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\chapter{\label{chapt:methodology}METHODOLOGY} |
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|
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\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
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|
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In order to mimic the experiments, which are usually performed under |
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constant temperature and/or pressure, extended Hamiltonian system |
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methods have been developed to generate statistical ensemble, such |
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as canonical ensemble and isobaric-isothermal ensemble \textit{etc}. |
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In addition to the standard ensemble, specific ensembles have been |
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developed to account for the anisotropy between the lateral and |
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normal directions of membranes. The $NPAT$ ensemble, in which the |
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normal pressure and the lateral surface area of the membrane are |
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kept constant, and the $NP\gamma T$ ensemble, in which the normal |
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pressure and the lateral surface tension are kept constant were |
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proposed to address this issue. |
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|
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Integration schemes for rotational motion of the rigid molecules in |
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microcanonical ensemble have been extensively studied in the last |
| 19 |
two decades. Matubayasi and Nakahara developed a time-reversible |
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integrator for rigid bodies in quaternion representation. Although |
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it is not symplectic, this integrator still demonstrates a better |
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long-time energy conservation than traditional methods because of |
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the time-reversible nature. Extending Trotter-Suzuki to general |
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system with a flat phase space, Miller and his colleagues devised an |
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novel symplectic, time-reversible and volume-preserving integrator |
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in quaternion representation, which was shown to be superior to the |
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time-reversible integrator of Matubayasi and Nakahara. However, all |
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of the integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved |
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the same structural properties of the Hamiltonian flow. In this |
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section, the integration scheme of DLM method will be reviewed and |
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extended to other ensembles. |
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|
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\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
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DLM method} |
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|
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The DLM method uses a Trotter factorization of the orientational |
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propagator. This has three effects: |
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\begin{enumerate} |
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\item the integrator is area-preserving in phase space (i.e. it is |
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{\it symplectic}), |
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\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
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Monte Carlo applications, and |
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\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
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for timesteps of length $h$. |
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\end{enumerate} |
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|
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The integration of the equations of motion is carried out in a |
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velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
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|
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{\tt moveA:} |
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\begin{align*} |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
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+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
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% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
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+ h {\bf v}\left(t + h / 2 \right), \\ |
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% |
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{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
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+ \frac{h}{2} {\bf \tau}^b(t), \\ |
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% |
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\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
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(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
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\end{align*} |
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|
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In this context, the $\mathrm{rotate}$ function is the reversible |
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product of the three body-fixed rotations, |
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\begin{equation} |
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\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
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\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
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/ 2) \cdot \mathsf{G}_x(a_x /2), |
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\end{equation} |
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where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
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rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
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angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
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axis $\alpha$, |
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\begin{equation} |
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\mathsf{G}_\alpha( \theta ) = \left\{ |
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\begin{array}{lcl} |
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\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
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{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
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j}(0). |
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\end{array} |
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\right. |
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\end{equation} |
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$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
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rotation matrix. For example, in the small-angle limit, the |
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rotation matrix around the body-fixed x-axis can be approximated as |
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\begin{equation} |
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\mathsf{R}_x(\theta) \approx \left( |
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\begin{array}{ccc} |
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1 & 0 & 0 \\ |
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0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
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\theta^2 / 4} \\ |
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0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
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\theta^2 / 4} |
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\end{array} |
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\right). |
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\end{equation} |
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All other rotations follow in a straightforward manner. |
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|
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After the first part of the propagation, the forces and body-fixed |
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torques are calculated at the new positions and orientations |
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|
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{\tt doForces:} |
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\begin{align*} |
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{\bf f}(t + h) &\leftarrow |
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- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
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% |
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{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
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\times \frac{\partial V}{\partial {\bf u}}, \\ |
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% |
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{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
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\cdot {\bf \tau}^s(t + h). |
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\end{align*} |
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|
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{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
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$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
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torques have been obtained at the new time step, the velocities can |
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be advanced to the same time value. |
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|
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{\tt moveB:} |
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\begin{align*} |
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{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
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\right) |
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+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
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% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
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\right) |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
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\end{align*} |
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|
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The matrix rotations used in the DLM method end up being more costly |
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computationally than the simpler arithmetic quaternion propagation. |
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With the same time step, a 1000-molecule water simulation shows an |
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average 7\% increase in computation time using the DLM method in |
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place of quaternions. This cost is more than justified when |
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comparing the energy conservation of the two methods as illustrated |
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in Fig.~\ref{timestep}. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{timeStep.eps} |
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\caption[Energy conservation for quaternion versus DLM |
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dynamics]{Energy conservation using quaternion based integration |
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versus the method proposed by Dullweber \emph{et al.} with |
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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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\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 |
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compared. All of the 1000 molecule water simulations started with |
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the same configuration, and the only difference was the method for |
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handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
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methods for propagating molecule rotation conserve energy fairly |
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well, with the quaternion method showing a slight energy drift over |
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time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
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fs, the energy conservation benefits of the DLM method are clearly |
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demonstrated. Thus, while maintaining the same degree of energy |
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conservation, one can take considerably longer time steps, leading |
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to an overall reduction in computation time. |
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|
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\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} |
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\begin{eqnarray} |
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\dot{{\bf r}} & = & {\bf v}, \\ |
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\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
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\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
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\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
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\dot{{\bf j}} & = & {\bf j} \times \left( |
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\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j} \right) - \mbox{ |
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rot}\left(\mathsf{A}^{T} \cdot \frac{\partial V}{\partial |
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\mathsf{A}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
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\end{eqnarray} |
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|
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$\chi$ is an ``extra'' variable included in the extended system, and |
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it is propagated using the first order equation of motion |
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\begin{equation} |
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\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
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\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
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\end{equation} |
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|
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The instantaneous temperature $T$ is proportional to the total |
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kinetic energy (both translational and orientational) and is given |
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by |
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\begin{equation} |
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T = \frac{2 K}{f k_B} |
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\end{equation} |
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Here, $f$ is the total number of degrees of freedom in the system, |
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\begin{equation} |
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f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
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\end{equation} |
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and $K$ is the total kinetic energy, |
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\begin{equation} |
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K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
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\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
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\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
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\end{equation} |
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|
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In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
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relaxation of the temperature to the target value. To set values |
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for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
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the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
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{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
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units for the {\tt targetTemperature} are degrees K. The |
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integration of the equations of motion is carried out in a |
| 214 |
velocity-Verlet style 2 part algorithm: |
| 215 |
|
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{\tt moveA:} |
| 217 |
\begin{align*} |
| 218 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 219 |
% |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 221 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 222 |
\chi(t)\right), \\ |
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% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
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+ h {\bf v}\left(t + h / 2 \right) ,\\ |
| 226 |
% |
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{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
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+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 229 |
\chi(t) \right) ,\\ |
| 230 |
% |
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\mathsf{A}(t + h) &\leftarrow \mathrm{rotate} |
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\left(h * {\bf j}(t + h / 2) |
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\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
| 234 |
% |
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\chi\left(t + h / 2 \right) &\leftarrow \chi(t) |
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+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
| 237 |
{T_{\mathrm{target}}} - 1 \right) . |
| 238 |
\end{align*} |
| 239 |
|
| 240 |
Here $\mathrm{rotate}(h * {\bf j} |
| 241 |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic |
| 242 |
Trotter factorization of the three rotation operations that was |
| 243 |
discussed in the section on the DLM integrator. Note that this |
| 244 |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 245 |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
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half time step, and positional degrees of freedom by a full time |
| 247 |
step. The new positions (and orientations) are then used to |
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calculate a new set of forces and torques in exactly the same way |
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they are calculated in the {\tt doForces} portion of the DLM |
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integrator. |
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|
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Once the forces and torques have been obtained at the new time step, |
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the temperature, velocities, and the extended system variable can be |
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advanced to the same time value. |
| 255 |
|
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{\tt moveB:} |
| 257 |
\begin{align*} |
| 258 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 259 |
\left\{{\bf j}(t + h)\right\}, \\ |
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% |
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\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
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2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
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{T_{\mathrm{target}}} - 1 \right), \\ |
| 264 |
% |
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{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 266 |
+ h / 2 \right) + \frac{h}{2} \left( |
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\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 268 |
\chi(t h)\right) ,\\ |
| 269 |
% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 271 |
+ h / 2 \right) + \frac{h}{2} |
| 272 |
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
| 273 |
\chi(t + h) \right) . |
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\end{align*} |
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|
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Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to |
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caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
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depend on their own values at time $t + h$. {\tt moveB} is |
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therefore done in an iterative fashion until $\chi(t + h)$ becomes |
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self-consistent. The relative tolerance for the self-consistency |
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check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
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terminate the iteration after 4 loops even if the consistency check |
| 283 |
has not been satisfied. |
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|
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The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 286 |
the extended system that is, to within a constant, identical to the |
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Helmholtz free energy,\cite{melchionna93} |
| 288 |
\begin{equation} |
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H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 290 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 291 |
dt^\prime \right). |
| 292 |
\end{equation} |
| 293 |
Poor choices of $h$ or $\tau_T$ can result in non-conservation of |
| 294 |
$H_{\mathrm{NVT}}$, 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 |
| 296 |
of the integration. |
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|
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Bond constraints are applied at the end of both the {\tt moveA} and |
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{\tt moveB} portions of the algorithm. Details on the constraint |
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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)} |
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|
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To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 306 |
implements the Melchionna modifications to the |
| 307 |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
| 308 |
|
| 309 |
\begin{eqnarray} |
| 310 |
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 311 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
| 312 |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 313 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
| 314 |
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 315 |
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 316 |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 317 |
V}{\partial \mathsf{A}} \right) - \chi {\bf j}, \\ |
| 318 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 319 |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 320 |
\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V |
| 321 |
\left( P - |
| 322 |
P_{\mathrm{target}} \right), \\ |
| 323 |
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
| 324 |
\end{eqnarray} |
| 325 |
|
| 326 |
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
| 327 |
extended system. $\chi$ is a thermostat, and it has the same |
| 328 |
function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
| 329 |
a barostat which controls changes to the volume of the simulation |
| 330 |
box. ${\bf R}_0$ is the location of the center of mass for the |
| 331 |
entire system, and $\mathcal{V}$ is the volume of the simulation |
| 332 |
box. At any time, the volume can be calculated from the determinant |
| 333 |
of the matrix which describes the box shape: |
| 334 |
\begin{equation} |
| 335 |
\mathcal{V} = \det(\mathsf{H}). |
| 336 |
\end{equation} |
| 337 |
|
| 338 |
The NPTi integrator requires an instantaneous pressure. This |
| 339 |
quantity is calculated via the pressure tensor, |
| 340 |
\begin{equation} |
| 341 |
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{\mathcal{V}(t)} \left( |
| 342 |
\sum_{i=1}^{N} m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
| 343 |
\overleftrightarrow{\mathsf{W}}(t). |
| 344 |
\end{equation} |
| 345 |
The kinetic contribution to the pressure tensor utilizes the {\it |
| 346 |
outer} product of the velocities denoted by the $\otimes$ symbol. |
| 347 |
The stress tensor is calculated from another outer product of the |
| 348 |
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
| 349 |
r}_i$) with the forces between the same two atoms, |
| 350 |
\begin{equation} |
| 351 |
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf |
| 352 |
r}_{ij}(t) \otimes {\bf f}_{ij}(t). |
| 353 |
\end{equation} |
| 354 |
The instantaneous pressure is then simply obtained from the trace of |
| 355 |
the Pressure tensor, |
| 356 |
\begin{equation} |
| 357 |
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
| 358 |
\overleftrightarrow{\mathsf{P}}(t). \right) |
| 359 |
\end{equation} |
| 360 |
|
| 361 |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 362 |
relaxation of the pressure to the target value. To set values for |
| 363 |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
| 364 |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
| 365 |
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
| 366 |
for the {\tt targetPressure} are atmospheres. Like in the NVT |
| 367 |
integrator, the integration of the equations of motion is carried |
| 368 |
out in a velocity-Verlet style 2 part algorithm: |
| 369 |
|
| 370 |
{\tt moveA:} |
| 371 |
\begin{align*} |
| 372 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 373 |
% |
| 374 |
P(t) &\leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,\\ |
| 375 |
% |
| 376 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 377 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 378 |
\left(\chi(t) + \eta(t) \right) \right), \\ |
| 379 |
% |
| 380 |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 381 |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 382 |
\chi(t) \right), \\ |
| 383 |
% |
| 384 |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 385 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 386 |
\right) ,\\ |
| 387 |
% |
| 388 |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 389 |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 390 |
\right) ,\\ |
| 391 |
% |
| 392 |
\eta(t + h / 2) &\leftarrow \eta(t) + \frac{h |
| 393 |
\mathcal{V}(t)}{2 N k_B T(t) \tau_B^2} \left( P(t) |
| 394 |
- P_{\mathrm{target}} \right), \\ |
| 395 |
% |
| 396 |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h |
| 397 |
\left\{ {\bf v}\left(t + h / 2 \right) |
| 398 |
+ \eta(t + h / 2)\left[ {\bf r}(t + h) |
| 399 |
- {\bf R}_0 \right] \right\} ,\\ |
| 400 |
% |
| 401 |
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
| 402 |
\mathsf{H}(t). |
| 403 |
\end{align*} |
| 404 |
|
| 405 |
Most of these equations are identical to their counterparts in the |
| 406 |
NVT integrator, but the propagation of positions to time $t + h$ |
| 407 |
depends on the positions at the same time. {\sc oopse} carries out |
| 408 |
this step iteratively (with a limit of 5 passes through the |
| 409 |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
| 410 |
uniformly for one full time step by an exponential factor that |
| 411 |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
| 412 |
box uniformly also scales the volume of the box by |
| 413 |
\begin{equation} |
| 414 |
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 415 |
\mathcal{V}(t) |
| 416 |
\end{equation} |
| 417 |
|
| 418 |
The {\tt doForces} step for the NPTi integrator is exactly the same |
| 419 |
as in both the DLM and NVT integrators. Once the forces and torques |
| 420 |
have been obtained at the new time step, the velocities can be |
| 421 |
advanced to the same time value. |
| 422 |
|
| 423 |
{\tt moveB:} |
| 424 |
\begin{align*} |
| 425 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 426 |
\left\{{\bf j}(t + h)\right\} ,\\ |
| 427 |
% |
| 428 |
P(t + h) &\leftarrow \left\{{\bf r}(t + h)\right\}, |
| 429 |
\left\{{\bf v}(t + h)\right\}, \\ |
| 430 |
% |
| 431 |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 432 |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
| 433 |
{T_{\mathrm{target}}} - 1 \right), \\ |
| 434 |
% |
| 435 |
\eta(t + h) &\leftarrow \eta(t + h / 2) + |
| 436 |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 437 |
\tau_B^2} \left( P(t + h) - P_{\mathrm{target}} \right), \\ |
| 438 |
% |
| 439 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 440 |
+ h / 2 \right) + \frac{h}{2} \left( |
| 441 |
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 442 |
(\chi(t + h) + \eta(t + h)) \right) ,\\ |
| 443 |
% |
| 444 |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 445 |
+ h / 2 \right) + \frac{h}{2} \left( {\bf |
| 446 |
\tau}^b(t + h) - {\bf j}(t + h) |
| 447 |
\chi(t + h) \right) . |
| 448 |
\end{align*} |
| 449 |
|
| 450 |
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
| 451 |
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 452 |
h)$, they indirectly depend on their own values at time $t + h$. |
| 453 |
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
| 454 |
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
| 455 |
tolerance for the self-consistency check defaults to a value of |
| 456 |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
| 457 |
4 loops even if the consistency check has not been satisfied. |
| 458 |
|
| 459 |
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
| 460 |
is known to conserve a Hamiltonian for the extended system that is, |
| 461 |
to within a constant, identical to the Gibbs free energy, |
| 462 |
\begin{equation} |
| 463 |
H_{\mathrm{NPTi}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 464 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 465 |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t). |
| 466 |
\end{equation} |
| 467 |
Poor choices of $\delta t$, $\tau_T$, or $\tau_B$ can result in |
| 468 |
non-conservation of $H_{\mathrm{NPTi}}$, so the conserved quantity |
| 469 |
is maintained in the last column of the {\tt .stat} file to allow |
| 470 |
checks on the quality of the integration. It is also known that |
| 471 |
this algorithm samples the equilibrium distribution for the enthalpy |
| 472 |
(including contributions for the thermostat and barostat), |
| 473 |
\begin{equation} |
| 474 |
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} |
| 475 |
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
| 476 |
P_{\mathrm{target}} \mathcal{V}(t). |
| 477 |
\end{equation} |
| 478 |
|
| 479 |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 480 |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 481 |
algorithms are given in section \ref{oopseSec:rattle}. |
| 482 |
|
| 483 |
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
| 484 |
flexible box (NPTf)} |
| 485 |
|
| 486 |
There is a relatively simple generalization of the |
| 487 |
Nos\'e-Hoover-Andersen method to include changes in the simulation |
| 488 |
box {\it shape} as well as in the volume of the box. This method |
| 489 |
utilizes the full $3 \times 3$ pressure tensor and introduces a |
| 490 |
tensor of extended variables ($\overleftrightarrow{\eta}$) to |
| 491 |
control changes to the box shape. The equations of motion for this |
| 492 |
method are |
| 493 |
\begin{eqnarray} |
| 494 |
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
| 495 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
| 496 |
\chi \cdot \mathsf{1}) {\bf v}, \\ |
| 497 |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 498 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
| 499 |
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 500 |
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 501 |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 502 |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} ,\\ |
| 503 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 504 |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 505 |
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
| 506 |
T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 507 |
\dot{\mathsf{H}} & = & \overleftrightarrow{\eta} \cdot \mathsf{H} . |
| 508 |
\label{eq:melchionna2} |
| 509 |
\end{eqnarray} |
| 510 |
|
| 511 |
Here, $\mathsf{1}$ is the unit matrix and |
| 512 |
$\overleftrightarrow{\mathsf{P}}$ is the pressure tensor. Again, |
| 513 |
the volume, $\mathcal{V} = \det \mathsf{H}$. |
| 514 |
|
| 515 |
The propagation of the equations of motion is nearly identical to |
| 516 |
the NPTi integration: |
| 517 |
|
| 518 |
{\tt moveA:} |
| 519 |
\begin{align*} |
| 520 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 521 |
% |
| 522 |
\overleftrightarrow{\mathsf{P}}(t) &\leftarrow \left\{{\bf |
| 523 |
r}(t)\right\}, |
| 524 |
\left\{{\bf v}(t)\right\} ,\\ |
| 525 |
% |
| 526 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 527 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - |
| 528 |
\left(\chi(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot |
| 529 |
{\bf v}(t) \right), \\ |
| 530 |
% |
| 531 |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 532 |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 533 |
\chi(t) \right), \\ |
| 534 |
% |
| 535 |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 536 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 537 |
\right), \\ |
| 538 |
% |
| 539 |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 540 |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} |
| 541 |
- 1 \right), \\ |
| 542 |
% |
| 543 |
\overleftrightarrow{\eta}(t + h / 2) &\leftarrow |
| 544 |
\overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}{2 N k_B |
| 545 |
T(t) \tau_B^2} \left( \overleftrightarrow{\mathsf{P}}(t) |
| 546 |
- P_{\mathrm{target}}\mathsf{1} \right), \\ |
| 547 |
% |
| 548 |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h \left\{ {\bf v} |
| 549 |
\left(t + h / 2 \right) + \overleftrightarrow{\eta}(t + |
| 550 |
h / 2) \cdot \left[ {\bf r}(t + h) |
| 551 |
- {\bf R}_0 \right] \right\}, \\ |
| 552 |
% |
| 553 |
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
| 554 |
\overleftrightarrow{\eta}(t + h / 2)} . |
| 555 |
\end{align*} |
| 556 |
{\sc oopse} uses a power series expansion truncated at second order |
| 557 |
for the exponential operation which scales the simulation box. |
| 558 |
|
| 559 |
The {\tt moveB} portion of the algorithm is largely unchanged from |
| 560 |
the NPTi integrator: |
| 561 |
|
| 562 |
{\tt moveB:} |
| 563 |
\begin{align*} |
| 564 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 565 |
\left\{{\bf j}(t + h)\right\}, \\ |
| 566 |
% |
| 567 |
\overleftrightarrow{\mathsf{P}}(t + h) &\leftarrow \left\{{\bf r} |
| 568 |
(t + h)\right\}, \left\{{\bf v}(t |
| 569 |
+ h)\right\}, \left\{{\bf f}(t + h)\right\} ,\\ |
| 570 |
% |
| 571 |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 572 |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+ |
| 573 |
h)}{T_{\mathrm{target}}} - 1 \right), \\ |
| 574 |
% |
| 575 |
\overleftrightarrow{\eta}(t + h) &\leftarrow |
| 576 |
\overleftrightarrow{\eta}(t + h / 2) + |
| 577 |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 578 |
\tau_B^2} \left( \overleftrightarrow{P}(t + h) |
| 579 |
- P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 580 |
% |
| 581 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 582 |
+ h / 2 \right) + \frac{h}{2} \left( |
| 583 |
\frac{{\bf f}(t + h)}{m} - |
| 584 |
(\chi(t + h)\mathsf{1} + \overleftrightarrow{\eta}(t |
| 585 |
+ h)) \right) \cdot {\bf v}(t + h), \\ |
| 586 |
% |
| 587 |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 588 |
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
| 589 |
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
| 590 |
\end{align*} |
| 591 |
|
| 592 |
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
| 593 |
identical to those described for the NPTi integrator. |
| 594 |
|
| 595 |
The NPTf integrator is known to conserve the following Hamiltonian: |
| 596 |
\begin{equation} |
| 597 |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 598 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 599 |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
| 600 |
T_{\mathrm{target}}}{2} |
| 601 |
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 602 |
\end{equation} |
| 603 |
|
| 604 |
This integrator must be used with care, particularly in liquid |
| 605 |
simulations. Liquids have very small restoring forces in the |
| 606 |
off-diagonal directions, and the simulation box can very quickly |
| 607 |
form elongated and sheared geometries which become smaller than the |
| 608 |
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
| 609 |
finds most use in simulating crystals or liquid crystals which |
| 610 |
assume non-orthorhombic geometries. |
| 611 |
|
| 612 |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
| 613 |
|
| 614 |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
| 615 |
|
| 616 |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
| 617 |
|
| 618 |
There is one additional extended system integrator which is somewhat |
| 619 |
simpler than the NPTf method described above. In this case, the |
| 620 |
three axes have independent barostats which each attempt to preserve |
| 621 |
the target pressure along the box walls perpendicular to that |
| 622 |
particular axis. The lengths of the box axes are allowed to |
| 623 |
fluctuate independently, but the angle between the box axes does not |
| 624 |
change. The equations of motion are identical to those described |
| 625 |
above, but only the {\it diagonal} elements of |
| 626 |
$\overleftrightarrow{\eta}$ are computed. The off-diagonal elements |
| 627 |
are set to zero (even when the pressure tensor has non-zero |
| 628 |
off-diagonal elements). It should be noted that the NPTxyz |
| 629 |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 630 |
$\gamma$ is set to zero. |
| 631 |
|
| 632 |
|
| 633 |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
| 634 |
|
| 635 |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
| 636 |
Constraints} |
| 637 |
|
| 638 |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
| 639 |
|
| 640 |
Based on the fluctuation-dissipation theorem, a force |
| 641 |
auto-correlation method was developed by Roux and Karplus to |
| 642 |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
| 643 |
The time-dependent friction coefficient can be calculated from the |
| 644 |
deviation of the instantaneous force from its mean force. |
| 645 |
\begin{equation} |
| 646 |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 647 |
\end{equation} |
| 648 |
where% |
| 649 |
\begin{equation} |
| 650 |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 651 |
\end{equation} |
| 652 |
|
| 653 |
If the time-dependent friction decays rapidly, the static friction |
| 654 |
coefficient can be approximated by |
| 655 |
\begin{equation} |
| 656 |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 657 |
F(z,0)\rangle dt. |
| 658 |
\end{equation} |
| 659 |
Allowing diffusion constant to then be calculated through the |
| 660 |
Einstein relation:\cite{Marrink94} |
| 661 |
\begin{equation} |
| 662 |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 663 |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 664 |
\end{equation} |
| 665 |
|
| 666 |
The Z-Constraint method, which fixes the z coordinates of the |
| 667 |
molecules with respect to the center of the mass of the system, has |
| 668 |
been a method suggested to obtain the forces required for the force |
| 669 |
auto-correlation calculation.\cite{Marrink94} However, simply |
| 670 |
resetting the coordinate will move the center of the mass of the |
| 671 |
whole system. To avoid this problem, a new method was used in {\sc |
| 672 |
oopse}. Instead of resetting the coordinate, we reset the forces of |
| 673 |
z-constrained molecules as well as subtract the total constraint |
| 674 |
forces from the rest of the system after the force calculation at |
| 675 |
each time step. |
| 676 |
|
| 677 |
After the force calculation, define $G_\alpha$ as |
| 678 |
\begin{equation} |
| 679 |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 680 |
\end{equation} |
| 681 |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 682 |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 683 |
molecule are then set to: |
| 684 |
\begin{equation} |
| 685 |
F_{\alpha i} = F_{\alpha i} - |
| 686 |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 687 |
\end{equation} |
| 688 |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 689 |
molecule. Having rescaled the forces, the velocities must also be |
| 690 |
rescaled to subtract out any center of mass velocity in the z |
| 691 |
direction. |
| 692 |
\begin{equation} |
| 693 |
v_{\alpha i} = v_{\alpha i} - |
| 694 |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 695 |
\end{equation} |
| 696 |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 697 |
Lastly, all of the accumulated z constrained forces must be |
| 698 |
subtracted from the system to keep the system center of mass from |
| 699 |
drifting. |
| 700 |
\begin{equation} |
| 701 |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 702 |
G_{\alpha}} |
| 703 |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 704 |
\end{equation} |
| 705 |
where $\beta$ are all of the unconstrained molecules in the system. |
| 706 |
Similarly, the velocities of the unconstrained molecules must also |
| 707 |
be scaled. |
| 708 |
\begin{equation} |
| 709 |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 710 |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 711 |
\end{equation} |
| 712 |
|
| 713 |
At the very beginning of the simulation, the molecules may not be at |
| 714 |
their constrained positions. To move a z-constrained molecule to its |
| 715 |
specified position, a simple harmonic potential is used |
| 716 |
\begin{equation} |
| 717 |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 718 |
\end{equation} |
| 719 |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 720 |
is the current $z$ coordinate of the center of mass of the |
| 721 |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 722 |
position. The harmonic force operating on the z-constrained molecule |
| 723 |
at time $t$ can be calculated by |
| 724 |
\begin{equation} |
| 725 |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 726 |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 727 |
\end{equation} |
| 728 |
|
| 729 |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 730 |
|
| 731 |
\subsection{\label{methodSection:temperature}Temperature Control} |
| 732 |
|
| 733 |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
| 734 |
|
| 735 |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
| 736 |
|
| 737 |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
| 738 |
|
| 739 |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |
