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 <[email protected]> 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), 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/ >> >> iD8DBQFTNAz0UxlJ7aRr7hoRAj7IAKDs/J0L/XCYPpQSyB2BPJ2uWV2lVgCeKD72 >> 0DemwU57v6fekF6iOC4/5IA= >> =PeT9 >> -----END PGP SIGNATURE----- >> >>
