| 48 |
|
\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
| 49 |
|
for timesteps of length $h$. |
| 50 |
|
\end{enumerate} |
| 51 |
– |
|
| 51 |
|
The integration of the equations of motion is carried out in a |
| 52 |
|
velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
| 53 |
|
|
| 65 |
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 66 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 67 |
|
\end{align*} |
| 69 |
– |
|
| 68 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 69 |
|
product of the three body-fixed rotations, |
| 70 |
|
\begin{equation} |
| 99 |
|
\end{array} |
| 100 |
|
\right). |
| 101 |
|
\end{equation} |
| 102 |
< |
All other rotations follow in a straightforward manner. |
| 102 |
> |
All other rotations follow in a straightforward manner. After the |
| 103 |
> |
first part of the propagation, the forces and body-fixed torques are |
| 104 |
> |
calculated at the new positions and orientations |
| 105 |
|
|
| 106 |
– |
After the first part of the propagation, the forces and body-fixed |
| 107 |
– |
torques are calculated at the new positions and orientations |
| 108 |
– |
|
| 106 |
|
{\tt doForces:} |
| 107 |
|
\begin{align*} |
| 108 |
|
{\bf f}(t + h) &\leftarrow |
| 114 |
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 115 |
|
\cdot {\bf \tau}^s(t + h). |
| 116 |
|
\end{align*} |
| 120 |
– |
|
| 117 |
|
${\bf u}$ is automatically updated when the rotation matrix |
| 118 |
|
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 119 |
|
torques have been obtained at the new time step, the velocities can |
| 129 |
|
\right) |
| 130 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 131 |
|
\end{align*} |
| 136 |
– |
|
| 132 |
|
The matrix rotations used in the DLM method end up being more costly |
| 133 |
|
computationally than the simpler arithmetic quaternion propagation. |
| 134 |
|
With the same time step, a 1000-molecule water simulation shows an |
| 135 |
|
average 7\% increase in computation time using the DLM method in |
| 136 |
|
place of quaternions. This cost is more than justified when |
| 137 |
|
comparing the energy conservation of the two methods as illustrated |
| 138 |
< |
in Fig.~\ref{methodFig:timestep}. |
| 138 |
> |
in Fig.~\ref{methodFig:timestep} where the resulting energy drift at |
| 139 |
> |
various time steps for both the DLM and quaternion integration |
| 140 |
> |
schemes is compared. All of the 1000 molecule water simulations |
| 141 |
> |
started with the same configuration, and the only difference was the |
| 142 |
> |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 143 |
> |
fs, both methods for propagating molecule rotation conserve energy |
| 144 |
> |
fairly well, with the quaternion method showing a slight energy |
| 145 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of |
| 146 |
> |
1 and 2 fs, the energy conservation benefits of the DLM method are |
| 147 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
| 148 |
> |
energy conservation, one can take considerably longer time steps, |
| 149 |
> |
leading to an overall reduction in computation time. |
| 150 |
|
|
| 151 |
|
\begin{figure} |
| 152 |
|
\centering |
| 161 |
|
\label{methodFig:timestep} |
| 162 |
|
\end{figure} |
| 163 |
|
|
| 158 |
– |
In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
| 159 |
– |
various time steps for both the DLM and quaternion integration |
| 160 |
– |
schemes is compared. All of the 1000 molecule water simulations |
| 161 |
– |
started with the same configuration, and the only difference was the |
| 162 |
– |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 163 |
– |
fs, both methods for propagating molecule rotation conserve energy |
| 164 |
– |
fairly well, with the quaternion method showing a slight energy |
| 165 |
– |
drift over time in the 0.5 fs time step simulation. At time steps of |
| 166 |
– |
1 and 2 fs, the energy conservation benefits of the DLM method are |
| 167 |
– |
clearly demonstrated. Thus, while maintaining the same degree of |
| 168 |
– |
energy conservation, one can take considerably longer time steps, |
| 169 |
– |
leading to an overall reduction in computation time. |
| 170 |
– |
|
| 164 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
| 165 |
|
|
| 166 |
|
The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
| 174 |
|
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial V}{\partial |
| 175 |
|
\mathsf{A}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
| 176 |
|
\end{eqnarray} |
| 184 |
– |
|
| 177 |
|
$\chi$ is an ``extra'' variable included in the extended system, and |
| 178 |
|
it is propagated using the first order equation of motion |
| 179 |
|
\begin{equation} |
| 180 |
|
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
| 181 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
| 182 |
|
\end{equation} |
| 191 |
– |
|
| 183 |
|
The instantaneous temperature $T$ is proportional to the total |
| 184 |
|
kinetic energy (both translational and orientational) and is given |
| 185 |
|
by |
| 186 |
|
\begin{equation} |
| 187 |
< |
T = \frac{2 K}{f k_B} |
| 187 |
> |
T = \frac{2 K}{f k_B}. |
| 188 |
|
\end{equation} |
| 189 |
|
Here, $f$ is the total number of degrees of freedom in the system, |
| 190 |
|
\begin{equation} |
| 198 |
|
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
| 199 |
|
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
| 200 |
|
\end{equation} |
| 201 |
< |
|
| 211 |
< |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
| 201 |
> |
In Eq.~\ref{eq:nosehooverext}, $\tau_T$ is the time constant for |
| 202 |
|
relaxation of the temperature to the target value. The integration |
| 203 |
|
of the equations of motion is carried out in a velocity-Verlet style |
| 204 |
|
2 part algorithm: |
| 226 |
|
+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
| 227 |
|
{T_{\mathrm{target}}} - 1 \right) . |
| 228 |
|
\end{align*} |
| 239 |
– |
|
| 229 |
|
Here $\mathrm{rotate}(h * {\bf j} |
| 230 |
|
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic |
| 231 |
|
Trotter factorization of the three rotation operations that was |
| 236 |
|
step. The new positions (and orientations) are then used to |
| 237 |
|
calculate a new set of forces and torques in exactly the same way |
| 238 |
|
they are calculated in the {\tt doForces} portion of the DLM |
| 239 |
< |
integrator. |
| 239 |
> |
integrator. Once the forces and torques have been obtained at the |
| 240 |
> |
new time step, the temperature, velocities, and the extended system |
| 241 |
> |
variable can be advanced to the same time value. |
| 242 |
|
|
| 252 |
– |
Once the forces and torques have been obtained at the new time step, |
| 253 |
– |
the temperature, velocities, and the extended system variable can be |
| 254 |
– |
advanced to the same time value. |
| 255 |
– |
|
| 243 |
|
{\tt moveB:} |
| 244 |
|
\begin{align*} |
| 245 |
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 259 |
|
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
| 260 |
|
\chi(t + h) \right) . |
| 261 |
|
\end{align*} |
| 275 |
– |
|
| 262 |
|
Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to |
| 263 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
| 264 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
| 265 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
| 266 |
< |
self-consistent. |
| 267 |
< |
|
| 268 |
< |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 283 |
< |
the extended system that is, to within a constant, identical to the |
| 284 |
< |
Helmholtz free energy,\cite{Melchionna1993} |
| 266 |
> |
self-consistent. The Nos\'e-Hoover algorithm is known to conserve a |
| 267 |
> |
Hamiltonian for the extended system that is, to within a constant, |
| 268 |
> |
identical to the Helmholtz free energy,\cite{Melchionna1993} |
| 269 |
|
\begin{equation} |
| 270 |
|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 271 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 281 |
|
|
| 282 |
|
We can used an isobaric-isothermal ensemble integrator which is |
| 283 |
|
implemented using the Melchionna modifications to the |
| 284 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
| 301 |
< |
|
| 284 |
> |
Nos\'e-Hoover-Andersen equations of motion\cite{Melchionna1993} |
| 285 |
|
\begin{eqnarray} |
| 286 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 287 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
| 298 |
|
P_{\mathrm{target}} \right), \\ |
| 299 |
|
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
| 300 |
|
\end{eqnarray} |
| 318 |
– |
|
| 301 |
|
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
| 302 |
|
extended system. $\chi$ is a thermostat, and it has the same |
| 303 |
|
function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
| 330 |
|
the Pressure tensor, |
| 331 |
|
\begin{equation} |
| 332 |
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
| 333 |
< |
\overleftrightarrow{\mathsf{P}}(t). \right) |
| 333 |
> |
\overleftrightarrow{\mathsf{P}}(t) \right) . |
| 334 |
|
\end{equation} |
| 335 |
< |
|
| 354 |
< |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 335 |
> |
In Eq.~\ref{eq:melchionna1}, $\tau_B$ is the time constant for |
| 336 |
|
relaxation of the pressure to the target value. Like in the NVT |
| 337 |
|
integrator, the integration of the equations of motion is carried |
| 338 |
|
out in a velocity-Verlet style 2 part algorithm: |
| 371 |
|
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
| 372 |
|
\mathsf{H}(t). |
| 373 |
|
\end{align*} |
| 393 |
– |
|
| 374 |
|
Most of these equations are identical to their counterparts in the |
| 375 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
| 376 |
|
depends on the positions at the same time. The simulation box |
| 382 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 383 |
|
\mathcal{V}(t) |
| 384 |
|
\end{equation} |
| 405 |
– |
|
| 385 |
|
The {\tt doForces} step for the NPTi integrator is exactly the same |
| 386 |
|
as in both the DLM and NVT integrators. Once the forces and torques |
| 387 |
|
have been obtained at the new time step, the velocities can be |
| 413 |
|
\tau}^b(t + h) - {\bf j}(t + h) |
| 414 |
|
\chi(t + h) \right) . |
| 415 |
|
\end{align*} |
| 437 |
– |
|
| 416 |
|
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
| 417 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 418 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
| 513 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
| 514 |
|
\end{align*} |
| 515 |
|
Here, a power series expansion truncated at second order for the |
| 516 |
< |
exponential operation is used to scale the simulation box. |
| 516 |
> |
exponential operation is used to scale the simulation box. The {\tt |
| 517 |
> |
moveB} portion of the algorithm is largely unchanged from the NPTi |
| 518 |
> |
integrator: |
| 519 |
|
|
| 540 |
– |
The {\tt moveB} portion of the algorithm is largely unchanged from |
| 541 |
– |
the NPTi integrator: |
| 542 |
– |
|
| 520 |
|
{\tt moveB:} |
| 521 |
|
\begin{align*} |
| 522 |
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 546 |
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
| 547 |
|
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
| 548 |
|
\end{align*} |
| 572 |
– |
|
| 549 |
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
| 550 |
< |
identical to those described for the NPTi integrator. |
| 551 |
< |
|
| 576 |
< |
The NPTf integrator is known to conserve the following Hamiltonian: |
| 550 |
> |
identical to those described for the NPTi integrator. The NPTf |
| 551 |
> |
integrator is known to conserve the following Hamiltonian: |
| 552 |
|
\begin{eqnarray*} |
| 553 |
|
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
| 554 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 557 |
|
T_{\mathrm{target}}}{2} |
| 558 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 559 |
|
\end{eqnarray*} |
| 585 |
– |
|
| 560 |
|
This integrator must be used with care, particularly in liquid |
| 561 |
|
simulations. Liquids have very small restoring forces in the |
| 562 |
|
off-diagonal directions, and the simulation box can very quickly |
| 594 |
|
|
| 595 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 596 |
|
membrane system should be zero since its surface free energy $G$ is |
| 597 |
< |
minimum with respect to surface area $A$ |
| 598 |
< |
\[ |
| 599 |
< |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 600 |
< |
\] |
| 601 |
< |
However, a surface tension of zero is not appropriate for relatively |
| 602 |
< |
small patches of membrane. In order to eliminate the edge effect of |
| 603 |
< |
the membrane simulation, a special ensemble, NP$\gamma$T, has been |
| 630 |
< |
proposed to maintain the lateral surface tension and normal |
| 631 |
< |
pressure. The equation of motion for the cell size control tensor, |
| 632 |
< |
$\eta$, in $NP\gamma T$ is |
| 597 |
> |
minimum with respect to surface area $A$, $\gamma = \frac{{\partial |
| 598 |
> |
G}}{{\partial A}}.$ However, a surface tension of zero is not |
| 599 |
> |
appropriate for relatively small patches of membrane. In order to |
| 600 |
> |
eliminate the edge effect of the membrane simulation, a special |
| 601 |
> |
ensemble, NP$\gamma$T, has been proposed to maintain the lateral |
| 602 |
> |
surface tension and normal pressure. The equation of motion for the |
| 603 |
> |
cell size control tensor, $\eta$, in $NP\gamma T$ is |
| 604 |
|
\begin{equation} |
| 605 |
|
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 606 |
|
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
| 616 |
|
- P_{{\rm{target}}} ) |
| 617 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
| 618 |
|
\end{equation} |
| 648 |
– |
|
| 619 |
|
There is one additional extended system integrator (NPTxyz), in |
| 620 |
|
which each attempt to preserve the target pressure along the box |
| 621 |
|
walls perpendicular to that particular axis. The lengths of the box |
| 638 |
|
\begin{equation} |
| 639 |
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 640 |
|
\end{equation} |
| 671 |
– |
|
| 641 |
|
If the time-dependent friction decays rapidly, the static friction |
| 642 |
|
coefficient can be approximated by |
| 643 |
|
\begin{equation} |
| 650 |
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 651 |
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 652 |
|
\end{equation} |
| 684 |
– |
|
| 653 |
|
The Z-Constraint method, which fixes the z coordinates of the |
| 654 |
|
molecules with respect to the center of the mass of the system, has |
| 655 |
|
been a method suggested to obtain the forces required for the force |
| 658 |
|
whole system. To avoid this problem, we reset the forces of |
| 659 |
|
z-constrained molecules as well as subtract the total constraint |
| 660 |
|
forces from the rest of the system after the force calculation at |
| 661 |
< |
each time step instead of resetting the coordinate. |
| 662 |
< |
|
| 695 |
< |
After the force calculation, we define $G_\alpha$ as |
| 661 |
> |
each time step instead of resetting the coordinate. After the force |
| 662 |
> |
calculation, we define $G_\alpha$ as |
| 663 |
|
\begin{equation} |
| 664 |
|
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 665 |
|
\end{equation} |
| 694 |
|
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 695 |
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 696 |
|
\end{equation} |
| 730 |
– |
|
| 697 |
|
At the very beginning of the simulation, the molecules may not be at |
| 698 |
|
their constrained positions. To move a z-constrained molecule to its |
| 699 |
|
specified position, a simple harmonic potential is used |
