0.52032 ); |
| 209 |
> |
} |
| 210 |
> |
|
| 211 |
|
nRigidBodies = 1; |
| 212 |
< |
rigidBody[0]{ |
| 213 |
< |
nAtoms = 3; |
| 214 |
< |
atom[0]{ |
| 202 |
< |
type = "O_TIP3P"; |
| 203 |
< |
position( 0.0, 0.0, -0.06556 ); |
| 204 |
< |
} |
| 205 |
< |
atom[1]{ |
| 206 |
< |
type = "H_TIP3P"; |
| 207 |
< |
position( 0.0, 0.75695, 0.52032 ); |
| 208 |
< |
} |
| 209 |
< |
atom[2]{ |
| 210 |
< |
type = "H_TIP3P"; |
| 211 |
< |
position( 0.0, -0.75695, 0.52032 ); |
| 212 |
< |
} |
| 213 |
< |
position( 0.0, 0.0, 0.0 ); |
| 214 |
< |
orientation( 0.0, 0.0, 1.0 ); |
| 212 |
> |
rigidBody[0]{ |
| 213 |
> |
nMembers = 3; |
| 214 |
> |
members(0, 1, 2); |
| 215 |
|
} |
| 216 |
|
} |
| 217 |
|
\end{lstlisting} |
| 227 |
|
4\epsilon_{ij} \biggl[ |
| 228 |
|
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
| 229 |
|
- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
| 230 |
< |
\biggr] |
| 230 |
> |
\biggr], |
| 231 |
|
\label{eq:lennardJonesPot} |
| 232 |
|
\end{equation} |
| 233 |
< |
Where $r_{ij}$ is the distance between particles $i$ and $j$, |
| 233 |
> |
where $r_{ij}$ is the distance between particles $i$ and $j$, |
| 234 |
|
$\sigma_{ij}$ scales the length of the interaction, and |
| 235 |
|
$\epsilon_{ij}$ scales the well depth of the potential. Scheme |
| 236 |
< |
\ref{sch:LJFF} gives and example \texttt{.bass} file that |
| 236 |
> |
\ref{sch:LJFF} gives an example \texttt{.bass} file that |
| 237 |
|
sets up a system of 108 Ar particles to be simulated using the |
| 238 |
|
Lennard-Jones force field. |
| 239 |
|
|
| 264 |
|
the energy value at $r_{\text{cut}}$ is subtracted from the |
| 265 |
|
potential. This causes the potential to go to zero smoothly at the |
| 266 |
|
cut-off radius, and preserves conservation of energy in integrating |
| 267 |
< |
the equations of motion. |
| 267 |
> |
the equations of motion. There still remains a discontinuity in the derivative (the forces), however, this does not significantly affect the dynamics. |
| 268 |
|
|
| 269 |
|
Interactions between dissimilar particles requires the generation of |
| 270 |
|
cross term parameters for $\sigma$ and $\epsilon$. These are |
| 271 |
|
calculated through the Lorentz-Berthelot mixing |
| 272 |
|
rules:\cite{allen87:csl} |
| 273 |
|
\begin{equation} |
| 274 |
< |
\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}] |
| 274 |
> |
\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}], |
| 275 |
|
\label{eq:sigmaMix} |
| 276 |
|
\end{equation} |
| 277 |
|
and |
| 278 |
|
\begin{equation} |
| 279 |
< |
\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}} |
| 279 |
> |
\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}. |
| 280 |
|
\label{eq:epsilonMix} |
| 281 |
|
\end{equation} |
| 282 |
|
|
| 283 |
– |
|
| 284 |
– |
|
| 283 |
|
\subsection{\label{oopseSec:DUFF}Dipolar Unified-Atom Force Field} |
| 284 |
|
|
| 285 |
|
The dipolar unified-atom force field ({\sc duff}) was developed to |
| 299 |
|
include a reaction field to mimic larger range interactions. |
| 300 |
|
|
| 301 |
|
As an example, lipid head-groups in {\sc duff} are represented as |
| 302 |
< |
point dipole interaction sites. By placing a dipole at the head group |
| 303 |
< |
center of mass, our model mimics the charge separation found in common |
| 304 |
< |
phospholipids such as phosphatidylcholine.\cite{Cevc87} Additionally, |
| 305 |
< |
a large Lennard-Jones site is located at the pseudoatom's center of |
| 306 |
< |
mass. The model is illustrated by the red atom in |
| 307 |
< |
Fig.~\ref{oopseFig:lipidModel}. The water model we use to complement |
| 308 |
< |
the dipoles of the lipids is our reparameterization of the soft sticky |
| 309 |
< |
dipole (SSD) model of Ichiye |
| 302 |
> |
point dipole interaction sites. By placing a dipole at the head |
| 303 |
> |
group's center of mass, our model mimics the charge separation found |
| 304 |
> |
in common phospholipid head groups such as |
| 305 |
> |
phosphatidylcholine.\cite{Cevc87} Additionally, a large Lennard-Jones |
| 306 |
> |
site is located at the pseudoatom's center of mass. The model is |
| 307 |
> |
illustrated by the red atom in Fig.~\ref{oopseFig:lipidModel}. The |
| 308 |
> |
water model we use to complement the dipoles of the lipids is our |
| 309 |
> |
reparameterization of the soft sticky dipole (SSD) model of Ichiye |
| 310 |
|
\emph{et al.}\cite{liu96:new_model} |
| 311 |
|
|
| 312 |
|
\begin{figure} |
| 313 |
|
\centering |
| 314 |
< |
\includegraphics[width=\linewidth]{lipidModel.eps} |
| 315 |
< |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 316 |
< |
is the bend angle, $\mu$ is the dipole moment of the head group, and n |
| 319 |
< |
is the chain length.} |
| 314 |
> |
\includegraphics[width=\linewidth]{twoChainFig.eps} |
| 315 |
> |
\caption[A representation of a lipid model in {\sc duff}]{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 316 |
> |
is the bend angle, and $\mu$ is the dipole moment of the head group.} |
| 317 |
|
\label{oopseFig:lipidModel} |
| 318 |
|
\end{figure} |
| 319 |
|
|
| 337 |
|
integrating the equations of motion. A simulation using {\sc duff} is |
| 338 |
|
illustrated in Scheme \ref{sch:DUFF}. |
| 339 |
|
|
| 340 |
< |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
| 340 |
> |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]A portion of a \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
| 341 |
|
|
| 342 |
|
#include "water.mdl" |
| 343 |
|
#include "lipid.mdl" |
| 364 |
|
The total potential energy function in {\sc duff} is |
| 365 |
|
\begin{equation} |
| 366 |
|
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 367 |
< |
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
| 367 |
> |
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}}, |
| 368 |
|
\label{eq:totalPotential} |
| 369 |
|
\end{equation} |
| 370 |
< |
Where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
| 370 |
> |
where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
| 371 |
|
\begin{equation} |
| 372 |
|
V^{I}_{\text{Internal}} = |
| 373 |
|
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 375 |
|
+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 376 |
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 377 |
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 378 |
< |
\biggr] |
| 378 |
> |
\biggr]. |
| 379 |
|
\label{eq:internalPotential} |
| 380 |
|
\end{equation} |
| 381 |
|
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 382 |
|
within the molecule $I$, and $V_{\text{torsion}}$ is the torsion potential |
| 383 |
|
for all 1, 4 bonded pairs. The pairwise portions of the internal |
| 384 |
< |
potential are excluded for pairs that are closer than three bonds, |
| 388 |
< |
i.e.~atom pairs farther away than a torsion are included in the |
| 389 |
< |
pair-wise loop. |
| 384 |
> |
potential are excluded for atom pairs that are involved in the same bond, bend, or torsion. All other atom pairs within the molecule are subject to the LJ pair potential. |
| 385 |
|
|
| 386 |
|
|
| 387 |
|
The bend potential of a molecule is represented by the following function: |
| 388 |
|
\begin{equation} |
| 389 |
< |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 389 |
> |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2, \label{eq:bendPot} |
| 390 |
|
\end{equation} |
| 391 |
< |
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 391 |
> |
where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 392 |
|
(see Fig.~\ref{oopseFig:lipidModel}), $\theta_0$ is the equilibrium |
| 393 |
|
bond angle, and $k_{\theta}$ is the force constant which determines the |
| 394 |
|
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 399 |
|
\begin{equation} |
| 400 |
|
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
| 401 |
|
+ c_2[1 + \cos(2\phi)] |
| 402 |
< |
+ c_3[1 + \cos(3\phi)] |
| 402 |
> |
+ c_3[1 + \cos(3\phi)], |
| 403 |
|
\label{eq:origTorsionPot} |
| 404 |
|
\end{equation} |
| 405 |
< |
Where: |
| 405 |
> |
where: |
| 406 |
|
\begin{equation} |
| 407 |
|
\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 408 |
< |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}) |
| 408 |
> |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}). |
| 409 |
|
\label{eq:torsPhi} |
| 410 |
|
\end{equation} |
| 411 |
|
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 415 |
|
a power series of the form: |
| 416 |
|
\begin{equation} |
| 417 |
|
V_{\text{torsion}}(\phi) = |
| 418 |
< |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 418 |
> |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 419 |
|
\label{eq:torsionPot} |
| 420 |
|
\end{equation} |
| 421 |
< |
Where: |
| 421 |
> |
where: |
| 422 |
|
\begin{align*} |
| 423 |
< |
k_0 &= c_1 + c_3 \\ |
| 424 |
< |
k_1 &= c_1 - 3c_3 \\ |
| 425 |
< |
k_2 &= 2 c_2 \\ |
| 426 |
< |
k_3 &= 4c_3 |
| 423 |
> |
k_0 &= c_1 + c_3, \\ |
| 424 |
> |
k_1 &= c_1 - 3c_3, \\ |
| 425 |
> |
k_2 &= 2 c_2, \\ |
| 426 |
> |
k_3 &= 4c_3. |
| 427 |
|
\end{align*} |
| 428 |
|
By recasting the potential as a power series, repeated trigonometric |
| 429 |
|
evaluations are avoided during the calculation of the potential energy. |
| 438 |
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 439 |
|
+ V_{\text{sticky}} |
| 440 |
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 441 |
< |
\biggr] |
| 441 |
> |
\biggr], |
| 442 |
|
\label{eq:crossPotentail} |
| 443 |
|
\end{equation} |
| 444 |
< |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 444 |
> |
where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 445 |
|
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 446 |
|
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
| 447 |
|
(Sec.~\ref{oopseSec:SSD}). Note that not all atom types include all |
| 454 |
|
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 455 |
|
- |
| 456 |
|
3(\boldsymbol{\hat{u}}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 457 |
< |
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr] |
| 457 |
> |
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr]. |
| 458 |
|
\label{eq:dipolePot} |
| 459 |
|
\end{equation} |
| 460 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 492 |
|
|
| 493 |
|
In the interest of computational efficiency, the default solvent used |
| 494 |
|
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
| 495 |
< |
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
| 495 |
> |
model.\cite{fennell04} The original SSD was developed by Ichiye |
| 496 |
|
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
| 497 |
|
water model proposed by Bratko, Blum, and |
| 498 |
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 566 |
|
|
| 567 |
|
Recent constant pressure simulations revealed issues in the original |
| 568 |
|
SSD model that led to lower than expected densities at all target |
| 569 |
< |
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
| 569 |
> |
pressures.\cite{Ichiye03,fennell04} The default model in {\sc oopse} |
| 570 |
|
is therefore SSD/E, a density corrected derivative of SSD that |
| 571 |
|
exhibits improved liquid structure and transport behavior. If the use |
| 572 |
|
of a reaction field long-range interaction correction is desired, it |
| 573 |
|
is recommended that the parameters be modified to those of the SSD/RF |
| 574 |
< |
model. Solvent parameters can be easily modified in an accompanying |
| 574 |
> |
model (an SSD variant parameterized for reaction field). Solvent parameters can be easily modified in an accompanying |
| 575 |
|
\texttt{.bass} file as illustrated in the scheme below. A table of the |
| 576 |
|
parameter values and the drawbacks and benefits of the different |
| 577 |
|
density corrected SSD models can be found in |
| 578 |
< |
reference~\cite{Gezelter04}. |
| 578 |
> |
reference~\cite{fennell04}. |
| 579 |
|
|
| 580 |
< |
\begin{lstlisting}[float,caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
| 580 |
> |
\begin{lstlisting}[float,caption={[A simulation of {\sc ssd} water]A portion of a \texttt{.bass} file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
| 581 |
|
|
| 582 |
|
#include "water.mdl" |
| 583 |
|
|
| 625 |
|
The {\sc eam} potential has the form: |
| 626 |
|
\begin{eqnarray} |
| 627 |
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 628 |
< |
\phi_{ij}({\bf r}_{ij}) \\ |
| 629 |
< |
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 628 |
> |
\phi_{ij}({\bf r}_{ij}), \\ |
| 629 |
> |
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}), |
| 630 |
|
\end{eqnarray} |
| 631 |
< |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 632 |
< |
positively-charged core ion $i$ into a linear superposition of |
| 633 |
< |
spherically averaged atomic electron densities given by |
| 634 |
< |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 635 |
< |
between atoms $i$ and $j$. In the original formulation of |
| 636 |
< |
{\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 637 |
< |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 638 |
< |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 639 |
< |
There is a cutoff distance, $r_{cut}$, which limits the |
| 640 |
< |
summations in the {\sc eam} equation to the few dozen atoms |
| 631 |
> |
where $F_{i} $ is the embedding function that equates the energy |
| 632 |
> |
required to embed a positively-charged core ion $i$ into a linear |
| 633 |
> |
superposition of spherically averaged atomic electron densities given |
| 634 |
> |
by $\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise |
| 635 |
> |
interaction between atoms $i$ and $j$. In the original formulation of |
| 636 |
> |
{\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, |
| 637 |
> |
however in later refinements to {\sc eam} have shown that non-uniqueness |
| 638 |
> |
between $F$ and $\phi$ allow for more general forms for |
| 639 |
> |
$\phi$.\cite{Daw89} There is a cutoff distance, $r_{cut}$, which |
| 640 |
> |
limits the summations in the {\sc eam} equation to the few dozen atoms |
| 641 |
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 642 |
< |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FBD86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 642 |
> |
interactions. Foiles \emph{et al}.~fit {\sc eam} potentials for the fcc |
| 643 |
> |
metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals.\cite{FBD86} |
| 644 |
> |
These fits are included in {\sc oopse}. |
| 645 |
|
|
| 649 |
– |
|
| 646 |
|
\subsection{\label{oopseSec:pbc}Periodic Boundary Conditions} |
| 647 |
|
|
| 648 |
|
\newcommand{\roundme}{\operatorname{round}} |
| 656 |
|
simulation box is large enough to avoid ``feeling'' the symmetries of |
| 657 |
|
the periodic lattice, surface effects can be ignored. The available |
| 658 |
|
periodic cells in OOPSE are cubic, orthorhombic and parallelepiped. We |
| 659 |
< |
use a $3 \times 3$ matrix, $\mathbf{H}$, to describe the shape and |
| 660 |
< |
size of the simulation box. $\mathbf{H}$ is defined: |
| 659 |
> |
use a $3 \times 3$ matrix, $\mathsf{H}$, to describe the shape and |
| 660 |
> |
size of the simulation box. $\mathsf{H}$ is defined: |
| 661 |
|
\begin{equation} |
| 662 |
< |
\mathbf{H} = ( \mathbf{h}_x, \mathbf{h}_y, \mathbf{h}_z ) |
| 662 |
> |
\mathsf{H} = ( \mathbf{h}_x, \mathbf{h}_y, \mathbf{h}_z ), |
| 663 |
|
\end{equation} |
| 664 |
< |
Where $\mathbf{h}_j$ is the column vector of the $j$th axis of the |
| 664 |
> |
where $\mathbf{h}_{\alpha}$ is the column vector of the $\alpha$ axis of the |
| 665 |
|
box. During the course of the simulation both the size and shape of |
| 666 |
< |
the box can be changed to allow volume fluctations when constraining |
| 666 |
> |
the box can be changed to allow volume fluctuations when constraining |
| 667 |
|
the pressure. |
| 668 |
|
|
| 669 |
|
A real space vector, $\mathbf{r}$ can be transformed in to a box space |
| 670 |
|
vector, $\mathbf{s}$, and back through the following transformations: |
| 671 |
|
\begin{align} |
| 672 |
< |
\mathbf{s} &= \mathbf{H}^{-1} \mathbf{r} \\ |
| 673 |
< |
\mathbf{r} &= \mathbf{H} \mathbf{s} |
| 672 |
> |
\mathbf{s} &= \mathsf{H}^{-1} \mathbf{r}, \\ |
| 673 |
> |
\mathbf{r} &= \mathsf{H} \mathbf{s}. |
| 674 |
|
\end{align} |
| 675 |
|
The vector $\mathbf{s}$ is now a vector expressed as the number of box |
| 676 |
|
lengths in the $\mathbf{h}_x$, $\mathbf{h}_y$, and $\mathbf{h}_z$ |
| 677 |
|
directions. To find the minimum image of a vector $\mathbf{r}$, we |
| 678 |
|
first convert it to its corresponding vector in box space, and then, |
| 679 |
< |
cast each element to lie on the in the range $[-0.5,0.5]$: |
| 679 |
> |
cast each element to lie in the range $[-0.5,0.5]$: |
| 680 |
|
\begin{equation} |
| 681 |
< |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 681 |
> |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}), |
| 682 |
|
\end{equation} |
| 683 |
< |
Where $s_i$ is the $i$th element of $\mathbf{s}$, and |
| 684 |
< |
$\roundme(s_i)$is given by |
| 683 |
> |
where $s_i$ is the $i$th element of $\mathbf{s}$, and |
| 684 |
> |
$\roundme(s_i)$ is given by |
| 685 |
|
\begin{equation} |
| 686 |
|
\roundme(x) = |
| 687 |
|
\begin{cases} |
| 688 |
< |
\lfloor x+0.5 \rfloor & \text{if $x \ge 0$} \\ |
| 689 |
< |
\lceil x-0.5 \rceil & \text{if $x < 0$ } |
| 688 |
> |
\lfloor x+0.5 \rfloor & \text{if $x \ge 0$,} \\ |
| 689 |
> |
\lceil x-0.5 \rceil & \text{if $x < 0$.} |
| 690 |
|
\end{cases} |
| 691 |
|
\end{equation} |
| 692 |
|
Here $\lfloor x \rfloor$ is the floor operator, and gives the largest |
| 698 |
|
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
| 699 |
|
transforming back to real space, |
| 700 |
|
\begin{equation} |
| 701 |
< |
\mathbf{r}^{\prime}=\mathbf{H}^{-1}\mathbf{s}^{\prime}% |
| 701 |
> |
\mathbf{r}^{\prime}=\mathsf{H}^{-1}\mathbf{s}^{\prime}.% |
| 702 |
|
\end{equation} |
| 703 |
|
In this way, particles are allowed to diffuse freely in $\mathbf{r}$, |
| 704 |
|
but their minimum images, $\mathbf{r}^{\prime}$ are used to compute |
| 705 |
< |
the interatomic forces. |
| 705 |
> |
the inter-atomic forces. |
| 706 |
|
|
| 707 |
|
|
| 708 |
|
\section{\label{oopseSec:IOfiles}Input and Output Files} |
| 709 |
|
|
| 710 |
|
\subsection{{\sc bass} and Model Files} |
| 711 |
|
|
| 712 |
< |
Every {\sc oopse} simulation begins with a {\sc bass} file. {\sc bass} |
| 713 |
< |
(\underline{B}izarre \underline{A}tom \underline{S}imulation |
| 714 |
< |
\underline{S}yntax) is a script syntax that is parsed by {\sc oopse} at |
| 715 |
< |
runtime. The {\sc bass} file allows for the user to completely describe the |
| 716 |
< |
system they are to simulate, as well as tailor {\sc oopse}'s behavior during |
| 717 |
< |
the simulation. {\sc bass} files are denoted with the extension |
| 712 |
> |
Every {\sc oopse} simulation begins with a Bizarre Atom Simulation |
| 713 |
> |
Syntax ({\sc bass}) file. {\sc bass} is a script syntax that is parsed |
| 714 |
> |
by {\sc oopse} at runtime. The {\sc bass} file allows for the user to |
| 715 |
> |
completely describe the system they wish to simulate, as well as tailor |
| 716 |
> |
{\sc oopse}'s behavior during the simulation. {\sc bass} files are |
| 717 |
> |
denoted with the extension |
| 718 |
|
\texttt{.bass}, an example file is shown in |
| 719 |
< |
Fig.~\ref{fig:bassExample}. |
| 719 |
> |
Scheme~\ref{sch:bassExample}. |
| 720 |
|
|
| 721 |
< |
\begin{figure} |
| 722 |
< |
\centering |
| 723 |
< |
\framebox[\linewidth]{\rule{0cm}{0.75\linewidth}I'm a {\sc bass} file!} |
| 724 |
< |
\caption{Here is an example \texttt{.bass} file} |
| 725 |
< |
\label{fig:bassExample} |
| 726 |
< |
\end{figure} |
| 721 |
> |
\begin{lstlisting}[float,caption={[An example of a complete {\sc bass} file] An example showing a complete {\sc bass} file.},label={sch:bassExample}] |
| 722 |
> |
|
| 723 |
> |
molecule{ |
| 724 |
> |
name = "Ar"; |
| 725 |
> |
nAtoms = 1; |
| 726 |
> |
atom[0]{ |
| 727 |
> |
type="Ar"; |
| 728 |
> |
position( 0.0, 0.0, 0.0 ); |
| 729 |
> |
} |
| 730 |
> |
} |
| 731 |
|
|
| 732 |
+ |
nComponents = 1; |
| 733 |
+ |
component{ |
| 734 |
+ |
type = "Ar"; |
| 735 |
+ |
nMol = 108; |
| 736 |
+ |
} |
| 737 |
+ |
|
| 738 |
+ |
initialConfig = "./argon.init"; |
| 739 |
+ |
|
| 740 |
+ |
forceField = "LJ"; |
| 741 |
+ |
ensemble = "NVE"; // specify the simulation ensemble |
| 742 |
+ |
dt = 1.0; // the time step for integration |
| 743 |
+ |
runTime = 1e3; // the total simulation run time |
| 744 |
+ |
sampleTime = 100; // trajectory file frequency |
| 745 |
+ |
statusTime = 50; // statistics file frequency |
| 746 |
+ |
|
| 747 |
+ |
\end{lstlisting} |
| 748 |
+ |
|
| 749 |
|
Within the \texttt{.bass} file it is necessary to provide a complete |
| 750 |
|
description of the molecule before it is actually placed in the |
| 751 |
< |
simulation. The {\sc bass} syntax was originally developed with this goal in |
| 752 |
< |
mind, and allows for the specification of all the atoms in a molecular |
| 753 |
< |
prototype, as well as any bonds, bends, or torsions. These |
| 751 |
> |
simulation. The {\sc bass} syntax was originally developed with this |
| 752 |
> |
goal in mind, and allows for the specification of all the atoms in a |
| 753 |
> |
molecular prototype, as well as any bonds, bends, or torsions. These |
| 754 |
|
descriptions can become lengthy for complex molecules, and it would be |
| 755 |
< |
inconvenient to duplicate the simulation at the beginning of each {\sc bass} |
| 756 |
< |
script. Addressing this issue {\sc bass} allows for the inclusion of model |
| 757 |
< |
files at the top of a \texttt{.bass} file. These model files, denoted |
| 758 |
< |
with the \texttt{.mdl} extension, allow the user to describe a |
| 759 |
< |
molecular prototype once, then simply include it into each simulation |
| 760 |
< |
containing that molecule. |
| 755 |
> |
inconvenient to duplicate the simulation at the beginning of each {\sc |
| 756 |
> |
bass} script. Addressing this issue {\sc bass} allows for the |
| 757 |
> |
inclusion of model files at the top of a \texttt{.bass} file. These |
| 758 |
> |
model files, denoted with the \texttt{.mdl} extension, allow the user |
| 759 |
> |
to describe a molecular prototype once, then simply include it into |
| 760 |
> |
each simulation containing that molecule. Returning to the example in |
| 761 |
> |
Scheme~\ref{sch:bassExample}, the \texttt{.mdl} file's contents would |
| 762 |
> |
be Scheme~\ref{sch:mdlExample}, and the new \texttt{.bass} file would |
| 763 |
> |
become Scheme~\ref{sch:bassExPrime}. |
| 764 |
|
|
| 765 |
+ |
\begin{lstlisting}[float,caption={An example \texttt{.mdl} file.},label={sch:mdlExample}] |
| 766 |
+ |
|
| 767 |
+ |
molecule{ |
| 768 |
+ |
name = "Ar"; |
| 769 |
+ |
nAtoms = 1; |
| 770 |
+ |
atom[0]{ |
| 771 |
+ |
type="Ar"; |
| 772 |
+ |
position( 0.0, 0.0, 0.0 ); |
| 773 |
+ |
} |
| 774 |
+ |
} |
| 775 |
+ |
|
| 776 |
+ |
\end{lstlisting} |
| 777 |
+ |
|
| 778 |
+ |
\begin{lstlisting}[float,caption={Revised {\sc bass} example.},label={sch:bassExPrime}] |
| 779 |
+ |
|
| 780 |
+ |
#include "argon.mdl" |
| 781 |
+ |
|
| 782 |
+ |
nComponents = 1; |
| 783 |
+ |
component{ |
| 784 |
+ |
type = "Ar"; |
| 785 |
+ |
nMol = 108; |
| 786 |
+ |
} |
| 787 |
+ |
|
| 788 |
+ |
initialConfig = "./argon.init"; |
| 789 |
+ |
|
| 790 |
+ |
forceField = "LJ"; |
| 791 |
+ |
ensemble = "NVE"; |
| 792 |
+ |
dt = 1.0; |
| 793 |
+ |
runTime = 1e3; |
| 794 |
+ |
sampleTime = 100; |
| 795 |
+ |
statusTime = 50; |
| 796 |
+ |
|
| 797 |
+ |
\end{lstlisting} |
| 798 |
+ |
|
| 799 |
|
\subsection{\label{oopseSec:coordFiles}Coordinate Files} |
| 800 |
|
|
| 801 |
|
The standard format for storage of a systems coordinates is a modified |
| 802 |
|
xyz-file syntax, the exact details of which can be seen in |
| 803 |
< |
App.~\ref{appCoordFormat}. As all bonding and molecular information is |
| 804 |
< |
stored in the \texttt{.bass} and \texttt{.mdl} files, the coordinate |
| 805 |
< |
files are simply the complete set of coordinates for each atom at a |
| 806 |
< |
given simulation time. |
| 803 |
> |
Scheme~\ref{sch:dumpFormat}. As all bonding and molecular information |
| 804 |
> |
is stored in the \texttt{.bass} and \texttt{.mdl} files, the |
| 805 |
> |
coordinate files are simply the complete set of coordinates for each |
| 806 |
> |
atom at a given simulation time. One important note, although the |
| 807 |
> |
simulation propagates the complete rotation matrix, directional |
| 808 |
> |
entities are written out using quanternions, to save space in the |
| 809 |
> |
output files. |
| 810 |
|
|
| 811 |
< |
There are three major files used by {\sc oopse} written in the coordinate |
| 812 |
< |
format, they are as follows: the initialization file, the simulation |
| 813 |
< |
trajectory file, and the final coordinates of the simulation. The |
| 814 |
< |
initialization file is necessary for {\sc oopse} to start the simulation |
| 815 |
< |
with the proper coordinates. It is typically denoted with the |
| 816 |
< |
extension \texttt{.init}. The trajectory file is created at the |
| 817 |
< |
beginning of the simulation, and is used to store snapshots of the |
| 818 |
< |
simulation at regular intervals. The first frame is a duplication of |
| 819 |
< |
the \texttt{.init} file, and each subsequent frame is appended to the |
| 820 |
< |
file at an interval specified in the \texttt{.bass} file. The |
| 821 |
< |
trajectory file is given the extension \texttt{.dump}. The final |
| 822 |
< |
coordinate file is the end of run or \texttt{.eor} file. The |
| 811 |
> |
\begin{lstlisting}[float,caption={[The format of the coordinate files]Shows the format of the coordinate files. The fist line is the number of atoms. The second line begins with the time stamp followed by the three $\mathsf{H}$ column vectors. It is important to note, that for extended system ensembles, additional information pertinent to the integrators may be stored on this line as well. The next lines are the atomic coordinates for all atoms in the system. First is the name followed by position, velocity, quanternions, and lastly angular velocities.},label=sch:dumpFormat] |
| 812 |
> |
|
| 813 |
> |
nAtoms |
| 814 |
> |
time; Hxx Hyx Hzx; Hxy Hyy Hzy; Hxz Hyz Hzz; |
| 815 |
> |
Name1 x y z vx vy vz q0 q1 q2 q3 jx jy jz |
| 816 |
> |
Name2 x y z vx vy vz q0 q1 q2 q3 jx jy jz |
| 817 |
> |
etc... |
| 818 |
> |
|
| 819 |
> |
\end{lstlisting} |
| 820 |
> |
|
| 821 |
> |
|
| 822 |
> |
There are three major files used by {\sc oopse} written in the |
| 823 |
> |
coordinate format, they are as follows: the initialization file |
| 824 |
> |
(\texttt{.init}), the simulation trajectory file (\texttt{.dump}), and |
| 825 |
> |
the final coordinates of the simulation. The initialization file is |
| 826 |
> |
necessary for {\sc oopse} to start the simulation with the proper |
| 827 |
> |
coordinates, and is generated before the simulation run. The |
| 828 |
> |
trajectory file is created at the beginning of the simulation, and is |
| 829 |
> |
used to store snapshots of the simulation at regular intervals. The |
| 830 |
> |
first frame is a duplication of the |
| 831 |
> |
\texttt{.init} file, and each subsequent frame is appended to the file |
| 832 |
> |
at an interval specified in the \texttt{.bass} file with the |
| 833 |
> |
\texttt{sampleTime} flag. The final coordinate file is the end of run file. The |
| 834 |
|
\texttt{.eor} file stores the final configuration of the system for a |
| 835 |
|
given simulation. The file is updated at the same time as the |
| 836 |
< |
\texttt{.dump} file. However, it only contains the most recent |
| 836 |
> |
\texttt{.dump} file, however, it only contains the most recent |
| 837 |
|
frame. In this way, an \texttt{.eor} file may be used as the |
| 838 |
< |
initialization file to a second simulation in order to continue or |
| 839 |
< |
recover the previous simulation. |
| 838 |
> |
initialization file to a second simulation in order to continue a |
| 839 |
> |
simulation or recover one from a processor that has crashed during the |
| 840 |
> |
course of the run. |
| 841 |
|
|
| 842 |
|
\subsection{\label{oopseSec:initCoords}Generation of Initial Coordinates} |
| 843 |
|
|
| 844 |
< |
As was stated in Sec.~\ref{oopseSec:coordFiles}, an initialization file |
| 845 |
< |
is needed to provide the starting coordinates for a simulation. The |
| 846 |
< |
{\sc oopse} package provides a program called \texttt{sysBuilder} to aid in |
| 847 |
< |
the creation of the \texttt{.init} file. \texttt{sysBuilder} is {\sc bass} |
| 848 |
< |
aware, and will recognize arguments and parameters in the |
| 849 |
< |
\texttt{.bass} file that would otherwise be ignored by the |
| 850 |
< |
simulation. The program itself is under continual development, and is |
| 782 |
< |
offered here as a helper tool only. |
| 844 |
> |
As was stated in Sec.~\ref{oopseSec:coordFiles}, an initialization |
| 845 |
> |
file is needed to provide the starting coordinates for a |
| 846 |
> |
simulation. The {\sc oopse} package provides several system building |
| 847 |
> |
programs to aid in the creation of the \texttt{.init} |
| 848 |
> |
file. The programs use {\sc bass}, and will recognize |
| 849 |
> |
arguments and parameters in the \texttt{.bass} file that would |
| 850 |
> |
otherwise be ignored by the simulation. |
| 851 |
|
|
| 852 |
|
\subsection{The Statistics File} |
| 853 |
|
|
| 854 |
< |
The last output file generated by {\sc oopse} is the statistics file. This |
| 855 |
< |
file records such statistical quantities as the instantaneous |
| 856 |
< |
temperature, volume, pressure, etc. It is written out with the |
| 857 |
< |
frequency specified in the \texttt{.bass} file. The file allows the |
| 858 |
< |
user to observe the system variables as a function of simulation time |
| 859 |
< |
while the simulation is in progress. One useful function the |
| 860 |
< |
statistics file serves is to monitor the conserved quantity of a given |
| 861 |
< |
simulation ensemble, this allows the user to observe the stability of |
| 862 |
< |
the integrator. The statistics file is denoted with the \texttt{.stat} |
| 863 |
< |
file extension. |
| 854 |
> |
The last output file generated by {\sc oopse} is the statistics |
| 855 |
> |
file. This file records such statistical quantities as the |
| 856 |
> |
instantaneous temperature, volume, pressure, etc. It is written out |
| 857 |
> |
with the frequency specified in the \texttt{.bass} file with the |
| 858 |
> |
\texttt{statusTime} keyword. The file allows the user to observe the |
| 859 |
> |
system variables as a function of simulation time while the simulation |
| 860 |
> |
is in progress. One useful function the statistics file serves is to |
| 861 |
> |
monitor the conserved quantity of a given simulation ensemble, this |
| 862 |
> |
allows the user to observe the stability of the integrator. The |
| 863 |
> |
statistics file is denoted with the \texttt{.stat} file extension. |
| 864 |
|
|
| 865 |
|
\section{\label{oopseSec:mechanics}Mechanics} |
| 866 |
|
|
| 867 |
< |
\subsection{\label{oopseSec:integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
| 867 |
> |
\subsection{\label{oopseSec:integrate}Integrating the Equations of Motion: the |
| 868 |
> |
DLM method} |
| 869 |
|
|
| 870 |
< |
Integration of the equations of motion was carried out using the |
| 871 |
< |
symplectic splitting method proposed by Dullweber \emph{et |
| 872 |
< |
al.}.\cite{Dullweber1997} The reason for this integrator selection |
| 873 |
< |
deals with poor energy conservation of rigid body systems using |
| 874 |
< |
quaternions. While quaternions work well for orientational motion in |
| 875 |
< |
alternate ensembles, the microcanonical ensemble has a constant energy |
| 876 |
< |
requirement that is quite sensitive to errors in the equations of |
| 808 |
< |
motion. The original implementation of this code utilized quaternions |
| 809 |
< |
for rotational motion propagation; however, a detailed investigation |
| 810 |
< |
showed that they resulted in a steady drift in the total energy, |
| 811 |
< |
something that has been observed by others.\cite{Laird97} |
| 870 |
> |
The default method for integrating the equations of motion in {\sc |
| 871 |
> |
oopse} is a velocity-Verlet version of the symplectic splitting method |
| 872 |
> |
proposed by Dullweber, Leimkuhler and McLachlan |
| 873 |
> |
(DLM).\cite{Dullweber1997} When there are no directional atoms or |
| 874 |
> |
rigid bodies present in the simulation, this integrator becomes the |
| 875 |
> |
standard velocity-Verlet integrator which is known to sample the |
| 876 |
> |
microcanonical (NVE) ensemble.\cite{Frenkel1996} |
| 877 |
|
|
| 878 |
+ |
Previous integration methods for orientational motion have problems |
| 879 |
+ |
that are avoided in the DLM method. Direct propagation of the Euler |
| 880 |
+ |
angles has a known $1/\sin\theta$ divergence in the equations of |
| 881 |
+ |
motion for $\phi$ and $\psi$,\cite{allen87:csl} leading to |
| 882 |
+ |
numerical instabilities any time one of the directional atoms or rigid |
| 883 |
+ |
bodies has an orientation near $\theta=0$ or $\theta=\pi$. More |
| 884 |
+ |
modern quaternion-based integration methods have relatively poor |
| 885 |
+ |
energy conservation. While quaternions work well for orientational |
| 886 |
+ |
motion in other ensembles, the microcanonical ensemble has a |
| 887 |
+ |
constant energy requirement that is quite sensitive to errors in the |
| 888 |
+ |
equations of motion. An earlier implementation of {\sc oopse} |
| 889 |
+ |
utilized quaternions for propagation of rotational motion; however, a |
| 890 |
+ |
detailed investigation showed that they resulted in a steady drift in |
| 891 |
+ |
the total energy, something that has been observed by |
| 892 |
+ |
Laird {\it et al.}\cite{Laird97} |
| 893 |
+ |
|
| 894 |
|
The key difference in the integration method proposed by Dullweber |
| 895 |
< |
\emph{et al.} is that the entire rotation matrix is propagated from |
| 896 |
< |
one time step to the next. In the past, this would not have been as |
| 897 |
< |
feasible a option, being that the rotation matrix for a single body is |
| 898 |
< |
nine elements long as opposed to 3 or 4 elements for Euler angles and |
| 899 |
< |
quaternions respectively. System memory has become much less of an |
| 900 |
< |
issue in recent times, and this has resulted in substantial benefits |
| 901 |
< |
in energy conservation. There is still the issue of 5 or 6 additional |
| 821 |
< |
elements for describing the orientation of each particle, which will |
| 822 |
< |
increase dump files substantially. Simply translating the rotation |
| 823 |
< |
matrix into its component Euler angles or quaternions for storage |
| 824 |
< |
purposes relieves this burden. |
| 895 |
> |
\emph{et al.} is that the entire $3 \times 3$ rotation matrix is |
| 896 |
> |
propagated from one time step to the next. In the past, this would not |
| 897 |
> |
have been feasible, since the rotation matrix for a single body has |
| 898 |
> |
nine elements compared with the more memory-efficient methods (using |
| 899 |
> |
three Euler angles or 4 quaternions). Computer memory has become much |
| 900 |
> |
less costly in recent years, and this can be translated into |
| 901 |
> |
substantial benefits in energy conservation. |
| 902 |
|
|
| 903 |
< |
The symplectic splitting method allows for Verlet style integration of |
| 904 |
< |
both linear and angular motion of rigid bodies. In the integration |
| 905 |
< |
method, the orientational propagation involves a sequence of matrix |
| 906 |
< |
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
| 907 |
< |
matrix rotations end up being more costly computationally than the |
| 908 |
< |
simpler arithmetic quaternion propagation. With the same time step, a |
| 909 |
< |
1000 SSD particle simulation shows an average 7\% increase in |
| 910 |
< |
computation time using the symplectic step method in place of |
| 911 |
< |
quaternions. This cost is more than justified when comparing the |
| 912 |
< |
energy conservation of the two methods as illustrated in figure |
| 913 |
< |
\ref{timestep}. |
| 903 |
> |
The basic equations of motion being integrated are derived from the |
| 904 |
> |
Hamiltonian for conservative systems containing rigid bodies, |
| 905 |
> |
\begin{equation} |
| 906 |
> |
H = \sum_{i} \left( \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 907 |
> |
\frac{1}{2} {\bf j}_i^T \cdot \overleftrightarrow{\mathsf{I}}_i^{-1} \cdot |
| 908 |
> |
{\bf j}_i \right) + |
| 909 |
> |
V\left(\left\{{\bf r}\right\}, \left\{\mathsf{A}\right\}\right), |
| 910 |
> |
\end{equation} |
| 911 |
> |
where ${\bf r}_i$ and ${\bf v}_i$ are the cartesian position vector |
| 912 |
> |
and velocity of the center of mass of particle $i$, and ${\bf j}_i$, |
| 913 |
> |
$\overleftrightarrow{\mathsf{I}}_i$ are the body-fixed angular |
| 914 |
> |
momentum and moment of inertia tensor respectively, and the |
| 915 |
> |
superscript $T$ denotes the transpose of the vector. $\mathsf{A}_i$ |
| 916 |
> |
is the $3 \times 3$ rotation matrix describing the instantaneous |
| 917 |
> |
orientation of the particle. $V$ is the potential energy function |
| 918 |
> |
which may depend on both the positions $\left\{{\bf r}\right\}$ and |
| 919 |
> |
orientations $\left\{\mathsf{A}\right\}$ of all particles. The |
| 920 |
> |
equations of motion for the particle centers of mass are derived from |
| 921 |
> |
Hamilton's equations and are quite simple, |
| 922 |
> |
\begin{eqnarray} |
| 923 |
> |
\dot{{\bf r}} & = & {\bf v}, \\ |
| 924 |
> |
\dot{{\bf v}} & = & \frac{{\bf f}}{m}, |
| 925 |
> |
\end{eqnarray} |
| 926 |
> |
where ${\bf f}$ is the instantaneous force on the center of mass |
| 927 |
> |
of the particle, |
| 928 |
> |
\begin{equation} |
| 929 |
> |
{\bf f} = - \frac{\partial}{\partial |
| 930 |
> |
{\bf r}} V(\left\{{\bf r}(t)\right\}, \left\{\mathsf{A}(t)\right\}). |
| 931 |
> |
\end{equation} |
| 932 |
|
|
| 933 |
+ |
The equations of motion for the orientational degrees of freedom are |
| 934 |
+ |
\begin{eqnarray} |
| 935 |
+ |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 936 |
+ |
\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right),\\ |
| 937 |
+ |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{\mathsf{I}}^{-1} |
| 938 |
+ |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 939 |
+ |
V}{\partial \mathsf{A}} \right). |
| 940 |
+ |
\end{eqnarray} |
| 941 |
+ |
In these equations of motion, the $\mbox{skew}$ matrix of a vector |
| 942 |
+ |
${\bf v} = \left( v_1, v_2, v_3 \right)$ is defined: |
| 943 |
+ |
\begin{equation} |
| 944 |
+ |
\mbox{skew}\left( {\bf v} \right) := \left( |
| 945 |
+ |
\begin{array}{ccc} |
| 946 |
+ |
0 & v_3 & - v_2 \\ |
| 947 |
+ |
-v_3 & 0 & v_1 \\ |
| 948 |
+ |
v_2 & -v_1 & 0 |
| 949 |
+ |
\end{array} |
| 950 |
+ |
\right). |
| 951 |
+ |
\end{equation} |
| 952 |
+ |
The $\mbox{rot}$ notation refers to the mapping of the $3 \times 3$ |
| 953 |
+ |
rotation matrix to a vector of orientations by first computing the |
| 954 |
+ |
skew-symmetric part $\left(\mathsf{A} - \mathsf{A}^{T}\right)$ and |
| 955 |
+ |
then associating this with a length 3 vector by inverting the |
| 956 |
+ |
$\mbox{skew}$ function above: |
| 957 |
+ |
\begin{equation} |
| 958 |
+ |
\mbox{rot}\left(\mathsf{A}\right) := \mbox{ skew}^{-1}\left(\mathsf{A} |
| 959 |
+ |
- \mathsf{A}^{T} \right). |
| 960 |
+ |
\end{equation} |
| 961 |
+ |
Written this way, the $\mbox{rot}$ operation creates a set of |
| 962 |
+ |
conjugate angle coordinates to the body-fixed angular momenta |
| 963 |
+ |
represented by ${\bf j}$. This equation of motion for angular momenta |
| 964 |
+ |
is equivalent to the more familiar body-fixed forms, |
| 965 |
+ |
\begin{eqnarray} |
| 966 |
+ |
\dot{j_{x}} & = & \tau^b_x(t) + |
| 967 |
+ |
\left(\overleftrightarrow{\mathsf{I}}_{yy} - \overleftrightarrow{\mathsf{I}}_{zz} \right) j_y j_z, \\ |
| 968 |
+ |
\dot{j_{y}} & = & \tau^b_y(t) + |
| 969 |
+ |
\left(\overleftrightarrow{\mathsf{I}}_{zz} - \overleftrightarrow{\mathsf{I}}_{xx} \right) j_z j_x,\\ |
| 970 |
+ |
\dot{j_{z}} & = & \tau^b_z(t) + |
| 971 |
+ |
\left(\overleftrightarrow{\mathsf{I}}_{xx} - \overleftrightarrow{\mathsf{I}}_{yy} \right) j_x j_y, |
| 972 |
+ |
\end{eqnarray} |
| 973 |
+ |
which utilize the body-fixed torques, ${\bf \tau}^b$. Torques are |
| 974 |
+ |
most easily derived in the space-fixed frame, |
| 975 |
+ |
\begin{equation} |
| 976 |
+ |
{\bf \tau}^b(t) = \mathsf{A}(t) \cdot {\bf \tau}^s(t), |
| 977 |
+ |
\end{equation} |
| 978 |
+ |
where the torques are either derived from the forces on the |
| 979 |
+ |
constituent atoms of the rigid body, or for directional atoms, |
| 980 |
+ |
directly from derivatives of the potential energy, |
| 981 |
+ |
\begin{equation} |
| 982 |
+ |
{\bf \tau}^s(t) = - \hat{\bf u}(t) \times \left( \frac{\partial} |
| 983 |
+ |
{\partial \hat{\bf u}} V\left(\left\{ {\bf r}(t) \right\}, \left\{ |
| 984 |
+ |
\mathsf{A}(t) \right\}\right) \right). |
| 985 |
+ |
\end{equation} |
| 986 |
+ |
Here $\hat{\bf u}$ is a unit vector pointing along the principal axis |
| 987 |
+ |
of the particle in the space-fixed frame. |
| 988 |
+ |
|
| 989 |
+ |
The DLM method uses a Trotter factorization of the orientational |
| 990 |
+ |
propagator. This has three effects: |
| 991 |
+ |
\begin{enumerate} |
| 992 |
+ |
\item the integrator is area-preserving in phase space (i.e. it is |
| 993 |
+ |
{\it symplectic}), |
| 994 |
+ |
\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 995 |
+ |
Monte Carlo applications, and |
| 996 |
+ |
\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
| 997 |
+ |
for timesteps of length $h$. |
| 998 |
+ |
\end{enumerate} |
| 999 |
+ |
|
| 1000 |
+ |
The integration of the equations of motion is carried out in a |
| 1001 |
+ |
velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
| 1002 |
+ |
|
| 1003 |
+ |
{\tt moveA:} |
| 1004 |
+ |
\begin{align*} |
| 1005 |
+ |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 1006 |
+ |
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
| 1007 |
+ |
% |
| 1008 |
+ |
{\bf r}(t + h) &\leftarrow {\bf r}(t) |
| 1009 |
+ |
+ h {\bf v}\left(t + h / 2 \right), \\ |
| 1010 |
+ |
% |
| 1011 |
+ |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 1012 |
+ |
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 1013 |
+ |
% |
| 1014 |
+ |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 1015 |
+ |
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 1016 |
+ |
\end{align*} |
| 1017 |
+ |
|
| 1018 |
+ |
In this context, the $\mathrm{rotate}$ function is the reversible product |
| 1019 |
+ |
of the three body-fixed rotations, |
| 1020 |
+ |
\begin{equation} |
| 1021 |
+ |
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
| 1022 |
+ |
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y / |
| 1023 |
+ |
2) \cdot \mathsf{G}_x(a_x /2), |
| 1024 |
+ |
\end{equation} |
| 1025 |
+ |
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, rotates |
| 1026 |
+ |
both the rotation matrix ($\mathsf{A}$) and the body-fixed angular |
| 1027 |
+ |
momentum (${\bf j}$) by an angle $\theta$ around body-fixed axis |
| 1028 |
+ |
$\alpha$, |
| 1029 |
+ |
\begin{equation} |
| 1030 |
+ |
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 1031 |
+ |
\begin{array}{lcl} |
| 1032 |
+ |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 1033 |
+ |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf j}(0). |
| 1034 |
+ |
\end{array} |
| 1035 |
+ |
\right. |
| 1036 |
+ |
\end{equation} |
| 1037 |
+ |
$\mathsf{R}_\alpha$ is a quadratic approximation to |
| 1038 |
+ |
the single-axis rotation matrix. For example, in the small-angle |
| 1039 |
+ |
limit, the rotation matrix around the body-fixed x-axis can be |
| 1040 |
+ |
approximated as |
| 1041 |
+ |
\begin{equation} |
| 1042 |
+ |
\mathsf{R}_x(\theta) \approx \left( |
| 1043 |
+ |
\begin{array}{ccc} |
| 1044 |
+ |
1 & 0 & 0 \\ |
| 1045 |
+ |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1046 |
+ |
\theta^2 / 4} \\ |
| 1047 |
+ |
0 & \frac{\theta}{1+ |
| 1048 |
+ |
\theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} |
| 1049 |
+ |
\end{array} |
| 1050 |
+ |
\right). |
| 1051 |
+ |
\end{equation} |
| 1052 |
+ |
All other rotations follow in a straightforward manner. |
| 1053 |
+ |
|
| 1054 |
+ |
After the first part of the propagation, the forces and body-fixed |
| 1055 |
+ |
torques are calculated at the new positions and orientations |
| 1056 |
+ |
|
| 1057 |
+ |
{\tt doForces:} |
| 1058 |
+ |
\begin{align*} |
| 1059 |
+ |
{\bf f}(t + h) &\leftarrow |
| 1060 |
+ |
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
| 1061 |
+ |
% |
| 1062 |
+ |
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 1063 |
+ |
\times \frac{\partial V}{\partial {\bf u}}, \\ |
| 1064 |
+ |
% |
| 1065 |
+ |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 1066 |
+ |
\cdot {\bf \tau}^s(t + h). |
| 1067 |
+ |
\end{align*} |
| 1068 |
+ |
|
| 1069 |
+ |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
| 1070 |
+ |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 1071 |
+ |
torques have been obtained at the new time step, the velocities can be |
| 1072 |
+ |
advanced to the same time value. |
| 1073 |
+ |
|
| 1074 |
+ |
{\tt moveB:} |
| 1075 |
+ |
\begin{align*} |
| 1076 |
+ |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 \right) |
| 1077 |
+ |
+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
| 1078 |
+ |
% |
| 1079 |
+ |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 \right) |
| 1080 |
+ |
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 1081 |
+ |
\end{align*} |
| 1082 |
+ |
|
| 1083 |
+ |
The matrix rotations used in the DLM method end up being more costly |
| 1084 |
+ |
computationally than the simpler arithmetic quaternion |
| 1085 |
+ |
propagation. With the same time step, a 1000-molecule water simulation |
| 1086 |
+ |
shows an average 7\% increase in computation time using the DLM method |
| 1087 |
+ |
in place of quaternions. This cost is more than justified when |
| 1088 |
+ |
comparing the energy conservation of the two methods as illustrated in |
| 1089 |
+ |
Fig.~\ref{timestep}. |
| 1090 |
+ |
|
| 1091 |
|
\begin{figure} |
| 1092 |
|
\centering |
| 1093 |
|
\includegraphics[width=\linewidth]{timeStep.eps} |
| 1094 |
< |
\caption{Energy conservation using quaternion based integration versus |
| 1095 |
< |
the symplectic step method proposed by Dullweber \emph{et al.} with |
| 1096 |
< |
increasing time step. For each time step, the dotted line is total |
| 1097 |
< |
energy using the symplectic step integrator, and the solid line comes |
| 1098 |
< |
from the quaternion integrator. The larger time step plots are shifted |
| 1099 |
< |
up from the true energy baseline for clarity.} |
| 1094 |
> |
\caption[Energy conservation for quaternion versus DLM dynamics]{Energy conservation using quaternion based integration versus |
| 1095 |
> |
the method proposed by Dullweber \emph{et al.} with increasing time |
| 1096 |
> |
step. For each time step, the dotted line is total energy using the |
| 1097 |
> |
DLM integrator, and the solid line comes from the quaternion |
| 1098 |
> |
integrator. The larger time step plots are shifted up from the true |
| 1099 |
> |
energy baseline for clarity.} |
| 1100 |
|
\label{timestep} |
| 1101 |
|
\end{figure} |
| 1102 |
|
|
| 1103 |
< |
In figure \ref{timestep}, the resulting energy drift at various time |
| 1104 |
< |
steps for both the symplectic step and quaternion integration schemes |
| 1105 |
< |
is compared. All of the 1000 SSD particle simulations started with the |
| 1103 |
> |
In Fig.~\ref{timestep}, the resulting energy drift at various time |
| 1104 |
> |
steps for both the DLM and quaternion integration schemes is |
| 1105 |
> |
compared. All of the 1000 molecule water simulations started with the |
| 1106 |
|
same configuration, and the only difference was the method for |
| 1107 |
|
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
| 1108 |
< |
methods for propagating particle rotation conserve energy fairly well, |
| 1108 |
> |
methods for propagating molecule rotation conserve energy fairly well, |
| 1109 |
|
with the quaternion method showing a slight energy drift over time in |
| 1110 |
|
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
| 1111 |
< |
energy conservation benefits of the symplectic step method are clearly |
| 1111 |
> |
energy conservation benefits of the DLM method are clearly |
| 1112 |
|
demonstrated. Thus, while maintaining the same degree of energy |
| 1113 |
|
conservation, one can take considerably longer time steps, leading to |
| 1114 |
|
an overall reduction in computation time. |
| 1115 |
|
|
| 1116 |
< |
Energy drift in these SSD particle simulations was unnoticeable for |
| 1117 |
< |
time steps up to three femtoseconds. A slight energy drift on the |
| 865 |
< |
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
| 866 |
< |
four femtoseconds, and as expected, this drift increases dramatically |
| 867 |
< |
with increasing time step. To insure accuracy in the constant energy |
| 868 |
< |
simulations, time steps were set at 2 fs and kept at this value for |
| 869 |
< |
constant pressure simulations as well. |
| 1116 |
> |
There is only one specific keyword relevant to the default integrator, |
| 1117 |
> |
and that is the time step for integrating the equations of motion. |
| 1118 |
|
|
| 1119 |
+ |
\begin{center} |
| 1120 |
+ |
\begin{tabular}{llll} |
| 1121 |
+ |
{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf |
| 1122 |
+ |
default value} \\ |
| 1123 |
+ |
$h$ & {\tt dt = 2.0;} & fs & none |
| 1124 |
+ |
\end{tabular} |
| 1125 |
+ |
\end{center} |
| 1126 |
|
|
| 1127 |
|
\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
| 1128 |
|
|
| 1129 |
+ |
{\sc oopse} implements a number of extended system integrators for |
| 1130 |
+ |
sampling from other ensembles relevant to chemical physics. The |
| 1131 |
+ |
integrator can selected with the {\tt ensemble} keyword in the |
| 1132 |
+ |
{\tt .bass} file: |
| 1133 |
|
|
| 1134 |
< |
{\sc oopse} implements a |
| 1134 |
> |
\begin{center} |
| 1135 |
> |
\begin{tabular}{lll} |
| 1136 |
> |
{\bf Integrator} & {\bf Ensemble} & {\bf {\tt .bass} line} \\ |
| 1137 |
> |
NVE & microcanonical & {\tt ensemble = NVE; } \\ |
| 1138 |
> |
NVT & canonical & {\tt ensemble = NVT; } \\ |
| 1139 |
> |
NPTi & isobaric-isothermal & {\tt ensemble = NPTi;} \\ |
| 1140 |
> |
& (with isotropic volume changes) & \\ |
| 1141 |
> |
NPTf & isobaric-isothermal & {\tt ensemble = NPTf;} \\ |
| 1142 |
> |
& (with changes to box shape) & \\ |
| 1143 |
> |
NPTxyz & approximate isobaric-isothermal & {\tt ensemble = NPTxyz;} \\ |
| 1144 |
> |
& (with separate barostats on each box dimension) & \\ |
| 1145 |
> |
\end{tabular} |
| 1146 |
> |
\end{center} |
| 1147 |
|
|
| 1148 |
+ |
The relatively well-known Nos\'e-Hoover thermostat\cite{Hoover85} is |
| 1149 |
+ |
implemented in {\sc oopse}'s NVT integrator. This method couples an |
| 1150 |
+ |
extra degree of freedom (the thermostat) to the kinetic energy of the |
| 1151 |
+ |
system, and has been shown to sample the canonical distribution in the |
| 1152 |
+ |
system degrees of freedom while conserving a quantity that is, to |
| 1153 |
+ |
within a constant, the Helmholtz free energy.\cite{melchionna93} |
| 1154 |
|
|
| 1155 |
< |
\subsubsection{\label{oopseSec:noseHooverThermo}Nose-Hoover Thermostatting} |
| 1155 |
> |
NPT algorithms attempt to maintain constant pressure in the system by |
| 1156 |
> |
coupling the volume of the system to a barostat. {\sc oopse} contains |
| 1157 |
> |
three different constant pressure algorithms. The first two, NPTi and |
| 1158 |
> |
NPTf have been shown to conserve a quantity that is, to within a |
| 1159 |
> |
constant, the Gibbs free energy.\cite{melchionna93} The Melchionna |
| 1160 |
> |
modification to the Hoover barostat is implemented in both NPTi and |
| 1161 |
> |
NPTf. NPTi allows only isotropic changes in the simulation box, while |
| 1162 |
> |
box {\it shape} variations are allowed in NPTf. The NPTxyz integrator |
| 1163 |
> |
has {\it not} been shown to sample from the isobaric-isothermal |
| 1164 |
> |
ensemble. It is useful, however, in that it maintains orthogonality |
| 1165 |
> |
for the axes of the simulation box while attempting to equalize |
| 1166 |
> |
pressure along the three perpendicular directions in the box. |
| 1167 |
|
|
| 1168 |
< |
To mimic the effects of being in a constant temperature ({\sc nvt}) |
| 1169 |
< |
ensemble, {\sc oopse} uses the Nose-Hoover extended system |
| 1170 |
< |
approach.\cite{Hoover85} In this method, the equations of motion for |
| 1171 |
< |
the particle positions and velocities are |
| 1168 |
> |
Each of the extended system integrators requires additional keywords |
| 1169 |
> |
to set target values for the thermodynamic state variables that are |
| 1170 |
> |
being held constant. Keywords are also required to set the |
| 1171 |
> |
characteristic decay times for the dynamics of the extended |
| 1172 |
> |
variables. |
| 1173 |
> |
|
| 1174 |
> |
\begin{center} |
| 1175 |
> |
\begin{tabular}{llll} |
| 1176 |
> |
{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf |
| 1177 |
> |
default value} \\ |
| 1178 |
> |
$T_{\mathrm{target}}$ & {\tt targetTemperature = 300;} & K & none \\ |
| 1179 |
> |
$P_{\mathrm{target}}$ & {\tt targetPressure = 1;} & atm & none \\ |
| 1180 |
> |
$\tau_T$ & {\tt tauThermostat = 1e3;} & fs & none \\ |
| 1181 |
> |
$\tau_B$ & {\tt tauBarostat = 5e3;} & fs & none \\ |
| 1182 |
> |
& {\tt resetTime = 200;} & fs & none \\ |
| 1183 |
> |
& {\tt useInitialExtendedSystemState = true;} & logical & |
| 1184 |
> |
true |
| 1185 |
> |
\end{tabular} |
| 1186 |
> |
\end{center} |
| 1187 |
> |
|
| 1188 |
> |
Two additional keywords can be used to either clear the extended |
| 1189 |
> |
system variables periodically ({\tt resetTime}), or to maintain the |
| 1190 |
> |
state of the extended system variables between simulations ({\tt |
| 1191 |
> |
useInitialExtendedSystemState}). More details on these variables |
| 1192 |
> |
and their use in the integrators follows below. |
| 1193 |
> |
|
| 1194 |
> |
\subsection{\label{oopseSec:noseHooverThermo}Nos\'{e}-Hoover Thermostatting} |
| 1195 |
> |
|
| 1196 |
> |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover85} |
| 1197 |
|
\begin{eqnarray} |
| 1198 |
< |
\dot{{\bf r}} & = & {\bf v} \\ |
| 1199 |
< |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} |
| 1198 |
> |
\dot{{\bf r}} & = & {\bf v}, \\ |
| 1199 |
> |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
| 1200 |
> |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 1201 |
> |
\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
| 1202 |
> |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{\mathsf{I}}^{-1} |
| 1203 |
> |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 1204 |
> |
V}{\partial \mathsf{A}} \right) - \chi {\bf j}. |
| 1205 |
|
\label{eq:nosehoovereom} |
| 1206 |
|
\end{eqnarray} |
| 1207 |
|
|
| 1208 |
|
$\chi$ is an ``extra'' variable included in the extended system, and |
| 1209 |
|
it is propagated using the first order equation of motion |
| 1210 |
|
\begin{equation} |
| 1211 |
< |
\dot{\chi} = \frac{1}{\tau_{T}} \left( \frac{T}{T_{target}} - 1 \right) |
| 1211 |
> |
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( \frac{T}{T_{\mathrm{target}}} - 1 \right). |
| 1212 |
|
\label{eq:nosehooverext} |
| 1213 |
|
\end{equation} |
| 896 |
– |
where $T_{target}$ is the target temperature for the simulation, and |
| 897 |
– |
$\tau_{T}$ is a time constant for the thermostat. |
| 1214 |
|
|
| 1215 |
< |
To select the Nose-Hoover {\sc nvt} ensemble, the {\tt ensemble = NVT;} |
| 1216 |
< |
command would be used in the simulation's {\sc bass} file. There is |
| 1217 |
< |
some subtlety in choosing values for $\tau_{T}$, and it is usually set |
| 1218 |
< |
to values of a few ps. Within a {\sc bass} file, $\tau_{T}$ could be |
| 1219 |
< |
set to 1 ps using the {\tt tauThermostat = 1000; } command. |
| 1215 |
> |
The instantaneous temperature $T$ is proportional to the total kinetic |
| 1216 |
> |
energy (both translational and orientational) and is given by |
| 1217 |
> |
\begin{equation} |
| 1218 |
> |
T = \frac{2 K}{f k_B} |
| 1219 |
> |
\end{equation} |
| 1220 |
> |
Here, $f$ is the total number of degrees of freedom in the system, |
| 1221 |
> |
\begin{equation} |
| 1222 |
> |
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
| 1223 |
> |
\end{equation} |
| 1224 |
> |
and $K$ is the total kinetic energy, |
| 1225 |
> |
\begin{equation} |
| 1226 |
> |
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 1227 |
> |
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
| 1228 |
> |
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
| 1229 |
> |
\end{equation} |
| 1230 |
|
|
| 1231 |
+ |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
| 1232 |
+ |
relaxation of the temperature to the target value. To set values for |
| 1233 |
+ |
$\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use the |
| 1234 |
+ |
{\tt tauThermostat} and {\tt targetTemperature} keywords in the {\tt |
| 1235 |
+ |
.bass} file. The units for {\tt tauThermostat} are fs, and the units |
| 1236 |
+ |
for the {\tt targetTemperature} are degrees K. The integration of |
| 1237 |
+ |
the equations of motion is carried out in a velocity-Verlet style 2 |
| 1238 |
+ |
part algorithm: |
| 1239 |
|
|
| 1240 |
< |
\subsection{\label{oopseSec:zcons}Z-Constraint Method} |
| 1240 |
> |
{\tt moveA:} |
| 1241 |
> |
\begin{align*} |
| 1242 |
> |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 1243 |
> |
% |
| 1244 |
> |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 1245 |
> |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 1246 |
> |
\chi(t)\right), \\ |
| 1247 |
> |
% |
| 1248 |
> |
{\bf r}(t + h) &\leftarrow {\bf r}(t) |
| 1249 |
> |
+ h {\bf v}\left(t + h / 2 \right) ,\\ |
| 1250 |
> |
% |
| 1251 |
> |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 1252 |
> |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 1253 |
> |
\chi(t) \right) ,\\ |
| 1254 |
> |
% |
| 1255 |
> |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate} |
| 1256 |
> |
\left(h * {\bf j}(t + h / 2) |
| 1257 |
> |
\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
| 1258 |
> |
% |
| 1259 |
> |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) |
| 1260 |
> |
+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
| 1261 |
> |
{T_{\mathrm{target}}} - 1 \right) . |
| 1262 |
> |
\end{align*} |
| 1263 |
|
|
| 1264 |
< |
Based on fluctuation-dissipation theorem,\bigskip\ force auto-correlation |
| 1265 |
< |
method was developed to investigate the dynamics of ions inside the ion |
| 1266 |
< |
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated |
| 1267 |
< |
from the deviation of the instantaneous force from its mean force. |
| 1264 |
> |
Here $\mathrm{rotate}(h * {\bf j} |
| 1265 |
> |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic Trotter |
| 1266 |
> |
factorization of the three rotation operations that was discussed in |
| 1267 |
> |
the section on the DLM integrator. Note that this operation modifies |
| 1268 |
> |
both the rotation matrix $\mathsf{A}$ and the angular momentum ${\bf |
| 1269 |
> |
j}$. {\tt moveA} propagates velocities by a half time step, and |
| 1270 |
> |
positional degrees of freedom by a full time step. The new positions |
| 1271 |
> |
(and orientations) are then used to calculate a new set of forces and |
| 1272 |
> |
torques in exactly the same way they are calculated in the {\tt |
| 1273 |
> |
doForces} portion of the DLM integrator. |
| 1274 |
|
|
| 1275 |
+ |
Once the forces and torques have been obtained at the new time step, |
| 1276 |
+ |
the temperature, velocities, and the extended system variable can be |
| 1277 |
+ |
advanced to the same time value. |
| 1278 |
+ |
|
| 1279 |
+ |
{\tt moveB:} |
| 1280 |
+ |
\begin{align*} |
| 1281 |
+ |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 1282 |
+ |
\left\{{\bf j}(t + h)\right\}, \\ |
| 1283 |
|
% |
| 1284 |
+ |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 1285 |
+ |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
| 1286 |
+ |
{T_{\mathrm{target}}} - 1 \right), \\ |
| 1287 |
+ |
% |
| 1288 |
+ |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 1289 |
+ |
+ h / 2 \right) + \frac{h}{2} \left( |
| 1290 |
+ |
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 1291 |
+ |
\chi(t h)\right) ,\\ |
| 1292 |
+ |
% |
| 1293 |
+ |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 1294 |
+ |
+ h / 2 \right) + \frac{h}{2} |
| 1295 |
+ |
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
| 1296 |
+ |
\chi(t + h) \right) . |
| 1297 |
+ |
\end{align*} |
| 1298 |
|
|
| 1299 |
+ |
Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to caclculate |
| 1300 |
+ |
$T(t + h)$ as well as $\chi(t + h)$, they indirectly depend on their |
| 1301 |
+ |
own values at time $t + h$. {\tt moveB} is therefore done in an |
| 1302 |
+ |
iterative fashion until $\chi(t + h)$ becomes self-consistent. The |
| 1303 |
+ |
relative tolerance for the self-consistency check defaults to a value |
| 1304 |
+ |
of $\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration |
| 1305 |
+ |
after 4 loops even if the consistency check has not been satisfied. |
| 1306 |
+ |
|
| 1307 |
+ |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for the |
| 1308 |
+ |
extended system that is, to within a constant, identical to the |
| 1309 |
+ |
Helmholtz free energy,\cite{melchionna93} |
| 1310 |
|
\begin{equation} |
| 1311 |
< |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
| 1311 |
> |
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 1312 |
> |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) dt^\prime |
| 1313 |
> |
\right). |
| 1314 |
> |
\end{equation} |
| 1315 |
> |
Poor choices of $h$ or $\tau_T$ can result in non-conservation |
| 1316 |
> |
of $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 1317 |
> |
last column of the {\tt .stat} file to allow checks on the quality of |
| 1318 |
> |
the integration. |
| 1319 |
> |
|
| 1320 |
> |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 1321 |
> |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 1322 |
> |
algorithms are given in section \ref{oopseSec:rattle}. |
| 1323 |
> |
|
| 1324 |
> |
\subsection{\label{sec:NPTi}Constant-pressure integration with |
| 1325 |
> |
isotropic box deformations (NPTi)} |
| 1326 |
> |
|
| 1327 |
> |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 1328 |
> |
implements the Melchionna modifications to the Nos\'e-Hoover-Andersen |
| 1329 |
> |
equations of motion,\cite{melchionna93} |
| 1330 |
> |
|
| 1331 |
> |
\begin{eqnarray} |
| 1332 |
> |
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 1333 |
> |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
| 1334 |
> |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 1335 |
> |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
| 1336 |
> |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 1337 |
> |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 1338 |
> |
V}{\partial \mathsf{A}} \right) - \chi {\bf j}, \\ |
| 1339 |
> |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 1340 |
> |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 1341 |
> |
\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V \left( P - |
| 1342 |
> |
P_{\mathrm{target}} \right), \\ |
| 1343 |
> |
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . |
| 1344 |
> |
\label{eq:melchionna1} |
| 1345 |
> |
\end{eqnarray} |
| 1346 |
> |
|
| 1347 |
> |
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the extended |
| 1348 |
> |
system. $\chi$ is a thermostat, and it has the same function as it |
| 1349 |
> |
does in the Nos\'e-Hoover NVT integrator. $\eta$ is a barostat which |
| 1350 |
> |
controls changes to the volume of the simulation box. ${\bf R}_0$ is |
| 1351 |
> |
the location of the center of mass for the entire system, and |
| 1352 |
> |
$\mathcal{V}$ is the volume of the simulation box. At any time, the |
| 1353 |
> |
volume can be calculated from the determinant of the matrix which |
| 1354 |
> |
describes the box shape: |
| 1355 |
> |
\begin{equation} |
| 1356 |
> |
\mathcal{V} = \det(\mathsf{H}). |
| 1357 |
|
\end{equation} |
| 1358 |
< |
where% |
| 1358 |
> |
|
| 1359 |
> |
The NPTi integrator requires an instantaneous pressure. This quantity |
| 1360 |
> |
is calculated via the pressure tensor, |
| 1361 |
|
\begin{equation} |
| 1362 |
< |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
| 1362 |
> |
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{\mathcal{V}(t)} \left( |
| 1363 |
> |
\sum_{i=1}^{N} m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
| 1364 |
> |
\overleftrightarrow{\mathsf{W}}(t). |
| 1365 |
|
\end{equation} |
| 1366 |
+ |
The kinetic contribution to the pressure tensor utilizes the {\it |
| 1367 |
+ |
outer} product of the velocities denoted by the $\otimes$ symbol. The |
| 1368 |
+ |
stress tensor is calculated from another outer product of the |
| 1369 |
+ |
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
| 1370 |
+ |
r}_i$) with the forces between the same two atoms, |
| 1371 |
+ |
\begin{equation} |
| 1372 |
+ |
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf r}_{ij}(t) |
| 1373 |
+ |
\otimes {\bf f}_{ij}(t). |
| 1374 |
+ |
\end{equation} |
| 1375 |
+ |
The instantaneous pressure is then simply obtained from the trace of |
| 1376 |
+ |
the Pressure tensor, |
| 1377 |
+ |
\begin{equation} |
| 1378 |
+ |
P(t) = \frac{1}{3} \mathrm{Tr} \left( \overleftrightarrow{\mathsf{P}}(t). |
| 1379 |
+ |
\right) |
| 1380 |
+ |
\end{equation} |
| 1381 |
|
|
| 1382 |
+ |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 1383 |
+ |
relaxation of the pressure to the target value. To set values for |
| 1384 |
+ |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
| 1385 |
+ |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt .bass} |
| 1386 |
+ |
file. The units for {\tt tauBarostat} are fs, and the units for the |
| 1387 |
+ |
{\tt targetPressure} are atmospheres. Like in the NVT integrator, the |
| 1388 |
+ |
integration of the equations of motion is carried out in a |
| 1389 |
+ |
velocity-Verlet style 2 part algorithm: |
| 1390 |
|
|
| 1391 |
< |
If the time-dependent friction decay rapidly, static friction coefficient can |
| 1392 |
< |
be approximated by% |
| 1391 |
> |
{\tt moveA:} |
| 1392 |
> |
\begin{align*} |
| 1393 |
> |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 1394 |
> |
% |
| 1395 |
> |
P(t) &\leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,\\ |
| 1396 |
> |
% |
| 1397 |
> |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 1398 |
> |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 1399 |
> |
\left(\chi(t) + \eta(t) \right) \right), \\ |
| 1400 |
> |
% |
| 1401 |
> |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 1402 |
> |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 1403 |
> |
\chi(t) \right), \\ |
| 1404 |
> |
% |
| 1405 |
> |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 1406 |
> |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 1407 |
> |
\right) ,\\ |
| 1408 |
> |
% |
| 1409 |
> |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 1410 |
> |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 1411 |
> |
\right) ,\\ |
| 1412 |
> |
% |
| 1413 |
> |
\eta(t + h / 2) &\leftarrow \eta(t) + \frac{h |
| 1414 |
> |
\mathcal{V}(t)}{2 N k_B T(t) \tau_B^2} \left( P(t) |
| 1415 |
> |
- P_{\mathrm{target}} \right), \\ |
| 1416 |
> |
% |
| 1417 |
> |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h |
| 1418 |
> |
\left\{ {\bf v}\left(t + h / 2 \right) |
| 1419 |
> |
+ \eta(t + h / 2)\left[ {\bf r}(t + h) |
| 1420 |
> |
- {\bf R}_0 \right] \right\} ,\\ |
| 1421 |
> |
% |
| 1422 |
> |
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
| 1423 |
> |
\mathsf{H}(t). |
| 1424 |
> |
\end{align*} |
| 1425 |
|
|
| 1426 |
+ |
Most of these equations are identical to their counterparts in the NVT |
| 1427 |
+ |
integrator, but the propagation of positions to time $t + h$ |
| 1428 |
+ |
depends on the positions at the same time. {\sc oopse} carries out |
| 1429 |
+ |
this step iteratively (with a limit of 5 passes through the iterative |
| 1430 |
+ |
loop). Also, the simulation box $\mathsf{H}$ is scaled uniformly for |
| 1431 |
+ |
one full time step by an exponential factor that depends on the value |
| 1432 |
+ |
of $\eta$ at time $t + |
| 1433 |
+ |
h / 2$. Reshaping the box uniformly also scales the volume of |
| 1434 |
+ |
the box by |
| 1435 |
|
\begin{equation} |
| 1436 |
< |
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt |
| 1436 |
> |
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 1437 |
> |
\mathcal{V}(t) |
| 1438 |
|
\end{equation} |
| 1439 |
|
|
| 1440 |
+ |
The {\tt doForces} step for the NPTi integrator is exactly the same as |
| 1441 |
+ |
in both the DLM and NVT integrators. Once the forces and torques have |
| 1442 |
+ |
been obtained at the new time step, the velocities can be advanced to |
| 1443 |
+ |
the same time value. |
| 1444 |
|
|
| 1445 |
< |
Hence, diffusion constant can be estimated by |
| 1445 |
> |
{\tt moveB:} |
| 1446 |
> |
\begin{align*} |
| 1447 |
> |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 1448 |
> |
\left\{{\bf j}(t + h)\right\} ,\\ |
| 1449 |
> |
% |
| 1450 |
> |
P(t + h) &\leftarrow \left\{{\bf r}(t + h)\right\}, |
| 1451 |
> |
\left\{{\bf v}(t + h)\right\}, \\ |
| 1452 |
> |
% |
| 1453 |
> |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 1454 |
> |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
| 1455 |
> |
{T_{\mathrm{target}}} - 1 \right), \\ |
| 1456 |
> |
% |
| 1457 |
> |
\eta(t + h) &\leftarrow \eta(t + h / 2) + |
| 1458 |
> |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 1459 |
> |
\tau_B^2} \left( P(t + h) - P_{\mathrm{target}} \right), \\ |
| 1460 |
> |
% |
| 1461 |
> |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 1462 |
> |
+ h / 2 \right) + \frac{h}{2} \left( |
| 1463 |
> |
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 1464 |
> |
(\chi(t + h) + \eta(t + h)) \right) ,\\ |
| 1465 |
> |
% |
| 1466 |
> |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 1467 |
> |
+ h / 2 \right) + \frac{h}{2} \left( {\bf |
| 1468 |
> |
\tau}^b(t + h) - {\bf j}(t + h) |
| 1469 |
> |
\chi(t + h) \right) . |
| 1470 |
> |
\end{align*} |
| 1471 |
> |
|
| 1472 |
> |
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
| 1473 |
> |
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 1474 |
> |
h)$, they indirectly depend on their own values at time $t + h$. {\tt |
| 1475 |
> |
moveB} is therefore done in an iterative fashion until $\chi(t + h)$ |
| 1476 |
> |
and $\eta(t + h)$ become self-consistent. The relative tolerance for |
| 1477 |
> |
the self-consistency check defaults to a value of $\mbox{10}^{-6}$, |
| 1478 |
> |
but {\sc oopse} will terminate the iteration after 4 loops even if the |
| 1479 |
> |
consistency check has not been satisfied. |
| 1480 |
> |
|
| 1481 |
> |
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm is |
| 1482 |
> |
known to conserve a Hamiltonian for the extended system that is, to |
| 1483 |
> |
within a constant, identical to the Gibbs free energy, |
| 1484 |
|
\begin{equation} |
| 1485 |
< |
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 1486 |
< |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}% |
| 1485 |
> |
H_{\mathrm{NPTi}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 1486 |
> |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) dt^\prime |
| 1487 |
> |
\right) + P_{\mathrm{target}} \mathcal{V}(t). |
| 1488 |
|
\end{equation} |
| 1489 |
+ |
Poor choices of $\delta t$, $\tau_T$, or $\tau_B$ can result in |
| 1490 |
+ |
non-conservation of $H_{\mathrm{NPTi}}$, so the conserved quantity is |
| 1491 |
+ |
maintained in the last column of the {\tt .stat} file to allow checks |
| 1492 |
+ |
on the quality of the integration. It is also known that this |
| 1493 |
+ |
algorithm samples the equilibrium distribution for the enthalpy |
| 1494 |
+ |
(including contributions for the thermostat and barostat), |
| 1495 |
+ |
\begin{equation} |
| 1496 |
+ |
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} \left( |
| 1497 |
+ |
\chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + P_{\mathrm{target}} |
| 1498 |
+ |
\mathcal{V}(t). |
| 1499 |
+ |
\end{equation} |
| 1500 |
|
|
| 1501 |
+ |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 1502 |
+ |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 1503 |
+ |
algorithms are given in section \ref{oopseSec:rattle}. |
| 1504 |
|
|
| 1505 |
< |
\bigskip Z-Constraint method, which fixed the z coordinates of the molecules |
| 1506 |
< |
with respect to the center of the mass of the system, was proposed to obtain |
| 1507 |
< |
the forces required in force auto-correlation method.\cite{Marrink94} However, |
| 1508 |
< |
simply resetting the coordinate will move the center of the mass of the whole |
| 1509 |
< |
system. To avoid this problem, a new method was used at {\sc oopse}. Instead of |
| 1510 |
< |
resetting the coordinate, we reset the forces of z-constraint molecules as |
| 1511 |
< |
well as subtract the total constraint forces from the rest of the system after |
| 1512 |
< |
force calculation at each time step. |
| 1505 |
> |
\subsection{\label{sec:NPTf}Constant-pressure integration with a |
| 1506 |
> |
flexible box (NPTf)} |
| 1507 |
> |
|
| 1508 |
> |
There is a relatively simple generalization of the |
| 1509 |
> |
Nos\'e-Hoover-Andersen method to include changes in the simulation box |
| 1510 |
> |
{\it shape} as well as in the volume of the box. This method utilizes |
| 1511 |
> |
the full $3 \times 3$ pressure tensor and introduces a tensor of |
| 1512 |
> |
extended variables ($\overleftrightarrow{\eta}$) to control changes to |
| 1513 |
> |
the box shape. The equations of motion for this method are |
| 1514 |
> |
\begin{eqnarray} |
| 1515 |
> |
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
| 1516 |
> |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
| 1517 |
> |
\chi \cdot \mathsf{1}) {\bf v}, \\ |
| 1518 |
> |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 1519 |
> |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
| 1520 |
> |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 1521 |
> |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 1522 |
> |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} ,\\ |
| 1523 |
> |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 1524 |
> |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 1525 |
> |
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
| 1526 |
> |
T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 1527 |
> |
\dot{\mathsf{H}} & = & \overleftrightarrow{\eta} \cdot \mathsf{H} . |
| 1528 |
> |
\label{eq:melchionna2} |
| 1529 |
> |
\end{eqnarray} |
| 1530 |
> |
|
| 1531 |
> |
Here, $\mathsf{1}$ is the unit matrix and $\overleftrightarrow{\mathsf{P}}$ |
| 1532 |
> |
is the pressure tensor. Again, the volume, $\mathcal{V} = \det |
| 1533 |
> |
\mathsf{H}$. |
| 1534 |
> |
|
| 1535 |
> |
The propagation of the equations of motion is nearly identical to the |
| 1536 |
> |
NPTi integration: |
| 1537 |
> |
|
| 1538 |
> |
{\tt moveA:} |
| 1539 |
> |
\begin{align*} |
| 1540 |
> |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 1541 |
> |
% |
| 1542 |
> |
\overleftrightarrow{\mathsf{P}}(t) &\leftarrow \left\{{\bf r}(t)\right\}, |
| 1543 |
> |
\left\{{\bf v}(t)\right\} ,\\ |
| 1544 |
> |
% |
| 1545 |
> |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 1546 |
> |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - |
| 1547 |
> |
\left(\chi(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot |
| 1548 |
> |
{\bf v}(t) \right), \\ |
| 1549 |
> |
% |
| 1550 |
> |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 1551 |
> |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 1552 |
> |
\chi(t) \right), \\ |
| 1553 |
> |
% |
| 1554 |
> |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 1555 |
> |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 1556 |
> |
\right), \\ |
| 1557 |
> |
% |
| 1558 |
> |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 1559 |
> |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} |
| 1560 |
> |
- 1 \right), \\ |
| 1561 |
> |
% |
| 1562 |
> |
\overleftrightarrow{\eta}(t + h / 2) &\leftarrow |
| 1563 |
> |
\overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}{2 N k_B |
| 1564 |
> |
T(t) \tau_B^2} \left( \overleftrightarrow{\mathsf{P}}(t) |
| 1565 |
> |
- P_{\mathrm{target}}\mathsf{1} \right), \\ |
| 1566 |
> |
% |
| 1567 |
> |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h \left\{ {\bf v} |
| 1568 |
> |
\left(t + h / 2 \right) + \overleftrightarrow{\eta}(t + |
| 1569 |
> |
h / 2) \cdot \left[ {\bf r}(t + h) |
| 1570 |
> |
- {\bf R}_0 \right] \right\}, \\ |
| 1571 |
> |
% |
| 1572 |
> |
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
| 1573 |
> |
\overleftrightarrow{\eta}(t + h / 2)} . |
| 1574 |
> |
\end{align*} |
| 1575 |
> |
{\sc oopse} uses a power series expansion truncated at second order |
| 1576 |
> |
for the exponential operation which scales the simulation box. |
| 1577 |
> |
|
| 1578 |
> |
The {\tt moveB} portion of the algorithm is largely unchanged from the |
| 1579 |
> |
NPTi integrator: |
| 1580 |
> |
|
| 1581 |
> |
{\tt moveB:} |
| 1582 |
> |
\begin{align*} |
| 1583 |
> |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 1584 |
> |
\left\{{\bf j}(t + h)\right\}, \\ |
| 1585 |
> |
% |
| 1586 |
> |
\overleftrightarrow{\mathsf{P}}(t + h) &\leftarrow \left\{{\bf r} |
| 1587 |
> |
(t + h)\right\}, \left\{{\bf v}(t |
| 1588 |
> |
+ h)\right\}, \left\{{\bf f}(t + h)\right\} ,\\ |
| 1589 |
> |
% |
| 1590 |
> |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 1591 |
> |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+ |
| 1592 |
> |
h)}{T_{\mathrm{target}}} - 1 \right), \\ |
| 1593 |
> |
% |
| 1594 |
> |
\overleftrightarrow{\eta}(t + h) &\leftarrow |
| 1595 |
> |
\overleftrightarrow{\eta}(t + h / 2) + |
| 1596 |
> |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 1597 |
> |
\tau_B^2} \left( \overleftrightarrow{P}(t + h) |
| 1598 |
> |
- P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 1599 |
> |
% |
| 1600 |
> |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 1601 |
> |
+ h / 2 \right) + \frac{h}{2} \left( |
| 1602 |
> |
\frac{{\bf f}(t + h)}{m} - |
| 1603 |
> |
(\chi(t + h)\mathsf{1} + \overleftrightarrow{\eta}(t |
| 1604 |
> |
+ h)) \right) \cdot {\bf v}(t + h), \\ |
| 1605 |
> |
% |
| 1606 |
> |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 1607 |
> |
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
| 1608 |
> |
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
| 1609 |
> |
\end{align*} |
| 1610 |
> |
|
| 1611 |
> |
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
| 1612 |
> |
identical to those described for the NPTi integrator. |
| 1613 |
> |
|
| 1614 |
> |
The NPTf integrator is known to conserve the following Hamiltonian: |
| 1615 |
> |
\begin{equation} |
| 1616 |
> |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 1617 |
> |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) dt^\prime |
| 1618 |
> |
\right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
| 1619 |
> |
T_{\mathrm{target}}}{2} |
| 1620 |
> |
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 1621 |
> |
\end{equation} |
| 1622 |
> |
|
| 1623 |
> |
This integrator must be used with care, particularly in liquid |
| 1624 |
> |
simulations. Liquids have very small restoring forces in the |
| 1625 |
> |
off-diagonal directions, and the simulation box can very quickly form |
| 1626 |
> |
elongated and sheared geometries which become smaller than the |
| 1627 |
> |
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
| 1628 |
> |
finds most use in simulating crystals or liquid crystals which assume |
| 1629 |
> |
non-orthorhombic geometries. |
| 1630 |
> |
|
| 1631 |
> |
\subsection{\label{nptxyz}Constant pressure in 3 axes (NPTxyz)} |
| 1632 |
> |
|
| 1633 |
> |
There is one additional extended system integrator which is somewhat |
| 1634 |
> |
simpler than the NPTf method described above. In this case, the three |
| 1635 |
> |
axes have independent barostats which each attempt to preserve the |
| 1636 |
> |
target pressure along the box walls perpendicular to that particular |
| 1637 |
> |
axis. The lengths of the box axes are allowed to fluctuate |
| 1638 |
> |
independently, but the angle between the box axes does not change. |
| 1639 |
> |
The equations of motion are identical to those described above, but |
| 1640 |
> |
only the {\it diagonal} elements of $\overleftrightarrow{\eta}$ are |
| 1641 |
> |
computed. The off-diagonal elements are set to zero (even when the |
| 1642 |
> |
pressure tensor has non-zero off-diagonal elements). |
| 1643 |
> |
|
| 1644 |
> |
It should be noted that the NPTxyz integrator is {\it not} known to |
| 1645 |
> |
preserve any Hamiltonian of interest to the chemical physics |
| 1646 |
> |
community. The integrator is extremely useful, however, in generating |
| 1647 |
> |
initial conditions for other integration methods. It {\it is} suitable |
| 1648 |
> |
for use with liquid simulations, or in cases where there is |
| 1649 |
> |
orientational anisotropy in the system (i.e. in lipid bilayer |
| 1650 |
> |
simulations). |
| 1651 |
> |
|
| 1652 |
> |
\subsection{\label{oopseSec:rattle}The {\sc rattle} Method for Bond |
| 1653 |
> |
Constraints} |
| 1654 |
> |
|
| 1655 |
> |
In order to satisfy the constraints of fixed bond lengths within {\sc |
| 1656 |
> |
oopse}, we have implemented the {\sc rattle} algorithm of |
| 1657 |
> |
Andersen.\cite{andersen83} The algorithm is a velocity verlet |
| 1658 |
> |
formulation of the {\sc shake} method\cite{ryckaert77} of iteratively |
| 1659 |
> |
solving the Lagrange multipliers of constraint. The system of Lagrange |
| 1660 |
> |
multipliers allows one to reformulate the equations of motion with |
| 1661 |
> |
explicit constraint forces.\cite{fowles99:lagrange} |
| 1662 |
> |
|
| 1663 |
> |
Consider a system described by coordinates $q_1$ and $q_2$ subject to an |
| 1664 |
> |
equation of constraint: |
| 1665 |
> |
\begin{equation} |
| 1666 |
> |
\sigma(q_1, q_2,t) = 0 |
| 1667 |
> |
\label{oopseEq:lm1} |
| 1668 |
> |
\end{equation} |
| 1669 |
> |
The Lagrange formulation of the equations of motion can be written: |
| 1670 |
> |
\begin{equation} |
| 1671 |
> |
\delta\int_{t_1}^{t_2}L\, dt = |
| 1672 |
> |
\int_{t_1}^{t_2} \sum_i \biggl [ \frac{\partial L}{\partial q_i} |
| 1673 |
> |
- \frac{d}{dt}\biggl(\frac{\partial L}{\partial \dot{q}_i} |
| 1674 |
> |
\biggr ) \biggr] \delta q_i \, dt = 0. |
| 1675 |
> |
\label{oopseEq:lm2} |
| 1676 |
> |
\end{equation} |
| 1677 |
> |
Here, $\delta q_i$ is not independent for each $q$, as $q_1$ and $q_2$ |
| 1678 |
> |
are linked by $\sigma$. However, $\sigma$ is fixed at any given |
| 1679 |
> |
instant of time, giving: |
| 1680 |
|
\begin{align} |
| 1681 |
< |
F_{\alpha i}&=0\\ |
| 1682 |
< |
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}\\ |
| 1683 |
< |
F_{\alpha i}&=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}\\ |
| 1684 |
< |
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}} |
| 1681 |
> |
\delta\sigma &= \biggl( \frac{\partial\sigma}{\partial q_1} \delta q_1 % |
| 1682 |
> |
+ \frac{\partial\sigma}{\partial q_2} \delta q_2 \biggr) = 0 ,\\ |
| 1683 |
> |
% |
| 1684 |
> |
\frac{\partial\sigma}{\partial q_1} \delta q_1 &= % |
| 1685 |
> |
- \frac{\partial\sigma}{\partial q_2} \delta q_2, \\ |
| 1686 |
> |
% |
| 1687 |
> |
\delta q_2 &= - \biggl(\frac{\partial\sigma}{\partial q_1} \bigg / % |
| 1688 |
> |
\frac{\partial\sigma}{\partial q_2} \biggr) \delta q_1. |
| 1689 |
|
\end{align} |
| 1690 |
+ |
Substituted back into Eq.~\ref{oopseEq:lm2}, |
| 1691 |
+ |
\begin{equation} |
| 1692 |
+ |
\int_{t_1}^{t_2}\biggl [ \biggl(\frac{\partial L}{\partial q_1} |
| 1693 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
| 1694 |
+ |
\biggr) |
| 1695 |
+ |
- \biggl( \frac{\partial L}{\partial q_1} |
| 1696 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
| 1697 |
+ |
\biggr) \biggl(\frac{\partial\sigma}{\partial q_1} \bigg / % |
| 1698 |
+ |
\frac{\partial\sigma}{\partial q_2} \biggr)\biggr] \delta q_1 \, dt = 0. |
| 1699 |
+ |
\label{oopseEq:lm3} |
| 1700 |
+ |
\end{equation} |
| 1701 |
+ |
Leading to, |
| 1702 |
+ |
\begin{equation} |
| 1703 |
+ |
\frac{\biggl(\frac{\partial L}{\partial q_1} |
| 1704 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
| 1705 |
+ |
\biggr)}{\frac{\partial\sigma}{\partial q_1}} = |
| 1706 |
+ |
\frac{\biggl(\frac{\partial L}{\partial q_2} |
| 1707 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_2} |
| 1708 |
+ |
\biggr)}{\frac{\partial\sigma}{\partial q_2}}. |
| 1709 |
+ |
\label{oopseEq:lm4} |
| 1710 |
+ |
\end{equation} |
| 1711 |
+ |
This relation can only be statisfied, if both are equal to a single |
| 1712 |
+ |
function $-\lambda(t)$, |
| 1713 |
+ |
\begin{align} |
| 1714 |
+ |
\frac{\biggl(\frac{\partial L}{\partial q_1} |
| 1715 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
| 1716 |
+ |
\biggr)}{\frac{\partial\sigma}{\partial q_1}} &= -\lambda(t), \\ |
| 1717 |
+ |
% |
| 1718 |
+ |
\frac{\partial L}{\partial q_1} |
| 1719 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} &= |
| 1720 |
+ |
-\lambda(t)\,\frac{\partial\sigma}{\partial q_1} ,\\ |
| 1721 |
+ |
% |
| 1722 |
+ |
\frac{\partial L}{\partial q_1} |
| 1723 |
+ |
- \frac{d}{dt}\,\frac{\partial L}{\partial \dot{q}_1} |
| 1724 |
+ |
+ \mathcal{G}_i &= 0, |
| 1725 |
+ |
\end{align} |
| 1726 |
+ |
where $\mathcal{G}_i$, the force of constraint on $i$, is: |
| 1727 |
+ |
\begin{equation} |
| 1728 |
+ |
\mathcal{G}_i = \lambda(t)\,\frac{\partial\sigma}{\partial q_1}. |
| 1729 |
+ |
\label{oopseEq:lm5} |
| 1730 |
+ |
\end{equation} |
| 1731 |
|
|
| 1732 |
< |
At the very beginning of the simulation, the molecules may not be at its |
| 1733 |
< |
constraint position. To move the z-constraint molecule to the specified |
| 1734 |
< |
position, a simple harmonic potential is used% |
| 1732 |
> |
In a simulation, this would involve the solution of a set of $(m + n)$ |
| 1733 |
> |
number of equations. Where $m$ is the number of constraints, and $n$ |
| 1734 |
> |
is the number of constrained coordinates. In practice, this is not |
| 1735 |
> |
done, as the matrix inversion necessary to solve the system of |
| 1736 |
> |
equations would be very time consuming to solve. Additionally, the |
| 1737 |
> |
numerical error in the solution of the set of $\lambda$'s would be |
| 1738 |
> |
compounded by the error inherent in propagating by the Velocity Verlet |
| 1739 |
> |
algorithm ($\Delta t^4$). The Verlet propagation error is negligible |
| 1740 |
> |
in an unconstrained system, as one is interested in the statistics of |
| 1741 |
> |
the run, and not that the run be numerically exact to the ``true'' |
| 1742 |
> |
integration. This relates back to the ergodic hypothesis that a time |
| 1743 |
> |
integral of a valid trajectory will still give the correct ensemble |
| 1744 |
> |
average. However, in the case of constraints, if the equations of |
| 1745 |
> |
motion leave the ``true'' trajectory, they are departing from the |
| 1746 |
> |
constrained surface. The method that is used, is to iteratively solve |
| 1747 |
> |
for $\lambda(t)$ at each time step. |
| 1748 |
|
|
| 1749 |
+ |
In {\sc rattle} the equations of motion are modified subject to the |
| 1750 |
+ |
following two constraints: |
| 1751 |
+ |
\begin{align} |
| 1752 |
+ |
\sigma_{ij}[\mathbf{r}(t)] \equiv |
| 1753 |
+ |
[ \mathbf{r}_i(t) - \mathbf{r}_j(t)]^2 - d_{ij}^2 &= 0 % |
| 1754 |
+ |
\label{oopseEq:c1}, \\ |
| 1755 |
+ |
% |
| 1756 |
+ |
[\mathbf{\dot{r}}_i(t) - \mathbf{\dot{r}}_j(t)] \cdot |
| 1757 |
+ |
[\mathbf{r}_i(t) - \mathbf{r}_j(t)] &= 0 .\label{oopseEq:c2} |
| 1758 |
+ |
\end{align} |
| 1759 |
+ |
Eq.~\ref{oopseEq:c1} is the set of bond constraints, where $d_{ij}$ is |
| 1760 |
+ |
the constrained distance between atom $i$ and |
| 1761 |
+ |
$j$. Eq.~\ref{oopseEq:c2} constrains the velocities of $i$ and $j$ to |
| 1762 |
+ |
be perpendicular to the bond vector, so that the bond can neither grow |
| 1763 |
+ |
nor shrink. The constrained dynamics equations become: |
| 1764 |
|
\begin{equation} |
| 1765 |
< |
U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}% |
| 1765 |
> |
m_i \mathbf{\ddot{r}}_i = \mathbf{F}_i + \mathbf{\mathcal{G}}_i, |
| 1766 |
> |
\label{oopseEq:r1} |
| 1767 |
|
\end{equation} |
| 1768 |
< |
where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is |
| 1769 |
< |
current z coordinate of the center of mass of the z-constraint molecule, and |
| 963 |
< |
$z_{cons}$ is the restraint position. Therefore, the harmonic force operated |
| 964 |
< |
on the z-constraint molecule at time $t$ can be calculated by% |
| 1768 |
> |
where,$\mathbf{\mathcal{G}}_i$ are the forces of constraint on $i$, |
| 1769 |
> |
and are defined: |
| 1770 |
|
\begin{equation} |
| 1771 |
< |
F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}% |
| 1772 |
< |
(z(t)-z_{cons}) |
| 1771 |
> |
\mathbf{\mathcal{G}}_i = - \sum_j \lambda_{ij}(t)\,\nabla \sigma_{ij}. |
| 1772 |
> |
\label{oopseEq:r2} |
| 1773 |
|
\end{equation} |
| 969 |
– |
Worthy of mention, other kinds of potential functions can also be used to |
| 970 |
– |
drive the z-constraint molecule. |
| 1774 |
|
|
| 1775 |
+ |
In Velocity Verlet, if $\Delta t = h$, the propagation can be written: |
| 1776 |
+ |
\begin{align} |
| 1777 |
+ |
\mathbf{r}_i(t+h) &= |
| 1778 |
+ |
\mathbf{r}_i(t) + h\mathbf{\dot{r}}(t) + |
| 1779 |
+ |
\frac{h^2}{2m_i}\,\Bigl[ \mathbf{F}_i(t) + |
| 1780 |
+ |
\mathbf{\mathcal{G}}_{Ri}(t) \Bigr] \label{oopseEq:vv1}, \\ |
| 1781 |
+ |
% |
| 1782 |
+ |
\mathbf{\dot{r}}_i(t+h) &= |
| 1783 |
+ |
\mathbf{\dot{r}}_i(t) + \frac{h}{2m_i} |
| 1784 |
+ |
\Bigl[ \mathbf{F}_i(t) + \mathbf{\mathcal{G}}_{Ri}(t) + |
| 1785 |
+ |
\mathbf{F}_i(t+h) + \mathbf{\mathcal{G}}_{Vi}(t+h) \Bigr] ,% |
| 1786 |
+ |
\label{oopseEq:vv2} |
| 1787 |
+ |
\end{align} |
| 1788 |
+ |
where: |
| 1789 |
+ |
\begin{align} |
| 1790 |
+ |
\mathbf{\mathcal{G}}_{Ri}(t) &= |
| 1791 |
+ |
-2 \sum_j \lambda_{Rij}(t) \mathbf{r}_{ij}(t) ,\\ |
| 1792 |
+ |
% |
| 1793 |
+ |
\mathbf{\mathcal{G}}_{Vi}(t+h) &= |
| 1794 |
+ |
-2 \sum_j \lambda_{Vij}(t+h) \mathbf{r}(t+h). |
| 1795 |
+ |
\end{align} |
| 1796 |
+ |
Next, define: |
| 1797 |
+ |
\begin{align} |
| 1798 |
+ |
g_{ij} &= h \lambda_{Rij}(t) ,\\ |
| 1799 |
+ |
k_{ij} &= h \lambda_{Vij}(t+h), \\ |
| 1800 |
+ |
\mathbf{q}_i &= \mathbf{\dot{r}}_i(t) + \frac{h}{2m_i} \mathbf{F}_i(t) |
| 1801 |
+ |
- \frac{1}{m_i}\sum_j g_{ij}\mathbf{r}_{ij}(t). |
| 1802 |
+ |
\end{align} |
| 1803 |
+ |
Using these definitions, Eq.~\ref{oopseEq:vv1} and \ref{oopseEq:vv2} |
| 1804 |
+ |
can be rewritten as, |
| 1805 |
+ |
\begin{align} |
| 1806 |
+ |
\mathbf{r}_i(t+h) &= \mathbf{r}_i(t) + h \mathbf{q}_i ,\\ |
| 1807 |
+ |
% |
| 1808 |
+ |
\mathbf{\dot{r}}(t+h) &= \mathbf{q}_i + \frac{h}{2m_i}\mathbf{F}_i(t+h) |
| 1809 |
+ |
-\frac{1}{m_i}\sum_j k_{ij} \mathbf{r}_{ij}(t+h). |
| 1810 |
+ |
\end{align} |
| 1811 |
+ |
|
| 1812 |
+ |
To integrate the equations of motion, the {\sc rattle} algorithm first |
| 1813 |
+ |
solves for $\mathbf{r}(t+h)$. Let, |
| 1814 |
+ |
\begin{equation} |
| 1815 |
+ |
\mathbf{q}_i = \mathbf{\dot{r}}(t) + \frac{h}{2m_i}\mathbf{F}_i(t). |
| 1816 |
+ |
\end{equation} |
| 1817 |
+ |
Here $\mathbf{q}_i$ corresponds to an initial unconstrained move. Next |
| 1818 |
+ |
pick a constraint $j$, and let, |
| 1819 |
+ |
\begin{equation} |
| 1820 |
+ |
\mathbf{s} = \mathbf{r}_i(t) + h\mathbf{q}_i(t) |
| 1821 |
+ |
- \mathbf{r}_j(t) + h\mathbf{q}_j(t). |
| 1822 |
+ |
\label{oopseEq:ra1} |
| 1823 |
+ |
\end{equation} |
| 1824 |
+ |
If |
| 1825 |
+ |
\begin{equation} |
| 1826 |
+ |
\Big| |\mathbf{s}|^2 - d_{ij}^2 \Big| > \text{tolerance}, |
| 1827 |
+ |
\end{equation} |
| 1828 |
+ |
then the constraint is unsatisfied, and corrections are made to the |
| 1829 |
+ |
positions. First we define a test corrected configuration as, |
| 1830 |
+ |
\begin{align} |
| 1831 |
+ |
\mathbf{r}_i^T(t+h) = \mathbf{r}_i(t) + h\biggl[\mathbf{q}_i - |
| 1832 |
+ |
g_{ij}\,\frac{\mathbf{r}_{ij}(t)}{m_i} \biggr] ,\\ |
| 1833 |
+ |
% |
| 1834 |
+ |
\mathbf{r}_j^T(t+h) = \mathbf{r}_j(t) + h\biggl[\mathbf{q}_j + |
| 1835 |
+ |
g_{ij}\,\frac{\mathbf{r}_{ij}(t)}{m_j} \biggr]. |
| 1836 |
+ |
\end{align} |
| 1837 |
+ |
And we chose $g_{ij}$ such that, $|\mathbf{r}_i^T - \mathbf{r}_j^T|^2 |
| 1838 |
+ |
= d_{ij}^2$. Solving the quadratic for $g_{ij}$ we obtain the |
| 1839 |
+ |
approximation, |
| 1840 |
+ |
\begin{equation} |
| 1841 |
+ |
g_{ij} = \frac{(s^2 - d^2)}{2h[\mathbf{s}\cdot\mathbf{r}_{ij}(t)] |
| 1842 |
+ |
(\frac{1}{m_i} + \frac{1}{m_j})}. |
| 1843 |
+ |
\end{equation} |
| 1844 |
+ |
Although not an exact solution for $g_{ij}$, as this is an iterative |
| 1845 |
+ |
scheme overall, the eventual solution will converge. With a trial |
| 1846 |
+ |
$g_{ij}$, the new $\mathbf{q}$'s become, |
| 1847 |
+ |
\begin{align} |
| 1848 |
+ |
\mathbf{q}_i &= \mathbf{q}^{\text{old}}_i - g_{ij}\, |
| 1849 |
+ |
\frac{\mathbf{r}_{ij}(t)}{m_i} ,\\ |
| 1850 |
+ |
% |
| 1851 |
+ |
\mathbf{q}_j &= \mathbf{q}^{\text{old}}_j + g_{ij}\, |
| 1852 |
+ |
\frac{\mathbf{r}_{ij}(t)}{m_j} . |
| 1853 |
+ |
\end{align} |
| 1854 |
+ |
The whole algorithm is then repeated from Eq.~\ref{oopseEq:ra1} until |
| 1855 |
+ |
all constraints are satisfied. |
| 1856 |
+ |
|
| 1857 |
+ |
The second step of {\sc rattle}, is to then update the velocities. The |
| 1858 |
+ |
step starts with, |
| 1859 |
+ |
\begin{equation} |
| 1860 |
+ |
\mathbf{\dot{r}}_i(t+h) = \mathbf{q}_i + \frac{h}{2m_i}\mathbf{F}_i(t+h). |
| 1861 |
+ |
\end{equation} |
| 1862 |
+ |
Next we pick a constraint $j$, and calculate the dot product $\ell$. |
| 1863 |
+ |
\begin{equation} |
| 1864 |
+ |
\ell = \mathbf{r}_{ij}(t+h) \cdot \mathbf{\dot{r}}_{ij}(t+h). |
| 1865 |
+ |
\label{oopseEq:rv1} |
| 1866 |
+ |
\end{equation} |
| 1867 |
+ |
Here if constraint Eq.~\ref{oopseEq:c2} holds, $\ell$ should be |
| 1868 |
+ |
zero. Therefore if $\ell$ is greater than some tolerance, then |
| 1869 |
+ |
corrections are made to the $i$ and $j$ velocities. |
| 1870 |
+ |
\begin{align} |
| 1871 |
+ |
\mathbf{\dot{r}}_i^T &= \mathbf{\dot{r}}_i(t+h) - k_{ij} |
| 1872 |
+ |
\frac{\mathbf{\dot{r}}_{ij}(t+h)}{m_i}, \\ |
| 1873 |
+ |
% |
| 1874 |
+ |
\mathbf{\dot{r}}_j^T &= \mathbf{\dot{r}}_j(t+h) + k_{ij} |
| 1875 |
+ |
\frac{\mathbf{\dot{r}}_{ij}(t+h)}{m_j}. |
| 1876 |
+ |
\end{align} |
| 1877 |
+ |
Like in the previous step, we select a value for $k_{ij}$ such that |
| 1878 |
+ |
$\ell$ is zero. |
| 1879 |
+ |
\begin{equation} |
| 1880 |
+ |
k_{ij} = \frac{\ell}{d^2_{ij}(\frac{1}{m_i} + \frac{1}{m_j})}. |
| 1881 |
+ |
\end{equation} |
| 1882 |
+ |
The test velocities, $\mathbf{\dot{r}}^T_i$ and |
| 1883 |
+ |
$\mathbf{\dot{r}}^T_j$, then replace their respective velocities, and |
| 1884 |
+ |
the algorithm is iterated from Eq.~\ref{oopseEq:rv1} until all |
| 1885 |
+ |
constraints are satisfied. |
| 1886 |
+ |
|
| 1887 |
+ |
|
| 1888 |
+ |
\subsection{\label{oopseSec:zcons}Z-Constraint Method} |
| 1889 |
+ |
|
| 1890 |
+ |
Based on the fluctuation-dissipation theorem, a force auto-correlation |
| 1891 |
+ |
method was developed by Roux and Karplus to investigate the dynamics |
| 1892 |
+ |
of ions inside ion channels.\cite{Roux91} The time-dependent friction |
| 1893 |
+ |
coefficient can be calculated from the deviation of the instantaneous |
| 1894 |
+ |
force from its mean force. |
| 1895 |
+ |
\begin{equation} |
| 1896 |
+ |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 1897 |
+ |
\end{equation} |
| 1898 |
+ |
where% |
| 1899 |
+ |
\begin{equation} |
| 1900 |
+ |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 1901 |
+ |
\end{equation} |
| 1902 |
+ |
|
| 1903 |
+ |
|
| 1904 |
+ |
If the time-dependent friction decays rapidly, the static friction |
| 1905 |
+ |
coefficient can be approximated by |
| 1906 |
+ |
\begin{equation} |
| 1907 |
+ |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt. |
| 1908 |
+ |
\end{equation} |
| 1909 |
+ |
Allowing diffusion constant to then be calculated through the |
| 1910 |
+ |
Einstein relation:\cite{Marrink94} |
| 1911 |
+ |
\begin{equation} |
| 1912 |
+ |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 1913 |
+ |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 1914 |
+ |
\end{equation} |
| 1915 |
+ |
|
| 1916 |
+ |
The Z-Constraint method, which fixes the z coordinates of the |
| 1917 |
+ |
molecules with respect to the center of the mass of the system, has |
| 1918 |
+ |
been a method suggested to obtain the forces required for the force |
| 1919 |
+ |
auto-correlation calculation.\cite{Marrink94} However, simply resetting the |
| 1920 |
+ |
coordinate will move the center of the mass of the whole system. To |
| 1921 |
+ |
avoid this problem, a new method was used in {\sc oopse}. Instead of |
| 1922 |
+ |
resetting the coordinate, we reset the forces of z-constrained |
| 1923 |
+ |
molecules as well as subtract the total constraint forces from the |
| 1924 |
+ |
rest of the system after the force calculation at each time step. |
| 1925 |
+ |
|
| 1926 |
+ |
After the force calculation, define $G_\alpha$ as |
| 1927 |
+ |
\begin{equation} |
| 1928 |
+ |
G_{\alpha} = \sum_i F_{\alpha i}, |
| 1929 |
+ |
\label{oopseEq:zc1} |
| 1930 |
+ |
\end{equation} |
| 1931 |
+ |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 1932 |
+ |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 1933 |
+ |
molecule are then set to: |
| 1934 |
+ |
\begin{equation} |
| 1935 |
+ |
F_{\alpha i} = F_{\alpha i} - |
| 1936 |
+ |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 1937 |
+ |
\end{equation} |
| 1938 |
+ |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 1939 |
+ |
molecule. Having rescaled the forces, the velocities must also be |
| 1940 |
+ |
rescaled to subtract out any center of mass velocity in the z |
| 1941 |
+ |
direction. |
| 1942 |
+ |
\begin{equation} |
| 1943 |
+ |
v_{\alpha i} = v_{\alpha i} - |
| 1944 |
+ |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 1945 |
+ |
\end{equation} |
| 1946 |
+ |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 1947 |
+ |
Lastly, all of the accumulated z constrained forces must be subtracted |
| 1948 |
+ |
from the system to keep the system center of mass from drifting. |
| 1949 |
+ |
\begin{equation} |
| 1950 |
+ |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} G_{\alpha}} |
| 1951 |
+ |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 1952 |
+ |
\end{equation} |
| 1953 |
+ |
where $\beta$ are all of the unconstrained molecules in the |
| 1954 |
+ |
system. Similarly, the velocities of the unconstrained molecules must |
| 1955 |
+ |
also be scaled. |
| 1956 |
+ |
\begin{equation} |
| 1957 |
+ |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 1958 |
+ |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 1959 |
+ |
\end{equation} |
| 1960 |
+ |
|
| 1961 |
+ |
At the very beginning of the simulation, the molecules may not be at their |
| 1962 |
+ |
constrained positions. To move a z-constrained molecule to its specified |
| 1963 |
+ |
position, a simple harmonic potential is used |
| 1964 |
+ |
\begin{equation} |
| 1965 |
+ |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 1966 |
+ |
\end{equation} |
| 1967 |
+ |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ is the |
| 1968 |
+ |
current $z$ coordinate of the center of mass of the constrained molecule, and |
| 1969 |
+ |
$z_{\text{cons}}$ is the constrained position. The harmonic force operating |
| 1970 |
+ |
on the z-constrained molecule at time $t$ can be calculated by |
| 1971 |
+ |
\begin{equation} |
| 1972 |
+ |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 1973 |
+ |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 1974 |
+ |
\end{equation} |
| 1975 |
+ |
|
| 1976 |
|
\section{\label{oopseSec:props}Trajectory Analysis} |
| 1977 |
|
|
| 1978 |
|
\subsection{\label{oopseSec:staticProps}Static Property Analysis} |
| 1979 |
|
|
| 1980 |
|
The static properties of the trajectories are analyzed with the |
| 1981 |
< |
program \texttt{staticProps}. The code is capable of calculating the following |
| 1982 |
< |
pair correlations between species A and B: |
| 1983 |
< |
\begin{itemize} |
| 1984 |
< |
\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr} |
| 981 |
< |
\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta} |
| 982 |
< |
\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega} |
| 983 |
< |
\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ} |
| 984 |
< |
\item $\langle \cos \omega \rangle_{\text{AB}}(r)$: |
| 985 |
< |
Eq.~\ref{eq:cosOmegaOfR} |
| 986 |
< |
\end{itemize} |
| 1981 |
> |
program \texttt{staticProps}. The code is capable of calculating a |
| 1982 |
> |
number of pair correlations between species A and B. Some of which |
| 1983 |
> |
only apply to directional entities. The summary of pair correlations |
| 1984 |
> |
can be found in Table~\ref{oopseTb:gofrs} |
| 1985 |
|
|
| 1986 |
+ |
\begin{table} |
| 1987 |
+ |
\caption{THE DIFFERENT PAIR CORRELATIONS IN \texttt{staticProps}} |
| 1988 |
+ |
\label{oopseTb:gofrs} |
| 1989 |
+ |
\begin{center} |
| 1990 |
+ |
\begin{tabular}{|l|c|c|} |
| 1991 |
+ |
\hline |
| 1992 |
+ |
Name & Equation & Directional Atom \\ \hline |
| 1993 |
+ |
$g_{\text{AB}}(r)$ & Eq.~\ref{eq:gofr} & neither \\ \hline |
| 1994 |
+ |
$g_{\text{AB}}(r, \cos \theta)$ & Eq.~\ref{eq:gofrCosTheta} & A \\ \hline |
| 1995 |
+ |
$g_{\text{AB}}(r, \cos \omega)$ & Eq.~\ref{eq:gofrCosOmega} & both \\ \hline |
| 1996 |
+ |
$g_{\text{AB}}(x, y, z)$ & Eq.~\ref{eq:gofrXYZ} & neither \\ \hline |
| 1997 |
+ |
$\langle \cos \omega \rangle_{\text{AB}}(r)$ & Eq.~\ref{eq:cosOmegaOfR} &% |
| 1998 |
+ |
both \\ \hline |
| 1999 |
+ |
\end{tabular} |
| 2000 |
+ |
\begin{minipage}{\linewidth} |
| 2001 |
+ |
\centering |
| 2002 |
+ |
\vspace{2mm} |
| 2003 |
+ |
The third column specifies which atom, if any, need be a directional entity. |
| 2004 |
+ |
\end{minipage} |
| 2005 |
+ |
\end{center} |
| 2006 |
+ |
\end{table} |
| 2007 |
+ |
|
| 2008 |
|
The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
| 2009 |
|
\begin{equation} |
| 2010 |
|
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %% |
| 2011 |
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} %% |
| 2012 |
< |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr} |
| 2012 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle, \label{eq:gofr} |
| 2013 |
|
\end{equation} |
| 2014 |
< |
Where $\mathbf{r}_{ij}$ is the vector |
| 2014 |
> |
where $\mathbf{r}_{ij}$ is the vector |
| 2015 |
|
\begin{equation*} |
| 2016 |
< |
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag |
| 2016 |
> |
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i, \notag |
| 2017 |
|
\end{equation*} |
| 2018 |
|
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over |
| 2019 |
|
the expected pair density at a given $r$. |
| 2030 |
|
g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
| 2031 |
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
| 2032 |
|
\delta( \cos \theta - \cos \theta_{ij}) |
| 2033 |
< |
\delta( r - |\mathbf{r}_{ij}|) \rangle |
| 2033 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle. |
| 2034 |
|
\label{eq:gofrCosTheta} |
| 2035 |
|
\end{equation} |
| 2036 |
< |
Where |
| 2036 |
> |
Here |
| 2037 |
|
\begin{equation*} |
| 2038 |
< |
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
| 2038 |
> |
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij}, |
| 2039 |
|
\end{equation*} |
| 2040 |
< |
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
| 2040 |
> |
where $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
| 2041 |
|
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector |
| 2042 |
|
$\mathbf{r}_{ij}$. |
| 2043 |
|
|
| 2047 |
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
| 2048 |
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
| 2049 |
|
\delta( \cos \omega - \cos \omega_{ij}) |
| 2050 |
< |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
| 2050 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle. \label{eq:gofrCosOmega} |
| 2051 |
|
\end{equation} |
| 2052 |
|
Here |
| 2053 |
|
\begin{equation*} |
| 2054 |
< |
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
| 2054 |
> |
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}}. |
| 2055 |
|
\end{equation*} |
| 2056 |
|
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit |
| 2057 |
|
directional vectors of species $i$ and $j$. |
| 2064 |
|
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
| 2065 |
|
\delta( x - x_{ij}) |
| 2066 |
|
\delta( y - y_{ij}) |
| 2067 |
< |
\delta( z - z_{ij}) \rangle |
| 2067 |
> |
\delta( z - z_{ij}) \rangle, |
| 2068 |
|
\end{equation} |
| 2069 |
< |
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
| 2069 |
> |
where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
| 2070 |
|
components respectively of vector $\mathbf{r}_{ij}$. |
| 2071 |
|
|
| 2072 |
|
The final pair correlation is similar to |
| 2075 |
|
\begin{equation}\label{eq:cosOmegaOfR} |
| 2076 |
|
\langle \cos \omega \rangle_{\text{AB}}(r) = |
| 2077 |
|
\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}} |
| 2078 |
< |
(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle |
| 2078 |
> |
(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle. |
| 2079 |
|
\end{equation} |
| 2080 |
|
Here $\cos \omega_{ij}$ is defined in the same way as in |
| 2081 |
|
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair |
| 2082 |
|
correlation that gives the average correlation of two directional |
| 2083 |
|
entities as a function of their distance from each other. |
| 2084 |
|
|
| 1065 |
– |
All static properties are calculated on a frame by frame basis. The |
| 1066 |
– |
trajectory is read a single frame at a time, and the appropriate |
| 1067 |
– |
calculations are done on each frame. Once one frame is finished, the |
| 1068 |
– |
next frame is read in, and a running average of the property being |
| 1069 |
– |
calculated is accumulated in each frame. The program allows for the |
| 1070 |
– |
user to specify more than one property be calculated in single run, |
| 1071 |
– |
preventing the need to read a file multiple times. |
| 1072 |
– |
|
| 2085 |
|
\subsection{\label{dynamicProps}Dynamic Property Analysis} |
| 2086 |
|
|
| 2087 |
|
The dynamic properties of a trajectory are calculated with the program |
| 2088 |
< |
\texttt{dynamicProps}. The program will calculate the following properties: |
| 2088 |
> |
\texttt{dynamicProps}. The program calculates the following properties: |
| 2089 |
|
\begin{gather} |
| 2090 |
< |
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\ |
| 2091 |
< |
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\ |
| 2092 |
< |
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr} |
| 2090 |
> |
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle, \label{eq:rms}\\ |
| 2091 |
> |
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle, \label{eq:velCorr} \\ |
| 2092 |
> |
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle. \label{eq:angularVelCorr} |
| 2093 |
|
\end{gather} |
| 2094 |
|
|
| 2095 |
< |
Eq.~\ref{eq:rms} is the root mean square displacement |
| 2096 |
< |
function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the |
| 2097 |
< |
velocity and angular velocity correlation functions respectively. The |
| 2098 |
< |
latter is only applicable to directional species in the simulation. |
| 2095 |
> |
Eq.~\ref{eq:rms} is the root mean square displacement function. Which |
| 2096 |
> |
allows one to observe the average displacement of an atom as a |
| 2097 |
> |
function of time. The quantity is useful when calculating diffusion |
| 2098 |
> |
coefficients because of the Einstein Relation, which is valid at long |
| 2099 |
> |
times.\cite{allen87:csl} |
| 2100 |
> |
\begin{equation} |
| 2101 |
> |
2tD = \langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle. |
| 2102 |
> |
\label{oopseEq:einstein} |
| 2103 |
> |
\end{equation} |
| 2104 |
|
|
| 2105 |
< |
The \texttt{dynamicProps} program handles he file in a manner different from |
| 2106 |
< |
\texttt{staticProps}. As the properties calculated by this program are time |
| 2107 |
< |
dependent, multiple frames must be read in simultaneously by the |
| 2108 |
< |
program. For small trajectories this is no problem, and the entire |
| 2109 |
< |
trajectory is read into memory. However, for long trajectories of |
| 1093 |
< |
large systems, the files can be quite large. In order to accommodate |
| 1094 |
< |
large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory |
| 1095 |
< |
are allocated to read in several frames each. |
| 1096 |
< |
|
| 1097 |
< |
In this two block scheme, the correlation functions are first |
| 1098 |
< |
calculated within each memory block, then the cross correlations |
| 1099 |
< |
between the frames contained within the two blocks are |
| 1100 |
< |
calculated. Once completed, the memory blocks are incremented, and the |
| 1101 |
< |
process is repeated. A diagram illustrating the process is shown in |
| 1102 |
< |
Fig.~\ref{oopseFig:dynamicPropsMemory}. As was the case with |
| 1103 |
< |
\texttt{staticProps}, multiple properties may be calculated in a |
| 1104 |
< |
single run to avoid multiple reads on the same file. |
| 1105 |
< |
|
| 1106 |
< |
|
| 2105 |
> |
Eq.~\ref{eq:velCorr} and \ref{eq:angularVelCorr} are the translational |
| 2106 |
> |
velocity and angular velocity correlation functions respectively. The |
| 2107 |
> |
latter is only applicable to directional species in the |
| 2108 |
> |
simulation. The velocity autocorrelation functions are useful when |
| 2109 |
> |
determining vibrational information about the system of interest. |
| 2110 |
|
|
| 2111 |
|
\section{\label{oopseSec:design}Program Design} |
| 2112 |
|
|
| 2146 |
|
developed to utilize the routines provided by \texttt{libBASS} and |
| 2147 |
|
\texttt{libmdtools}. The main program of the package is \texttt{oopse} |
| 2148 |
|
and the corresponding parallel version \texttt{oopse\_MPI}. These two |
| 2149 |
< |
programs will take the \texttt{.bass} file, and create then integrate |
| 2149 |
> |
programs will take the \texttt{.bass} file, and create (and integrate) |
| 2150 |
|
the simulation specified in the script. The two analysis programs |
| 2151 |
|
\texttt{staticProps} and \texttt{dynamicProps} utilize the core |
| 2152 |
|
libraries to initialize and read in trajectories from previously |
| 2158 |
|
|
| 2159 |
|
\subsection{\label{oopseSec:parallelization} Parallelization of {\sc oopse}} |
| 2160 |
|
|
| 2161 |
< |
Although processor power is continually growing month by month, it is |
| 2162 |
< |
still unreasonable to simulate systems of more then a 1000 atoms on a |
| 2163 |
< |
single processor. To facilitate study of larger system sizes or |
| 2164 |
< |
smaller systems on long time scales in a reasonable period of time, |
| 2165 |
< |
parallel methods were developed allowing multiple CPU's to share the |
| 2166 |
< |
simulation workload. Three general categories of parallel |
| 2167 |
< |
decomposition method's have been developed including atomic, spatial |
| 2168 |
< |
and force decomposition methods. |
| 2161 |
> |
Although processor power is continually growing roughly following |
| 2162 |
> |
Moore's Law, it is still unreasonable to simulate systems of more then |
| 2163 |
> |
a 1000 atoms on a single processor. To facilitate study of larger |
| 2164 |
> |
system sizes or smaller systems on long time scales in a reasonable |
| 2165 |
> |
period of time, parallel methods were developed allowing multiple |
| 2166 |
> |
CPU's to share the simulation workload. Three general categories of |
| 2167 |
> |
parallel decomposition methods have been developed including atomic, |
| 2168 |
> |
spatial and force decomposition methods. |
| 2169 |
|
|
| 2170 |
< |
Algorithmically simplest of the three method's is atomic decomposition |
| 2170 |
> |
Algorithmically simplest of the three methods is atomic decomposition |
| 2171 |
|
where N particles in a simulation are split among P processors for the |
| 2172 |
|
duration of the simulation. Computational cost scales as an optimal |
| 2173 |
< |
$O(N/P)$ for atomic decomposition. Unfortunately all processors must |
| 2174 |
< |
communicate positions and forces with all other processors leading |
| 2175 |
< |
communication to scale as an unfavorable $O(N)$ independent of the |
| 2176 |
< |
number of processors. This communication bottleneck led to the |
| 2173 |
> |
$\mathcal{O}(N/P)$ for atomic decomposition. Unfortunately all |
| 2174 |
> |
processors must communicate positions and forces with all other |
| 2175 |
> |
processors at every force evaluation, leading communication costs to |
| 2176 |
> |
scale as an unfavorable $\mathcal{O}(N)$, \emph{independent of the |
| 2177 |
> |
number of processors}. This communication bottleneck led to the |
| 2178 |
|
development of spatial and force decomposition methods in which |
| 2179 |
|
communication among processors scales much more favorably. Spatial or |
| 2180 |
|
domain decomposition divides the physical spatial domain into 3D boxes |
| 2185 |
|
positions of particles within some cutoff radius located on nearby |
| 2186 |
|
processors instead of the positions of particles on all |
| 2187 |
|
processors. Both communication between processors and computation |
| 2188 |
< |
scale as $O(N/P)$ in the spatial method. However, spatial |
| 2188 |
> |
scale as $\mathcal{O}(N/P)$ in the spatial method. However, spatial |
| 2189 |
|
decomposition adds algorithmic complexity to the simulation code and |
| 2190 |
|
is not very efficient for small N since the overall communication |
| 2191 |
< |
scales as the surface to volume ratio $(N/P)^{2/3}$ in three |
| 2192 |
< |
dimensions. |
| 2191 |
> |
scales as the surface to volume ratio $\mathcal{O}(N/P)^{2/3}$ in |
| 2192 |
> |
three dimensions. |
| 2193 |
|
|
| 2194 |
< |
Force decomposition assigns particles to processors based on a block |
| 2195 |
< |
decomposition of the force matrix. Processors are split into a |
| 2196 |
< |
optimally square grid forming row and column processor groups. Forces |
| 2197 |
< |
are calculated on particles in a given row by particles located in |
| 2198 |
< |
that processors column assignment. Force decomposition is less complex |
| 2199 |
< |
to implement then the spatial method but still scales computationally |
| 2200 |
< |
as $O(N/P)$ and scales as $(N/\sqrt{p})$ in communication |
| 2201 |
< |
cost. Plimpton also found that force decompositions scales more |
| 2202 |
< |
favorably then spatial decomposition up to 10,000 atoms and favorably |
| 2203 |
< |
competes with spatial methods for up to 100,000 atoms. |
| 2194 |
> |
The parallelization method used in {\sc oopse} is the force |
| 2195 |
> |
decomposition method. Force decomposition assigns particles to |
| 2196 |
> |
processors based on a block decomposition of the force |
| 2197 |
> |
matrix. Processors are split into an optimally square grid forming row |
| 2198 |
> |
and column processor groups. Forces are calculated on particles in a |
| 2199 |
> |
given row by particles located in that processors column |
| 2200 |
> |
assignment. Force decomposition is less complex to implement than the |
| 2201 |
> |
spatial method but still scales computationally as $\mathcal{O}(N/P)$ |
| 2202 |
> |
and scales as $\mathcal{O}(N/\sqrt{P})$ in communication |
| 2203 |
> |
cost. Plimpton has also found that force decompositions scale more |
| 2204 |
> |
favorably than spatial decompositions for systems up to 10,000 atoms |
| 2205 |
> |
and favorably compete with spatial methods up to 100,000 |
| 2206 |
> |
atoms.\cite{plimpton95} |
| 2207 |
|
|
| 2208 |
|
\subsection{\label{oopseSec:memAlloc}Memory Issues in Trajectory Analysis} |
| 2209 |
|
|
| 2210 |
|
For large simulations, the trajectory files can sometimes reach sizes |
| 2211 |
|
in excess of several gigabytes. In order to effectively analyze that |
| 2212 |
< |
amount of data+, two memory management schemes have been devised for |
| 2212 |
> |
amount of data, two memory management schemes have been devised for |
| 2213 |
|
\texttt{staticProps} and for \texttt{dynamicProps}. The first scheme, |
| 2214 |
|
developed for \texttt{staticProps}, is the simplest. As each frame's |
| 2215 |
|
statistics are calculated independent of each other, memory is |
| 2217 |
|
complete for the snapshot. To prevent multiple passes through a |
| 2218 |
|
potentially large file, \texttt{staticProps} is capable of calculating |
| 2219 |
|
all requested correlations per frame with only a single pair loop in |
| 2220 |
< |
each frame and a single read through of the file. |
| 2220 |
> |
each frame and a single read of the file. |
| 2221 |
|
|
| 2222 |
|
The second, more advanced memory scheme, is used by |
| 2223 |
|
\texttt{dynamicProps}. Here, the program must have multiple frames in |
| 2227 |
|
in blocks. The number of frames in each block is specified by the |
| 2228 |
|
user, and upon reading a block of the trajectory, |
| 2229 |
|
\texttt{dynamicProps} will calculate all of the time correlation frame |
| 2230 |
< |
pairs within the block. After in block correlations are complete, a |
| 2230 |
> |
pairs within the block. After in-block correlations are complete, a |
| 2231 |
|
second block of the trajectory is read, and the cross correlations are |
| 2232 |
< |
calculated between the two blocks. this second block is then freed and |
| 2232 |
> |
calculated between the two blocks. This second block is then freed and |
| 2233 |
|
then incremented and the process repeated until the end of the |
| 2234 |
|
trajectory. Once the end is reached, the first block is freed then |
| 2235 |
|
incremented, and the again the internal time correlations are |
| 2245 |
|
\label{oopseFig:dynamicPropsMemory} |
| 2246 |
|
\end{figure} |
| 2247 |
|
|
| 1241 |
– |
\subsection{\label{openSource}Open Source and Distribution License} |
| 1242 |
– |
|
| 2248 |
|
\section{\label{oopseSec:conclusion}Conclusion} |
| 2249 |
|
|
| 2250 |
|
We have presented the design and implementation of our open source |
| 2251 |
< |
simulation package {\sc oopse}. The package offers novel |
| 2252 |
< |
capabilities to the field of Molecular Dynamics simulation packages in |
| 2253 |
< |
the form of dipolar force fields, and symplectic integration of rigid |
| 2254 |
< |
body dynamics. It is capable of scaling across multiple processors |
| 2255 |
< |
through the use of MPI. It also implements several integration |
| 2256 |
< |
ensembles allowing the end user control over temperature and |
| 2257 |
< |
pressure. In addition, it is capable of integrating constrained |
| 2258 |
< |
dynamics through both the {\sc rattle} algorithm and the z-constraint |
| 2259 |
< |
method. |
| 2251 |
> |
simulation package {\sc oopse}. The package offers novel capabilities |
| 2252 |
> |
to the field of Molecular Dynamics simulation packages in the form of |
| 2253 |
> |
dipolar force fields, and symplectic integration of rigid body |
| 2254 |
> |
dynamics. It is capable of scaling across multiple processors through |
| 2255 |
> |
the use of force based decomposition using MPI. It also implements |
| 2256 |
> |
several advanced integrators allowing the end user control over |
| 2257 |
> |
temperature and pressure. In addition, it is capable of integrating |
| 2258 |
> |
constrained dynamics through both the {\sc rattle} algorithm and the |
| 2259 |
> |
z-constraint method. |
| 2260 |
|
|
| 2261 |
|
These features are all brought together in a single open-source |
| 2262 |
< |
development package. This allows researchers to not only benefit from |
| 2262 |
> |
program. This allows researchers to not only benefit from |
| 2263 |
|
{\sc oopse}, but also contribute to {\sc oopse}'s development as |
| 2264 |
< |
well.Documentation and source code for {\sc oopse} can be downloaded |
| 1260 |
< |
from \texttt{http://www.openscience.org/oopse/}. |
| 2264 |
> |
well. |
| 2265 |
|
|
