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.
HTH! Cheers -- Ian On 29 March 2014 21:22, Edward Berry <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> 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----- >>> >>> >
