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|
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\section{\label{sec:mechanics}Mechanics} |
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|
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\subsection{\label{integrate}Integrating the Equations of Motion: the |
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DLM method} |
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|
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The default method for integrating the equations of motion in {\sc |
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oopse} is a velocity-Verlet version of the symplectic splitting method |
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proposed by Dullweber, Leimkuhler and McLachlan |
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(DLM).\cite{Dullweber1997} When there are no directional atoms or |
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rigid bodies present in the simulation, this integrator becomes the |
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standard velocity-Verlet integrator which is known to sample the |
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microcanonical (NVE) ensemble.\cite{} |
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|
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Previous integration methods for orientational motion have problems |
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that are avoided in the DLM method. Direct propagation of the Euler |
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angles has a known $1/\sin\theta$ divergence in the equations of |
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motion for $\phi$ and $\psi$,\cite{AllenTildesley} leading to |
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numerical instabilities any time one of the directional atoms or rigid |
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bodies has an orientation near $\theta=0$ or $\theta=\pi$. More |
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modern quaternion-based integration methods have relatively poor |
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energy conservation. While quaternions work well for orientational |
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motion in other ensembles, the microcanonical ensemble has a |
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constant energy requirement that is quite sensitive to errors in the |
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equations of motion. An earlier implementation of {\sc oopse} |
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utilized quaternions for propagation of rotational motion; however, a |
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detailed investigation showed that they resulted in a steady drift in |
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the total energy, something that has been observed by |
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others.\cite{Laird97} |
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|
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The key difference in the integration method proposed by Dullweber |
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\emph{et al.} is that the entire rotation matrix is propagated from |
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one time step to the next. In the past, this would not have been |
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feasible, since that the rotation matrix for a single body has nine |
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elements compared to 3 or 4 elements for Euler angles and quaternions |
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respectively. System memory has become less costly in recent times, |
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and this can be translated into substantial benefits in energy |
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conservation. |
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|
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The basic equations of motion being integrated are derived from the |
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Hamiltonian for conservative systems containing rigid bodies, |
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\begin{equation} |
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H = \sum_{i} \left( \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
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\frac{1}{2} {\bf j}_i^T \cdot \overleftrightarrow{\mathsf{I}}_i^{-1} \cdot |
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{\bf j}_i \right) + |
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V\left(\left\{{\bf r}\right\}, \left\{\mathsf{A}\right\}\right) |
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\end{equation} |
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where ${\bf r}_i$ and ${\bf v}_i$ are the cartesian position vector |
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and velocity of the center of mass of particle $i$, and ${\bf j}_i$ |
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and $\overleftrightarrow{\mathsf{I}}_i$ are the body-fixed angular |
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momentum and moment of inertia tensor, respectively. $\mathsf{A}_i$ |
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is the $3 \times 3$ rotation matrix describing the instantaneous |
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orientation of the particle. $V$ is the potential energy function |
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which may depend on both the positions $\left\{{\bf r}\right\}$ and |
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orientations $\left\{\mathsf{A}\right\}$ of all particles. The |
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equations of motion for the particle centers of mass are derived from |
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Hamilton's equations and are quite simple, |
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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} |
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\end{eqnarray} |
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where ${\bf f}$ is the instantaneous force on the center of mass |
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of the particle, |
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\begin{equation} |
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{\bf f} = - \frac{\partial}{\partial |
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{\bf r}} V(\left\{{\bf r}(t)\right\}, \left\{\mathsf{A}(t)\right\}). |
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\end{equation} |
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|
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The equations of motion for the orientational degrees of freedom are |
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\begin{eqnarray} |
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\dot{\mathsf{A}} & = & \mathsf{A} \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( \overleftrightarrow{I}^{-1} |
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\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
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V}{\partial \mathsf{A}} \right) |
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\end{eqnarray} |
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In these equations of motion, the $\mbox{skew}$ matrix of a vector |
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${\bf v} = \left( v_1, v_2, v_3 \right)$ is defined: |
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\begin{equation} |
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\mbox{skew}\left( {\bf v} \right) := \left( |
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\begin{array}{ccc} |
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0 & v_3 & - v_2 \\ |
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-v_3 & 0 & v_1 \\ |
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v_2 & -v_1 & 0 |
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\end{array} |
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\right) |
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\end{equation} |
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The $\mbox{rot}$ notation refers to the mapping of the $3 \times 3$ |
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rotation matrix to a vector of orientations by first computing the |
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skew-symmetric part $\left(\mathsf{A} - \mathsf{A}^{T}\right)$ and |
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then associating this with a length 3 vector by inverting the |
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$\mbox{skew}$ function above: |
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\begin{equation} |
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\mbox{rot}\left(\mathsf{A}\right) := \mbox{ skew}^{-1}\left(\mathsf{A} |
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- \mathsf{A}^{T} \right) |
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\end{equation} |
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Written this way, the $\mbox{rot}$ operation creates a set of |
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conjugate angle coordinates to the body-fixed angular momenta |
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represented by ${\bf j}$. This equation of motion for angular momenta |
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is equivalent to the more familiar body-fixed forms, |
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\begin{eqnarray} |
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\dot{j_{x}} & = & \tau^b_x(t) + |
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\left(\overleftrightarrow{\mathsf{I}}_{yy} - \overleftrightarrow{\mathsf{I}}_{zz} \right) j_y j_z \\ |
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\dot{j_{y}} & = & \tau^b_y(t) + |
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\left(\overleftrightarrow{\mathsf{I}}_{zz} - \overleftrightarrow{\mathsf{I}}_{xx} \right) j_z j_x \\ |
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\dot{j_{z}} & = & \tau^b_z(t) + |
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\left(\overleftrightarrow{\mathsf{I}}_{xx} - \overleftrightarrow{\mathsf{I}}_{yy} \right) j_x j_y |
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\end{eqnarray} |
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which utilize the body-fixed torques, ${\bf \tau}^b$. Torques are |
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most easily derived in the space-fixed frame, |
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\begin{equation} |
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{\bf \tau}^b(t) = \mathsf{A}(t) \cdot {\bf \tau}^s(t) |
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\end{equation} |
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where the torques are either derived from the forces on the |
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constituent atoms of the rigid body, or for directional atoms, |
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directly from derivatives of the potential energy, |
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\begin{equation} |
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{\bf \tau}^s(t) = - \hat{\bf u}(t) \times \left( \frac{\partial} |
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{\partial \hat{\bf u}} V\left(\left\{ {\bf r}(t) \right\}, \left\{ |
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\mathsf{A}(t) \right\}\right) \right). |
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\end{equation} |
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Here $\hat{\bf u}$ is a unit vector pointing along the principal axis |
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of the particle in the space-fixed frame. |
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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 $O\left(h^3\right)$ |
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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: |
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|
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{\tt moveA:} |
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\begin{eqnarray} |
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{\bf v}\left(t + \delta t / 2\right) & \leftarrow & {\bf |
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v}(t) + \frac{\delta t}{2} \left( {\bf f}(t) / m \right) \\ |
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{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf |
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v}\left(t + \delta t / 2 \right) \\ |
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{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
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j}(t) + \frac{\delta t}{2} {\bf \tau}^b(t) \\ |
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\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left( \delta t |
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{\bf j}(t + \delta t / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} |
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\right) |
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\end{eqnarray} |
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|
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In this context, the $\mathrm{rot}$ function is the reversible product |
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of the three body-fixed rotations, |
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\begin{equation} |
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\mathrm{rot}({\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, $\mathsf{G}_\alpha(\theta)$, rotates |
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both the rotation matrix ($\mathsf{A}$) and the body-fixed angular |
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momentum (${\bf j}$) by an angle $\theta$ around body-fixed axis |
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$\alpha$, |
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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 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 |
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the single-axis rotation matrix. For example, in the small-angle |
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limit, the rotation matrix around the body-fixed x-axis can be |
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approximated as |
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\begin{equation} |
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\mathsf{R}_x(\theta) = \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+ |
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\theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + \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{eqnarray} |
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{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf |
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r}(t + \delta t)} \\ |
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{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t) |
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\times \frac{\partial V}{\partial {\bf u}} \\ |
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{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t) |
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\cdot {\bf \tau}^s(t + \delta t) |
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\end{eqnarray} |
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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 be |
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advanced to the same time value. |
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|
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{\tt moveB:} |
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\begin{eqnarray} |
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{\bf v}\left(t + \delta t \right) & \leftarrow & {\bf |
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v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( |
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{\bf f}(t + \delta t) / m \right) \\ |
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{\bf j}\left(t + \delta t \right) & \leftarrow & {\bf |
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j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} {\bf |
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\tau}^b(t + \delta t) |
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\end{eqnarray} |
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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 |
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propagation. With the same time step, a 1000-molecule water simulation |
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shows an average 7\% increase in computation time using the DLM method |
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in place of quaternions. This cost is more than justified when |
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comparing the energy conservation of the two methods as illustrated in |
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figure \ref{timestep}. |
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|
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\begin{figure} |
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\epsfxsize=6in |
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\epsfbox{timeStep.epsi} |
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\caption{Energy conservation using quaternion based integration versus |
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the method proposed by Dullweber \emph{et al.} with increasing time |
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step. For each time step, the dotted line is total energy using the |
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symplectic step integrator, and the solid line comes from the |
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quaternion integrator. The larger time step plots are shifted up from |
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the true energy baseline for clarity.} |
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\label{timestep} |
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\end{figure} |
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|
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In figure \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 the |
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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 well, |
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with the quaternion method showing a slight energy drift over time in |
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the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
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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 to |
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an overall reduction in computation time. |
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|
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|
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There is only one specific keyword relevant to the default integrator, |
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and that is the time step for integrating the equations of motion. |
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|
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\begin{tabular}{llll} |
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{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf |
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default value} \\ |
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$\delta t$ & {\tt dt = 2.0;} & fs & none |
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\end{tabular} |
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|
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\subsection{\label{sec:extended}Extended Systems for other Ensembles} |
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|
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{\sc oopse} implements a number of extended system integrators for |
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sampling from other ensembles relevant to chemical physics. The |
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integrator can selected with the {\tt ensemble} keyword in the |
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{\tt .bass} file: |
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|
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\begin{center} |
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\begin{tabular}{lll} |
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{\bf Integrator} & {\bf Ensemble} & {\bf {\tt .bass} line} \\ |
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NVE & microcanonical & {\tt ensemble = ``NVE''; } \\ |
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NVT & canonical & {\tt ensemble = ``NVT''; } \\ |
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NPTi & isobaric-isothermal (with isotropic volume changes) & {\tt |
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ensemble = ``NPTi'';} \\ |
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NPTf & isobaric-isothermal (with changes to box shape) & {\tt |
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ensemble = ``NPTf'';} \\ |
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NPTxyz & approximate isobaric-isothermal & {\tt ensemble = |
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``NPTxyz'';} \\ |
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& (with separate barostats on each box dimension) & |
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\end{tabular} |
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\end{center} |
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|
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The relatively well-known Nos\'e-Hoover thermostat is implemented in |
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{\sc oopse}'s NVT integrator. This method couples an extra degree of |
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freedom (the thermostat) to the kinetic energy of the system, and has |
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been shown to sample the canonical distribution in the system degrees |
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of freedom while conserving a quantity that is, to within a constant, |
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the Helmholtz free energy. |
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|
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NPT algorithms attempt to maintain constant pressure in the system by |
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coupling the volume of the system to a barostat. {\sc oopse} contains |
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three different constant pressure algorithms. The first two, NPTi and |
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NPTf have been shown to conserve a quantity that is, to within a |
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constant, the Gibbs free energy. The Melchionna modification to the |
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Hoover barostat is implemented in both NPTi and NPTf. NPTi allows |
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only isotropic changes in the simulation box, while box {\it shape} |
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variations are allowed in NPTf. The NPTxyz integrator has {\it not} |
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been shown to sample from the isobaric-isothermal ensemble. It is |
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useful, however, in that it maintains orthogonality for the axes of |
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the simulation box while attempting to equalize pressure along the |
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three perpendicular directions in the box. |
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|
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Each of the extended system integrators requires additional keywords |
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to set target values for the thermodynamic state variables that are |
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being held constant. Keywords are also required to set the |
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characteristic decay times for the dynamics of the extended |
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variables. |
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|
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\begin{tabular}{llll} |
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{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf |
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default value} \\ |
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$T_{\mathrm{target}}$ & {\tt targetTemperature = 300;} & K & none \\ |
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$P_{\mathrm{target}}$ & {\tt targetPressure = 1;} & atm & none \\ |
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$\tau_T$ & {\tt tauThermostat = 1e3;} & fs & none \\ |
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$\tau_B$ & {\tt tauBarostat = 5e3;} & fs & none \\ |
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& {\tt resetTime = 200;} & fs & none \\ |
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& {\tt useInitialExtendedSystemState = ``true'';} & logical & |
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false |
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\end{tabular} |
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|
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Two additional keywords can be used to either clear the extended |
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system variables periodically ({\tt resetTime}), or to maintain the |
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state of the extended system variables between simulations ({\tt |
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useInitialExtendedSystemState}). More details on these variables |
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and their use in the integrators follows below. |
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|
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\subsubsection{\label{sec:noseHooverThermo}Nos\'{e}-Hoover Thermostatting} |
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|
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The Nos\'e-Hoover equations of motion are given by\cite{Hoover85} |
| 328 |
\begin{eqnarray} |
| 329 |
\dot{{\bf r}} & = & {\bf v} \\ |
| 330 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} \\ |
| 331 |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 332 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\ |
| 333 |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 334 |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 335 |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} |
| 336 |
\label{eq:nosehoovereom} |
| 337 |
\end{eqnarray} |
| 338 |
|
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$\chi$ is an ``extra'' variable included in the extended system, and |
| 340 |
it is propagated using the first order equation of motion |
| 341 |
\begin{equation} |
| 342 |
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( \frac{T}{T_{\mathrm{target}}} - 1 \right). |
| 343 |
\label{eq:nosehooverext} |
| 344 |
\end{equation} |
| 345 |
|
| 346 |
The instantaneous temperature $T$ is proportional to the total kinetic |
| 347 |
energy (both translational and orientational) and is given by |
| 348 |
\begin{equation} |
| 349 |
T = \frac{2 K}{f k_B} |
| 350 |
\end{equation} |
| 351 |
Here, $f$ is the total number of degrees of freedom in the system, |
| 352 |
\begin{equation} |
| 353 |
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}} |
| 354 |
\end{equation} |
| 355 |
and $K$ is the total kinetic energy, |
| 356 |
\begin{equation} |
| 357 |
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 358 |
\sum_{i=1}^{N_{\mathrm{orient}}} \sum_{\alpha=x,y,z} \frac{{\bf |
| 359 |
j}_{i\alpha}^T \cdot {\bf j}_{i\alpha}}{2 |
| 360 |
\overleftrightarrow{\mathsf{I}}_{i,\alpha \alpha}} |
| 361 |
\end{equation} |
| 362 |
|
| 363 |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
| 364 |
relaxation of the temperature to the target value. To set values for |
| 365 |
$\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use the |
| 366 |
{\tt tauThermostat} and {\tt targetTemperature} keywords in the {\tt |
| 367 |
.bass} file. The units for {\tt tauThermostat} are fs, and the units |
| 368 |
for the {\tt targetTemperature} are degrees K. The integration of |
| 369 |
the equations of motion is carried out in a velocity-Verlet style 2 |
| 370 |
part algorithm: |
| 371 |
|
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{\tt moveA:} |
| 373 |
\begin{eqnarray} |
| 374 |
T(t) & \leftarrow & \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} \\ |
| 375 |
{\bf v}\left(t + \delta t / 2\right) & \leftarrow & {\bf |
| 376 |
v}(t) + \frac{\delta t}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 377 |
\chi(t)\right) \\ |
| 378 |
{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf |
| 379 |
v}\left(t + \delta t / 2 \right) \\ |
| 380 |
{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
| 381 |
j}(t) + \frac{\delta t}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 382 |
\chi(t) \right) \\ |
| 383 |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left(\delta t * |
| 384 |
{\bf j}(t + \delta t / 2) \overleftrightarrow{\mathsf{I}}^{b} \right) \\ |
| 385 |
\chi\left(t + \delta t / 2 \right) & \leftarrow & \chi(t) + |
| 386 |
\frac{\delta t}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 387 |
\right) |
| 388 |
\end{eqnarray} |
| 389 |
|
| 390 |
Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic |
| 391 |
Trotter factorization of the three rotation operations that was |
| 392 |
discussed in the section on the DLM integrator. Note that this |
| 393 |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 394 |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 395 |
half time step, and positional degrees of freedom by a full time step. |
| 396 |
The new positions (and orientations) are then used to calculate a new |
| 397 |
set of forces and torques. |
| 398 |
|
| 399 |
{\tt doForces:} |
| 400 |
\begin{eqnarray} |
| 401 |
{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf |
| 402 |
r}(t + \delta t)} \\ |
| 403 |
{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t) |
| 404 |
\times \frac{\partial V}{\partial {\bf u}} \\ |
| 405 |
{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t) |
| 406 |
\cdot {\bf \tau}^s(t + \delta t) |
| 407 |
\end{eqnarray} |
| 408 |
|
| 409 |
Here ${\bf u}$ is a unit vector pointing along the principal axis of |
| 410 |
the rigid body being propagated, ${\bf \tau}^s$ is the torque in the |
| 411 |
space-fixed (laboratory) frame, and ${\bf \tau}^b$ is the torque in |
| 412 |
the body-fixed frame. ${\bf u}$ is automatically calculated when the |
| 413 |
rotation matrix $\mathsf{A}$ is calculated in {\tt moveA}. |
| 414 |
|
| 415 |
Once the forces and torques have been obtained at the new time step, |
| 416 |
the velocities can be advanced to the same time value. |
| 417 |
|
| 418 |
{\tt moveB:} |
| 419 |
\begin{eqnarray} |
| 420 |
T(t + \delta t) & \leftarrow & \left\{{\bf v}(t + \delta t)\right\}, |
| 421 |
\left\{{\bf j}(t + \delta t)\right\} \\ |
| 422 |
\chi\left(t + \delta t \right) & \leftarrow & \chi\left(t + \delta t / |
| 423 |
2 \right) + \frac{\delta t}{2 \tau_T^2} \left( \frac{T(t+\delta |
| 424 |
t)}{T_{\mathrm{target}}} - 1 \right) \\ |
| 425 |
{\bf v}\left(t + \delta t \right) & \leftarrow & {\bf |
| 426 |
v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( |
| 427 |
\frac{{\bf f}(t + \delta t)}{m} - {\bf v}(t + \delta t) |
| 428 |
\chi(t \delta t)\right) \\ |
| 429 |
{\bf j}\left(t + \delta t \right) & \leftarrow & {\bf |
| 430 |
j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( {\bf |
| 431 |
\tau}^b(t + \delta t) - {\bf j}(t + \delta t) |
| 432 |
\chi(t + \delta t) \right) |
| 433 |
\end{eqnarray} |
| 434 |
|
| 435 |
Since ${\bf v}(t + \delta t)$ and ${\bf j}(t + \delta t)$ are required |
| 436 |
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, they |
| 437 |
indirectly depend on their own values at time $t + \delta t$. {\tt |
| 438 |
moveB} is therefore done in an iterative fashion until $\chi(t + |
| 439 |
\delta t)$ becomes self-consistent. The relative tolerance for the |
| 440 |
self-consistency check defaults to a value of $\mbox{10}^{-6}$, but |
| 441 |
{\sc oopse} will terminate the iteration after 4 loops even if the |
| 442 |
consistency check has not been satisfied. |
| 443 |
|
| 444 |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for the |
| 445 |
extended system that is, to within a constant, identical to the |
| 446 |
Helmholtz free energy, |
| 447 |
\begin{equation} |
| 448 |
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 449 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t\prime) dt\prime |
| 450 |
\right) |
| 451 |
\end{equation} |
| 452 |
Poor choices of $\delta t$ or $\tau_T$ can result in non-conservation |
| 453 |
of $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 454 |
last column of the {\tt .stat} file to allow checks on the quality of |
| 455 |
the integration. |
| 456 |
|
| 457 |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 458 |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 459 |
algorithms are given in section \ref{sec:rattle}. |
| 460 |
|
| 461 |
\subsubsection{\label{sec:NPTi}Constant-pressure integration (isotropic box)} |
| 462 |
|
| 463 |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 464 |
implements the Melchionna modifications to the Nos\'e-Hoover-Andersen |
| 465 |
equations of motion,\cite{Melchionna93} |
| 466 |
|
| 467 |
\begin{eqnarray} |
| 468 |
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right) \\ |
| 469 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v} \\ |
| 470 |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 471 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\ |
| 472 |
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1} |
| 473 |
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 474 |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} \\ |
| 475 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 476 |
\frac{T}{T_{\mathrm{target}}} - 1 \right) \\ |
| 477 |
\dot{\eta} & = & \frac{1}{\tau_{B}^2} f k_B T_{\mathrm{target}} V \left( P - |
| 478 |
P_{\mathrm{target}} \right) \\ |
| 479 |
\dot{V} & = & 3 V \eta |
| 480 |
\label{eq:melchionna1} |
| 481 |
\end{eqnarray} |
| 482 |
|
| 483 |
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the extended |
| 484 |
system. $\chi$ is a thermostat, and it has the same function as it |
| 485 |
does in the Nos\'e-Hoover NVT integrator. $\eta$ is a barostat which |
| 486 |
controls changes to the volume of the simulation box. ${\bf R}_0$ is |
| 487 |
the location of the center of mass for the entire system. |
| 488 |
|
| 489 |
The NPTi integrator requires an instantaneous pressure. This quantity |
| 490 |
is calculated via the pressure tensor, |
| 491 |
\begin{equation} |
| 492 |
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{V(t)} \left( \sum_{i=1}^{N} |
| 493 |
m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
| 494 |
\overleftrightarrow{\mathsf{W}}(t) |
| 495 |
\end{equation} |
| 496 |
The kinetic contribution to the pressure tensor utilizes the {\it |
| 497 |
outer} product of the velocities denoted by the $\otimes$ symbol. The |
| 498 |
stress tensor is calculated from another outer product of the |
| 499 |
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
| 500 |
r}_i$) with the forces between the same two atoms, |
| 501 |
\begin{equation} |
| 502 |
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf r}_{ij}(t) |
| 503 |
\otimes {\bf f}_{ij}(t) |
| 504 |
\end{equation} |
| 505 |
The instantaneous pressure is then simply obtained from the trace of |
| 506 |
the Pressure tensor, |
| 507 |
\begin{equation} |
| 508 |
P(t) = \frac{1}{3} \mathrm{Tr} \left( \overleftrightarrow{\mathsf{P}}(t) |
| 509 |
\right) |
| 510 |
\end{equation} |
| 511 |
|
| 512 |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 513 |
relaxation of the pressure to the target value. To set values for |
| 514 |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
| 515 |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt .bass} |
| 516 |
file. The units for {\tt tauBarostat} are fs, and the units for the |
| 517 |
{\tt targetPressure} are atmospheres. Like in the NVT integrator, the |
| 518 |
integration of the equations of motion is carried out in a |
| 519 |
velocity-Verlet style 2 part algorithm: |
| 520 |
|
| 521 |
{\tt moveA:} |
| 522 |
\begin{eqnarray} |
| 523 |
T(t) & \leftarrow & \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} \\ |
| 524 |
{\bf v}\left(t + \delta t / 2\right) & \leftarrow & {\bf |
| 525 |
v}(t) + \frac{\delta t}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 526 |
\chi(t)\right) \\ |
| 527 |
{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf |
| 528 |
v}\left(t + \delta t / 2 \right) \\ |
| 529 |
{\bf j}\left(t + \delta t / 2 \right) & \leftarrow & {\bf |
| 530 |
j}(t) + \frac{\delta t}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 531 |
\chi(t) \right) \\ |
| 532 |
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}\left(\delta t * |
| 533 |
{\bf j}(t + \delta t / 2) \overleftrightarrow{\mathsf{I}}^{b} \right) \\ |
| 534 |
\chi\left(t + \delta t / 2 \right) & \leftarrow & \chi(t) + |
| 535 |
\frac{\delta t}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 536 |
\right) |
| 537 |
\end{eqnarray} |
| 538 |
|
| 539 |
Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic |
| 540 |
Trotter factorization of the three rotation operations that was |
| 541 |
discussed in the section on the DLM integrator. Note that this |
| 542 |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 543 |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 544 |
half time step, and positional degrees of freedom by a full time step. |
| 545 |
The new positions (and orientations) are then used to calculate a new |
| 546 |
set of forces and torques. |
| 547 |
|
| 548 |
{\tt doForces:} |
| 549 |
\begin{eqnarray} |
| 550 |
{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf |
| 551 |
r}(t + \delta t)} \\ |
| 552 |
{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t) |
| 553 |
\times \frac{\partial V}{\partial {\bf u}} \\ |
| 554 |
{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t) |
| 555 |
\cdot {\bf \tau}^s(t + \delta t) |
| 556 |
\end{eqnarray} |
| 557 |
|
| 558 |
Here ${\bf u}$ is a unit vector pointing along the principal axis of |
| 559 |
the rigid body being propagated, ${\bf \tau}^s$ is the torque in the |
| 560 |
space-fixed (laboratory) frame, and ${\bf \tau}^b$ is the torque in |
| 561 |
the body-fixed frame. ${\bf u}$ is automatically calculated when the |
| 562 |
rotation matrix $\mathsf{A}$ is calculated in {\tt moveA}. |
| 563 |
|
| 564 |
Once the forces and torques have been obtained at the new time step, |
| 565 |
the velocities can be advanced to the same time value. |
| 566 |
|
| 567 |
{\tt moveB:} |
| 568 |
\begin{eqnarray} |
| 569 |
T(t + \delta t) & \leftarrow & \left\{{\bf v}(t + \delta t)\right\}, |
| 570 |
\left\{{\bf j}(t + \delta t)\right\} \\ |
| 571 |
\chi\left(t + \delta t \right) & \leftarrow & \chi\left(t + \delta t / |
| 572 |
2 \right) + \frac{\delta t}{2 \tau_T^2} \left( \frac{T(t+\delta |
| 573 |
t)}{T_{\mathrm{target}}} - 1 \right) \\ |
| 574 |
{\bf v}\left(t + \delta t \right) & \leftarrow & {\bf |
| 575 |
v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( |
| 576 |
\frac{{\bf f}(t + \delta t)}{m} - {\bf v}(t + \delta t) |
| 577 |
\chi(t \delta t)\right) \\ |
| 578 |
{\bf j}\left(t + \delta t \right) & \leftarrow & {\bf |
| 579 |
j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( {\bf |
| 580 |
\tau}^b(t + \delta t) - {\bf j}(t + \delta t) |
| 581 |
\chi(t + \delta t) \right) |
| 582 |
\end{eqnarray} |
| 583 |
|
| 584 |
Since ${\bf v}(t + \delta t)$ and ${\bf j}(t + \delta t)$ are required |
| 585 |
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, they |
| 586 |
indirectly depend on their own values at time $t + \delta t$. {\tt |
| 587 |
moveB} is therefore done in an iterative fashion until $\chi(t + |
| 588 |
\delta t)$ becomes self-consistent. The relative tolerance for the |
| 589 |
self-consistency check defaults to a value of $\mbox{10}^{-6}$, but |
| 590 |
{\sc oopse} will terminate the iteration after 4 loops even if the |
| 591 |
consistency check has not been satisfied. |
| 592 |
|
| 593 |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for the |
| 594 |
extended system that is, to within a constant, identical to the |
| 595 |
Helmholtz free energy, |
| 596 |
\begin{equation} |
| 597 |
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 598 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t\prime) dt\prime |
| 599 |
\right) |
| 600 |
\end{equation} |
| 601 |
Poor choices of $\delta t$ or $\tau_T$ can result in non-conservation |
| 602 |
of $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 603 |
last column of the {\tt .stat} file to allow checks on the quality of |
| 604 |
the integration. |
| 605 |
|
| 606 |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 607 |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 608 |
algorithms are given in section \ref{sec:rattle}. |
| 609 |
|
| 610 |
|
| 611 |
|
| 612 |
\subsection{\label{Sec:zcons}Z-Constraint Method} |
| 613 |
|
| 614 |
Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation |
| 615 |
method was developed to investigate the dynamics of ions inside the ion |
| 616 |
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated |
| 617 |
from the deviation of the instaneous force from its mean force. |
| 618 |
|
| 619 |
% |
| 620 |
|
| 621 |
\begin{equation} |
| 622 |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
| 623 |
\end{equation} |
| 624 |
|
| 625 |
|
| 626 |
where% |
| 627 |
\begin{equation} |
| 628 |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
| 629 |
\end{equation} |
| 630 |
|
| 631 |
|
| 632 |
If the time-dependent friction decay rapidly, static friction coefficient can |
| 633 |
be approximated by% |
| 634 |
|
| 635 |
\begin{equation} |
| 636 |
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt |
| 637 |
\end{equation} |
| 638 |
|
| 639 |
|
| 640 |
Hence, diffusion constant can be estimated by |
| 641 |
\begin{equation} |
| 642 |
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 643 |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}% |
| 644 |
\end{equation} |
| 645 |
|
| 646 |
|
| 647 |
\bigskip Z-Constraint method, which fixed the z coordinates of the molecules |
| 648 |
with respect to the center of the mass of the system, was proposed to obtain |
| 649 |
the forces required in force auto-correlation method.\cite{Marrink94} However, |
| 650 |
simply resetting the coordinate will move the center of the mass of the whole |
| 651 |
system. To avoid this problem, a new method was used at {\sc oopse}. Instead of |
| 652 |
resetting the coordinate, we reset the forces of z-constraint molecules as |
| 653 |
well as subtract the total constraint forces from the rest of the system after |
| 654 |
force calculation at each time step. |
| 655 |
\begin{verbatim} |
| 656 |
$F_{\alpha i}=0$ |
| 657 |
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}$ |
| 658 |
$F_{\alpha i}=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}$ |
| 659 |
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}$ |
| 660 |
\end{verbatim} |
| 661 |
|
| 662 |
At the very beginning of the simulation, the molecules may not be at its |
| 663 |
constraint position. To move the z-constraint molecule to the specified |
| 664 |
position, a simple harmonic potential is used% |
| 665 |
|
| 666 |
\begin{equation} |
| 667 |
U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}% |
| 668 |
\end{equation} |
| 669 |
where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is |
| 670 |
current z coordinate of the center of mass of the z-constraint molecule, and |
| 671 |
$z_{cons}$ is the restraint position. Therefore, the harmonic force operated |
| 672 |
on the z-constraint molecule at time $t$ can be calculated by% |
| 673 |
\begin{equation} |
| 674 |
F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}% |
| 675 |
(z(t)-z_{cons}) |
| 676 |
\end{equation} |
| 677 |
Worthy of mention, other kinds of potential functions can also be used to |
| 678 |
drive the z-constraint molecule. |
