trunk/oopsePaper/Mechanics.tex


\section{\label{sec:mechanics}Mechanics}

\subsection{\label{integrate}Integrating the Equations of Motion: the
DLM method}

The default method for integrating the equations of motion in {\sc
oopse} is a velocity-Verlet version of the symplectic splitting method
proposed by Dullweber, Leimkuhler and McLachlan
(DLM).\cite{Dullweber1997} When there are no directional atoms or
rigid bodies present in the simulation, this integrator becomes the
standard velocity-Verlet integrator which is known to sample the
microcanonical (NVE) ensemble.\cite{}

Previous integration methods for orientational motion have problems
that are avoided in the DLM method.  Direct propagation of the Euler
angles has a known $1/\sin\theta$ divergence in the equations of
motion for $\phi$ and $\psi$,\cite{AllenTildesley} leading to
numerical instabilities any time one of the directional atoms or rigid
bodies has an orientation near $\theta=0$ or $\theta=\pi$.  More
modern quaternion-based integration methods have relatively poor
energy conservation.  While quaternions work well for orientational
motion in other ensembles, the microcanonical ensemble has a
constant energy requirement that is quite sensitive to errors in the
equations of motion.  An earlier implementation of {\sc oopse}
utilized quaternions for propagation of rotational motion; however, a
detailed investigation showed that they resulted in a steady drift in
the total energy, something that has been observed by
others.\cite{Laird97}      

The key difference in the integration method proposed by Dullweber
\emph{et al.} is that the entire rotation matrix is propagated from
one time step to the next. In the past, this would not have been
feasible, since that the rotation matrix for a single body has nine
elements compared to 3 or 4 elements for Euler angles and quaternions
respectively. System memory has become less costly in recent times,
and this can be translated into substantial benefits in energy
conservation.

The basic equations of motion being integrated are derived from the
Hamiltonian for conservative systems containing rigid bodies,
\begin{equation}
H = \sum_{i} \left( \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i +
\frac{1}{2} {\bf j}_i^T \cdot \overleftrightarrow{\mathsf{I}}_i^{-1} \cdot
{\bf j}_i \right) +
V\left(\left\{{\bf r}\right\}, \left\{\mathsf{A}\right\}\right)
\end{equation}
where ${\bf r}_i$ and ${\bf v}_i$ are the cartesian position vector
and velocity of the center of mass of particle $i$, and ${\bf j}_i$
and $\overleftrightarrow{\mathsf{I}}_i$ are the body-fixed angular
momentum and moment of inertia tensor, respectively.  $\mathsf{A}_i$
is the $3 \times 3$ rotation matrix describing the instantaneous
orientation of the particle.  $V$ is the potential energy function
which may depend on both the positions $\left\{{\bf r}\right\}$ and
orientations $\left\{\mathsf{A}\right\}$ of all particles.  The
equations of motion for the particle centers of mass are derived from
Hamilton's equations and are quite simple,
\begin{eqnarray}
\dot{{\bf r}} & = & {\bf v} \\
\dot{{\bf v}} & = & \frac{{\bf f}}{m}
\end{eqnarray}
where ${\bf f}$ is the instantaneous force on the center of mass
of the particle,
\begin{equation}
{\bf f} = - \frac{\partial}{\partial
{\bf r}} V(\left\{{\bf r}(t)\right\}, \left\{\mathsf{A}(t)\right\}).
\end{equation}

The equations of motion for the orientational degrees of freedom are
\begin{eqnarray}
\dot{\mathsf{A}} & = & \mathsf{A} \cdot
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) \\
\dot{{\bf j}} & = & {\bf j} \times \left( \overleftrightarrow{I}^{-1}
\cdot {\bf j} \right) - \mbox{ rot}\left(\mathsf{A}^{T} \cdot \frac{\partial
V}{\partial \mathsf{A}} \right)
\end{eqnarray}
In these equations of motion, the $\mbox{skew}$ matrix of a vector
${\bf v} = \left( v_1, v_2, v_3 \right)$ is defined:
\begin{equation}
\mbox{skew}\left( {\bf v} \right) := \left( 
\begin{array}{ccc}
0 & v_3 & - v_2 \\
-v_3 & 0 & v_1 \\
v_2 & -v_1 & 0 
\end{array}
\right)
\end{equation}
The $\mbox{rot}$ notation refers to the mapping of the $3 \times 3$
rotation matrix to a vector of orientations by first computing the
skew-symmetric part $\left(\mathsf{A} - \mathsf{A}^{T}\right)$ and
then associating this with a length 3 vector by inverting the
$\mbox{skew}$ function above:
\begin{equation}
\mbox{rot}\left(\mathsf{A}\right) := \mbox{ skew}^{-1}\left(\mathsf{A}
- \mathsf{A}^{T} \right)
\end{equation}
Written this way, the $\mbox{rot}$ operation creates a set of
conjugate angle coordinates to the body-fixed angular momenta
represented by ${\bf j}$.  This equation of motion for angular momenta
is equivalent to the more familiar body-fixed form:
\begin{eqnarray}
\dot{j_{x}} & = & \tau^b_x(t)  +
\left(\overleftrightarrow{\mathsf{I}}_{yy} - \overleftrightarrow{\mathsf{I}}_{zz} \right) j_y j_z \\
\dot{j_{y}} & = & \tau^b_y(t) +
\left(\overleftrightarrow{\mathsf{I}}_{zz} - \overleftrightarrow{\mathsf{I}}_{xx} \right) j_z j_x \\
\dot{j_{z}} & = & \tau^b_z(t) +
\left(\overleftrightarrow{\mathsf{I}}_{xx} - \overleftrightarrow{\mathsf{I}}_{yy} \right) j_x j_y 
\end{eqnarray}
which utilize the body-fixed torques, ${\bf \tau}^b$.  Torques are
most easily derived in the space-fixed frame, 
\begin{equation}
{\bf \tau}^b(t) = \mathsf{A}(t) \cdot {\bf \tau}^s(t)
\end{equation}
where 
\begin{equation}
{\bf \tau}^s(t) = - \hat{\bf u}(t) \times \left( \frac{\partial}
{\partial \hat{\bf u}} V\left(\left\{ {\bf r}(t) \right\}, \left\{
\mathsf{A}(t) \right\}\right) \right)
\end{equation}
where $\hat{\bf u}$ is a unit vector pointing along the principal axis
of the particle in the space-fixed frame.

The DLM method uses a Trotter factorization of the orientational
propagator.  This has three effects:
\begin{enumerate}
\item the integrator is area preserving in phase space (i.e. it is
{\it symplectic}),
\item the integrator is time-{\it reversible}, making it suitable for Hybrid
Monte Carlo applications, and
\item the error for a single time step is of order $O\left(h^3\right)$
for timesteps of length $h$.
\end{enumerate}

In the integration method, the
orientational propagation involves a sequence of matrix evaluations to
update the rotation matrix.\cite{Dullweber1997} These matrix rotations
end up being more costly computationally than the simpler arithmetic
quaternion propagation. With the same time step, a 1000 SSD particle
simulation shows an average 7\% increase in computation time using the
symplectic step method in place of quaternions. This cost is more than
justified when comparing the energy conservation of the two methods as
illustrated in figure
\ref{timestep}.

\begin{figure}
\epsfxsize=6in
\epsfbox{timeStep.epsi}
\caption{Energy conservation using quaternion based integration versus 
the symplectic step method proposed by Dullweber \emph{et al.} with
increasing time step. For each time step, the dotted line is total
energy using the symplectic step integrator, and the solid line comes
from the quaternion integrator. The larger time step plots are shifted
up from the true energy baseline for clarity.}
\label{timestep}
\end{figure}

In figure \ref{timestep}, the resulting energy drift at various time
steps for both the symplectic step and quaternion integration schemes
is compared. All of the 1000 SSD particle simulations started with the
same configuration, and the only difference was the method for
handling rotational motion. At time steps of 0.1 and 0.5 fs, both
methods for propagating particle rotation conserve energy fairly well,
with the quaternion method showing a slight energy drift over time in
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the
energy conservation benefits of the symplectic step method are clearly
demonstrated. Thus, while maintaining the same degree of energy
conservation, one can take considerably longer time steps, leading to
an overall reduction in computation time.

Energy drift in these SSD particle simulations was unnoticeable for
time steps up to three femtoseconds. A slight energy drift on the
order of 0.012 kcal/mol per nanosecond was observed at a time step of
four femtoseconds, and as expected, this drift increases dramatically
with increasing time step. To insure accuracy in the constant energy
simulations, time steps were set at 2 fs and kept at this value for
constant pressure simulations as well.


\subsection{\label{sec:extended}Extended Systems for other Ensembles}

{\sc oopse} implements a number of extended system integrators for
sampling from other ensembles relevant to chemical physics.  The
integrator can selected with the {\tt ensemble} keyword in the
{\tt .bass} file:

\begin{center}
\begin{tabular}{lll}
{\bf Integrator} & {\bf Ensemble} & {\bf {\tt .bass} line} \\
NVE & microcanonical & {\tt ensemble = ``NVE''; } \\
NVT & canonical & {\tt ensemble = ``NVT''; } \\
NPTi & isobaric-isothermal (with isotropic volume changes) & {\tt
ensemble = ``NPTi'';} \\
NPTf & isobaric-isothermal (with changes to box shape) & {\tt
ensemble = ``NPTf'';} \\
NPTxyz & approximate isobaric-isothermal & {\tt ensemble =
``NPTxyz'';} \\
 &  (with separate barostats on each box dimension) &
\end{tabular}
\end{center}

The relatively well-known Nos\'e-Hoover thermostat is implemented in
{\sc oopse}'s NVT integrator.  This method couples an extra degree of
freedom (the thermostat) to the kinetic energy of the system, and has
been shown to sample the canonical distribution in the system degrees
of freedom while conserving a quantity that is, to within a constant,
the Helmholtz free energy.

NPT algorithms attempt to maintain constant pressure in the system by
coupling the volume of the system to a barostat.  {\sc oopse} contains
three different constant pressure algorithms.  The first two, NPTi and
NPTf have been shown to conserve a quantity that is, to within a
constant, the Gibbs free energy.  The Melchionna modification to the
Hoover barostat is implemented in both NPTi and NPTf.  NPTi allows
only isotropic changes in the simulation box, while box {\it shape}
variations are allowed in NPTf.  The NPTxyz integrator has {\it not}
been shown to sample from the isobaric-isothermal ensemble.  It is
useful, however, in that it maintains orthogonality for the axes of
the simulation box while attempting to equalize pressure along the
three perpendicular directions in the box.

Each of the extended system integrators requires additional keywords
to set target values for the thermodynamic state variables that are
being held constant.  Keywords are also required to set the
characteristic decay times for the dynamics of the extended
variables.

\begin{tabular}{llll}
{\bf variable} & {\bf {\tt .bass} keyword} & {\bf units} & {\bf
default value} \\  
$T_{\mathrm{target}}$ & {\tt targetTemperature = 300;} &  K & none \\
$P_{\mathrm{target}}$ & {\tt targetPressure = 1;} & atm & none \\
$\tau_T$ & {\tt tauThermostat = 1e3;} & fs & none \\
$\tau_B$ & {\tt tauBarostat = 5e3;} & fs  & none \\
         & {\tt resetTime = 200;} & fs & none \\
         & {\tt useInitialExtendedSystemState = ``true'';} & logical &
false
\end{tabular}

Two additional keywords can be used to either clear the extended
system variables periodically ({\tt resetTime}), or to maintain the
state of the extended system variables between simulations ({\tt
useInitialExtendedSystemState}).  More details on these variables
and their use in the integrators follows below.

\subsubsection{\label{sec:noseHooverThermo}Nos\'{e}-Hoover Thermostatting}

The Nos\'e-Hoover equations of motion are given by\cite{Hoover85}
\begin{eqnarray}
\dot{{\bf r}} & = & {\bf v} \\
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} \\
\dot{\mathsf{A}} & = & \\
\dot{{\bf j}} & = & - \chi {\bf j}
\label{eq:nosehoovereom}
\end{eqnarray}

$\chi$ is an ``extra'' variable included in the extended system, and
it is propagated using the first order equation of motion
\begin{equation}
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( \frac{T}{T_{\mathrm{target}}} - 1 \right).
\label{eq:nosehooverext}
\end{equation}

The instantaneous temperature $T$ is proportional to the total kinetic
energy (both translational and orientational) and is given by
\begin{equation}
T = \frac{2 K}{f k_B}
\end{equation}
Here, $f$ is the total number of degrees of freedom in the system,
\begin{equation}
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}
\end{equation}
and $K$ is the total kinetic energy,
\begin{equation}
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i +
\sum_{i=1}^{N_{\mathrm{orient}}} \sum_{\alpha=x,y,z} \frac{{\bf
j}_{i\alpha}^T \cdot {\bf j}_{i\alpha}}{2
\overleftrightarrow{\mathsf{I}}_{i,\alpha \alpha}}
\end{equation}

In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for
relaxation of the temperature to the target value.  To set values for
$\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use the
{\tt tauThermostat} and {\tt targetTemperature} keywords in the {\tt
.bass} file.  The units for {\tt tauThermostat} are fs, and the units
for the {\tt targetTemperature} are degrees K.   The integration of
the equations of motion is carried out in a velocity-Verlet style 2
part algorithm:

{\tt moveA:}
\begin{eqnarray}
T(t) & \leftarrow & \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} \\
{\bf v}\left(t + \delta t / 2\right)  & \leftarrow & {\bf
v}(t) + \frac{\delta t}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t)
\chi(t)\right) \\
{\bf r}(t + \delta t) & \leftarrow & {\bf r}(t) + \delta t {\bf
v}\left(t + \delta t / 2 \right) \\
{\bf j}\left(t + \delta t / 2 \right)  & \leftarrow & {\bf
j}(t) + \frac{\delta t}{2} \left( {\bf \tau}^b(t) - {\bf j}(t)
\chi(t) \right) \\
\mathsf{A}(t + \delta t) & \leftarrow & \mathrm{rot}({\bf j}(t +
\delta t / 2) \overleftrightarrow{\mathsf{I}}^{b},
\delta t) \\
\chi\left(t + \delta t / 2 \right) & \leftarrow & \chi(t) +
\frac{\delta t}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1
\right) 
\end{eqnarray}

Here $\mathrm{rot}( {\bf j} / {\bf I} )$ is the same symplectic Trotter
factorization of the three rotation operations that was discussed in
the section on the DLM integrator.  Note that this operation modifies
both the rotation matrix $\mathsf{A}$ and the angular momentum ${\bf
j}$.  {\tt moveA} propagates velocities by a half time step, and
positional degrees of freedom by a full time step.  The new positions
(and orientations) are then used to calculate a new set of forces and
torques.

{\tt doForces:}
\begin{eqnarray}
{\bf f}(t + \delta t) & \leftarrow & - \frac{\partial V}{\partial {\bf
r}(t + \delta t)} \\
{\bf \tau}^{s}(t + \delta t) & \leftarrow & {\bf u}(t + \delta t)
\times \frac{\partial V}{\partial {\bf u}} \\
{\bf \tau}^{b}(t + \delta t) & \leftarrow & \mathsf{A}(t + \delta t)
\cdot {\bf \tau}^s(t + \delta t) 
\end{eqnarray}

Here ${\bf u}$ is a unit vector pointing along the principal axis of
the rigid body being propagated, ${\bf \tau}^s$  is the torque in the
space-fixed (laboratory) frame, and ${\bf \tau}^b$ is the torque in
the body-fixed frame.  ${\bf u}$ is automatically calculated when the
rotation matrix $\mathsf{A}$ is calculated in {\tt moveA}.  

Once the forces and torques have been obtained at the new time step,
the velocities can be advanced to the same time value.

{\tt moveB:}
\begin{eqnarray}
T(t + \delta t) & \leftarrow & \left\{{\bf v}(t + \delta t)\right\},
\left\{{\bf j}(t + \delta t)\right\} \\
\chi\left(t + \delta t \right) & \leftarrow & \chi\left(t + \delta t /
2 \right) + \frac{\delta t}{2 \tau_T^2} \left( \frac{T(t+\delta
t)}{T_{\mathrm{target}}} - 1 \right) \\
{\bf v}\left(t + \delta t \right)  & \leftarrow & {\bf
v}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left(
\frac{{\bf f}(t + \delta t)}{m} - {\bf v}(t + \delta t)
\chi(t \delta t)\right) \\
{\bf j}\left(t + \delta t \right)  & \leftarrow & {\bf
j}\left(t + \delta t / 2 \right) + \frac{\delta t}{2} \left( {\bf
\tau}^b(t + \delta t) - {\bf j}(t + \delta t)
\chi(t + \delta t) \right) 
\end{eqnarray}

Since ${\bf v}(t + \delta t)$ and ${\bf j}(t + \delta t)$ are required
to caclculate $T(t + \delta t)$ as well as $\chi(t + \delta t)$, the
indirectly depend on their own values at time $t + \delta t$.  {\tt
moveB} is therefore done in an iterative fashion until $\chi(t +
\delta t)$ becomes self-consistent.  The relative tolerance for the
self-consistency check defaults to a value of $\mbox{10}^{-6}$, but
{\sc oopse} will terminate the iteration after 4 loops even if the
consistency check has not been satisfied.

The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for the
extended system that is, to within a constant, identical to the
Helmholtz free energy,
\begin{equation}
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left(
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t\prime) dt\prime
\right)
\end{equation}
Poor choices of $\delta t$ or $\tau_T$ can result in non-conservation
of $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the
last column of the {\tt .stat} file to allow checks on the quality of
the integration.

Bond constraints are applied at the end of both the {\tt moveA} and
{\tt moveB} portions of the algorithm.  Details on the constraint
algorithms are given in section \ref{sec:rattle}.

\subsubsection{\label{sec:NPTi}Constant-pressure integration (isotropic box)}


\subsection{\label{Sec:zcons}Z-Constraint Method}

Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation
method was developed to investigate the dynamics of ions inside the ion
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated
from the deviation of the instaneous force from its mean force.

%

\begin{equation}
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T
\end{equation}


where%
\begin{equation}
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle
\end{equation}


If the time-dependent friction decay rapidly, static friction coefficient can
be approximated by%

\begin{equation}
\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt
\end{equation}


Hence, diffusion constant can be estimated by
\begin{equation}
D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}%
\end{equation}


\bigskip Z-Constraint method, which fixed the z coordinates of the molecules
with respect to the center of the mass of the system, was proposed to obtain
the forces required in force auto-correlation method.\cite{Marrink94} However,
simply resetting the coordinate will move the center of the mass of the whole
system. To avoid this problem,  a new method was used at {\sc oopse}. Instead of
resetting the coordinate, we reset the forces of z-constraint molecules as
well as subtract the total constraint forces from the rest of the system after
force calculation at each time step. 
\begin{verbatim}
$F_{\alpha i}=0$
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}$
$F_{\alpha i}=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}$
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}$
\end{verbatim}

At the very beginning of the simulation, the molecules may not be at its
constraint position. To move the z-constraint molecule to the specified
position, a simple harmonic potential is used%

\begin{equation}
U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}%
\end{equation}
where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is
current z coordinate of the center of mass of the z-constraint molecule, and
$z_{cons}$ is the restraint position. Therefore, the harmonic force operated
on the z-constraint molecule at time $t$ can be calculated by%
\begin{equation}
F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}%
(z(t)-z_{cons})
\end{equation}
Worthy of mention, other kinds of potential functions can also be used to
drive the z-constraint molecule.
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