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increasing time step. For each time step, the dotted line is total |
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|
energy using the DLM integrator, and the solid line comes from the |
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|
quaternion integrator. The larger time step plots are shifted up |
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< |
from the true energy baseline for clarity.} \label{timestep} |
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> |
from the true energy baseline for clarity.} |
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> |
\label{methodFig:timestep} |
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\end{figure} |
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|
|
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< |
In Fig.~\ref{timestep}, the resulting energy drift at various time |
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< |
steps for both the DLM and quaternion integration schemes is |
159 |
< |
compared. All of the 1000 molecule water simulations started with |
160 |
< |
the same configuration, and the only difference was the method for |
161 |
< |
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
162 |
< |
methods for propagating molecule rotation conserve energy fairly |
163 |
< |
well, with the quaternion method showing a slight energy drift over |
164 |
< |
time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
165 |
< |
fs, the energy conservation benefits of the DLM method are clearly |
166 |
< |
demonstrated. Thus, while maintaining the same degree of energy |
167 |
< |
conservation, one can take considerably longer time steps, leading |
168 |
< |
to an overall reduction in computation time. |
157 |
> |
In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
158 |
> |
various time steps for both the DLM and quaternion integration |
159 |
> |
schemes is compared. All of the 1000 molecule water simulations |
160 |
> |
started with the same configuration, and the only difference was the |
161 |
> |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
162 |
> |
fs, both methods for propagating molecule rotation conserve energy |
163 |
> |
fairly well, with the quaternion method showing a slight energy |
164 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of |
165 |
> |
1 and 2 fs, the energy conservation benefits of the DLM method are |
166 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
167 |
> |
energy conservation, one can take considerably longer time steps, |
168 |
> |
leading to an overall reduction in computation time. |
169 |
|
|
170 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
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|
|
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< |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover85} |
172 |
> |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
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|
\begin{eqnarray} |
174 |
|
\dot{{\bf r}} & = & {\bf v}, \\ |
175 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
285 |
|
|
286 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
287 |
|
the extended system that is, to within a constant, identical to the |
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< |
Helmholtz free energy,\cite{melchionna93} |
288 |
> |
Helmholtz free energy,\cite{Melchionna1993} |
289 |
|
\begin{equation} |
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|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
291 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
295 |
|
$H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
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|
last column of the {\tt .stat} file to allow checks on the quality |
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|
of the integration. |
297 |
– |
|
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– |
Bond constraints are applied at the end of both the {\tt moveA} and |
299 |
– |
{\tt moveB} portions of the algorithm. Details on the constraint |
300 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
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|
|
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|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
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|
isotropic box deformations (NPTi)} |
301 |
|
|
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|
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
303 |
|
implements the Melchionna modifications to the |
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< |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
304 |
> |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
305 |
|
|
306 |
|
\begin{eqnarray} |
307 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |