Ian Tickle wrote:
Ed, the screen z axis is not the same axis in the molecule for the first and
last
rotations, except in the special case beta = 0 or 180. The fallacy in your
argument is
that you're implicitly assuming that rotations commute, whereas of course they
don't i.e.
Rz.Ry.Rz is not the same as Rz.Rz.Ry unless Ry = unit matrix or 2-fold. The
first and
last rotations are both indeed around the screen z axis but the orientation of
the
molecule has changed because of the intervening y rotation, so the two z
rotations are not
additive unless beta = 0. Indeed if beta = 180 the net effect is the
difference of the
two z rotations. For other values of beta the net z rotation is a more
complicated
function of the Eulerian angles.
OK, I need to think about this more when I have time, but at this point
I think it is a semantic difference- For me the first and last rotation are
about the same Z axis because as you say they are both around the screen Z axis
and both operators look like cos,sin, 0, -sin, cos, 0, 0, 0, 1; i.e. rotation
about "THE" z axis; and it is not helpful to consider it a different z axis just
because the atoms moved.
We come up with the same conclusions with our different ways of thinking about
it:
for one, deriving the concatenated simple operators to represent a general
rotation,
and the commutativity: I would say the operators do not commute as long as the
axes
they rotate about are kept fixed, but if the axes rotate the same as the
molecule
then the z axis will always be passing through the atoms the same way.
Then rotations would commute, because the z axis would always represent the same molecular
axis. Which I am sure is NOT what you meant by saying "new z axis".
Thanks,
eab
HTH!
Cheers
-- Ian
On 29 March 2014 21:22, Edward Berry <ber...@upstate.edu
<mailto:ber...@upstate.edu>> wrote:
Thanks, Ian!
I agree it may have to do with being used to computer graphics, where x,y,z
are fixed
and the coordinates rotate. But it still doesn't make sense:
If the axes rotate along with the molecule, in the catenated operators of
the polar
angles, after the first two operators the z axis would still be passing
through the
molecule in the same way it did originally, so rotation about z in the
third step
would have the same effect as rotating about z in the original orientation.
Or in eulerian angles, if the axes rotate along with the molecule at each
step, the z
axis in the third step passes through the molecule in the same way it did
in the first
step, so alpha and gamma would have the same effect and be additive. In
other words
if the axes we are rotating about rotate themselves in lock step with the
molecule, we
can never rotate about any molecular axes except those that were originally
along x,
y, and z (because they will always be alng x,y,z) (I mean using simple
rotations about
principle axes: cos sin -sin cos).
Maybe I need to think about the concept of molecular axes as opposed to lab
axes. The
lab axes are defined relative to the world and never change. The molecular
axis is
defined by how the lab axis passes through the molecule, and changes as the
molecule
rotates relative to the lab axis. But then the molecular axis seems
redundant, since
I can understand the operator fine just in terms of the rotating
coordinates and the
fixed lab axes. Except the "desired rotation axis" of the polar angles
would be a
molecular axis, since it is defined by a line through the atoms that we
want to rotate
about. So it rotates along with the coordinates during the first two
operations, which
align it with the old lab Z axis (which is the new molecular z axis?) . . .
You see
my confusion.
Or think about the math one step at a time, and suppose we look at the
coordinates
after each step with a graphics program keeping the x axis horizontal, y
axis
vertical, and z axis coming out of the plane. For Eulerian angles, the
first rotation
will be about Z. This will leave the z coordinate of each atom unchanged
and change
the x,y coordinates. If we give the new coordnates to the graphics
program, it will
display the atoms rotated in the plane of the screen (about the z axis
perpendicular
to the screen). The next rotation will be about y, will leave the y
coordinates
unchanged, and we see rotation about the vertical axis. Final rotation
about z is in
the plane of the screen again, although this represents rotation about a
different
axis of the molecule. My view would be to say the first and final rotation
are
rotating about the perpendicular to the screen which we have kept equal to
the z axis,
and it is the same z axis.
Ed
>>> Ian Tickle __ 03/29/14 1:39 PM >>>
Hi Edward
As far as Eulerian rotations go, in the 'Crowther' description the 2nd
rotation can
occur either about the new (rotated) Y axis or about the old (unrotated) Y
axis, and
similarly for the 3rd rotation about the new or old Z. Obviously the same
thing
applies to polar angles since they can also be described in terms of a
concatenation
of rotations (5 instead of 3). So in the 'new' description the rotation
axes do
change: they are rotating with the molecule.
For reasons I find hard to fathom virtually all program documentation seems
to
describe it in terms of rotations about already-rotated angles. If as you
say you
find this confusing then you are not alone! However it's very easy to
change from a
description involving 'new' axes to one involving 'old' axes: you just
reverse the
order of the angles. So in the Eulerian case a rotation of alpha around Z,
then beta
around new Y, then gamma around new Z (i.e. 'Crowther' convention) is
completely
equivalent to a rotation of gamma around Z, then beta around _old_ Y, then
alpha
around _old_ Z.
So if you're used to computer graphics where the molecules rotate around
the fixed
screen axes (rotation around the rotating molecular axes would be very
confusing!)
then it seems to me that the 'old' description is much more intuitive.
Cheers
-- Ian
On 27 March 2014 22:18, Edward A. Berry <ber...@upstate.edu
<mailto:ber...@upstate.edu>> wrote:
According to the html-side the 'visualisation' includes two
back-rotations in addition to what you copied here, so there is
at
least one difference to the visualisation of the Eulerian
angles.
Right- it says:
"This can also be visualised as
rotation ϕ about Z,
rotation ω about the new Y,
rotation κ about the new Z,
rotation (-ω) about the new Y,
rotation (-ϕ) about the new Z."
The first two and the last two rotations can be seen as a "wrapper"
which
first transforms the coordinates so the rotation axis lies along z,
then after
the actual kappa rotation is carried out (by rotation about z),
transforms the
rotated molecule back to the otherwise original position.
Or which transforms the coordinate system to put Z along the rotation
axis, then after
the rotation by kappa about z transforms back to the original
coordinate system.
Specifically,
rotation ϕ about Z brings the axis into the x-z plane so that
rotation ω about the Y brings the axis onto the z axis, so that
rotation κ about Z is doing the desired rotation about a line that
passes through
the atoms in the same way the desired lmn axis did in the
original orientation;
Then the 4'th and 5'th operations are the inverse of the 2nd and
first,
bringing the rotated molecule back to its otherwise original
position
I think all the emphasis on "new" y and "new" z is confusing. If we are
rotating
the molecule (coordinates), then the axes don't change. They pass
through the molecule
in a different way because the molecule is rotated, but the axes are
the same.
After the first two rotations the Z axis passes along the desired
rotation axis,
but the Z axis has not moved, the coordinates (molecules) have.
Of course there is the alternate interpretation that we are doing a
change of
coordinates and expressing the unmoved molecular coordinates relative
to new
principle axes. but if we are rotating the coordinates about the axes
then the
axes should remain the same, shouldn't they? Or maybe there is yet
another way of
looking at it.
Tim Gruene wrote:
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
Dear Qixu Cai,
maybe the confusion is due to that your quote seems incomplete.
According to the html-side the 'visualisation' includes two
back-rotations in addition to what you copied here, so there is at
least one difference to the visualisation of the Eulerian angles.
Best,
Tim
On 03/27/2014 07:11 AM, Qixu Cai wrote:
Dear all,
From the definition of CCP4
(http://www.ccp4.ac.uk/html/__rotationmatrices.html
<http://www.ccp4.ac.uk/html/rotationmatrices.html>), the polar
angle
(ϕ, ω, κ) can be visualised as rotation ϕ about Z, rotation ω
about
the new Y, rotation κ about the new Z. It seems the same as the
ZXZ
convention of eulerian angle definition. What's the difference
between the CCP4 polar angle definition and eulerian angle ZXZ
definition?
And what's the definition of polar angle XYK convention in GLRF
program?
Thank you very much!
Best wishes,
- --
- --
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen
GPG Key ID = A46BEE1A
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.4.12 (GNU/Linux)
Comment: Using GnuPG with Icedove - http://www.enigmail.net/
iD8DBQFTNAz0UxlJ7aRr7hoRAj7IAK__Ds/J0L/__XCYPpQSyB2BPJ2uWV2lVgCeKD72
0DemwU57v6fekF6iOC4/5IA=
=PeT9
-----END PGP SIGNATURE-----
__
