contributions of atoms are weighted as a function of distance (in
addition to the mass weighting):
-.. math::
-
- \begin{aligned}
- w(r < r_\mathrm{cyl}) & = &
- 1-2 \left(\frac{r}{r_\mathrm{cyl}}\right)^2 + \left(\frac{r}{r_\mathrm{cyl}}\right)^4 \\
- w(r \geq r_\mathrm{cyl}) & = & 0\end{aligned}
+.. math:: \begin{aligned}
+ w(r < r_\mathrm{cyl}) & = &
+ 1-2 \left(\frac{r}{r_\mathrm{cyl}}\right)^2 + \left(\frac{r}{r_\mathrm{cyl}}\right)^4 \\
+ w(r \geq r_\mathrm{cyl}) & = & 0\end{aligned}
+ :label: eqnpulldistmassweight
Note that the radial dependence on the weight causes a radial force on
both cylinder group and the other pull group. This is an undesirable,
due to cosine weighting, the weights need to be scaled to conserve
momentum:
-.. math::
-
- w'_i = w_i
- \left. \sum_{j=1}^N w_j \, m_j \right/ \sum_{j=1}^N w_j^2 \, m_j
+.. math:: w'_i = w_i
+ \left. \sum_{j=1}^N w_j \, m_j \right/ \sum_{j=1}^N w_j^2 \, m_j
+ :label: eqnpullmassscale
where :math:`m_j` is the mass of atom :math:`j` of the group. The mass
of the group, required for calculating the constraint force, is:
.. math:: M = \sum_{i=1}^N w'_i \, m_i
+ :label: eqnpullconstraint
The definition of the weighted center of mass is:
.. math:: \mathbf{r}_{com} = \left. \sum_{i=1}^N w'_i \, m_i \, \mathbf{r}_i \right/ M
+ :label: eqnpullcom
From the centers of mass the AFM, constraint, or umbrella force
:math:`\mathbf{F}_{\!com}` on each group can be
to the atoms as follows:
.. math:: \mathbf{F}_{\!i} = \frac{w'_i \, m_i}{M} \, \mathbf{F}_{\!com}
+ :label: eqnpullcomforce
Definition of the pull direction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^