The types of virtual sites supported in |Gromacs| are given in the list
below. Constructing atoms in virtual sites can be virtual sites
themselves, but only if they are higher in the list, i.e. virtual sites
-can be constructed from “particles” that are simpler virtual sites.
+can be constructed from “particles” that are simpler virtual sites. The
+virtual site velocities are reported, but not used in the integration
+of the virtual site positions.
-- On top of an atom. This allows giving an atom multiple atom types and
+On top of an atom
+~~~~~~~~~~~~~~~~~
+
+- This allows giving an atom multiple atom types and
with that also assigned multiple, different bonded interactions. This
- can espically be of use in free-energy calculations.
+ can especially be of use in free-energy calculations.
- The coordinates of the virtual site equal that of the constructing atom:
.. math:: \mathbf{F}_i ~=~ \mathbf{F}_{s}
:label: eqnvsite1force
-- As a linear combination of two atoms
- (:numref:`Fig. %s <fig-vsites>` 2):
+- The velocity of the virtual site equals that of the constructing atom:
+
+ .. math:: \mathbf{v}_s ~=~ \mathbf{v}_i
+ :label: eqnvsite1vel
+
+As a linear combination of two atoms (:numref:`Fig. %s <fig-vsites>` 2)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The weights are calculated as
.. math:: w_i = 1 - a ~,~~ w_j = a
:label: eqnvsitelin2atom
- In this case the virtual site is on the line through atoms :math:`i`
and :math:`j`.
-- On the line through two atoms, with a fixed distance
- (:numref:`Fig. %s <fig-vsites>` 2fd):
+- The velocity of the virtual site is a linear combination of the
+ velocities of the constructing atoms
+
+On the line through two atoms, with a fixed distance (:numref:`Fig. %s <fig-vsites>` 2fd)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The position is calculated as:
.. math:: \mathbf{r}_s ~=~ \mathbf{r}_i + a \frac{ \mathbf{r}_{ij} }
{ | \mathbf{r}_{ij} | }
\end{array}
:label: eqnvsite2fdforce
-- As a linear combination of three atoms
- (:numref:`Fig. %s <fig-vsites>` 3):
+- The velocity is calculated as:
+
+ .. math:: \mathbf{v}_{s} = \mathbf{v}_{i} + \frac{a}{|\mathbf{r}_{ij}|}
+ \left(\mathbf{v}_{ij} - \mathbf{r}_{ij}
+ \frac{\mathbf{v}_{ij}\cdot\mathbf{r}_{ij}}
+ {|\mathbf{r}_{ij}|^2}\right)
+ :label: eqnvsite2fdatomvel
+
+As a linear combination of three atoms (:numref:`Fig. %s <fig-vsites>` 3)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The weights are calculated as:
.. math:: w_i = 1 - a - b ~,~~ w_j = a ~,~~ w_k = b
:label: eqnvsitelin3atom
- In this case the virtual site is in the plane of the other three
particles.
-- In the plane of three atoms, with a fixed distance
- (:numref:`Fig. %s <fig-vsites>` 3fd):
+In the plane of three atoms, with a fixed distance (:numref:`Fig. %s <fig-vsites>` 3fd)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- .. math:: \mathbf{r}_s ~=~ \mathbf{r}_i + b \frac{ (1 - a) \mathbf{r}_{ij} + a \mathbf{r}_{jk} }
- { | (1 - a) \mathbf{r}_{ij} + a \mathbf{r}_{jk} | }
+- The position is calculated as:
+
+ .. math:: \mathbf{r}_s ~=~ \mathbf{r}_i + b \frac{ \mathbf{r}_{ijk} } { | \mathbf{r}_{ijk} | }
+ ~\mbox{ where }~
+ \mathbf{r}_{ijk} ~=~ (1 - a) \mathbf{r}_{ij} + a \mathbf{r}_{jk}
:label: eqnvsiteplane3atom
- In this case the virtual site is in the plane of the other three
\end{array}
:label: eqnvsiteplane3atomforce
-- In the plane of three atoms, with a fixed angle and
- distance (:numref:`Fig. %s <fig-vsites>` 3fad):
+- The velocity is calculated as:
+
+ .. math:: \mathbf{v}_{s} ~=~ \mathbf{v}_{i} +
+ \frac{b}{|\mathbf{r}_{ijk}|}
+ \left(\dot{\mathbf{r}}_{ijk} -
+ \mathbf{r}_{ijk}\frac{\dot{\mathbf{r}}_{ijk}\cdot\mathbf{r}_{ijk}}
+ {|\mathbf{r}_{ijk}|^2}\right)
+ :label: eqnvsiteplane3atomvel
+
+In the plane of three atoms, with a fixed angle and distance (:numref:`Fig. %s <fig-vsites>` 3fad)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The position is calculated as:
.. math:: \mathbf{r}_s ~=~ \mathbf{r}_i +
d \cos \theta \frac{\mathbf{r}_{ij}}{ | \mathbf{r}_{ij} | } +
\end{array}
:label: eqnvsite2fadFforce
-- As a non-linear combination of three atoms, out of
- plane (:numref:`Fig. %s <fig-vsites>` 3out):
+- The velocity is calculated as:
+
+ .. math:: \mathbf{v}_{s} &= \mathbf{v}_{i} + d\cos\theta\ \frac{\delta}{\delta t}\frac{\mathbf{r}_{ij}}{|\mathbf{r}_{ij}|} +
+ d\sin\theta\ \frac{\delta}{\delta t}\frac{\mathbf{r}_{\perp}}{|\mathbf{r}_{\perp}|} \\
+ ~\mbox{where}~&\\
+ \frac{\delta}{\delta t}\frac{\mathbf{r}_{ij}}{|\mathbf{r}_{ij}|} &=
+ \frac{1}{|\mathbf{r}_{ij}|}\left(\mathbf{v}_{ij} - \mathbf{r}_{ij}
+ \frac{\mathbf{v}_{ij}\cdot\mathbf{r}_{ij}}{|\mathbf{r}_{ij}|^2}\right)\\
+ \frac{\delta}{\delta t}\frac{\mathbf{r}_{\perp}}{|\mathbf{r}_{\perp}|} &=
+ \frac{1}{|\mathbf{r}_{\perp}|}
+ \left(\dot{\mathbf{r}}_{\perp} - \mathbf{r}_{\perp}\frac{\dot{\mathbf{r}}_{\perp}\cdot\mathbf{r}_{\perp}}{|\mathbf{r}_{\perp}|^2}\right)\\
+ \dot{\mathbf{r}}_\perp &= \mathbf{v}_{jk} - \mathbf{r}_{ij}
+ \frac{|\mathbf{r}_{ij}|^2(\mathbf{v}_{ij}\cdot\mathbf{r}_{jk} + \mathbf{r}_{ij}\cdot\mathbf{v}_{jk}) -
+ (\mathbf{r}_{ij}\cdot\mathbf{r}_{jk})(2\mathbf{r}_{ij}\cdot\mathbf{v}_{ij})} {|\mathbf{r}_{ij}|^4} -
+ \frac{\mathbf{r}_{ij}\cdot\mathbf{r}_{jk}}{|\mathbf{r}_{ij}|^2}\ \mathbf{v}_{ij}
+ :label: eqnvsite2fadvel
+
+As a non-linear combination of three atoms, out of plane (:numref:`Fig. %s <fig-vsites>` 3out)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The position is calculated as:
.. math:: \mathbf{r}_s ~=~ \mathbf{r}_i + a \mathbf{r}_{ij} + b \mathbf{r}_{ik} +
c (\mathbf{r}_{ij} \times \mathbf{r}_{ik})
\end{array}
:label: eqnvsitenonlin3atomforce
-- From four atoms, with a fixed distance, see
- separate :numref:`Fig. %s <fig-vsite4fdn>`. This construction is a bit complex,
+- The velocity is calculated as:
+
+ .. math:: \mathbf{v}_{s} ~=~ \mathbf{v}_{i} + \frac{c}{|\mathbf{r}_{m}|}\left(\dot{\mathbf{r}}_{m} -
+ \mathbf{r}_{m} \frac{\dot{\mathbf{r}}_{m}\cdot\mathbf{r}_{m}}{|\mathbf{r}_{m}|^2}\right)
+ :label: eqnvsitenonlin3atomvel
+
+From four atoms, with a fixed distance, see separate :numref:`Fig. %s <fig-vsite4fdn>`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+- This construction is a bit complex,
in particular since the previous type (4fd) could be unstable which
forced us to introduce a more elaborate construction:
The new 4fdn virtual site construction, which is stable even when
all constructing atoms are in the same plane.
--
+- The position is calculated as
+
.. math:: \begin{aligned}
\mathbf{r}_{ja} &=& a\, \mathbf{r}_{ik} - \mathbf{r}_{ij} = a\, (\mathbf{x}_k - \mathbf{x}_i) - (\mathbf{x}_j - \mathbf{x}_i) \nonumber \\
\mathbf{r}_{jb} &=& b\, \mathbf{r}_{il} - \mathbf{r}_{ij} = b\, (\mathbf{x}_l - \mathbf{x}_i) - (\mathbf{x}_j - \mathbf{x}_i) \nonumber \\
\mathbf{r}_m &=& \mathbf{r}_{ja} \times \mathbf{r}_{jb} \nonumber \\
- \mathbf{x}_s &=& \mathbf{x}_i + c \frac{\mathbf{r}_m}{ | \mathbf{r}_m | }
+ \mathbf{r}_s &=& \mathbf{r}_i + c \frac{\mathbf{r}_m}{ | \mathbf{r}_m | }
\end{aligned}
:label: eqnvsite
+- The velocity is calculated as:
+
+ .. math:: \mathbf{v}_{s} = \mathbf{v}_{i} + \frac{c}{|\mathbf{r}_{m}|}\left(\dot{\mathbf{r}}_{m} - \mathbf{r}_{m} \frac{\dot{\mathbf{r}}_{m}\cdot\mathbf{r}_{m}}{|\mathbf{r}_{m}|^2}\right)\\
+ ~\mbox{where}~&\\
+ \dot{\mathbf{r}}_{m} &= \dot{\mathbf{r}}_{ja} \times \mathbf{r}_{jb} + \mathbf{r}_{ja} \times \dot{\mathbf{r}}_{jb}
+ :label: eqnvsitevel
+
- In this case the virtual site is at a distance of :math:`|c|` from
:math:`i`, while :math:`a` and :math:`b` are parameters. **Note**
that the vectors :math:`\mathbf{r}_{ik}` and :math:`\mathbf{r}_{ij}`
value 1), but it should not be used for new simulations. All current
|Gromacs| tools will automatically generate type 4fdn instead.
-- A linear combination of :math:`N` atoms with relative
- weights :math:`a_i`. The weight for atom :math:`i` is:
+A linear combination of :math:`N` atoms with relative weights :math:`a_i`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- The weight for atom :math:`i` is:
.. math:: w_i = a_i \left(\sum_{j=1}^N a_j \right)^{-1}
:label: eqnvsiterelweight
biod.pnpi.spb.ru / alexxy / gromacs.git / blobdiff