total mass :math:`M = \sum_{i=1}^N m_i`. The rotation matrix
:math:`\mathbf{\Omega}(t)` is
-.. math::
-
- \mathbf{\Omega}(t) =
- \left(
- \begin{array}{ccc}
- \cos\omega t + v_x^2{\,\xi\,}& v_x v_y{\,\xi\,}- v_z\sin\omega t & v_x v_z{\,\xi\,}+ v_y\sin\omega t\\
- v_x v_y{\,\xi\,}+ v_z\sin\omega t & \cos\omega t + v_y^2{\,\xi\,}& v_y v_z{\,\xi\,}- v_x\sin\omega t\\
- v_x v_z{\,\xi\,}- v_y\sin\omega t & v_y v_z{\,\xi\,}+ v_x\sin\omega t & \cos\omega t + v_z^2{\,\xi\,}\\
- \end{array}
- \right)
+.. math:: \mathbf{\Omega}(t) =
+ \left(
+ \begin{array}{ccc}
+ \cos\omega t + v_x^2{\,\xi\,}& v_x v_y{\,\xi\,}- v_z\sin\omega t & v_x v_z{\,\xi\,}+ v_y\sin\omega t\\
+ v_x v_y{\,\xi\,}+ v_z\sin\omega t & \cos\omega t + v_y^2{\,\xi\,}& v_y v_z{\,\xi\,}- v_x\sin\omega t\\
+ v_x v_z{\,\xi\,}- v_y\sin\omega t & v_y v_z{\,\xi\,}+ v_x\sin\omega t & \cos\omega t + v_z^2{\,\xi\,}\\
+ \end{array}
+ \right)
+ :label: eqnrotmat
where :math:`v_x`, :math:`v_y`, and :math:`v_z` are the components of
the normalized rotation vector :math:`\hat{\mathbf{v}}`,
(eqns. :eq:`%s <eqnpotiso>` and :eq:`%s <eqnpotisopf>`) also contain components parallel to the
rotation axis and thereby restrain motions along the axis of either the
whole rotation group (in case of :math:`V^\mathrm{iso}`) or within the
-rotation group, in case of
-
-.. math:: V^\mathrm{iso-pf}
+rotation group, in case of :math:`V^\mathrm{iso-pf}`.
-For cases where
-unrestrained motion along the axis is preferred, we have implemented a
+For cases where unrestrained motion along the axis is preferred, we have implemented a
“parallel motion” variant by eliminating all components parallel to the
rotation axis for the potential. This is achieved by projecting the
distance vectors between reference and actual positions
.. math:: \mathbf{r}_i = \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{u}) - (\mathbf{x}_i - \mathbf{u})
+ :label: eqnrotdistvectors
onto the plane perpendicular to the rotation vector,
parallel motion potential. With
.. math:: \mathbf{s}_i = \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{y}_c^0) - (\mathbf{x}_i - \mathbf{x}_c)
+ :label: eqnparrallelpotential
the respective potential and forces are
with
-.. math::
-
- \mathbf{p}_i :=
- \frac{\hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{u})} {\| \hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{u})\|} \ .
+.. math:: \mathbf{p}_i :=
+ \frac{\hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{u})} {\| \hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{u})\|} \ .
+ :label: eqnpotrmpart2
This variant depends only on the distance
:math:`\mathbf{p}_i \cdot (\mathbf{x}_i -
Proceeding similar to the pivot-free isotropic potential yields a
pivot-free version of the above potential. With
-.. math::
-
- \mathbf{q}_i :=
- \frac{\hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{y}_c^0)} {\| \hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{y}_c^0)\|} \, ,
+.. math:: \mathbf{q}_i :=
+ \frac{\hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{y}_c^0)} {\| \hat{\mathbf{v}}\times \mathbf{\Omega}(t) (\mathbf{y}_i^0 - \mathbf{y}_c^0)\|} \, ,
+ :label: eqnpotrmpfpart1
the potential and force for the pivot-free variant of the radial motion
potential read
for :math:`\epsilon'\mathrm{ = }0\mathrm{nm}^2`
(:numref:`Fig. %s C <fig-equipotential>`). With
-.. math::
-
- \begin{aligned}
- \mathbf{r}_i & := & \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{u})\\
- \mathbf{s}_i & := & \frac{\hat{\mathbf{v}} \times (\mathbf{x}_i -
- \mathbf{u} ) }{ \| \hat{\mathbf{v}} \times (\mathbf{x}_i - \mathbf{u})
- \| } \equiv \; \Psi_{i} \;\; {\hat{\mathbf{v}} \times
- (\mathbf{x}_i-\mathbf{u} ) }\\
- \Psi_i^{*} & := & \frac{1}{ \| \hat{\mathbf{v}} \times
- (\mathbf{x}_i-\mathbf{u}) \|^2 + \epsilon'}\end{aligned}
+.. math:: \begin{aligned}
+ \mathbf{r}_i & := & \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{u})\\
+ \mathbf{s}_i & := & \frac{\hat{\mathbf{v}} \times (\mathbf{x}_i -
+ \mathbf{u} ) }{ \| \hat{\mathbf{v}} \times (\mathbf{x}_i - \mathbf{u})
+ \| } \equiv \; \Psi_{i} \;\; {\hat{\mathbf{v}} \times
+ (\mathbf{x}_i-\mathbf{u} ) }\\
+ \Psi_i^{*} & := & \frac{1}{ \| \hat{\mathbf{v}} \times
+ (\mathbf{x}_i-\mathbf{u}) \|^2 + \epsilon'}\end{aligned}
+ :label: eqnpotrm2forcepart1
the force on atom :math:`j` reads
With
-.. math::
-
- \begin{aligned}
- \mathbf{r}_i & := & \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c)\\
- \mathbf{s}_i & := & \frac{\hat{\mathbf{v}} \times (\mathbf{x}_i -
- \mathbf{x}_c ) }{ \| \hat{\mathbf{v}} \times (\mathbf{x}_i - \mathbf{x}_c)
- \| } \equiv \; \Psi_{i} \;\; {\hat{\mathbf{v}} \times
- (\mathbf{x}_i-\mathbf{x}_c ) }\\ \Psi_i^{*} & := & \frac{1}{ \| \hat{\mathbf{v}} \times
- (\mathbf{x}_i-\mathbf{x}_c) \|^2 + \epsilon'}\end{aligned}
+.. math:: \begin{aligned}
+ \mathbf{r}_i & := & \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c)\\
+ \mathbf{s}_i & := & \frac{\hat{\mathbf{v}} \times (\mathbf{x}_i -
+ \mathbf{x}_c ) }{ \| \hat{\mathbf{v}} \times (\mathbf{x}_i - \mathbf{x}_c)
+ \| } \equiv \; \Psi_{i} \;\; {\hat{\mathbf{v}} \times
+ (\mathbf{x}_i-\mathbf{x}_c ) }\\ \Psi_i^{*} & := & \frac{1}{ \| \hat{\mathbf{v}} \times
+ (\mathbf{x}_i-\mathbf{x}_c) \|^2 + \epsilon'}\end{aligned}
+ :label: eqnpotrm2pfpart2
the force on atom :math:`j` reads
between adjacent slabs, and
.. math:: \beta_n(\mathbf{x}_i) := \mathbf{x}_i \cdot \hat{\mathbf{v}} - n \, \Delta x \, .
+ :label: eqngaussianpart2
.. _fig-gaussian:
A most convenient choice is :math:`\sigma = 0.7 \Delta x` and
-.. math::
-
- 1/\Gamma = \sum_{n \in Z}
- \mbox{exp}
- \left(-\frac{(n - \frac{1}{4})^2}{2\cdot 0.7^2}\right)
- \approx 1.75464 \, ,
+.. math:: 1/\Gamma = \sum_{n \in Z}
+ \mbox{exp}
+ \left(-\frac{(n - \frac{1}{4})^2}{2\cdot 0.7^2}\right)
+ \approx 1.75464 \, ,
+ :label: eqngaussianpart3
which yields a nearly constant sum, essentially independent of
:math:`\mathbf{x}_i` (dashed line in
:math:`\mathbf{x}_c^n`, the center of mass of the slab.
With
-.. math::
-
- \begin{aligned}
- \mathbf{q}_i^n & := & \frac{\hat{\mathbf{v}} \times
- \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c^n) }{ \| \hat{\mathbf{v}}
- \times \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c^n) \| } \\
- b_i^n & := & \mathbf{q}_i^n \cdot (\mathbf{x}_i - \mathbf{x}_c^n) \, ,\end{aligned}
+.. math:: \begin{aligned}
+ \mathbf{q}_i^n & := & \frac{\hat{\mathbf{v}} \times
+ \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c^n) }{ \| \hat{\mathbf{v}}
+ \times \mathbf{\Omega}(t)(\mathbf{y}_i^0 - \mathbf{y}_c^n) \| } \\
+ b_i^n & := & \mathbf{q}_i^n \cdot (\mathbf{x}_i - \mathbf{x}_c^n) \, ,\end{aligned}
+ :label: eqnflexpotpart2
the resulting force on atom :math:`j` reads
\partial \mathbf{x}_c / \partial y = \partial \mathbf{x}_c / \partial z = 0`.
The resulting force error is small (of order :math:`O(1/N)` or
:math:`O(m_j/M)` if mass-weighting is applied) and can therefore be
-tolerated. With this assumption, the forces
-
-.. math::
- \mathbf{F}^\mathrm{flex-t}
-
-have the same form as
-eqn. :eq:`%s <eqnpotflexforce>`.
+tolerated. With this assumption, the forces :math:`\mathbf{F}^\mathrm{flex-t}`
+have the same form as eqn. :eq:`%s <eqnpotflexforce>`.
Flexible Axis 2 Alternative Potential
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
eqn. :eq:`%s <eqnproject>` for the definition of
:math:`\perp`),
-.. math::
-
- \cos \theta_i =
- \frac{(\mathbf{y}_i-\mathbf{u})^\perp \cdot (\mathbf{x}_i-\mathbf{u})^\perp}
- { \| (\mathbf{y}_i-\mathbf{u})^\perp \cdot (\mathbf{x}_i-\mathbf{u})^\perp
- \| } \ .
+.. math:: \cos \theta_i =
+ \frac{(\mathbf{y}_i-\mathbf{u})^\perp \cdot (\mathbf{x}_i-\mathbf{u})^\perp}
+ { \| (\mathbf{y}_i-\mathbf{u})^\perp \cdot (\mathbf{x}_i-\mathbf{u})^\perp
+ \| } \ .
+ :label: eqnavanglepart2
The sign of :math:`\theta_\mathrm{av}` is chosen such that
:math:`\theta_\mathrm{av} > 0` if the actual structure rotates ahead of
biod.pnpi.spb.ru / alexxy / gromacs.git / blobdiff