Rotation of crystal structure to match another structure of the same compound/polymorph

Question

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 Eulerian 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.

Answer 1

If you know which atoms correspond to each other in the two structures, you can use a structural superposition method. Least-squares superposition methods find the rotation matrix and translation that minimizes the RMSD between given points.

There are a few well-established methods. Recently, I had to use one and I picked QCP (because it comes with BSD-licensed C code), but for a small set of points like in your problem any method should work.

Answer 2

If you're familiar with Python, this can be done using pymatgen fairly easily.

To illustrate, let's first define our problem:

from pymatgen.core.lattice import Lattice
from pymatgen.core.structure import Structure

malnac_lattice = Lattice.from_parameters(5.33, 5.14, 11.25, 102.7, 135.17, 85.17)
malnac02_lattice = Lattice.from_parameters(5.156, 5.341, 8.407, 71.48, 76.12, 85.09)

# here, because we're only interested in the lattices, 
# we can define a "dummy structure" with a single H atom at the origin

malnac_struct = Structure(malnac_lattice, ["H"], [[0.0, 0.0, 0.0]])
malnac02_struct = Structure(malnac02_lattice, ["H"], [[0.0, 0.0, 0.0]])

Now we have the structures defined, we can use StructureMatcher to compare them and find the transformation to change one into the other.

from pymatgen.analysis.structure_matcher import StructureMatcher

# StructureMatcher can accept different tolerances for judging equivalence
matcher = StructureMatcher(primitive_cell=False)

# first, we can verify these lattices are equivalent
matcher.fit(malnac_struct, malnac02_struct)  # returns True

# and we can get the transformation matrix from one to the other
# this returns the supercell matrix (e.g. change of basis), 
# as well as any relevant translation, and mapping of atoms from one
# crystal to the other
matcher.get_transformation(malnac_struct, malnac02_struct)
# returns (array([[ 0, -1,  0], [-1,  0,  0], [ 0,  1,  1]]), array([0., 0., 0.]), [0])

This gives your answer:

$$\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}$$

This matrix defines the transformation from the one set of lattice vectors (a, b, c) into the other, and the angles can be obtained from here.

More information can be found in the pymatgen docs, or you can ask the developers (myself included) at matsci.org/pymatgen.

Hope this helps!

Answer 3

It seems to me that there is a discrepancy about the c-axis in the 2 unit cells, producing a significant difference in the unit cell volume (213.1 and 210.8, respectively).

If you apply the transformation matrix [-b,-a,c] to the first unit cell 5.156 5.341 8.407 71.48 76.12 85.09 and compare with the second one you obtain:

5.341 5.156 8.407 103.88 108.52 85.09 Transformed Unit Cell 5.33 5.14 11.25 102.7 135.17 85.17 Second Unit Cell

You can see that a and b axes, as well as gamma (the angle between a and b) are in good agreement in both cases (within experimental differences), whereas the parameters where the c-axis plays a role diverge (c-axis and alfa and beta angles). I would suggest to draw both structures using the corresponding structural data (if available) as well as to compare the occupation of the special positions (if any) for the different atoms.