Say, I have two crystal structures of a particular organic molecule, the crystal structures are basically identical, apart from a rotation and redefinition of the lattice vectors/angles, a simple result of different choices during the structure refinement process after the xray or neutron experiment for structure determination.
As a practical example consider triclinic (P-1) malonic acid for which two structures have been published:
one (MALNAC02) with: a,b,c, alpha,beta,gamma=5.156,5.341,8.407,71.48,76.12,85.09
the other (MALNAC) with a,b,c, alpha,beta,gamma=5.33,5.14,11.25,102.7,135.17,85.17
If I wanted to convert one structure to the other so that they perfectly overlap (apart from
a trivial translation step), by rotating it successively through one or more of the cartesian (x,y,z) axis through appropriate angles, how would I find these angles?
I understand there are different ways to do this, other than the successive rotation around the cartesian axes (use a rotation matrix or Euler angles) but here I need to know these angles (around the cartesian axes), also I know there is not only one unique version of these angles - but I am fine with having just one of them.
In literature you find two different approaches to compare crystal structures. One is t make a Fourier transform and to compare the distances, .i.e XPRD. The approach is described at Hofmann, D. W. M., and L. N. Kuleshova. "A method for automated determination of the crystal@ structures from X-ray powder diffraction data." Crystallography Reports 51.3 (2006): 419-427. The other approach is to match one molecule and to compare step by step the surrounding molecules. This approach is implemented in Mercury of CCDC.
Have a close look at:
Article 5-Fluorocytosine/Isocytosine Monohydrate. The First Example ...
If I understand your question correctly, there are a number of programs (free ) to do just that. The CCDC provides the program "Mercury" and that allows you to overlap two structures easily. Similarly, Olex2 allows you to do the same. Both are free and run on multiple computer platforms.
You have to transform the unit cells. Although there is probably some kind of mathematical equivalent in terms of the rotations you describe, transforming the unit cell is the proper way to do it. For the general case, this is not always trivial (ADPs are tough, the symmetry operators may have to be transformed, there will be translations in addition to the unit-cell transformation and don't get me started on handling multiple chemical species in the asymmetric unit) but for the case you describe it will be relatively straightforward. I have some C++ code that can do this, feel free to contact me by e-mail (it should be trivial to find my e-mail address online).
This can be done in Python using scipy.spatial.transform.Rotation.align_vectors which has implemented the Kabsch algorithm. Although, there will still be issues to overcome such as symmetry operations and chemical identities. I have such an algorithm implemented in Python specifically for molecular crystals that will be able to obtain the Euler angles and visualize your problem. Let me know if there's any interest.
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