| 97 |
|
Written this way, the $\mbox{rot}$ operation creates a set of |
| 98 |
|
conjugate angle coordinates to the body-fixed angular momenta |
| 99 |
|
represented by ${\bf j}$. This equation of motion for angular momenta |
| 100 |
< |
is equivalent to the more familiar body-fixed form: |
| 100 |
> |
is equivalent to the more familiar body-fixed forms, |
| 101 |
|
\begin{eqnarray} |
| 102 |
|
\dot{j_{x}} & = & \tau^b_x(t) + |
| 103 |
|
\left(\overleftrightarrow{\mathsf{I}}_{yy} - \overleftrightarrow{\mathsf{I}}_{zz} \right) j_y j_z \\ |
| 106 |
|
\dot{j_{z}} & = & \tau^b_z(t) + |
| 107 |
|
\left(\overleftrightarrow{\mathsf{I}}_{xx} - \overleftrightarrow{\mathsf{I}}_{yy} \right) j_x j_y |
| 108 |
|
\end{eqnarray} |
| 109 |
< |
which utilize the body-fixed torques, ${\bf \tau}^b$. Torques are |
| 109 |
> |
which utilize the body-fixed torques, ${\bf \tau}^b$. Torques are |
| 110 |
|
most easily derived in the space-fixed frame, |
| 111 |
|
\begin{equation} |
| 112 |
|
{\bf \tau}^b(t) = \mathsf{A}(t) \cdot {\bf \tau}^s(t) |
| 113 |
|
\end{equation} |
| 114 |
< |
where |
| 114 |
> |
where the torques are either derived from the forces on the |
| 115 |
> |
constituent atoms of the rigid body, or for directional atoms, |
| 116 |
> |
directly from derivatives of the potential energy, |
| 117 |
|
\begin{equation} |
| 118 |
|
{\bf \tau}^s(t) = - \hat{\bf u}(t) \times \left( \frac{\partial} |
| 119 |
|
{\partial \hat{\bf u}} V\left(\left\{ {\bf r}(t) \right\}, \left\{ |
| 120 |
< |
\mathsf{A}(t) \right\}\right) \right) |
| 120 |
> |
\mathsf{A}(t) \right\}\right) \right). |
| 121 |
|
\end{equation} |
| 122 |
< |
where $\hat{\bf u}$ is a unit vector pointing along the principal axis |
| 122 |
> |
Here $\hat{\bf u}$ is a unit vector pointing along the principal axis |
| 123 |
|
of the particle in the space-fixed frame. |
| 124 |
|
|
| 125 |
|
The DLM method uses a Trotter factorization of the orientational |
| 126 |
|
propagator. This has three effects: |
| 127 |
|
\begin{enumerate} |
| 128 |
< |
\item the integrator is area preserving in phase space (i.e. it is |
| 128 |
> |
\item the integrator is area-preserving in phase space (i.e. it is |
| 129 |
|
{\it symplectic}), |
| 130 |
|
\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 131 |
|
Monte Carlo applications, and |
| 133 |
|
for timesteps of length $h$. |
| 134 |
|
\end{enumerate} |
| 135 |
|
|
| 136 |
< |
In the integration method, the |
| 137 |
< |
orientational propagation involves a sequence of matrix evaluations to |
| 138 |
< |
update the rotation matrix.\cite{Dullweber1997} These matrix rotations |
| 139 |
< |
end up being more costly computationally than the simpler arithmetic |
| 140 |
< |
quaternion propagation. With the same time step, a 1000 SSD particle |
| 141 |
< |
simulation shows an average 7\% increase in computation time using the |
| 142 |
< |
symplectic step method in place of quaternions. This cost is more than |
| 143 |
< |
justified when comparing the energy conservation of the two methods as |
| 144 |
< |
illustrated in figure |
| 145 |
< |
\ref{timestep}. |
| 136 |
> |
The integration of the equations of motion is carried out in a |
| 137 |
> |
velocity-Verlet style 2 part algorithm: |
| 138 |
> |
|
| 139 |
> |
{\tt moveA:} |
| 140 |
> |
\begin{eqnarray} |
| 141 |
> |
{\bf v}\left(t + \delta t / 2\right) & \leftarrow & {\bf |
| 142 |
> |
v}(t) + \frac{\delta t}{2} \left( {\bf f}(t) / m \right) \\ |
| 143 |
> |
{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf |
| 144 |
> |
v}\left(t + \delta t / 2 \right) \\ |
| 145 |
> |
{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
| 146 |
> |
j}(t) + \frac{\delta t}{2} {\bf \tau}^b(t) \\ |
| 147 |
> |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left( \delta t |
| 148 |
> |
{\bf j}(t + \delta t / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} |
| 149 |
> |
\right) |
| 150 |
> |
\end{eqnarray} |
| 151 |
> |
|
| 152 |
> |
In this context, the $\mathrm{rot}$ function is the reversible product |
| 153 |
> |
of the three body-fixed rotations, |
| 154 |
> |
\begin{equation} |
| 155 |
> |
\mathrm{rot}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
| 156 |
> |
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y / |
| 157 |
> |
2) \cdot \mathsf{G}_x(a_x /2) |
| 158 |
> |
\end{equation} |
| 159 |
> |
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, rotates |
| 160 |
> |
both the rotation matrix ($\mathsf{A}$) and the body-fixed angular |
| 161 |
> |
momentum (${\bf j}$) by an angle $\theta$ around body-fixed axis |
| 162 |
> |
$\alpha$, |
| 163 |
> |
\begin{equation} |
| 164 |
> |
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 165 |
> |
\begin{array}{lcl} |
| 166 |
> |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T \\ |
| 167 |
> |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf j}(0) |
| 168 |
> |
\end{array} |
| 169 |
> |
\right. |
| 170 |
> |
\end{equation} |
| 171 |
> |
$\mathsf{R}_\alpha$ is a quadratic approximation to |
| 172 |
> |
the single-axis rotation matrix. For example, in the small-angle |
| 173 |
> |
limit, the rotation matrix around the body-fixed x-axis can be |
| 174 |
> |
approximated as |
| 175 |
> |
\begin{equation} |
| 176 |
> |
\mathsf{R}_x(\theta) = \left( |
| 177 |
> |
\begin{array}{ccc} |
| 178 |
> |
1 & 0 & 0 \\ |
| 179 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 180 |
> |
\theta^2 / 4} \\ |
| 181 |
> |
0 & \frac{\theta}{1+ |
| 182 |
> |
\theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} |
| 183 |
> |
\end{array} |
| 184 |
> |
\right). |
| 185 |
> |
\end{equation} |
| 186 |
> |
All other rotations follow in a straightforward manner. |
| 187 |
> |
|
| 188 |
> |
After the first part of the propagation, the forces and body-fixed |
| 189 |
> |
torques are calculated at the new positions and orientations |
| 190 |
> |
|
| 191 |
> |
{\tt doForces:} |
| 192 |
> |
\begin{eqnarray} |
| 193 |
> |
{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf |
| 194 |
> |
r}(t + \delta t)} \\ |
| 195 |
> |
{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t) |
| 196 |
> |
\times \frac{\partial V}{\partial {\bf u}} \\ |
| 197 |
> |
{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t) |
| 198 |
> |
\cdot {\bf \tau}^s(t + \delta t) |
| 199 |
> |
\end{eqnarray} |
| 200 |
|
|
| 201 |
+ |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
| 202 |
+ |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 203 |
+ |
torques have been obtained at the new time step, the velocities can be |
| 204 |
+ |
advanced to the same time value. |
| 205 |
+ |
|
| 206 |
+ |
{\tt moveB:} |
| 207 |
+ |
\begin{eqnarray} |
| 208 |
+ |
{\bf v}\left(t + \delta t \right) & \leftarrow & {\bf |
| 209 |
+ |
v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( |
| 210 |
+ |
{\bf f}(t + \delta t) / m \right) \\ |
| 211 |
+ |
{\bf j}\left(t + \delta t \right) & \leftarrow & {\bf |
| 212 |
+ |
j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} {\bf |
| 213 |
+ |
\tau}^b(t + \delta t) |
| 214 |
+ |
\end{eqnarray} |
| 215 |
+ |
|
| 216 |
+ |
The matrix rotations used in the DLM method end up being more costly |
| 217 |
+ |
computationally than the simpler arithmetic quaternion |
| 218 |
+ |
propagation. With the same time step, a 1000-molecule water simulation |
| 219 |
+ |
shows an average 7\% increase in computation time using the DLM method |
| 220 |
+ |
in place of quaternions. This cost is more than justified when |
| 221 |
+ |
comparing the energy conservation of the two methods as illustrated in |
| 222 |
+ |
figure \ref{timestep}. |
| 223 |
+ |
|
| 224 |
|
\begin{figure} |
| 225 |
|
\epsfxsize=6in |
| 226 |
|
\epsfbox{timeStep.epsi} |
| 227 |
|
\caption{Energy conservation using quaternion based integration versus |
| 228 |
< |
the symplectic step method proposed by Dullweber \emph{et al.} with |
| 229 |
< |
increasing time step. For each time step, the dotted line is total |
| 230 |
< |
energy using the symplectic step integrator, and the solid line comes |
| 231 |
< |
from the quaternion integrator. The larger time step plots are shifted |
| 232 |
< |
up from the true energy baseline for clarity.} |
| 228 |
> |
the method proposed by Dullweber \emph{et al.} with increasing time |
| 229 |
> |
step. For each time step, the dotted line is total energy using the |
| 230 |
> |
symplectic step integrator, and the solid line comes from the |
| 231 |
> |
quaternion integrator. The larger time step plots are shifted up from |
| 232 |
> |
the true energy baseline for clarity.} |
| 233 |
|
\label{timestep} |
| 234 |
|
\end{figure} |
| 235 |
|
|
| 236 |
|
In figure \ref{timestep}, the resulting energy drift at various time |
| 237 |
< |
steps for both the symplectic step and quaternion integration schemes |
| 238 |
< |
is compared. All of the 1000 SSD particle simulations started with the |
| 237 |
> |
steps for both the DLM and quaternion integration schemes is |
| 238 |
> |
compared. All of the 1000 molecule water simulations started with the |
| 239 |
|
same configuration, and the only difference was the method for |
| 240 |
|
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
| 241 |
< |
methods for propagating particle rotation conserve energy fairly well, |
| 241 |
> |
methods for propagating molecule rotation conserve energy fairly well, |
| 242 |
|
with the quaternion method showing a slight energy drift over time in |
| 243 |
|
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
| 244 |
< |
energy conservation benefits of the symplectic step method are clearly |
| 244 |
> |
energy conservation benefits of the DLM method are clearly |
| 245 |
|
demonstrated. Thus, while maintaining the same degree of energy |
| 246 |
|
conservation, one can take considerably longer time steps, leading to |
| 247 |
|
an overall reduction in computation time. |
| 248 |
|
|
| 170 |
– |
Energy drift in these SSD particle simulations was unnoticeable for |
| 171 |
– |
time steps up to three femtoseconds. A slight energy drift on the |
| 172 |
– |
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
| 173 |
– |
four femtoseconds, and as expected, this drift increases dramatically |
| 174 |
– |
with increasing time step. To insure accuracy in the constant energy |
| 175 |
– |
simulations, time steps were set at 2 fs and kept at this value for |
| 176 |
– |
constant pressure simulations as well. |
| 249 |
|
|
| 250 |
+ |
There is only one specific keyword relevant to the default integrator, |
| 251 |
+ |
and that is the time step for integrating the equations of motion. |
| 252 |
|
|
| 253 |
+ |
\begin{tabular}{llll} |
| 254 |
+ |
{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf |
| 255 |
+ |
default value} \\ |
| 256 |
+ |
$\delta t$ & {\tt dt = 2.0;} & fs & none |
| 257 |
+ |
\end{tabular} |
| 258 |
+ |
|
| 259 |
|
\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
| 260 |
|
|
| 261 |
|
{\sc oopse} implements a number of extended system integrators for |
| 328 |
|
\begin{eqnarray} |
| 329 |
|
\dot{{\bf r}} & = & {\bf v} \\ |
| 330 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} \\ |
| 331 |
< |
\dot{\mathsf{A}} & = & \\ |
| 332 |
< |
\dot{{\bf j}} & = & - \chi {\bf j} |
| 331 |
> |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 332 |
> |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\ |
| 333 |
> |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 334 |
> |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 335 |
> |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} |
| 336 |
|
\label{eq:nosehoovereom} |
| 337 |
|
\end{eqnarray} |
| 338 |
|
|
| 380 |
|
{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
| 381 |
|
j}(t) + \frac{\delta t}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 382 |
|
\chi(t) \right) \\ |
| 383 |
< |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}({\bf j}(t + |
| 384 |
< |
\delta t / 2) \overleftrightarrow{\mathsf{I}}^{b}, |
| 302 |
< |
\delta t) \\ |
| 383 |
> |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left(\delta t * |
| 384 |
> |
{\bf j}(t + \delta t / 2) \overleftrightarrow{\mathsf{I}}^{b} \right) \\ |
| 385 |
|
\chi\left(t + \delta t / 2 \right) & \leftarrow & \chi(t) + |
| 386 |
|
\frac{\delta t}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 387 |
|
\right) |
| 388 |
|
\end{eqnarray} |
| 389 |
|
|
| 390 |
< |
Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic Trotter |
| 391 |
< |
factorization of the three rotation operations that was discussed in |
| 392 |
< |
the section on the DLM integrator. Note that this operation modifies |
| 393 |
< |
both the rotation matrix $\mathsf{A}$ and the angular momentum ${\bf |
| 394 |
< |
j}$. {\tt moveA} propagates velocities by a half time step, and |
| 395 |
< |
positional degrees of freedom by a full time step. The new positions |
| 396 |
< |
(and orientations) are then used to calculate a new set of forces and |
| 397 |
< |
torques. |
| 390 |
> |
Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic |
| 391 |
> |
Trotter factorization of the three rotation operations that was |
| 392 |
> |
discussed in the section on the DLM integrator. Note that this |
| 393 |
> |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 394 |
> |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 395 |
> |
half time step, and positional degrees of freedom by a full time step. |
| 396 |
> |
The new positions (and orientations) are then used to calculate a new |
| 397 |
> |
set of forces and torques. |
| 398 |
|
|
| 399 |
|
{\tt doForces:} |
| 400 |
|
\begin{eqnarray} |
| 433 |
|
\end{eqnarray} |
| 434 |
|
|
| 435 |
|
Since ${\bf v}(t + \delta t)$ and ${\bf j}(t + \delta t)$ are required |
| 436 |
< |
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, the |
| 436 |
> |
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, they |
| 437 |
|
indirectly depend on their own values at time $t + \delta t$. {\tt |
| 438 |
|
moveB} is therefore done in an iterative fashion until $\chi(t + |
| 439 |
|
\delta t)$ becomes self-consistent. The relative tolerance for the |
| 460 |
|
|
| 461 |
|
\subsubsection{\label{sec:NPTi}Constant-pressure integration (isotropic box)} |
| 462 |
|
|
| 463 |
+ |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 464 |
+ |
implements the Melchionna modifications to the Nos\'e-Hoover-Andersen |
| 465 |
+ |
equations of motion,\cite{Melchionna93} |
| 466 |
|
|
| 467 |
+ |
\begin{eqnarray} |
| 468 |
+ |
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right) \\ |
| 469 |
+ |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v} \\ |
| 470 |
+ |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 471 |
+ |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\ |
| 472 |
+ |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 473 |
+ |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 474 |
+ |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} \\ |
| 475 |
+ |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 476 |
+ |
\frac{T}{T_{\mathrm{target}}} - 1 \right) \\ |
| 477 |
+ |
\dot{\eta} & = & \frac{1}{\tau_{B}^2} f k_B T_{\mathrm{target}} V \left( P - |
| 478 |
+ |
P_{\mathrm{target}} \right) \\ |
| 479 |
+ |
\dot{V} & = & 3 V \eta |
| 480 |
+ |
\label{eq:melchionna1} |
| 481 |
+ |
\end{eqnarray} |
| 482 |
+ |
|
| 483 |
+ |
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the extended |
| 484 |
+ |
system. $\chi$ is a thermostat, and it has the same function as it |
| 485 |
+ |
does in the Nos\'e-Hoover NVT integrator. $\eta$ is a barostat which |
| 486 |
+ |
controls changes to the volume of the simulation box. ${\bf R}_0$ is |
| 487 |
+ |
the location of the center of mass for the entire system. |
| 488 |
+ |
|
| 489 |
+ |
The NPTi integrator requires an instantaneous pressure. This quantity |
| 490 |
+ |
is calculated via the pressure tensor, |
| 491 |
+ |
\begin{equation} |
| 492 |
+ |
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{V(t)} \left( \sum_{i=1}^{N} |
| 493 |
+ |
m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
| 494 |
+ |
\overleftrightarrow{\mathsf{W}}(t) |
| 495 |
+ |
\end{equation} |
| 496 |
+ |
The kinetic contribution to the pressure tensor utilizes the {\it |
| 497 |
+ |
outer} product of the velocities denoted by the $\otimes$ symbol. The |
| 498 |
+ |
stress tensor is calculated from another outer product of the |
| 499 |
+ |
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
| 500 |
+ |
r}_i$) with the forces between the same two atoms, |
| 501 |
+ |
\begin{equation} |
| 502 |
+ |
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf r}_{ij}(t) |
| 503 |
+ |
\otimes {\bf f}_{ij}(t) |
| 504 |
+ |
\end{equation} |
| 505 |
+ |
The instantaneous pressure is then simply obtained from the trace of |
| 506 |
+ |
the Pressure tensor, |
| 507 |
+ |
\begin{equation} |
| 508 |
+ |
P(t) = \frac{1}{3} \mathrm{Tr} \left( \overleftrightarrow{\mathsf{P}}(t) |
| 509 |
+ |
\right) |
| 510 |
+ |
\end{equation} |
| 511 |
+ |
|
| 512 |
+ |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 513 |
+ |
relaxation of the pressure to the target value. To set values for |
| 514 |
+ |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
| 515 |
+ |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt .bass} |
| 516 |
+ |
file. The units for {\tt tauBarostat} are fs, and the units for the |
| 517 |
+ |
{\tt targetPressure} are atmospheres. Like in the NVT integrator, the |
| 518 |
+ |
integration of the equations of motion is carried out in a |
| 519 |
+ |
velocity-Verlet style 2 part algorithm: |
| 520 |
+ |
|
| 521 |
+ |
{\tt moveA:} |
| 522 |
+ |
\begin{eqnarray} |
| 523 |
+ |
T(t) & \leftarrow & \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} \\ |
| 524 |
+ |
{\bf v}\left(t + \delta t / 2\right) & \leftarrow & {\bf |
| 525 |
+ |
v}(t) + \frac{\delta t}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 526 |
+ |
\chi(t)\right) \\ |
| 527 |
+ |
{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf |
| 528 |
+ |
v}\left(t + \delta t / 2 \right) \\ |
| 529 |
+ |
{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
| 530 |
+ |
j}(t) + \frac{\delta t}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 531 |
+ |
\chi(t) \right) \\ |
| 532 |
+ |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left(\delta t * |
| 533 |
+ |
{\bf j}(t + \delta t / 2) \overleftrightarrow{\mathsf{I}}^{b} \right) \\ |
| 534 |
+ |
\chi\left(t + \delta t / 2 \right) & \leftarrow & \chi(t) + |
| 535 |
+ |
\frac{\delta t}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 536 |
+ |
\right) |
| 537 |
+ |
\end{eqnarray} |
| 538 |
+ |
|
| 539 |
+ |
Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic |
| 540 |
+ |
Trotter factorization of the three rotation operations that was |
| 541 |
+ |
discussed in the section on the DLM integrator. Note that this |
| 542 |
+ |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 543 |
+ |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 544 |
+ |
half time step, and positional degrees of freedom by a full time step. |
| 545 |
+ |
The new positions (and orientations) are then used to calculate a new |
| 546 |
+ |
set of forces and torques. |
| 547 |
+ |
|
| 548 |
+ |
{\tt doForces:} |
| 549 |
+ |
\begin{eqnarray} |
| 550 |
+ |
{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf |
| 551 |
+ |
r}(t + \delta t)} \\ |
| 552 |
+ |
{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t) |
| 553 |
+ |
\times \frac{\partial V}{\partial {\bf u}} \\ |
| 554 |
+ |
{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t) |
| 555 |
+ |
\cdot {\bf \tau}^s(t + \delta t) |
| 556 |
+ |
\end{eqnarray} |
| 557 |
+ |
|
| 558 |
+ |
Here ${\bf u}$ is a unit vector pointing along the principal axis of |
| 559 |
+ |
the rigid body being propagated, ${\bf \tau}^s$ is the torque in the |
| 560 |
+ |
space-fixed (laboratory) frame, and ${\bf \tau}^b$ is the torque in |
| 561 |
+ |
the body-fixed frame. ${\bf u}$ is automatically calculated when the |
| 562 |
+ |
rotation matrix $\mathsf{A}$ is calculated in {\tt moveA}. |
| 563 |
+ |
|
| 564 |
+ |
Once the forces and torques have been obtained at the new time step, |
| 565 |
+ |
the velocities can be advanced to the same time value. |
| 566 |
+ |
|
| 567 |
+ |
{\tt moveB:} |
| 568 |
+ |
\begin{eqnarray} |
| 569 |
+ |
T(t + \delta t) & \leftarrow & \left\{{\bf v}(t + \delta t)\right\}, |
| 570 |
+ |
\left\{{\bf j}(t + \delta t)\right\} \\ |
| 571 |
+ |
\chi\left(t + \delta t \right) & \leftarrow & \chi\left(t + \delta t / |
| 572 |
+ |
2 \right) + \frac{\delta t}{2 \tau_T^2} \left( \frac{T(t+\delta |
| 573 |
+ |
t)}{T_{\mathrm{target}}} - 1 \right) \\ |
| 574 |
+ |
{\bf v}\left(t + \delta t \right) & \leftarrow & {\bf |
| 575 |
+ |
v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( |
| 576 |
+ |
\frac{{\bf f}(t + \delta t)}{m} - {\bf v}(t + \delta t) |
| 577 |
+ |
\chi(t \delta t)\right) \\ |
| 578 |
+ |
{\bf j}\left(t + \delta t \right) & \leftarrow & {\bf |
| 579 |
+ |
j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( {\bf |
| 580 |
+ |
\tau}^b(t + \delta t) - {\bf j}(t + \delta t) |
| 581 |
+ |
\chi(t + \delta t) \right) |
| 582 |
+ |
\end{eqnarray} |
| 583 |
+ |
|
| 584 |
+ |
Since ${\bf v}(t + \delta t)$ and ${\bf j}(t + \delta t)$ are required |
| 585 |
+ |
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, they |
| 586 |
+ |
indirectly depend on their own values at time $t + \delta t$. {\tt |
| 587 |
+ |
moveB} is therefore done in an iterative fashion until $\chi(t + |
| 588 |
+ |
\delta t)$ becomes self-consistent. The relative tolerance for the |
| 589 |
+ |
self-consistency check defaults to a value of $\mbox{10}^{-6}$, but |
| 590 |
+ |
{\sc oopse} will terminate the iteration after 4 loops even if the |
| 591 |
+ |
consistency check has not been satisfied. |
| 592 |
+ |
|
| 593 |
+ |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for the |
| 594 |
+ |
extended system that is, to within a constant, identical to the |
| 595 |
+ |
Helmholtz free energy, |
| 596 |
+ |
\begin{equation} |
| 597 |
+ |
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 598 |
+ |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t\prime) dt\prime |
| 599 |
+ |
\right) |
| 600 |
+ |
\end{equation} |
| 601 |
+ |
Poor choices of $\delta t$ or $\tau_T$ can result in non-conservation |
| 602 |
+ |
of $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 603 |
+ |
last column of the {\tt .stat} file to allow checks on the quality of |
| 604 |
+ |
the integration. |
| 605 |
+ |
|
| 606 |
+ |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 607 |
+ |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 608 |
+ |
algorithms are given in section \ref{sec:rattle}. |
| 609 |
+ |
|
| 610 |
+ |
|
| 611 |
+ |
|
| 612 |
|
\subsection{\label{Sec:zcons}Z-Constraint Method} |
| 613 |
|
|
| 614 |
|
Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation |
