Re: [ccp4bb] Merging Data from Multiple Crystals
Dear Tassos, Do you know by chance whether BLEND is available? Best wishes, Alex On 27 Mar 2014, at 16:06, Tassos Papageorgiou tassos.papageorg...@btk.fi wrote: Hi, You may also try BLEND to choose the optimal data sets before scaling and merging Foadi J, Aller P, Alguel Y, Cameron A, Axford D, Owen RL, Armour W, Waterman DG, Iwata S Evans G (2013) Clustering procedures for the optimal selection of data sets from multiple crystals in macromolecular crystallography. Acta Crystallogr D Biol Crystallogr 69: 1617–1632 Tassos Papageorgiou Jarrod Mousa wrote: Hi, I am trying to solve the structure of a membrane protein. The protein has 12 helices and I have a good molecular replacement model that seems to work for about half of the structure. I used chainsaw to convert the amino acid residues to that of my protein sequence, and the density fits the structure well on one side of the protein, but on the other side (about 5 helices), there doesn't seem to be any density for the side chains. Has anyone had experience with this? The completeness is high ~99% for 3.2 angstroms. The data was collected from fairly small crystals ~ 20um. Thanks. -- Alex Batyuk The Plueckthun Lab www.bioc.uzh.ch/plueckthun
Re: [ccp4bb] Merging Data from Multiple Crystals
Contact James Foadi. http://diamond.ac.uk/Beamlines/Mx/I24/Staff/Foadi.html Andreas On 29/03/2014 1:21, Alexander Batyuk wrote: Dear Tassos, Do you know by chance whether BLEND is available? Best wishes, Alex
Re: [ccp4bb] Merging Data from Multiple Crystals
Dear Alex BLEND will be released soon through the CCP4 Updates. In the meantime, the easiest way to try it out is to install one of the nightly builds from here: http://www.ccp4.ac.uk/dev/nightly/. The usual warnings apply. What you obtain that way may not be (fully) tested, and may differ slightly from what is finally released. Your comments are welcome. I hope this helps. Cheers, -- David On 29 March 2014 13:38, Andreas Förster docandr...@gmail.com wrote: Contact James Foadi. http://diamond.ac.uk/Beamlines/Mx/I24/Staff/Foadi.html Andreas On 29/03/2014 1:21, Alexander Batyuk wrote: Dear Tassos, Do you know by chance whether BLEND is available? Best wishes, Alex
Re: [ccp4bb] difference between polar angle and eulerian angle
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-
[ccp4bb] Usage of gels in protein crystallography
Dear CCP4BBers, I would very much appreciate any information or resources regarding usage of gels in order to achieve supersaturation/crystallization through liquid diffusion. It appears to me that, although this crystallization method is usually claimed as a powerful technique, it is very limited in practice. For instance, PEG solutions display little to no diffusion through gel matrices, and some gels may undergo significant changes in volume (and thereby their average pore size) depending on the medium ionic strength and salt composition, leading to significant loss of protein. Thanks in advance, Javier -- Javier M. Gonzalez, PhD. Protein Crystallography Station Bioscience Division Bioenergy and Biome Sciences Group (B-11) Los Alamos National Laboratory TA-3, Building 4200, Room 202B Mailstop T007 Los Alamos, NM 87545 Phone: +1 (505) 667-9376 LinkedIn http://www.linkedin.com/pub/javier-gonz%C3%A1lez/22/7b/83a Emailbio...@gmail.com
Re: [ccp4bb] difference between polar angle and eulerian angle
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 bmuch 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
Re: [ccp4bb] difference between polar angle and eulerian angle
There are good arguiments for using quaternions rather than Eulerian (or other) angles anyway, this is very well explained in the paper *Quaternions *in *molecular modeling* http://scholar.google.de/scholar_url?hl=enq=http://arxiv.org/pdf/physics/0506177sa=Xscisig=AAGBfm13yuMgR9JJ3LvihnDJIoFFejNTrgoi=scholarrei=Zzs3U__QEYKw7AaL4IDYAwved=0CCoQgAMoADAA' by Karney. George On 03/29/2014 10:22 PM, Edward Berry 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
Re: [ccp4bb] difference between polar angle and eulerian angle
Edward Berry 03/29/14 5:22 PM 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: -My mistake- in computer graphics x,y,z rotates with the atomic coordinates relative to screen coordintes, or the viewpoint changes 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. Maybe in my thinking I am going in reverse order- didn't pay attention to sign of the angles. If you think of the Eulerian navigator at http://sb20.lbl.gov/berry/Euler2.gif it is obvious that the same setting on each of the three angles will give the same orientation. Now assuming the outside frame is fixed (bolted to the bench) and you adjust the angles starting with the inside ring, you will be using lab axes all the way. If you first adjust the outside ring, then the next two rotation will be about new axes. Computationally it must be much easier to use old or Lab axes. In the case of polar coordinates, the whole problem involves rotation by kappa about an axis at odd angles to x,y,z. If in order to do that, we introduce 3 more rotations about non-standard axes, and the same for each of them, we will never get there! 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- HasAccording 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/
Re: [ccp4bb] difference between polar angle and eulerian angle
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
Re: [ccp4bb] difference between polar angle and eulerian angle
The main reason for using Eulerian (or polar) angles is speed (not for nothing is Crowther's implementation called the Fast Rotation Function). Expression of the rotation in terms of Eulerian or polar angles makes it possible to express the Patterson functions in terms of orthogonal spherical harmonics and thus decompose the problem into a set of simpler problems involving spherical Bessel functions for the radial variable and Fourier summations for the angular variables of the spherical harmonics, which can then take advantage of the FFT algorithm. I'm not aware of any algorithm that makes use of quaternions which can decompose the problem similarly in order to take advantage of the speed of FFT to do most of the work. Cheers -- Ian On 29 March 2014 21:41, George Sheldrick gshe...@shelx.uni-ac.gwdg.dewrote: There are good arguiments for using quaternions rather than Eulerian (or other) angles anyway, this is very well explained in the paper *Quaternions *in *molecular modeling*http://scholar.google.de/scholar_url?hl=enq=http://arxiv.org/pdf/physics/0506177sa=Xscisig=AAGBfm13yuMgR9JJ3LvihnDJIoFFejNTrgoi=scholarrei=Zzs3U__QEYKw7AaL4IDYAwved=0CCoQgAMoADAA' by Karney. George On 03/29/2014 10:22 PM, Edward Berry 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
