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\chapter{\label{chapt:methodology}METHODOLOGY} |
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\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
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In order to mimic 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 ensembles, such |
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as the canonical and isobaric-isothermal ensembles. In addition to |
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the standard ensemble, specific ensembles have been developed to |
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account for the anisotropy between the lateral and normal directions |
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of membranes. The $NPAT$ ensemble, in which the normal pressure and |
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the lateral surface area of the membrane are kept constant, and the |
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$NP\gamma T$ ensemble, in which the normal pressure and the lateral |
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surface tension are kept constant were proposed to address the |
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issues. |
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Integration schemes for the rotational motion of the rigid molecules |
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in the microcanonical ensemble have been extensively studied over |
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the last two decades. Matubayasi\cite{Matubayasi1999} developed a |
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time-reversible integrator for rigid bodies in quaternion |
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representation. Although it is not symplectic, this integrator still |
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demonstrates a better long-time energy conservation than Euler angle |
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methods because of the time-reversible nature. Extending the |
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Trotter-Suzuki factorization to general system with a flat phase |
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space, Miller\cite{Miller2002} and his colleagues devised a novel |
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symplectic, time-reversible and volume-preserving integrator in the |
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quaternion representation, which was shown to be superior to the |
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Matubayasi's time-reversible integrator. However, all of the |
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integrators in the 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 the rotation matrix |
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directly proposed by Dullweber, Leimkuhler and McLachlan (DLM) also |
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preserved the same structural properties of the Hamiltonian |
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propagator\cite{Dullweber1997}. In this section, the integration |
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scheme of DLM method will be reviewed and extended to other |
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ensembles. |
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\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
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DLM method} |
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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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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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{\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{Q}(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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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 ($\mathsf{Q}$) 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{Q}(t) & \leftarrow & \mathsf{Q}(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. After the |
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first part of the propagation, the forces and body-fixed torques are |
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calculated at the new positions and orientations |
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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}})_{u(t+h)}, \\ |
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% |
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{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
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\cdot {\bf \tau}^s(t + h). |
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\end{align*} |
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${\bf u}$ is automatically updated when the rotation matrix |
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$\mathsf{Q}$ 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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{\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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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{methodFig:timestep} where the resulting energy drift at |
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various time steps for both the DLM and quaternion integration |
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schemes is compared. All of the 1000 molecule water simulations |
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started with the same configuration, and the only difference was the |
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method for handling rotational motion. At time steps of 0.1 and 0.5 |
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fs, both methods for propagating molecule rotation conserve energy |
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fairly well, with the quaternion method showing a slight energy |
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drift over time in the 0.5 fs time step simulation. At time steps of |
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1 and 2 fs, the energy conservation benefits of the DLM method are |
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clearly demonstrated. Thus, while maintaining the same degree of |
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energy conservation, one can take considerably longer time steps, |
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leading to an overall reduction in computation time. |
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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.} |
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\label{methodFig:timestep} |
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\end{figure} |
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\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
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The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
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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{Q}} & = & \mathsf{Q} \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{Q}^{T} \cdot \frac{\partial V}{\partial |
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\mathsf{Q}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
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\end{eqnarray} |
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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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where $\tau_T$ is the time constant for relaxation of the |
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temperature to the target value, and the instantaneous temperature |
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$T$ is given 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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where $N_{\mathrm{orient}}$ is the number of molecules with |
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orientational degrees of freedom. The integration of the equations of motion |
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is carried out in a velocity-Verlet style 2 part algorithm: |
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|
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{\tt moveA:} |
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\begin{align*} |
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T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
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% |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
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+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
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\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) ,\\ |
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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} \left( {\bf \tau}^b(t) - {\bf j}(t) |
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\chi(t) \right) ,\\ |
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% |
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\mathsf{Q}(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) ,\\ |
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% |
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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)} |
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{T_{\mathrm{target}}} - 1 \right) . |
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\end{align*} |
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Here $\mathrm{rotate}(h * {\bf j} |
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\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic Strang |
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factorization of the three rotation operations that was discussed in |
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the section on the DLM integrator. Note that this operation |
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modifies both the rotation matrix $\mathsf{Q}$ and the angular |
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momentum ${\bf j}$. {\tt moveA} propagates velocities by a half |
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time step, and positional degrees of freedom by a full time step. |
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The new positions (and orientations) are then used to calculate a |
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new set of forces and torques in exactly the same way they are |
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calculated in the {\tt doForces} portion of the DLM integrator. Once |
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the forces and torques have been obtained at the new time step, the |
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temperature, velocities, and the extended system variable can be |
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advanced to the same time value. |
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{\tt moveB:} |
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\begin{align*} |
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T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
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\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), \\ |
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% |
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{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
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+ h / 2 \right) + \frac{h}{2} \left( |
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\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
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\chi(t h)\right) ,\\ |
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% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
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+ h / 2 \right) + \frac{h}{2} |
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\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
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\chi(t + h) \right) . |
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\end{align*} |
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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 Nos\'e-Hoover algorithm is known to conserve a |
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Hamiltonian for the extended system that is, to within a constant, |
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identical to the Helmholtz free energy,\cite{Melchionna1993} |
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\begin{equation} |
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H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
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\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
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dt^\prime \right). |
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\end{equation} |
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Poor choices of $h$ or $\tau_T$ can result in non-conservation of |
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$H_{\mathrm{NVT}}$, so the conserved quantity should be checked |
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periodically to verify the quality of the integration. |
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|
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\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
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isotropic box (NPTi)} |
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We can used an isobaric-isothermal ensemble integrator which is |
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implemented using the Melchionna modifications to the |
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Nos\'e-Hoover-Andersen equations of motion\cite{Melchionna1993} |
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\begin{eqnarray} |
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\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
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\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
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\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
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\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
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\dot{{\bf j}} & = & {\bf j} \times \left( |
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\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
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rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
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V}{\partial \mathsf{Q}} \right) - \chi {\bf j}, \\ |
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\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
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\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
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\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V |
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\left( P - |
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P_{\mathrm{target}} \right), \\ |
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\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
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\end{eqnarray} |
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$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
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extended system. $\chi$ is a thermostat, and it has the same |
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function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
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a barostat which controls changes to the volume of the simulation |
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box. ${\bf R}_0$ is the location of the center of mass for the |
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entire system, and $\mathcal{V}$ is the volume of the simulation |
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box. At any time, the volume can be calculated from the determinant |
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of the matrix which describes the box shape: |
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\begin{equation} |
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\mathcal{V} = \det(\mathsf{H}). |
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\end{equation} |
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The NPTi integrator requires an instantaneous pressure. This |
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quantity is calculated via the pressure tensor, |
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\begin{equation} |
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\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{\mathcal{V}(t)} \left( |
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\sum_{i=1}^{N} m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
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\overleftrightarrow{\mathsf{W}}(t). |
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\end{equation} |
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The kinetic contribution to the pressure tensor utilizes the {\it |
312 |
|
|
outer} product of the velocities denoted by the $\otimes$ symbol. |
313 |
|
|
The stress tensor is calculated from another outer product of the |
314 |
|
|
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
315 |
|
|
r}_i$) with the forces between the same two atoms, |
316 |
|
|
\begin{equation} |
317 |
|
|
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf |
318 |
|
|
r}_{ij}(t) \otimes {\bf f}_{ij}(t). |
319 |
|
|
\end{equation} |
320 |
|
|
The instantaneous pressure is then simply obtained from the trace of |
321 |
|
|
the Pressure tensor, |
322 |
|
|
\begin{equation} |
323 |
|
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
324 |
tim |
2905 |
\overleftrightarrow{\mathsf{P}}(t) \right) . |
325 |
tim |
2729 |
\end{equation} |
326 |
tim |
2905 |
In Eq.~\ref{eq:melchionna1}, $\tau_B$ is the time constant for |
327 |
tim |
2853 |
relaxation of the pressure to the target value. Like in the NVT |
328 |
tim |
2729 |
integrator, the integration of the equations of motion is carried |
329 |
|
|
out in a velocity-Verlet style 2 part algorithm: |
330 |
|
|
|
331 |
|
|
{\tt moveA:} |
332 |
|
|
\begin{align*} |
333 |
|
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
334 |
|
|
% |
335 |
|
|
P(t) &\leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,\\ |
336 |
|
|
% |
337 |
|
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
338 |
|
|
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
339 |
|
|
\left(\chi(t) + \eta(t) \right) \right), \\ |
340 |
|
|
% |
341 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
342 |
|
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
343 |
|
|
\chi(t) \right), \\ |
344 |
|
|
% |
345 |
tim |
2909 |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
346 |
tim |
2729 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
347 |
|
|
\right) ,\\ |
348 |
|
|
% |
349 |
|
|
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
350 |
|
|
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
351 |
|
|
\right) ,\\ |
352 |
|
|
% |
353 |
|
|
\eta(t + h / 2) &\leftarrow \eta(t) + \frac{h |
354 |
|
|
\mathcal{V}(t)}{2 N k_B T(t) \tau_B^2} \left( P(t) |
355 |
|
|
- P_{\mathrm{target}} \right), \\ |
356 |
|
|
% |
357 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h |
358 |
|
|
\left\{ {\bf v}\left(t + h / 2 \right) |
359 |
|
|
+ \eta(t + h / 2)\left[ {\bf r}(t + h) |
360 |
|
|
- {\bf R}_0 \right] \right\} ,\\ |
361 |
|
|
% |
362 |
|
|
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
363 |
|
|
\mathsf{H}(t). |
364 |
|
|
\end{align*} |
365 |
|
|
Most of these equations are identical to their counterparts in the |
366 |
|
|
NVT integrator, but the propagation of positions to time $t + h$ |
367 |
tim |
2854 |
depends on the positions at the same time. The simulation box |
368 |
|
|
$\mathsf{H}$ is scaled uniformly for one full time step by an |
369 |
|
|
exponential factor that depends on the value of $\eta$ at time $t + |
370 |
|
|
h / 2$. Reshaping the box uniformly also scales the volume of the |
371 |
|
|
box by |
372 |
tim |
2729 |
\begin{equation} |
373 |
|
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
374 |
|
|
\mathcal{V}(t) |
375 |
|
|
\end{equation} |
376 |
|
|
The {\tt doForces} step for the NPTi integrator is exactly the same |
377 |
|
|
as in both the DLM and NVT integrators. Once the forces and torques |
378 |
|
|
have been obtained at the new time step, the velocities can be |
379 |
|
|
advanced to the same time value. |
380 |
|
|
|
381 |
|
|
{\tt moveB:} |
382 |
|
|
\begin{align*} |
383 |
|
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
384 |
|
|
\left\{{\bf j}(t + h)\right\} ,\\ |
385 |
|
|
% |
386 |
|
|
P(t + h) &\leftarrow \left\{{\bf r}(t + h)\right\}, |
387 |
|
|
\left\{{\bf v}(t + h)\right\}, \\ |
388 |
|
|
% |
389 |
|
|
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
390 |
|
|
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
391 |
|
|
{T_{\mathrm{target}}} - 1 \right), \\ |
392 |
|
|
% |
393 |
|
|
\eta(t + h) &\leftarrow \eta(t + h / 2) + |
394 |
|
|
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
395 |
|
|
\tau_B^2} \left( P(t + h) - P_{\mathrm{target}} \right), \\ |
396 |
|
|
% |
397 |
|
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
398 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( |
399 |
|
|
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
400 |
|
|
(\chi(t + h) + \eta(t + h)) \right) ,\\ |
401 |
|
|
% |
402 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
403 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf |
404 |
|
|
\tau}^b(t + h) - {\bf j}(t + h) |
405 |
|
|
\chi(t + h) \right) . |
406 |
|
|
\end{align*} |
407 |
|
|
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
408 |
|
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
409 |
|
|
h)$, they indirectly depend on their own values at time $t + h$. |
410 |
|
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
411 |
tim |
2854 |
+ h)$ and $\eta(t + h)$ become self-consistent. |
412 |
tim |
2729 |
|
413 |
|
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
414 |
|
|
is known to conserve a Hamiltonian for the extended system that is, |
415 |
|
|
to within a constant, identical to the Gibbs free energy, |
416 |
|
|
\begin{equation} |
417 |
|
|
H_{\mathrm{NPTi}} = V + K + f k_B T_{\mathrm{target}} \left( |
418 |
|
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
419 |
|
|
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t). |
420 |
|
|
\end{equation} |
421 |
tim |
2909 |
It is also known that this algorithm samples the equilibrium |
422 |
|
|
distribution for the enthalpy (including contributions for the |
423 |
|
|
thermostat and barostat), |
424 |
tim |
2729 |
\begin{equation} |
425 |
|
|
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} |
426 |
|
|
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
427 |
|
|
P_{\mathrm{target}} \mathcal{V}(t). |
428 |
|
|
\end{equation} |
429 |
|
|
|
430 |
|
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
431 |
|
|
flexible box (NPTf)} |
432 |
|
|
|
433 |
|
|
There is a relatively simple generalization of the |
434 |
|
|
Nos\'e-Hoover-Andersen method to include changes in the simulation |
435 |
|
|
box {\it shape} as well as in the volume of the box. This method |
436 |
|
|
utilizes the full $3 \times 3$ pressure tensor and introduces a |
437 |
|
|
tensor of extended variables ($\overleftrightarrow{\eta}$) to |
438 |
|
|
control changes to the box shape. The equations of motion for this |
439 |
|
|
method are |
440 |
|
|
\begin{eqnarray} |
441 |
|
|
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
442 |
|
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
443 |
|
|
\chi \cdot \mathsf{1}) {\bf v}, \\ |
444 |
tim |
2909 |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
445 |
tim |
2729 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
446 |
|
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
447 |
|
|
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
448 |
tim |
2909 |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
449 |
|
|
V}{\partial \mathsf{Q}} \right) - \chi {\bf j} ,\\ |
450 |
tim |
2729 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
451 |
|
|
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
452 |
|
|
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
453 |
|
|
T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
454 |
|
|
\dot{\mathsf{H}} & = & \overleftrightarrow{\eta} \cdot \mathsf{H} . |
455 |
|
|
\label{eq:melchionna2} |
456 |
|
|
\end{eqnarray} |
457 |
|
|
|
458 |
|
|
Here, $\mathsf{1}$ is the unit matrix and |
459 |
|
|
$\overleftrightarrow{\mathsf{P}}$ is the pressure tensor. Again, |
460 |
|
|
the volume, $\mathcal{V} = \det \mathsf{H}$. |
461 |
|
|
|
462 |
|
|
The propagation of the equations of motion is nearly identical to |
463 |
|
|
the NPTi integration: |
464 |
|
|
|
465 |
|
|
{\tt moveA:} |
466 |
|
|
\begin{align*} |
467 |
|
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
468 |
|
|
% |
469 |
|
|
\overleftrightarrow{\mathsf{P}}(t) &\leftarrow \left\{{\bf |
470 |
|
|
r}(t)\right\}, |
471 |
|
|
\left\{{\bf v}(t)\right\} ,\\ |
472 |
|
|
% |
473 |
|
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
474 |
|
|
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - |
475 |
|
|
\left(\chi(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot |
476 |
|
|
{\bf v}(t) \right), \\ |
477 |
|
|
% |
478 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
479 |
|
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
480 |
|
|
\chi(t) \right), \\ |
481 |
|
|
% |
482 |
tim |
2909 |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
483 |
tim |
2729 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
484 |
|
|
\right), \\ |
485 |
|
|
% |
486 |
|
|
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
487 |
|
|
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} |
488 |
|
|
- 1 \right), \\ |
489 |
|
|
% |
490 |
|
|
\overleftrightarrow{\eta}(t + h / 2) &\leftarrow |
491 |
|
|
\overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}{2 N k_B |
492 |
|
|
T(t) \tau_B^2} \left( \overleftrightarrow{\mathsf{P}}(t) |
493 |
|
|
- P_{\mathrm{target}}\mathsf{1} \right), \\ |
494 |
|
|
% |
495 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h \left\{ {\bf v} |
496 |
|
|
\left(t + h / 2 \right) + \overleftrightarrow{\eta}(t + |
497 |
|
|
h / 2) \cdot \left[ {\bf r}(t + h) |
498 |
|
|
- {\bf R}_0 \right] \right\}, \\ |
499 |
|
|
% |
500 |
|
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
501 |
|
|
\overleftrightarrow{\eta}(t + h / 2)} . |
502 |
|
|
\end{align*} |
503 |
tim |
2854 |
Here, a power series expansion truncated at second order for the |
504 |
tim |
2905 |
exponential operation is used to scale the simulation box. The {\tt |
505 |
|
|
moveB} portion of the algorithm is largely unchanged from the NPTi |
506 |
|
|
integrator: |
507 |
tim |
2729 |
|
508 |
|
|
{\tt moveB:} |
509 |
|
|
\begin{align*} |
510 |
|
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
511 |
|
|
\left\{{\bf j}(t + h)\right\}, \\ |
512 |
|
|
% |
513 |
|
|
\overleftrightarrow{\mathsf{P}}(t + h) &\leftarrow \left\{{\bf r} |
514 |
|
|
(t + h)\right\}, \left\{{\bf v}(t |
515 |
|
|
+ h)\right\}, \left\{{\bf f}(t + h)\right\} ,\\ |
516 |
|
|
% |
517 |
|
|
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
518 |
|
|
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+ |
519 |
|
|
h)}{T_{\mathrm{target}}} - 1 \right), \\ |
520 |
|
|
% |
521 |
|
|
\overleftrightarrow{\eta}(t + h) &\leftarrow |
522 |
|
|
\overleftrightarrow{\eta}(t + h / 2) + |
523 |
|
|
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
524 |
|
|
\tau_B^2} \left( \overleftrightarrow{P}(t + h) |
525 |
|
|
- P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
526 |
|
|
% |
527 |
|
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
528 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( |
529 |
|
|
\frac{{\bf f}(t + h)}{m} - |
530 |
|
|
(\chi(t + h)\mathsf{1} + \overleftrightarrow{\eta}(t |
531 |
|
|
+ h)) \right) \cdot {\bf v}(t + h), \\ |
532 |
|
|
% |
533 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
534 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
535 |
|
|
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
536 |
|
|
\end{align*} |
537 |
|
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
538 |
tim |
2905 |
identical to those described for the NPTi integrator. The NPTf |
539 |
|
|
integrator is known to conserve the following Hamiltonian: |
540 |
tim |
2854 |
\begin{eqnarray*} |
541 |
|
|
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
542 |
tim |
2729 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
543 |
tim |
2854 |
dt^\prime \right) \\ |
544 |
tim |
2882 |
& & + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
545 |
tim |
2856 |
T_{\mathrm{target}}}{2} |
546 |
tim |
2729 |
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
547 |
tim |
2854 |
\end{eqnarray*} |
548 |
tim |
2729 |
This integrator must be used with care, particularly in liquid |
549 |
|
|
simulations. Liquids have very small restoring forces in the |
550 |
|
|
off-diagonal directions, and the simulation box can very quickly |
551 |
|
|
form elongated and sheared geometries which become smaller than the |
552 |
|
|
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
553 |
|
|
finds most use in simulating crystals or liquid crystals which |
554 |
|
|
assume non-orthorhombic geometries. |
555 |
|
|
|
556 |
tim |
2855 |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
557 |
tim |
2729 |
|
558 |
tim |
2881 |
A comprehensive understanding of relations between structures and |
559 |
|
|
functions in biological membrane system ultimately relies on |
560 |
|
|
structure and dynamics of lipid bilayers, which are strongly |
561 |
|
|
affected by the interfacial interaction between lipid molecules and |
562 |
tim |
2909 |
surrounding media. One quantity used to describe the interfacial |
563 |
|
|
interaction is the average surface area per lipid. |
564 |
tim |
2881 |
Constant area and constant lateral pressure simulations can be |
565 |
|
|
achieved by extending the standard NPT ensemble with a different |
566 |
|
|
pressure control strategy |
567 |
tim |
2798 |
|
568 |
tim |
2729 |
\begin{equation} |
569 |
tim |
2799 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
570 |
tim |
2798 |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
571 |
tim |
2799 |
& \mbox{if $ \alpha = \beta = z$}\\ |
572 |
tim |
2798 |
0 & \mbox{otherwise}\\ |
573 |
|
|
\end{array} |
574 |
|
|
\right. |
575 |
tim |
2729 |
\end{equation} |
576 |
tim |
2798 |
|
577 |
tim |
2739 |
Note that the iterative schemes for NPAT are identical to those |
578 |
|
|
described for the NPTi integrator. |
579 |
tim |
2729 |
|
580 |
tim |
2854 |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
581 |
|
|
Ensemble} |
582 |
tim |
2729 |
|
583 |
tim |
2739 |
Theoretically, the surface tension $\gamma$ of a stress free |
584 |
|
|
membrane system should be zero since its surface free energy $G$ is |
585 |
tim |
2905 |
minimum with respect to surface area $A$, $\gamma = \frac{{\partial |
586 |
|
|
G}}{{\partial A}}.$ However, a surface tension of zero is not |
587 |
|
|
appropriate for relatively small patches of membrane. In order to |
588 |
tim |
2909 |
eliminate the edge effect of membrane simulations, a special |
589 |
tim |
2905 |
ensemble, NP$\gamma$T, has been proposed to maintain the lateral |
590 |
|
|
surface tension and normal pressure. The equation of motion for the |
591 |
|
|
cell size control tensor, $\eta$, in $NP\gamma T$ is |
592 |
tim |
2729 |
\begin{equation} |
593 |
tim |
2799 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
594 |
tim |
2798 |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
595 |
|
|
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
596 |
|
|
0 & \mbox{$\alpha \ne \beta$} \\ |
597 |
tim |
2799 |
\end{array} |
598 |
tim |
2798 |
\right. |
599 |
tim |
2729 |
\end{equation} |
600 |
tim |
2739 |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
601 |
|
|
the instantaneous surface tensor $\gamma _\alpha$ is given by |
602 |
tim |
2729 |
\begin{equation} |
603 |
tim |
2800 |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
604 |
|
|
- P_{{\rm{target}}} ) |
605 |
tim |
2739 |
\label{methodEquation:instantaneousSurfaceTensor} |
606 |
tim |
2729 |
\end{equation} |
607 |
tim |
2739 |
There is one additional extended system integrator (NPTxyz), in |
608 |
|
|
which each attempt to preserve the target pressure along the box |
609 |
|
|
walls perpendicular to that particular axis. The lengths of the box |
610 |
|
|
axes are allowed to fluctuate independently, but the angle between |
611 |
|
|
the box axes does not change. It should be noted that the NPTxyz |
612 |
|
|
integrator is a special case of $NP\gamma T$ if the surface tension |
613 |
tim |
2881 |
$\gamma$ is set to zero, and if $x$ and $y$ can move independently. |
614 |
tim |
2729 |
|
615 |
tim |
2881 |
\section{\label{methodSection:zcons}The Z-Constraint Method} |
616 |
tim |
2776 |
|
617 |
tim |
2804 |
Based on the fluctuation-dissipation theorem, a force |
618 |
|
|
auto-correlation method was developed by Roux and Karplus to |
619 |
|
|
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
620 |
|
|
The time-dependent friction coefficient can be calculated from the |
621 |
|
|
deviation of the instantaneous force from its mean force. |
622 |
|
|
\begin{equation} |
623 |
|
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
624 |
|
|
\end{equation} |
625 |
|
|
where% |
626 |
|
|
\begin{equation} |
627 |
|
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
628 |
|
|
\end{equation} |
629 |
|
|
If the time-dependent friction decays rapidly, the static friction |
630 |
|
|
coefficient can be approximated by |
631 |
|
|
\begin{equation} |
632 |
|
|
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
633 |
|
|
F(z,0)\rangle dt. |
634 |
|
|
\end{equation} |
635 |
|
|
Allowing diffusion constant to then be calculated through the |
636 |
|
|
Einstein relation:\cite{Marrink1994} |
637 |
|
|
\begin{equation} |
638 |
|
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
639 |
|
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
640 |
|
|
\end{equation} |
641 |
|
|
The Z-Constraint method, which fixes the z coordinates of the |
642 |
|
|
molecules with respect to the center of the mass of the system, has |
643 |
|
|
been a method suggested to obtain the forces required for the force |
644 |
|
|
auto-correlation calculation.\cite{Marrink1994} However, simply |
645 |
|
|
resetting the coordinate will move the center of the mass of the |
646 |
|
|
whole system. To avoid this problem, we reset the forces of |
647 |
|
|
z-constrained molecules as well as subtract the total constraint |
648 |
|
|
forces from the rest of the system after the force calculation at |
649 |
tim |
2905 |
each time step instead of resetting the coordinate. After the force |
650 |
|
|
calculation, we define $G_\alpha$ as |
651 |
tim |
2804 |
\begin{equation} |
652 |
|
|
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
653 |
|
|
\end{equation} |
654 |
|
|
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
655 |
|
|
z-constrained molecule $\alpha$. The forces of the z constrained |
656 |
|
|
molecule are then set to: |
657 |
|
|
\begin{equation} |
658 |
|
|
F_{\alpha i} = F_{\alpha i} - |
659 |
|
|
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
660 |
|
|
\end{equation} |
661 |
|
|
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
662 |
|
|
molecule. Having rescaled the forces, the velocities must also be |
663 |
|
|
rescaled to subtract out any center of mass velocity in the z |
664 |
|
|
direction. |
665 |
|
|
\begin{equation} |
666 |
|
|
v_{\alpha i} = v_{\alpha i} - |
667 |
|
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
668 |
|
|
\end{equation} |
669 |
|
|
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
670 |
|
|
Lastly, all of the accumulated z constrained forces must be |
671 |
|
|
subtracted from the system to keep the system center of mass from |
672 |
|
|
drifting. |
673 |
|
|
\begin{equation} |
674 |
|
|
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
675 |
|
|
G_{\alpha}} |
676 |
|
|
{\sum_{\beta}\sum_i m_{\beta i}}, |
677 |
|
|
\end{equation} |
678 |
|
|
where $\beta$ are all of the unconstrained molecules in the system. |
679 |
|
|
Similarly, the velocities of the unconstrained molecules must also |
680 |
|
|
be scaled. |
681 |
|
|
\begin{equation} |
682 |
|
|
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
683 |
|
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
684 |
|
|
\end{equation} |
685 |
|
|
At the very beginning of the simulation, the molecules may not be at |
686 |
|
|
their constrained positions. To move a z-constrained molecule to its |
687 |
|
|
specified position, a simple harmonic potential is used |
688 |
|
|
\begin{equation} |
689 |
|
|
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
690 |
|
|
\end{equation} |
691 |
|
|
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
692 |
|
|
is the current $z$ coordinate of the center of mass of the |
693 |
|
|
constrained molecule, and $z_{\text{cons}}$ is the constrained |
694 |
|
|
position. The harmonic force operating on the z-constrained molecule |
695 |
|
|
at time $t$ can be calculated by |
696 |
|
|
\begin{equation} |
697 |
|
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
698 |
|
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
699 |
|
|
\end{equation} |