From: Giacomo Fiorin (giacomo.fiorin_at_gmail.com)
Date: Thu Feb 21 2013 - 10:10:24 CST
Yeah, that distance is simply the angle w (in radians) between Q1 and Q2 as
unit vectors in R^4. Because of the symmetry, the actual distance is the
lower number between w and (pi - w), so as Jérôme already said, values of
the distance range from 0 to pi/2.
Benjamin: apart from the lack of grids, no problems using quaternions for
metadynamics, if you need a completely isotropic variable. Check however
if you could simplify it by ignoring e.g. one axis of rotation and using
tilt and/or spinAngle.
Regarding the grid implementation of the PMF: a grid for the energy (scalar
function of a quaternion variable) is a bit complex, but doable. For
instance, you can find in the literature that people use multiple 3-d
projections of the 4-d unit sphere, switching from one projection to the
other to preserve the resolution. Also, because the metadynamics PMF is
essentially a histogram, the symmetry is automatically taken care of, by
incrementing the grid's value at both locations, Q and (-Q).
However, things start getting messier with the grid for the energy
gradients (quaternion function of a quaternion variable...).
Essentially, if you can live with the performance loss of using explicit
gaussian functions instead of the grid, I think your best bet is to get the
quaternion values from the colvars.traj file, and calculate the histogram
(i.e. the PMF) choosing the 3-d projection that it's best for your case. I
can send you a python script for that, if you like.
Bests
Giacomo
On Thu, Feb 21, 2013 at 10:21 AM, Jérôme Hénin <jerome.henin_at_ibpc.fr> wrote:
> ----- Original Message -----
> > Thanks for the answers Jérôme. One last question, if I may - in
> > quaternion
> > space, is the biasing Gaussian function 'isotropic' (ie, same width
> > applied to all 4 dimensions), or are the scalar and vector parts
> > treated
> > differently?
>
> The four components are treated in a symmetric way, but not as a Euclidean
> distance: since the orientation quaternions are always unit, their distance
> is defined as the geodesic distance on the unit sphere (4D hypersphere).
> More precisely, the distance between Q1 and Q2 is the min of d(Q1, Q2) and
> d(Q1, -Q2), as Q2 and -Q2 have the same meaning in terms of rotations. So
> unless I am mistaken, it should be between 0 and pi/2.
>
> Best,
> Jerome
>
>
>
> > On Thu, 21 Feb 2013 13:53:28 +0100, Jérôme Hénin
> > <jerome.henin_at_ibpc.fr>
> > wrote:
> >
> > > Hi Benjamin,
> > >
> > > ----- Original Message -----
> > >> Hello all,
> > >>
> > >> I'm attempting to use quaternions (colvar 'orientation') as a
> > >> biasing
> > >> coordinate for a metadynamics simulation, to control the
> > >> orientation
> > >> of a
> > >> molecule with respect to another. I have a few questions that
> > >> neither
> > >> the
> > >> documentation nor my first results have completely answered:
> > >>
> > >> - Am I correct in assuming that the 'orientation' colvar is
> > >> affected
> > >> by
> > >> the overall rotation of the system, meaning that if I want to
> > >> control
> > >> the
> > >> orientation of B relative to A, I have to restrain the rotation of
> > >> A
> > >> (using, e.g., another 'orientation' colvar and a 'harmonic'
> > >> restraint)?
> > >
> > > You are correct that the 'orientation' colvar is, in general,
> > > sensitive
> > > to overall rotations of the system, but incorrect in concluding
> > > that you
> > > need restraints. As with all colvars, you can define this
> > > orientation
> > > based on rotated coordinates, in the frame of reference of group A.
> > > to
> > > that effect, use the rotateReference and refPositionsGroup
> > > keywords,
> > > providing group A as the the refPositionsGroup.
> > >
> > >
> > >> - How exactly is the width of the hill Gaussians (in R^4) defined
> > >> from the
> > >> scalar 'width' parameter? I could find no trivial relationship
> > >> between
> > >> this parameter and the width of the hills (also scalar) given in
> > >> the
> > >> colvars.state file.
> > >
> > > The width of each Gaussian is considered to be twice its standard
> > > deviation, and it is obtained as the hillWidth parameter of
> > > metadynamics, times that colvar's width parameter. So the std. dev.
> > > is:
> > > 1/2 * hillWidth * colvar->width
> > >
> > >
> > >> - Since 'orientation' is not grid-compatible, PMF output seems
> > >> disabled.
> > >> Is there a utility program to convert the hills file to a PMF, or
> > >> should I
> > >> write my own (which would require me to understand the
> > >> aforementioned
> > >> hill-width thing...)?
> > >
> > > I don't think there is such a utility at this point, but I hope the
> > > info
> > > above is enough for you to compute the PMF.
> > >
> > > Best,
> > > Jerome
> >
>
>
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